# Stability of Two Weakly Coupled Elastic Beams with Partially Local Damping

Authors: Caihong Zhang; Yinuo Huang; Licheng Wang; Chongxiong Duan; Tiezhu Zhang; Kai Wang
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this paper, the stability of two weakly coupled elastic beams connected vertically by a spring is investigated via the frequency domain method and the multiplier technique. When the two beams have partially local damping, the operatorA is obtained via variable conversion, and it generating a semigroup is proved, then we obtain that the semigroup is exponentially stable by reduction to absurdity.


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## Body

## 1. Introduction

Artificial intelligence (AI) has been around and penetrated into all fields, such as research, production, and life [1–5]. Scientists pay more attention to energy, materials, and environment [6–10]. The stability study of the coupling control system in space vehicles is one of the important research subjects in the control field in recent years. Much attention has been paid to research the stability of control systems using semigroup theory [11]. Reference [12] is for coupled heat-wave system. References [13, 14] are for wave equations. Reference [15] is for second-order hyperbolic operators. References [16–19] are about polynomial stability of systems, and references [20–33] are about exponential stability. The authors of [33–35] considered weakly coupled evolution equations of wave-Petrowsky, wave-wave, and Kirchhoff–Petrowsky for its asymptotic stability and boundary controllability. The case of strongly coupled system was studied by Lasiecka [36], and she obtained the strong stability for the open-loop systems with polynomial energy decay rate.A viscoelastic microcomposite beam reinforced by various distributions of boron nitride nanotubes with initial geometrical imperfection has been described in [37], and the nonlinear static, buckling, and vibration are analyzed by using the finite element method. The bending and dynamic behavior of functionally graded plates resting on visco-Pasternak foundations is studied in [38]. Using a simple quasi-3D hyperbolic theory, the dynamic behavior of functionally graded plates is concerned in [39]. Using a hyperbolic shear deformation theory, the static and dynamic behaviors of functionally graded beams is studied in [40]. A dynamic study of functionally graded plates resting on elastic foundation is considered in [41]. Thermomechanical analysis of functionally graded sandwich plates resting on a two-parameter elastic foundation is studied in [42]. The free vibration of FG plates resting on elastic foundations is modeled by two-dimensional (2D) and quasi-three-dimensional (quasi-3D) shear deformation theories in [43]. Input-to-state stability for a class of discrete-time nonlinear input-saturated switched descriptor systems with unstable subsystems is discussed in [44]. In [45], nonlinear dynamic behavior of the winding hoisting rope under head sheave axial wobbles is concerned. A dynamic model of a mine hoisting system with constant length and variable length is analyzed in [46]. Liu and Rao [47] studied the stability of a weakly coupled and partially damped system. They obtained a sharp polynomial decay rate, when compared with Alabau-Boussouira’s results. Recently, they also obtained the exact boundary controllability of this system with the control acted only on one equation. The Timoshenko beam equation with locally distributed Kelvin–Voigt damping is considered in [48]. A wave equation with local Kelvin–Voigt damping is proposed in [49], and the semigroup corresponding to the system is eventually differentiable. The behavior of slow or exponential decay is analyzed about elastic material with voids in [50]. The stability of an elastic string system with local Kelvin–Voigt damping is studied in [51].The above research leads us to study a new problem. Consider a system of two beams connected vertically by a spring. In engineering construction, there are elastic beams everywhere, and it is of great significance to study the elastic beam system. If two beams subject to local damping or only one beam subjects to local damping, what is the energy decay rate of every system? The coupling terms are local which model the location of the spring in this case. Such a coupling is even weaker than the one studied by Liu and Rao. Hence, finding the energy decay rate is a challenge. For reader’s convenience, we include these two frequency domain conditions here. The first one is the frequency domain characterization of exponential decay, which was obtained by Prss [52]. The second one is of polynomial energy decay rate for a C0-semigroup of contraction, which was obtained by Liu and Rao [53].In this paper, the stability of weakly coupled elastic beam system with damping control by using the semigroup theoretical frequency domain multiplier method is studied. By variable conversion, the elastic beam control system is transformed into first-order evolution equations and a linear operator is obtained, and the linear operator-producing semigroup is proved. When the two beams have local damping control, using reduction to absurdity, from the local dissipation to the global dissipation, the exponential stability of the semigroup generated from the linear operator is proved. The method in this paper can be employed to handle other elastic beam systems in the future.In this paper, we need some definitions and lemmas which are as follows.Definition 1.
(see [19]). Let H be a real or complex Hilbert space and we define , is the inner product of H and ⋅ is the norm of H. Let A be a dense linear operator on H, that is, DA⊆H⟶H, then A is dissipative, and for any x∈DA, we get ReAx,x≤0.Definition 2.
(see [19]). eAt is exponentially stable if there are normal numbers M and α which make(1)eAt≤Me−αt,∀t≥0.Lemma 1.
(see [19]). Linear operator A can generate C0 semigroup St on Hilbert space H if it satisfies the following:(1)
DA is dense on Hilbert space H(2)
A is dissipative(3)
0∈ρALemma 2.
(see [19]). A C0 semigroup eAt of contractions on a Hilbert space H is exponentially stable if and only if(2)ρA⊇iβ,β∈R≡iR,limβ⟶+∞¯iβI−A−1<+∞.


