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Research ArticleOn the Uniform Exponential Stability ofTime-Varying Systems Subject to Discrete Time-VaryingDelays and Nonlinear Delayed Perturbations
Maher Hammami1 Mohamed Ali Hammami1 and Manuel De la Sen2
1Faculty of Sciences of Sfax University of Sfax Route Soukra BP 1171 3000 Sfax Tunisia2Departamento de Electricidad y Electronica Facultad de Ciencias Universidad del Pais Vasco Leioa (Bizkaia)Apartado 644 480809 Bilbao Spain
Correspondence should be addressed to Maher Hammami hammami maheryahoofr
Received 9 November 2014 Accepted 19 February 2015
Academic Editor Qingling Zhang
Copyright copy 2015 Maher Hammami et al 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
This paper addresses the problem of stability analysis of systems with delayed time-varying perturbations Some sufficientconditions for a class of linear time-varying systems with nonlinear delayed perturbations are derived by using an improvedLyapunov-Krasovskii functionalTheuniformglobal asymptotic stability of the solutions is obtained in terms of convergence towarda neighborhood of the origin
1 Introduction
Theproblemof robust stability analysis of linear time-varyingsystems subject to time-varying perturbations has attractedthe attention of many researchers Explicit bounds for thestructured time-varying perturbations have been derived [1ndash6] where the stability problem of linear systems subject todelayed time-varying perturbations has been studied whileonly few papers [7ndash11] give stability conditions for lineartime-varying delay systems among those [10] dealing withthe exponential stability of perturbed systems In [5] a newsufficient delay dependent exponential stability for a class oflinear time-varying systemswith nonlinear delayed perturba-tions is obtained based on a Lyapunov-Krasovskii functionalTime delay systems can include mixed neutral discrete (orpoint) delays and distributed delays including Volterra-typedistributed dynamics [12 13] Also delayed dynamics oftenappears in real-life problems like for instance epidemicpropagation models [14 15] since they affect the illnesspropagation via the incubation process in the studied pop-ulation and the vaccination period Delays are also useful todescribe single-species population evolution models [16] andare related to certain diffusion and competition predator-prey
models [17] Conditions to preserve the asymptotic stabilitycompared to a delay-free nominal model description havebeen widely studied in the literature including the case ofpresence of possibly delayed perturbation dynamics See forinstance [1ndash5 7 8 11 12 18ndash22] and references therein Themain novelty of this paper relies on the fact that the proposedapproach for stability analysis allows for the computationof the bounds which characterize the exponential rate ofconvergence of the solution towards a closed ball centeredat the origin by extending the complexity of the system byconsidering at the same time time-varying dynamics withtime-varying time differentiable in the delays in the nominalpart by considering nonnecessarily zero lower-bounds forthe delays and by considering more general conditions thanjust to be Lipschitz for the delayed in general nonlineardynamics Note for instance that the nominal part of thesystem has no delays in [5] the lower-bound of the delaysof the perturbations is assumed to be zero while those per-turbations are assumed to be Lipschitz in the state-variablesIn this paper the nominal part is time-varying with time-varying delays the lower-bounds of the delays can exceedzero and the perturbations norms incorporate a time-varyingupper-bound apart from the Lipschiptz type one In [9]
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 641268 12 pageshttpdxdoiorg1011552015641268
2 Mathematical Problems in Engineering
a global-null controllability is required while the asymptoticstability is not guaranteed to be of exponential type In thesame way the asymptotic stability is not guaranteed to beexponential in [11] Another novelty is that the delays aretime-varying time-differentiable and they are not requiredto be known Only lower and upper-bound of the delayfunctions and their time-derivatives are required for stabilityanalysis We will study a class of nonlinear system such thatthe nonlinearity is bounded by some integrable functionswhich are bounded where the origin is not necessarily anequilibrium point We deal with the practical stability of theorigin (see [23]) The asymptotic stability is more importantthan stability also the desired system may be unstable andyet the system may oscillate sufficiently near this state thatits performance is acceptable thus the notion of practicalstability is more suitable in several situations than Lyapunovstability In this case all state trajectories are bounded andapproach a sufficiently small neighborhood of the origin Onealso desires that the state approaches the origin (or some suffi-ciently small neighborhood of it) in a sufficiently fast mannerThis notion of practical stability was introduced by [24] fornonlinear time-varying systems and studied for differentialequations with delays by [25] (see also the references therein)Moreover the authors in [26 27] constructed stabilizingcontrollers to obtain global convergence of solutions towarda small ball for some classes of uncertain control systemsIn this paper some sufficient conditions are given to obtainthe exponential uniform stability of the solutions toward aneighborhood of the origin based on a suitable Lyapunov-Krasovskii functional Two illustrative examples are givento demonstrate the validity of the main result where weestablish a table of comparison with other results
2 Preliminaries
We start by introducing some notations and definitions thatwill be employed throughout the paper
R+ denotes the set of all nonnegative real numbersR119899 denotes the 119899-dimensional Euclidean space 119909denotes the Euclidean vector norm of 119909 isin R119899 119909119879119910denotes the scalar product of two vectors 119909 119910
R119899times119903 denotes the space of all (119899 times 119903)-matrices
119860119879 denotes the transpose of the matrix 119860 119860 is
symmetric if 119860 = 119860119879
119868 denotes the identity matrix
120582(119860) denotes the set of eigenvalues of 119860 120582max(119860) =maxR119890(120582) 120582 isin 120582(119860)
120583(119860(119905)) denotes the matrix measure of the matrix 119860defined by
120583 (119860 (119905)) =
1
2
120582max (119860 (119905) + 119860119879
(119905)) (1)
1198712([minus120591119889 0]R119899) denotes the Hilbert space of all 119871
2-
integrable andR119899-valued functions on [0 119905]
119862([minus120591119889 0]R119899) denotes the Banach space of all R119899-
valued continuous functions mapping [minus120591119889 0] into
10038171003817100381710038171199101003817100381710038171003817+ 1205751198942(119905) forall119905 ge 0 forall119910 isin R
119899
(6)
where 1205751198941gt 0 and 120575
1198942(sdot) are nonnegative continuous bounded
functions for 119894 = 0 1 119898
Definition 1 The system (3) is said to be globally uniformlyexponentially practically stable toward a ball 119861(0 119903) of radius119903which is a neighborhood of the origin if there exist positivenumbers 120572 120574 and 119903 such that every solution 119909(119905 120601) of thesystem satisfies
10038171003817100381710038171206011003817100381710038171003817+ 119903 forall119905 ge 119905
0ge 0 (7)
The following technical proposition is needed for theproof of the main result
Mathematical Problems in Engineering 3
Proposition 2 Let 119876 119878 be symmetric matrices of appropriatedimensions and 119878 gt 0 Then
2119909119879
119876119910 minus 119910119879
119878119910 le 119909119879
119876119878minus1
119876119879
119909 forall (119909 119910) isin R119899
timesR119899
(8)
3 Main Result
Theorem3 Suppose that there exist some positive constants 120572120573 ] 120581
1 12058121119894 12058122119894 12058123119894 12058124119894 12058131119894 12058132119894 and a symmetric bounded
positive semidefinite differentiable matrix function 119875(119905) for all119905 ge 0 satisfying the following Lyapunov differential matrixequation (see [18])
In this case the solution converges to the ball 119861(0 1199032)
Remark that from (45) if we suppose that 1205752(119905) tends
to zero when 119905 goes to infinity then 119903(119905) rarr 0 as 119905 rarr
+infin hence the solution of (7) will converge uniformlyexponentially to zero when 119905 tends to infinity Also note thatwe can estimate 119881(0 119909
Remark 4 If the delayed nonlinear disturbances are allowedto be of large size in the sense that the constants 120575
(sdot)2char-
acterizing the upper-bounding functions are large enoughin (6) then the radius 119903 of the closed ball 119861(0 119903) becomesaccordingly larger according to their values provided inthe statement of Theorem 3 That means that if the systemis globally exponentially practically stable then the radiusof the residual ball 119861(0 119903) increases as the constants 120575
(sdot)2
increase As a result then the uncertainty about how faris the state-trajectory solution from zero becomes larger asthose constants increase Thus to a larger disturbance itcorresponds to a larger uncertainty about the final deviationof the trajectory from the origin
Remark 5 On the other hand if the size of the nonlinearperturbations is allowed to be large in the sense that theconstants 120575
(sdot)1are large enough then there is trade-off
between the values of 1205811and the maximum matrix measure
120583(119860) of 119860 so as to ensure that 120578 gt 0 in Theorem 3 Howevernote that if the constant 120581
1is large then the constant 120598 is
requested to be accordingly large As a result 119860(119905) shouldhave a sufficiently large absolute stability abscissa for all timein order to compensate for the effects of the perturbationswhile satisfying the Lyapunov-like matrix equality (9)
Remark 6 Note also from Theorem 3 that the radius of theresidual ball 119861(0 119903) also increase with the squared upper-bounds of the delays and the squared differences betweenthose upper-bounds and the corresponding delay lower-bounds as well as on certain exponential functions of themaximum delay sizes
4 Examples
Example 1 Consider the nonautonomous system with non-linear time-delay perturbation (3) with time-varying delay1205911(119905) = 120591
1cos2(045119905)
1198911(119905 119909 (119905 minus 120591
1(119905)))
=[
[
minus120575 sin (119905) 1199092(119905 minus 1205911(119905)) + 120575
2lowast cos( 119905
1 + 1199092
1
)
120575 cos (119905) 1199091(119905 minus 1205911(119905))
]
]
(67)where 120591
1 120575 gt 0 120575
2ge 0 will be chosen later and
119860 (119905) = [
119886 (119905) 1
minus1 119886 (119905)
] (68)
where 119886(119905) = minus05 cos(119905)minus1011989010minussin(119905)minus51119890minus sin(119905)minus1119860119894(119905) = 0
and 119902119894= 0 for (119894 = 1 119898) By computation we obtain
120583(119860) asymp minus81033741
Let120573 = 001 120572 = 1 120583 = 09 ] = 11
1205911119867
= 1205911 120591
1119871= 0 997904rArr 120591
1119867119871= 1205911119867
1205751= 12057511= 120575 120575
12= 1205752
1205811= 11989010
120581211
= 0 120581221
= 0 120581231
= 11989010
120581241
= 0 997904rArr 12058121= 11989010
120581311
=
1
1205912
1119867exp (2120572120591
1119867)
120581321
= 0
997904rArr 12058131=
1
exp (21205721205911119867)
(69)
Then120598 = 2 (119901 + 120573) 120575
01+ 2120572120573 + 120581
21
+ 1205813111989021205721205911119867
+ 1205811= 102 + 2119890
10
gt 0
(70)
10 Mathematical Problems in Engineering
Table 1
Delay bound (1205911119867) 2 4 6 8 10
Perturbation bound 1205751
3465679 469028 63476 8591 1163Estimated value of 120574 2098897 2968276 3635376 4197767 4693245
We can verify that a solution 119875(119905) is given by
119875 (119905) =
[
[
[
[
119890sin(119905)
10
0
0
119890sin(119905)
10
]
]
]
]
(71)
We have 119901 = sup119905isinR+119875(119905) = 11989010
1205751lt
radic120581231119890minus21205721205911119867 (1 minus 120583) (120581
1minus 2120573120583 (119860))
(119901 + 120573)
1205811= 11989010
gt 2120573120583 (119860) = minus162067
120578 = 1205811minus 2120573120583 (119860) minus
(119901 + 120573)2
1205752
120581231119890minus21205721205911119867 (1 minus 120583)
gt 0
120578120577= (1 minus 120577
2
) (1205811minus 2120573120583 (119860)) gt 0 where 120575 = 120577120575
0 lt 120598 le minus10120573 + 2120572120573 + 1199021(120572 minus 5)
2
120572 = 1 120573 = 0001 1199021= 00010489
1205811= 002 gt minus10120573 = minus001
120581211
= 0 120581221
= 0
120581241
= 0 997904rArr 12058121= 120581231
= 0001282
120581311
= 016374 120581321
= 0 997904rArr 12058131= 00016374
11990111= 11990122= 000319
11990112= 000125 997904rArr 119901 = 000445
1205751= 002 lt 00206
120578 = 68310minus4
gt 0
120574 = radic
119901 + 120573 + 1205812311205911119867
+ 12058131111988811198671205913
1119867
120573
= 2416
with 1198881119867
=
2120572120591111986711989021205721205911119867
+ 119890minus21205721205911119867
minus 1
(21205721205911119867)2
119903 =
(119901 + 120573)119872
radic2120572120573120578
= 513 with 119872 = 1205752
(77)
in such a way that condition (12) in Theorem 3 is satisfiedThe result of the simulation of this example is depicted
in Figure 3 The evolution of states 1199091and 119909
2is given It is
shown in Figure 1 that the time-delay perturbed system is
0 1 2 3 4 5minus02
0
02
04
06
08
1
12
x1
x2
Solu
tion x
Time t
Figure 3 Convergence of solutions
globally uniformly practically exponentially stable toward aneighborhood of the origin
5 Conclusion
Based on improved Lyapunov-Krasovskii functional for per-turbed systems with time-varying delay we have presentednew sufficient conditions for global uniformly exponentialpractical stability toward a certain ball neighborhood of theorigin The perturbations are assumed to be nonlinear ingeneral with delayed contributions The delayed contribu-tions of such perturbations are not necessarily boundedwhilethey are upper-bounded by known nonnegative integrablefunctions which are linear functions of the various time-delayed state norms The point delays are assumed to beunknown bounded time-differentiable functions of timewithknown lower- and upper-bounds and known upper-boundsof their time-derivatives
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors wish to thank the editor and the anonymousreviewers for their valuable and careful comments The thirdauthor is grateful to the Spanish Ministry of Educationfor its partial support of this work through Grant DPIDPI2012-30651 and is also grateful to the BasqueGovernmentfor its support through Grants IT378-1 and SAIOTEK S-PE13UN039
12 Mathematical Problems in Engineering
References
[1] K Zhou and P P Khargonekar ldquoStability robustness boundsfor linear state-space models with structured uncertaintyrdquo IEEETransactions on Automatic Control vol 32 no 7 pp 621ndash6231987
[2] E Cheres Z J Palmor and S Gutman ldquoQuantitative measuresof robustness for systems including delayed perturbationsrdquoIEEE Transactions on Automatic Control vol 34 no 11 pp1203ndash1204 1989
[3] H Trinh and M Aldeen ldquoOn the stability of linear systemswith delayed perturbationsrdquo IEEE Transactions on AutomaticControl vol 39 no 9 pp 1948ndash1951 1994
[4] H Trinh and M Aldeen ldquoOn robustness and stabilizationof linear systems with delayed nonlinear perturbationsrdquo IEEETransactions on Automatic Control vol 42 no 7 pp 1005ndash10071997
[5] P Niamsup K Mukdasai and V N Phat ldquoImproved exponen-tial stability for time-varying systems with nonlinear delayedperturbationsrdquoAppliedMathematics andComputation vol 204no 1 pp 490ndash495 2008
[6] M Hammami and I Ellouze ldquoA robust detector for a class ofuncertain systemsrdquo Nonlinear Dynamics and Systems Theoryvol 8 no 4 pp 349ndash358 2008
[7] M de la Sen ldquoOn the stability of a certain class of linear time-varying systemsrdquoThe American Journal of Applied Sciences vol2 no 8 pp 1240ndash1245 2005
[8] M de la Sen ldquoRobust stability of a class of linear time-varying systemsrdquo IMA Journal of Mathematical Control andInformation vol 19 no 4 pp 399ndash418 2002
[9] V N Phat ldquoGlobal stabilization for linear continuous time-varying systemsrdquo Applied Mathematics and Computation vol175 no 2 pp 1730ndash1743 2006
[10] A Zevin and M Pinsky ldquoExponential stability and solutionbounds for systems with bounded nonlinearitiesrdquo IEEE Trans-actions on Automatic Control vol 48 no 10 pp 1799ndash18042003
[11] X-L Zhu and G-H Yang ldquoDelay-dependent stability criteriafor systems with differentiable time delaysrdquo Acta AutomaticaSinica vol 34 no 7 pp 765ndash771 2008
[12] M de la Sen and N S Luo ldquoOn the uniform exponentialstability of a wide class of linear time-delay systemsrdquo Journalof Mathematical Analysis and Applications vol 289 no 2 pp456ndash476 2004
