Volume 10 No. 4 2005, 75-82 EXPONENTIAL ...

International Journal of Differential Equations and Applications

—————————————————————————— Volume 10 No. 4 2005, 75-82

EXPONENTIAL STABILITY FOR THE TIMOSHENKO BEAM BY A LOCALLY DISTRIBUTED DAMPING C.A. Raposo1 , J. Ferreira2 § 1,2 Departamento

de Matemática Universidade Federal de S˜ ao Jo˜ ao Del Rei — UFSJ Pra¸ca Frei Orlando 170, Cep 363307-352 S˜ ao Jo˜ ao del Rei, Minas Gerais, MG, BRAZIL 1 e-mail: [email protected] 2 e-mail: [email protected] Abstract: In this paper we generalize the result obtained by C.A. Raposo et al ([13]), where two dissipative terms distributed in the whole beam was considered. In practice, it is desirable to consider the problem with locally distributed damping. In this sense we prove the exponential stability for the Timoshenko beam by one control force, that is, one locally distributed damping in the transverse displacement of the beam. AMS Subject Classification: 34D20 Key Words: exponential stability, dissipative term of friction, linear systems of Timoshenko, locally distributed damping 1. Introduction The transverse vibration of beam is given by two coupled partial differential equations ρutt = (K(ux − ψ))x ,

in

]0, L[×]0, ∞[,

(1.1)

Iρ ψtt (x, t) = (EIψx )x − K(ux − ψ),

in

]0, L[×]0, ∞[.

(1.2)

Here, t is the time variable and x is the space coordinate along the beam, the length of which is L, in its equilibrium position. The function u is the transverse displacement of the beam and ψ is the rotation angle of a filament of the beam. The coefficients ρ, Iρ , E, I, and K are the mass per unit length, Received:

March 8, 2006

§ Correspondence

author

c 2005 Academic Publications

76

C.A. Raposo, J. Ferreira

the polar moment of inertia of a cross section, Young’s modulus of elasticity, the moment of inertia of a cross section and the shear modulus respectively. We denote ρ1 = ρ, ρ2 = Iρ , b = EI, k = K, and we obtain directly from (1.1)-(1.2) the following system ρ1 utt − k(ux − ψ)x = 0,

in

]0, L[×]0, ∞[,

(1.3)

ρ2 ψtt − bψxx + k(ux − ψ) = 0,

in

]0, L[×]0, ∞[.

(1.4)

The energy of the beam is Z 1 L ρ1 |ut |2 + ρ2 |ψt |2 + b|ψx |2 + k |ux + ψ|2 dx E(t) = 2 0

(1.5)

Let us mention some results about the stabilization of this model. The case of two boundary control force has already been considered by Kin and Renardy [8] for the Timoshenko beam, they proved the exponential decay of the energy E(t) by using a multiplier technique and provided numerical estimates of the eigenvalues of the operator associated to this system, and by Lagnese and Lions [9] for the study of the exact controllability. Taylor [10] studied the boundary control of system with variable physical characteristics. Rivera et al [4], showed the exponential decay of the associated energy for Timoshenko systems with memory and Raposo [6] studied the transmition problem for Timoshenko systems of memory type. Shi and Feng [11] established the exponential decay of the energy with locally distributes feedback (two feedback). Recently Soufyany and Wehbe [12] proved the uniform stabilization using a locally distributive feedback under the rotation angle ψ of a filament of the beam. We prove that it is possible to obtain the exponential stability for the beam, which is assumed to be clamped at the two ends of (0, L), by using a unique locally distributed feedback under the transverse displacement ρ1 utt − k(ux − ψ)x + α(x)ut = 0,

in

]0, L[×]0, ∞[, (1.6)

ρ2 ψtt − bψxx + k(ux − ψ) = 0,

in

]0, L[×]0, ∞[, (1.7)

u(0, t) = u(L, t) = ψ(0, t) = ψ(L, t) = 0,

t > 0,

(1.8)

where α is a positive continuous function of the space variable, such that α(x) ≥ a > 0. Our formulation for the system of Timoshenko is very important because it can be seen in [9] that when k → ∞ the solution u converges (in an appropriate topology) to the solution of the model of Kirchhoff with the appropriate boundary conditions.

EXPONENTIAL STABILITY FOR THE TIMOSHENKO...

