Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 521052, 8 pages http://dx.doi.org/10.1155/2013/521052 Research Article On Partial Complete Controllability of Semilinear Systems Agamirza E. Bashirov and Maher Jneid Eastern Mediterranean University, Gazimagusa, North Cyprus, P.O. Box 95, Mersin 10, Turkey Correspondence should be addressed to Agamirza E. Bashirov; [email protected] Received 27 March 2013; Revised 28 May 2013; Accepted 11 June 2013 Academic Editor: Sakthivel Rathinasamy Copyright © 2013 A. E. Bashirov and M. Jneid. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Many control systems can be written as a first-order differential equation if the state space enlarged. erefore, general conditions on controllability, stated for the first-order differential equations, are too strong for these systems. For such systems partial controllability concepts, which assume the original state space, are more suitable. In this paper, a sufficient condition for the partial complete controllability of semilinear control system is proved. e result is demonstrated through examples. 1. Introduction A concept of controllability, defined by Kalman [1] in 1960 for finite dimensional control systems, is a property of attaining every point in the state space from every initial state point for a finite time. Further studies on this concept in infinite dimensional spaces demonstrated that it is suitable to con- sider its two versions: a stronger version of complete control- lability and a weaker version of approximate controllability. e reason for these versions was the fact that many infinite dimensional control systems are not completely controllable while they are approximately controllable (see Fattorini [2] and Russell [3]). e necessary and sufficient conditions for complete and approximate controllability concepts are almost completely studied and presented in, for example, Curtain and Zwart [4], Bensoussan [5], Bensoussan et al. [6], Zabczyk [7], Bashirov [8], Klamka [9], and so forth for linear systems; Balachandran and Dauer [10, 11], Klamka [12], Mahmudov [13], Li and Yong [14], and so forth for nonlinear systems; Sakthivel et al. [15–17], Yan [18], and so forth for fractional differential systems; and Ren et al. [19] for differential inclusions. Recently, in Bashirov et al. [20, 21] the partial controlla- bility concepts were initiated. e idea of these concepts is that some control systems, including higher order differential equations, wave equations, and delay equations, can be written as a first-order differential equation only by enlarging the dimension of the state space. erefore, the theorems on controllability, which are formulated for control systems in the form of first-order differential equation, are too strong for them because they involve the enlarged state space. In such cases the partial controllability concepts became preferable, which assume the original state space. e basic controllability conditions for linear systems, including resol- vent conditions from Bashirov and Mahmudov [22] and Bashirov and Kerimov [23] (see also [24–26]), are extended to partial controllability concepts by just a replacement of the controllability operator by its partial version. In this paper our aim is to study the partial complete con- trollability of semilinear systems. e controllability concepts for semilinear systems are intensively discussed in the litera- ture (see Balachandran and Dauer [10, 11], Klamka [12], Mah- mudov [13], Sakthivel et al. [15, 17], and references therein). A basic tool of study in these works is fixed point theorems. In this paper, we also use one of the fixed point theorems, a contraction mapping theorem, and find a sufficient condition for the partial complete controllability of a semilinear control system. e rest of this paper is organised in the following way. In Section 2 we set the problem, give basic definitions, and moti- vate the partial controllability concepts by considering a higher order differential equation, a wave equation, and a delay equation. Section 3 contains the proof of the main result. In Section 4, we demonstrate the main result in the examples. Finally, Section 5 contains directions of further research regarding partial controllability concepts.
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Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 521052 8 pageshttpdxdoiorg1011552013521052
Research ArticleOn Partial Complete Controllability of Semilinear Systems
Agamirza E Bashirov and Maher Jneid
Eastern Mediterranean University Gazimagusa North Cyprus PO Box 95 Mersin 10 Turkey
Correspondence should be addressed to Agamirza E Bashirov agamirzabashirovemuedutr
Received 27 March 2013 Revised 28 May 2013 Accepted 11 June 2013
Academic Editor Sakthivel Rathinasamy
Copyright copy 2013 A E Bashirov and M Jneid This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
Many control systems can be written as a first-order differential equation if the state space enlarged Therefore general conditionson controllability stated for the first-order differential equations are too strong for these systems For such systems partialcontrollability concepts which assume the original state space are more suitable In this paper a sufficient condition for the partialcomplete controllability of semilinear control system is proved The result is demonstrated through examples
1 Introduction
A concept of controllability defined by Kalman [1] in 1960 forfinite dimensional control systems is a property of attainingevery point in the state space from every initial state pointfor a finite time Further studies on this concept in infinitedimensional spaces demonstrated that it is suitable to con-sider its two versions a stronger version of complete control-lability and a weaker version of approximate controllabilityThe reason for these versions was the fact that many infinitedimensional control systems are not completely controllablewhile they are approximately controllable (see Fattorini [2]and Russell [3]) The necessary and sufficient conditionsfor complete and approximate controllability concepts arealmost completely studied and presented in for exampleCurtain and Zwart [4] Bensoussan [5] Bensoussan et al[6] Zabczyk [7] Bashirov [8] Klamka [9] and so forth forlinear systems Balachandran and Dauer [10 11] Klamka[12] Mahmudov [13] Li and Yong [14] and so forth fornonlinear systems Sakthivel et al [15ndash17] Yan [18] and soforth for fractional differential systems and Ren et al [19] fordifferential inclusions
Recently in Bashirov et al [20 21] the partial controlla-bility concepts were initiated The idea of these concepts isthat some control systems including higher order differentialequations wave equations and delay equations can bewritten as a first-order differential equation only by enlargingthe dimension of the state space Therefore the theorems on
controllability which are formulated for control systems inthe form of first-order differential equation are too strongfor them because they involve the enlarged state spaceIn such cases the partial controllability concepts becamepreferable which assume the original state space The basiccontrollability conditions for linear systems including resol-vent conditions from Bashirov and Mahmudov [22] andBashirov and Kerimov [23] (see also [24ndash26]) are extendedto partial controllability concepts by just a replacement of thecontrollability operator by its partial version
In this paper our aim is to study the partial complete con-trollability of semilinear systemsThe controllability conceptsfor semilinear systems are intensively discussed in the litera-ture (see Balachandran andDauer [10 11] Klamka [12] Mah-mudov [13] Sakthivel et al [15 17] and references therein)A basic tool of study in these works is fixed point theoremsIn this paper we also use one of the fixed point theorems acontractionmapping theorem and find a sufficient conditionfor the partial complete controllability of a semilinear controlsystem
The rest of this paper is organised in the following way InSection 2we set the problem give basic definitions andmoti-vate the partial controllability concepts by considering ahigher order differential equation a wave equation and adelay equation Section 3 contains the proof of the mainresult In Section 4 we demonstrate the main result in theexamples Finally Section 5 contains directions of furtherresearch regarding partial controllability concepts
2 Abstract and Applied Analysis
2 Setting the Problem and Motivation
Consider the basic semilinear control system
1199091015840
119905= 119860119909119905+ 119861119906119905+ 119891 (119905 119909
119905 119906119905) 0 lt 119905 le 119879 119909
0isin 119883 (1)
on the interval [0 119879] with 119879 gt 0 where 119909 and 119906 are state andcontrol processes We assume that the following conditionshold
(A) 119883 and 119880 are separable Hilbert spaces 119867 is a closedsubspace of 119883 and 119871 is a projection operator from 119883
to119867(B) 119860 is a densely defined closed linear operator on 119883
generating a strongly continuous semigroup 119890119860119905 119905 ge
0(C) 119861 is a bounded linear operator from 119880 to119883(D) 119891 is a nonlinear function from [0 119879] times 119883 times 119880 to 119883
satisfying that
(i) 119891 is continuous on [0 119879] times 119883 times 119880(ii) 119891 is Lipschitz continuous with respect to 119909 and
119906 that is for all 119905 isin [0 119879] 119906 V isin 119880 and 119909 119910 isin 1198831003817100381710038171003817119891 (119905 119909 119906) minus 119891 (119905 119910 V)100381710038171003817
1003817le 119870 (
1003817100381710038171003817119909 minus 119910
1003817100381710038171003817+ 119906 minus V) (2)
for some119870 ge 0
(E) 119880ad = 119862(0 119879 119880) is the space of all continuous fun-ctions from [0 119879] to 119880Define the controllability and 119871-partial controllabilityoperators 119876
119905and 119876
119905by
119876119905= int
119905
0
119890119860119904119861119861lowast119890119860lowast
119904119889119904 119876
119905= 119871119876119905119871lowast (3)
where 119871lowast is the adjoint of 119871 The 119871-partial controlla-bility operator becomes the controllability operator if119871 = 119868 (the identity operator) We will also assume thefollowing condition
(F) 119876119879is coercive that is there is 120574 gt 0 such that
⟨119876119879ℎ ℎ⟩ ge 120574ℎ
2 for all ℎ isin 119867
Note that this condition implies the existence of 119876minus1119879
as abounded linear operator and 119876
minus1
119879 le 1120574 Respectively the
linear system associated with (1) (the case when 119891 = 0) is119871-partially complete controllable on the interval [0 119879] (seeBashirov et al [20 21])
The above conditions imply the existence of a uniquecontinuous solution of (1) in the mild sense for every 119906 isin 119880adand 119909
0isin 119883 (see Li and Yong [14] and Byszewski [27]) that is
there is a unique continuous function 119909 from [0 119879] to119883 suchthat
119909119905= 1198901198601199051199090+ int
119905
0
119890119860(119905minus119904)
(119861119906119904+ 119891 (119904 119909
119904 119906119904)) 119889119904 (4)
Let
1198631199090
119879= ℎ isin 119867 exist119906 isin 119880ad such that ℎ = 119871119909
119879 (5)
Following Bashirov et al [21] the semilinear control system(1) is said to be 119871-partially complete controllable on 119880ad if1198631199090
119879= 119867 for all 119909
0isin 119883 Similarly the semilinear system in
(1) is said to be 119871-partially approximate controllable on119880ad if1198631199090
119879= 119867 for all 119909
0isin 119883 where 1198631199090
119879is the closure of 1198631199090
119879 If
119867 = 119883 these are just well-known complete and approximatecontrollability concepts respectively In this paper we studythe concept of 119871-partial complete controllability
The reason for studying 119871-partial controllability conceptsis that many systems can be written in the form of (1)if the original state space is enlarged Therefore suitablecontrollability concepts for such systems are the 119871-partialcontrollability concepts with the operator 119871 projecting theenlarged state space to the original one Here are someexamples of such systems which are discussed in Bashirov[8] Section 311 in more details
Example 1 Consider the system
119909(119899)
119905= 119891 (119905 119909
119905 1199091015840
119905 119909
(119899minus1)
119905 119906119905) (6)
assuming that its state space is the one-dimensional spaceR The ordinary controllability concepts for this system arethe equality to or denseness in R of the respective attainableset We can write this system as the first-order differentialequation
1199101015840
119905= 119860119910119905+ 119865 (119905 119910
119905 119906119905) (7)
if
119910119905=
[
[
[
[
[
[
[
119909119905
1199091015840
119905
119909(119899minus2)
119905
119909(119899minus1)
119905
]
]
]
]
]
]
]
119860 =
[
[
[
[
[
[
[
0 1 sdot sdot sdot 0 0
0 0 sdot sdot sdot 0 0
d
0 0 sdot sdot sdot 0 1
0 0 sdot sdot sdot 0 0
]
]
]
]
]
]
]
119865 (119905 119910 119906) =
[
[
[
[
[
[
[
0
0
0
119891 (119905 119909 1199091015840 119909
(119899minus1) 119906)
]
]
]
]
]
]
]
(8)
The state space of this system is the 119899-dimensional Euclideanspace R119899 and respectively its attainable set is a subset of R119899Therefore the controllability concepts of the system for 119910 arestronger than those of the system for 119909 But if we define theprojection operator 119871 by
119871 = [1 0 sdot sdot sdot 0 0] R119899997888rarr R (9)
then the 119871-partial controllability concepts of the system for 119910become the same as the ordinary controllability concepts ofthe system for 119909
Example 2 Consider the nonlinear wave equation
1205972119909119905120579
1205971199052
=
1205972119909119905120579
1205971205792
+ 119891(119905 119909119905120579
120597119909119905120579
120597119905
119906119905) (10)
Abstract and Applied Analysis 3
where 119909 is a real-valued function of two variables 119905 ge 0 and0 le 120579 le 1 The state space of this system is 119871
2(0 1) (the space
of square integrable functions on [0 1]) This system can bewritten as the first-order abstract differential equation
1199101015840
119905= 119860119910119905+ 119865 (119905 119910
119905 119906119905) (11)
if
119910119905=[
[
119909119905120579
120597119909119905120579
120597119905
]
]
119860 =[
[
0 119868
1198892
1198891205792
0
]
]
119865 (119905 119910 119906) = [
0
119891 (119905 1199101 1199102 119906)
]
(12)
where 119910 isin 1198712(0 1) times 119871
2(0 1) The state space 119871
2(0 1) times
1198712(0 1) of the system for 119910 is the enlargement of the state
space 1198712(0 1) of the system for 119909 This is a cost that is
paid to bring the wave equation to the form of first-orderdifferential equation The ordinary controllability conceptsfor the system (11) are too strong for the system (10) If
119871 = [119868 0] 1198712(0 1) times 119871
2(0 1) 997888rarr 119871
2(0 1) (13)
then 119871-partially controllability concepts of the system for 119910become ordinary controllability concepts of the system for 119909
Example 3 Consider the system
1199091015840
119905= 119891(119905 119909
119905 int
0
minus120576
119909119905+120579
119889120579 119906119905) (14)
which contains a simple distributed delay in the nonlinearterm assuming that 119909 is a real-valued function Then thestate space is R To bring this system to a system withoutdelay enlargeR toR times 119871
2(minus120576 0) and define 119871
2(minus120576 0)-valued
function
[119909119905]120579= 119909119905+120579
119905 ge 0 minus120576 le 120579 le 0 (15)
Then for
119910119905= [
119909119905
119909119905
] 119860 =[
[
0 0
0
119889
119889120579
]
]
119865 (119905 119910 119906) = [
119891 (119905 119909 119909 119906)
0]
(16)
the above system can be written as the abstract system
1199101015840
119905= 119860119910119905+ 119891 (119905 119910
119905 119906119905) (17)
Similar to the previous examples one can easily observe thatthe ordinary controllability concepts for the system (17) aretoo strong for the system (14) but the 119871-partial controllabilityconcepts of the system for 119910 with
119871 = [119868 0] R times 1198712(0 1) 997888rarr R (18)
are exactly the ordinary controllability concepts of the systemfor 119909
These examples motivate a study of the partial control-lability concepts In this paper it is proved that under theconditions (A)ndash(F) the system in (1) is 119871-partially completecontrollable
3 Main Result
Denote119883 = 119862(0 119879119883) Then119883 times 119880ad is a Banach space withthe norm
(sdot sdot)times119880ad
= sdot+ sdot119880ad
(19)
Lemma 4 Under the conditions (A) (B) and (C)1003817100381710038171003817119876119905
1003817100381710038171003817le1003817100381710038171003817119876119879
1003817100381710038171003817
10038171003817100381710038171003817119876119905
10038171003817100381710038171003817le
10038171003817100381710038171003817119876119879
10038171003817100381710038171003817 0 le 119905 le 119879 (20)
Proof It is easy to see that 119876119905= 119876lowast
119905and ⟨119876
119905119909 119909⟩ ge 0 for all
119909 isin 119883 Hence1003817100381710038171003817119876119905
1003817100381710038171003817= sup119909=1
⟨119876119905119909 119909⟩ (21)
Then
⟨119876119879119909 119909⟩ = int
119879
0
⟨119890119860119904119861119861lowast119890119860lowast
119904119909 119909⟩119889119904
= ⟨119876119905119909 119909⟩ + int
119879
119905
⟨119890119860119904119861119861lowast119890119860lowast
119904119909 119909⟩119889119904
= ⟨119876119905119909 119909⟩ + int
119879
119905
100381710038171003817100381710038171003817
119861lowast119890119860lowast
119904119909
100381710038171003817100381710038171003817
2
119889119904
ge ⟨119876119905119909 119909⟩
(22)
This implies 119876119905 le 119876
119879 The conclusion of the lemma
regarding 119876119879follows from ⟨119876
119905119909 119909⟩ = ⟨119876
119905119871lowast119909 119871lowast119909⟩
The proof of the following lemma appears in differentforms in several papers for example Mahmudov [13] Ourproof is a minor modification of them
Lemma 5 Assume that the conditions (A)ndash(F) hold and takearbitrary ℎ isin 119867 Then for the operator 119866 119883 times 119880
119886119889rarr 119883 times
119880119886119889 defined by
119866 (119910 V) (119905) = (119884 (119905) 119881 (119905)) 0 le 119905 le 119879 (23)
where
119884 (119905) = minus119876119905119890119860lowast
(119879minus119905)119871lowast119876minus1
119879119871int
119879
0
119890119860(119879minus119904)
119891 (119904 119910119904 V119904) 119889119904
+ int
119905
0
119890119860(119905minus119904)
119891 (119904 119910119904 V119904) 119889119904
119881 (119905) = 119861lowast119890119860lowast
(119879minus119905)119871lowast119876minus1
119879
times (ℎ minus 1198711198901198601198791199090minus 119871int
119879
0
119890119860(119879minus119904)
119891 (119904 119910119904 V119904) 119889119904)
(24)
the following inequality holds
1003817100381710038171003817119866 (119910 V) minus 119866 (119911 119908)
1003817100381710038171003817le (1 +
1003817100381710038171003817119876119879
1003817100381710038171003817119872
120574
+
119861119872
120574
)
times119872119870119879 (1003817100381710038171003817119910 minus 119911
1003817100381710038171003817+ V minus 119908)
(25)
where
119872 = sup0le119905le119879
1003817100381710038171003817100381711989011986011990510038171003817100381710038171003817 (26)
4 Abstract and Applied Analysis
Proof Let (119910 V) and (119911 119908) be two functions in 119883 times 119880ad suchthat 119866(119910 V) = (119884 119881) and 119866(119911 119908) = (119885119882) Then
1003817100381710038171003817119866(119910 V) minus 119866(119911 119908)
1003817100381710038171003817times119880ad
= 119884 minus 119885+ 119881 minus119882
119880ad (27)
Here 119884 minus 119885can be estimated as follows
119884 minus 119885 = max119905isin[0119879]
10038171003817100381710038171003817100381710038171003817
int
119905
0
119890119860(119905minus119904)
(119891 (119904 119910119904 V119904) minus 119891 (119904 119911
119904 119908119904)) 119889119904
minus 119876119905119890119860lowast
(119879minus119905)119871lowast119876minus1
119879119871
timesint
119879
0
119890119860(119879minus119904)
(119891 (119904 119910119904 V119904)minus119891 (119904 119911
119904 119908119904)) 119889119904
100381710038171003817100381710038171003817100381710038171003817