[13] H R Karimi M Zapateiro and N Luo ldquoNew delay-dependentstability criteria for uncertain neutral systems with mixed time-varying delays and nonlinear perturbationsrdquo MathematicalProblems in Engineering vol 2009 Article ID 759248 22 pages2009
[14] J-Z Zhang Z Jin Q-X Liu and Z-Y Zhang ldquoAnalysis ofa delayed SIR model with nonlinear incidence raterdquo DiscreteDynamics in Nature and Society vol 2008 Article ID 63615316 pages 2008
[15] C Wei and L Chen ldquoA delayed epidemic model with pulsevaccinationrdquoDiscreteDynamics inNature and Society vol 2008Article ID 746951 12 pages 2008
[16] Y-H Fan and L-L Wang ldquoPermanence for a discrete modelwith feedback control and delayrdquo Discrete Dynamics in Natureand Society vol 2008 Article ID 945109 8 pages 2008
[17] Q Liu ldquoAlmost periodic solution of a diffusive mixed systemwith time delay and type III functional responserdquo Discrete
Dynamics in Nature and Society vol 2008 Article ID 70615413 pages 2008
[18] V N Phat and P T Nam ldquoExponential stability criteria of linearnon-autonomous systems withmultiple delaysrdquo Electronic Jour-nal of Differential Equations vol 2005 no 58 pp 1ndash8 2005
[19] M de la Sen and A Ibeas ldquoOn the global asymptotic stabilityof switched linear time-varying systems with constant pointdelaysrdquo Discrete Dynamics in Nature and Society vol 2008Article ID 231710 31 pages 2008
[20] Y Chen W Bi and Y Wu ldquoDelay-dependent exponentialstability for discrete-time BAM neural networks with time-varying delaysrdquo Discrete Dynamics in Nature and Society vol2008 Article ID 421614 14 pages 2008
[21] J H Park and O Kwon ldquoMatrix inequality approach to anovel stability criterion for time-delay systems with nonlinearuncertaintiesrdquo Journal of OptimizationTheory and Applicationsvol 126 no 3 pp 643ndash656 2005
[22] V N Phat and P Niamsup ldquoStability analysis for a class offunctional differential equations and applicationsrdquo NonlinearAnalysis Theory Methodsamp Applications vol 71 no 12 pp6265ndash6275 2009
[23] A Benabdallah I Ellouze and M A Hammami ldquoPracticalstability of nonlinear time-varying cascade systemsrdquo Journal ofDynamical and Control Systems vol 15 no 1 pp 45ndash62 2009
[24] V Lakshmikantham S Leela and A A Martynyuk PracticalStability of Nonlinear Systems World Scientific 1990
[25] V Lakshmikantham and Y Zhang ldquoStrict practical stability ofdelay differential equationrdquo Applied Mathematics and Compu-tation vol 118 no 2-3 pp 275ndash285 2001
[26] M Corless ldquoGuaranteed rates of exponential convergencefor uncertain systemsrdquo Journal of Optimization Theory andApplications vol 64 no 3 pp 481ndash494 1990
[27] M Corless and G Leitmann ldquoBounded controllers for robustexponential convergencerdquo Journal of Optimization Theory andApplications vol 76 no 1 pp 1ndash12 1993
a global-null controllability is required while the asymptoticstability is not guaranteed to be of exponential type In thesame way the asymptotic stability is not guaranteed to beexponential in [11] Another novelty is that the delays aretime-varying time-differentiable and they are not requiredto be known Only lower and upper-bound of the delayfunctions and their time-derivatives are required for stabilityanalysis We will study a class of nonlinear system such thatthe nonlinearity is bounded by some integrable functionswhich are bounded where the origin is not necessarily anequilibrium point We deal with the practical stability of theorigin (see [23]) The asymptotic stability is more importantthan stability also the desired system may be unstable andyet the system may oscillate sufficiently near this state thatits performance is acceptable thus the notion of practicalstability is more suitable in several situations than Lyapunovstability In this case all state trajectories are bounded andapproach a sufficiently small neighborhood of the origin Onealso desires that the state approaches the origin (or some suffi-ciently small neighborhood of it) in a sufficiently fast mannerThis notion of practical stability was introduced by [24] fornonlinear time-varying systems and studied for differentialequations with delays by [25] (see also the references therein)Moreover the authors in [26 27] constructed stabilizingcontrollers to obtain global convergence of solutions towarda small ball for some classes of uncertain control systemsIn this paper some sufficient conditions are given to obtainthe exponential uniform stability of the solutions toward aneighborhood of the origin based on a suitable Lyapunov-Krasovskii functional Two illustrative examples are givento demonstrate the validity of the main result where weestablish a table of comparison with other results
2 Preliminaries
We start by introducing some notations and definitions thatwill be employed throughout the paper
R+ denotes the set of all nonnegative real numbersR119899 denotes the 119899-dimensional Euclidean space 119909denotes the Euclidean vector norm of 119909 isin R119899 119909119879119910denotes the scalar product of two vectors 119909 119910
R119899times119903 denotes the space of all (119899 times 119903)-matrices
119860119879 denotes the transpose of the matrix 119860 119860 is
symmetric if 119860 = 119860119879
119868 denotes the identity matrix
120582(119860) denotes the set of eigenvalues of 119860 120582max(119860) =maxR119890(120582) 120582 isin 120582(119860)
120583(119860(119905)) denotes the matrix measure of the matrix 119860defined by
120583 (119860 (119905)) =
1
2
120582max (119860 (119905) + 119860119879
(119905)) (1)
1198712([minus120591119889 0]R119899) denotes the Hilbert space of all 119871
2-
integrable andR119899-valued functions on [0 119905]
119862([minus120591119889 0]R119899) denotes the Banach space of all R119899-
valued continuous functions mapping [minus120591119889 0] into
10038171003817100381710038171199101003817100381710038171003817+ 1205751198942(119905) forall119905 ge 0 forall119910 isin R
119899
(6)
where 1205751198941gt 0 and 120575
1198942(sdot) are nonnegative continuous bounded
functions for 119894 = 0 1 119898
Definition 1 The system (3) is said to be globally uniformlyexponentially practically stable toward a ball 119861(0 119903) of radius119903which is a neighborhood of the origin if there exist positivenumbers 120572 120574 and 119903 such that every solution 119909(119905 120601) of thesystem satisfies
10038171003817100381710038171206011003817100381710038171003817+ 119903 forall119905 ge 119905
0ge 0 (7)
The following technical proposition is needed for theproof of the main result
Mathematical Problems in Engineering 3
Proposition 2 Let 119876 119878 be symmetric matrices of appropriatedimensions and 119878 gt 0 Then
2119909119879
119876119910 minus 119910119879
119878119910 le 119909119879
119876119878minus1
119876119879
119909 forall (119909 119910) isin R119899
timesR119899
(8)
3 Main Result
Theorem3 Suppose that there exist some positive constants 120572120573 ] 120581
1 12058121119894 12058122119894 12058123119894 12058124119894 12058131119894 12058132119894 and a symmetric bounded
positive semidefinite differentiable matrix function 119875(119905) for all119905 ge 0 satisfying the following Lyapunov differential matrixequation (see [18])
In this case the solution converges to the ball 119861(0 1199032)
Remark that from (45) if we suppose that 1205752(119905) tends
to zero when 119905 goes to infinity then 119903(119905) rarr 0 as 119905 rarr
+infin hence the solution of (7) will converge uniformlyexponentially to zero when 119905 tends to infinity Also note thatwe can estimate 119881(0 119909
Remark 4 If the delayed nonlinear disturbances are allowedto be of large size in the sense that the constants 120575
(sdot)2char-
acterizing the upper-bounding functions are large enoughin (6) then the radius 119903 of the closed ball 119861(0 119903) becomesaccordingly larger according to their values provided inthe statement of Theorem 3 That means that if the systemis globally exponentially practically stable then the radiusof the residual ball 119861(0 119903) increases as the constants 120575
(sdot)2
increase As a result then the uncertainty about how faris the state-trajectory solution from zero becomes larger asthose constants increase Thus to a larger disturbance itcorresponds to a larger uncertainty about the final deviationof the trajectory from the origin
Remark 5 On the other hand if the size of the nonlinearperturbations is allowed to be large in the sense that theconstants 120575
(sdot)1are large enough then there is trade-off
between the values of 1205811and the maximum matrix measure
120583(119860) of 119860 so as to ensure that 120578 gt 0 in Theorem 3 Howevernote that if the constant 120581
1is large then the constant 120598 is
requested to be accordingly large As a result 119860(119905) shouldhave a sufficiently large absolute stability abscissa for all timein order to compensate for the effects of the perturbationswhile satisfying the Lyapunov-like matrix equality (9)
Remark 6 Note also from Theorem 3 that the radius of theresidual ball 119861(0 119903) also increase with the squared upper-bounds of the delays and the squared differences betweenthose upper-bounds and the corresponding delay lower-bounds as well as on certain exponential functions of themaximum delay sizes
4 Examples
Example 1 Consider the nonautonomous system with non-linear time-delay perturbation (3) with time-varying delay1205911(119905) = 120591
1cos2(045119905)
1198911(119905 119909 (119905 minus 120591
1(119905)))
=[
[
minus120575 sin (119905) 1199092(119905 minus 1205911(119905)) + 120575
2lowast cos( 119905
1 + 1199092
1
)
120575 cos (119905) 1199091(119905 minus 1205911(119905))
]
]
(67)where 120591
1 120575 gt 0 120575
2ge 0 will be chosen later and
119860 (119905) = [
119886 (119905) 1
minus1 119886 (119905)
] (68)
where 119886(119905) = minus05 cos(119905)minus1011989010minussin(119905)minus51119890minus sin(119905)minus1119860119894(119905) = 0
and 119902119894= 0 for (119894 = 1 119898) By computation we obtain
120583(119860) asymp minus81033741
Let120573 = 001 120572 = 1 120583 = 09 ] = 11
1205911119867
= 1205911 120591
1119871= 0 997904rArr 120591
1119867119871= 1205911119867
1205751= 12057511= 120575 120575
12= 1205752
1205811= 11989010
120581211
= 0 120581221
= 0 120581231
= 11989010
120581241
= 0 997904rArr 12058121= 11989010
120581311
=
1
1205912
1119867exp (2120572120591
1119867)
120581321
= 0
997904rArr 12058131=
1
exp (21205721205911119867)
(69)
Then120598 = 2 (119901 + 120573) 120575
01+ 2120572120573 + 120581
21
+ 1205813111989021205721205911119867
+ 1205811= 102 + 2119890
10
gt 0
(70)
10 Mathematical Problems in Engineering
Table 1
Delay bound (1205911119867) 2 4 6 8 10
Perturbation bound 1205751
3465679 469028 63476 8591 1163Estimated value of 120574 2098897 2968276 3635376 4197767 4693245
We can verify that a solution 119875(119905) is given by
119875 (119905) =
[
[
[
[
119890sin(119905)
10
0
0
119890sin(119905)
10
]
]
]
]
(71)
We have 119901 = sup119905isinR+119875(119905) = 11989010
1205751lt
radic120581231119890minus21205721205911119867 (1 minus 120583) (120581
1minus 2120573120583 (119860))
(119901 + 120573)
1205811= 11989010
gt 2120573120583 (119860) = minus162067
120578 = 1205811minus 2120573120583 (119860) minus
(119901 + 120573)2
1205752
120581231119890minus21205721205911119867 (1 minus 120583)
gt 0
120578120577= (1 minus 120577
2
) (1205811minus 2120573120583 (119860)) gt 0 where 120575 = 120577120575
0 lt 120598 le minus10120573 + 2120572120573 + 1199021(120572 minus 5)
2
120572 = 1 120573 = 0001 1199021= 00010489
1205811= 002 gt minus10120573 = minus001
120581211
= 0 120581221
= 0
120581241
= 0 997904rArr 12058121= 120581231
= 0001282
120581311
= 016374 120581321
= 0 997904rArr 12058131= 00016374
11990111= 11990122= 000319
11990112= 000125 997904rArr 119901 = 000445
1205751= 002 lt 00206
120578 = 68310minus4
gt 0
120574 = radic
119901 + 120573 + 1205812311205911119867
+ 12058131111988811198671205913
1119867
120573
= 2416
with 1198881119867
=
2120572120591111986711989021205721205911119867
+ 119890minus21205721205911119867
minus 1
(21205721205911119867)2
119903 =
(119901 + 120573)119872
radic2120572120573120578
= 513 with 119872 = 1205752
(77)
in such a way that condition (12) in Theorem 3 is satisfiedThe result of the simulation of this example is depicted
in Figure 3 The evolution of states 1199091and 119909
2is given It is
shown in Figure 1 that the time-delay perturbed system is
0 1 2 3 4 5minus02
0
02
04
06
08
1
12
x1
x2
Solu
tion x
Time t
Figure 3 Convergence of solutions
globally uniformly practically exponentially stable toward aneighborhood of the origin
5 Conclusion
Based on improved Lyapunov-Krasovskii functional for per-turbed systems with time-varying delay we have presentednew sufficient conditions for global uniformly exponentialpractical stability toward a certain ball neighborhood of theorigin The perturbations are assumed to be nonlinear ingeneral with delayed contributions The delayed contribu-tions of such perturbations are not necessarily boundedwhilethey are upper-bounded by known nonnegative integrablefunctions which are linear functions of the various time-delayed state norms The point delays are assumed to beunknown bounded time-differentiable functions of timewithknown lower- and upper-bounds and known upper-boundsof their time-derivatives
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors wish to thank the editor and the anonymousreviewers for their valuable and careful comments The thirdauthor is grateful to the Spanish Ministry of Educationfor its partial support of this work through Grant DPIDPI2012-30651 and is also grateful to the BasqueGovernmentfor its support through Grants IT378-1 and SAIOTEK S-PE13UN039
12 Mathematical Problems in Engineering
References
[1] K Zhou and P P Khargonekar ldquoStability robustness boundsfor linear state-space models with structured uncertaintyrdquo IEEETransactions on Automatic Control vol 32 no 7 pp 621ndash6231987
[2] E Cheres Z J Palmor and S Gutman ldquoQuantitative measuresof robustness for systems including delayed perturbationsrdquoIEEE Transactions on Automatic Control vol 34 no 11 pp1203ndash1204 1989
[3] H Trinh and M Aldeen ldquoOn the stability of linear systemswith delayed perturbationsrdquo IEEE Transactions on AutomaticControl vol 39 no 9 pp 1948ndash1951 1994
[4] H Trinh and M Aldeen ldquoOn robustness and stabilizationof linear systems with delayed nonlinear perturbationsrdquo IEEETransactions on Automatic Control vol 42 no 7 pp 1005ndash10071997
[5] P Niamsup K Mukdasai and V N Phat ldquoImproved exponen-tial stability for time-varying systems with nonlinear delayedperturbationsrdquoAppliedMathematics andComputation vol 204no 1 pp 490ndash495 2008
[6] M Hammami and I Ellouze ldquoA robust detector for a class ofuncertain systemsrdquo Nonlinear Dynamics and Systems Theoryvol 8 no 4 pp 349ndash358 2008
[7] M de la Sen ldquoOn the stability of a certain class of linear time-varying systemsrdquoThe American Journal of Applied Sciences vol2 no 8 pp 1240ndash1245 2005
[8] M de la Sen ldquoRobust stability of a class of linear time-varying systemsrdquo IMA Journal of Mathematical Control andInformation vol 19 no 4 pp 399ndash418 2002
[9] V N Phat ldquoGlobal stabilization for linear continuous time-varying systemsrdquo Applied Mathematics and Computation vol175 no 2 pp 1730ndash1743 2006
[10] A Zevin and M Pinsky ldquoExponential stability and solutionbounds for systems with bounded nonlinearitiesrdquo IEEE Trans-actions on Automatic Control vol 48 no 10 pp 1799ndash18042003
[11] X-L Zhu and G-H Yang ldquoDelay-dependent stability criteriafor systems with differentiable time delaysrdquo Acta AutomaticaSinica vol 34 no 7 pp 765ndash771 2008
[12] M de la Sen and N S Luo ldquoOn the uniform exponentialstability of a wide class of linear time-delay systemsrdquo Journalof Mathematical Analysis and Applications vol 289 no 2 pp456ndash476 2004
[13] H R Karimi M Zapateiro and N Luo ldquoNew delay-dependentstability criteria for uncertain neutral systems with mixed time-varying delays and nonlinear perturbationsrdquo MathematicalProblems in Engineering vol 2009 Article ID 759248 22 pages2009
[14] J-Z Zhang Z Jin Q-X Liu and Z-Y Zhang ldquoAnalysis ofa delayed SIR model with nonlinear incidence raterdquo DiscreteDynamics in Nature and Society vol 2008 Article ID 63615316 pages 2008
[15] C Wei and L Chen ldquoA delayed epidemic model with pulsevaccinationrdquoDiscreteDynamics inNature and Society vol 2008Article ID 746951 12 pages 2008
[16] Y-H Fan and L-L Wang ldquoPermanence for a discrete modelwith feedback control and delayrdquo Discrete Dynamics in Natureand Society vol 2008 Article ID 945109 8 pages 2008
[17] Q Liu ldquoAlmost periodic solution of a diffusive mixed systemwith time delay and type III functional responserdquo Discrete
Dynamics in Nature and Society vol 2008 Article ID 70615413 pages 2008
[18] V N Phat and P T Nam ldquoExponential stability criteria of linearnon-autonomous systems withmultiple delaysrdquo Electronic Jour-nal of Differential Equations vol 2005 no 58 pp 1ndash8 2005
