Stability of Two Weakly Coupled Elastic Beams with ...

Research ArticleStability of Two Weakly Coupled Elastic Beams with PartiallyLocal Damping

Caihong Zhang 1 Yinuo Huang 2 Licheng Wang 3 Chongxiong Duan4

Tiezhu Zhang5 and Kai Wang 2

1College of Automation Qingdao University Qingdao 266071 China2School of Electrical Engineering Qingdao University Qingdao 266071 China3School of Information Engineering Zhejiang University of Technology Hangzhou 310023 China4School of Materials Science and Energy Engineering Foshan University Foshan 528231 China5Shandong University of Technology Zibo 255000 China

Correspondence should be addressed to Caihong Zhang rainbow823163com and Kai Wang wkwj888163com

Received 18 November 2019 Accepted 4 February 2020 Published 6 May 2020

Academic Editor Kishin Sadarangani

Copyright copy 2020 Caihong Zhang et al is 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 isproperly cited

In this paper the stability of two weakly coupled elastic beams connected vertically by a spring is investigated via the frequencydomain method and the multiplier technique When the two beams have partially local damping the operator A is obtained viavariable conversion and it generating a semigroup is proved then we obtain that the semigroup is exponentially stable byreduction to absurdity

1 Introduction

Artificial intelligence (AI) has been around and penetratedinto all fields such as research production and life [1ndash5]Scientists pay more attention to energy materials and en-vironment [6ndash10] e stability study of the coupling controlsystem in space vehicles is one of the important researchsubjects in the control field in recent years Much attentionhas been paid to research the stability of control systems usingsemigroup theory [11] Reference [12] is for coupled heat-wave system References [13 14] are for wave equationsReference [15] is for second-order hyperbolic operatorsReferences [16ndash19] are about polynomial stability of systemsand references [20ndash33] are about exponential stability eauthors of [33ndash35] considered weakly coupled evolutionequations of wave-Petrowsky wave-wave and Kirch-hoffndashPetrowsky for its asymptotic stability and boundarycontrollability e case of strongly coupled system wasstudied by Lasiecka [36] and she obtained the strong stabilityfor the open-loop systems with polynomial energy decay rate

A viscoelastic microcomposite beam reinforced byvarious distributions of boron nitride nanotubes with initialgeometrical imperfection has been described in [37] and thenonlinear static buckling and vibration are analyzed byusing the finite element method e bending and dynamicbehavior of functionally graded plates resting on visco-Pasternak foundations is studied in [38] Using a simplequasi-3D hyperbolic theory the dynamic behavior offunctionally graded plates is concerned in [39] Using ahyperbolic shear deformation theory the static and dynamicbehaviors of functionally graded beams is studied in [40] Adynamic study of functionally graded plates resting onelastic foundation is considered in [41] ermomechanicalanalysis of functionally graded sandwich plates resting on atwo-parameter elastic foundation is studied in [42] e freevibration of FG plates resting on elastic foundations ismodeled by two-dimensional (2D) and quasi-three-di-mensional (quasi-3D) shear deformation theories in [43]Input-to-state stability for a class of discrete-time nonlinearinput-saturated switched descriptor systems with unstable

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 7169526 9 pageshttpsdoiorg10115520207169526

subsystems is discussed in [44] In [45] nonlinear dynamicbehavior of the winding hoisting rope under head sheaveaxial wobbles is concerned A dynamic model of a minehoisting system with constant length and variable length isanalyzed in [46] Liu and Rao [47] studied the stability of aweakly coupled and partially damped system ey obtaineda sharp polynomial decay rate when compared with Alabau-Boussouirarsquos results Recently they also obtained the exactboundary controllability of this system with the controlacted only on one equatione Timoshenko beam equationwith locally distributed KelvinndashVoigt damping is consideredin [48] A wave equation with local KelvinndashVoigt damping isproposed in [49] and the semigroup corresponding to thesystem is eventually differentiable e behavior of slow orexponential decay is analyzed about elastic material withvoids in [50] e stability of an elastic string system withlocal KelvinndashVoigt damping is studied in [51]

e above research leads us to study a new problemConsider a system of two beams connected vertically by aspring In engineering construction there are elastic beamseverywhere and it is of great significance to study the elasticbeam system If two beams subject to local damping or onlyone beam subjects to local damping what is the energy decayrate of every system e coupling terms are local whichmodel the location of the spring in this case Such a couplingis even weaker than the one studied by Liu and Rao Hencefinding the energy decay rate is a challenge For readerrsquosconvenience we include these two frequency domainconditions here e first one is the frequency domaincharacterization of exponential decay which was obtainedby Prss [52] e second one is of polynomial energy decayrate for a C0-semigroup of contraction which was obtainedby Liu and Rao [53]