le max119905isin[0119879]
(119872 +1198722 1003817100381710038171003817119876119905
1003817100381710038171003817
10038171003817100381710038171003817119876minus1
119879
10038171003817100381710038171003817)
times int
119879
0
1003817100381710038171003817119891 (119904 119910
119904 V119904) minus 119891 (119904 119911
119904 119908119904)1003817100381710038171003817119889119904
le 119872(1 +1198721003817100381710038171003817119876119879
1003817100381710038171003817
10038171003817100381710038171003817119876minus1
119879
10038171003817100381710038171003817)
times int
119879
0
1003817100381710038171003817119891 (119904 119910
119904 V119904) minus 119891 (119904 119911
119904 119908119904)1003817100381710038171003817119889119904
le (1 +
1003817100381710038171003817119876119879
1003817100381710038171003817119872
120574
)119872119870int
119879
0
(1003817100381710038171003817119910119904minus 119911119904
1003817100381710038171003817+1003817100381710038171003817V119904minus 119908119904
1003817100381710038171003817) 119889119904
le (1 +
1003817100381710038171003817119876119879
1003817100381710038171003817119872
120574
)119872119870119879 (1003817100381710038171003817119910 minus 119911
1003817100381710038171003817+ V minus 119908)
(28)
Similarly for 119881 minus119882119880ad
we have
119881 minus119882 = max119905isin[0119879]
100381710038171003817100381710038171003817100381710038171003817
119861lowast119890119860lowast
(119879minus119905)119871lowast119876minus1
119879119871
timesint
119879
0
119890119860(119879minus119904)
(119891 (119904 119910119904 V119904)minus119891 (119904 119911
119904 119908119904)) 119889119904
100381710038171003817100381710038171003817100381710038171003817
le 1198722
119861
10038171003817100381710038171003817119876minus1
119879
10038171003817100381710038171003817int
119879
0
1003817100381710038171003817119891 (119904 119910
119904 V119904) minus 119891 (119904 119911
119904 119908119904)1003817100381710038171003817119889119904
le 1198722
119861
1
120574
119870int
119879
0
(1003817100381710038171003817119910119904minus 119911119904
1003817100381710038171003817+1003817100381710038171003817V119904minus 119908119904
1003817100381710038171003817) 119889119904
le
119861119872
120574
119872119870119879 (1003817100381710038171003817119910 minus 119911
1003817100381710038171003817+ V minus 119908)
(29)
Combining (28) and (29) we obtain the demanded inequal-ity
Lemma 6 Under the conditions (A)ndash(F) if
(1 +
1003817100381710038171003817119876119879
1003817100381710038171003817119872
120574
+
119861119872
120574
)119872119870119879 lt 1 (30)
then the operator 119866 mapping 119883 times 119880119886119889
into 119883 times 119880119886119889 has a
unique fixed point (119909 119906) isin 119883 times 119880119886119889
Proof By Lemma 5 119866 is a contraction mapping Also thespace119883 times 119880ad is complete Hence 119866 has a fixed point
Theorem 7 Under the conditions (A)ndash(F) and (30) the semi-linear system (1) is 119871-partially complete controllable on [0 119879]
Proof Take any 1199090isin 119883 and ℎ isin 119867 Show that there is 119906 isin 119880ad
such that ℎ = 119871119909119879 To this end consider 119906 defined as follows
119906119905= 119861lowast119890119860lowast
(119879minus119905)119871lowast119876minus1
119879
times (ℎ minus 1198711198901198601198791199090minus 119871int
119879
0
119890119860(119879minus119904)
119891 (119904 119909119904 119906119904) 119889119904)
(31)
Substituting (31) in (4) and applying Fubinirsquos theorem (seeBashirov [8] p 45) we obtain
119909119905= 1198901198601199051199090
+ int
119905
0
119890119860(119905minus119904)
119861119861lowast119890119860lowast
(119905minus119904)119890119860lowast
(119879minus119905)119871lowast119876minus1
119879(ℎ minus 119871119890
1198601198791199090) 119889119904
minus int
119905
0
int
119879
0
119890119860(119905minus119904)
119861119861lowast119890119860lowast
(119905minus119904)119890119860lowast
(119879minus119905)119871lowast119876minus1
119879119871119890119860(119879minus119903)
times 119891 (119903 119909119903 119906119903) 119889119903 119889119904
+ int
119905
0
119890119860(119905minus119904)
119891 (119904 119909119904 119906119904) 119889119904
= 1198901198601199051199090+ 119876119905119890119860lowast
(119879minus119905)119871lowast119876minus1
119879(ℎ minus 119871119890
1198601198791199090)
+ int
119905
0
119890119860(119905minus119904)
119891 (119904 119909119904 119906119904) 119889119904
minus int
119879
0
int
119905
0
119890119860(119905minus119904)
119861119861lowast119890119860lowast
(119905minus119904)119890119860lowast
(119879minus119905)119871lowast119876minus1
119879119871119890119860(119879minus119903)
times 119891 (119903 119909119903 119906119903) 119889119904 119889119903
Abstract and Applied Analysis 5
= 1198901198601199051199090+ 119876119905119890119860lowast
(119879minus119905)119871lowast119876minus1
119879(ℎ minus 119871119890
1198601198791199090)
+ int
119905
0
119890119860(119905minus119904)
119891 (119904 119909119904 119906119904) 119889119904
minus int
119879
0
119876119905119890119860lowast
(119879minus119905)119871lowast119876minus1
119879119871119890119860(119879minus119903)
119891 (119903 119909119903 119906119903) 119889119903
= 1198901198601199051199090+ 119876119905119890119860lowast
(119879minus119905)119871lowast119876minus1
119879(ℎ minus 119871119890
1198601198791199090)
+ int
119905
0
119890119860(119905minus119904)
119891 (119904 119909119904 119906119904) 119889119904
minus 119876119905119890119860lowast
(119879minus119905)119871lowast119876minus1
119879119871int
119879
0
119890119860(119879minus119904)
119891 (119904 119909119904 119906119904) 119889119904
(32)
According to Lemma 6 there is a unique pair (119909 119906) isin 119883 times
119880ad satisfying (31) and (32) So 119906 isin 119880ad Furthermore wehave
119871119909119879= 119871(119890
1198601198791199090+ 119876119879119871lowast119876minus1
119879(ℎ minus 119871119890
1198601198791199090)
+ int
119879
0
119890119860(119879minus119904)
119891 (119904 119909119904 119906119904) 119889119904
minus 119876119879119871lowast119876minus1
119879119871int
119879
0
119890119860(119879minus119904)
119891 (119904 119909119904 119906119904) 119889119904)
= 1198711198901198601198791199090+ 119871119876119879119871lowast119876minus1
119879(ℎ minus 119871119890
1198601198791199090)
+ 119871int
119879
0
119890119860(119879minus119904)
119891 (119904 119909119904 119906119904) 119889119904
minus 119871119876119879119871lowast119876minus1
119879119871int
119879
0
119890119860(119879minus119904)
119891 (119904 119909119904 119906119904) 119889119904 = ℎ
(33)
Thus there is119906 isin 119880ad which steers1199090 to119909119879with119871119909119879 = ℎThismeans that the semilinear system (1) is 119871-partially completecontrollable on [0 119879] as desired
Remark 8 Decomposing 119876119879in the form
119876119879= [
119876119879
119877119879
119877lowast
119877119875119879
] (34)
where 119877119879 119867perprarr 119867 and 119875
119879 119867perprarr 119867perp are other compo-
nents of119876119879besides119876
119879and119867perp is an orthogonal complement
of119867 in119883 one can calculate
⟨119876119879ℎ ℎ⟩ = ⟨119876
119879ℎ1 ℎ1⟩ + 2⟨119877
119879ℎ2 ℎ1⟩ + ⟨119875
119879ℎ2 ℎ2⟩ (35)
where ℎ1= 119871ℎ isin 119867 and ℎ
2= ℎ minus 119871ℎ isin 119867
perp Therefore thecoercivity of 119876
119879implies the same of 119876
119879 But the converse
is not true Theorem 7 is powerful in the cases when 119876119879is
coercive but 119875119879is not
Example 9 Theorem 7 establishes just sufficient conditionof 119871-partial complete controllability In this example we will
demonstrate that this is not a necessary condition We willconsider a simple case of 119871 = 119868 when 119871-partial completecontrollability reduces to complete controllability Considerthe one-dimensional control system
1199091015840
119905= 2119909119905+ 2119906119905 1199090isin R (36)
This is a linear system and the controllability operator of thissystem is equal to
int
119879
0
41198904119905119889119905 = 119890
4119879minus 1 gt 0 for every 119879 gt 0 (37)
According to the theory of controllability for linear systemsthis system is controllable (completely) for every 119879 gt 0
Have another look at this system by writing it as
1199091015840
119905= 119909119905+ 119906119905+ 119891 (119909
119905 119906119905) 119909
0isin R (38)
where
119891 (119909 119906) = 119909 + 119906 (39)
Here 119891 satisfies the Lipschitz condition with 119870 = 1 Also119860 = 119861 = 1 implying 119861 = 1 and 119872 = sup
[0119879]119890119860119905 = 119890119879
Furthermore
119876119879= int
119879
0
1198902119905119889119905 =
1198902119879
minus 1
2
(40)
So 119876119879 = 120574 = (119890
2119879minus1)2 Then the inequality (30) becomes
(1 + 119890119879+
2119890119879
119890119879minus 1
) 119890119879119879 lt 1 (41)
The limit of the left-hand side in this inequality when 119879 rarr
infin is equal toinfin This means that there is a sufficiently large119879 such that the conditions of Theorem 7 do not hold for this119879 although the system under consideration is completelycontrollable Thus Theorem 7 states a sufficient conditionwhich is not a necessary condition
4 Examples
We demonstrate the features of 119871-partial complete controlla-bility in the following examples of control systems
Example 1 Consider the system of differential equations
1199091015840
119905= 119910119905+ 119887119906119905 1199090isin R
1199101015840
119905= 119891 (119905 119909
119905 119910119905 119906119905) 119910
0isin R
(42)
on [0 119879] where 119906 isin 119880ad = 119862(0 119879R) Besides the completecontrollability property that is
(119909 119910) isin R2 exist119906 isin 119880ad such that (119909
119879 119910119879) = (119909 119910) = R
2
(43)
we can investigate the partial complete controllability prop-erty that is
119909 isin R exist119906 isin 119880ad such that 119909119879= 119909 = R (44)
6 Abstract and Applied Analysis
We can write this system in R2 as the following semilinearsystem
1199111015840
119905= 119860119911119905+ 119865 (119905 119911
119905 119906119905) + 119861119906
119905 (45)
where
119911119905= [
119909119905
119910119905
] 119860 = [
0 1
0 0] 119861 = [
119887
0]
119865 (119905 119911 119906) = [
0
119891 (119905 119909 119910 119906)]
(46)
assuming that
119911 = [
119909
119910] (47)
It can be calculated that
119890119860119905
= [
1 119905
0 1] = [
1 0
0 1] + [
0 119905
0 0] (48)
Hence1003817100381710038171003817100381711989011986011990510038171003817100381710038171003817le 1 + 119905 le 1 + 119879 0 le 119905 le 119879 (49)
The controllability operator is
119876119879= int
119879
0
119890119860119905119861119861lowast119890119860lowast
119905119889119905 = 119887
2119879[
1 0
0 0] (50)
Hence 119876119879is not coercive and the conditions for complete
controllability based on coercivity of119876119879 fail for this example
Although system (42) can still be complete controllable forproperly selected functions 119891 we can investigate the partialcomplete controllability for this system being interested injust the first component 119909
119905of 119911119905
Let 119871 = [1 0] Then
119876119879= 119871119876119879119871lowast= 1198872119879 gt 0 (51)
This means that the linear system associated with the semili-near system (45) is119871-partially complete controllable Further-more the inequality (30) becomes
(1 +
1198872119879 (1 + 119879)
1198872119879
+
119887 (1 + 119879)
1198872119879
) (1 + 119879) 119879119870 lt 1 (52)
or simplifying
119870 lt
119887
(1 + 119879) (1 + 119879 + 2119887119879 + 1198871198792)
(53)
This establishes a relation between Lipschitz coefficient 119870and terminal time moment 119879 Depending on 119870 119879 must betaken sufficiently large to satisfy (53) So the system (42) is 119871-partially complete controllable for the time 119879 if the Lipschitzcoefficient 119870 related to 119891 satisfies (53)
Example 2 Delay equations are typical for application ofpartial controllability concepts Consider a nonlinear delayequation
1199091015840
119905= 119886119909119905+ 119887119906119905+ 119891(119905 119909
119905 int
0
minus120576
119909119905+120579
119889120579 119906119905)
1199090= 120585 119909
120579= 120578120579 minus120576 le 120579 le 0
(54)
on [0 119879] where 0 lt 120576 lt 119879 120578 isin 1198712(minus120576 0R) and 119906 isin 119880ad =
119862(0 119879R)Similar to Example 3 introduce the function119909 [0 119879] rarr
1198712(minus120576 0R) by
[119909119905]120579= 119909119905+120579
0 le 119905 le 119879 minus120576 le 120579 le 0 (55)
This function satisfies
1199091015840
119905= (
119889
119889120579
)119909119905 119909
0= 120578 0 lt 119905 le 119879 (56)
Denote by T119905 119905 ge 0 the semigroup generated by the
differential operator 119889119889120579 and let Γ be the integral operatorfrom 119871
2(minus120576 0R) to R defined by
Γℎ = int
0
minus120576
ℎ120579119889120579 ℎ isin 119871
2(minus120576 0R) (57)
noticing that Γ le radic120576 Then for
119910119905= [
119909119905
119909119905
] 120577 = [
120585
120578] isin R times 119871
2(minus120576 0R) (58)
we can write system (54) as
1199101015840
119905= 119860119910119905+ 119865 (119905 119910
119905 119906119905) + 119861119906
119905 1199100= 120577 (59)
where
119860 =[
[
119886 0
0
119889
119889120579
]
]
119865 (119905 119910 119906) = [
119891 (119905 119909 Γ119909 119906)
0]
119861 = [
119887
0]
(60)
assuming that
119910 = [
119909
119909] isin R times 119871
2(minus120576 0R) (61)
The semigroup generated by 119860 has the form
119890119860119905
= [119890119886119905
0
0 T119905
] 119905 ge 0 (62)
Therefore the controllability operator for system (54) can becalculated as
119876119879= int
119879
0
119890119860119905119861lowast119861119890119860lowast
119905119889119905 = int
119879
0
[11988721198902119886119905
0
0 0
] 119889119905
=[
[
[
1198872(1198902119886119879
minus 1)
2119886
0
0 0
]
]
]
(63)
This is definitely not a coercive operator
Abstract and Applied Analysis 7
Taking into account that the original system is given by(54) and (59) is just representation of (54) in the standardform which enlarges the original state space R to R times
1198712(minus120576 0R) we observe that the complete controllability for
system (54) is in fact 119871-partial complete controllability forsystem (59) if
119871 = [1 0] R times 1198712(minus120576 0R) 997888rarr R (64)
Calculating partial controllability operator we obtain
119876119879= 119871119876119879119871lowast=
1198872(1198902119886119879
minus 1)
2119886
gt 0(65)
which is coerciveFurthermore using
1003817100381710038171003817100381711989011986011990510038171003817100381710038171003817le 1 + 119890
119886119879
1003817100381710038171003817119876119879
1003817100381710038171003817= 120574 =
1198872(1198902119886119879
minus 1)
2119886
(66)
we write the inequality (30) in the form
(1 + 1 + 119890119886119879
+
2119886
119887 (119890119886119879
minus 1)
) (1 + 119890119886119879) 119879119870 lt 1 (67)
If the Lipschitz coefficient 119870 of the function 119865 and terminaltime moment 119879 satisfy this inequality then system (54) iscompletely controllable which in turnmeans that system (59)is 119871-partially complete controllable
5 Conclusion
In this paper a sufficient condition for partial complete con-trollability of a semilinear control system is proved This is acontinuation of the pioneering research that has been done byBashirov et al [20 21] about partial controllability conceptsA research in this way concerning partial complete andapproximate controllability for semilinear deterministic andstochastic systems has already been done and awaitingfor publication There are other kinds of systems whichbesides semilinearity include other features for exampleimpulsiveness fractional derivative issue and so forth Theresult of this paper can be extended to these systems as well
References
[1] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering D Transactions ofASME vol 82 pp 35ndash45 1960
[2] H O Fattorini ldquoSome remarks on complete controllabilityrdquoSIAM Journal on Control vol 4 pp 686ndash694 1966
[3] D L Russell ldquoNonharmonic Fourier series in the control theoryof distributed parameter systemsrdquo Journal of MathematicalAnalysis and Applications vol 18 pp 542ndash560 1967
[4] R F Curtain and H J ZwartAn Introduction to Infinite Dimen-sional Linear Systems Theory Springer Berlin Germany 1995
[5] A Bensoussan Stochastic Control of Partially Observable Sys-tems Cambridge University Press Cambridge UK 1992
[6] A Bensoussan G Da Prato M C Delfour and S K MitterRepresentation and Control of Infinite Dimensional SystemsSystems amp Control Foundations amp Applications BirkhauserBoston Mass USA 2nd edition 2007
[7] J Zabczyk Mathematical Control Theory An IntroductionSystems amp Control Foundations amp Applications BirkhauserBoston Mass USA 1995
[8] A E Bashirov Partially Observable Linear Systems underDependent Noises Systems amp Control Foundations amp Applica-tions Birkhauser Basel Switzerland 2003
[9] J Klamka Controllability of Dynamical Systems vol 48 ofMathematics and Its Applications Kluwer Academic PublishersDordrecht The Netherlands 1991
[10] K Balachandran and J P Dauer ldquoControllability of nonlinearsystems in Banach spaces a surveyrdquo Journal of OptimizationTheory and Applications vol 115 no 1 pp 7ndash28 2002
[11] K Balachandran and J P Dauer ldquoLocal controllability of semili-near evolution systems in Banach spacesrdquo Indian Journal of Pureand Applied Mathematics vol 29 no 3 pp 311ndash320 1998
[12] J Klamka ldquoSchauderrsquos fixed-point theorem in nonlinear con-trollability problemsrdquo Control and Cybernetics vol 29 no 1 pp153ndash165 2000
[13] N I Mahmudov ldquoControllability of semilinear stochastic sys-tems in Hilbert spacesrdquo Journal of Mathematical Analysis andApplications vol 288 no 1 pp 197ndash211 2003
[14] X J Li and J M Yong Optimal Control Theory for Infinite-Dimensional Systems SystemsampControl Foundations ampAppli-cations Birkhauser Boston Mass USA 1995
[15] R Sakthivel N I Mahmudov and J J Nieto ldquoControllabilityfor a class of fractional-order neutral evolution control systemsrdquoApplied Mathematics and Computation vol 218 no 20 pp10334ndash10340 2012
[16] R Sakthivel R Ganesh and S Suganya ldquoApproximate con-trollability of fractional neutral stochastic system with infinitedelayrdquo Reports on Mathematical Physics vol 70 no 3 pp 291ndash311 2012
[17] R Sakthivel Y Ren and N I Mahmudov ldquoOn the approximatecontrollability of semilinear fractional differential systemsrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1451ndash1459 2011
[18] Z Yan ldquoApproximate controllability of partial neutral func-tional differential systems of fractional order with state-dependent delayrdquo International Journal of Control vol 85 no8 pp 1051ndash1062 2012
[19] Y Ren L Hu and R Sakthivel ldquoControllability of impulsiveneutral stochastic functional differential inclusions with infinitedelayrdquo Journal of Computational and Applied Mathematics vol235 no 8 pp 2603ndash2614 2011
[20] A E Bashirov H Etikan andN Semi ldquoPartial controllability ofstochastic linear systemsrdquo International Journal of Control vol83 no 12 pp 2564ndash2572 2010
[21] A E Bashirov N Mahmudov N Semi and H Etikan ldquoPartialcontrollability conceptsrdquo International Journal of Control vol80 no 1 pp 1ndash7 2007
[22] A E Bashirov and N I Mahmudov ldquoOn concepts of control-lability for deterministic and stochastic systemsrdquo SIAM Journalon Control and Optimization vol 37 no 6 pp 1808ndash1821 1999
[23] A E Bashirov and K R Kerimov ldquoOn controllability concep-tion for stochastic systemsrdquo SIAM Journal on Control and Opti-mization vol 35 no 2 pp 384ndash398 1997
8 Abstract and Applied Analysis
[24] A E Bashirov ldquoOn weakening of the controllability conceptsrdquoin Proceedings of the 35th Conference on Decission and Controlpp 640ndash645 Kobe Japan 1996
[25] A E Bashirov and N I Mahmudov ldquoControllability of lineardeterministic and stochastic systemsrdquo in Proceedings of the 38thConference on Decission and Control pp 3196ndash3201 PhoenixAris USA 1999
[26] A E Bashirov and N I Mahmudov ldquoSome new results in the-ory of controllabilityrdquo in Proceedings of the 7th MediterraneanConference on Control and Automation pp 323ndash343 HaifaIsrael 1999