## 2. Model Description

Consider the system of two beams connected vertically by a spring. When both upper and lower beams have local damping control. The physical model of weakly coupled elastic beam control system is given in Figure1.Figure 1
The figure of two weakly coupled elastic beams.The system is governed by the following equations:(3)ρ1utt=−a1uxxxx−kxu−y−δkxut,(4)ρ2ytt=−a2yxxxx+kxu−y−δkxyt,where u and y are the displacement of upper and lower beams. x∈0,l and t∈0,+∞; ρ1, ρ2, a1, and a2 are positive physical constants; kx≥0 in a,b⊂0,l is a locally supported smooth function, which represents the position and elasticity of the spring. We assume kx∈C2, and there exists a constant c such that k″x≤ckx and k′x≤ckx; δ>0 is the damping coefficient of the system.The two beams (3) and (4) have local damping and their boundary conditions are(5)uxx0,t=uxxl,t=uxxx0,t=uxxxl,t=0,y0,t=yl,t=yx0,t=yxl,t=0.To convert the system into a first-order evolution equation, here we denote(6)v=ut,w=yt,and the state variable vector is z≡zt=u,v,y,wT; then, systems (3) and (4) can be rewritten as the following form:(7)ut=v,vt=1ρ1−a1D4u−kxu−y−δkxv,yt=w,wt=1ρ2−a2D4y+kxu−y−δkxw.Here, we have used the notationDi=∂i/∂xi, and the state space is(8)H=H20,l×L20,l×H020,l×L20,l.The Hilbert spaceH is equipped with the inner product which induces the energy norm:(9)zH2=a1u″2+a2y″2+ρ1v2+ρ2w2+k1/2xu−y2.Here and after,⋅, ′, and ⋅,⋅ denote the L20,l norm, derivative, and inner product, respectively.Define a linear operatorA:H⟶H by(10)A=0I00−a1D4+kxρ1−δkxρ1kxρ10000Ikxρ20−a2D4+kxρ2−δkxρ2,with(11)DA=z∈H|u,y∈H4,v∈H2,w∈H02,u″∈H02.Thus, equations (3) and (4) are transformed into a first-order evolution on the Hilbert space:(12)dzdt=Az,z0=z0.