In this paper the stability of weakly coupled elastic beamsystem with damping control by using the semigroup the-oretical frequency domain multiplier method is studied Byvariable conversion the elastic beam control system istransformed into first-order evolution equations and a linearoperator is obtained and the linear operator-producingsemigroup is proved When the two beams have localdamping control using reduction to absurdity from thelocal dissipation to the global dissipation the exponentialstability of the semigroup generated from the linear operatoris proved e method in this paper can be employed tohandle other elastic beam systems in the future

In this paper we need some definitions and lemmaswhich are as follows

Definition 1 (see [19]) Let H be a real or complex Hilbertspace and we define ( ) is the inner product of H and middot isthe norm ofH LetA be a dense linear operator onH thatis D(A)subeH⟶H then A is dissipative and for anyx isin D(A) we get Re(Ax x)le 0

Definition 2 (see [19]) eAt is exponentially stable if there arenormal numbers M and α which make

eAt

leMe

minus αt foralltge 0 (1)

Lemma 1 (see [19]) Linear operator A can generate C0semigroup S(t) on Hilbert spaceH if it satisfies the following

(1) D(A) is dense on Hilbert space H(2) A is dissipative(3) 0 isin ρ(A)

Lemma 2 (see [19]) AC0 semigroup eAt of contractions on aHilbert space H is exponentially stable if and only if

ρ(A)supe iβ β isin R1113864 1113865 equiv iR

lim|β|⟶+infin

(iβI minus A)minus 1

lt +infin (2)

2 Model Description

Consider the system of two beams connected vertically by aspring When both upper and lower beams have localdamping control e physical model of weakly coupledelastic beam control system is given in Figure 1

e system is governed by the following equations

ρ1utt minus a1uxxxx minus k(x)(u minus y) minus δk(x)ut (3)

ρ2ytt minus a2yxxxx + k(x)(u minus y) minus δk(x)yt (4)

where u and y are the displacement of upper and lowerbeams x isin (0 l) and t isin [0 +infin) ρ1 ρ2 a1 and a2 arepositive physical constants k(x)ge 0 in (a b) sub (0 l) is alocally supported smooth function which represents theposition and elasticity of the spring We assume k(x) isin C2and there exists a constant c such that |kPrime(x)|le ck(x) and|kprime(x)|le ck(x) δ gt 0 is the damping coefficient of thesystem

e two beams (3) and (4) have local damping and theirboundary conditions are

uxx(0 t) uxx(l t) uxxx(0 t) uxxx(l t) 0

y(0 t) y(l t) yx(0 t) yx(l t) 0(5)

To convert the system into a first-order evolutionequation here we denote

v ut

w yt(6)

and the state variable vector is z equiv z(t) (u v y w)T thensystems (3) and (4) can be rewritten as the following form

ut v

vt 1ρ1

minus a1D4u minus k(x)(u minus y) minus δk(x)v1113960 1113961

yt w

wt 1ρ2

minus a2D4y + k(x)(u minus y) minus δk(x)w1113960 1113961

(7)

2 Mathematical Problems in Engineering

Here we have used the notation Di zizxi and thestate space is

H H2(0 l) times L

2(0 l) times H

20(0 l) times L

2(0 l) (8)

e Hilbert space H is equipped with the inner productwhich induces the energy norm

z2H a1 uPrime

2

+ a2 yPrime

2

+ ρ1v2

+ ρ2w2

+ k12

(x)(u minus y)

2

(9)

Here and after middot prime and langmiddot middotrang denote the L2(0 l) normderivative and inner product respectively

Define a linear operator A H⟶H by

A

0 I 0 0

minusa1D

4 + k(x)

ρ1minusδk(x)

ρ1

k(x)

ρ10

0 0 0 I

k(x)

ρ20 minus

a2D4 + k(x)

ρ2minusδk(x)

ρ2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(10)

with

D(A) z isinH | u y isin H4 v isin H

2 w isin H

20 uPrime isin H

201113966 1113967

(11)

us equations (3) and (4) are transformed into a first-order evolution on the Hilbert space

dzdt

Az z(0) z0 (12)

3 Main Result

Theorem 1 e operatorA generates aC0 semigroup S(t) ofcontractions on H

Proof It is clear that D(A) is dense in H By a straightforward calculation

RelangAz zrangH a1 1113946l

0vPrimeuPrimedx + 1113946

l

0minus a1D

4u minus k(x)(u minus y) minus δk(x)v1113960 1113961v dx

+ a2 1113946l

0yPrimewPrimedx + 1113946

l

0minus a2D

4y + k(x)(u minus y) minus δk(x)w1113960 1113961w dx

minus δ k12

(x)v

2

+ k12

(x)w

2

1113874 1113875le 0

(13)