[19] M de la Sen and A Ibeas ldquoOn the global asymptotic stabilityof switched linear time-varying systems with constant pointdelaysrdquo Discrete Dynamics in Nature and Society vol 2008Article ID 231710 31 pages 2008
[20] Y Chen W Bi and Y Wu ldquoDelay-dependent exponentialstability for discrete-time BAM neural networks with time-varying delaysrdquo Discrete Dynamics in Nature and Society vol2008 Article ID 421614 14 pages 2008
[21] J H Park and O Kwon ldquoMatrix inequality approach to anovel stability criterion for time-delay systems with nonlinearuncertaintiesrdquo Journal of OptimizationTheory and Applicationsvol 126 no 3 pp 643ndash656 2005
[22] V N Phat and P Niamsup ldquoStability analysis for a class offunctional differential equations and applicationsrdquo NonlinearAnalysis Theory Methodsamp Applications vol 71 no 12 pp6265ndash6275 2009
[23] A Benabdallah I Ellouze and M A Hammami ldquoPracticalstability of nonlinear time-varying cascade systemsrdquo Journal ofDynamical and Control Systems vol 15 no 1 pp 45ndash62 2009
[24] V Lakshmikantham S Leela and A A Martynyuk PracticalStability of Nonlinear Systems World Scientific 1990
[25] V Lakshmikantham and Y Zhang ldquoStrict practical stability ofdelay differential equationrdquo Applied Mathematics and Compu-tation vol 118 no 2-3 pp 275ndash285 2001
[26] M Corless ldquoGuaranteed rates of exponential convergencefor uncertain systemsrdquo Journal of Optimization Theory andApplications vol 64 no 3 pp 481ndash494 1990
[27] M Corless and G Leitmann ldquoBounded controllers for robustexponential convergencerdquo Journal of Optimization Theory andApplications vol 76 no 1 pp 1ndash12 1993
Proposition 2 Let 119876 119878 be symmetric matrices of appropriatedimensions and 119878 gt 0 Then
2119909119879
119876119910 minus 119910119879
119878119910 le 119909119879
119876119878minus1
119876119879
119909 forall (119909 119910) isin R119899
timesR119899
(8)
3 Main Result
Theorem3 Suppose that there exist some positive constants 120572120573 ] 120581
1 12058121119894 12058122119894 12058123119894 12058124119894 12058131119894 12058132119894 and a symmetric bounded
positive semidefinite differentiable matrix function 119875(119905) for all119905 ge 0 satisfying the following Lyapunov differential matrixequation (see [18])
In this case the solution converges to the ball 119861(0 1199032)
Remark that from (45) if we suppose that 1205752(119905) tends
to zero when 119905 goes to infinity then 119903(119905) rarr 0 as 119905 rarr
+infin hence the solution of (7) will converge uniformlyexponentially to zero when 119905 tends to infinity Also note thatwe can estimate 119881(0 119909
Remark 4 If the delayed nonlinear disturbances are allowedto be of large size in the sense that the constants 120575
(sdot)2char-
acterizing the upper-bounding functions are large enoughin (6) then the radius 119903 of the closed ball 119861(0 119903) becomesaccordingly larger according to their values provided inthe statement of Theorem 3 That means that if the systemis globally exponentially practically stable then the radiusof the residual ball 119861(0 119903) increases as the constants 120575
(sdot)2
increase As a result then the uncertainty about how faris the state-trajectory solution from zero becomes larger asthose constants increase Thus to a larger disturbance itcorresponds to a larger uncertainty about the final deviationof the trajectory from the origin
Remark 5 On the other hand if the size of the nonlinearperturbations is allowed to be large in the sense that theconstants 120575
(sdot)1are large enough then there is trade-off
between the values of 1205811and the maximum matrix measure
120583(119860) of 119860 so as to ensure that 120578 gt 0 in Theorem 3 Howevernote that if the constant 120581
1is large then the constant 120598 is
requested to be accordingly large As a result 119860(119905) shouldhave a sufficiently large absolute stability abscissa for all timein order to compensate for the effects of the perturbationswhile satisfying the Lyapunov-like matrix equality (9)
Remark 6 Note also from Theorem 3 that the radius of theresidual ball 119861(0 119903) also increase with the squared upper-bounds of the delays and the squared differences betweenthose upper-bounds and the corresponding delay lower-bounds as well as on certain exponential functions of themaximum delay sizes
4 Examples
Example 1 Consider the nonautonomous system with non-linear time-delay perturbation (3) with time-varying delay1205911(119905) = 120591
1cos2(045119905)
1198911(119905 119909 (119905 minus 120591
1(119905)))
=[
[
minus120575 sin (119905) 1199092(119905 minus 1205911(119905)) + 120575
2lowast cos( 119905
1 + 1199092
1
)
120575 cos (119905) 1199091(119905 minus 1205911(119905))
]
]
(67)where 120591
1 120575 gt 0 120575
2ge 0 will be chosen later and
119860 (119905) = [
119886 (119905) 1
minus1 119886 (119905)
] (68)
where 119886(119905) = minus05 cos(119905)minus1011989010minussin(119905)minus51119890minus sin(119905)minus1119860119894(119905) = 0
and 119902119894= 0 for (119894 = 1 119898) By computation we obtain
120583(119860) asymp minus81033741
Let120573 = 001 120572 = 1 120583 = 09 ] = 11
1205911119867
= 1205911 120591
1119871= 0 997904rArr 120591
1119867119871= 1205911119867
1205751= 12057511= 120575 120575
12= 1205752
1205811= 11989010
120581211
= 0 120581221
= 0 120581231
= 11989010
120581241
= 0 997904rArr 12058121= 11989010
120581311
=
1
1205912
1119867exp (2120572120591
1119867)
120581321
= 0
997904rArr 12058131=
1
exp (21205721205911119867)
(69)
Then120598 = 2 (119901 + 120573) 120575
01+ 2120572120573 + 120581
21
+ 1205813111989021205721205911119867
+ 1205811= 102 + 2119890
10
gt 0
(70)
10 Mathematical Problems in Engineering
Table 1
Delay bound (1205911119867) 2 4 6 8 10
Perturbation bound 1205751
3465679 469028 63476 8591 1163Estimated value of 120574 2098897 2968276 3635376 4197767 4693245
We can verify that a solution 119875(119905) is given by
119875 (119905) =
[
[
[
[
119890sin(119905)
10
0
0
119890sin(119905)
10
]
]
]
]
(71)
We have 119901 = sup119905isinR+119875(119905) = 11989010
1205751lt
radic120581231119890minus21205721205911119867 (1 minus 120583) (120581
1minus 2120573120583 (119860))
(119901 + 120573)
1205811= 11989010
gt 2120573120583 (119860) = minus162067
120578 = 1205811minus 2120573120583 (119860) minus
(119901 + 120573)2
1205752
120581231119890minus21205721205911119867 (1 minus 120583)
gt 0
120578120577= (1 minus 120577
2
) (1205811minus 2120573120583 (119860)) gt 0 where 120575 = 120577120575
0 lt 120598 le minus10120573 + 2120572120573 + 1199021(120572 minus 5)
2
120572 = 1 120573 = 0001 1199021= 00010489
1205811= 002 gt minus10120573 = minus001
120581211
= 0 120581221
= 0
120581241
= 0 997904rArr 12058121= 120581231
= 0001282
120581311
= 016374 120581321
= 0 997904rArr 12058131= 00016374
11990111= 11990122= 000319
11990112= 000125 997904rArr 119901 = 000445
1205751= 002 lt 00206
120578 = 68310minus4
gt 0
120574 = radic
119901 + 120573 + 1205812311205911119867
+ 12058131111988811198671205913
1119867
120573
= 2416
with 1198881119867
=
2120572120591111986711989021205721205911119867
+ 119890minus21205721205911119867
minus 1
(21205721205911119867)2
119903 =
(119901 + 120573)119872
radic2120572120573120578
= 513 with 119872 = 1205752
(77)
in such a way that condition (12) in Theorem 3 is satisfiedThe result of the simulation of this example is depicted
in Figure 3 The evolution of states 1199091and 119909
2is given It is
shown in Figure 1 that the time-delay perturbed system is
0 1 2 3 4 5minus02
0
02
04
06
08
1
12
x1
x2
Solu
tion x
Time t
Figure 3 Convergence of solutions
globally uniformly practically exponentially stable toward aneighborhood of the origin
5 Conclusion
Based on improved Lyapunov-Krasovskii functional for per-turbed systems with time-varying delay we have presentednew sufficient conditions for global uniformly exponentialpractical stability toward a certain ball neighborhood of theorigin The perturbations are assumed to be nonlinear ingeneral with delayed contributions The delayed contribu-tions of such perturbations are not necessarily boundedwhilethey are upper-bounded by known nonnegative integrablefunctions which are linear functions of the various time-delayed state norms The point delays are assumed to beunknown bounded time-differentiable functions of timewithknown lower- and upper-bounds and known upper-boundsof their time-derivatives
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors wish to thank the editor and the anonymousreviewers for their valuable and careful comments The thirdauthor is grateful to the Spanish Ministry of Educationfor its partial support of this work through Grant DPIDPI2012-30651 and is also grateful to the BasqueGovernmentfor its support through Grants IT378-1 and SAIOTEK S-PE13UN039
12 Mathematical Problems in Engineering
References
[1] K Zhou and P P Khargonekar ldquoStability robustness boundsfor linear state-space models with structured uncertaintyrdquo IEEETransactions on Automatic Control vol 32 no 7 pp 621ndash6231987
[2] E Cheres Z J Palmor and S Gutman ldquoQuantitative measuresof robustness for systems including delayed perturbationsrdquoIEEE Transactions on Automatic Control vol 34 no 11 pp1203ndash1204 1989
[3] H Trinh and M Aldeen ldquoOn the stability of linear systemswith delayed perturbationsrdquo IEEE Transactions on AutomaticControl vol 39 no 9 pp 1948ndash1951 1994
[4] H Trinh and M Aldeen ldquoOn robustness and stabilizationof linear systems with delayed nonlinear perturbationsrdquo IEEETransactions on Automatic Control vol 42 no 7 pp 1005ndash10071997
[5] P Niamsup K Mukdasai and V N Phat ldquoImproved exponen-tial stability for time-varying systems with nonlinear delayedperturbationsrdquoAppliedMathematics andComputation vol 204no 1 pp 490ndash495 2008
[6] M Hammami and I Ellouze ldquoA robust detector for a class ofuncertain systemsrdquo Nonlinear Dynamics and Systems Theoryvol 8 no 4 pp 349ndash358 2008
[7] M de la Sen ldquoOn the stability of a certain class of linear time-varying systemsrdquoThe American Journal of Applied Sciences vol2 no 8 pp 1240ndash1245 2005
[8] M de la Sen ldquoRobust stability of a class of linear time-varying systemsrdquo IMA Journal of Mathematical Control andInformation vol 19 no 4 pp 399ndash418 2002
[9] V N Phat ldquoGlobal stabilization for linear continuous time-varying systemsrdquo Applied Mathematics and Computation vol175 no 2 pp 1730ndash1743 2006
[10] A Zevin and M Pinsky ldquoExponential stability and solutionbounds for systems with bounded nonlinearitiesrdquo IEEE Trans-actions on Automatic Control vol 48 no 10 pp 1799ndash18042003
[11] X-L Zhu and G-H Yang ldquoDelay-dependent stability criteriafor systems with differentiable time delaysrdquo Acta AutomaticaSinica vol 34 no 7 pp 765ndash771 2008
[12] M de la Sen and N S Luo ldquoOn the uniform exponentialstability of a wide class of linear time-delay systemsrdquo Journalof Mathematical Analysis and Applications vol 289 no 2 pp456ndash476 2004
[13] H R Karimi M Zapateiro and N Luo ldquoNew delay-dependentstability criteria for uncertain neutral systems with mixed time-varying delays and nonlinear perturbationsrdquo MathematicalProblems in Engineering vol 2009 Article ID 759248 22 pages2009
[14] J-Z Zhang Z Jin Q-X Liu and Z-Y Zhang ldquoAnalysis ofa delayed SIR model with nonlinear incidence raterdquo DiscreteDynamics in Nature and Society vol 2008 Article ID 63615316 pages 2008
[15] C Wei and L Chen ldquoA delayed epidemic model with pulsevaccinationrdquoDiscreteDynamics inNature and Society vol 2008Article ID 746951 12 pages 2008
[16] Y-H Fan and L-L Wang ldquoPermanence for a discrete modelwith feedback control and delayrdquo Discrete Dynamics in Natureand Society vol 2008 Article ID 945109 8 pages 2008
[17] Q Liu ldquoAlmost periodic solution of a diffusive mixed systemwith time delay and type III functional responserdquo Discrete
Dynamics in Nature and Society vol 2008 Article ID 70615413 pages 2008
[18] V N Phat and P T Nam ldquoExponential stability criteria of linearnon-autonomous systems withmultiple delaysrdquo Electronic Jour-nal of Differential Equations vol 2005 no 58 pp 1ndash8 2005
[19] M de la Sen and A Ibeas ldquoOn the global asymptotic stabilityof switched linear time-varying systems with constant pointdelaysrdquo Discrete Dynamics in Nature and Society vol 2008Article ID 231710 31 pages 2008
[20] Y Chen W Bi and Y Wu ldquoDelay-dependent exponentialstability for discrete-time BAM neural networks with time-varying delaysrdquo Discrete Dynamics in Nature and Society vol2008 Article ID 421614 14 pages 2008
[21] J H Park and O Kwon ldquoMatrix inequality approach to anovel stability criterion for time-delay systems with nonlinearuncertaintiesrdquo Journal of OptimizationTheory and Applicationsvol 126 no 3 pp 643ndash656 2005
[22] V N Phat and P Niamsup ldquoStability analysis for a class offunctional differential equations and applicationsrdquo NonlinearAnalysis Theory Methodsamp Applications vol 71 no 12 pp6265ndash6275 2009
[23] A Benabdallah I Ellouze and M A Hammami ldquoPracticalstability of nonlinear time-varying cascade systemsrdquo Journal ofDynamical and Control Systems vol 15 no 1 pp 45ndash62 2009
[24] V Lakshmikantham S Leela and A A Martynyuk PracticalStability of Nonlinear Systems World Scientific 1990
[25] V Lakshmikantham and Y Zhang ldquoStrict practical stability ofdelay differential equationrdquo Applied Mathematics and Compu-tation vol 118 no 2-3 pp 275ndash285 2001
[26] M Corless ldquoGuaranteed rates of exponential convergencefor uncertain systemsrdquo Journal of Optimization Theory andApplications vol 64 no 3 pp 481ndash494 1990
[27] M Corless and G Leitmann ldquoBounded controllers for robustexponential convergencerdquo Journal of Optimization Theory andApplications vol 76 no 1 pp 1ndash12 1993
In this case the solution converges to the ball 119861(0 1199032)
Remark that from (45) if we suppose that 1205752(119905) tends
to zero when 119905 goes to infinity then 119903(119905) rarr 0 as 119905 rarr
+infin hence the solution of (7) will converge uniformlyexponentially to zero when 119905 tends to infinity Also note thatwe can estimate 119881(0 119909
Remark 4 If the delayed nonlinear disturbances are allowedto be of large size in the sense that the constants 120575
(sdot)2char-
acterizing the upper-bounding functions are large enoughin (6) then the radius 119903 of the closed ball 119861(0 119903) becomesaccordingly larger according to their values provided inthe statement of Theorem 3 That means that if the systemis globally exponentially practically stable then the radiusof the residual ball 119861(0 119903) increases as the constants 120575
(sdot)2
increase As a result then the uncertainty about how faris the state-trajectory solution from zero becomes larger asthose constants increase Thus to a larger disturbance itcorresponds to a larger uncertainty about the final deviationof the trajectory from the origin
Remark 5 On the other hand if the size of the nonlinearperturbations is allowed to be large in the sense that theconstants 120575
(sdot)1are large enough then there is trade-off
between the values of 1205811and the maximum matrix measure
120583(119860) of 119860 so as to ensure that 120578 gt 0 in Theorem 3 Howevernote that if the constant 120581
1is large then the constant 120598 is
requested to be accordingly large As a result 119860(119905) shouldhave a sufficiently large absolute stability abscissa for all timein order to compensate for the effects of the perturbationswhile satisfying the Lyapunov-like matrix equality (9)
Remark 6 Note also from Theorem 3 that the radius of theresidual ball 119861(0 119903) also increase with the squared upper-bounds of the delays and the squared differences betweenthose upper-bounds and the corresponding delay lower-bounds as well as on certain exponential functions of themaximum delay sizes
4 Examples
Example 1 Consider the nonautonomous system with non-linear time-delay perturbation (3) with time-varying delay1205911(119905) = 120591
1cos2(045119905)
1198911(119905 119909 (119905 minus 120591
1(119905)))
=[
[
minus120575 sin (119905) 1199092(119905 minus 1205911(119905)) + 120575
2lowast cos( 119905
1 + 1199092
1
)
120575 cos (119905) 1199091(119905 minus 1205911(119905))
]
]
(67)where 120591
1 120575 gt 0 120575
2ge 0 will be chosen later and
119860 (119905) = [
119886 (119905) 1
minus1 119886 (119905)
] (68)
where 119886(119905) = minus05 cos(119905)minus1011989010minussin(119905)minus51119890minus sin(119905)minus1119860119894(119905) = 0
and 119902119894= 0 for (119894 = 1 119898) By computation we obtain
120583(119860) asymp minus81033741
Let120573 = 001 120572 = 1 120583 = 09 ] = 11
1205911119867
= 1205911 120591
1119871= 0 997904rArr 120591
1119867119871= 1205911119867
1205751= 12057511= 120575 120575
12= 1205752
1205811= 11989010
120581211
= 0 120581221
= 0 120581231
= 11989010
120581241
= 0 997904rArr 12058121= 11989010
120581311
=
1
1205912
1119867exp (2120572120591
1119867)
120581321
= 0
997904rArr 12058131=
1
exp (21205721205911119867)