[27] L Byszewski ldquoTheorems about the existence and uniqueness ofsolutions of a semilinear evolution nonlocal Cauchy problemrdquoJournal of Mathematical Analysis and Applications vol 162 no2 pp 494ndash505 1991
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DifferentialEquations
International Journal of
Volume 2013
2 Abstract and Applied Analysis
2 Setting the Problem and Motivation
Consider the basic semilinear control system
1199091015840
119905= 119860119909119905+ 119861119906119905+ 119891 (119905 119909
119905 119906119905) 0 lt 119905 le 119879 119909
0isin 119883 (1)
on the interval [0 119879] with 119879 gt 0 where 119909 and 119906 are state andcontrol processes We assume that the following conditionshold
(A) 119883 and 119880 are separable Hilbert spaces 119867 is a closedsubspace of 119883 and 119871 is a projection operator from 119883
to119867(B) 119860 is a densely defined closed linear operator on 119883
generating a strongly continuous semigroup 119890119860119905 119905 ge
0(C) 119861 is a bounded linear operator from 119880 to119883(D) 119891 is a nonlinear function from [0 119879] times 119883 times 119880 to 119883
satisfying that
(i) 119891 is continuous on [0 119879] times 119883 times 119880(ii) 119891 is Lipschitz continuous with respect to 119909 and
119906 that is for all 119905 isin [0 119879] 119906 V isin 119880 and 119909 119910 isin 1198831003817100381710038171003817119891 (119905 119909 119906) minus 119891 (119905 119910 V)100381710038171003817
1003817le 119870 (
1003817100381710038171003817119909 minus 119910
1003817100381710038171003817+ 119906 minus V) (2)
for some119870 ge 0
(E) 119880ad = 119862(0 119879 119880) is the space of all continuous fun-ctions from [0 119879] to 119880Define the controllability and 119871-partial controllabilityoperators 119876
119905and 119876
119905by
119876119905= int
119905
0
119890119860119904119861119861lowast119890119860lowast
119904119889119904 119876
119905= 119871119876119905119871lowast (3)
where 119871lowast is the adjoint of 119871 The 119871-partial controlla-bility operator becomes the controllability operator if119871 = 119868 (the identity operator) We will also assume thefollowing condition
(F) 119876119879is coercive that is there is 120574 gt 0 such that
⟨119876119879ℎ ℎ⟩ ge 120574ℎ
2 for all ℎ isin 119867
Note that this condition implies the existence of 119876minus1119879
as abounded linear operator and 119876
minus1
119879 le 1120574 Respectively the
linear system associated with (1) (the case when 119891 = 0) is119871-partially complete controllable on the interval [0 119879] (seeBashirov et al [20 21])
The above conditions imply the existence of a uniquecontinuous solution of (1) in the mild sense for every 119906 isin 119880adand 119909
0isin 119883 (see Li and Yong [14] and Byszewski [27]) that is
there is a unique continuous function 119909 from [0 119879] to119883 suchthat
119909119905= 1198901198601199051199090+ int
119905
0
119890119860(119905minus119904)
(119861119906119904+ 119891 (119904 119909
119904 119906119904)) 119889119904 (4)
Let
1198631199090
119879= ℎ isin 119867 exist119906 isin 119880ad such that ℎ = 119871119909
119879 (5)
Following Bashirov et al [21] the semilinear control system(1) is said to be 119871-partially complete controllable on 119880ad if1198631199090
119879= 119867 for all 119909
0isin 119883 Similarly the semilinear system in
(1) is said to be 119871-partially approximate controllable on119880ad if1198631199090
119879= 119867 for all 119909
0isin 119883 where 1198631199090
119879is the closure of 1198631199090
119879 If
119867 = 119883 these are just well-known complete and approximatecontrollability concepts respectively In this paper we studythe concept of 119871-partial complete controllability
The reason for studying 119871-partial controllability conceptsis that many systems can be written in the form of (1)if the original state space is enlarged Therefore suitablecontrollability concepts for such systems are the 119871-partialcontrollability concepts with the operator 119871 projecting theenlarged state space to the original one Here are someexamples of such systems which are discussed in Bashirov[8] Section 311 in more details
Example 1 Consider the system
119909(119899)
119905= 119891 (119905 119909
119905 1199091015840
119905 119909
(119899minus1)
119905 119906119905) (6)
assuming that its state space is the one-dimensional spaceR The ordinary controllability concepts for this system arethe equality to or denseness in R of the respective attainableset We can write this system as the first-order differentialequation
1199101015840
119905= 119860119910119905+ 119865 (119905 119910
119905 119906119905) (7)
if
119910119905=
[
[
[
[
[
[
[
119909119905
1199091015840
119905
119909(119899minus2)
119905
119909(119899minus1)
119905
]
]
]
]
]
]
]
119860 =
[
[
[
[
[
[
[
0 1 sdot sdot sdot 0 0
0 0 sdot sdot sdot 0 0
d
0 0 sdot sdot sdot 0 1
0 0 sdot sdot sdot 0 0
]
]
]
]
]
]
]
119865 (119905 119910 119906) =
[
[
[
[
[
[
[
0
0
0
119891 (119905 119909 1199091015840 119909
(119899minus1) 119906)
]
]
]
]
]
]
]
(8)
The state space of this system is the 119899-dimensional Euclideanspace R119899 and respectively its attainable set is a subset of R119899Therefore the controllability concepts of the system for 119910 arestronger than those of the system for 119909 But if we define theprojection operator 119871 by
119871 = [1 0 sdot sdot sdot 0 0] R119899997888rarr R (9)
then the 119871-partial controllability concepts of the system for 119910become the same as the ordinary controllability concepts ofthe system for 119909
Example 2 Consider the nonlinear wave equation
1205972119909119905120579
1205971199052
=
1205972119909119905120579
1205971205792
+ 119891(119905 119909119905120579
120597119909119905120579
120597119905
119906119905) (10)
Abstract and Applied Analysis 3
where 119909 is a real-valued function of two variables 119905 ge 0 and0 le 120579 le 1 The state space of this system is 119871
2(0 1) (the space
of square integrable functions on [0 1]) This system can bewritten as the first-order abstract differential equation
1199101015840
119905= 119860119910119905+ 119865 (119905 119910
119905 119906119905) (11)
if
119910119905=[
[
119909119905120579
120597119909119905120579
120597119905
]
]
119860 =[
[
0 119868
1198892
1198891205792
0
]
]
119865 (119905 119910 119906) = [
0
119891 (119905 1199101 1199102 119906)
]
(12)
where 119910 isin 1198712(0 1) times 119871
2(0 1) The state space 119871
2(0 1) times
1198712(0 1) of the system for 119910 is the enlargement of the state
space 1198712(0 1) of the system for 119909 This is a cost that is
paid to bring the wave equation to the form of first-orderdifferential equation The ordinary controllability conceptsfor the system (11) are too strong for the system (10) If
119871 = [119868 0] 1198712(0 1) times 119871
2(0 1) 997888rarr 119871
2(0 1) (13)
then 119871-partially controllability concepts of the system for 119910become ordinary controllability concepts of the system for 119909
Example 3 Consider the system
1199091015840
119905= 119891(119905 119909
119905 int
0
minus120576
119909119905+120579
119889120579 119906119905) (14)
which contains a simple distributed delay in the nonlinearterm assuming that 119909 is a real-valued function Then thestate space is R To bring this system to a system withoutdelay enlargeR toR times 119871
2(minus120576 0) and define 119871
2(minus120576 0)-valued
function
[119909119905]120579= 119909119905+120579
119905 ge 0 minus120576 le 120579 le 0 (15)
Then for
119910119905= [
119909119905
119909119905
] 119860 =[
[
0 0
0
119889
119889120579
]
]
119865 (119905 119910 119906) = [
119891 (119905 119909 119909 119906)
0]
(16)
the above system can be written as the abstract system
1199101015840
119905= 119860119910119905+ 119891 (119905 119910
119905 119906119905) (17)
Similar to the previous examples one can easily observe thatthe ordinary controllability concepts for the system (17) aretoo strong for the system (14) but the 119871-partial controllabilityconcepts of the system for 119910 with
119871 = [119868 0] R times 1198712(0 1) 997888rarr R (18)
are exactly the ordinary controllability concepts of the systemfor 119909
These examples motivate a study of the partial control-lability concepts In this paper it is proved that under theconditions (A)ndash(F) the system in (1) is 119871-partially completecontrollable
3 Main Result
Denote119883 = 119862(0 119879119883) Then119883 times 119880ad is a Banach space withthe norm
(sdot sdot)times119880ad
= sdot+ sdot119880ad
(19)
Lemma 4 Under the conditions (A) (B) and (C)1003817100381710038171003817119876119905
1003817100381710038171003817le1003817100381710038171003817119876119879
1003817100381710038171003817
10038171003817100381710038171003817119876119905
10038171003817100381710038171003817le
10038171003817100381710038171003817119876119879
10038171003817100381710038171003817 0 le 119905 le 119879 (20)
Proof It is easy to see that 119876119905= 119876lowast
119905and ⟨119876
119905119909 119909⟩ ge 0 for all
119909 isin 119883 Hence1003817100381710038171003817119876119905
1003817100381710038171003817= sup119909=1
⟨119876119905119909 119909⟩ (21)
Then
⟨119876119879119909 119909⟩ = int
119879
0
⟨119890119860119904119861119861lowast119890119860lowast
119904119909 119909⟩119889119904
= ⟨119876119905119909 119909⟩ + int
119879
119905
⟨119890119860119904119861119861lowast119890119860lowast
119904119909 119909⟩119889119904
= ⟨119876119905119909 119909⟩ + int
119879
119905
100381710038171003817100381710038171003817
119861lowast119890119860lowast
119904119909
100381710038171003817100381710038171003817
2
119889119904
ge ⟨119876119905119909 119909⟩
(22)
This implies 119876119905 le 119876
119879 The conclusion of the lemma
regarding 119876119879follows from ⟨119876
119905119909 119909⟩ = ⟨119876
119905119871lowast119909 119871lowast119909⟩
The proof of the following lemma appears in differentforms in several papers for example Mahmudov [13] Ourproof is a minor modification of them
Lemma 5 Assume that the conditions (A)ndash(F) hold and takearbitrary ℎ isin 119867 Then for the operator 119866 119883 times 119880
119886119889rarr 119883 times
119880119886119889 defined by
119866 (119910 V) (119905) = (119884 (119905) 119881 (119905)) 0 le 119905 le 119879 (23)
where
119884 (119905) = minus119876119905119890119860lowast
(119879minus119905)119871lowast119876minus1
119879119871int
119879
0
119890119860(119879minus119904)
119891 (119904 119910119904 V119904) 119889119904
+ int
119905
0
119890119860(119905minus119904)
119891 (119904 119910119904 V119904) 119889119904
119881 (119905) = 119861lowast119890119860lowast
(119879minus119905)119871lowast119876minus1
119879
times (ℎ minus 1198711198901198601198791199090minus 119871int
119879
0
119890119860(119879minus119904)
119891 (119904 119910119904 V119904) 119889119904)
(24)
the following inequality holds
1003817100381710038171003817119866 (119910 V) minus 119866 (119911 119908)
1003817100381710038171003817le (1 +
1003817100381710038171003817119876119879
1003817100381710038171003817119872
120574
+
119861119872
120574
)
times119872119870119879 (1003817100381710038171003817119910 minus 119911
1003817100381710038171003817+ V minus 119908)
(25)
where
119872 = sup0le119905le119879
1003817100381710038171003817100381711989011986011990510038171003817100381710038171003817 (26)
4 Abstract and Applied Analysis
Proof Let (119910 V) and (119911 119908) be two functions in 119883 times 119880ad suchthat 119866(119910 V) = (119884 119881) and 119866(119911 119908) = (119885119882) Then
1003817100381710038171003817119866(119910 V) minus 119866(119911 119908)
1003817100381710038171003817times119880ad
= 119884 minus 119885+ 119881 minus119882
119880ad (27)
Here 119884 minus 119885can be estimated as follows
119884 minus 119885 = max119905isin[0119879]
10038171003817100381710038171003817100381710038171003817
int
119905
0
119890119860(119905minus119904)
(119891 (119904 119910119904 V119904) minus 119891 (119904 119911
119904 119908119904)) 119889119904
minus 119876119905119890119860lowast
(119879minus119905)119871lowast119876minus1
119879119871
timesint
119879
0
119890119860(119879minus119904)
(119891 (119904 119910119904 V119904)minus119891 (119904 119911
119904 119908119904)) 119889119904
100381710038171003817100381710038171003817100381710038171003817
le max119905isin[0119879]
(119872 +1198722 1003817100381710038171003817119876119905
1003817100381710038171003817
10038171003817100381710038171003817119876minus1
119879
10038171003817100381710038171003817)
times int
119879
0
1003817100381710038171003817119891 (119904 119910
119904 V119904) minus 119891 (119904 119911
119904 119908119904)1003817100381710038171003817119889119904
le 119872(1 +1198721003817100381710038171003817119876119879
1003817100381710038171003817
10038171003817100381710038171003817119876minus1
119879
10038171003817100381710038171003817)
times int
119879
0
1003817100381710038171003817119891 (119904 119910
119904 V119904) minus 119891 (119904 119911
119904 119908119904)1003817100381710038171003817119889119904
le (1 +
1003817100381710038171003817119876119879
1003817100381710038171003817119872
120574
)119872119870int
119879
0
(1003817100381710038171003817119910119904minus 119911119904
1003817100381710038171003817+1003817100381710038171003817V119904minus 119908119904
1003817100381710038171003817) 119889119904
le (1 +
1003817100381710038171003817119876119879
1003817100381710038171003817119872
120574
)119872119870119879 (1003817100381710038171003817119910 minus 119911
1003817100381710038171003817+ V minus 119908)
(28)
Similarly for 119881 minus119882119880ad
we have
119881 minus119882 = max119905isin[0119879]
100381710038171003817100381710038171003817100381710038171003817
119861lowast119890119860lowast
(119879minus119905)119871lowast119876minus1
119879119871
timesint
119879
0
119890119860(119879minus119904)
(119891 (119904 119910119904 V119904)minus119891 (119904 119911
119904 119908119904)) 119889119904
100381710038171003817100381710038171003817100381710038171003817
le 1198722
119861
10038171003817100381710038171003817119876minus1
119879
10038171003817100381710038171003817int
119879
0
1003817100381710038171003817119891 (119904 119910
119904 V119904) minus 119891 (119904 119911
119904 119908119904)1003817100381710038171003817119889119904
le 1198722
119861
1
120574
119870int
119879
0
(1003817100381710038171003817119910119904minus 119911119904
1003817100381710038171003817+1003817100381710038171003817V119904minus 119908119904
1003817100381710038171003817) 119889119904
le
119861119872
120574
119872119870119879 (1003817100381710038171003817119910 minus 119911
1003817100381710038171003817+ V minus 119908)
(29)
Combining (28) and (29) we obtain the demanded inequal-ity
Lemma 6 Under the conditions (A)ndash(F) if
(1 +
1003817100381710038171003817119876119879
1003817100381710038171003817119872
120574
+
119861119872
120574
)119872119870119879 lt 1 (30)
then the operator 119866 mapping 119883 times 119880119886119889
into 119883 times 119880119886119889 has a
unique fixed point (119909 119906) isin 119883 times 119880119886119889
Proof By Lemma 5 119866 is a contraction mapping Also thespace119883 times 119880ad is complete Hence 119866 has a fixed point
Theorem 7 Under the conditions (A)ndash(F) and (30) the semi-linear system (1) is 119871-partially complete controllable on [0 119879]
Proof Take any 1199090isin 119883 and ℎ isin 119867 Show that there is 119906 isin 119880ad
such that ℎ = 119871119909119879 To this end consider 119906 defined as follows
119906119905= 119861lowast119890119860lowast
(119879minus119905)119871lowast119876minus1
119879
times (ℎ minus 1198711198901198601198791199090minus 119871int
119879
0
119890119860(119879minus119904)
119891 (119904 119909119904 119906119904) 119889119904)
(31)
Substituting (31) in (4) and applying Fubinirsquos theorem (seeBashirov [8] p 45) we obtain
119909119905= 1198901198601199051199090
+ int
119905
0
119890119860(119905minus119904)
119861119861lowast119890119860lowast
(119905minus119904)119890119860lowast
(119879minus119905)119871lowast119876minus1
119879(ℎ minus 119871119890
1198601198791199090) 119889119904
minus int
119905
0
int
119879
0
119890119860(119905minus119904)
119861119861lowast119890119860lowast
(119905minus119904)119890119860lowast
(119879minus119905)119871lowast119876minus1
119879119871119890119860(119879minus119903)
times 119891 (119903 119909119903 119906119903) 119889119903 119889119904
+ int
119905
0
119890119860(119905minus119904)
119891 (119904 119909119904 119906119904) 119889119904
= 1198901198601199051199090+ 119876119905119890119860lowast
(119879minus119905)119871lowast119876minus1
119879(ℎ minus 119871119890
1198601198791199090)
+ int
119905
0
119890119860(119905minus119904)
119891 (119904 119909119904 119906119904) 119889119904
minus int
119879
0
int
119905
0
119890119860(119905minus119904)
119861119861lowast119890119860lowast
(119905minus119904)119890119860lowast
(119879minus119905)119871lowast119876minus1
119879119871119890119860(119879minus119903)
times 119891 (119903 119909119903 119906119903) 119889119904 119889119903
Abstract and Applied Analysis 5
= 1198901198601199051199090+ 119876119905119890119860lowast
(119879minus119905)119871lowast119876minus1
119879(ℎ minus 119871119890
1198601198791199090)
+ int
119905
0
119890119860(119905minus119904)
119891 (119904 119909119904 119906119904) 119889119904
minus int
119879
0
119876119905119890119860lowast
(119879minus119905)119871lowast119876minus1
119879119871119890119860(119879minus119903)
119891 (119903 119909119903 119906119903) 119889119903
= 1198901198601199051199090+ 119876119905119890119860lowast
(119879minus119905)119871lowast119876minus1
119879(ℎ minus 119871119890
1198601198791199090)
+ int
119905
0
119890119860(119905minus119904)
119891 (119904 119909119904 119906119904) 119889119904
minus 119876119905119890119860lowast
(119879minus119905)119871lowast119876minus1
119879119871int
119879
0
119890119860(119879minus119904)
119891 (119904 119909119904 119906119904) 119889119904
(32)
According to Lemma 6 there is a unique pair (119909 119906) isin 119883 times
119880ad satisfying (31) and (32) So 119906 isin 119880ad Furthermore wehave
119871119909119879= 119871(119890
1198601198791199090+ 119876119879119871lowast119876minus1
119879(ℎ minus 119871119890
1198601198791199090)
+ int
119879
0
119890119860(119879minus119904)
119891 (119904 119909119904 119906119904) 119889119904
minus 119876119879119871lowast119876minus1
119879119871int
119879
0
119890119860(119879minus119904)
119891 (119904 119909119904 119906119904) 119889119904)
= 1198711198901198601198791199090+ 119871119876119879119871lowast119876minus1
119879(ℎ minus 119871119890
1198601198791199090)
+ 119871int
119879
0
119890119860(119879minus119904)
119891 (119904 119909119904 119906119904) 119889119904
minus 119871119876119879119871lowast119876minus1
119879119871int
119879
0
119890119860(119879minus119904)
119891 (119904 119909119904 119906119904) 119889119904 = ℎ
(33)
Thus there is119906 isin 119880ad which steers1199090 to119909119879with119871119909119879 = ℎThismeans that the semilinear system (1) is 119871-partially completecontrollable on [0 119879] as desired
Remark 8 Decomposing 119876119879in the form
119876119879= [
119876119879
119877119879
119877lowast
119877119875119879
] (34)
where 119877119879 119867perprarr 119867 and 119875
119879 119867perprarr 119867perp are other compo-
nents of119876119879besides119876
119879and119867perp is an orthogonal complement
of119867 in119883 one can calculate
⟨119876119879ℎ ℎ⟩ = ⟨119876
119879ℎ1 ℎ1⟩ + 2⟨119877
119879ℎ2 ℎ1⟩ + ⟨119875
119879ℎ2 ℎ2⟩ (35)
where ℎ1= 119871ℎ isin 119867 and ℎ
2= ℎ minus 119871ℎ isin 119867
perp Therefore thecoercivity of 119876
119879implies the same of 119876
119879 But the converse
is not true Theorem 7 is powerful in the cases when 119876119879is
coercive but 119875119879is not
Example 9 Theorem 7 establishes just sufficient conditionof 119871-partial complete controllability In this example we will
demonstrate that this is not a necessary condition We willconsider a simple case of 119871 = 119868 when 119871-partial completecontrollability reduces to complete controllability Considerthe one-dimensional control system
1199091015840
119905= 2119909119905+ 2119906119905 1199090isin R (36)
This is a linear system and the controllability operator of thissystem is equal to
int
119879
0
41198904119905119889119905 = 119890
4119879minus 1 gt 0 for every 119879 gt 0 (37)
