STABILIZATION OF 2-D MINDLIN-TIMOSHENKO PLATES WITH LOCALIZED ACOUSTIC BOUNDARY FEEDBACK

. In this paper, we investigate the well-posedness and the asymptotic stability of a two dimensional Mindlin-Timoshenko plate imposed the so-called acoustic control by a part of the boundary and a Dirichlet boundary condition on the remainder. We ﬁrst establish the well-posedness results of our model based on the theory of linear operator semigroup and then prove that the system is not exponentially stable by using the frequency domain approach. Finally, we show that the system is polynomially stable with the aid of the exponential or polynomial stability of a system with standard damping acting on a part of the boundary.


(Communicated by Chao Xu)
Abstract. In this paper, we investigate the well-posedness and the asymptotic stability of a two dimensional Mindlin-Timoshenko plate imposed the so-called acoustic control by a part of the boundary and a Dirichlet boundary condition on the remainder. We first establish the well-posedness results of our model based on the theory of linear operator semigroup and then prove that the system is not exponentially stable by using the frequency domain approach. Finally, we show that the system is polynomially stable with the aid of the exponential or polynomial stability of a system with standard damping acting on a part of the boundary.
1. Introduction. The stabilization of boundary value problem of deformed structures has been considerably stimulated during the past few decades by the wide applications of flexible materials in space technology and robot components (see [12,13,17,19,22,33]). In particular, structures of interest are largely carried out in problems which require appropriate feedback mechanisms to stabilize the vibrating materials that may be inherently unstable without external control or that only have very weak natural damping. We refer the reader to ( [6,8,10,16,24,35,36]) for some classic results on the well posedness and stabilizability by feedback controls. As a class of more comprehensive distributed parameter systems (Mindlin-Timoshenko Plate), the model contains shear effects in addition to displacement and rotational inertia effects, which reflects the vibration characteristics of thin plates in general case more accurately. Therefore, there is an extensive literature (such as Lagnese [20], Sare [34], Dalsen [11] and Messaoudi [25]) on the well-posedness and the asymptotic stabilization of Mindlin-Timoshenko beam or plate systems. The conservative two dimensional model is described as follows ψ tt , ϕ tt , ω tt − ρ 1 1 , ρ 1 2 , ρ 2 3 = 0, in Ω × (0, +∞), (1) where Ω ⊂ R 2 is an open bounded set and 0 denotes a three dimensional zero column vector. stands for the transpose symbol. 1 := 1 (ψ, ϕ, ω), 2 := 2 (ψ, ϕ, ω), and 3 := 3 (ψ, ϕ, ω) are given by (2) ρ 1 := 12 ρh 3 , ρ 2 := 1 ρh , and the positive constants ρ, h, K, D denote the density of the thin plate, thickness of the thin plate, the shear modulus and the modulus of flexural rigidity, respectively. µ represents the Poisson's ratio 0 < µ < 1 2 . The functions ω, ψ and ϕ, depending on (x, y, t) ∈ Ω×[0, +∞), symbolize the transversal displacement of the midplane of the thin plate and two shear angles respectively, (see [20], [21] for details).