(69)
Then120598 = 2 (119901 + 120573) 120575
01+ 2120572120573 + 120581
21
+ 1205813111989021205721205911119867
+ 1205811= 102 + 2119890
10
gt 0
(70)
10 Mathematical Problems in Engineering
Table 1
Delay bound (1205911119867) 2 4 6 8 10
Perturbation bound 1205751
3465679 469028 63476 8591 1163Estimated value of 120574 2098897 2968276 3635376 4197767 4693245
We can verify that a solution 119875(119905) is given by
119875 (119905) =
[
[
[
[
119890sin(119905)
10
0
0
119890sin(119905)
10
]
]
]
]
(71)
We have 119901 = sup119905isinR+119875(119905) = 11989010
1205751lt
radic120581231119890minus21205721205911119867 (1 minus 120583) (120581
1minus 2120573120583 (119860))
(119901 + 120573)
1205811= 11989010
gt 2120573120583 (119860) = minus162067
120578 = 1205811minus 2120573120583 (119860) minus
(119901 + 120573)2
1205752
120581231119890minus21205721205911119867 (1 minus 120583)
gt 0
120578120577= (1 minus 120577
2
) (1205811minus 2120573120583 (119860)) gt 0 where 120575 = 120577120575
0 lt 120598 le minus10120573 + 2120572120573 + 1199021(120572 minus 5)
2
120572 = 1 120573 = 0001 1199021= 00010489
1205811= 002 gt minus10120573 = minus001
120581211
= 0 120581221
= 0
120581241
= 0 997904rArr 12058121= 120581231
= 0001282
120581311
= 016374 120581321
= 0 997904rArr 12058131= 00016374
11990111= 11990122= 000319
11990112= 000125 997904rArr 119901 = 000445
1205751= 002 lt 00206
120578 = 68310minus4
gt 0
120574 = radic
119901 + 120573 + 1205812311205911119867
+ 12058131111988811198671205913
1119867
120573
= 2416
with 1198881119867
=
2120572120591111986711989021205721205911119867
+ 119890minus21205721205911119867
minus 1
(21205721205911119867)2
119903 =
(119901 + 120573)119872
radic2120572120573120578
= 513 with 119872 = 1205752
(77)
in such a way that condition (12) in Theorem 3 is satisfiedThe result of the simulation of this example is depicted
in Figure 3 The evolution of states 1199091and 119909
2is given It is
shown in Figure 1 that the time-delay perturbed system is
0 1 2 3 4 5minus02
0
02
04
06
08
1
12
x1
x2
Solu
tion x
Time t
Figure 3 Convergence of solutions
globally uniformly practically exponentially stable toward aneighborhood of the origin
5 Conclusion
Based on improved Lyapunov-Krasovskii functional for per-turbed systems with time-varying delay we have presentednew sufficient conditions for global uniformly exponentialpractical stability toward a certain ball neighborhood of theorigin The perturbations are assumed to be nonlinear ingeneral with delayed contributions The delayed contribu-tions of such perturbations are not necessarily boundedwhilethey are upper-bounded by known nonnegative integrablefunctions which are linear functions of the various time-delayed state norms The point delays are assumed to beunknown bounded time-differentiable functions of timewithknown lower- and upper-bounds and known upper-boundsof their time-derivatives
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors wish to thank the editor and the anonymousreviewers for their valuable and careful comments The thirdauthor is grateful to the Spanish Ministry of Educationfor its partial support of this work through Grant DPIDPI2012-30651 and is also grateful to the BasqueGovernmentfor its support through Grants IT378-1 and SAIOTEK S-PE13UN039
12 Mathematical Problems in Engineering
References
[1] K Zhou and P P Khargonekar ldquoStability robustness boundsfor linear state-space models with structured uncertaintyrdquo IEEETransactions on Automatic Control vol 32 no 7 pp 621ndash6231987
[2] E Cheres Z J Palmor and S Gutman ldquoQuantitative measuresof robustness for systems including delayed perturbationsrdquoIEEE Transactions on Automatic Control vol 34 no 11 pp1203ndash1204 1989
[3] H Trinh and M Aldeen ldquoOn the stability of linear systemswith delayed perturbationsrdquo IEEE Transactions on AutomaticControl vol 39 no 9 pp 1948ndash1951 1994
[4] H Trinh and M Aldeen ldquoOn robustness and stabilizationof linear systems with delayed nonlinear perturbationsrdquo IEEETransactions on Automatic Control vol 42 no 7 pp 1005ndash10071997
[5] P Niamsup K Mukdasai and V N Phat ldquoImproved exponen-tial stability for time-varying systems with nonlinear delayedperturbationsrdquoAppliedMathematics andComputation vol 204no 1 pp 490ndash495 2008
[6] M Hammami and I Ellouze ldquoA robust detector for a class ofuncertain systemsrdquo Nonlinear Dynamics and Systems Theoryvol 8 no 4 pp 349ndash358 2008
[7] M de la Sen ldquoOn the stability of a certain class of linear time-varying systemsrdquoThe American Journal of Applied Sciences vol2 no 8 pp 1240ndash1245 2005
[8] M de la Sen ldquoRobust stability of a class of linear time-varying systemsrdquo IMA Journal of Mathematical Control andInformation vol 19 no 4 pp 399ndash418 2002
[9] V N Phat ldquoGlobal stabilization for linear continuous time-varying systemsrdquo Applied Mathematics and Computation vol175 no 2 pp 1730ndash1743 2006
[10] A Zevin and M Pinsky ldquoExponential stability and solutionbounds for systems with bounded nonlinearitiesrdquo IEEE Trans-actions on Automatic Control vol 48 no 10 pp 1799ndash18042003
[11] X-L Zhu and G-H Yang ldquoDelay-dependent stability criteriafor systems with differentiable time delaysrdquo Acta AutomaticaSinica vol 34 no 7 pp 765ndash771 2008
[12] M de la Sen and N S Luo ldquoOn the uniform exponentialstability of a wide class of linear time-delay systemsrdquo Journalof Mathematical Analysis and Applications vol 289 no 2 pp456ndash476 2004
[13] H R Karimi M Zapateiro and N Luo ldquoNew delay-dependentstability criteria for uncertain neutral systems with mixed time-varying delays and nonlinear perturbationsrdquo MathematicalProblems in Engineering vol 2009 Article ID 759248 22 pages2009
[14] J-Z Zhang Z Jin Q-X Liu and Z-Y Zhang ldquoAnalysis ofa delayed SIR model with nonlinear incidence raterdquo DiscreteDynamics in Nature and Society vol 2008 Article ID 63615316 pages 2008
[15] C Wei and L Chen ldquoA delayed epidemic model with pulsevaccinationrdquoDiscreteDynamics inNature and Society vol 2008Article ID 746951 12 pages 2008
[16] Y-H Fan and L-L Wang ldquoPermanence for a discrete modelwith feedback control and delayrdquo Discrete Dynamics in Natureand Society vol 2008 Article ID 945109 8 pages 2008
[17] Q Liu ldquoAlmost periodic solution of a diffusive mixed systemwith time delay and type III functional responserdquo Discrete
Dynamics in Nature and Society vol 2008 Article ID 70615413 pages 2008
[18] V N Phat and P T Nam ldquoExponential stability criteria of linearnon-autonomous systems withmultiple delaysrdquo Electronic Jour-nal of Differential Equations vol 2005 no 58 pp 1ndash8 2005
[19] M de la Sen and A Ibeas ldquoOn the global asymptotic stabilityof switched linear time-varying systems with constant pointdelaysrdquo Discrete Dynamics in Nature and Society vol 2008Article ID 231710 31 pages 2008
[20] Y Chen W Bi and Y Wu ldquoDelay-dependent exponentialstability for discrete-time BAM neural networks with time-varying delaysrdquo Discrete Dynamics in Nature and Society vol2008 Article ID 421614 14 pages 2008
[21] J H Park and O Kwon ldquoMatrix inequality approach to anovel stability criterion for time-delay systems with nonlinearuncertaintiesrdquo Journal of OptimizationTheory and Applicationsvol 126 no 3 pp 643ndash656 2005
[22] V N Phat and P Niamsup ldquoStability analysis for a class offunctional differential equations and applicationsrdquo NonlinearAnalysis Theory Methodsamp Applications vol 71 no 12 pp6265ndash6275 2009
[23] A Benabdallah I Ellouze and M A Hammami ldquoPracticalstability of nonlinear time-varying cascade systemsrdquo Journal ofDynamical and Control Systems vol 15 no 1 pp 45ndash62 2009
[24] V Lakshmikantham S Leela and A A Martynyuk PracticalStability of Nonlinear Systems World Scientific 1990
[25] V Lakshmikantham and Y Zhang ldquoStrict practical stability ofdelay differential equationrdquo Applied Mathematics and Compu-tation vol 118 no 2-3 pp 275ndash285 2001
[26] M Corless ldquoGuaranteed rates of exponential convergencefor uncertain systemsrdquo Journal of Optimization Theory andApplications vol 64 no 3 pp 481ndash494 1990
[27] M Corless and G Leitmann ldquoBounded controllers for robustexponential convergencerdquo Journal of Optimization Theory andApplications vol 76 no 1 pp 1ndash12 1993
In this case the solution converges to the ball 119861(0 1199032)
Remark that from (45) if we suppose that 1205752(119905) tends
to zero when 119905 goes to infinity then 119903(119905) rarr 0 as 119905 rarr
+infin hence the solution of (7) will converge uniformlyexponentially to zero when 119905 tends to infinity Also note thatwe can estimate 119881(0 119909
Remark 4 If the delayed nonlinear disturbances are allowedto be of large size in the sense that the constants 120575
(sdot)2char-
acterizing the upper-bounding functions are large enoughin (6) then the radius 119903 of the closed ball 119861(0 119903) becomesaccordingly larger according to their values provided inthe statement of Theorem 3 That means that if the systemis globally exponentially practically stable then the radiusof the residual ball 119861(0 119903) increases as the constants 120575
(sdot)2
increase As a result then the uncertainty about how faris the state-trajectory solution from zero becomes larger asthose constants increase Thus to a larger disturbance itcorresponds to a larger uncertainty about the final deviationof the trajectory from the origin
Remark 5 On the other hand if the size of the nonlinearperturbations is allowed to be large in the sense that theconstants 120575
(sdot)1are large enough then there is trade-off
between the values of 1205811and the maximum matrix measure
120583(119860) of 119860 so as to ensure that 120578 gt 0 in Theorem 3 Howevernote that if the constant 120581
1is large then the constant 120598 is
requested to be accordingly large As a result 119860(119905) shouldhave a sufficiently large absolute stability abscissa for all timein order to compensate for the effects of the perturbationswhile satisfying the Lyapunov-like matrix equality (9)
Remark 6 Note also from Theorem 3 that the radius of theresidual ball 119861(0 119903) also increase with the squared upper-bounds of the delays and the squared differences betweenthose upper-bounds and the corresponding delay lower-bounds as well as on certain exponential functions of themaximum delay sizes
4 Examples
Example 1 Consider the nonautonomous system with non-linear time-delay perturbation (3) with time-varying delay1205911(119905) = 120591
1cos2(045119905)
1198911(119905 119909 (119905 minus 120591
1(119905)))
=[
[
minus120575 sin (119905) 1199092(119905 minus 1205911(119905)) + 120575
2lowast cos( 119905
1 + 1199092
1
)
120575 cos (119905) 1199091(119905 minus 1205911(119905))
]
]
(67)where 120591
1 120575 gt 0 120575
2ge 0 will be chosen later and
119860 (119905) = [
119886 (119905) 1
minus1 119886 (119905)
] (68)
where 119886(119905) = minus05 cos(119905)minus1011989010minussin(119905)minus51119890minus sin(119905)minus1119860119894(119905) = 0
and 119902119894= 0 for (119894 = 1 119898) By computation we obtain
120583(119860) asymp minus81033741
Let120573 = 001 120572 = 1 120583 = 09 ] = 11
1205911119867
= 1205911 120591
1119871= 0 997904rArr 120591
1119867119871= 1205911119867
1205751= 12057511= 120575 120575
12= 1205752
1205811= 11989010
120581211
= 0 120581221
= 0 120581231
= 11989010
120581241
= 0 997904rArr 12058121= 11989010
120581311
=
1
1205912
1119867exp (2120572120591
1119867)
120581321
= 0
997904rArr 12058131=
1
exp (21205721205911119867)
(69)
Then120598 = 2 (119901 + 120573) 120575
01+ 2120572120573 + 120581
21
+ 1205813111989021205721205911119867
+ 1205811= 102 + 2119890
10
gt 0
(70)
10 Mathematical Problems in Engineering
Table 1
Delay bound (1205911119867) 2 4 6 8 10
Perturbation bound 1205751
3465679 469028 63476 8591 1163Estimated value of 120574 2098897 2968276 3635376 4197767 4693245
We can verify that a solution 119875(119905) is given by
119875 (119905) =
[
[
[
[
119890sin(119905)
10
0
0
119890sin(119905)
10
]
]
]
]
(71)
We have 119901 = sup119905isinR+119875(119905) = 11989010
1205751lt
radic120581231119890minus21205721205911119867 (1 minus 120583) (120581
1minus 2120573120583 (119860))
(119901 + 120573)
1205811= 11989010
gt 2120573120583 (119860) = minus162067
120578 = 1205811minus 2120573120583 (119860) minus
(119901 + 120573)2
1205752
120581231119890minus21205721205911119867 (1 minus 120583)
gt 0
120578120577= (1 minus 120577
2
) (1205811minus 2120573120583 (119860)) gt 0 where 120575 = 120577120575
0 lt 120598 le minus10120573 + 2120572120573 + 1199021(120572 minus 5)
2
120572 = 1 120573 = 0001 1199021= 00010489
1205811= 002 gt minus10120573 = minus001
120581211
= 0 120581221
= 0
120581241
= 0 997904rArr 12058121= 120581231
= 0001282
120581311
= 016374 120581321
= 0 997904rArr 12058131= 00016374
11990111= 11990122= 000319
11990112= 000125 997904rArr 119901 = 000445
1205751= 002 lt 00206
120578 = 68310minus4
gt 0
120574 = radic
119901 + 120573 + 1205812311205911119867
+ 12058131111988811198671205913
1119867
120573
= 2416
with 1198881119867
=
2120572120591111986711989021205721205911119867
+ 119890minus21205721205911119867
minus 1
(21205721205911119867)2
119903 =
(119901 + 120573)119872
radic2120572120573120578
= 513 with 119872 = 1205752
(77)
in such a way that condition (12) in Theorem 3 is satisfiedThe result of the simulation of this example is depicted
in Figure 3 The evolution of states 1199091and 119909
2is given It is
shown in Figure 1 that the time-delay perturbed system is
0 1 2 3 4 5minus02
0
02
04
06
08
1
12
x1
x2
Solu
tion x
Time t
Figure 3 Convergence of solutions
globally uniformly practically exponentially stable toward aneighborhood of the origin
5 Conclusion
Based on improved Lyapunov-Krasovskii functional for per-turbed systems with time-varying delay we have presentednew sufficient conditions for global uniformly exponentialpractical stability toward a certain ball neighborhood of theorigin The perturbations are assumed to be nonlinear ingeneral with delayed contributions The delayed contribu-tions of such perturbations are not necessarily boundedwhilethey are upper-bounded by known nonnegative integrablefunctions which are linear functions of the various time-delayed state norms The point delays are assumed to beunknown bounded time-differentiable functions of timewithknown lower- and upper-bounds and known upper-boundsof their time-derivatives
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors wish to thank the editor and the anonymousreviewers for their valuable and careful comments The thirdauthor is grateful to the Spanish Ministry of Educationfor its partial support of this work through Grant DPIDPI2012-30651 and is also grateful to the BasqueGovernmentfor its support through Grants IT378-1 and SAIOTEK S-PE13UN039
12 Mathematical Problems in Engineering
References
[1] K Zhou and P P Khargonekar ldquoStability robustness boundsfor linear state-space models with structured uncertaintyrdquo IEEETransactions on Automatic Control vol 32 no 7 pp 621ndash6231987
[2] E Cheres Z J Palmor and S Gutman ldquoQuantitative measuresof robustness for systems including delayed perturbationsrdquoIEEE Transactions on Automatic Control vol 34 no 11 pp1203ndash1204 1989
[3] H Trinh and M Aldeen ldquoOn the stability of linear systemswith delayed perturbationsrdquo IEEE Transactions on AutomaticControl vol 39 no 9 pp 1948ndash1951 1994
[4] H Trinh and M Aldeen ldquoOn robustness and stabilizationof linear systems with delayed nonlinear perturbationsrdquo IEEETransactions on Automatic Control vol 42 no 7 pp 1005ndash10071997
[5] P Niamsup K Mukdasai and V N Phat ldquoImproved exponen-tial stability for time-varying systems with nonlinear delayedperturbationsrdquoAppliedMathematics andComputation vol 204no 1 pp 490ndash495 2008
[6] M Hammami and I Ellouze ldquoA robust detector for a class ofuncertain systemsrdquo Nonlinear Dynamics and Systems Theoryvol 8 no 4 pp 349ndash358 2008
[7] M de la Sen ldquoOn the stability of a certain class of linear time-varying systemsrdquoThe American Journal of Applied Sciences vol2 no 8 pp 1240ndash1245 2005
[8] M de la Sen ldquoRobust stability of a class of linear time-varying systemsrdquo IMA Journal of Mathematical Control andInformation vol 19 no 4 pp 399ndash418 2002
[9] V N Phat ldquoGlobal stabilization for linear continuous time-varying systemsrdquo Applied Mathematics and Computation vol175 no 2 pp 1730ndash1743 2006
[10] A Zevin and M Pinsky ldquoExponential stability and solutionbounds for systems with bounded nonlinearitiesrdquo IEEE Trans-actions on Automatic Control vol 48 no 10 pp 1799ndash18042003
[11] X-L Zhu and G-H Yang ldquoDelay-dependent stability criteriafor systems with differentiable time delaysrdquo Acta AutomaticaSinica vol 34 no 7 pp 765ndash771 2008
[12] M de la Sen and N S Luo ldquoOn the uniform exponentialstability of a wide class of linear time-delay systemsrdquo Journalof Mathematical Analysis and Applications vol 289 no 2 pp456ndash476 2004