According to the theory of controllability for linear systemsthis system is controllable (completely) for every 119879 gt 0
Have another look at this system by writing it as
1199091015840
119905= 119909119905+ 119906119905+ 119891 (119909
119905 119906119905) 119909
0isin R (38)
where
119891 (119909 119906) = 119909 + 119906 (39)
Here 119891 satisfies the Lipschitz condition with 119870 = 1 Also119860 = 119861 = 1 implying 119861 = 1 and 119872 = sup
[0119879]119890119860119905 = 119890119879
Furthermore
119876119879= int
119879
0
1198902119905119889119905 =
1198902119879
minus 1
2
(40)
So 119876119879 = 120574 = (119890
2119879minus1)2 Then the inequality (30) becomes
(1 + 119890119879+
2119890119879
119890119879minus 1
) 119890119879119879 lt 1 (41)
The limit of the left-hand side in this inequality when 119879 rarr
infin is equal toinfin This means that there is a sufficiently large119879 such that the conditions of Theorem 7 do not hold for this119879 although the system under consideration is completelycontrollable Thus Theorem 7 states a sufficient conditionwhich is not a necessary condition
4 Examples
We demonstrate the features of 119871-partial complete controlla-bility in the following examples of control systems
Example 1 Consider the system of differential equations
1199091015840
119905= 119910119905+ 119887119906119905 1199090isin R
1199101015840
119905= 119891 (119905 119909
119905 119910119905 119906119905) 119910
0isin R
(42)
on [0 119879] where 119906 isin 119880ad = 119862(0 119879R) Besides the completecontrollability property that is
(119909 119910) isin R2 exist119906 isin 119880ad such that (119909
119879 119910119879) = (119909 119910) = R
2
(43)
we can investigate the partial complete controllability prop-erty that is
119909 isin R exist119906 isin 119880ad such that 119909119879= 119909 = R (44)
6 Abstract and Applied Analysis
We can write this system in R2 as the following semilinearsystem
1199111015840
119905= 119860119911119905+ 119865 (119905 119911
119905 119906119905) + 119861119906
119905 (45)
where
119911119905= [
119909119905
119910119905
] 119860 = [
0 1
0 0] 119861 = [
119887
0]
119865 (119905 119911 119906) = [
0
119891 (119905 119909 119910 119906)]
(46)
assuming that
119911 = [
119909
119910] (47)
It can be calculated that
119890119860119905
= [
1 119905
0 1] = [
1 0
0 1] + [
0 119905
0 0] (48)
Hence1003817100381710038171003817100381711989011986011990510038171003817100381710038171003817le 1 + 119905 le 1 + 119879 0 le 119905 le 119879 (49)
The controllability operator is
119876119879= int
119879
0
119890119860119905119861119861lowast119890119860lowast
119905119889119905 = 119887
2119879[
1 0
0 0] (50)
Hence 119876119879is not coercive and the conditions for complete
controllability based on coercivity of119876119879 fail for this example
Although system (42) can still be complete controllable forproperly selected functions 119891 we can investigate the partialcomplete controllability for this system being interested injust the first component 119909
119905of 119911119905
Let 119871 = [1 0] Then
119876119879= 119871119876119879119871lowast= 1198872119879 gt 0 (51)
This means that the linear system associated with the semili-near system (45) is119871-partially complete controllable Further-more the inequality (30) becomes
(1 +
1198872119879 (1 + 119879)
1198872119879
+
119887 (1 + 119879)
1198872119879
) (1 + 119879) 119879119870 lt 1 (52)
or simplifying
119870 lt
119887
(1 + 119879) (1 + 119879 + 2119887119879 + 1198871198792)
(53)
This establishes a relation between Lipschitz coefficient 119870and terminal time moment 119879 Depending on 119870 119879 must betaken sufficiently large to satisfy (53) So the system (42) is 119871-partially complete controllable for the time 119879 if the Lipschitzcoefficient 119870 related to 119891 satisfies (53)
Example 2 Delay equations are typical for application ofpartial controllability concepts Consider a nonlinear delayequation
1199091015840
119905= 119886119909119905+ 119887119906119905+ 119891(119905 119909
119905 int
0
minus120576
119909119905+120579
119889120579 119906119905)
1199090= 120585 119909
120579= 120578120579 minus120576 le 120579 le 0
(54)
on [0 119879] where 0 lt 120576 lt 119879 120578 isin 1198712(minus120576 0R) and 119906 isin 119880ad =
119862(0 119879R)Similar to Example 3 introduce the function119909 [0 119879] rarr
1198712(minus120576 0R) by
[119909119905]120579= 119909119905+120579
0 le 119905 le 119879 minus120576 le 120579 le 0 (55)
This function satisfies
1199091015840
119905= (
119889
119889120579
)119909119905 119909
0= 120578 0 lt 119905 le 119879 (56)
Denote by T119905 119905 ge 0 the semigroup generated by the
differential operator 119889119889120579 and let Γ be the integral operatorfrom 119871
2(minus120576 0R) to R defined by
Γℎ = int
0
minus120576
ℎ120579119889120579 ℎ isin 119871
2(minus120576 0R) (57)
noticing that Γ le radic120576 Then for
119910119905= [
119909119905
119909119905
] 120577 = [
120585
120578] isin R times 119871
2(minus120576 0R) (58)
we can write system (54) as
1199101015840
119905= 119860119910119905+ 119865 (119905 119910
119905 119906119905) + 119861119906
119905 1199100= 120577 (59)
where
119860 =[
[
119886 0
0
119889
119889120579
]
]
119865 (119905 119910 119906) = [
119891 (119905 119909 Γ119909 119906)
0]
119861 = [
119887
0]
(60)
assuming that
119910 = [
119909
119909] isin R times 119871
2(minus120576 0R) (61)
The semigroup generated by 119860 has the form
119890119860119905
= [119890119886119905
0
0 T119905
] 119905 ge 0 (62)
Therefore the controllability operator for system (54) can becalculated as
119876119879= int
119879
0
119890119860119905119861lowast119861119890119860lowast
119905119889119905 = int
119879
0
[11988721198902119886119905
0
0 0
] 119889119905
=[
[
[
1198872(1198902119886119879
minus 1)
2119886
0
0 0
]
]
]
(63)
This is definitely not a coercive operator
Abstract and Applied Analysis 7
Taking into account that the original system is given by(54) and (59) is just representation of (54) in the standardform which enlarges the original state space R to R times
1198712(minus120576 0R) we observe that the complete controllability for
system (54) is in fact 119871-partial complete controllability forsystem (59) if
119871 = [1 0] R times 1198712(minus120576 0R) 997888rarr R (64)
Calculating partial controllability operator we obtain
119876119879= 119871119876119879119871lowast=
1198872(1198902119886119879
minus 1)
2119886
gt 0(65)
which is coerciveFurthermore using
1003817100381710038171003817100381711989011986011990510038171003817100381710038171003817le 1 + 119890
119886119879
1003817100381710038171003817119876119879
1003817100381710038171003817= 120574 =
1198872(1198902119886119879
minus 1)
2119886
(66)
we write the inequality (30) in the form
(1 + 1 + 119890119886119879
+
2119886
119887 (119890119886119879
minus 1)
) (1 + 119890119886119879) 119879119870 lt 1 (67)
If the Lipschitz coefficient 119870 of the function 119865 and terminaltime moment 119879 satisfy this inequality then system (54) iscompletely controllable which in turnmeans that system (59)is 119871-partially complete controllable
5 Conclusion
In this paper a sufficient condition for partial complete con-trollability of a semilinear control system is proved This is acontinuation of the pioneering research that has been done byBashirov et al [20 21] about partial controllability conceptsA research in this way concerning partial complete andapproximate controllability for semilinear deterministic andstochastic systems has already been done and awaitingfor publication There are other kinds of systems whichbesides semilinearity include other features for exampleimpulsiveness fractional derivative issue and so forth Theresult of this paper can be extended to these systems as well
References
[1] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering D Transactions ofASME vol 82 pp 35ndash45 1960
[2] H O Fattorini ldquoSome remarks on complete controllabilityrdquoSIAM Journal on Control vol 4 pp 686ndash694 1966
[3] D L Russell ldquoNonharmonic Fourier series in the control theoryof distributed parameter systemsrdquo Journal of MathematicalAnalysis and Applications vol 18 pp 542ndash560 1967
[4] R F Curtain and H J ZwartAn Introduction to Infinite Dimen-sional Linear Systems Theory Springer Berlin Germany 1995
[5] A Bensoussan Stochastic Control of Partially Observable Sys-tems Cambridge University Press Cambridge UK 1992
[6] A Bensoussan G Da Prato M C Delfour and S K MitterRepresentation and Control of Infinite Dimensional SystemsSystems amp Control Foundations amp Applications BirkhauserBoston Mass USA 2nd edition 2007
[7] J Zabczyk Mathematical Control Theory An IntroductionSystems amp Control Foundations amp Applications BirkhauserBoston Mass USA 1995
[8] A E Bashirov Partially Observable Linear Systems underDependent Noises Systems amp Control Foundations amp Applica-tions Birkhauser Basel Switzerland 2003
[9] J Klamka Controllability of Dynamical Systems vol 48 ofMathematics and Its Applications Kluwer Academic PublishersDordrecht The Netherlands 1991
[10] K Balachandran and J P Dauer ldquoControllability of nonlinearsystems in Banach spaces a surveyrdquo Journal of OptimizationTheory and Applications vol 115 no 1 pp 7ndash28 2002
[11] K Balachandran and J P Dauer ldquoLocal controllability of semili-near evolution systems in Banach spacesrdquo Indian Journal of Pureand Applied Mathematics vol 29 no 3 pp 311ndash320 1998
[12] J Klamka ldquoSchauderrsquos fixed-point theorem in nonlinear con-trollability problemsrdquo Control and Cybernetics vol 29 no 1 pp153ndash165 2000
[13] N I Mahmudov ldquoControllability of semilinear stochastic sys-tems in Hilbert spacesrdquo Journal of Mathematical Analysis andApplications vol 288 no 1 pp 197ndash211 2003
[14] X J Li and J M Yong Optimal Control Theory for Infinite-Dimensional Systems SystemsampControl Foundations ampAppli-cations Birkhauser Boston Mass USA 1995
[15] R Sakthivel N I Mahmudov and J J Nieto ldquoControllabilityfor a class of fractional-order neutral evolution control systemsrdquoApplied Mathematics and Computation vol 218 no 20 pp10334ndash10340 2012
[16] R Sakthivel R Ganesh and S Suganya ldquoApproximate con-trollability of fractional neutral stochastic system with infinitedelayrdquo Reports on Mathematical Physics vol 70 no 3 pp 291ndash311 2012
[17] R Sakthivel Y Ren and N I Mahmudov ldquoOn the approximatecontrollability of semilinear fractional differential systemsrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1451ndash1459 2011
[18] Z Yan ldquoApproximate controllability of partial neutral func-tional differential systems of fractional order with state-dependent delayrdquo International Journal of Control vol 85 no8 pp 1051ndash1062 2012
[19] Y Ren L Hu and R Sakthivel ldquoControllability of impulsiveneutral stochastic functional differential inclusions with infinitedelayrdquo Journal of Computational and Applied Mathematics vol235 no 8 pp 2603ndash2614 2011
[20] A E Bashirov H Etikan andN Semi ldquoPartial controllability ofstochastic linear systemsrdquo International Journal of Control vol83 no 12 pp 2564ndash2572 2010
[21] A E Bashirov N Mahmudov N Semi and H Etikan ldquoPartialcontrollability conceptsrdquo International Journal of Control vol80 no 1 pp 1ndash7 2007
[22] A E Bashirov and N I Mahmudov ldquoOn concepts of control-lability for deterministic and stochastic systemsrdquo SIAM Journalon Control and Optimization vol 37 no 6 pp 1808ndash1821 1999
[23] A E Bashirov and K R Kerimov ldquoOn controllability concep-tion for stochastic systemsrdquo SIAM Journal on Control and Opti-mization vol 35 no 2 pp 384ndash398 1997
8 Abstract and Applied Analysis
[24] A E Bashirov ldquoOn weakening of the controllability conceptsrdquoin Proceedings of the 35th Conference on Decission and Controlpp 640ndash645 Kobe Japan 1996
[25] A E Bashirov and N I Mahmudov ldquoControllability of lineardeterministic and stochastic systemsrdquo in Proceedings of the 38thConference on Decission and Control pp 3196ndash3201 PhoenixAris USA 1999
[26] A E Bashirov and N I Mahmudov ldquoSome new results in the-ory of controllabilityrdquo in Proceedings of the 7th MediterraneanConference on Control and Automation pp 323ndash343 HaifaIsrael 1999
[27] L Byszewski ldquoTheorems about the existence and uniqueness ofsolutions of a semilinear evolution nonlocal Cauchy problemrdquoJournal of Mathematical Analysis and Applications vol 162 no2 pp 494ndash505 1991
Submit your manuscripts athttpwwwhindawicom
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Abstract and Applied Analysis
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Stochastic AnalysisInternational Journal of
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DifferentialEquations
International Journal of
Volume 2013
Abstract and Applied Analysis 3
where 119909 is a real-valued function of two variables 119905 ge 0 and0 le 120579 le 1 The state space of this system is 119871
2(0 1) (the space
of square integrable functions on [0 1]) This system can bewritten as the first-order abstract differential equation
1199101015840
119905= 119860119910119905+ 119865 (119905 119910
119905 119906119905) (11)
if
119910119905=[
[
119909119905120579
120597119909119905120579
120597119905
]
]
119860 =[
[
0 119868
1198892
1198891205792
0
]
]
119865 (119905 119910 119906) = [
0
119891 (119905 1199101 1199102 119906)
]
(12)
where 119910 isin 1198712(0 1) times 119871
2(0 1) The state space 119871
2(0 1) times
1198712(0 1) of the system for 119910 is the enlargement of the state
space 1198712(0 1) of the system for 119909 This is a cost that is
paid to bring the wave equation to the form of first-orderdifferential equation The ordinary controllability conceptsfor the system (11) are too strong for the system (10) If
119871 = [119868 0] 1198712(0 1) times 119871
2(0 1) 997888rarr 119871
2(0 1) (13)
then 119871-partially controllability concepts of the system for 119910become ordinary controllability concepts of the system for 119909
Example 3 Consider the system
1199091015840
119905= 119891(119905 119909
119905 int
0
minus120576
119909119905+120579
119889120579 119906119905) (14)
which contains a simple distributed delay in the nonlinearterm assuming that 119909 is a real-valued function Then thestate space is R To bring this system to a system withoutdelay enlargeR toR times 119871
2(minus120576 0) and define 119871
2(minus120576 0)-valued
function
[119909119905]120579= 119909119905+120579
119905 ge 0 minus120576 le 120579 le 0 (15)
Then for
119910119905= [
119909119905
119909119905
] 119860 =[
[
0 0
0
119889
119889120579
]
]
119865 (119905 119910 119906) = [
119891 (119905 119909 119909 119906)
0]
(16)
the above system can be written as the abstract system
1199101015840
119905= 119860119910119905+ 119891 (119905 119910
119905 119906119905) (17)
Similar to the previous examples one can easily observe thatthe ordinary controllability concepts for the system (17) aretoo strong for the system (14) but the 119871-partial controllabilityconcepts of the system for 119910 with
119871 = [119868 0] R times 1198712(0 1) 997888rarr R (18)
are exactly the ordinary controllability concepts of the systemfor 119909
These examples motivate a study of the partial control-lability concepts In this paper it is proved that under theconditions (A)ndash(F) the system in (1) is 119871-partially completecontrollable
3 Main Result
Denote119883 = 119862(0 119879119883) Then119883 times 119880ad is a Banach space withthe norm
(sdot sdot)times119880ad
= sdot+ sdot119880ad
(19)
Lemma 4 Under the conditions (A) (B) and (C)1003817100381710038171003817119876119905
1003817100381710038171003817le1003817100381710038171003817119876119879
1003817100381710038171003817
10038171003817100381710038171003817119876119905
10038171003817100381710038171003817le
10038171003817100381710038171003817119876119879
10038171003817100381710038171003817 0 le 119905 le 119879 (20)
Proof It is easy to see that 119876119905= 119876lowast
119905and ⟨119876
119905119909 119909⟩ ge 0 for all
119909 isin 119883 Hence1003817100381710038171003817119876119905
1003817100381710038171003817= sup119909=1
⟨119876119905119909 119909⟩ (21)
Then
⟨119876119879119909 119909⟩ = int
119879
0
⟨119890119860119904119861119861lowast119890119860lowast
119904119909 119909⟩119889119904
= ⟨119876119905119909 119909⟩ + int
119879
119905
⟨119890119860119904119861119861lowast119890119860lowast
119904119909 119909⟩119889119904
= ⟨119876119905119909 119909⟩ + int
119879
119905
100381710038171003817100381710038171003817
119861lowast119890119860lowast
119904119909
100381710038171003817100381710038171003817
2
119889119904
ge ⟨119876119905119909 119909⟩
(22)
This implies 119876119905 le 119876
119879 The conclusion of the lemma
regarding 119876119879follows from ⟨119876
119905119909 119909⟩ = ⟨119876
119905119871lowast119909 119871lowast119909⟩
The proof of the following lemma appears in differentforms in several papers for example Mahmudov [13] Ourproof is a minor modification of them
Lemma 5 Assume that the conditions (A)ndash(F) hold and takearbitrary ℎ isin 119867 Then for the operator 119866 119883 times 119880
119886119889rarr 119883 times
119880119886119889 defined by
119866 (119910 V) (119905) = (119884 (119905) 119881 (119905)) 0 le 119905 le 119879 (23)
where
119884 (119905) = minus119876119905119890119860lowast
(119879minus119905)119871lowast119876minus1
119879119871int
119879
0
119890119860(119879minus119904)
119891 (119904 119910119904 V119904) 119889119904
+ int
119905
0
119890119860(119905minus119904)
119891 (119904 119910119904 V119904) 119889119904
119881 (119905) = 119861lowast119890119860lowast
(119879minus119905)119871lowast119876minus1
119879
times (ℎ minus 1198711198901198601198791199090minus 119871int
119879
0
119890119860(119879minus119904)
119891 (119904 119910119904 V119904) 119889119904)
(24)
the following inequality holds
1003817100381710038171003817119866 (119910 V) minus 119866 (119911 119908)
1003817100381710038171003817le (1 +
1003817100381710038171003817119876119879
1003817100381710038171003817119872
120574
+
119861119872
120574
)
times119872119870119879 (1003817100381710038171003817119910 minus 119911
1003817100381710038171003817+ V minus 119908)
(25)
where
119872 = sup0le119905le119879
1003817100381710038171003817100381711989011986011990510038171003817100381710038171003817 (26)
4 Abstract and Applied Analysis
Proof Let (119910 V) and (119911 119908) be two functions in 119883 times 119880ad suchthat 119866(119910 V) = (119884 119881) and 119866(119911 119908) = (119885119882) Then
1003817100381710038171003817119866(119910 V) minus 119866(119911 119908)
1003817100381710038171003817times119880ad
= 119884 minus 119885+ 119881 minus119882
119880ad (27)
Here 119884 minus 119885can be estimated as follows
119884 minus 119885 = max119905isin[0119879]
10038171003817100381710038171003817100381710038171003817
int
119905
0
119890119860(119905minus119904)
(119891 (119904 119910119904 V119904) minus 119891 (119904 119911
119904 119908119904)) 119889119904
minus 119876119905119890119860lowast
(119879minus119905)119871lowast119876minus1
119879119871
timesint
119879
0
119890119860(119879minus119904)
(119891 (119904 119910119904 V119904)minus119891 (119904 119911
119904 119908119904)) 119889119904
100381710038171003817100381710038171003817100381710038171003817
le max119905isin[0119879]
(119872 +1198722 1003817100381710038171003817119876119905
1003817100381710038171003817
10038171003817100381710038171003817119876minus1
119879
10038171003817100381710038171003817)
times int
119879
0
1003817100381710038171003817119891 (119904 119910
119904 V119904) minus 119891 (119904 119911
119904 119908119904)1003817100381710038171003817119889119904
le 119872(1 +1198721003817100381710038171003817119876119879
1003817100381710038171003817
10038171003817100381710038171003817119876minus1
119879
10038171003817100381710038171003817)
times int
119879
0
1003817100381710038171003817119891 (119904 119910
119904 V119904) minus 119891 (119904 119911
119904 119908119904)1003817100381710038171003817119889119904
le (1 +
1003817100381710038171003817119876119879
1003817100381710038171003817119872
120574
)119872119870int
119879
0
(1003817100381710038171003817119910119904minus 119911119904
1003817100381710038171003817+1003817100381710038171003817V119904minus 119908119904
1003817100381710038171003817) 119889119904
le (1 +
1003817100381710038171003817119876119879
1003817100381710038171003817119872
120574