Hence A is dissipative It is easy to show that for anyF (f1 middot middot middot f4)

T isin H

Az F (14)

has unique solution z isin D(A) and

F2H a1 f1Prime

2

+ a2 f3Prime

2

+ ρ1 f2

2

+ ρ2 f4

2

+ k12

(x) f1 minus f3( 1113857

2

(15)

In fact from the first and third equations of (14) we getv f1 isinH

2 andw f3 isinH20

Substituting them into the second and fourth equationsin (14) we have

minus a1D4u minus k(x)(u minus y) minus δk(x)f1 ρ1f2 (16)

minus a2D4y + k(x)(u minus y) minus δk(x)f3 ρ2f4 (17)

Taking the inner product with (16) and minus u and with (17)and minus y respectively we obtain

a1langD4u urang +langk(x)(u minus y) urang + δlangk(x)f1 urang ρ1langf2 minus urang

(18)

a2langD4yyrang minus langk(x)(u minus y)yrang +δlangk(x)f3yrang ρ2langf4 minus yrang

(19)

Figure 1 e figure of two weakly coupled elastic beams

Mathematical Problems in Engineering 3

Suppose there exist infinitesimal constants ε ε1 and ε2and

ε min ε1 ε21113864 1113865 (20)

Taking the boundary conditions to (18) and (19) we have

a1 uPrime

2

+langk(x)(u minus y) urang + δlangk(x)f1 urang

ρ1langf2 minus urang le ρ1 f2

uleρ12ε1

f2

2

+ρ1ε12

u2

(21)

a2 yPrime

2

minus langk(x)(u minus y) yrang + δlangk(x)f3 yrang

ρ2langf4 minus yrang le ρ2 f4

yleρ22ε2

f4

2

+ρ2ε22

y2

(22)

Adding (21) and (22) and because ε is an infinitesimalconstant we have

a1 uPrime

2

+ a2 yPrime

2

+ k12

(x)(u minus y)

2

+ δlangk(x)f1 urang

+ δlangk(x)f3 yrang

leρ12ε

f2

2

+ρ22ε

f4

2

(23)

Also we have

δlangk(x)f1 urang leδ2ε

f1

2

+δε2

k(x)u2 (24)

δlangk(x)f3 yrang leδ2ε

f3

2

+δε2

k(x)y2 (25)

Suppose there exist positive constants c1 c2 c3 c4 and C

independent of z and F and

C max c1 c2 c3 c41113864 1113865 (26)

by Poincarersquos inequalities we have

ρ1v2 le c1a1 f1Prime

2 (27)

ρ2w2 le c2a2 f3Prime

2 (28)

Combining (23)ndash(28) yields

a1 uPrime

2

+ a2 yPrime

2

+ k12

(x)(u minus y)

2

+ ρ1v2

+ ρ2w2

+ δlangk(x)f1 urang + δlangk(x)f3 yrang

le c3 a1 f1Prime

2

+ a2 f3Prime

2

+ ρ1 f2

2

+ ρ2 f4

2

1113874 1113875

(29)

By Poincarersquos inequalities (29) yields

z2H le c4F

2H minus δlangk(x)f1 urang minus δlangk(x)f3 yrang (30)

en by (24) (25) and Poincarersquos inequalities we have

z2H leCF

2H (31)

thus 0 isin ρ(A) By Lemma 1 the proof is completed

Theorem 2 e semigroup S(t) generated by the operatorA defined in (10) is exponentially stable ie there exist twopositive constants α andM such that

S(t) leMeminus αt

foralltgt 0 (32)

Proof By Lemma 2 it suffices to verify

ρ(A)supe iβ β isin R1113864 1113865 equiv iR (33)

lim|β|⟶+infin

(iβI minus A)minus 1

lt +infin (34)

We use reduction to absurdity to prove (33) If (33) isfalse then there exists β isin R and βne 0 iβ is the spectral pointof A Because Aminus 1 is dense iβ is the eigenvalue of operatorA then there exists vector

z (u v y w)T isin D(A) zH 1 (35)

such that

(iβI minus A)zH 0 (36)

ie

iβu minus v 0 inH2 (37)

iβρ1v + a1D4u + k(x)(u minus y) + δk(x)v 0 inL2

(38)

iβy minus w 0 inH20 (39)

iβρ2w + a2D4y minus k(x)(u minus y) + δk(x)w 0 inL2

(40)

Taking the inner product of (36) with z in H and takingits real part

Relang(iβI minus A)z zrangH minus δ k12

(x)v

2

minus δ k12

(x)w

2

0

(41)

yields that

k12

(x)v

0 (42)

k12

(x)w

0 (43)

Taking (42) and (43) into (37) and (39) we obtain

k12

(x)u

0

k12

(x)y

0(44)