In [20], Lagnese assumed the domain Ω satisfies Lipschitz continuity condition on the boundary Γ such that Γ = Γ 0 Γ 1 for two disjoint and relatively open sets Γ 0 , Γ 1 , where Γ 0 , Γ 1 are the closure of Γ 0 , Γ 1 , respectively. In this regard, the system (1) with the following boundary conditions was mainly discussed provided that Γ 1 = ∅ , i.e., on Γ 0 × (0, +∞), Here, ν := (ν 1 , ν 2 ) is referred to as the unit outward normal vector of the boundary Γ and the linear boundary feedbacks were defined as with F = [f ij ] 3×3 , a matrix of real L ∞ (Γ 1 ) functions, which is symmetric and positive semidefinite on Γ 1 . Based on the above conditions, it was proved that the system (1) is exponentially stable in the absence of any restrictions on the system coefficients. The same conclusion was obtained by Dalsen in [11], where the stability of the magnetoelastic Mindlin-Timoshenko plate model was investigated by introducing nonlinear locally supported damping p(x, y, ψ t , ϕ t , ω t ) = p 1 (x, y, ψ t ), p 2 (x, y, ϕ t ), p 3 (x, y, ω t ) into the interior of the plate, viz., (4) Provided that the nonlinear function p (x, y, ψ t , ϕ t , ω t ) satisfying the so-called "dissipativity assumptions" [11], the exponential stability of the energy of system (4) was obtained by means of Nakao's lemma [28]. However, this model is polynomially stable rather than exponentially stable, proved by Sare [34] who mainly considered putting the terms with frictional dissipation effects d 1 ψ t , d 2 ϕ t into rotation angle equations of this model with Dirichlet boundary conditions, namely, Since Russell and Zhang [32] first proposed the concepts of indirect damping mechanisms into the research on system stability, a large number of academic literatures have published in various journals(see [1,2,5,23,26,27,31,37] and the references therein). However, the models discussed in the previous literatures are either a one spatial dimension or a relatively simple one such that a spectrum method or Lyapunov method can be relatively easy to obtain the desired stability. Hence, the research method used in the above literatures is mainly based on the precise eigenvalue calculations or the concepts in system theory together with the theoretical achievements proposed by Huang, Prüss and Gearhart independently (see [15], [29], [30]). As we can see from (1) and (2), the Mindlin-Timoshenko plate system, a highly coupled partial differential equations (PDEs), is more complex than previous models. The stability of system (1) can hardly be obtained by the spectral method resulting from the impossibility of decoupling for this model.
In recent years, with the increasing demand of modern vehicles in the speed and comfort of riding, the vibration and acoustic radiation of plates and shell structures has attracted the widespread attention in engineering fields, such as vehicle, ship, aircraft cabin structure, etc. Therefore, the design of acoustic materials or structures for noise absorption, noise elimination and noise isolation has become an important necessity. Acoustic boundary controls may consist of a large number of acoustic point sources such as speakers, or a distributed acoustic actuation mechanism made of polyvinylidene fluoride (PVDF) or other advanced smart film materials. As a special kind of indirect dampings, they have been widely studied by Barucq et al. [4], Rivera1 [31] et al. and Beale [5] and others. To our knowledge, the stability of Mindlin-Timoshenko plate systems with such boundary conditions has not been studied by some author. Based on this case, the main goal of this paper is to apply the acoustic boundary conditions to the two dimensional Mindlin-Timoshenko plate and investigates its stability. In other words, we consider the following initial-boundary-value problem in Ω × (0, +∞), where 3. An n-order matrix over complex fields is expressed by M n×n (C). For the boundary Γ 1 , the symbol (·, ·) C n denotes an inner product in C n , which is defined as q 1 , q 2 C n =q 2 M (x, y) q 1 , for every q 1 , q 2 ∈ C n .
Here, the matrix function M (x, y) ∈ M n×n (C) is Hermitian positive-definite and satisfies Lipschitz condition on Γ 1 . The associated norm is denoted by · C n . The n-dimensional column vector function δ (j) := δ (j) (x, y, t) (j = 1, 2, 3) is referred as acoustic control variables. In this work, we assume that the matrix B j (x, y) holds on the boundary Γ 1 B j x, y q 1 , q 2 C n ≤ 0, j = 1, 2, 3, for every q 1 , q 2 ∈ C n .