## 3. Main Result

Theorem 1.
The operatorA generates a C0 semigroup St of contractions on H.Proof.
It is clear thatDA is dense in H. By a straight forward calculation,(13)ReAz,zH=a1∫0lv″u¯″dx+∫0l−a1D4u−kxu−y−δkxvv¯dx+a2∫0ly″w¯″dx+∫0l−a2D4y+kxu−y−δkxww¯dx=−δk1/2xv2+k1/2xw2≤0.
Hence,A is dissipative. It is easy to show that, for any F=f1,⋯,f4T∈H,(14)Az=F,has unique solution z∈DA, and(15)FH2=a1f1″2+a2f3″2+ρ1f22+ρ2f42+k1/2xf1−f32.
In fact, from the first and third equations of (14), we get v=f1∈H2andw=f3∈H02.
Substituting them into the second and fourth equations in (14), we have(16)−a1D4u−kxu−y−δkxf1=ρ1f2,(17)−a2D4y+kxu−y−δkxf3=ρ2f4.
Taking the inner product with (16) and −u and with (17) and −y, respectively, we obtain(18)a1D4u,u+kxu−y,u+δkxf1,u=ρ1f2,−u,(19)a2D4y,y−kxu−y,y+δkxf3,y=ρ2f4,−y.
Suppose there exist infinitesimal constantsε,ε1,andε2 and(20)ε=minε1,ε2.
Taking the boundary conditions to (18) and (19), we have(21)a1u″2+kxu−y,u+δkxf1,u=ρ1f2,−u≤ρ1f2u≤ρ12ε1f22+ρ1ε12u2,(22)a2y″2−kxu−y,y+δkxf3,y=ρ2f4,−y≤ρ2f4y≤ρ22ε2f42+ρ2ε22y2.
Adding (21) and (22) and because ε is an infinitesimal constant, we have(23)a1u″2+a2y″2+k1/2xu−y2+δkxf1,u+δkxf3,y≤ρ12εf22+ρ22εf42.
Also, we have(24)δkxf1,u≤δ2εf12+δε2kxu2,(25)δkxf3,y≤δ2εf32+δε2kxy2.
Suppose there exist positive constantsc1,c2,c3,c4, and C independent of z and F and(26)C=maxc1,c2,c3,c4,by Poincare’s inequalities, we have(27)ρ1v2≤c1a1f1″2,(28)ρ2w2≤c2a2f3″2.
Combining (23)–(28) yields(29)a1u″2+a2y″2+k1/2xu−y2+ρ1v2+ρ2w2+δkxf1,u+δkxf3,y≤c3a1f1″2+a2f3″2+ρ1f22+ρ2f42.
By Poincare’s inequalities, (29) yields(30)zH2≤c4FH2−δkxf1,u−δkxf3,y.
Then, by (24), (25), and Poincare’s inequalities, we have(31)zH2≤CFH2,thus 0∈ρA. By Lemma 1, the proof is completed.Theorem 2.
The semigroupSt, generated by the operator A, defined in (10), is exponentially stable, i.e., there exist two positive constants αandM such that(32)St≤Me−αt,∀t>0.Proof.
By Lemma2, it suffices to verify(33)ρA⊇iβ,β∈R≡iR,(34)limβ⟶+∞¯iβI−A−1<+∞.
We use reduction to absurdity to prove (33). If (33) is false, then there exists β∈R and β≠0, iβ is the spectral point of A. Because A−1 is dense, iβ is the eigenvalue of operator A; then, there exists vector(35)z=u,v,y,wT∈DA,zH=1,such that(36)iβI-AzH=0,i.e.,(37)iβu−v=0,inH2,(38)iβρ1v+a1D4u+kxu−y+δkxv=0,inL2,(39)iβy−w=0,inH02,(40)iβρ2w+a2D4y−kxu−y+δkxw=0,inL2.
Taking the inner product of (36) with z in H and taking its real part(41)ReiβI-Az,zH=−δk1/2xv2−δk1/2xw2=0,yields that(42)k1/2xv=0,(43)k1/2xw=0.
Taking (42) and (43) into (37) and (39), we obtain(44)k1/2xu=0,k1/2xy=0.
Taking (37) into (38) and (39) into (40), we can easily deduce from (38) and (40) that(45)a1D4u−β2ρ1u=0,(46)a2D4y−β2ρ2y=0.
If there existsx0∈a,b⊂0,l,(47)ux0,t=0,yx0,t=0.
According to the existence-uniqueness theorem of solutions to ordinary differential equations, (45) and (46) have unique solution, respectively, as follows:(48)ux,t=0,x∈0,l,yx,t=0,x∈0,l.
Then, we obtain(49)vx,t=0,x∈0,l,wx,t=0,x∈0,l,i.e.,(50)zH=0.which contradicts with zH=1; thus, the proof of iR⊂ρA is completed.
Now, we use reduction to absurdity to prove (34). If (34) is false, then there exists a sequence zn∈DA, zn=un,vn,yn,wnT with znH=1 and a sequence βn∈R with βn⟶∞ as n⟶∞ such that(51)iβnI−AznH⟶0,i.e.,(52)iβnun−vn=fn⟶0,inH2,(53)iβnρ1vn+a1D4un+kxun−yn+δkxvn=gn⟶0,inL2,(54)iβnyn−wn=Tn⟶0,inH02,(55)iβnρ2wn+a2D4yn−kxun−yn+δkxwn=Sn⟶0,inL2.
Our goal is to proveznH2=0, which contradicts with znH2=1.Step 1.
Local attenuation
Taking the inner product of (51) with zn in H and then taking its real part(56)ReiβnI−Azn,znH=−δk1/2xvn2−δk1/2xwn2⟶0.yields that(57)k1/2xvn⟶0,k1/2xwn⟶0.