[13] H R Karimi M Zapateiro and N Luo ldquoNew delay-dependentstability criteria for uncertain neutral systems with mixed time-varying delays and nonlinear perturbationsrdquo MathematicalProblems in Engineering vol 2009 Article ID 759248 22 pages2009
[14] J-Z Zhang Z Jin Q-X Liu and Z-Y Zhang ldquoAnalysis ofa delayed SIR model with nonlinear incidence raterdquo DiscreteDynamics in Nature and Society vol 2008 Article ID 63615316 pages 2008
[15] C Wei and L Chen ldquoA delayed epidemic model with pulsevaccinationrdquoDiscreteDynamics inNature and Society vol 2008Article ID 746951 12 pages 2008
[16] Y-H Fan and L-L Wang ldquoPermanence for a discrete modelwith feedback control and delayrdquo Discrete Dynamics in Natureand Society vol 2008 Article ID 945109 8 pages 2008
[17] Q Liu ldquoAlmost periodic solution of a diffusive mixed systemwith time delay and type III functional responserdquo Discrete
Dynamics in Nature and Society vol 2008 Article ID 70615413 pages 2008
[18] V N Phat and P T Nam ldquoExponential stability criteria of linearnon-autonomous systems withmultiple delaysrdquo Electronic Jour-nal of Differential Equations vol 2005 no 58 pp 1ndash8 2005
[19] M de la Sen and A Ibeas ldquoOn the global asymptotic stabilityof switched linear time-varying systems with constant pointdelaysrdquo Discrete Dynamics in Nature and Society vol 2008Article ID 231710 31 pages 2008
[20] Y Chen W Bi and Y Wu ldquoDelay-dependent exponentialstability for discrete-time BAM neural networks with time-varying delaysrdquo Discrete Dynamics in Nature and Society vol2008 Article ID 421614 14 pages 2008
[21] J H Park and O Kwon ldquoMatrix inequality approach to anovel stability criterion for time-delay systems with nonlinearuncertaintiesrdquo Journal of OptimizationTheory and Applicationsvol 126 no 3 pp 643ndash656 2005
[22] V N Phat and P Niamsup ldquoStability analysis for a class offunctional differential equations and applicationsrdquo NonlinearAnalysis Theory Methodsamp Applications vol 71 no 12 pp6265ndash6275 2009
[23] A Benabdallah I Ellouze and M A Hammami ldquoPracticalstability of nonlinear time-varying cascade systemsrdquo Journal ofDynamical and Control Systems vol 15 no 1 pp 45ndash62 2009
[24] V Lakshmikantham S Leela and A A Martynyuk PracticalStability of Nonlinear Systems World Scientific 1990
[25] V Lakshmikantham and Y Zhang ldquoStrict practical stability ofdelay differential equationrdquo Applied Mathematics and Compu-tation vol 118 no 2-3 pp 275ndash285 2001
[26] M Corless ldquoGuaranteed rates of exponential convergencefor uncertain systemsrdquo Journal of Optimization Theory andApplications vol 64 no 3 pp 481ndash494 1990
[27] M Corless and G Leitmann ldquoBounded controllers for robustexponential convergencerdquo Journal of Optimization Theory andApplications vol 76 no 1 pp 1ndash12 1993
In this case the solution converges to the ball 119861(0 1199032)
Remark that from (45) if we suppose that 1205752(119905) tends
to zero when 119905 goes to infinity then 119903(119905) rarr 0 as 119905 rarr
+infin hence the solution of (7) will converge uniformlyexponentially to zero when 119905 tends to infinity Also note thatwe can estimate 119881(0 119909
Remark 4 If the delayed nonlinear disturbances are allowedto be of large size in the sense that the constants 120575
(sdot)2char-
acterizing the upper-bounding functions are large enoughin (6) then the radius 119903 of the closed ball 119861(0 119903) becomesaccordingly larger according to their values provided inthe statement of Theorem 3 That means that if the systemis globally exponentially practically stable then the radiusof the residual ball 119861(0 119903) increases as the constants 120575
(sdot)2
increase As a result then the uncertainty about how faris the state-trajectory solution from zero becomes larger asthose constants increase Thus to a larger disturbance itcorresponds to a larger uncertainty about the final deviationof the trajectory from the origin
Remark 5 On the other hand if the size of the nonlinearperturbations is allowed to be large in the sense that theconstants 120575
(sdot)1are large enough then there is trade-off
between the values of 1205811and the maximum matrix measure
120583(119860) of 119860 so as to ensure that 120578 gt 0 in Theorem 3 Howevernote that if the constant 120581
1is large then the constant 120598 is
requested to be accordingly large As a result 119860(119905) shouldhave a sufficiently large absolute stability abscissa for all timein order to compensate for the effects of the perturbationswhile satisfying the Lyapunov-like matrix equality (9)
Remark 6 Note also from Theorem 3 that the radius of theresidual ball 119861(0 119903) also increase with the squared upper-bounds of the delays and the squared differences betweenthose upper-bounds and the corresponding delay lower-bounds as well as on certain exponential functions of themaximum delay sizes
4 Examples
Example 1 Consider the nonautonomous system with non-linear time-delay perturbation (3) with time-varying delay1205911(119905) = 120591
1cos2(045119905)
1198911(119905 119909 (119905 minus 120591
1(119905)))
=[
[
minus120575 sin (119905) 1199092(119905 minus 1205911(119905)) + 120575
2lowast cos( 119905
1 + 1199092
1
)
120575 cos (119905) 1199091(119905 minus 1205911(119905))
]
]
(67)where 120591
1 120575 gt 0 120575
2ge 0 will be chosen later and
119860 (119905) = [
119886 (119905) 1
minus1 119886 (119905)
] (68)
where 119886(119905) = minus05 cos(119905)minus1011989010minussin(119905)minus51119890minus sin(119905)minus1119860119894(119905) = 0
and 119902119894= 0 for (119894 = 1 119898) By computation we obtain
120583(119860) asymp minus81033741
Let120573 = 001 120572 = 1 120583 = 09 ] = 11
1205911119867
= 1205911 120591
1119871= 0 997904rArr 120591
1119867119871= 1205911119867
1205751= 12057511= 120575 120575
12= 1205752
1205811= 11989010
120581211
= 0 120581221
= 0 120581231
= 11989010
120581241
= 0 997904rArr 12058121= 11989010
120581311
=
1
1205912
1119867exp (2120572120591
1119867)
120581321
= 0
997904rArr 12058131=
1
exp (21205721205911119867)
(69)
Then120598 = 2 (119901 + 120573) 120575
01+ 2120572120573 + 120581
21
+ 1205813111989021205721205911119867
+ 1205811= 102 + 2119890
10
gt 0
(70)
10 Mathematical Problems in Engineering
Table 1
Delay bound (1205911119867) 2 4 6 8 10
Perturbation bound 1205751
3465679 469028 63476 8591 1163Estimated value of 120574 2098897 2968276 3635376 4197767 4693245
We can verify that a solution 119875(119905) is given by
119875 (119905) =
[
[
[
[
119890sin(119905)
10
0
0
119890sin(119905)
10
]
]
]
]
(71)
We have 119901 = sup119905isinR+119875(119905) = 11989010
1205751lt
radic120581231119890minus21205721205911119867 (1 minus 120583) (120581
1minus 2120573120583 (119860))
(119901 + 120573)
1205811= 11989010
gt 2120573120583 (119860) = minus162067
120578 = 1205811minus 2120573120583 (119860) minus
(119901 + 120573)2
1205752
120581231119890minus21205721205911119867 (1 minus 120583)
gt 0
120578120577= (1 minus 120577
2
) (1205811minus 2120573120583 (119860)) gt 0 where 120575 = 120577120575
0 lt 120598 le minus10120573 + 2120572120573 + 1199021(120572 minus 5)
2
120572 = 1 120573 = 0001 1199021= 00010489
1205811= 002 gt minus10120573 = minus001
120581211
= 0 120581221
= 0
120581241
= 0 997904rArr 12058121= 120581231
= 0001282
120581311
= 016374 120581321
= 0 997904rArr 12058131= 00016374
11990111= 11990122= 000319
11990112= 000125 997904rArr 119901 = 000445
1205751= 002 lt 00206
120578 = 68310minus4
gt 0
120574 = radic
119901 + 120573 + 1205812311205911119867
+ 12058131111988811198671205913
1119867
120573
= 2416
with 1198881119867
=
2120572120591111986711989021205721205911119867
+ 119890minus21205721205911119867
minus 1
(21205721205911119867)2
119903 =
(119901 + 120573)119872
radic2120572120573120578
= 513 with 119872 = 1205752
(77)
in such a way that condition (12) in Theorem 3 is satisfiedThe result of the simulation of this example is depicted
in Figure 3 The evolution of states 1199091and 119909
2is given It is
shown in Figure 1 that the time-delay perturbed system is
0 1 2 3 4 5minus02
0
02
04
06
08
1
12
x1
x2
Solu
tion x
Time t
Figure 3 Convergence of solutions
globally uniformly practically exponentially stable toward aneighborhood of the origin
5 Conclusion
Based on improved Lyapunov-Krasovskii functional for per-turbed systems with time-varying delay we have presentednew sufficient conditions for global uniformly exponentialpractical stability toward a certain ball neighborhood of theorigin The perturbations are assumed to be nonlinear ingeneral with delayed contributions The delayed contribu-tions of such perturbations are not necessarily boundedwhilethey are upper-bounded by known nonnegative integrablefunctions which are linear functions of the various time-delayed state norms The point delays are assumed to beunknown bounded time-differentiable functions of timewithknown lower- and upper-bounds and known upper-boundsof their time-derivatives
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors wish to thank the editor and the anonymousreviewers for their valuable and careful comments The thirdauthor is grateful to the Spanish Ministry of Educationfor its partial support of this work through Grant DPIDPI2012-30651 and is also grateful to the BasqueGovernmentfor its support through Grants IT378-1 and SAIOTEK S-PE13UN039
12 Mathematical Problems in Engineering
References
[1] K Zhou and P P Khargonekar ldquoStability robustness boundsfor linear state-space models with structured uncertaintyrdquo IEEETransactions on Automatic Control vol 32 no 7 pp 621ndash6231987
[2] E Cheres Z J Palmor and S Gutman ldquoQuantitative measuresof robustness for systems including delayed perturbationsrdquoIEEE Transactions on Automatic Control vol 34 no 11 pp1203ndash1204 1989
[3] H Trinh and M Aldeen ldquoOn the stability of linear systemswith delayed perturbationsrdquo IEEE Transactions on AutomaticControl vol 39 no 9 pp 1948ndash1951 1994
[4] H Trinh and M Aldeen ldquoOn robustness and stabilizationof linear systems with delayed nonlinear perturbationsrdquo IEEETransactions on Automatic Control vol 42 no 7 pp 1005ndash10071997
[5] P Niamsup K Mukdasai and V N Phat ldquoImproved exponen-tial stability for time-varying systems with nonlinear delayedperturbationsrdquoAppliedMathematics andComputation vol 204no 1 pp 490ndash495 2008
[6] M Hammami and I Ellouze ldquoA robust detector for a class ofuncertain systemsrdquo Nonlinear Dynamics and Systems Theoryvol 8 no 4 pp 349ndash358 2008
[7] M de la Sen ldquoOn the stability of a certain class of linear time-varying systemsrdquoThe American Journal of Applied Sciences vol2 no 8 pp 1240ndash1245 2005
[8] M de la Sen ldquoRobust stability of a class of linear time-varying systemsrdquo IMA Journal of Mathematical Control andInformation vol 19 no 4 pp 399ndash418 2002
[9] V N Phat ldquoGlobal stabilization for linear continuous time-varying systemsrdquo Applied Mathematics and Computation vol175 no 2 pp 1730ndash1743 2006
[10] A Zevin and M Pinsky ldquoExponential stability and solutionbounds for systems with bounded nonlinearitiesrdquo IEEE Trans-actions on Automatic Control vol 48 no 10 pp 1799ndash18042003
[11] X-L Zhu and G-H Yang ldquoDelay-dependent stability criteriafor systems with differentiable time delaysrdquo Acta AutomaticaSinica vol 34 no 7 pp 765ndash771 2008
[12] M de la Sen and N S Luo ldquoOn the uniform exponentialstability of a wide class of linear time-delay systemsrdquo Journalof Mathematical Analysis and Applications vol 289 no 2 pp456ndash476 2004
[13] H R Karimi M Zapateiro and N Luo ldquoNew delay-dependentstability criteria for uncertain neutral systems with mixed time-varying delays and nonlinear perturbationsrdquo MathematicalProblems in Engineering vol 2009 Article ID 759248 22 pages2009
[14] J-Z Zhang Z Jin Q-X Liu and Z-Y Zhang ldquoAnalysis ofa delayed SIR model with nonlinear incidence raterdquo DiscreteDynamics in Nature and Society vol 2008 Article ID 63615316 pages 2008
[15] C Wei and L Chen ldquoA delayed epidemic model with pulsevaccinationrdquoDiscreteDynamics inNature and Society vol 2008Article ID 746951 12 pages 2008
[16] Y-H Fan and L-L Wang ldquoPermanence for a discrete modelwith feedback control and delayrdquo Discrete Dynamics in Natureand Society vol 2008 Article ID 945109 8 pages 2008
[17] Q Liu ldquoAlmost periodic solution of a diffusive mixed systemwith time delay and type III functional responserdquo Discrete
Dynamics in Nature and Society vol 2008 Article ID 70615413 pages 2008
[18] V N Phat and P T Nam ldquoExponential stability criteria of linearnon-autonomous systems withmultiple delaysrdquo Electronic Jour-nal of Differential Equations vol 2005 no 58 pp 1ndash8 2005
[19] M de la Sen and A Ibeas ldquoOn the global asymptotic stabilityof switched linear time-varying systems with constant pointdelaysrdquo Discrete Dynamics in Nature and Society vol 2008Article ID 231710 31 pages 2008
[20] Y Chen W Bi and Y Wu ldquoDelay-dependent exponentialstability for discrete-time BAM neural networks with time-varying delaysrdquo Discrete Dynamics in Nature and Society vol2008 Article ID 421614 14 pages 2008
[21] J H Park and O Kwon ldquoMatrix inequality approach to anovel stability criterion for time-delay systems with nonlinearuncertaintiesrdquo Journal of OptimizationTheory and Applicationsvol 126 no 3 pp 643ndash656 2005
[22] V N Phat and P Niamsup ldquoStability analysis for a class offunctional differential equations and applicationsrdquo NonlinearAnalysis Theory Methodsamp Applications vol 71 no 12 pp6265ndash6275 2009
[23] A Benabdallah I Ellouze and M A Hammami ldquoPracticalstability of nonlinear time-varying cascade systemsrdquo Journal ofDynamical and Control Systems vol 15 no 1 pp 45ndash62 2009
[24] V Lakshmikantham S Leela and A A Martynyuk PracticalStability of Nonlinear Systems World Scientific 1990
[25] V Lakshmikantham and Y Zhang ldquoStrict practical stability ofdelay differential equationrdquo Applied Mathematics and Compu-tation vol 118 no 2-3 pp 275ndash285 2001
[26] M Corless ldquoGuaranteed rates of exponential convergencefor uncertain systemsrdquo Journal of Optimization Theory andApplications vol 64 no 3 pp 481ndash494 1990
[27] M Corless and G Leitmann ldquoBounded controllers for robustexponential convergencerdquo Journal of Optimization Theory andApplications vol 76 no 1 pp 1ndash12 1993
In this case the solution converges to the ball 119861(0 1199032)
Remark that from (45) if we suppose that 1205752(119905) tends
to zero when 119905 goes to infinity then 119903(119905) rarr 0 as 119905 rarr
+infin hence the solution of (7) will converge uniformlyexponentially to zero when 119905 tends to infinity Also note thatwe can estimate 119881(0 119909
Remark 4 If the delayed nonlinear disturbances are allowedto be of large size in the sense that the constants 120575
(sdot)2char-
acterizing the upper-bounding functions are large enoughin (6) then the radius 119903 of the closed ball 119861(0 119903) becomesaccordingly larger according to their values provided inthe statement of Theorem 3 That means that if the systemis globally exponentially practically stable then the radiusof the residual ball 119861(0 119903) increases as the constants 120575
(sdot)2
increase As a result then the uncertainty about how faris the state-trajectory solution from zero becomes larger asthose constants increase Thus to a larger disturbance itcorresponds to a larger uncertainty about the final deviationof the trajectory from the origin
Remark 5 On the other hand if the size of the nonlinearperturbations is allowed to be large in the sense that theconstants 120575
(sdot)1are large enough then there is trade-off
between the values of 1205811and the maximum matrix measure
120583(119860) of 119860 so as to ensure that 120578 gt 0 in Theorem 3 Howevernote that if the constant 120581
1is large then the constant 120598 is
requested to be accordingly large As a result 119860(119905) shouldhave a sufficiently large absolute stability abscissa for all timein order to compensate for the effects of the perturbationswhile satisfying the Lyapunov-like matrix equality (9)
Remark 6 Note also from Theorem 3 that the radius of theresidual ball 119861(0 119903) also increase with the squared upper-bounds of the delays and the squared differences betweenthose upper-bounds and the corresponding delay lower-bounds as well as on certain exponential functions of themaximum delay sizes
4 Examples
Example 1 Consider the nonautonomous system with non-linear time-delay perturbation (3) with time-varying delay1205911(119905) = 120591
1cos2(045119905)
1198911(119905 119909 (119905 minus 120591
1(119905)))
=[
[
minus120575 sin (119905) 1199092(119905 minus 1205911(119905)) + 120575
2lowast cos( 119905
1 + 1199092
1
)
120575 cos (119905) 1199091(119905 minus 1205911(119905))
]
]
(67)where 120591
1 120575 gt 0 120575
2ge 0 will be chosen later and
119860 (119905) = [
119886 (119905) 1
minus1 119886 (119905)
] (68)
where 119886(119905) = minus05 cos(119905)minus1011989010minussin(119905)minus51119890minus sin(119905)minus1119860119894(119905) = 0