In addition, the energy of system (5) is defined as and For all sufficiently smooth [ψ, ϕ, ω] and [ψ * , ϕ * , ω * ], the following equation holds using integration by parts Based on the complexity of the model studied in this paper, the spectral analysis method can not be applied to the following discussion. Therefore, our approach, avoiding spectrum calculations, is based on a contradiction argument occurring in a special analysis for the resolvent on the imaginary axis. Furthermore, we deduce some estimations of energy components from the resolvent equation of system (1). Combining the equivalent conditions for the polynomial stabilization theorem of semigroup given by Borichev et al. [7] and the above estimates, the polynomial decay of energy (8) can be proved. As far as we know these estimates were not given in the literature. Compared with other related works (see [6,10,16,24,34,35,37]), our main contributions are summarized as follows: 1) A class of acoustic boundary controls derived from theoretical acoustics are applied to the stability research of a Mindlin-Timoshenko plate. This acoustic boundary controller introduces extra degrees of freedom in designing controllers which could be exploited in solving a variety of control problems, such as disturbance rejection, pole assignment, etc.
2) Based on the high degree of coupling of this model, we only use the frequency domain methods rather than Lyapunov methods or spectral methods to obtain many intensive estimations which are applied to analyse the stability of Mindlin-Timoshenko plate.
3) With some reasonable hypotheses, we prove that the Mindlin-Timoshenko plate system with a localized acoustic boundary condition is not exponentially stable but polynomially stable.
The remaining part of this paper is organized as follows: the existence and uniqueness of the solution of system (5) using the contraction semigroup theory are given in Section 2. In Section 3, we shall prove the non-exponential stability of system (5) via the theory originated from Huang [18] and Prüss [30] in their research of the generalized first order linear evolution equation. Finally, the system (5) is demonstrated to be polynomial stability subject to acoustic control conditions in Section 4. In section 5, we present some concluding remarks.
2. Statement of the well-posedness of global solution. In this section we study the existence and uniqueness of strong and global solutions for the system (5) by using the semigroup theories. We use the usual notation H k (Ω) or H k 0 (Ω) as indication of general Sobolev spaces of order k on a regular domain. For the purpose of endowing with a norm associated with the energy (8), we use the the following estimates derived from Korn's inequality in [20], i.e., Moreover, for every θ 0 > 0, there exists ζ := ζ (θ 0 ) > 0 such that for all θ > θ 0 and for every [ψ, ϕ, ω] ∈ H 1 0 (Ω) 3 , Remark 1. From the results of Lemma 2.1, we can define a complex Hilbert space W whose norm is equivalent to the usual norm in H 1 (Ω) 3 , that is and the inner product (2) , ϕ (2) , ω (2) ] ∈ W . In the sequel, we omit the transposed mark for simplicity of notation if there is no ambiguity in calculation.
We first need to establish the following function spaces and partial derivative operators for investigating the well-posedness and stability of the solution of system (5).
In the sequel, we prove that there exists a positive number λ such that λI − A is surjective. That is to say that there is U = [ψ, ϕ, ω; φ, υ, η; Suppose that there exists a U ∈ D (A ) such that (22) holds. Hence, when λ / ∈ σ(B j ) (j = 1, 2, 3), then we successively obtain Hence, we only look for [ψ, ϕ, ω] ∈ W satisfying and the boundary conditions defined in the domain D(A ).
To complete the proof of Proposition 1, it remains to show that the equation (24) has a unique weak solution.
Remark 2. From the above proof, it can be seen that if 0 is not included in the eigenvalue of B j for all (x, y) ∈ Γ 1 , then A is a one-to-one mapping and A −1 is bounded which implies that A is a closed operator. Thus, we reach the conclusion that A generates a C 0 -semigroup e tA of contractions on H by using the Lumer-Phillips theorem [29]. 3. The Lack of Exponential Stability. In this section, we show that the system (5) is not exponentially stable by using the equivalent conditions for exponential stability of C 0 -semigroups in a Hilbert space. This correlative results were mainly described as follows (see Huang [18], Prüss [30] and Gearhart [15]).
Corollary 2. According to the self-adjointness of A, it is easy to verify, by (37), that A 1 is skew-self-adjoint in W × V , i.e., Lemma 3.3. The resolvent of A is compact.