Taking (37) into (38) and (39) into (40) we can easilydeduce from (38) and (40) that

a1D4u minus β2ρ1u 0 (45)

a2D4y minus β2ρ2y 0 (46)

If there exists x0 isin [a b] sub (0 l)

u x0 t( 1113857 0 y x0 t( 1113857 0 (47)

4 Mathematical Problems in Engineering

According to the existence-uniqueness theorem of so-lutions to ordinary differential equations (45) and (46) haveunique solution respectively as follows

u(x t) 0 x isin (0 l)

y(x t) 0 x isin (0 l)(48)

en we obtain

v(x t) 0 x isin (0 l)

w(x t) 0 x isin (0 l)(49)

ie

zH 0 (50)

which contradicts with zH 1 thus the proof ofiR sub ρ(A) is completed

Now we use reduction to absurdity to prove (34) If (34)is false then there exists a sequence zn isin D(A)zn (un vn yn wn)T with znH 1 and a sequence βn isin R

with βn⟶infin as n⟶infin such that

iβnI minus A( 1113857zn

H⟶ 0 (51)

ie

iβnun minus vn fn⟶ 0 inH2 (52)

iβnρ1vn + a1D4un + k(x) un minus yn( 1113857 + δk(x)vn gn⟶ 0 inL2

(53)

iβnyn minus wn Tn⟶ 0 inH20 (54)

iβnρ2wn + a2D4yn minus k(x) un minus yn( 1113857 + δk(x)wn Sn⟶ 0 inL2

(55)

Our goal is to prove zn2H 0 which contradicts withzn2H 1

Step 1 Local attenuationTaking the inner product of (51) with zn in H and then

taking its real part

Relang iβnI minus A( 1113857zn znrangH minus δ k12

(x)vn

2

minus δ k12

(x)wn

2⟶ 0

(56)

yields that

k12

(x)vn

⟶ 0

k12

(x)wn

⟶ 0

(57)

From (52) and (54) we obtain

k12

(x)βnun

⟶ 0

k12

(x)βnyn

⟶ 0

(58)

ie

k12

(x)un

⟶ 0

k12

(x)yn

⟶ 0

(59)

Taking the inner product of (53) with k6(x)un and (55)with k6(x)yn respectively because of k(x) isin C2 k6(x)un

and k6(x)yn are bounded we obtain

langiβnρ1vn + a1D4un + k(x) un minus yn( 1113857 + δk(x)vn k

6(x)unrang⟶ 0

(60)

langiβnρ2wn + a2D4yn minus k(x) un minus yn( 1113857 + δk(x)wn k

6(x)ynrang⟶ 0

(61)

by k(x) and we can easily deduce from (57) and (59) that

langk(x)un k6(x)unrang leC k

12(x)un

⟶ 0

langk(x)yn k6(x)ynrangleC k

12(x)yn

⟶ 0

langδk(x)vn k6(x)unrang⟶ 0

langδk(x)wn k6(x)ynrang⟶ 0

langk(x)yn k6(x)unrang

11138681113868111386811138681113868111386811138681113868 langk(x)un k

6(x)ynrang

11138681113868111386811138681113868111386811138681113868⟶ 0

(62)

By (60) and (61) we can obtain that

langiβnρ1vn + a1D4un k

6(x)unrang⟶ 0

langiβnρ2wn + a2D4yn k

6(x)ynrang⟶ 0

(63)

Because

langiβnρ1vn k6(x)unrang minus langiρ1vn minus ik

6(x)vnrang minus ρ1 k

3(x)vn

2⟶ 0

(64)

langiβnρ2wn k6(x)ynrang langiρ2wn minus ik

6(x)wnrang minus ρ2 k

3(x)wn

2⟶ 0

(65)

us

langa1D4un k

6(x)unrang a1 langun

Prime k6

1113872 1113873Prime(x)unrang + 2RelangunPrime k

61113872 1113873prime(x)unn

prime rang +langunPrime k

6(x)unPrimerang1113960 1113961⟶ 0 (66)

langa2D4yn k

6(x)ynrang a2[ langyn

Prime k6

1113872 1113873Prime(x)ynrang + 2RelangynPrime k

61113872 1113873prime(x)yn

primerang +langynPrime k

6(x)ynPrimerang1113960 1113961⟶ 0 (67)

Mathematical Problems in Engineering 5

Because

langunPrime k

61113872 1113873Prime(x)unrang leC k

3(x)unPrime

k

12(x)un

⟶ 0 (68)

langynPrime k

61113872 1113873Prime(x)ynrang leC k

3(x)ynPrime

k

12(x)yn

⟶ 0 (69)

integrating by part we obtain that

RelangunPrime k

61113872 1113873prime(x)un

primerang11138681113868111386811138681113868

11138681113868111386811138681113868le c k3(x)unPrime

k

12(x)un

+ k

12(x)un

2

1113874 1113875⟶ 0 (70)