and 119902119894= 0 for (119894 = 1 119898) By computation we obtain
120583(119860) asymp minus81033741
Let120573 = 001 120572 = 1 120583 = 09 ] = 11
1205911119867
= 1205911 120591
1119871= 0 997904rArr 120591
1119867119871= 1205911119867
1205751= 12057511= 120575 120575
12= 1205752
1205811= 11989010
120581211
= 0 120581221
= 0 120581231
= 11989010
120581241
= 0 997904rArr 12058121= 11989010
120581311
=
1
1205912
1119867exp (2120572120591
1119867)
120581321
= 0
997904rArr 12058131=
1
exp (21205721205911119867)
(69)
Then120598 = 2 (119901 + 120573) 120575
01+ 2120572120573 + 120581
21
+ 1205813111989021205721205911119867
+ 1205811= 102 + 2119890
10
gt 0
(70)
10 Mathematical Problems in Engineering
Table 1
Delay bound (1205911119867) 2 4 6 8 10
Perturbation bound 1205751
3465679 469028 63476 8591 1163Estimated value of 120574 2098897 2968276 3635376 4197767 4693245
We can verify that a solution 119875(119905) is given by
119875 (119905) =
[
[
[
[
119890sin(119905)
10
0
0
119890sin(119905)
10
]
]
]
]
(71)
We have 119901 = sup119905isinR+119875(119905) = 11989010
1205751lt
radic120581231119890minus21205721205911119867 (1 minus 120583) (120581
1minus 2120573120583 (119860))
(119901 + 120573)
1205811= 11989010
gt 2120573120583 (119860) = minus162067
120578 = 1205811minus 2120573120583 (119860) minus
(119901 + 120573)2
1205752
120581231119890minus21205721205911119867 (1 minus 120583)
gt 0
120578120577= (1 minus 120577
2
) (1205811minus 2120573120583 (119860)) gt 0 where 120575 = 120577120575
0 lt 120598 le minus10120573 + 2120572120573 + 1199021(120572 minus 5)
2
120572 = 1 120573 = 0001 1199021= 00010489
1205811= 002 gt minus10120573 = minus001
120581211
= 0 120581221
= 0
120581241
= 0 997904rArr 12058121= 120581231
= 0001282
120581311
= 016374 120581321
= 0 997904rArr 12058131= 00016374
11990111= 11990122= 000319
11990112= 000125 997904rArr 119901 = 000445
1205751= 002 lt 00206
120578 = 68310minus4
gt 0
120574 = radic
119901 + 120573 + 1205812311205911119867
+ 12058131111988811198671205913
1119867
120573
= 2416
with 1198881119867
=
2120572120591111986711989021205721205911119867
+ 119890minus21205721205911119867
minus 1
(21205721205911119867)2
119903 =
(119901 + 120573)119872
radic2120572120573120578
= 513 with 119872 = 1205752
(77)
in such a way that condition (12) in Theorem 3 is satisfiedThe result of the simulation of this example is depicted
in Figure 3 The evolution of states 1199091and 119909
2is given It is
shown in Figure 1 that the time-delay perturbed system is
0 1 2 3 4 5minus02
0
02
04
06
08
1
12
x1
x2
Solu
tion x
Time t
Figure 3 Convergence of solutions
globally uniformly practically exponentially stable toward aneighborhood of the origin
5 Conclusion
Based on improved Lyapunov-Krasovskii functional for per-turbed systems with time-varying delay we have presentednew sufficient conditions for global uniformly exponentialpractical stability toward a certain ball neighborhood of theorigin The perturbations are assumed to be nonlinear ingeneral with delayed contributions The delayed contribu-tions of such perturbations are not necessarily boundedwhilethey are upper-bounded by known nonnegative integrablefunctions which are linear functions of the various time-delayed state norms The point delays are assumed to beunknown bounded time-differentiable functions of timewithknown lower- and upper-bounds and known upper-boundsof their time-derivatives
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors wish to thank the editor and the anonymousreviewers for their valuable and careful comments The thirdauthor is grateful to the Spanish Ministry of Educationfor its partial support of this work through Grant DPIDPI2012-30651 and is also grateful to the BasqueGovernmentfor its support through Grants IT378-1 and SAIOTEK S-PE13UN039
12 Mathematical Problems in Engineering
References
[1] K Zhou and P P Khargonekar ldquoStability robustness boundsfor linear state-space models with structured uncertaintyrdquo IEEETransactions on Automatic Control vol 32 no 7 pp 621ndash6231987
[2] E Cheres Z J Palmor and S Gutman ldquoQuantitative measuresof robustness for systems including delayed perturbationsrdquoIEEE Transactions on Automatic Control vol 34 no 11 pp1203ndash1204 1989
[3] H Trinh and M Aldeen ldquoOn the stability of linear systemswith delayed perturbationsrdquo IEEE Transactions on AutomaticControl vol 39 no 9 pp 1948ndash1951 1994
[4] H Trinh and M Aldeen ldquoOn robustness and stabilizationof linear systems with delayed nonlinear perturbationsrdquo IEEETransactions on Automatic Control vol 42 no 7 pp 1005ndash10071997
[5] P Niamsup K Mukdasai and V N Phat ldquoImproved exponen-tial stability for time-varying systems with nonlinear delayedperturbationsrdquoAppliedMathematics andComputation vol 204no 1 pp 490ndash495 2008
[6] M Hammami and I Ellouze ldquoA robust detector for a class ofuncertain systemsrdquo Nonlinear Dynamics and Systems Theoryvol 8 no 4 pp 349ndash358 2008
[7] M de la Sen ldquoOn the stability of a certain class of linear time-varying systemsrdquoThe American Journal of Applied Sciences vol2 no 8 pp 1240ndash1245 2005
[8] M de la Sen ldquoRobust stability of a class of linear time-varying systemsrdquo IMA Journal of Mathematical Control andInformation vol 19 no 4 pp 399ndash418 2002
[9] V N Phat ldquoGlobal stabilization for linear continuous time-varying systemsrdquo Applied Mathematics and Computation vol175 no 2 pp 1730ndash1743 2006
[10] A Zevin and M Pinsky ldquoExponential stability and solutionbounds for systems with bounded nonlinearitiesrdquo IEEE Trans-actions on Automatic Control vol 48 no 10 pp 1799ndash18042003
[11] X-L Zhu and G-H Yang ldquoDelay-dependent stability criteriafor systems with differentiable time delaysrdquo Acta AutomaticaSinica vol 34 no 7 pp 765ndash771 2008
[12] M de la Sen and N S Luo ldquoOn the uniform exponentialstability of a wide class of linear time-delay systemsrdquo Journalof Mathematical Analysis and Applications vol 289 no 2 pp456ndash476 2004
[13] H R Karimi M Zapateiro and N Luo ldquoNew delay-dependentstability criteria for uncertain neutral systems with mixed time-varying delays and nonlinear perturbationsrdquo MathematicalProblems in Engineering vol 2009 Article ID 759248 22 pages2009
[14] J-Z Zhang Z Jin Q-X Liu and Z-Y Zhang ldquoAnalysis ofa delayed SIR model with nonlinear incidence raterdquo DiscreteDynamics in Nature and Society vol 2008 Article ID 63615316 pages 2008
[15] C Wei and L Chen ldquoA delayed epidemic model with pulsevaccinationrdquoDiscreteDynamics inNature and Society vol 2008Article ID 746951 12 pages 2008
[16] Y-H Fan and L-L Wang ldquoPermanence for a discrete modelwith feedback control and delayrdquo Discrete Dynamics in Natureand Society vol 2008 Article ID 945109 8 pages 2008
[17] Q Liu ldquoAlmost periodic solution of a diffusive mixed systemwith time delay and type III functional responserdquo Discrete
Dynamics in Nature and Society vol 2008 Article ID 70615413 pages 2008
[18] V N Phat and P T Nam ldquoExponential stability criteria of linearnon-autonomous systems withmultiple delaysrdquo Electronic Jour-nal of Differential Equations vol 2005 no 58 pp 1ndash8 2005
[19] M de la Sen and A Ibeas ldquoOn the global asymptotic stabilityof switched linear time-varying systems with constant pointdelaysrdquo Discrete Dynamics in Nature and Society vol 2008Article ID 231710 31 pages 2008
[20] Y Chen W Bi and Y Wu ldquoDelay-dependent exponentialstability for discrete-time BAM neural networks with time-varying delaysrdquo Discrete Dynamics in Nature and Society vol2008 Article ID 421614 14 pages 2008
[21] J H Park and O Kwon ldquoMatrix inequality approach to anovel stability criterion for time-delay systems with nonlinearuncertaintiesrdquo Journal of OptimizationTheory and Applicationsvol 126 no 3 pp 643ndash656 2005
[22] V N Phat and P Niamsup ldquoStability analysis for a class offunctional differential equations and applicationsrdquo NonlinearAnalysis Theory Methodsamp Applications vol 71 no 12 pp6265ndash6275 2009
[23] A Benabdallah I Ellouze and M A Hammami ldquoPracticalstability of nonlinear time-varying cascade systemsrdquo Journal ofDynamical and Control Systems vol 15 no 1 pp 45ndash62 2009
[24] V Lakshmikantham S Leela and A A Martynyuk PracticalStability of Nonlinear Systems World Scientific 1990
[25] V Lakshmikantham and Y Zhang ldquoStrict practical stability ofdelay differential equationrdquo Applied Mathematics and Compu-tation vol 118 no 2-3 pp 275ndash285 2001
[26] M Corless ldquoGuaranteed rates of exponential convergencefor uncertain systemsrdquo Journal of Optimization Theory andApplications vol 64 no 3 pp 481ndash494 1990
[27] M Corless and G Leitmann ldquoBounded controllers for robustexponential convergencerdquo Journal of Optimization Theory andApplications vol 76 no 1 pp 1ndash12 1993
In this case the solution converges to the ball 119861(0 1199032)
Remark that from (45) if we suppose that 1205752(119905) tends
to zero when 119905 goes to infinity then 119903(119905) rarr 0 as 119905 rarr
+infin hence the solution of (7) will converge uniformlyexponentially to zero when 119905 tends to infinity Also note thatwe can estimate 119881(0 119909
Remark 4 If the delayed nonlinear disturbances are allowedto be of large size in the sense that the constants 120575
(sdot)2char-
acterizing the upper-bounding functions are large enoughin (6) then the radius 119903 of the closed ball 119861(0 119903) becomesaccordingly larger according to their values provided inthe statement of Theorem 3 That means that if the systemis globally exponentially practically stable then the radiusof the residual ball 119861(0 119903) increases as the constants 120575
(sdot)2
increase As a result then the uncertainty about how faris the state-trajectory solution from zero becomes larger asthose constants increase Thus to a larger disturbance itcorresponds to a larger uncertainty about the final deviationof the trajectory from the origin
Remark 5 On the other hand if the size of the nonlinearperturbations is allowed to be large in the sense that theconstants 120575
(sdot)1are large enough then there is trade-off
between the values of 1205811and the maximum matrix measure
120583(119860) of 119860 so as to ensure that 120578 gt 0 in Theorem 3 Howevernote that if the constant 120581
1is large then the constant 120598 is
requested to be accordingly large As a result 119860(119905) shouldhave a sufficiently large absolute stability abscissa for all timein order to compensate for the effects of the perturbationswhile satisfying the Lyapunov-like matrix equality (9)
Remark 6 Note also from Theorem 3 that the radius of theresidual ball 119861(0 119903) also increase with the squared upper-bounds of the delays and the squared differences betweenthose upper-bounds and the corresponding delay lower-bounds as well as on certain exponential functions of themaximum delay sizes
4 Examples
Example 1 Consider the nonautonomous system with non-linear time-delay perturbation (3) with time-varying delay1205911(119905) = 120591
1cos2(045119905)
1198911(119905 119909 (119905 minus 120591
1(119905)))
=[
[
minus120575 sin (119905) 1199092(119905 minus 1205911(119905)) + 120575
2lowast cos( 119905
1 + 1199092
1
)
120575 cos (119905) 1199091(119905 minus 1205911(119905))
]
]
(67)where 120591
1 120575 gt 0 120575
2ge 0 will be chosen later and
119860 (119905) = [
119886 (119905) 1
minus1 119886 (119905)
] (68)
where 119886(119905) = minus05 cos(119905)minus1011989010minussin(119905)minus51119890minus sin(119905)minus1119860119894(119905) = 0
and 119902119894= 0 for (119894 = 1 119898) By computation we obtain
120583(119860) asymp minus81033741
Let120573 = 001 120572 = 1 120583 = 09 ] = 11
1205911119867
= 1205911 120591
1119871= 0 997904rArr 120591
1119867119871= 1205911119867
1205751= 12057511= 120575 120575
12= 1205752
1205811= 11989010
120581211
= 0 120581221
= 0 120581231
= 11989010
120581241
= 0 997904rArr 12058121= 11989010
120581311
=
1
1205912
1119867exp (2120572120591
1119867)
120581321
= 0
997904rArr 12058131=
1
exp (21205721205911119867)
(69)
Then120598 = 2 (119901 + 120573) 120575
01+ 2120572120573 + 120581
21
+ 1205813111989021205721205911119867
+ 1205811= 102 + 2119890
10
gt 0
(70)
10 Mathematical Problems in Engineering
Table 1
Delay bound (1205911119867) 2 4 6 8 10
Perturbation bound 1205751
3465679 469028 63476 8591 1163Estimated value of 120574 2098897 2968276 3635376 4197767 4693245
We can verify that a solution 119875(119905) is given by
119875 (119905) =
[
[
[
[
119890sin(119905)
10
0
0
119890sin(119905)
10
]
]
]
]
(71)
We have 119901 = sup119905isinR+119875(119905) = 11989010
1205751lt
radic120581231119890minus21205721205911119867 (1 minus 120583) (120581
1minus 2120573120583 (119860))
(119901 + 120573)
1205811= 11989010
gt 2120573120583 (119860) = minus162067
120578 = 1205811minus 2120573120583 (119860) minus
(119901 + 120573)2
1205752
120581231119890minus21205721205911119867 (1 minus 120583)
gt 0
120578120577= (1 minus 120577
2
) (1205811minus 2120573120583 (119860)) gt 0 where 120575 = 120577120575
0 lt 120598 le minus10120573 + 2120572120573 + 1199021(120572 minus 5)
2
120572 = 1 120573 = 0001 1199021= 00010489
1205811= 002 gt minus10120573 = minus001
120581211
= 0 120581221
= 0
120581241
= 0 997904rArr 12058121= 120581231
= 0001282
120581311
= 016374 120581321
= 0 997904rArr 12058131= 00016374
11990111= 11990122= 000319
11990112= 000125 997904rArr 119901 = 000445
1205751= 002 lt 00206
120578 = 68310minus4
gt 0
120574 = radic
119901 + 120573 + 1205812311205911119867
+ 12058131111988811198671205913
1119867
120573
= 2416
with 1198881119867
=
2120572120591111986711989021205721205911119867
+ 119890minus21205721205911119867
minus 1
(21205721205911119867)2
119903 =
(119901 + 120573)119872
radic2120572120573120578
= 513 with 119872 = 1205752
(77)
in such a way that condition (12) in Theorem 3 is satisfiedThe result of the simulation of this example is depicted
in Figure 3 The evolution of states 1199091and 119909
2is given It is
shown in Figure 1 that the time-delay perturbed system is
0 1 2 3 4 5minus02
0
02
04
06
08
1
12
x1
x2
Solu
tion x
Time t
Figure 3 Convergence of solutions
globally uniformly practically exponentially stable toward aneighborhood of the origin
5 Conclusion
Based on improved Lyapunov-Krasovskii functional for per-turbed systems with time-varying delay we have presentednew sufficient conditions for global uniformly exponentialpractical stability toward a certain ball neighborhood of theorigin The perturbations are assumed to be nonlinear ingeneral with delayed contributions The delayed contribu-tions of such perturbations are not necessarily boundedwhilethey are upper-bounded by known nonnegative integrablefunctions which are linear functions of the various time-delayed state norms The point delays are assumed to beunknown bounded time-differentiable functions of timewithknown lower- and upper-bounds and known upper-boundsof their time-derivatives
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors wish to thank the editor and the anonymousreviewers for their valuable and careful comments The thirdauthor is grateful to the Spanish Ministry of Educationfor its partial support of this work through Grant DPIDPI2012-30651 and is also grateful to the BasqueGovernmentfor its support through Grants IT378-1 and SAIOTEK S-PE13UN039
12 Mathematical Problems in Engineering
References
[1] K Zhou and P P Khargonekar ldquoStability robustness boundsfor linear state-space models with structured uncertaintyrdquo IEEETransactions on Automatic Control vol 32 no 7 pp 621ndash6231987
[2] E Cheres Z J Palmor and S Gutman ldquoQuantitative measuresof robustness for systems including delayed perturbationsrdquoIEEE Transactions on Automatic Control vol 34 no 11 pp1203ndash1204 1989
[3] H Trinh and M Aldeen ldquoOn the stability of linear systemswith delayed perturbationsrdquo IEEE Transactions on AutomaticControl vol 39 no 9 pp 1948ndash1951 1994
[4] H Trinh and M Aldeen ldquoOn robustness and stabilizationof linear systems with delayed nonlinear perturbationsrdquo IEEETransactions on Automatic Control vol 42 no 7 pp 1005ndash10071997
[5] P Niamsup K Mukdasai and V N Phat ldquoImproved exponen-tial stability for time-varying systems with nonlinear delayedperturbationsrdquoAppliedMathematics andComputation vol 204no 1 pp 490ndash495 2008
[6] M Hammami and I Ellouze ldquoA robust detector for a class ofuncertain systemsrdquo Nonlinear Dynamics and Systems Theoryvol 8 no 4 pp 349ndash358 2008
[7] M de la Sen ldquoOn the stability of a certain class of linear time-varying systemsrdquoThe American Journal of Applied Sciences vol2 no 8 pp 1240ndash1245 2005
[8] M de la Sen ldquoRobust stability of a class of linear time-varying systemsrdquo IMA Journal of Mathematical Control andInformation vol 19 no 4 pp 399ndash418 2002
[9] V N Phat ldquoGlobal stabilization for linear continuous time-varying systemsrdquo Applied Mathematics and Computation vol175 no 2 pp 1730ndash1743 2006