From (52) and (54), we obtain(58)k1/2xβnun⟶0,k1/2xβnyn⟶0.i.e.,(59)k1/2xun⟶0,k1/2xyn⟶0.
Taking the inner product of (53) with k6xun and (55) with k6xyn, respectively, because of kx∈C2, k6xun and k6xyn are bounded, we obtain(60)iβnρ1vn+a1D4un+kxun−yn+δkxvn,k6xun⟶0,(61)iβnρ2wn+a2D4yn−kxun−yn+δkxwn,k6xyn⟶0,by kx, and we can easily deduce from (57) and (59) that(62)kxun,k6xun≤Ck1/2xun⟶0,kxyn,k6xyn≤Ck1/2xyn⟶0.δkxvn,k6xun⟶0,δkxwn,k6xyn⟶0,kxyn,k6xun=kxun,k6xyn⟶0.
By (60) and (61), we can obtain that(63)iβnρ1vn+a1D4un,k6xun⟶0,iβnρ2wn+a2D4yn,k6xyn⟶0.
Because(64)iβnρ1vn,k6xun=−iρ1vn,−ik6xvn=−ρ1k3xvn2⟶0,(65)iβnρ2wn,k6xyn=iρ2wn,−ik6xwn=−ρ2k3xwn2⟶0.
Thus,(66)a1D4un,k6xun=a1un″,k6″xun+2Reun″,k6′xunn′+un″,k6xun″⟶0,(67)a2D4yn,k6xyn=a2yn″,k6″xyn+2Reyn″,k6′xyn′+yn″,k6xyn″⟶0.
Because(68)un″,k6″xun≤Ck3xun″k1/2xun⟶0,(69)yn″,k6″xyn≤Ck3xyn″k1/2xyn⟶0,integrating by part, we obtain that(70)Reun″,k6′xun′≤ck3xun″k1/2xun+k1/2xun2⟶0,(71)Reyn″,k6′xyn′≤ck3xyn″k1/2xyn+k1/2xyn2⟶0.
From (68) to (71), we now take them into (66) and (67) to obtain that(72)k3xun″⟶0,(73)k3xyn″⟶0.
Becausekx is continuous and kx≥0 in a,b⊂0,l and there exists a constant c such that k″x≤ckx and k′x≤ckx, we can easily deduce from (57) that(74)k3xvn⟶0,k3xwn⟶0.Step 2.
From local dissipation to global dissipation
Here, were going to use the multiplier method to prove(75)un″⟶0,vn⟶0,yn″⟶0,wn⟶0in0,l.
Taking (52) into (53) and (54) into (55), respectively, we can easily deduce from (53) and (55) that(76)a1D4un−βn2ρ1un=gn+iβnρ1fn,(77)a2D4yn−βn2ρ2yn=Sn+iβnρ2Tn.
Letqx∈C2 be a real function, which will be chosen later. Taking the inner product of (76) with qxun′ and (77) with qxyn′ in L2, respectively, integrating by part, we obtain that(78)Rea1D4un−βn2ρ1un,qxun′=3a1∫0lq′xun″2dx+2Rea1∫0lq″xun′u¯n″dx−βn2ρ1qxun20l+βn2ρ1∫0lq′xun2dx=2gn,qxun−2iβnρ1fnqx′,un,(79)Rea2D4yn−βn2ρ2yn,qxyn′=3a2∫0lq′xyn″2dx+2Rea2∫0lq″xyn′y¯n″dx−a2qxyn″20l+βn2ρ2∫0lq′xyn2dx=2Sn,qxyn′−2iβnρ2Tnqx′,yn.
Becauseun′ and βnun are uniformly bounded in L2 and yn′ and βnyn are also uniformly bounded in L2, the terms on the right-hand side of (78) and (79) converge to zero. Taking qx=x, we deduce from (78) and (79) that(80)3a1un″2+ρ1vn2−lβn2ρ1unl2⟶0,(81)3a2yn″2+ρ2wn2−la2yn″l2⟶0.
We now takeqx=∫0xk6sds into (78) and (79) to obtain that(82)3a1k3xun″2+ρ1k3xvn2+2Rea1∫0lk6′xun′u¯n″dx−qlβn2ρ1unl2⟶0,(83)3a2k3xyn″2+ρ2k3xwn2+2Rea2∫0lk6′xyn′y¯n″dx−qla2yn″l2⟶0.
Taking (70), (72), and (74) into (82) and taking (71), (73), and (76) into (83), we obtain(84)qlβn2ρ1unl2⟶0,(85)qla2yn″l2⟶0,i.e.,(86)βn2ρ1unl2⟶0,(87)a2yn″l2⟶0.
Taking (86) and (87) into (80) and (81), we obtain(88)a1un″2+ρ1vn2⟶0,(89)a2yn″2+ρ2wn2⟶0.
From (59), (88), and (89), we obtain znH2=0, which contradicts with znH2=1. Thus, the proof is completed.


## 4. Conclusion

In this paper, sufficient findings are provided for the exponential stability of weakly coupled elastic beam system with damping control by using the semigroup theoretical frequency domain multiplier method. By variable conversion, the elastic beam control system is transformed into first-order evolution equations and a linear operator is obtained, and the linear operator-producing semigroup is proved. When the two beams have local damping control, from the local dissipation to the global dissipation, the exponential stability of the semigroup generated from the linear operator is proved by reduction to absurdity. The method in this paper can be employed to handle other elastic beam systems in the future.



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