)119872119870119879 (1003817100381710038171003817119910 minus 119911
1003817100381710038171003817+ V minus 119908)
(28)
Similarly for 119881 minus119882119880ad
we have
119881 minus119882 = max119905isin[0119879]
100381710038171003817100381710038171003817100381710038171003817
119861lowast119890119860lowast
(119879minus119905)119871lowast119876minus1
119879119871
timesint
119879
0
119890119860(119879minus119904)
(119891 (119904 119910119904 V119904)minus119891 (119904 119911
119904 119908119904)) 119889119904
100381710038171003817100381710038171003817100381710038171003817
le 1198722
119861
10038171003817100381710038171003817119876minus1
119879
10038171003817100381710038171003817int
119879
0
1003817100381710038171003817119891 (119904 119910
119904 V119904) minus 119891 (119904 119911
119904 119908119904)1003817100381710038171003817119889119904
le 1198722
119861
1
120574
119870int
119879
0
(1003817100381710038171003817119910119904minus 119911119904
1003817100381710038171003817+1003817100381710038171003817V119904minus 119908119904
1003817100381710038171003817) 119889119904
le
119861119872
120574
119872119870119879 (1003817100381710038171003817119910 minus 119911
1003817100381710038171003817+ V minus 119908)
(29)
Combining (28) and (29) we obtain the demanded inequal-ity
Lemma 6 Under the conditions (A)ndash(F) if
(1 +
1003817100381710038171003817119876119879
1003817100381710038171003817119872
120574
+
119861119872
120574
)119872119870119879 lt 1 (30)
then the operator 119866 mapping 119883 times 119880119886119889
into 119883 times 119880119886119889 has a
unique fixed point (119909 119906) isin 119883 times 119880119886119889
Proof By Lemma 5 119866 is a contraction mapping Also thespace119883 times 119880ad is complete Hence 119866 has a fixed point
Theorem 7 Under the conditions (A)ndash(F) and (30) the semi-linear system (1) is 119871-partially complete controllable on [0 119879]
Proof Take any 1199090isin 119883 and ℎ isin 119867 Show that there is 119906 isin 119880ad
such that ℎ = 119871119909119879 To this end consider 119906 defined as follows
119906119905= 119861lowast119890119860lowast
(119879minus119905)119871lowast119876minus1
119879
times (ℎ minus 1198711198901198601198791199090minus 119871int
119879
0
119890119860(119879minus119904)
119891 (119904 119909119904 119906119904) 119889119904)
(31)
Substituting (31) in (4) and applying Fubinirsquos theorem (seeBashirov [8] p 45) we obtain
119909119905= 1198901198601199051199090
+ int
119905
0
119890119860(119905minus119904)
119861119861lowast119890119860lowast
(119905minus119904)119890119860lowast
(119879minus119905)119871lowast119876minus1
119879(ℎ minus 119871119890
1198601198791199090) 119889119904
minus int
119905
0
int
119879
0
119890119860(119905minus119904)
119861119861lowast119890119860lowast
(119905minus119904)119890119860lowast
(119879minus119905)119871lowast119876minus1
119879119871119890119860(119879minus119903)
times 119891 (119903 119909119903 119906119903) 119889119903 119889119904
+ int
119905
0
119890119860(119905minus119904)
119891 (119904 119909119904 119906119904) 119889119904
= 1198901198601199051199090+ 119876119905119890119860lowast
(119879minus119905)119871lowast119876minus1
119879(ℎ minus 119871119890
1198601198791199090)
+ int
119905
0
119890119860(119905minus119904)
119891 (119904 119909119904 119906119904) 119889119904
minus int
119879
0
int
119905
0
119890119860(119905minus119904)
119861119861lowast119890119860lowast
(119905minus119904)119890119860lowast
(119879minus119905)119871lowast119876minus1
119879119871119890119860(119879minus119903)
times 119891 (119903 119909119903 119906119903) 119889119904 119889119903
Abstract and Applied Analysis 5
= 1198901198601199051199090+ 119876119905119890119860lowast
(119879minus119905)119871lowast119876minus1
119879(ℎ minus 119871119890
1198601198791199090)
+ int
119905
0
119890119860(119905minus119904)
119891 (119904 119909119904 119906119904) 119889119904
minus int
119879
0
119876119905119890119860lowast
(119879minus119905)119871lowast119876minus1
119879119871119890119860(119879minus119903)
119891 (119903 119909119903 119906119903) 119889119903
= 1198901198601199051199090+ 119876119905119890119860lowast
(119879minus119905)119871lowast119876minus1
119879(ℎ minus 119871119890
1198601198791199090)
+ int
119905
0
119890119860(119905minus119904)
119891 (119904 119909119904 119906119904) 119889119904
minus 119876119905119890119860lowast
(119879minus119905)119871lowast119876minus1
119879119871int
119879
0
119890119860(119879minus119904)
119891 (119904 119909119904 119906119904) 119889119904
(32)
According to Lemma 6 there is a unique pair (119909 119906) isin 119883 times
119880ad satisfying (31) and (32) So 119906 isin 119880ad Furthermore wehave
119871119909119879= 119871(119890
1198601198791199090+ 119876119879119871lowast119876minus1
119879(ℎ minus 119871119890
1198601198791199090)
+ int
119879
0
119890119860(119879minus119904)
119891 (119904 119909119904 119906119904) 119889119904
minus 119876119879119871lowast119876minus1
119879119871int
119879
0
119890119860(119879minus119904)
119891 (119904 119909119904 119906119904) 119889119904)
= 1198711198901198601198791199090+ 119871119876119879119871lowast119876minus1
119879(ℎ minus 119871119890
1198601198791199090)
+ 119871int
119879
0
119890119860(119879minus119904)
119891 (119904 119909119904 119906119904) 119889119904
minus 119871119876119879119871lowast119876minus1
119879119871int
119879
0
119890119860(119879minus119904)
119891 (119904 119909119904 119906119904) 119889119904 = ℎ
(33)
Thus there is119906 isin 119880ad which steers1199090 to119909119879with119871119909119879 = ℎThismeans that the semilinear system (1) is 119871-partially completecontrollable on [0 119879] as desired
Remark 8 Decomposing 119876119879in the form
119876119879= [
119876119879
119877119879
119877lowast
119877119875119879
] (34)
where 119877119879 119867perprarr 119867 and 119875
119879 119867perprarr 119867perp are other compo-
nents of119876119879besides119876
119879and119867perp is an orthogonal complement
of119867 in119883 one can calculate
⟨119876119879ℎ ℎ⟩ = ⟨119876
119879ℎ1 ℎ1⟩ + 2⟨119877
119879ℎ2 ℎ1⟩ + ⟨119875
119879ℎ2 ℎ2⟩ (35)
where ℎ1= 119871ℎ isin 119867 and ℎ
2= ℎ minus 119871ℎ isin 119867
perp Therefore thecoercivity of 119876
119879implies the same of 119876
119879 But the converse
is not true Theorem 7 is powerful in the cases when 119876119879is
coercive but 119875119879is not
Example 9 Theorem 7 establishes just sufficient conditionof 119871-partial complete controllability In this example we will
demonstrate that this is not a necessary condition We willconsider a simple case of 119871 = 119868 when 119871-partial completecontrollability reduces to complete controllability Considerthe one-dimensional control system
1199091015840
119905= 2119909119905+ 2119906119905 1199090isin R (36)
This is a linear system and the controllability operator of thissystem is equal to
int
119879
0
41198904119905119889119905 = 119890
4119879minus 1 gt 0 for every 119879 gt 0 (37)
According to the theory of controllability for linear systemsthis system is controllable (completely) for every 119879 gt 0
Have another look at this system by writing it as
1199091015840
119905= 119909119905+ 119906119905+ 119891 (119909
119905 119906119905) 119909
0isin R (38)
where
119891 (119909 119906) = 119909 + 119906 (39)
Here 119891 satisfies the Lipschitz condition with 119870 = 1 Also119860 = 119861 = 1 implying 119861 = 1 and 119872 = sup
[0119879]119890119860119905 = 119890119879
Furthermore
119876119879= int
119879
0
1198902119905119889119905 =
1198902119879
minus 1
2
(40)
So 119876119879 = 120574 = (119890
2119879minus1)2 Then the inequality (30) becomes
(1 + 119890119879+
2119890119879
119890119879minus 1
) 119890119879119879 lt 1 (41)
The limit of the left-hand side in this inequality when 119879 rarr
infin is equal toinfin This means that there is a sufficiently large119879 such that the conditions of Theorem 7 do not hold for this119879 although the system under consideration is completelycontrollable Thus Theorem 7 states a sufficient conditionwhich is not a necessary condition
4 Examples
We demonstrate the features of 119871-partial complete controlla-bility in the following examples of control systems
Example 1 Consider the system of differential equations
1199091015840
119905= 119910119905+ 119887119906119905 1199090isin R
1199101015840
119905= 119891 (119905 119909
119905 119910119905 119906119905) 119910
0isin R
(42)
on [0 119879] where 119906 isin 119880ad = 119862(0 119879R) Besides the completecontrollability property that is
(119909 119910) isin R2 exist119906 isin 119880ad such that (119909
119879 119910119879) = (119909 119910) = R
2
(43)
we can investigate the partial complete controllability prop-erty that is
119909 isin R exist119906 isin 119880ad such that 119909119879= 119909 = R (44)
6 Abstract and Applied Analysis
We can write this system in R2 as the following semilinearsystem
1199111015840
119905= 119860119911119905+ 119865 (119905 119911
119905 119906119905) + 119861119906
119905 (45)
where
119911119905= [
119909119905
119910119905
] 119860 = [
0 1
0 0] 119861 = [
119887
0]
119865 (119905 119911 119906) = [
0
119891 (119905 119909 119910 119906)]
(46)
assuming that
119911 = [
119909
119910] (47)
It can be calculated that
119890119860119905
= [
1 119905
0 1] = [
1 0
0 1] + [
0 119905
0 0] (48)
Hence1003817100381710038171003817100381711989011986011990510038171003817100381710038171003817le 1 + 119905 le 1 + 119879 0 le 119905 le 119879 (49)
The controllability operator is
119876119879= int
119879
0
119890119860119905119861119861lowast119890119860lowast
119905119889119905 = 119887
2119879[
1 0
0 0] (50)
Hence 119876119879is not coercive and the conditions for complete
controllability based on coercivity of119876119879 fail for this example
Although system (42) can still be complete controllable forproperly selected functions 119891 we can investigate the partialcomplete controllability for this system being interested injust the first component 119909
119905of 119911119905
Let 119871 = [1 0] Then
119876119879= 119871119876119879119871lowast= 1198872119879 gt 0 (51)
This means that the linear system associated with the semili-near system (45) is119871-partially complete controllable Further-more the inequality (30) becomes
(1 +
1198872119879 (1 + 119879)
1198872119879
+
119887 (1 + 119879)
1198872119879
) (1 + 119879) 119879119870 lt 1 (52)
or simplifying
119870 lt
119887
(1 + 119879) (1 + 119879 + 2119887119879 + 1198871198792)
(53)
This establishes a relation between Lipschitz coefficient 119870and terminal time moment 119879 Depending on 119870 119879 must betaken sufficiently large to satisfy (53) So the system (42) is 119871-partially complete controllable for the time 119879 if the Lipschitzcoefficient 119870 related to 119891 satisfies (53)
Example 2 Delay equations are typical for application ofpartial controllability concepts Consider a nonlinear delayequation
1199091015840
119905= 119886119909119905+ 119887119906119905+ 119891(119905 119909
119905 int
0
minus120576
119909119905+120579
119889120579 119906119905)
1199090= 120585 119909
120579= 120578120579 minus120576 le 120579 le 0
(54)
on [0 119879] where 0 lt 120576 lt 119879 120578 isin 1198712(minus120576 0R) and 119906 isin 119880ad =
119862(0 119879R)Similar to Example 3 introduce the function119909 [0 119879] rarr
1198712(minus120576 0R) by
[119909119905]120579= 119909119905+120579
0 le 119905 le 119879 minus120576 le 120579 le 0 (55)
This function satisfies
1199091015840
119905= (
119889
119889120579
)119909119905 119909
0= 120578 0 lt 119905 le 119879 (56)
Denote by T119905 119905 ge 0 the semigroup generated by the
differential operator 119889119889120579 and let Γ be the integral operatorfrom 119871
2(minus120576 0R) to R defined by
Γℎ = int
0
minus120576
ℎ120579119889120579 ℎ isin 119871
2(minus120576 0R) (57)
noticing that Γ le radic120576 Then for
119910119905= [
119909119905
119909119905
] 120577 = [
120585
120578] isin R times 119871
2(minus120576 0R) (58)
we can write system (54) as
1199101015840
119905= 119860119910119905+ 119865 (119905 119910
119905 119906119905) + 119861119906
119905 1199100= 120577 (59)
where
119860 =[
[
119886 0
0
119889
119889120579
]
]
119865 (119905 119910 119906) = [
119891 (119905 119909 Γ119909 119906)
0]
119861 = [
119887
0]
(60)
assuming that
119910 = [
119909
119909] isin R times 119871
2(minus120576 0R) (61)
The semigroup generated by 119860 has the form
119890119860119905
= [119890119886119905
0
0 T119905
] 119905 ge 0 (62)
Therefore the controllability operator for system (54) can becalculated as
119876119879= int
119879
0
119890119860119905119861lowast119861119890119860lowast
119905119889119905 = int
119879
0
[11988721198902119886119905
0
0 0
] 119889119905
=[
[
[
1198872(1198902119886119879
minus 1)
2119886
0
0 0
]
]
]
(63)
This is definitely not a coercive operator
Abstract and Applied Analysis 7
Taking into account that the original system is given by(54) and (59) is just representation of (54) in the standardform which enlarges the original state space R to R times
1198712(minus120576 0R) we observe that the complete controllability for
system (54) is in fact 119871-partial complete controllability forsystem (59) if
119871 = [1 0] R times 1198712(minus120576 0R) 997888rarr R (64)
Calculating partial controllability operator we obtain
119876119879= 119871119876119879119871lowast=
1198872(1198902119886119879
minus 1)
2119886
gt 0(65)
which is coerciveFurthermore using
1003817100381710038171003817100381711989011986011990510038171003817100381710038171003817le 1 + 119890
119886119879
1003817100381710038171003817119876119879
1003817100381710038171003817= 120574 =
1198872(1198902119886119879
minus 1)
2119886
(66)
we write the inequality (30) in the form
(1 + 1 + 119890119886119879
+
2119886
119887 (119890119886119879
minus 1)
) (1 + 119890119886119879) 119879119870 lt 1 (67)
If the Lipschitz coefficient 119870 of the function 119865 and terminaltime moment 119879 satisfy this inequality then system (54) iscompletely controllable which in turnmeans that system (59)is 119871-partially complete controllable
5 Conclusion
In this paper a sufficient condition for partial complete con-trollability of a semilinear control system is proved This is acontinuation of the pioneering research that has been done byBashirov et al [20 21] about partial controllability conceptsA research in this way concerning partial complete andapproximate controllability for semilinear deterministic andstochastic systems has already been done and awaitingfor publication There are other kinds of systems whichbesides semilinearity include other features for exampleimpulsiveness fractional derivative issue and so forth Theresult of this paper can be extended to these systems as well
References
[1] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering D Transactions ofASME vol 82 pp 35ndash45 1960
[2] H O Fattorini ldquoSome remarks on complete controllabilityrdquoSIAM Journal on Control vol 4 pp 686ndash694 1966
[3] D L Russell ldquoNonharmonic Fourier series in the control theoryof distributed parameter systemsrdquo Journal of MathematicalAnalysis and Applications vol 18 pp 542ndash560 1967
[4] R F Curtain and H J ZwartAn Introduction to Infinite Dimen-sional Linear Systems Theory Springer Berlin Germany 1995
[5] A Bensoussan Stochastic Control of Partially Observable Sys-tems Cambridge University Press Cambridge UK 1992
[6] A Bensoussan G Da Prato M C Delfour and S K MitterRepresentation and Control of Infinite Dimensional SystemsSystems amp Control Foundations amp Applications BirkhauserBoston Mass USA 2nd edition 2007
[7] J Zabczyk Mathematical Control Theory An IntroductionSystems amp Control Foundations amp Applications BirkhauserBoston Mass USA 1995
[8] A E Bashirov Partially Observable Linear Systems underDependent Noises Systems amp Control Foundations amp Applica-tions Birkhauser Basel Switzerland 2003
[9] J Klamka Controllability of Dynamical Systems vol 48 ofMathematics and Its Applications Kluwer Academic PublishersDordrecht The Netherlands 1991
[10] K Balachandran and J P Dauer ldquoControllability of nonlinearsystems in Banach spaces a surveyrdquo Journal of OptimizationTheory and Applications vol 115 no 1 pp 7ndash28 2002
[11] K Balachandran and J P Dauer ldquoLocal controllability of semili-near evolution systems in Banach spacesrdquo Indian Journal of Pureand Applied Mathematics vol 29 no 3 pp 311ndash320 1998
[12] J Klamka ldquoSchauderrsquos fixed-point theorem in nonlinear con-trollability problemsrdquo Control and Cybernetics vol 29 no 1 pp153ndash165 2000
[13] N I Mahmudov ldquoControllability of semilinear stochastic sys-tems in Hilbert spacesrdquo Journal of Mathematical Analysis andApplications vol 288 no 1 pp 197ndash211 2003
[14] X J Li and J M Yong Optimal Control Theory for Infinite-Dimensional Systems SystemsampControl Foundations ampAppli-cations Birkhauser Boston Mass USA 1995
[15] R Sakthivel N I Mahmudov and J J Nieto ldquoControllabilityfor a class of fractional-order neutral evolution control systemsrdquoApplied Mathematics and Computation vol 218 no 20 pp10334ndash10340 2012
[16] R Sakthivel R Ganesh and S Suganya ldquoApproximate con-trollability of fractional neutral stochastic system with infinitedelayrdquo Reports on Mathematical Physics vol 70 no 3 pp 291ndash311 2012
[17] R Sakthivel Y Ren and N I Mahmudov ldquoOn the approximatecontrollability of semilinear fractional differential systemsrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1451ndash1459 2011
[18] Z Yan ldquoApproximate controllability of partial neutral func-tional differential systems of fractional order with state-dependent delayrdquo International Journal of Control vol 85 no8 pp 1051ndash1062 2012
[19] Y Ren L Hu and R Sakthivel ldquoControllability of impulsiveneutral stochastic functional differential inclusions with infinitedelayrdquo Journal of Computational and Applied Mathematics vol235 no 8 pp 2603ndash2614 2011
[20] A E Bashirov H Etikan andN Semi ldquoPartial controllability ofstochastic linear systemsrdquo International Journal of Control vol83 no 12 pp 2564ndash2572 2010
[21] A E Bashirov N Mahmudov N Semi and H Etikan ldquoPartialcontrollability conceptsrdquo International Journal of Control vol80 no 1 pp 1ndash7 2007
[22] A E Bashirov and N I Mahmudov ldquoOn concepts of control-lability for deterministic and stochastic systemsrdquo SIAM Journalon Control and Optimization vol 37 no 6 pp 1808ndash1821 1999
[23] A E Bashirov and K R Kerimov ldquoOn controllability concep-tion for stochastic systemsrdquo SIAM Journal on Control and Opti-mization vol 35 no 2 pp 384ndash398 1997
8 Abstract and Applied Analysis
[24] A E Bashirov ldquoOn weakening of the controllability conceptsrdquoin Proceedings of the 35th Conference on Decission and Controlpp 640ndash645 Kobe Japan 1996
[25] A E Bashirov and N I Mahmudov ldquoControllability of lineardeterministic and stochastic systemsrdquo in Proceedings of the 38thConference on Decission and Control pp 3196ndash3201 PhoenixAris USA 1999
[26] A E Bashirov and N I Mahmudov ldquoSome new results in the-ory of controllabilityrdquo in Proceedings of the 7th MediterraneanConference on Control and Automation pp 323ndash343 HaifaIsrael 1999
[27] L Byszewski ldquoTheorems about the existence and uniqueness ofsolutions of a semilinear evolution nonlocal Cauchy problemrdquoJournal of Mathematical Analysis and Applications vol 162 no2 pp 494ndash505 1991
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Volume 2013
Advances in
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Stochastic AnalysisInternational Journal of
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DifferentialEquations
International Journal of
Volume 2013
4 Abstract and Applied Analysis
Proof Let (119910 V) and (119911 119908) be two functions in 119883 times 119880ad suchthat 119866(119910 V) = (119884 119881) and 119866(119911 119908) = (119885119882) Then
1003817100381710038171003817119866(119910 V) minus 119866(119911 119908)
1003817100381710038171003817times119880ad
= 119884 minus 119885+ 119881 minus119882
119880ad (27)
Here 119884 minus 119885can be estimated as follows
119884 minus 119885 = max119905isin[0119879]
10038171003817100381710038171003817100381710038171003817
int
119905
0
119890119860(119905minus119904)
(119891 (119904 119910119904 V119904) minus 119891 (119904 119911
119904 119908119904)) 119889119904
minus 119876119905119890119860lowast