RelangynPrime k

61113872 1113873prime(x)yn

primerang11138681113868111386811138681113868

11138681113868111386811138681113868le c k3(x)ynPrime

k

12(x)yn

+ k

12(x)yn

2

1113874 1113875⟶ 0 (71)

From (68) to (71) we now take them into (66) and (67) toobtain that

k3(x)unPrime

⟶ 0 (72)

k3(x)ynPrime

⟶ 0 (73)

Because k(x) is continuous and k(x)ge 0 in (a b) sub (0 l)

and there exists a constant c such that |kPrime(x)|le ck(x) and|kprime(x)|le ck(x) we can easily deduce from (57) that

k3(x)vn

⟶ 0

k3(x)wn

⟶ 0

(74)

Step 2 From local dissipation to global dissipationHere were going to use the multiplier method to prove

unPrime⟶ 0

vn⟶ 0

ynPrime⟶ 0

wn⟶ 0 in (0 l)

(75)

Taking (52) into (53) and (54) into (55) respectively wecan easily deduce from (53) and (55) that

a1D4un minus β2nρ1un gn + iβnρ1fn (76)

a2D4yn minus β2nρ2yn Sn + iβnρ2Tn (77)

Let q(x) isin C2 be a real function which will be chosenlater Taking the inner product of (76) with q(x)un

prime and (77)

with q(x)ynprime in L2 respectively integrating by part we

obtain that

Relanga1D4un minus β2nρ1un q(x)un

primerang

3a1 1113946l

0qprime(x) un

Prime1113868111386811138681113868

11138681113868111386811138682dx + 2Re a1 1113946

l

0qPrime(x)un

primeunPrimedx1113888 1113889

minus β2nρ1q(x) un

1113868111386811138681113868111386811138681113868111386821113868111386811138681113868

l

0 + β2nρ1 1113946l

0qprime(x) un

111386811138681113868111386811138681113868111386811138682dx

2langgn q(x)unrang minus 2langiβnρ1 fnq(x)( 1113857prime unrang(78)

Relanga2D4yn minus β2nρ2yn q(x)yn

primerang

3a2 1113946l

0qprime(x) yn

Prime1113868111386811138681113868

11138681113868111386811138682dx + 2Re a2 1113946

l

0qPrime(x)yn

primeynPrimedx1113888 1113889

minus a2q(x) ynPrime

1113868111386811138681113868111386811138681113868111386821113868111386811138681113868

l

0 + β2nρ2 1113946l

0qprime(x) yn

111386811138681113868111386811138681113868111386811138682dx

2langSn q(x)ynprimerang minus 2langiβnρ2 Tnq(x)( 1113857prime ynrang

(79)

Because unprime and βnun are uniformly bounded in L2 and yn

primeand βnyn are also uniformly bounded in L2 the terms on theright-hand side of (78) and (79) converge to zero Takingq(x) x we deduce from (78) and (79) that

3a1 unPrime

2

+ ρ1 vn

2

minus lβ2nρ1 un(l)1113868111386811138681113868

11138681113868111386811138682⟶ 0 (80)

3a2 ynPrime

2

+ ρ2 wn

2

minus la2 ynPrime(l)

111386811138681113868111386811138681113868111386811138682⟶ 0 (81)

We now take q(x) 1113938x

0 k6(s)ds into (78) and (79) toobtain that

3a1 k3(x)unPrime

2

+ ρ1 k3(x)vn

2

+ 2Re a1 1113946l

0k6

1113872 1113873prime(x)unprimeunPrimedx1113888 1113889 minus q(l)β2nρ1 un(l)

111386811138681113868111386811138681113868111386811138682⟶ 0 (82)

3a2 k3(x)ynPrime

2

+ ρ2 k3(x)wn

2

+ 2Re a2 1113946l

0k6

1113872 1113873prime(x)ynprimeynPrimedx1113888 1113889 minus q(l)a2 yn

Prime(l)1113868111386811138681113868

11138681113868111386811138682⟶ 0 (83)

6 Mathematical Problems in Engineering

Taking (70) (72) and (74) into (82) and taking (71) (73)and (76) into (83) we obtain

q(l)β2nρ1 un(l)1113868111386811138681113868

11138681113868111386811138682⟶ 0 (84)

q(l)a2 ynPrime(l)

111386811138681113868111386811138681113868111386811138682⟶ 0 (85)

ie

β2nρ1 un(l)1113868111386811138681113868

11138681113868111386811138682⟶ 0 (86)

a2 ynPrime(l)

111386811138681113868111386811138681113868111386811138682⟶ 0 (87)