77

In the sequel we assume the existence and uniqueness of solution to the initial-boundary value problem under consideration, which can be prove by semigroup theory (see [5]) or by Faedo-Galerkin methods (see [6]). The problem is well-posed for data (φ0 , φ1 ), (ψ0 , ψ1 ) in the Sobolev space [H 2 (]0, L[)× H01 (]0, L[)]2 . Weak solutions and the energy are defined also in [H01 (]0, L[) × L2 (]0, L[)]2 .

2. The Method The problem of establishing an energy estimate of the form E(t) ≤ CE(0)e−wt ,

∀t ≥ 0,

or equivalently establishing the exponential stability kS(t)k ≤ Ce−wt ,

∀t ≥ 0,

of the semigroup S(t) remained open for some time. However, it is now clear (see for instance, Zheng [7]) that these two statements are equivalent. In this section we collect some results in the literature concerning the necessary and sufficient conditions for a C0 -semigroup being exponentially stable. We denote by σ(A) and ρ(A). We denote by the spectrum and resolvent set of the operator A, respectively. The first result we are going to state is about the necessary and sufficient conditions of exponential stability of a C0 -semigroup on a Hilbert space. The result was obtained by Gearhart (see Wyler [3]) and Huang [2], independently. The following statement is due to Huang. Theorem 2.1. Let S(t) = eAt be a C0 -semigroup on a Hilbert space. Then S(t) is exponentially stable if and only if sup{Reλ; λ ∈ σ(A)} ≤ 0 and sup k(λI − A)−1 k < ∞ Reλ≥0

hold. The following invariant of the result is due to Gearhart.

78


Theorem 2.2. Let S(t) = eAt be a C0 -semigroup of contractions on a Hilbert space. Then S(t) is exponentially stable if and only if ρ(A) ⊇ {iβ, β ∈ ℜ} and lim sup k(iβ − A)−1 k < ∞ |β|→∞

hold. Liu and Zheng [1] gave the proof of equivalence of these two results under the conditions that S(t) = eAt be a C0 -semigroup of contractions on a Hilbert space. In this work we use the method developed by Liu and Zheng [1], where the authors used a contradiction argument by combining Gearhart’s Theorem with a PDE technique.

3. Exponential Stability The energy space associated to system (1.6)-(1.8) is [H01 (0, L) × L2 (0, L)]2 . The inner product in the energy space is defined for Uj = (uj , v j , wj , y j )T ∈ H, j = 1, 2, as follows: Z L (U1 , U2 ) = [k(u1x − w1 )(u2x − w2 ) + ρ1 v 1 v 2 + ρ2 y 1 y 2 + bw1 w2 ]dx. 0

In the sequel we will denote by kU k2 = (U, U ), the norm in the energy space. The system (1.6)-(1.8) can be written as d U (t) − AU (t) = 0, dt where 

 u  ut   U (t) =   ψ , ψt



  A= 

0 I k k ρ1 (·)xx − ρ1 α(x) 0 0 k 0 ρ2 (·)x

0 1 ρ1 (·)x 0 k ρ2 (·)xx −

with D(A) = [(H 2 (]0, L[) ∩ H01 (]0, L[)) × H01 (]0, L[)]2 .

k ρ2

0 0 I 0

    


79

Proposition 3.1. The operator A generates a C0 -semigroup of contractions (eAt )t>0 on H. Proof. See [11]. Proposition 3.2. The operator A is dissipative. Proof. Let U = (u, ut , ψ, ψt )T be, we have (AU, U ) = −

Z

L 2

α(x)|ut | dx ≤ − a

0

Z

L

|ut |2 dx .

(3.1)

0

In this work our main result is the following theorem. Theorem 3.1. Assume that α ∈ C([0, L]) and α(x) ≥ a > 0. Then eAt is exponentially stable. Proof. To prove the exponential stability of eAt we will use Theorem 2.2. First we will to prove that ρ(A) ⊇ {iβ, β ∈ ℜ}.

(3.2)

Suppose the conclusion of (3.2) is false. There exist β ∈ ℜ such that iβ ∈ σ(A) and from the compactness of the imbedding D(A) in H, iβ is an eigenvalues. Let U = (u, ut , v, vt )T , U 6= 0 such that AU = iβU . Using the definition of A it follows that AU = iβU if and only if kuxx − kvx + ρ1 β 2 u = iβα(x)u, 2

bvxx − kux + kv = −ρ2 β v. We multiply the first equation of this system by u to obtain kuxx u − kvx u + ρ1 β 2 u2 = iβα(x)u2 . Performing integration by parts on ]0, L[ we get Z Z

L

[−k|ux |2 − kvx u + ρ1 β 2 |u|2 ]dx = 0 , 0 L

βα(x)u2 dx = 0 . 0

(3.3) (3.4)

80


Then u = 0 and ux = 0 in ]0, L[. From (3.3) we get vx = 0, and using the Poincaré’s inequality we obtain v = 0. Now, it is trivial to see that u = v = 0 is the solution of the system (3.3)-(3.4) and then we have a contradiction. This completes the proof of (3.2). Now we will prove that lim sup k(iβ − A)−1 k < ∞.