[10] A Zevin and M Pinsky ldquoExponential stability and solutionbounds for systems with bounded nonlinearitiesrdquo IEEE Trans-actions on Automatic Control vol 48 no 10 pp 1799ndash18042003
[11] X-L Zhu and G-H Yang ldquoDelay-dependent stability criteriafor systems with differentiable time delaysrdquo Acta AutomaticaSinica vol 34 no 7 pp 765ndash771 2008
[12] M de la Sen and N S Luo ldquoOn the uniform exponentialstability of a wide class of linear time-delay systemsrdquo Journalof Mathematical Analysis and Applications vol 289 no 2 pp456ndash476 2004
[13] H R Karimi M Zapateiro and N Luo ldquoNew delay-dependentstability criteria for uncertain neutral systems with mixed time-varying delays and nonlinear perturbationsrdquo MathematicalProblems in Engineering vol 2009 Article ID 759248 22 pages2009
[14] J-Z Zhang Z Jin Q-X Liu and Z-Y Zhang ldquoAnalysis ofa delayed SIR model with nonlinear incidence raterdquo DiscreteDynamics in Nature and Society vol 2008 Article ID 63615316 pages 2008
[15] C Wei and L Chen ldquoA delayed epidemic model with pulsevaccinationrdquoDiscreteDynamics inNature and Society vol 2008Article ID 746951 12 pages 2008
[16] Y-H Fan and L-L Wang ldquoPermanence for a discrete modelwith feedback control and delayrdquo Discrete Dynamics in Natureand Society vol 2008 Article ID 945109 8 pages 2008
[17] Q Liu ldquoAlmost periodic solution of a diffusive mixed systemwith time delay and type III functional responserdquo Discrete
Dynamics in Nature and Society vol 2008 Article ID 70615413 pages 2008
[18] V N Phat and P T Nam ldquoExponential stability criteria of linearnon-autonomous systems withmultiple delaysrdquo Electronic Jour-nal of Differential Equations vol 2005 no 58 pp 1ndash8 2005
[19] M de la Sen and A Ibeas ldquoOn the global asymptotic stabilityof switched linear time-varying systems with constant pointdelaysrdquo Discrete Dynamics in Nature and Society vol 2008Article ID 231710 31 pages 2008
[20] Y Chen W Bi and Y Wu ldquoDelay-dependent exponentialstability for discrete-time BAM neural networks with time-varying delaysrdquo Discrete Dynamics in Nature and Society vol2008 Article ID 421614 14 pages 2008
[21] J H Park and O Kwon ldquoMatrix inequality approach to anovel stability criterion for time-delay systems with nonlinearuncertaintiesrdquo Journal of OptimizationTheory and Applicationsvol 126 no 3 pp 643ndash656 2005
[22] V N Phat and P Niamsup ldquoStability analysis for a class offunctional differential equations and applicationsrdquo NonlinearAnalysis Theory Methodsamp Applications vol 71 no 12 pp6265ndash6275 2009
[23] A Benabdallah I Ellouze and M A Hammami ldquoPracticalstability of nonlinear time-varying cascade systemsrdquo Journal ofDynamical and Control Systems vol 15 no 1 pp 45ndash62 2009
[24] V Lakshmikantham S Leela and A A Martynyuk PracticalStability of Nonlinear Systems World Scientific 1990
[25] V Lakshmikantham and Y Zhang ldquoStrict practical stability ofdelay differential equationrdquo Applied Mathematics and Compu-tation vol 118 no 2-3 pp 275ndash285 2001
[26] M Corless ldquoGuaranteed rates of exponential convergencefor uncertain systemsrdquo Journal of Optimization Theory andApplications vol 64 no 3 pp 481ndash494 1990
[27] M Corless and G Leitmann ldquoBounded controllers for robustexponential convergencerdquo Journal of Optimization Theory andApplications vol 76 no 1 pp 1ndash12 1993
Remark 4 If the delayed nonlinear disturbances are allowedto be of large size in the sense that the constants 120575
(sdot)2char-
acterizing the upper-bounding functions are large enoughin (6) then the radius 119903 of the closed ball 119861(0 119903) becomesaccordingly larger according to their values provided inthe statement of Theorem 3 That means that if the systemis globally exponentially practically stable then the radiusof the residual ball 119861(0 119903) increases as the constants 120575
(sdot)2
increase As a result then the uncertainty about how faris the state-trajectory solution from zero becomes larger asthose constants increase Thus to a larger disturbance itcorresponds to a larger uncertainty about the final deviationof the trajectory from the origin
Remark 5 On the other hand if the size of the nonlinearperturbations is allowed to be large in the sense that theconstants 120575
(sdot)1are large enough then there is trade-off
between the values of 1205811and the maximum matrix measure
120583(119860) of 119860 so as to ensure that 120578 gt 0 in Theorem 3 Howevernote that if the constant 120581
1is large then the constant 120598 is
requested to be accordingly large As a result 119860(119905) shouldhave a sufficiently large absolute stability abscissa for all timein order to compensate for the effects of the perturbationswhile satisfying the Lyapunov-like matrix equality (9)
Remark 6 Note also from Theorem 3 that the radius of theresidual ball 119861(0 119903) also increase with the squared upper-bounds of the delays and the squared differences betweenthose upper-bounds and the corresponding delay lower-bounds as well as on certain exponential functions of themaximum delay sizes
4 Examples
Example 1 Consider the nonautonomous system with non-linear time-delay perturbation (3) with time-varying delay1205911(119905) = 120591
1cos2(045119905)
1198911(119905 119909 (119905 minus 120591
1(119905)))
=[
[
minus120575 sin (119905) 1199092(119905 minus 1205911(119905)) + 120575
2lowast cos( 119905
1 + 1199092
1
)
120575 cos (119905) 1199091(119905 minus 1205911(119905))
]
]
(67)where 120591
1 120575 gt 0 120575
2ge 0 will be chosen later and
119860 (119905) = [
119886 (119905) 1
minus1 119886 (119905)
] (68)
where 119886(119905) = minus05 cos(119905)minus1011989010minussin(119905)minus51119890minus sin(119905)minus1119860119894(119905) = 0
and 119902119894= 0 for (119894 = 1 119898) By computation we obtain
120583(119860) asymp minus81033741
Let120573 = 001 120572 = 1 120583 = 09 ] = 11
1205911119867
= 1205911 120591
1119871= 0 997904rArr 120591
1119867119871= 1205911119867
1205751= 12057511= 120575 120575
12= 1205752
1205811= 11989010
120581211
= 0 120581221
= 0 120581231
= 11989010
120581241
= 0 997904rArr 12058121= 11989010
120581311
=
1
1205912
1119867exp (2120572120591
1119867)
120581321
= 0
997904rArr 12058131=
1
exp (21205721205911119867)
(69)
Then120598 = 2 (119901 + 120573) 120575
01+ 2120572120573 + 120581
21
+ 1205813111989021205721205911119867
+ 1205811= 102 + 2119890
10
gt 0
(70)
10 Mathematical Problems in Engineering
Table 1
Delay bound (1205911119867) 2 4 6 8 10
Perturbation bound 1205751
3465679 469028 63476 8591 1163Estimated value of 120574 2098897 2968276 3635376 4197767 4693245
We can verify that a solution 119875(119905) is given by
119875 (119905) =
[
[
[
[
119890sin(119905)
10
0
0
119890sin(119905)
10
]
]
]
]
(71)
We have 119901 = sup119905isinR+119875(119905) = 11989010
1205751lt
radic120581231119890minus21205721205911119867 (1 minus 120583) (120581
1minus 2120573120583 (119860))
(119901 + 120573)
1205811= 11989010
gt 2120573120583 (119860) = minus162067
120578 = 1205811minus 2120573120583 (119860) minus
(119901 + 120573)2
1205752
120581231119890minus21205721205911119867 (1 minus 120583)
gt 0
120578120577= (1 minus 120577
2
) (1205811minus 2120573120583 (119860)) gt 0 where 120575 = 120577120575
0 lt 120598 le minus10120573 + 2120572120573 + 1199021(120572 minus 5)
2
120572 = 1 120573 = 0001 1199021= 00010489
1205811= 002 gt minus10120573 = minus001
120581211
= 0 120581221
= 0
120581241
= 0 997904rArr 12058121= 120581231
= 0001282
120581311
= 016374 120581321
= 0 997904rArr 12058131= 00016374
11990111= 11990122= 000319
11990112= 000125 997904rArr 119901 = 000445
1205751= 002 lt 00206
120578 = 68310minus4
gt 0
120574 = radic
119901 + 120573 + 1205812311205911119867
+ 12058131111988811198671205913
1119867
120573
= 2416
with 1198881119867
=
2120572120591111986711989021205721205911119867
+ 119890minus21205721205911119867
minus 1
(21205721205911119867)2
119903 =
(119901 + 120573)119872
radic2120572120573120578
= 513 with 119872 = 1205752
(77)
in such a way that condition (12) in Theorem 3 is satisfiedThe result of the simulation of this example is depicted
in Figure 3 The evolution of states 1199091and 119909
2is given It is
shown in Figure 1 that the time-delay perturbed system is
0 1 2 3 4 5minus02
0
02
04
06
08
1
12
x1
x2
Solu
tion x
Time t
Figure 3 Convergence of solutions
globally uniformly practically exponentially stable toward aneighborhood of the origin
5 Conclusion
Based on improved Lyapunov-Krasovskii functional for per-turbed systems with time-varying delay we have presentednew sufficient conditions for global uniformly exponentialpractical stability toward a certain ball neighborhood of theorigin The perturbations are assumed to be nonlinear ingeneral with delayed contributions The delayed contribu-tions of such perturbations are not necessarily boundedwhilethey are upper-bounded by known nonnegative integrablefunctions which are linear functions of the various time-delayed state norms The point delays are assumed to beunknown bounded time-differentiable functions of timewithknown lower- and upper-bounds and known upper-boundsof their time-derivatives
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors wish to thank the editor and the anonymousreviewers for their valuable and careful comments The thirdauthor is grateful to the Spanish Ministry of Educationfor its partial support of this work through Grant DPIDPI2012-30651 and is also grateful to the BasqueGovernmentfor its support through Grants IT378-1 and SAIOTEK S-PE13UN039
12 Mathematical Problems in Engineering
References
[1] K Zhou and P P Khargonekar ldquoStability robustness boundsfor linear state-space models with structured uncertaintyrdquo IEEETransactions on Automatic Control vol 32 no 7 pp 621ndash6231987
[2] E Cheres Z J Palmor and S Gutman ldquoQuantitative measuresof robustness for systems including delayed perturbationsrdquoIEEE Transactions on Automatic Control vol 34 no 11 pp1203ndash1204 1989
[3] H Trinh and M Aldeen ldquoOn the stability of linear systemswith delayed perturbationsrdquo IEEE Transactions on AutomaticControl vol 39 no 9 pp 1948ndash1951 1994
[4] H Trinh and M Aldeen ldquoOn robustness and stabilizationof linear systems with delayed nonlinear perturbationsrdquo IEEETransactions on Automatic Control vol 42 no 7 pp 1005ndash10071997
[5] P Niamsup K Mukdasai and V N Phat ldquoImproved exponen-tial stability for time-varying systems with nonlinear delayedperturbationsrdquoAppliedMathematics andComputation vol 204no 1 pp 490ndash495 2008
[6] M Hammami and I Ellouze ldquoA robust detector for a class ofuncertain systemsrdquo Nonlinear Dynamics and Systems Theoryvol 8 no 4 pp 349ndash358 2008
[7] M de la Sen ldquoOn the stability of a certain class of linear time-varying systemsrdquoThe American Journal of Applied Sciences vol2 no 8 pp 1240ndash1245 2005
[8] M de la Sen ldquoRobust stability of a class of linear time-varying systemsrdquo IMA Journal of Mathematical Control andInformation vol 19 no 4 pp 399ndash418 2002
[9] V N Phat ldquoGlobal stabilization for linear continuous time-varying systemsrdquo Applied Mathematics and Computation vol175 no 2 pp 1730ndash1743 2006
[10] A Zevin and M Pinsky ldquoExponential stability and solutionbounds for systems with bounded nonlinearitiesrdquo IEEE Trans-actions on Automatic Control vol 48 no 10 pp 1799ndash18042003
[11] X-L Zhu and G-H Yang ldquoDelay-dependent stability criteriafor systems with differentiable time delaysrdquo Acta AutomaticaSinica vol 34 no 7 pp 765ndash771 2008
[12] M de la Sen and N S Luo ldquoOn the uniform exponentialstability of a wide class of linear time-delay systemsrdquo Journalof Mathematical Analysis and Applications vol 289 no 2 pp456ndash476 2004
[13] H R Karimi M Zapateiro and N Luo ldquoNew delay-dependentstability criteria for uncertain neutral systems with mixed time-varying delays and nonlinear perturbationsrdquo MathematicalProblems in Engineering vol 2009 Article ID 759248 22 pages2009
[14] J-Z Zhang Z Jin Q-X Liu and Z-Y Zhang ldquoAnalysis ofa delayed SIR model with nonlinear incidence raterdquo DiscreteDynamics in Nature and Society vol 2008 Article ID 63615316 pages 2008
[15] C Wei and L Chen ldquoA delayed epidemic model with pulsevaccinationrdquoDiscreteDynamics inNature and Society vol 2008Article ID 746951 12 pages 2008
[16] Y-H Fan and L-L Wang ldquoPermanence for a discrete modelwith feedback control and delayrdquo Discrete Dynamics in Natureand Society vol 2008 Article ID 945109 8 pages 2008
[17] Q Liu ldquoAlmost periodic solution of a diffusive mixed systemwith time delay and type III functional responserdquo Discrete
Dynamics in Nature and Society vol 2008 Article ID 70615413 pages 2008
[18] V N Phat and P T Nam ldquoExponential stability criteria of linearnon-autonomous systems withmultiple delaysrdquo Electronic Jour-nal of Differential Equations vol 2005 no 58 pp 1ndash8 2005
[19] M de la Sen and A Ibeas ldquoOn the global asymptotic stabilityof switched linear time-varying systems with constant pointdelaysrdquo Discrete Dynamics in Nature and Society vol 2008Article ID 231710 31 pages 2008
[20] Y Chen W Bi and Y Wu ldquoDelay-dependent exponentialstability for discrete-time BAM neural networks with time-varying delaysrdquo Discrete Dynamics in Nature and Society vol2008 Article ID 421614 14 pages 2008
[21] J H Park and O Kwon ldquoMatrix inequality approach to anovel stability criterion for time-delay systems with nonlinearuncertaintiesrdquo Journal of OptimizationTheory and Applicationsvol 126 no 3 pp 643ndash656 2005
[22] V N Phat and P Niamsup ldquoStability analysis for a class offunctional differential equations and applicationsrdquo NonlinearAnalysis Theory Methodsamp Applications vol 71 no 12 pp6265ndash6275 2009
[23] A Benabdallah I Ellouze and M A Hammami ldquoPracticalstability of nonlinear time-varying cascade systemsrdquo Journal ofDynamical and Control Systems vol 15 no 1 pp 45ndash62 2009
[24] V Lakshmikantham S Leela and A A Martynyuk PracticalStability of Nonlinear Systems World Scientific 1990
[25] V Lakshmikantham and Y Zhang ldquoStrict practical stability ofdelay differential equationrdquo Applied Mathematics and Compu-tation vol 118 no 2-3 pp 275ndash285 2001
[26] M Corless ldquoGuaranteed rates of exponential convergencefor uncertain systemsrdquo Journal of Optimization Theory andApplications vol 64 no 3 pp 481ndash494 1990
[27] M Corless and G Leitmann ldquoBounded controllers for robustexponential convergencerdquo Journal of Optimization Theory andApplications vol 76 no 1 pp 1ndash12 1993
0 lt 120598 le minus10120573 + 2120572120573 + 1199021(120572 minus 5)
2
120572 = 1 120573 = 0001 1199021= 00010489
1205811= 002 gt minus10120573 = minus001
120581211
= 0 120581221
= 0
120581241
= 0 997904rArr 12058121= 120581231
= 0001282
120581311
= 016374 120581321
= 0 997904rArr 12058131= 00016374
11990111= 11990122= 000319
11990112= 000125 997904rArr 119901 = 000445
1205751= 002 lt 00206
120578 = 68310minus4
gt 0
120574 = radic
119901 + 120573 + 1205812311205911119867
+ 12058131111988811198671205913
1119867
120573
= 2416
with 1198881119867
=
2120572120591111986711989021205721205911119867
+ 119890minus21205721205911119867
minus 1
(21205721205911119867)2
119903 =
(119901 + 120573)119872
radic2120572120573120578
= 513 with 119872 = 1205752
(77)
in such a way that condition (12) in Theorem 3 is satisfiedThe result of the simulation of this example is depicted
in Figure 3 The evolution of states 1199091and 119909
2is given It is
shown in Figure 1 that the time-delay perturbed system is
0 1 2 3 4 5minus02
0
02
04
06
08
1
12
x1
x2
Solu
tion x
Time t
Figure 3 Convergence of solutions
globally uniformly practically exponentially stable toward aneighborhood of the origin
5 Conclusion
Based on improved Lyapunov-Krasovskii functional for per-turbed systems with time-varying delay we have presentednew sufficient conditions for global uniformly exponentialpractical stability toward a certain ball neighborhood of theorigin The perturbations are assumed to be nonlinear ingeneral with delayed contributions The delayed contribu-tions of such perturbations are not necessarily boundedwhilethey are upper-bounded by known nonnegative integrablefunctions which are linear functions of the various time-delayed state norms The point delays are assumed to beunknown bounded time-differentiable functions of timewithknown lower- and upper-bounds and known upper-boundsof their time-derivatives
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors wish to thank the editor and the anonymousreviewers for their valuable and careful comments The thirdauthor is grateful to the Spanish Ministry of Educationfor its partial support of this work through Grant DPIDPI2012-30651 and is also grateful to the BasqueGovernmentfor its support through Grants IT378-1 and SAIOTEK S-PE13UN039
12 Mathematical Problems in Engineering
References
[1] K Zhou and P P Khargonekar ldquoStability robustness boundsfor linear state-space models with structured uncertaintyrdquo IEEETransactions on Automatic Control vol 32 no 7 pp 621ndash6231987