Taking (86) and (87) into (80) and (81) we obtain

a1 unPrime

2

+ ρ1 vn

2⟶ 0 (88)

a2 ynPrime

2

+ ρ2 wn

2⟶ 0 (89)

From (59) (88) and (89) we obtain zn2H 0 whichcontradicts with zn2H 1 us the proof is completed

4 Conclusion

In this paper sufficient findings are provided for theexponential stability of weakly coupled elastic beamsystem with damping control by using the semigrouptheoretical frequency domain multiplier method Byvariable conversion the elastic beam control system istransformed into first-order evolution equations and alinear operator is obtained and the linear operator-producing semigroup is proved When the two beamshave local damping control from the local dissipation tothe global dissipation the exponential stability of thesemigroup generated from the linear operator is proved byreduction to absurdity e method in this paper can beemployed to handle other elastic beam systems in thefuture

Data Availability

e datasets used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (61473097) Qingdao PostdoctoralApplication Research Project (no 2015117) ShandongProvince Natural Science Foundation under grantZR2017QF011 Shandong Province Higher EducationalScience and Technology Program under grant J16LB10 andJ18KA316 the State Key Program of Natural ScienceFoundation of China (U1533202) the Shandong Science andTechnology Development Plan (No 2019GGX104019) and

Guangdong Basic and Applied Basic Research Foundation(2019A1515110706)

References

[1] K Wang L W Li Y Lan P Dong and G T Xia ldquoAp-plication research of chaotic carrier frequency modulationtechnology in two-stage matrix converterrdquo MathematicalProblems in Engineering vol 2019 Article ID 26143278 pages 2019

[2] Y T Zhou Y N Wang K Wang et al ldquoHybrid geneticalgorithm method for efficient and robust evaluation ofremaining useful life of supercapacitorsrdquo Applied Energyvol 260 Article ID 114169 2020

[3] K Wang L W Li H X Yin T Z Zhang and W B WanldquoermalModelling Analysis of SpiralWound Supercapacitorunder Constant-Current Cyclingrdquo PLoS One vol 10 ArticleID e0138672 2015

[4] G Xia Y Huang F Li et al ldquoA thermally flexible and multi-site tactile sensor for remote 3D dynamic sensing imagingrdquoFrontiers of Chemical Science and Engineering vol 14 2020

[5] Y T Zhou Y N Huang J B Pang and KWang ldquoRemaininguseful life prediction for supercapacitor based on long short-term memory neural networkrdquo Journal of Power Sourcesvol 440 Article ID 227149 2019

[6] G-T Xia C Li KWang and L-W Li ldquoStructural design andelectrochemical performance of PANICNTs and MnO2CNTs supercapacitorrdquo Science of Advanced Materials vol 11no 8 pp 1079ndash1086 2019

[7] L C Wang R F Yan F F Bai et al ldquoA Distributed Inter-Phase Coordination Algorithm for Voltage Control withUnbalanced PV Integration in LV Systemsrdquo IEEE Transac-tions on Sustainable Energy 2020

[8] K Wang J Pang L Li S Zhou Y Li and T ZhangldquoSynthesis of hydrophobic carbon nanotubesreduced gra-phene oxide composite films by flash light irradiationrdquoFrontiers of Chemical Science and Engineering vol 12 no 3pp 376ndash382 2018

[9] S Tang Z T Wang D L Yuan et al ldquoEnhanced photo-catalytic performance of BiVO4 for degradation of methyleneblue under LED visible light irradiation assisted by perox-ymonosulfaterdquo International Journal of Electrochemical Sci-ence vol 15 pp 2470ndash2480 2020

[10] K Wang C Li and B Ji ldquoPreparation of electrode based onplasma modification and its electrochemical applicationrdquoJournal of Materials Engineering and Performance vol 23no 2 pp 588ndash592 2014

[11] X Zhang and E Zuazua ldquoPolynomial decay and control of a1minus d hyperbolic-parabolic coupled systemrdquo Journal of Dif-ferential Equations vol 204 no 2 pp 380ndash438 2004

[12] X Zhang and E Zuazua ldquoControl observation and poly-nomial decay for a coupled heat-wave systemrdquo ComptesRendus Mathematique vol 336 no 10 pp 823ndash828 2003

[13] I Lasiecka and D Tataru ldquoUniform boundary stabilization ofsemilinear wave equations with nonlinear boundary damp-ingrdquo Differential and Integral Equations vol 6 no 3pp 507ndash533 1993

[14] I Lasiecka and D Toundykov ldquoEnergy decay rates for thesemilinear wave equation with nonlinear localized dampingand source termsrdquo Nonlinear Analysis eory Methods ampApplications vol 64 no 8 pp 1757ndash1797 2006