(3.5)

|β|→∞

Suppose that the conclusion of (3.5) is false, that is lim sup k(iβI − A)−1 k = ∞. |β|→∞

There exists a sequence (Vn ) ∈ H and βn ∈ ℜ such that k(iβn I − A)−1 Vn k ≥ nkVn k ∀ n > 0. Therefore iβn ∈ ρ(A) and then we have that exists a unique sequence (Un ) ∈ D(A) such that iβn Un − AUn = Vn ,

kUn k = 1,

that is Un = (iβn I − A)−1 Vn , and kUn k ≥ nkiβn Un − AUn k. Now we denote by Fn = iβn Un − AUn and it follows that kFn k ≤

1 n

and hence Fn → 0 (strongly) in H as n → ∞. Let ut = v, ψt = φ and Un = (un , v n , ψ n , φn )T . With our notation, taking the inner product of iβn Un − AUn = Fn with Un yields iβn kUn k2 − (AUn , Un ) = (Fn , Un ) , and taking the real part and using (3.1) we get Z L α(x)|v n |2 dx = Re(Fn , Un ). 0

Noticing that (Un ) is bounded and that Fn → 0 we get Z L α(x)|v n |2 dx → 0 . 0

Now consider the equation −βn kUn k2 − i(AUn , Un ) = i(Fn , Un )

(3.6)


81

and using (3.1) we obtain −βn kUn k2 + i

Z

L

α(x)|v n |2 dx = i(Fn , Un ).

0

Finally, using that (Un ) is bounded, Fn → 0 and (3.6) we get −βn kUn k2 → 0, and from βn 6= 0 we conclude that kUn k2 → 0. We again have a contradiction and the proof of the theorem is complete.

Acknowledgments The second author was partially supported by CNPq-Brasilia - Brazil under grant 301025/2003.

References [1] Z. Liu, S. Zheng, Semigroups Associated with Dissipative Systems, Chapman, Hall, CRC (1999). [2] F.L. Huang, Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces, Ann. of Diff. Eqs., 1, No. 1 (1985), 43-56. [3] A. Wyler, Stability of wave equations with dissipative boundary condition in a bounded domain, Differential and Integral Equations, 7, No. 2 (1994), 345-366. [4] F. Ammar-Khodja, A. Benabdallah, J.E.M. Rivera, R. Racke, Energy decay for Timoshenko systems of memory type, Journal Of Differential Equations, 194, No. 1 (2003), 82-115. [5] A. Soufyane, Stabilisation Dynamique et Approximation Numérique de Problèmes de Contrˆ ole, Thesis, Universyty of Franche-Conté, Bensa¸con (1975). [6] C.A. Raposo, The Transmition Problem for Timoshenko Systems of Memory Type, Doctoral Thesis, Federal University of Rio de Janeiro – IM-UFRJ, Rio de Janeiro-Brazil (2001).

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[7] S. Zheng, Global solution to the Cauchy problem of a class of quasilinear huperbolic-parabolic coupled systems, Scientia Sinica, 4A (1987), 357372. [8] J.U. Kim, Y. Renardy, Boundary control of the Timoshenko beam, SIAM, J. Control and Optimizatin, 25, No. 6 (1987), 1417-1429. [9] J.E. Lagnese, J.L. Lions, Modelling Analysis and Control of Thin Plates, Masson (1988). [10] S.W. Taylor, A smoothing property of a hyperbolic system and boundary controllability, control of partial differential equation, J. Comput. Appl. Math., 114, No. 1 (2000), 23-40. [11] D.H. Shi, D.X. Feng, Exponential decay of Timoshenko beam with locally distributed feedback, In: Proceeding of the 99’IFAC World Congress, Volume F, Beijing. [12] A. Soufyane, A. Wehbe, Uniform stabilization for the Timoshenko beam by a locally distributed damping, In: Proceeding of the 99’IFAC World Congress, Volume F, Beijing. [13] C.A. Raposo, J. Ferreira, M.L. Santos, N.N.O. Castro, Exponential stability for the Timoshenko system with two weak dampings, Applied Mathematics Letters, 18 (2005), 535-541.