(119879minus119905)119871lowast119876minus1
119879119871
timesint
119879
0
119890119860(119879minus119904)
(119891 (119904 119910119904 V119904)minus119891 (119904 119911
119904 119908119904)) 119889119904
100381710038171003817100381710038171003817100381710038171003817
le max119905isin[0119879]
(119872 +1198722 1003817100381710038171003817119876119905
1003817100381710038171003817
10038171003817100381710038171003817119876minus1
119879
10038171003817100381710038171003817)
times int
119879
0
1003817100381710038171003817119891 (119904 119910
119904 V119904) minus 119891 (119904 119911
119904 119908119904)1003817100381710038171003817119889119904
le 119872(1 +1198721003817100381710038171003817119876119879
1003817100381710038171003817
10038171003817100381710038171003817119876minus1
119879
10038171003817100381710038171003817)
times int
119879
0
1003817100381710038171003817119891 (119904 119910
119904 V119904) minus 119891 (119904 119911
119904 119908119904)1003817100381710038171003817119889119904
le (1 +
1003817100381710038171003817119876119879
1003817100381710038171003817119872
120574
)119872119870int
119879
0
(1003817100381710038171003817119910119904minus 119911119904
1003817100381710038171003817+1003817100381710038171003817V119904minus 119908119904
1003817100381710038171003817) 119889119904
le (1 +
1003817100381710038171003817119876119879
1003817100381710038171003817119872
120574
)119872119870119879 (1003817100381710038171003817119910 minus 119911
1003817100381710038171003817+ V minus 119908)
(28)
Similarly for 119881 minus119882119880ad
we have
119881 minus119882 = max119905isin[0119879]
100381710038171003817100381710038171003817100381710038171003817
119861lowast119890119860lowast
(119879minus119905)119871lowast119876minus1
119879119871
timesint
119879
0
119890119860(119879minus119904)
(119891 (119904 119910119904 V119904)minus119891 (119904 119911
119904 119908119904)) 119889119904
100381710038171003817100381710038171003817100381710038171003817
le 1198722
119861
10038171003817100381710038171003817119876minus1
119879
10038171003817100381710038171003817int
119879
0
1003817100381710038171003817119891 (119904 119910
119904 V119904) minus 119891 (119904 119911
119904 119908119904)1003817100381710038171003817119889119904
le 1198722
119861
1
120574
119870int
119879
0
(1003817100381710038171003817119910119904minus 119911119904
1003817100381710038171003817+1003817100381710038171003817V119904minus 119908119904
1003817100381710038171003817) 119889119904
le
119861119872
120574
119872119870119879 (1003817100381710038171003817119910 minus 119911
1003817100381710038171003817+ V minus 119908)
(29)
Combining (28) and (29) we obtain the demanded inequal-ity
Lemma 6 Under the conditions (A)ndash(F) if
(1 +
1003817100381710038171003817119876119879
1003817100381710038171003817119872
120574
+
119861119872
120574
)119872119870119879 lt 1 (30)
then the operator 119866 mapping 119883 times 119880119886119889
into 119883 times 119880119886119889 has a
unique fixed point (119909 119906) isin 119883 times 119880119886119889
Proof By Lemma 5 119866 is a contraction mapping Also thespace119883 times 119880ad is complete Hence 119866 has a fixed point
Theorem 7 Under the conditions (A)ndash(F) and (30) the semi-linear system (1) is 119871-partially complete controllable on [0 119879]
Proof Take any 1199090isin 119883 and ℎ isin 119867 Show that there is 119906 isin 119880ad
such that ℎ = 119871119909119879 To this end consider 119906 defined as follows
119906119905= 119861lowast119890119860lowast
(119879minus119905)119871lowast119876minus1
119879
times (ℎ minus 1198711198901198601198791199090minus 119871int
119879
0
119890119860(119879minus119904)
119891 (119904 119909119904 119906119904) 119889119904)
(31)
Substituting (31) in (4) and applying Fubinirsquos theorem (seeBashirov [8] p 45) we obtain
119909119905= 1198901198601199051199090
+ int
119905
0
119890119860(119905minus119904)
119861119861lowast119890119860lowast
(119905minus119904)119890119860lowast
(119879minus119905)119871lowast119876minus1
119879(ℎ minus 119871119890
1198601198791199090) 119889119904
minus int
119905
0
int
119879
0
119890119860(119905minus119904)
119861119861lowast119890119860lowast
(119905minus119904)119890119860lowast
(119879minus119905)119871lowast119876minus1
119879119871119890119860(119879minus119903)
times 119891 (119903 119909119903 119906119903) 119889119903 119889119904
+ int
119905
0
119890119860(119905minus119904)
119891 (119904 119909119904 119906119904) 119889119904
= 1198901198601199051199090+ 119876119905119890119860lowast
(119879minus119905)119871lowast119876minus1
119879(ℎ minus 119871119890
1198601198791199090)
+ int
119905
0
119890119860(119905minus119904)
119891 (119904 119909119904 119906119904) 119889119904
minus int
119879
0
int
119905
0
119890119860(119905minus119904)
119861119861lowast119890119860lowast
(119905minus119904)119890119860lowast
(119879minus119905)119871lowast119876minus1
119879119871119890119860(119879minus119903)
times 119891 (119903 119909119903 119906119903) 119889119904 119889119903
Abstract and Applied Analysis 5
= 1198901198601199051199090+ 119876119905119890119860lowast
(119879minus119905)119871lowast119876minus1
119879(ℎ minus 119871119890
1198601198791199090)
+ int
119905
0
119890119860(119905minus119904)
119891 (119904 119909119904 119906119904) 119889119904
minus int
119879
0
119876119905119890119860lowast
(119879minus119905)119871lowast119876minus1
119879119871119890119860(119879minus119903)
119891 (119903 119909119903 119906119903) 119889119903
= 1198901198601199051199090+ 119876119905119890119860lowast
(119879minus119905)119871lowast119876minus1
119879(ℎ minus 119871119890
1198601198791199090)
+ int
119905
0
119890119860(119905minus119904)
119891 (119904 119909119904 119906119904) 119889119904
minus 119876119905119890119860lowast
(119879minus119905)119871lowast119876minus1
119879119871int
119879
0
119890119860(119879minus119904)
119891 (119904 119909119904 119906119904) 119889119904
(32)
According to Lemma 6 there is a unique pair (119909 119906) isin 119883 times
119880ad satisfying (31) and (32) So 119906 isin 119880ad Furthermore wehave
119871119909119879= 119871(119890
1198601198791199090+ 119876119879119871lowast119876minus1
119879(ℎ minus 119871119890
1198601198791199090)
+ int
119879
0
119890119860(119879minus119904)
119891 (119904 119909119904 119906119904) 119889119904
minus 119876119879119871lowast119876minus1
119879119871int
119879
0
119890119860(119879minus119904)
119891 (119904 119909119904 119906119904) 119889119904)
= 1198711198901198601198791199090+ 119871119876119879119871lowast119876minus1
119879(ℎ minus 119871119890
1198601198791199090)
+ 119871int
119879
0
119890119860(119879minus119904)
119891 (119904 119909119904 119906119904) 119889119904
minus 119871119876119879119871lowast119876minus1
119879119871int
119879
0
119890119860(119879minus119904)
119891 (119904 119909119904 119906119904) 119889119904 = ℎ
(33)
Thus there is119906 isin 119880ad which steers1199090 to119909119879with119871119909119879 = ℎThismeans that the semilinear system (1) is 119871-partially completecontrollable on [0 119879] as desired
Remark 8 Decomposing 119876119879in the form
119876119879= [
119876119879
119877119879
119877lowast
119877119875119879
] (34)
where 119877119879 119867perprarr 119867 and 119875
119879 119867perprarr 119867perp are other compo-
nents of119876119879besides119876
119879and119867perp is an orthogonal complement
of119867 in119883 one can calculate
⟨119876119879ℎ ℎ⟩ = ⟨119876
119879ℎ1 ℎ1⟩ + 2⟨119877
119879ℎ2 ℎ1⟩ + ⟨119875
119879ℎ2 ℎ2⟩ (35)
where ℎ1= 119871ℎ isin 119867 and ℎ
2= ℎ minus 119871ℎ isin 119867
perp Therefore thecoercivity of 119876
119879implies the same of 119876
119879 But the converse
is not true Theorem 7 is powerful in the cases when 119876119879is
coercive but 119875119879is not
Example 9 Theorem 7 establishes just sufficient conditionof 119871-partial complete controllability In this example we will
demonstrate that this is not a necessary condition We willconsider a simple case of 119871 = 119868 when 119871-partial completecontrollability reduces to complete controllability Considerthe one-dimensional control system
1199091015840
119905= 2119909119905+ 2119906119905 1199090isin R (36)
This is a linear system and the controllability operator of thissystem is equal to
int
119879
0
41198904119905119889119905 = 119890
4119879minus 1 gt 0 for every 119879 gt 0 (37)
According to the theory of controllability for linear systemsthis system is controllable (completely) for every 119879 gt 0
Have another look at this system by writing it as
1199091015840
119905= 119909119905+ 119906119905+ 119891 (119909
119905 119906119905) 119909
0isin R (38)
where
119891 (119909 119906) = 119909 + 119906 (39)
Here 119891 satisfies the Lipschitz condition with 119870 = 1 Also119860 = 119861 = 1 implying 119861 = 1 and 119872 = sup
[0119879]119890119860119905 = 119890119879
Furthermore
119876119879= int
119879
0
1198902119905119889119905 =
1198902119879
minus 1
2
(40)
So 119876119879 = 120574 = (119890
2119879minus1)2 Then the inequality (30) becomes
(1 + 119890119879+
2119890119879
119890119879minus 1
) 119890119879119879 lt 1 (41)
The limit of the left-hand side in this inequality when 119879 rarr
infin is equal toinfin This means that there is a sufficiently large119879 such that the conditions of Theorem 7 do not hold for this119879 although the system under consideration is completelycontrollable Thus Theorem 7 states a sufficient conditionwhich is not a necessary condition
4 Examples
We demonstrate the features of 119871-partial complete controlla-bility in the following examples of control systems
Example 1 Consider the system of differential equations
1199091015840
119905= 119910119905+ 119887119906119905 1199090isin R
1199101015840
119905= 119891 (119905 119909
119905 119910119905 119906119905) 119910
0isin R
(42)
on [0 119879] where 119906 isin 119880ad = 119862(0 119879R) Besides the completecontrollability property that is
(119909 119910) isin R2 exist119906 isin 119880ad such that (119909
119879 119910119879) = (119909 119910) = R
2
(43)
we can investigate the partial complete controllability prop-erty that is
119909 isin R exist119906 isin 119880ad such that 119909119879= 119909 = R (44)
6 Abstract and Applied Analysis
We can write this system in R2 as the following semilinearsystem
1199111015840
119905= 119860119911119905+ 119865 (119905 119911
119905 119906119905) + 119861119906
119905 (45)
where
119911119905= [
119909119905
119910119905
] 119860 = [
0 1
0 0] 119861 = [
119887
0]
119865 (119905 119911 119906) = [
0
119891 (119905 119909 119910 119906)]
(46)
assuming that
119911 = [
119909
119910] (47)
It can be calculated that
119890119860119905
= [
1 119905
0 1] = [
1 0
0 1] + [
0 119905
0 0] (48)
Hence1003817100381710038171003817100381711989011986011990510038171003817100381710038171003817le 1 + 119905 le 1 + 119879 0 le 119905 le 119879 (49)
The controllability operator is
119876119879= int
119879
0
119890119860119905119861119861lowast119890119860lowast
119905119889119905 = 119887
2119879[
1 0
0 0] (50)
Hence 119876119879is not coercive and the conditions for complete
controllability based on coercivity of119876119879 fail for this example
Although system (42) can still be complete controllable forproperly selected functions 119891 we can investigate the partialcomplete controllability for this system being interested injust the first component 119909
119905of 119911119905
Let 119871 = [1 0] Then
119876119879= 119871119876119879119871lowast= 1198872119879 gt 0 (51)
This means that the linear system associated with the semili-near system (45) is119871-partially complete controllable Further-more the inequality (30) becomes
(1 +
1198872119879 (1 + 119879)
1198872119879
+
119887 (1 + 119879)
1198872119879
) (1 + 119879) 119879119870 lt 1 (52)
or simplifying
119870 lt
119887
(1 + 119879) (1 + 119879 + 2119887119879 + 1198871198792)
(53)
This establishes a relation between Lipschitz coefficient 119870and terminal time moment 119879 Depending on 119870 119879 must betaken sufficiently large to satisfy (53) So the system (42) is 119871-partially complete controllable for the time 119879 if the Lipschitzcoefficient 119870 related to 119891 satisfies (53)
Example 2 Delay equations are typical for application ofpartial controllability concepts Consider a nonlinear delayequation
1199091015840
119905= 119886119909119905+ 119887119906119905+ 119891(119905 119909
119905 int
0
minus120576
119909119905+120579
119889120579 119906119905)
1199090= 120585 119909
120579= 120578120579 minus120576 le 120579 le 0
(54)
on [0 119879] where 0 lt 120576 lt 119879 120578 isin 1198712(minus120576 0R) and 119906 isin 119880ad =
119862(0 119879R)Similar to Example 3 introduce the function119909 [0 119879] rarr
1198712(minus120576 0R) by
[119909119905]120579= 119909119905+120579
0 le 119905 le 119879 minus120576 le 120579 le 0 (55)
This function satisfies
1199091015840
119905= (
119889
119889120579
)119909119905 119909
0= 120578 0 lt 119905 le 119879 (56)
Denote by T119905 119905 ge 0 the semigroup generated by the
differential operator 119889119889120579 and let Γ be the integral operatorfrom 119871
2(minus120576 0R) to R defined by
Γℎ = int
0
minus120576
ℎ120579119889120579 ℎ isin 119871
2(minus120576 0R) (57)
noticing that Γ le radic120576 Then for
119910119905= [
119909119905
119909119905
] 120577 = [
120585
120578] isin R times 119871
2(minus120576 0R) (58)
we can write system (54) as
1199101015840
119905= 119860119910119905+ 119865 (119905 119910
119905 119906119905) + 119861119906
119905 1199100= 120577 (59)
where
119860 =[
[
119886 0
0
119889
119889120579
]
]
119865 (119905 119910 119906) = [
119891 (119905 119909 Γ119909 119906)
0]
119861 = [
119887
0]
(60)
assuming that
119910 = [
119909
119909] isin R times 119871
2(minus120576 0R) (61)
The semigroup generated by 119860 has the form
119890119860119905
= [119890119886119905
0
0 T119905
] 119905 ge 0 (62)
Therefore the controllability operator for system (54) can becalculated as
119876119879= int
119879
0
119890119860119905119861lowast119861119890119860lowast
119905119889119905 = int
119879
0
[11988721198902119886119905
0
0 0
] 119889119905
=[
[
[
1198872(1198902119886119879
minus 1)
2119886
0
0 0
]
]
]
(63)
This is definitely not a coercive operator
Abstract and Applied Analysis 7
Taking into account that the original system is given by(54) and (59) is just representation of (54) in the standardform which enlarges the original state space R to R times
1198712(minus120576 0R) we observe that the complete controllability for
system (54) is in fact 119871-partial complete controllability forsystem (59) if
119871 = [1 0] R times 1198712(minus120576 0R) 997888rarr R (64)
Calculating partial controllability operator we obtain
119876119879= 119871119876119879119871lowast=
1198872(1198902119886119879
minus 1)
2119886
gt 0(65)
which is coerciveFurthermore using
1003817100381710038171003817100381711989011986011990510038171003817100381710038171003817le 1 + 119890
119886119879
1003817100381710038171003817119876119879
1003817100381710038171003817= 120574 =
1198872(1198902119886119879
minus 1)
2119886
(66)
we write the inequality (30) in the form
(1 + 1 + 119890119886119879
+
2119886
119887 (119890119886119879
minus 1)
) (1 + 119890119886119879) 119879119870 lt 1 (67)
If the Lipschitz coefficient 119870 of the function 119865 and terminaltime moment 119879 satisfy this inequality then system (54) iscompletely controllable which in turnmeans that system (59)is 119871-partially complete controllable
5 Conclusion
In this paper a sufficient condition for partial complete con-trollability of a semilinear control system is proved This is acontinuation of the pioneering research that has been done byBashirov et al [20 21] about partial controllability conceptsA research in this way concerning partial complete andapproximate controllability for semilinear deterministic andstochastic systems has already been done and awaitingfor publication There are other kinds of systems whichbesides semilinearity include other features for exampleimpulsiveness fractional derivative issue and so forth Theresult of this paper can be extended to these systems as well
References
[1] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering D Transactions ofASME vol 82 pp 35ndash45 1960
[2] H O Fattorini ldquoSome remarks on complete controllabilityrdquoSIAM Journal on Control vol 4 pp 686ndash694 1966
[3] D L Russell ldquoNonharmonic Fourier series in the control theoryof distributed parameter systemsrdquo Journal of MathematicalAnalysis and Applications vol 18 pp 542ndash560 1967
[4] R F Curtain and H J ZwartAn Introduction to Infinite Dimen-sional Linear Systems Theory Springer Berlin Germany 1995
[5] A Bensoussan Stochastic Control of Partially Observable Sys-tems Cambridge University Press Cambridge UK 1992
[6] A Bensoussan G Da Prato M C Delfour and S K MitterRepresentation and Control of Infinite Dimensional SystemsSystems amp Control Foundations amp Applications BirkhauserBoston Mass USA 2nd edition 2007
[7] J Zabczyk Mathematical Control Theory An IntroductionSystems amp Control Foundations amp Applications BirkhauserBoston Mass USA 1995
[8] A E Bashirov Partially Observable Linear Systems underDependent Noises Systems amp Control Foundations amp Applica-tions Birkhauser Basel Switzerland 2003
[9] J Klamka Controllability of Dynamical Systems vol 48 ofMathematics and Its Applications Kluwer Academic PublishersDordrecht The Netherlands 1991
[10] K Balachandran and J P Dauer ldquoControllability of nonlinearsystems in Banach spaces a surveyrdquo Journal of OptimizationTheory and Applications vol 115 no 1 pp 7ndash28 2002
[11] K Balachandran and J P Dauer ldquoLocal controllability of semili-near evolution systems in Banach spacesrdquo Indian Journal of Pureand Applied Mathematics vol 29 no 3 pp 311ndash320 1998
[12] J Klamka ldquoSchauderrsquos fixed-point theorem in nonlinear con-trollability problemsrdquo Control and Cybernetics vol 29 no 1 pp153ndash165 2000
[13] N I Mahmudov ldquoControllability of semilinear stochastic sys-tems in Hilbert spacesrdquo Journal of Mathematical Analysis andApplications vol 288 no 1 pp 197ndash211 2003
[14] X J Li and J M Yong Optimal Control Theory for Infinite-Dimensional Systems SystemsampControl Foundations ampAppli-cations Birkhauser Boston Mass USA 1995
[15] R Sakthivel N I Mahmudov and J J Nieto ldquoControllabilityfor a class of fractional-order neutral evolution control systemsrdquoApplied Mathematics and Computation vol 218 no 20 pp10334ndash10340 2012
[16] R Sakthivel R Ganesh and S Suganya ldquoApproximate con-trollability of fractional neutral stochastic system with infinitedelayrdquo Reports on Mathematical Physics vol 70 no 3 pp 291ndash311 2012
[17] R Sakthivel Y Ren and N I Mahmudov ldquoOn the approximatecontrollability of semilinear fractional differential systemsrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1451ndash1459 2011
[18] Z Yan ldquoApproximate controllability of partial neutral func-tional differential systems of fractional order with state-dependent delayrdquo International Journal of Control vol 85 no8 pp 1051ndash1062 2012
[19] Y Ren L Hu and R Sakthivel ldquoControllability of impulsiveneutral stochastic functional differential inclusions with infinitedelayrdquo Journal of Computational and Applied Mathematics vol235 no 8 pp 2603ndash2614 2011
[20] A E Bashirov H Etikan andN Semi ldquoPartial controllability ofstochastic linear systemsrdquo International Journal of Control vol83 no 12 pp 2564ndash2572 2010
[21] A E Bashirov N Mahmudov N Semi and H Etikan ldquoPartialcontrollability conceptsrdquo International Journal of Control vol80 no 1 pp 1ndash7 2007
[22] A E Bashirov and N I Mahmudov ldquoOn concepts of control-lability for deterministic and stochastic systemsrdquo SIAM Journalon Control and Optimization vol 37 no 6 pp 1808ndash1821 1999
[23] A E Bashirov and K R Kerimov ldquoOn controllability concep-tion for stochastic systemsrdquo SIAM Journal on Control and Opti-mization vol 35 no 2 pp 384ndash398 1997
8 Abstract and Applied Analysis
[24] A E Bashirov ldquoOn weakening of the controllability conceptsrdquoin Proceedings of the 35th Conference on Decission and Controlpp 640ndash645 Kobe Japan 1996
[25] A E Bashirov and N I Mahmudov ldquoControllability of lineardeterministic and stochastic systemsrdquo in Proceedings of the 38thConference on Decission and Control pp 3196ndash3201 PhoenixAris USA 1999