[15] I Lasiecka J L Lions and R Triggiani ldquoNon homogeneousboundary value problems for second order hyperbolic

Mathematical Problems in Engineering 7

operatorsrdquo Journal de Mathematiques pures et Appliqueesvol 65 no 2 pp 149ndash192 1986

[16] A Batkai K-J Engel J Pruss and R Schnaubelt ldquoPolynomialstability of operator semigroupsrdquo Mathematische Nach-richten vol 279 no 13-14 pp 1425ndash1440 2006

[17] B Rao and A Wehbe ldquoPolynomial energy decay rate andstrong stability of Kirchhoff plates with non-compact resol-ventrdquo Journal of Evolution Equations vol 5 no 2 pp 137ndash152 2005

[18] J Rauch X Zhang and E Zuazua ldquoPolynomial decay for ahyperbolicndashparabolic coupled systemrdquo Journal demathematiques pures et appliquees vol 84 no 4 pp 407ndash4702005

[19] Z Liu and S Zheng Semigroups Associated with DissipativeSystems CRC Press Boca Raton FL USA 1999

[20] C D Benchimol ldquoA note on weak stabilizability of con-traction semigroupsrdquo SIAM Journal on Control and Opti-mization vol 16 no 3 pp 373ndash379 1978

[21] K Wang L W Li W Xue et al ldquoElectrodeposition synthesisof PANIMnO2graphene composite materials and its elec-trochemical performancerdquo International Journal of Electro-chemical Science vol 12 pp 8306ndash8314 2017

[22] K Wang L Li T Zhang and Z Liu ldquoNitrogen-dopedgraphene for supercapacitor with long-term electrochemicalstabilityrdquo Energy vol 70 pp 612ndash617 2014

[23] D L Yuan M T Sun S F Tang et al ldquoAll-solid-state BiVO4ZnIn2S4 Z-scheme composite with efficient charge separationsfor improved visible light photocatalytic organics degrada-tionrdquo Chinese Chemical Letters vol 31 pp 547ndash550 2019

[24] KWang S Z Zhou Y T Zhou J Ren L W Li and L YongldquoSynthesis of porous carbon by activation method and itselectrochemical performancerdquo International Journal of Elec-trochemical Science vol 13 no 11 pp 10766ndash10773 2018

[25] K Liu and Z Liu ldquoExponential decay of energy of vibratingstrings with local viscoelasticityrdquo Zeitschrift fur angewandteMathematik und Physik vol 53 no 2 pp 265ndash280 2009

[26] B Lazzari and R Nibbi ldquoOn the exponential decay inthermoelasticity without energy dissipation and of type III inpresence of an absorbing boundaryrdquo Journal of MathematicalAnalysis and Applications vol 338 no 1 pp 317ndash329 2008

[27] Z Liu R Quintanilla and Y Wang ldquoOn the phase-lag heatequation with spatial dependent lagsrdquo Journal of MathematicalAnalysis and Applications vol 455 no 1 pp 422ndash438 2017

[28] C Zhang Y Kao B Kao and T Zhang ldquoStability of Mar-kovian jump stochastic parabolic it o equations with generallyuncertain transition ratesrdquo Applied Mathematics and Com-putation vol 337 pp 399ndash407 2018

[29] Y Kao Q Zhu and W Qi ldquoExponential stability and in-stability of impulsive stochastic functional differential equa-tions with Markovian switchingrdquo Applied Mathematics andComputation vol 271 pp 795ndash804 2015

[30] Y Kao L Shi J Xie and H R Karimi ldquoGlobal exponentialstability of delayed Markovian jump fuzzy cellular neuralnetworks with generally incomplete transition probabilityrdquoNeural Networks vol 63 pp 18ndash30 2015

[31] Y Liu C Zhang Y Kao and C Hou ldquoExponential stability ofneutral-type impulsive markovian jump neural networks withgeneral incomplete transition ratesrdquo Neural Processing Let-ters vol 47 no 2 pp 325ndash345 2018

[32] Y Liu Y Kao H R Karimi and Z Gao ldquoInput-to-statestability for discrete-time nonlinear switched singular sys-temsrdquo Information Sciences vol 358-359 pp 18ndash28 2016

[33] F Alabau-Boussouira ldquoA two-level energy method for in-direct boundary observability and controllability of weakly

coupled hyperbolic systemsrdquo SIAM Journal on Control andOptimization vol 42 no 3 pp 871ndash906 2003

[34] F Alabau P Cannarsa and V Komornik ldquoIndirect internalstabilization of weakly coupled evolution equationsrdquo Journalof Evolution Equations vol 2 no 2 pp 127ndash150 2009

[35] F Alabau-Boussouira ldquoIndirect boundary stabilization ofweakly coupled hyperbolic systemsrdquo SIAM Journal on Controland Optimization vol 41 no 2 pp 511ndash541 2009