Here, (A 1 ) denotes the resolvent set of A 1 . Suppose that {X n } n≥1 ⊂ W × V is a bounded sequence, that is, there exists a positive constant M such that X n 2 W ×V ≤ M . Let Z n = (I − A 1 )X n , then we thus have By applying Sobolev embedding theorem, we know that W → V is a compact embedding. Hence, Z n has convergent subsequences, which indicates the compactness of (I − A 1 ) −1 .
From Lemma 3.4 and Theorem 3.1, we know that system (5) satisfies one of two equivalent conditions for exponential stability. To complete the proof of nonexponential stability of system (5), what we need to do next is to verify that (30) does not hold. In order to achieve this purpose, we first introduce a positive selfadjoint operator A 1 given by Denote by {λ 2 n } n≥1 the (discrete) spectral set of A 1 and let Φ n = [ψ n , ϕ n , ω n ] be the corresponding orthogonal eigenvectors. By applying Lemma 3.2, it can be seen that λ n is positive and goes to +∞, as n approaches infinity.

(59)
Clearly, it follows from the definition of Π n that Π n H ≥ Φ n V = 1.
Moreover, we also obtain Therefore, by using the boundness of B j and C j , we have where c 1 > 0 is independent of Φ n and Γ 1 .
From (53), the estimate (60) is simplified into which completes the proof.
Theorem 3.5. The C 0 -semigroup generated by A in H is not exponentially stable.
Proof. We let (iβ n − A )Π n = Ψ n . According to Proposition 3, we get This contradicts (30). We thus complete the proof of Theorem 3.5.
4. Polynomial stability. In this section, the polynomial stability of the semigroup solution of system (5) can be obtained by using a part of fundamental theories in [7].
Theorem 4.1. Let T (t) t≥0 be a bounded C 0 -semigroup on a Hilbert space H with generator A such that (A) ⊃ iR. Then for a fixed α > 0, the following conditions are equivalent: We consider the auxiliary system in Ω × (0, +∞), We let the space H 1 = W × V and introduce a new operator A 3 given by with the domain In what follows, we prove that the system (5) is polynomially stable with the aid of the exponential or polynomial stability of system (63). Moreover, the decay rate of system (5) depends on the type of stability of (63).
Proposition 4. Suppose that the energy of system (63) is exponentially stable and there exist constants p > 0 and l > 0 such that for s ∈ R with |s| large enough satisfying then the energy of system (5) has a polynomial decay estimation, i.e., where C is independent of U 0 .
Proposition 5. Assume that the energy of system (63) is polynomially stable and there exist constants p > 0 and l > 0 such that for s ∈ R with |s| large enough, (65) holds. Then, the energy of the solution of system (5) satisfies a polynomial decay, i.e., where C > 0 and η > 0 are all independent of U 0 .
Proceeding as in the proof of Proposition 4, we obtain With a slight modification of the estimate in (83) and (85)  This implies that the estimate (92) is proved based on the results of Theorem 4.1.
Finally, we will clarify the sufficient conditions for B j and C j such that (65) is tenable.
Proof. In fact, by the conditions of (93), we can deduce that F j (B r j C j ) = 0 for all r < d. On the other hand, by the definition of S j , we know that S j is self-adjoint and S j C j = S j F j C j . Moreover, there exists C 25 > 0 such that − (S j F j q, F j q) C n ≥ C 25 q 2 C n , for all q ∈ C n .
Indeed, by the continuity of B j together with (6) and (7), we obtain from (95) and (96), we thus have The proof is complete. 5. Conclusion. In this paper, we have presented the nonuniform stability in a 2-D Mindlin-Timoshenko plate with acoustic boundary control conditions. In this case, we have further discussed and proved that the energy of this model is polynomially stable under this boundary conditions. The continuing work is focused on the numerical simulation to verify the validity of the above conclusions or on the stability of the Mindlin-Timoshenko plate with acoustic boundary conditions of thermal effects. Future work will deal with other forms of thin plates, cylindrical shells and beams, with this type of control applied on a part of the boundary.