[26] A E Bashirov and N I Mahmudov ldquoSome new results in the-ory of controllabilityrdquo in Proceedings of the 7th MediterraneanConference on Control and Automation pp 323ndash343 HaifaIsrael 1999
[27] L Byszewski ldquoTheorems about the existence and uniqueness ofsolutions of a semilinear evolution nonlocal Cauchy problemrdquoJournal of Mathematical Analysis and Applications vol 162 no2 pp 494ndash505 1991
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Abstract and Applied Analysis
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Volume 2013
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DifferentialEquations
International Journal of
Volume 2013
Abstract and Applied Analysis 5
= 1198901198601199051199090+ 119876119905119890119860lowast
(119879minus119905)119871lowast119876minus1
119879(ℎ minus 119871119890
1198601198791199090)
+ int
119905
0
119890119860(119905minus119904)
119891 (119904 119909119904 119906119904) 119889119904
minus int
119879
0
119876119905119890119860lowast
(119879minus119905)119871lowast119876minus1
119879119871119890119860(119879minus119903)
119891 (119903 119909119903 119906119903) 119889119903
= 1198901198601199051199090+ 119876119905119890119860lowast
(119879minus119905)119871lowast119876minus1
119879(ℎ minus 119871119890
1198601198791199090)
+ int
119905
0
119890119860(119905minus119904)
119891 (119904 119909119904 119906119904) 119889119904
minus 119876119905119890119860lowast
(119879minus119905)119871lowast119876minus1
119879119871int
119879
0
119890119860(119879minus119904)
119891 (119904 119909119904 119906119904) 119889119904
(32)
According to Lemma 6 there is a unique pair (119909 119906) isin 119883 times
119880ad satisfying (31) and (32) So 119906 isin 119880ad Furthermore wehave
119871119909119879= 119871(119890
1198601198791199090+ 119876119879119871lowast119876minus1
119879(ℎ minus 119871119890
1198601198791199090)
+ int
119879
0
119890119860(119879minus119904)
119891 (119904 119909119904 119906119904) 119889119904
minus 119876119879119871lowast119876minus1
119879119871int
119879
0
119890119860(119879minus119904)
119891 (119904 119909119904 119906119904) 119889119904)
= 1198711198901198601198791199090+ 119871119876119879119871lowast119876minus1
119879(ℎ minus 119871119890
1198601198791199090)
+ 119871int
119879
0
119890119860(119879minus119904)
119891 (119904 119909119904 119906119904) 119889119904
minus 119871119876119879119871lowast119876minus1
119879119871int
119879
0
119890119860(119879minus119904)
119891 (119904 119909119904 119906119904) 119889119904 = ℎ
(33)
Thus there is119906 isin 119880ad which steers1199090 to119909119879with119871119909119879 = ℎThismeans that the semilinear system (1) is 119871-partially completecontrollable on [0 119879] as desired
Remark 8 Decomposing 119876119879in the form
119876119879= [
119876119879
119877119879
119877lowast
119877119875119879
] (34)
where 119877119879 119867perprarr 119867 and 119875
119879 119867perprarr 119867perp are other compo-
nents of119876119879besides119876
119879and119867perp is an orthogonal complement
of119867 in119883 one can calculate
⟨119876119879ℎ ℎ⟩ = ⟨119876
119879ℎ1 ℎ1⟩ + 2⟨119877
119879ℎ2 ℎ1⟩ + ⟨119875
119879ℎ2 ℎ2⟩ (35)
where ℎ1= 119871ℎ isin 119867 and ℎ
2= ℎ minus 119871ℎ isin 119867
perp Therefore thecoercivity of 119876
119879implies the same of 119876
119879 But the converse
is not true Theorem 7 is powerful in the cases when 119876119879is
coercive but 119875119879is not
Example 9 Theorem 7 establishes just sufficient conditionof 119871-partial complete controllability In this example we will
demonstrate that this is not a necessary condition We willconsider a simple case of 119871 = 119868 when 119871-partial completecontrollability reduces to complete controllability Considerthe one-dimensional control system
1199091015840
119905= 2119909119905+ 2119906119905 1199090isin R (36)
This is a linear system and the controllability operator of thissystem is equal to
int
119879
0
41198904119905119889119905 = 119890
4119879minus 1 gt 0 for every 119879 gt 0 (37)
According to the theory of controllability for linear systemsthis system is controllable (completely) for every 119879 gt 0
Have another look at this system by writing it as
1199091015840
119905= 119909119905+ 119906119905+ 119891 (119909
119905 119906119905) 119909
0isin R (38)
where
119891 (119909 119906) = 119909 + 119906 (39)
Here 119891 satisfies the Lipschitz condition with 119870 = 1 Also119860 = 119861 = 1 implying 119861 = 1 and 119872 = sup
[0119879]119890119860119905 = 119890119879
Furthermore
119876119879= int
119879
0
1198902119905119889119905 =
1198902119879
minus 1
2
(40)
So 119876119879 = 120574 = (119890
2119879minus1)2 Then the inequality (30) becomes
(1 + 119890119879+
2119890119879
119890119879minus 1
) 119890119879119879 lt 1 (41)
The limit of the left-hand side in this inequality when 119879 rarr
infin is equal toinfin This means that there is a sufficiently large119879 such that the conditions of Theorem 7 do not hold for this119879 although the system under consideration is completelycontrollable Thus Theorem 7 states a sufficient conditionwhich is not a necessary condition
4 Examples
We demonstrate the features of 119871-partial complete controlla-bility in the following examples of control systems
Example 1 Consider the system of differential equations
1199091015840
119905= 119910119905+ 119887119906119905 1199090isin R
1199101015840
119905= 119891 (119905 119909
119905 119910119905 119906119905) 119910
0isin R
(42)
on [0 119879] where 119906 isin 119880ad = 119862(0 119879R) Besides the completecontrollability property that is
(119909 119910) isin R2 exist119906 isin 119880ad such that (119909
119879 119910119879) = (119909 119910) = R
2
(43)
we can investigate the partial complete controllability prop-erty that is
119909 isin R exist119906 isin 119880ad such that 119909119879= 119909 = R (44)
6 Abstract and Applied Analysis
We can write this system in R2 as the following semilinearsystem
1199111015840
119905= 119860119911119905+ 119865 (119905 119911
119905 119906119905) + 119861119906
119905 (45)
where
119911119905= [
119909119905
119910119905
] 119860 = [
0 1
0 0] 119861 = [
119887
0]
119865 (119905 119911 119906) = [
0
119891 (119905 119909 119910 119906)]
(46)
assuming that
119911 = [
119909
119910] (47)
It can be calculated that
119890119860119905
= [
1 119905
0 1] = [
1 0
0 1] + [
0 119905
0 0] (48)
Hence1003817100381710038171003817100381711989011986011990510038171003817100381710038171003817le 1 + 119905 le 1 + 119879 0 le 119905 le 119879 (49)
The controllability operator is
119876119879= int
119879
0
119890119860119905119861119861lowast119890119860lowast
119905119889119905 = 119887
2119879[
1 0
0 0] (50)
Hence 119876119879is not coercive and the conditions for complete
controllability based on coercivity of119876119879 fail for this example
Although system (42) can still be complete controllable forproperly selected functions 119891 we can investigate the partialcomplete controllability for this system being interested injust the first component 119909
119905of 119911119905
Let 119871 = [1 0] Then
119876119879= 119871119876119879119871lowast= 1198872119879 gt 0 (51)
This means that the linear system associated with the semili-near system (45) is119871-partially complete controllable Further-more the inequality (30) becomes
(1 +
1198872119879 (1 + 119879)
1198872119879
+
119887 (1 + 119879)
1198872119879
) (1 + 119879) 119879119870 lt 1 (52)
or simplifying
119870 lt
119887
(1 + 119879) (1 + 119879 + 2119887119879 + 1198871198792)
(53)
This establishes a relation between Lipschitz coefficient 119870and terminal time moment 119879 Depending on 119870 119879 must betaken sufficiently large to satisfy (53) So the system (42) is 119871-partially complete controllable for the time 119879 if the Lipschitzcoefficient 119870 related to 119891 satisfies (53)
Example 2 Delay equations are typical for application ofpartial controllability concepts Consider a nonlinear delayequation
1199091015840
119905= 119886119909119905+ 119887119906119905+ 119891(119905 119909
119905 int
0
minus120576
119909119905+120579
119889120579 119906119905)
1199090= 120585 119909
120579= 120578120579 minus120576 le 120579 le 0
(54)
on [0 119879] where 0 lt 120576 lt 119879 120578 isin 1198712(minus120576 0R) and 119906 isin 119880ad =
119862(0 119879R)Similar to Example 3 introduce the function119909 [0 119879] rarr
1198712(minus120576 0R) by
[119909119905]120579= 119909119905+120579
0 le 119905 le 119879 minus120576 le 120579 le 0 (55)
This function satisfies
1199091015840
119905= (
119889
119889120579
)119909119905 119909
0= 120578 0 lt 119905 le 119879 (56)
Denote by T119905 119905 ge 0 the semigroup generated by the
differential operator 119889119889120579 and let Γ be the integral operatorfrom 119871
2(minus120576 0R) to R defined by
Γℎ = int
0
minus120576
ℎ120579119889120579 ℎ isin 119871
2(minus120576 0R) (57)
noticing that Γ le radic120576 Then for
119910119905= [
119909119905
119909119905
] 120577 = [
120585
120578] isin R times 119871
2(minus120576 0R) (58)
we can write system (54) as
1199101015840
119905= 119860119910119905+ 119865 (119905 119910
119905 119906119905) + 119861119906
119905 1199100= 120577 (59)
where
119860 =[
[
119886 0
0
119889
119889120579
]
]
119865 (119905 119910 119906) = [
119891 (119905 119909 Γ119909 119906)
0]
119861 = [
119887
0]
(60)
assuming that
119910 = [
119909
119909] isin R times 119871
2(minus120576 0R) (61)
The semigroup generated by 119860 has the form
119890119860119905
= [119890119886119905
0
0 T119905
] 119905 ge 0 (62)
Therefore the controllability operator for system (54) can becalculated as
119876119879= int
119879
0
119890119860119905119861lowast119861119890119860lowast
119905119889119905 = int
119879
0
[11988721198902119886119905
0
0 0
] 119889119905
=[
[
[
1198872(1198902119886119879
minus 1)
2119886
0
0 0
]
]
]
(63)
This is definitely not a coercive operator
Abstract and Applied Analysis 7
Taking into account that the original system is given by(54) and (59) is just representation of (54) in the standardform which enlarges the original state space R to R times
1198712(minus120576 0R) we observe that the complete controllability for
system (54) is in fact 119871-partial complete controllability forsystem (59) if
119871 = [1 0] R times 1198712(minus120576 0R) 997888rarr R (64)
Calculating partial controllability operator we obtain
119876119879= 119871119876119879119871lowast=
1198872(1198902119886119879
minus 1)
2119886
gt 0(65)
which is coerciveFurthermore using
1003817100381710038171003817100381711989011986011990510038171003817100381710038171003817le 1 + 119890
119886119879
1003817100381710038171003817119876119879
1003817100381710038171003817= 120574 =
1198872(1198902119886119879
minus 1)
2119886
(66)
we write the inequality (30) in the form
(1 + 1 + 119890119886119879
+
2119886
119887 (119890119886119879
minus 1)
) (1 + 119890119886119879) 119879119870 lt 1 (67)
If the Lipschitz coefficient 119870 of the function 119865 and terminaltime moment 119879 satisfy this inequality then system (54) iscompletely controllable which in turnmeans that system (59)is 119871-partially complete controllable
5 Conclusion
In this paper a sufficient condition for partial complete con-trollability of a semilinear control system is proved This is acontinuation of the pioneering research that has been done byBashirov et al [20 21] about partial controllability conceptsA research in this way concerning partial complete andapproximate controllability for semilinear deterministic andstochastic systems has already been done and awaitingfor publication There are other kinds of systems whichbesides semilinearity include other features for exampleimpulsiveness fractional derivative issue and so forth Theresult of this paper can be extended to these systems as well
References
[1] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering D Transactions ofASME vol 82 pp 35ndash45 1960
[2] H O Fattorini ldquoSome remarks on complete controllabilityrdquoSIAM Journal on Control vol 4 pp 686ndash694 1966
[3] D L Russell ldquoNonharmonic Fourier series in the control theoryof distributed parameter systemsrdquo Journal of MathematicalAnalysis and Applications vol 18 pp 542ndash560 1967
[4] R F Curtain and H J ZwartAn Introduction to Infinite Dimen-sional Linear Systems Theory Springer Berlin Germany 1995
[5] A Bensoussan Stochastic Control of Partially Observable Sys-tems Cambridge University Press Cambridge UK 1992
[6] A Bensoussan G Da Prato M C Delfour and S K MitterRepresentation and Control of Infinite Dimensional SystemsSystems amp Control Foundations amp Applications BirkhauserBoston Mass USA 2nd edition 2007
[7] J Zabczyk Mathematical Control Theory An IntroductionSystems amp Control Foundations amp Applications BirkhauserBoston Mass USA 1995
[8] A E Bashirov Partially Observable Linear Systems underDependent Noises Systems amp Control Foundations amp Applica-tions Birkhauser Basel Switzerland 2003
[9] J Klamka Controllability of Dynamical Systems vol 48 ofMathematics and Its Applications Kluwer Academic PublishersDordrecht The Netherlands 1991
[10] K Balachandran and J P Dauer ldquoControllability of nonlinearsystems in Banach spaces a surveyrdquo Journal of OptimizationTheory and Applications vol 115 no 1 pp 7ndash28 2002
[11] K Balachandran and J P Dauer ldquoLocal controllability of semili-near evolution systems in Banach spacesrdquo Indian Journal of Pureand Applied Mathematics vol 29 no 3 pp 311ndash320 1998
[12] J Klamka ldquoSchauderrsquos fixed-point theorem in nonlinear con-trollability problemsrdquo Control and Cybernetics vol 29 no 1 pp153ndash165 2000
[13] N I Mahmudov ldquoControllability of semilinear stochastic sys-tems in Hilbert spacesrdquo Journal of Mathematical Analysis andApplications vol 288 no 1 pp 197ndash211 2003
[14] X J Li and J M Yong Optimal Control Theory for Infinite-Dimensional Systems SystemsampControl Foundations ampAppli-cations Birkhauser Boston Mass USA 1995
[15] R Sakthivel N I Mahmudov and J J Nieto ldquoControllabilityfor a class of fractional-order neutral evolution control systemsrdquoApplied Mathematics and Computation vol 218 no 20 pp10334ndash10340 2012
[16] R Sakthivel R Ganesh and S Suganya ldquoApproximate con-trollability of fractional neutral stochastic system with infinitedelayrdquo Reports on Mathematical Physics vol 70 no 3 pp 291ndash311 2012
[17] R Sakthivel Y Ren and N I Mahmudov ldquoOn the approximatecontrollability of semilinear fractional differential systemsrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1451ndash1459 2011
[18] Z Yan ldquoApproximate controllability of partial neutral func-tional differential systems of fractional order with state-dependent delayrdquo International Journal of Control vol 85 no8 pp 1051ndash1062 2012
[19] Y Ren L Hu and R Sakthivel ldquoControllability of impulsiveneutral stochastic functional differential inclusions with infinitedelayrdquo Journal of Computational and Applied Mathematics vol235 no 8 pp 2603ndash2614 2011
[20] A E Bashirov H Etikan andN Semi ldquoPartial controllability ofstochastic linear systemsrdquo International Journal of Control vol83 no 12 pp 2564ndash2572 2010
[21] A E Bashirov N Mahmudov N Semi and H Etikan ldquoPartialcontrollability conceptsrdquo International Journal of Control vol80 no 1 pp 1ndash7 2007
[22] A E Bashirov and N I Mahmudov ldquoOn concepts of control-lability for deterministic and stochastic systemsrdquo SIAM Journalon Control and Optimization vol 37 no 6 pp 1808ndash1821 1999
[23] A E Bashirov and K R Kerimov ldquoOn controllability concep-tion for stochastic systemsrdquo SIAM Journal on Control and Opti-mization vol 35 no 2 pp 384ndash398 1997
8 Abstract and Applied Analysis
[24] A E Bashirov ldquoOn weakening of the controllability conceptsrdquoin Proceedings of the 35th Conference on Decission and Controlpp 640ndash645 Kobe Japan 1996
[25] A E Bashirov and N I Mahmudov ldquoControllability of lineardeterministic and stochastic systemsrdquo in Proceedings of the 38thConference on Decission and Control pp 3196ndash3201 PhoenixAris USA 1999
[26] A E Bashirov and N I Mahmudov ldquoSome new results in the-ory of controllabilityrdquo in Proceedings of the 7th MediterraneanConference on Control and Automation pp 323ndash343 HaifaIsrael 1999
[27] L Byszewski ldquoTheorems about the existence and uniqueness ofsolutions of a semilinear evolution nonlocal Cauchy problemrdquoJournal of Mathematical Analysis and Applications vol 162 no2 pp 494ndash505 1991
Submit your manuscripts athttpwwwhindawicom
OperationsResearch
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Abstract and Applied Analysis
ISRN Applied Mathematics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2013
International Journal of
Combinatorics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Journal of Function Spaces and Applications
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
ISRN Geometry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Discrete Dynamicsin Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2013
Advances in
Mathematical Physics
ISRN Algebra
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
ProbabilityandStatistics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
ISRN Mathematical Analysis
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Journal ofApplied Mathematics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Advances in
DecisionSciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Stochastic AnalysisInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013
The Scientific World Journal
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
ISRN Discrete Mathematics
Hindawi Publishing Corporationhttpwwwhindawicom
DifferentialEquations
International Journal of
Volume 2013
6 Abstract and Applied Analysis
We can write this system in R2 as the following semilinearsystem
1199111015840
119905= 119860119911119905+ 119865 (119905 119911
119905 119906119905) + 119861119906
119905 (45)
where
119911119905= [
119909119905
119910119905
] 119860 = [
0 1
0 0] 119861 = [
119887
0]
119865 (119905 119911 119906) = [
0
119891 (119905 119909 119910 119906)]
(46)
assuming that
119911 = [
119909
119910] (47)
It can be calculated that
119890119860119905
= [
1 119905
0 1] = [
1 0
0 1] + [
0 119905
0 0] (48)
Hence1003817100381710038171003817100381711989011986011990510038171003817100381710038171003817le 1 + 119905 le 1 + 119879 0 le 119905 le 119879 (49)
The controllability operator is
119876119879= int
119879
0
119890119860119905119861119861lowast119890119860lowast
119905119889119905 = 119887
2119879[
1 0
0 0] (50)
Hence 119876119879is not coercive and the conditions for complete
controllability based on coercivity of119876119879 fail for this example
Although system (42) can still be complete controllable forproperly selected functions 119891 we can investigate the partialcomplete controllability for this system being interested injust the first component 119909
119905of 119911119905
Let 119871 = [1 0] Then
119876119879= 119871119876119879119871lowast= 1198872119879 gt 0 (51)
This means that the linear system associated with the semili-near system (45) is119871-partially complete controllable Further-more the inequality (30) becomes
(1 +
1198872119879 (1 + 119879)
1198872119879
+
119887 (1 + 119879)
1198872119879
) (1 + 119879) 119879119870 lt 1 (52)
or simplifying
119870 lt
119887
(1 + 119879) (1 + 119879 + 2119887119879 + 1198871198792)
(53)
This establishes a relation between Lipschitz coefficient 119870and terminal time moment 119879 Depending on 119870 119879 must betaken sufficiently large to satisfy (53) So the system (42) is 119871-partially complete controllable for the time 119879 if the Lipschitzcoefficient 119870 related to 119891 satisfies (53)
Example 2 Delay equations are typical for application ofpartial controllability concepts Consider a nonlinear delayequation
1199091015840
119905= 119886119909119905+ 119887119906119905+ 119891(119905 119909
119905 int
0
minus120576
119909119905+120579
119889120579 119906119905)
1199090= 120585 119909
120579= 120578120579 minus120576 le 120579 le 0
(54)
on [0 119879] where 0 lt 120576 lt 119879 120578 isin 1198712(minus120576 0R) and 119906 isin 119880ad =
119862(0 119879R)Similar to Example 3 introduce the function119909 [0 119879] rarr
1198712(minus120576 0R) by
[119909119905]120579= 119909119905+120579
0 le 119905 le 119879 minus120576 le 120579 le 0 (55)
This function satisfies
1199091015840