[36] I Lasiecka Mathematical Control eory of Coupled PDErsquosCMBS-NSF Lecture Notes SIAM Publications PhiladelphiaPA USA 2001

[37] S Alimirzaei M Mohammadimehr and A Tounsi ldquoNon-linear analysis of viscoelastic micro-composite beam withgeometrical imperfection using FEM MSGT electro-mag-neto-elastic bending buckling and vibration solutionsrdquoStructural Engineering and Mechanics vol 71 no 5pp 485ndash502 2019

[38] L Boulefrakh H Hebali A Chikh A A Bousahla A Tounsiand SMahmoud ldquoe effect of parameters of visco-Pasternakfoundation on the bending and vibration properties of a thickFG platerdquo Geomechanics and Engineering vol 18 no 2pp 161ndash178 2019

[39] F Y Addou M Meradjah A Anis Bousahla andS R Mahmoud ldquoInfluences of porosity on dynamic responseof FG plates resting on WinklerPasternakKerr foundationusing quasi 3D HSDTrdquo Computers and Concrete vol 24no 4 pp 347ndash367 2019

[40] L A Chaabane F Bourada M Sekkal et al ldquoAnalytical studyof bending and free vibration responses of functionally gradedbeams resting on elastic foundationrdquo Structural Engineeringand Mechanics vol 71 no 2 pp 185ndash196 2019

[41] Z Boukhlif M Bouremana F Bourada et al ldquoA simple quasi-3D HSDT for the dynamics analysis of FG thick plate onelastic foundationrdquo Steel and Composite Structures vol 31no 5 pp 503ndash516 2019

[42] A Mahmoudi S Benyoucef A Tounsi A BenachourE A Adda Bedia and SMahmoud ldquoA refined quasi-3D sheardeformation theory for thermo-mechanical behavior offunctionally graded sandwich plates on elastic foundationsrdquoJournal of Sandwich Structures amp Materials vol 21 no 6pp 1906ndash1929 2019

[43] F Z Zaoui D Ouinas and A Tounsi ldquoNew 2D and quasi-3Dshear deformation theories for free vibration of functionallygraded plates on elastic foundationsrdquo Composites Part BEngineering vol 159 pp 231ndash247 2019

[44] Y Liu J Wang C Gao S Tang and Z Gao ldquoInput-to-statestability analysis for a class of discrete-time nonlinear input-saturated switched descriptor systems with unstable subsys-temsrdquo Neural Computing and Applications vol 29 pp 417ndash424 2016

[45] G Wang X Xiao and Y Liu ldquoDynamic modeling andanalysis of a mine hoisting system with constant length andvariable lengthrdquo Mathematical Problems in Engineeringvol 2019 Article ID 4185362 12 pages 2019

[46] G Wang X Xiao C Ma G Cheng and X Di ldquoNonlineardynamic behavior of winding hoisting rope under head sheaveaxial wobblesrdquo Shock and Vibration vol 2019 Article ID7026125 11 pages 2019

[47] Z Liu and B Rao ldquoFrequency domain approach for thepolynomial stability of a system of partially damped waveequationsrdquo Journal of Mathematical Analysis and Applica-tions vol 335 no 2 pp 860ndash881 2007

[48] Z Liu and Q Zhang ldquoStability and regularity of solution tothe timoshenko beam equation with local kelvin--voigt

8 Mathematical Problems in Engineering

dampingrdquo SIAM Journal on Control and Optimizationvol 56 no 6 pp 3919ndash3947 2018

[49] K Liu Z Liu and Q Zhang ldquoEventual differentiability of astring with local Kelvin-Voigt dampingrdquo ESAIM ControlOptimisation and Calculus of Variations vol 23 no 2pp 443ndash454 2017

[50] Z Liu A Magantildea and R Quintanilla ldquoOn the time decay ofsolutions for non-simple elasticity with voidsrdquo ZAMM -Journal of Applied Mathematics and MechanicsZeitschrift furAngewandte Mathematik und Mechanik vol 96 no 7pp 857ndash873 2016

[51] Z Liu and Q Zhang ldquoStability of a string with local kelvin--voigt damping and nonsmooth coefficient at interfacerdquo SIAMJournal on Control and Optimization vol 54 no 4pp 1859ndash1871 2016

[52] J Prss ldquoOn the spectrum of C0-semigroups Transrdquo Journal ofthe American Mathematical Society vol 284 no 2 pp 847ndash857 1984

[53] Z Liu and B Rao ldquoCharacterization of polynomial decay ratefor the solution of linear evolution equationrdquo Zeitschrift furangewandte Mathematik und Physik vol 56 no 4 pp 630ndash644 2005

Mathematical Problems in Engineering 9