[2] E Cheres Z J Palmor and S Gutman ldquoQuantitative measuresof robustness for systems including delayed perturbationsrdquoIEEE Transactions on Automatic Control vol 34 no 11 pp1203ndash1204 1989
[3] H Trinh and M Aldeen ldquoOn the stability of linear systemswith delayed perturbationsrdquo IEEE Transactions on AutomaticControl vol 39 no 9 pp 1948ndash1951 1994
[4] H Trinh and M Aldeen ldquoOn robustness and stabilizationof linear systems with delayed nonlinear perturbationsrdquo IEEETransactions on Automatic Control vol 42 no 7 pp 1005ndash10071997
[5] P Niamsup K Mukdasai and V N Phat ldquoImproved exponen-tial stability for time-varying systems with nonlinear delayedperturbationsrdquoAppliedMathematics andComputation vol 204no 1 pp 490ndash495 2008
[6] M Hammami and I Ellouze ldquoA robust detector for a class ofuncertain systemsrdquo Nonlinear Dynamics and Systems Theoryvol 8 no 4 pp 349ndash358 2008
[7] M de la Sen ldquoOn the stability of a certain class of linear time-varying systemsrdquoThe American Journal of Applied Sciences vol2 no 8 pp 1240ndash1245 2005
[8] M de la Sen ldquoRobust stability of a class of linear time-varying systemsrdquo IMA Journal of Mathematical Control andInformation vol 19 no 4 pp 399ndash418 2002
[9] V N Phat ldquoGlobal stabilization for linear continuous time-varying systemsrdquo Applied Mathematics and Computation vol175 no 2 pp 1730ndash1743 2006
[10] A Zevin and M Pinsky ldquoExponential stability and solutionbounds for systems with bounded nonlinearitiesrdquo IEEE Trans-actions on Automatic Control vol 48 no 10 pp 1799ndash18042003
[11] X-L Zhu and G-H Yang ldquoDelay-dependent stability criteriafor systems with differentiable time delaysrdquo Acta AutomaticaSinica vol 34 no 7 pp 765ndash771 2008
[12] M de la Sen and N S Luo ldquoOn the uniform exponentialstability of a wide class of linear time-delay systemsrdquo Journalof Mathematical Analysis and Applications vol 289 no 2 pp456ndash476 2004
[13] H R Karimi M Zapateiro and N Luo ldquoNew delay-dependentstability criteria for uncertain neutral systems with mixed time-varying delays and nonlinear perturbationsrdquo MathematicalProblems in Engineering vol 2009 Article ID 759248 22 pages2009
[14] J-Z Zhang Z Jin Q-X Liu and Z-Y Zhang ldquoAnalysis ofa delayed SIR model with nonlinear incidence raterdquo DiscreteDynamics in Nature and Society vol 2008 Article ID 63615316 pages 2008
[15] C Wei and L Chen ldquoA delayed epidemic model with pulsevaccinationrdquoDiscreteDynamics inNature and Society vol 2008Article ID 746951 12 pages 2008
[16] Y-H Fan and L-L Wang ldquoPermanence for a discrete modelwith feedback control and delayrdquo Discrete Dynamics in Natureand Society vol 2008 Article ID 945109 8 pages 2008
[17] Q Liu ldquoAlmost periodic solution of a diffusive mixed systemwith time delay and type III functional responserdquo Discrete
Dynamics in Nature and Society vol 2008 Article ID 70615413 pages 2008
[18] V N Phat and P T Nam ldquoExponential stability criteria of linearnon-autonomous systems withmultiple delaysrdquo Electronic Jour-nal of Differential Equations vol 2005 no 58 pp 1ndash8 2005
[19] M de la Sen and A Ibeas ldquoOn the global asymptotic stabilityof switched linear time-varying systems with constant pointdelaysrdquo Discrete Dynamics in Nature and Society vol 2008Article ID 231710 31 pages 2008
[20] Y Chen W Bi and Y Wu ldquoDelay-dependent exponentialstability for discrete-time BAM neural networks with time-varying delaysrdquo Discrete Dynamics in Nature and Society vol2008 Article ID 421614 14 pages 2008
[21] J H Park and O Kwon ldquoMatrix inequality approach to anovel stability criterion for time-delay systems with nonlinearuncertaintiesrdquo Journal of OptimizationTheory and Applicationsvol 126 no 3 pp 643ndash656 2005
[22] V N Phat and P Niamsup ldquoStability analysis for a class offunctional differential equations and applicationsrdquo NonlinearAnalysis Theory Methodsamp Applications vol 71 no 12 pp6265ndash6275 2009
[23] A Benabdallah I Ellouze and M A Hammami ldquoPracticalstability of nonlinear time-varying cascade systemsrdquo Journal ofDynamical and Control Systems vol 15 no 1 pp 45ndash62 2009
[24] V Lakshmikantham S Leela and A A Martynyuk PracticalStability of Nonlinear Systems World Scientific 1990
[25] V Lakshmikantham and Y Zhang ldquoStrict practical stability ofdelay differential equationrdquo Applied Mathematics and Compu-tation vol 118 no 2-3 pp 275ndash285 2001
[26] M Corless ldquoGuaranteed rates of exponential convergencefor uncertain systemsrdquo Journal of Optimization Theory andApplications vol 64 no 3 pp 481ndash494 1990
[27] M Corless and G Leitmann ldquoBounded controllers for robustexponential convergencerdquo Journal of Optimization Theory andApplications vol 76 no 1 pp 1ndash12 1993
0 lt 120598 le minus10120573 + 2120572120573 + 1199021(120572 minus 5)
2
120572 = 1 120573 = 0001 1199021= 00010489
1205811= 002 gt minus10120573 = minus001
120581211
= 0 120581221
= 0
120581241
= 0 997904rArr 12058121= 120581231
= 0001282
120581311
= 016374 120581321
= 0 997904rArr 12058131= 00016374
11990111= 11990122= 000319
11990112= 000125 997904rArr 119901 = 000445
1205751= 002 lt 00206
120578 = 68310minus4
gt 0
120574 = radic
119901 + 120573 + 1205812311205911119867
+ 12058131111988811198671205913
1119867
120573
= 2416
with 1198881119867
=
2120572120591111986711989021205721205911119867
+ 119890minus21205721205911119867
minus 1
(21205721205911119867)2
119903 =
(119901 + 120573)119872
radic2120572120573120578
= 513 with 119872 = 1205752
(77)
in such a way that condition (12) in Theorem 3 is satisfiedThe result of the simulation of this example is depicted
in Figure 3 The evolution of states 1199091and 119909
2is given It is
shown in Figure 1 that the time-delay perturbed system is
0 1 2 3 4 5minus02
0
02
04
06
08
1
12
x1
x2
Solu
tion x
Time t
Figure 3 Convergence of solutions
globally uniformly practically exponentially stable toward aneighborhood of the origin
5 Conclusion
Based on improved Lyapunov-Krasovskii functional for per-turbed systems with time-varying delay we have presentednew sufficient conditions for global uniformly exponentialpractical stability toward a certain ball neighborhood of theorigin The perturbations are assumed to be nonlinear ingeneral with delayed contributions The delayed contribu-tions of such perturbations are not necessarily boundedwhilethey are upper-bounded by known nonnegative integrablefunctions which are linear functions of the various time-delayed state norms The point delays are assumed to beunknown bounded time-differentiable functions of timewithknown lower- and upper-bounds and known upper-boundsof their time-derivatives
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors wish to thank the editor and the anonymousreviewers for their valuable and careful comments The thirdauthor is grateful to the Spanish Ministry of Educationfor its partial support of this work through Grant DPIDPI2012-30651 and is also grateful to the BasqueGovernmentfor its support through Grants IT378-1 and SAIOTEK S-PE13UN039
12 Mathematical Problems in Engineering
References
[1] K Zhou and P P Khargonekar ldquoStability robustness boundsfor linear state-space models with structured uncertaintyrdquo IEEETransactions on Automatic Control vol 32 no 7 pp 621ndash6231987
[2] E Cheres Z J Palmor and S Gutman ldquoQuantitative measuresof robustness for systems including delayed perturbationsrdquoIEEE Transactions on Automatic Control vol 34 no 11 pp1203ndash1204 1989
[3] H Trinh and M Aldeen ldquoOn the stability of linear systemswith delayed perturbationsrdquo IEEE Transactions on AutomaticControl vol 39 no 9 pp 1948ndash1951 1994
[4] H Trinh and M Aldeen ldquoOn robustness and stabilizationof linear systems with delayed nonlinear perturbationsrdquo IEEETransactions on Automatic Control vol 42 no 7 pp 1005ndash10071997
[5] P Niamsup K Mukdasai and V N Phat ldquoImproved exponen-tial stability for time-varying systems with nonlinear delayedperturbationsrdquoAppliedMathematics andComputation vol 204no 1 pp 490ndash495 2008
[6] M Hammami and I Ellouze ldquoA robust detector for a class ofuncertain systemsrdquo Nonlinear Dynamics and Systems Theoryvol 8 no 4 pp 349ndash358 2008
[7] M de la Sen ldquoOn the stability of a certain class of linear time-varying systemsrdquoThe American Journal of Applied Sciences vol2 no 8 pp 1240ndash1245 2005
[8] M de la Sen ldquoRobust stability of a class of linear time-varying systemsrdquo IMA Journal of Mathematical Control andInformation vol 19 no 4 pp 399ndash418 2002
[9] V N Phat ldquoGlobal stabilization for linear continuous time-varying systemsrdquo Applied Mathematics and Computation vol175 no 2 pp 1730ndash1743 2006
[10] A Zevin and M Pinsky ldquoExponential stability and solutionbounds for systems with bounded nonlinearitiesrdquo IEEE Trans-actions on Automatic Control vol 48 no 10 pp 1799ndash18042003
[11] X-L Zhu and G-H Yang ldquoDelay-dependent stability criteriafor systems with differentiable time delaysrdquo Acta AutomaticaSinica vol 34 no 7 pp 765ndash771 2008
[12] M de la Sen and N S Luo ldquoOn the uniform exponentialstability of a wide class of linear time-delay systemsrdquo Journalof Mathematical Analysis and Applications vol 289 no 2 pp456ndash476 2004
[13] H R Karimi M Zapateiro and N Luo ldquoNew delay-dependentstability criteria for uncertain neutral systems with mixed time-varying delays and nonlinear perturbationsrdquo MathematicalProblems in Engineering vol 2009 Article ID 759248 22 pages2009
[14] J-Z Zhang Z Jin Q-X Liu and Z-Y Zhang ldquoAnalysis ofa delayed SIR model with nonlinear incidence raterdquo DiscreteDynamics in Nature and Society vol 2008 Article ID 63615316 pages 2008
[15] C Wei and L Chen ldquoA delayed epidemic model with pulsevaccinationrdquoDiscreteDynamics inNature and Society vol 2008Article ID 746951 12 pages 2008
[16] Y-H Fan and L-L Wang ldquoPermanence for a discrete modelwith feedback control and delayrdquo Discrete Dynamics in Natureand Society vol 2008 Article ID 945109 8 pages 2008
[17] Q Liu ldquoAlmost periodic solution of a diffusive mixed systemwith time delay and type III functional responserdquo Discrete
Dynamics in Nature and Society vol 2008 Article ID 70615413 pages 2008
[18] V N Phat and P T Nam ldquoExponential stability criteria of linearnon-autonomous systems withmultiple delaysrdquo Electronic Jour-nal of Differential Equations vol 2005 no 58 pp 1ndash8 2005
[19] M de la Sen and A Ibeas ldquoOn the global asymptotic stabilityof switched linear time-varying systems with constant pointdelaysrdquo Discrete Dynamics in Nature and Society vol 2008Article ID 231710 31 pages 2008
[20] Y Chen W Bi and Y Wu ldquoDelay-dependent exponentialstability for discrete-time BAM neural networks with time-varying delaysrdquo Discrete Dynamics in Nature and Society vol2008 Article ID 421614 14 pages 2008
[21] J H Park and O Kwon ldquoMatrix inequality approach to anovel stability criterion for time-delay systems with nonlinearuncertaintiesrdquo Journal of OptimizationTheory and Applicationsvol 126 no 3 pp 643ndash656 2005
[22] V N Phat and P Niamsup ldquoStability analysis for a class offunctional differential equations and applicationsrdquo NonlinearAnalysis Theory Methodsamp Applications vol 71 no 12 pp6265ndash6275 2009
[23] A Benabdallah I Ellouze and M A Hammami ldquoPracticalstability of nonlinear time-varying cascade systemsrdquo Journal ofDynamical and Control Systems vol 15 no 1 pp 45ndash62 2009
[24] V Lakshmikantham S Leela and A A Martynyuk PracticalStability of Nonlinear Systems World Scientific 1990
[25] V Lakshmikantham and Y Zhang ldquoStrict practical stability ofdelay differential equationrdquo Applied Mathematics and Compu-tation vol 118 no 2-3 pp 275ndash285 2001
[26] M Corless ldquoGuaranteed rates of exponential convergencefor uncertain systemsrdquo Journal of Optimization Theory andApplications vol 64 no 3 pp 481ndash494 1990
[27] M Corless and G Leitmann ldquoBounded controllers for robustexponential convergencerdquo Journal of Optimization Theory andApplications vol 76 no 1 pp 1ndash12 1993
[1] K Zhou and P P Khargonekar ldquoStability robustness boundsfor linear state-space models with structured uncertaintyrdquo IEEETransactions on Automatic Control vol 32 no 7 pp 621ndash6231987
[2] E Cheres Z J Palmor and S Gutman ldquoQuantitative measuresof robustness for systems including delayed perturbationsrdquoIEEE Transactions on Automatic Control vol 34 no 11 pp1203ndash1204 1989
[3] H Trinh and M Aldeen ldquoOn the stability of linear systemswith delayed perturbationsrdquo IEEE Transactions on AutomaticControl vol 39 no 9 pp 1948ndash1951 1994
[4] H Trinh and M Aldeen ldquoOn robustness and stabilizationof linear systems with delayed nonlinear perturbationsrdquo IEEETransactions on Automatic Control vol 42 no 7 pp 1005ndash10071997
[5] P Niamsup K Mukdasai and V N Phat ldquoImproved exponen-tial stability for time-varying systems with nonlinear delayedperturbationsrdquoAppliedMathematics andComputation vol 204no 1 pp 490ndash495 2008
[6] M Hammami and I Ellouze ldquoA robust detector for a class ofuncertain systemsrdquo Nonlinear Dynamics and Systems Theoryvol 8 no 4 pp 349ndash358 2008
[7] M de la Sen ldquoOn the stability of a certain class of linear time-varying systemsrdquoThe American Journal of Applied Sciences vol2 no 8 pp 1240ndash1245 2005
[8] M de la Sen ldquoRobust stability of a class of linear time-varying systemsrdquo IMA Journal of Mathematical Control andInformation vol 19 no 4 pp 399ndash418 2002
[9] V N Phat ldquoGlobal stabilization for linear continuous time-varying systemsrdquo Applied Mathematics and Computation vol175 no 2 pp 1730ndash1743 2006
[10] A Zevin and M Pinsky ldquoExponential stability and solutionbounds for systems with bounded nonlinearitiesrdquo IEEE Trans-actions on Automatic Control vol 48 no 10 pp 1799ndash18042003
[11] X-L Zhu and G-H Yang ldquoDelay-dependent stability criteriafor systems with differentiable time delaysrdquo Acta AutomaticaSinica vol 34 no 7 pp 765ndash771 2008
[12] M de la Sen and N S Luo ldquoOn the uniform exponentialstability of a wide class of linear time-delay systemsrdquo Journalof Mathematical Analysis and Applications vol 289 no 2 pp456ndash476 2004
[13] H R Karimi M Zapateiro and N Luo ldquoNew delay-dependentstability criteria for uncertain neutral systems with mixed time-varying delays and nonlinear perturbationsrdquo MathematicalProblems in Engineering vol 2009 Article ID 759248 22 pages2009
[14] J-Z Zhang Z Jin Q-X Liu and Z-Y Zhang ldquoAnalysis ofa delayed SIR model with nonlinear incidence raterdquo DiscreteDynamics in Nature and Society vol 2008 Article ID 63615316 pages 2008
[15] C Wei and L Chen ldquoA delayed epidemic model with pulsevaccinationrdquoDiscreteDynamics inNature and Society vol 2008Article ID 746951 12 pages 2008
[16] Y-H Fan and L-L Wang ldquoPermanence for a discrete modelwith feedback control and delayrdquo Discrete Dynamics in Natureand Society vol 2008 Article ID 945109 8 pages 2008
[17] Q Liu ldquoAlmost periodic solution of a diffusive mixed systemwith time delay and type III functional responserdquo Discrete
Dynamics in Nature and Society vol 2008 Article ID 70615413 pages 2008
[18] V N Phat and P T Nam ldquoExponential stability criteria of linearnon-autonomous systems withmultiple delaysrdquo Electronic Jour-nal of Differential Equations vol 2005 no 58 pp 1ndash8 2005
[19] M de la Sen and A Ibeas ldquoOn the global asymptotic stabilityof switched linear time-varying systems with constant pointdelaysrdquo Discrete Dynamics in Nature and Society vol 2008Article ID 231710 31 pages 2008
[20] Y Chen W Bi and Y Wu ldquoDelay-dependent exponentialstability for discrete-time BAM neural networks with time-varying delaysrdquo Discrete Dynamics in Nature and Society vol2008 Article ID 421614 14 pages 2008
[21] J H Park and O Kwon ldquoMatrix inequality approach to anovel stability criterion for time-delay systems with nonlinearuncertaintiesrdquo Journal of OptimizationTheory and Applicationsvol 126 no 3 pp 643ndash656 2005
[22] V N Phat and P Niamsup ldquoStability analysis for a class offunctional differential equations and applicationsrdquo NonlinearAnalysis Theory Methodsamp Applications vol 71 no 12 pp6265ndash6275 2009
[23] A Benabdallah I Ellouze and M A Hammami ldquoPracticalstability of nonlinear time-varying cascade systemsrdquo Journal ofDynamical and Control Systems vol 15 no 1 pp 45ndash62 2009
[24] V Lakshmikantham S Leela and A A Martynyuk PracticalStability of Nonlinear Systems World Scientific 1990
[25] V Lakshmikantham and Y Zhang ldquoStrict practical stability ofdelay differential equationrdquo Applied Mathematics and Compu-tation vol 118 no 2-3 pp 275ndash285 2001
[26] M Corless ldquoGuaranteed rates of exponential convergencefor uncertain systemsrdquo Journal of Optimization Theory andApplications vol 64 no 3 pp 481ndash494 1990
[27] M Corless and G Leitmann ldquoBounded controllers for robustexponential convergencerdquo Journal of Optimization Theory andApplications vol 76 no 1 pp 1ndash12 1993