119905= (
119889
119889120579
)119909119905 119909
0= 120578 0 lt 119905 le 119879 (56)
Denote by T119905 119905 ge 0 the semigroup generated by the
differential operator 119889119889120579 and let Γ be the integral operatorfrom 119871
2(minus120576 0R) to R defined by
Γℎ = int
0
minus120576
ℎ120579119889120579 ℎ isin 119871
2(minus120576 0R) (57)
noticing that Γ le radic120576 Then for
119910119905= [
119909119905
119909119905
] 120577 = [
120585
120578] isin R times 119871
2(minus120576 0R) (58)
we can write system (54) as
1199101015840
119905= 119860119910119905+ 119865 (119905 119910
119905 119906119905) + 119861119906
119905 1199100= 120577 (59)
where
119860 =[
[
119886 0
0
119889
119889120579
]
]
119865 (119905 119910 119906) = [
119891 (119905 119909 Γ119909 119906)
0]
119861 = [
119887
0]
(60)
assuming that
119910 = [
119909
119909] isin R times 119871
2(minus120576 0R) (61)
The semigroup generated by 119860 has the form
119890119860119905
= [119890119886119905
0
0 T119905
] 119905 ge 0 (62)
Therefore the controllability operator for system (54) can becalculated as
119876119879= int
119879
0
119890119860119905119861lowast119861119890119860lowast
119905119889119905 = int
119879
0
[11988721198902119886119905
0
0 0
] 119889119905
=[
[
[
1198872(1198902119886119879
minus 1)
2119886
0
0 0
]
]
]
(63)
This is definitely not a coercive operator
Abstract and Applied Analysis 7
Taking into account that the original system is given by(54) and (59) is just representation of (54) in the standardform which enlarges the original state space R to R times
1198712(minus120576 0R) we observe that the complete controllability for
system (54) is in fact 119871-partial complete controllability forsystem (59) if
119871 = [1 0] R times 1198712(minus120576 0R) 997888rarr R (64)
Calculating partial controllability operator we obtain
119876119879= 119871119876119879119871lowast=
1198872(1198902119886119879
minus 1)
2119886
gt 0(65)
which is coerciveFurthermore using
1003817100381710038171003817100381711989011986011990510038171003817100381710038171003817le 1 + 119890
119886119879
1003817100381710038171003817119876119879
1003817100381710038171003817= 120574 =
1198872(1198902119886119879
minus 1)
2119886
(66)
we write the inequality (30) in the form
(1 + 1 + 119890119886119879
+
2119886
119887 (119890119886119879
minus 1)
) (1 + 119890119886119879) 119879119870 lt 1 (67)
If the Lipschitz coefficient 119870 of the function 119865 and terminaltime moment 119879 satisfy this inequality then system (54) iscompletely controllable which in turnmeans that system (59)is 119871-partially complete controllable
5 Conclusion
In this paper a sufficient condition for partial complete con-trollability of a semilinear control system is proved This is acontinuation of the pioneering research that has been done byBashirov et al [20 21] about partial controllability conceptsA research in this way concerning partial complete andapproximate controllability for semilinear deterministic andstochastic systems has already been done and awaitingfor publication There are other kinds of systems whichbesides semilinearity include other features for exampleimpulsiveness fractional derivative issue and so forth Theresult of this paper can be extended to these systems as well
References
[1] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering D Transactions ofASME vol 82 pp 35ndash45 1960
[2] H O Fattorini ldquoSome remarks on complete controllabilityrdquoSIAM Journal on Control vol 4 pp 686ndash694 1966
[3] D L Russell ldquoNonharmonic Fourier series in the control theoryof distributed parameter systemsrdquo Journal of MathematicalAnalysis and Applications vol 18 pp 542ndash560 1967
[4] R F Curtain and H J ZwartAn Introduction to Infinite Dimen-sional Linear Systems Theory Springer Berlin Germany 1995
[5] A Bensoussan Stochastic Control of Partially Observable Sys-tems Cambridge University Press Cambridge UK 1992
[6] A Bensoussan G Da Prato M C Delfour and S K MitterRepresentation and Control of Infinite Dimensional SystemsSystems amp Control Foundations amp Applications BirkhauserBoston Mass USA 2nd edition 2007
[7] J Zabczyk Mathematical Control Theory An IntroductionSystems amp Control Foundations amp Applications BirkhauserBoston Mass USA 1995
[8] A E Bashirov Partially Observable Linear Systems underDependent Noises Systems amp Control Foundations amp Applica-tions Birkhauser Basel Switzerland 2003
[9] J Klamka Controllability of Dynamical Systems vol 48 ofMathematics and Its Applications Kluwer Academic PublishersDordrecht The Netherlands 1991
[10] K Balachandran and J P Dauer ldquoControllability of nonlinearsystems in Banach spaces a surveyrdquo Journal of OptimizationTheory and Applications vol 115 no 1 pp 7ndash28 2002
[11] K Balachandran and J P Dauer ldquoLocal controllability of semili-near evolution systems in Banach spacesrdquo Indian Journal of Pureand Applied Mathematics vol 29 no 3 pp 311ndash320 1998
[12] J Klamka ldquoSchauderrsquos fixed-point theorem in nonlinear con-trollability problemsrdquo Control and Cybernetics vol 29 no 1 pp153ndash165 2000
[13] N I Mahmudov ldquoControllability of semilinear stochastic sys-tems in Hilbert spacesrdquo Journal of Mathematical Analysis andApplications vol 288 no 1 pp 197ndash211 2003
[14] X J Li and J M Yong Optimal Control Theory for Infinite-Dimensional Systems SystemsampControl Foundations ampAppli-cations Birkhauser Boston Mass USA 1995
[15] R Sakthivel N I Mahmudov and J J Nieto ldquoControllabilityfor a class of fractional-order neutral evolution control systemsrdquoApplied Mathematics and Computation vol 218 no 20 pp10334ndash10340 2012
[16] R Sakthivel R Ganesh and S Suganya ldquoApproximate con-trollability of fractional neutral stochastic system with infinitedelayrdquo Reports on Mathematical Physics vol 70 no 3 pp 291ndash311 2012
[17] R Sakthivel Y Ren and N I Mahmudov ldquoOn the approximatecontrollability of semilinear fractional differential systemsrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1451ndash1459 2011
[18] Z Yan ldquoApproximate controllability of partial neutral func-tional differential systems of fractional order with state-dependent delayrdquo International Journal of Control vol 85 no8 pp 1051ndash1062 2012
[19] Y Ren L Hu and R Sakthivel ldquoControllability of impulsiveneutral stochastic functional differential inclusions with infinitedelayrdquo Journal of Computational and Applied Mathematics vol235 no 8 pp 2603ndash2614 2011
[20] A E Bashirov H Etikan andN Semi ldquoPartial controllability ofstochastic linear systemsrdquo International Journal of Control vol83 no 12 pp 2564ndash2572 2010
[21] A E Bashirov N Mahmudov N Semi and H Etikan ldquoPartialcontrollability conceptsrdquo International Journal of Control vol80 no 1 pp 1ndash7 2007
[22] A E Bashirov and N I Mahmudov ldquoOn concepts of control-lability for deterministic and stochastic systemsrdquo SIAM Journalon Control and Optimization vol 37 no 6 pp 1808ndash1821 1999
[23] A E Bashirov and K R Kerimov ldquoOn controllability concep-tion for stochastic systemsrdquo SIAM Journal on Control and Opti-mization vol 35 no 2 pp 384ndash398 1997
8 Abstract and Applied Analysis
[24] A E Bashirov ldquoOn weakening of the controllability conceptsrdquoin Proceedings of the 35th Conference on Decission and Controlpp 640ndash645 Kobe Japan 1996
[25] A E Bashirov and N I Mahmudov ldquoControllability of lineardeterministic and stochastic systemsrdquo in Proceedings of the 38thConference on Decission and Control pp 3196ndash3201 PhoenixAris USA 1999
[26] A E Bashirov and N I Mahmudov ldquoSome new results in the-ory of controllabilityrdquo in Proceedings of the 7th MediterraneanConference on Control and Automation pp 323ndash343 HaifaIsrael 1999
[27] L Byszewski ldquoTheorems about the existence and uniqueness ofsolutions of a semilinear evolution nonlocal Cauchy problemrdquoJournal of Mathematical Analysis and Applications vol 162 no2 pp 494ndash505 1991
Submit your manuscripts athttpwwwhindawicom
OperationsResearch
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Abstract and Applied Analysis
ISRN Applied Mathematics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2013
International Journal of
Combinatorics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Journal of Function Spaces and Applications
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
ISRN Geometry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Discrete Dynamicsin Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2013
Advances in
Mathematical Physics
ISRN Algebra
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
ProbabilityandStatistics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
ISRN Mathematical Analysis
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Journal ofApplied Mathematics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Advances in
DecisionSciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Stochastic AnalysisInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013
The Scientific World Journal
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
ISRN Discrete Mathematics
Hindawi Publishing Corporationhttpwwwhindawicom
DifferentialEquations
International Journal of
Volume 2013
Abstract and Applied Analysis 7
Taking into account that the original system is given by(54) and (59) is just representation of (54) in the standardform which enlarges the original state space R to R times
1198712(minus120576 0R) we observe that the complete controllability for
system (54) is in fact 119871-partial complete controllability forsystem (59) if
119871 = [1 0] R times 1198712(minus120576 0R) 997888rarr R (64)
Calculating partial controllability operator we obtain
119876119879= 119871119876119879119871lowast=
1198872(1198902119886119879
minus 1)
2119886
gt 0(65)
which is coerciveFurthermore using
1003817100381710038171003817100381711989011986011990510038171003817100381710038171003817le 1 + 119890
119886119879
1003817100381710038171003817119876119879
1003817100381710038171003817= 120574 =
1198872(1198902119886119879
minus 1)
2119886
(66)
we write the inequality (30) in the form
(1 + 1 + 119890119886119879
+
2119886
119887 (119890119886119879
minus 1)
) (1 + 119890119886119879) 119879119870 lt 1 (67)
If the Lipschitz coefficient 119870 of the function 119865 and terminaltime moment 119879 satisfy this inequality then system (54) iscompletely controllable which in turnmeans that system (59)is 119871-partially complete controllable
5 Conclusion
In this paper a sufficient condition for partial complete con-trollability of a semilinear control system is proved This is acontinuation of the pioneering research that has been done byBashirov et al [20 21] about partial controllability conceptsA research in this way concerning partial complete andapproximate controllability for semilinear deterministic andstochastic systems has already been done and awaitingfor publication There are other kinds of systems whichbesides semilinearity include other features for exampleimpulsiveness fractional derivative issue and so forth Theresult of this paper can be extended to these systems as well
References
[1] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering D Transactions ofASME vol 82 pp 35ndash45 1960
[2] H O Fattorini ldquoSome remarks on complete controllabilityrdquoSIAM Journal on Control vol 4 pp 686ndash694 1966
[3] D L Russell ldquoNonharmonic Fourier series in the control theoryof distributed parameter systemsrdquo Journal of MathematicalAnalysis and Applications vol 18 pp 542ndash560 1967
[4] R F Curtain and H J ZwartAn Introduction to Infinite Dimen-sional Linear Systems Theory Springer Berlin Germany 1995
[5] A Bensoussan Stochastic Control of Partially Observable Sys-tems Cambridge University Press Cambridge UK 1992
[6] A Bensoussan G Da Prato M C Delfour and S K MitterRepresentation and Control of Infinite Dimensional SystemsSystems amp Control Foundations amp Applications BirkhauserBoston Mass USA 2nd edition 2007
[7] J Zabczyk Mathematical Control Theory An IntroductionSystems amp Control Foundations amp Applications BirkhauserBoston Mass USA 1995
[8] A E Bashirov Partially Observable Linear Systems underDependent Noises Systems amp Control Foundations amp Applica-tions Birkhauser Basel Switzerland 2003
[9] J Klamka Controllability of Dynamical Systems vol 48 ofMathematics and Its Applications Kluwer Academic PublishersDordrecht The Netherlands 1991
[10] K Balachandran and J P Dauer ldquoControllability of nonlinearsystems in Banach spaces a surveyrdquo Journal of OptimizationTheory and Applications vol 115 no 1 pp 7ndash28 2002
[11] K Balachandran and J P Dauer ldquoLocal controllability of semili-near evolution systems in Banach spacesrdquo Indian Journal of Pureand Applied Mathematics vol 29 no 3 pp 311ndash320 1998
[12] J Klamka ldquoSchauderrsquos fixed-point theorem in nonlinear con-trollability problemsrdquo Control and Cybernetics vol 29 no 1 pp153ndash165 2000
[13] N I Mahmudov ldquoControllability of semilinear stochastic sys-tems in Hilbert spacesrdquo Journal of Mathematical Analysis andApplications vol 288 no 1 pp 197ndash211 2003
[14] X J Li and J M Yong Optimal Control Theory for Infinite-Dimensional Systems SystemsampControl Foundations ampAppli-cations Birkhauser Boston Mass USA 1995
[15] R Sakthivel N I Mahmudov and J J Nieto ldquoControllabilityfor a class of fractional-order neutral evolution control systemsrdquoApplied Mathematics and Computation vol 218 no 20 pp10334ndash10340 2012
[16] R Sakthivel R Ganesh and S Suganya ldquoApproximate con-trollability of fractional neutral stochastic system with infinitedelayrdquo Reports on Mathematical Physics vol 70 no 3 pp 291ndash311 2012
[17] R Sakthivel Y Ren and N I Mahmudov ldquoOn the approximatecontrollability of semilinear fractional differential systemsrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1451ndash1459 2011
[18] Z Yan ldquoApproximate controllability of partial neutral func-tional differential systems of fractional order with state-dependent delayrdquo International Journal of Control vol 85 no8 pp 1051ndash1062 2012
[19] Y Ren L Hu and R Sakthivel ldquoControllability of impulsiveneutral stochastic functional differential inclusions with infinitedelayrdquo Journal of Computational and Applied Mathematics vol235 no 8 pp 2603ndash2614 2011
[20] A E Bashirov H Etikan andN Semi ldquoPartial controllability ofstochastic linear systemsrdquo International Journal of Control vol83 no 12 pp 2564ndash2572 2010
[21] A E Bashirov N Mahmudov N Semi and H Etikan ldquoPartialcontrollability conceptsrdquo International Journal of Control vol80 no 1 pp 1ndash7 2007
[22] A E Bashirov and N I Mahmudov ldquoOn concepts of control-lability for deterministic and stochastic systemsrdquo SIAM Journalon Control and Optimization vol 37 no 6 pp 1808ndash1821 1999
[23] A E Bashirov and K R Kerimov ldquoOn controllability concep-tion for stochastic systemsrdquo SIAM Journal on Control and Opti-mization vol 35 no 2 pp 384ndash398 1997
8 Abstract and Applied Analysis
[24] A E Bashirov ldquoOn weakening of the controllability conceptsrdquoin Proceedings of the 35th Conference on Decission and Controlpp 640ndash645 Kobe Japan 1996
[25] A E Bashirov and N I Mahmudov ldquoControllability of lineardeterministic and stochastic systemsrdquo in Proceedings of the 38thConference on Decission and Control pp 3196ndash3201 PhoenixAris USA 1999
[26] A E Bashirov and N I Mahmudov ldquoSome new results in the-ory of controllabilityrdquo in Proceedings of the 7th MediterraneanConference on Control and Automation pp 323ndash343 HaifaIsrael 1999
[27] L Byszewski ldquoTheorems about the existence and uniqueness ofsolutions of a semilinear evolution nonlocal Cauchy problemrdquoJournal of Mathematical Analysis and Applications vol 162 no2 pp 494ndash505 1991
Submit your manuscripts athttpwwwhindawicom
OperationsResearch
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Abstract and Applied Analysis
ISRN Applied Mathematics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2013
International Journal of
Combinatorics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Journal of Function Spaces and Applications
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
ISRN Geometry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Discrete Dynamicsin Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2013
Advances in
Mathematical Physics
ISRN Algebra
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
ProbabilityandStatistics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
ISRN Mathematical Analysis
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Journal ofApplied Mathematics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Advances in
DecisionSciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Stochastic AnalysisInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013
The Scientific World Journal
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
ISRN Discrete Mathematics
Hindawi Publishing Corporationhttpwwwhindawicom
DifferentialEquations
International Journal of
Volume 2013
8 Abstract and Applied Analysis
[24] A E Bashirov ldquoOn weakening of the controllability conceptsrdquoin Proceedings of the 35th Conference on Decission and Controlpp 640ndash645 Kobe Japan 1996
[25] A E Bashirov and N I Mahmudov ldquoControllability of lineardeterministic and stochastic systemsrdquo in Proceedings of the 38thConference on Decission and Control pp 3196ndash3201 PhoenixAris USA 1999
[26] A E Bashirov and N I Mahmudov ldquoSome new results in the-ory of controllabilityrdquo in Proceedings of the 7th MediterraneanConference on Control and Automation pp 323ndash343 HaifaIsrael 1999
[27] L Byszewski ldquoTheorems about the existence and uniqueness ofsolutions of a semilinear evolution nonlocal Cauchy problemrdquoJournal of Mathematical Analysis and Applications vol 162 no2 pp 494ndash505 1991
Submit your manuscripts athttpwwwhindawicom
OperationsResearch
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Abstract and Applied Analysis
ISRN Applied Mathematics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2013
International Journal of
Combinatorics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Journal of Function Spaces and Applications
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
ISRN Geometry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Discrete Dynamicsin Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2013
Advances in
Mathematical Physics
ISRN Algebra
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
ProbabilityandStatistics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
ISRN Mathematical Analysis
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Journal ofApplied Mathematics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Advances in
DecisionSciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Stochastic AnalysisInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013
The Scientific World Journal
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
ISRN Discrete Mathematics
Hindawi Publishing Corporationhttpwwwhindawicom
DifferentialEquations
International Journal of
Volume 2013
Submit your manuscripts athttpwwwhindawicom
OperationsResearch
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Abstract and Applied Analysis
ISRN Applied Mathematics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2013
International Journal of
Combinatorics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Journal of Function Spaces and Applications
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
ISRN Geometry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Discrete Dynamicsin Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2013
Advances in
Mathematical Physics
ISRN Algebra
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
ProbabilityandStatistics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
ISRN Mathematical Analysis
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Journal ofApplied Mathematics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Advances in
DecisionSciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
Stochastic AnalysisInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013
The Scientific World Journal
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013
ISRN Discrete Mathematics
Hindawi Publishing Corporationhttpwwwhindawicom
DifferentialEquations
International Journal of
Volume 2013
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