Uniform stabilization of a multilayer Rao-Nakra sandwich beam

We consider the problem of boundary feedback stabilization of a multilayer Rao-Nakra sandwich beam. We show that the eigenfunctions of the decoupled system form a Riesz basis. This allows us to deduce that the decoupled system is exponentially stable. Since the coupling terms are compact, the exponential stability of the coupled system follows from the strong stability of the coupled system, which is proved using a unique continuation result for the overdetermined homogenous system in the case of zero feedback.


Introduction.
A sandwich beam is an engineering model for a three-layer beam consisting of stiff outer face plates and a more compliant inner core layer. Sandwich beam models found in the literature include the models of Mead and Markus [18], Rao and Nakra (RN) [24], Yan and Dowell [30] and others. The RN model assumes continuous, piecewise linear displacements through the cross-sections, with the Kirchhoff hypothesis imposed on the face plates. Transverse, longitudinal and rotational inertial forces are included in the modeling. In [7] several possible multilayer generalizations of the basic three layer sandwich beam structure are derived and analyzed (in the form of multilayer plates). In this paper we consider a multilayer generalization of the RN model described in [7]. The model consists of 2m + 1 alternating stiff and complaint (core) layers, with stiff layers on the outside. The stiff layers assume the Kirchhoff hypothesis, while the compliant layers admit shear. The equations of motion for the associated beam model can be written: where Ω = (0, L), primes denote differentiation with respect to the spatial variable x and dots denote differentiation with respect to time t.
In the above, z represents the transverse displacement, φ i denotes the shear angle in the i th layer, φ E = [φ 2 , φ 4 , . . . , φ 2m ] T , v i denote the longitudinal displacement along the center of the i th layer, and v O = [v 1 , v 3 , . . . , v 2m+1 ] T . Throughout this paper we use the convention that quantities relating to the stiff layers have odd indices 1, 3, . . . 2m + 1 and quantities relating to the even layers have even indices 2, 4, . . . 2m. In addition, m, α, K are positive physical constants, and where ρ i , h i , E i , G i denote the density, thickness, Young's modulus, and shear modulus of the i th layer, respectively. The vector N is defined as The aim of this paper is to prove that the uniform exponential stability of the RN system with standard boundary damping applied at one end point. Consider (1) with the following boundary conditions and the initial conditions where Υ O = diag (γ 1 , γ 3 · · · , γ 2m+1 ), and γ i ∈ R + , i = 0, 1, 3, . . . , 2m + 1 denote constant positive feedback gains. Throughout the paper, we assume α K = γ 0 and ρ k E k = γ k for k = 1, 3, . . . , 2m + 1.
1.1. Background. Boundary controllability of (1) has been studied in several papers. For the three layer case, in [22] the multiplier method was used to prove exact controllability with a control for each equation applied at an end point. The moment method was used in [12] to obtain boundary controllability for the multilayer case, but with the condition that wave speeds of the layers be distinct. The same approach was used to prove simultaneous controllability (i.e., with one boundary control instead of three) for the three layer case in [11]. In [20] exact boundary controllability of the general multilayer system was proved for a variety of boundary conditions: clamped, hinged, clamped-hinged, and hinged-clamped. The results in [20] improve earlier results in that there are no restrictions on the wave speeds or the size of G and moreover, exact controllability is proved in the optimal time (determined by characteristics). In [9], [10] exact controllability results for the multilayer RN plate system analogous to (1) with locally distributed control in a neighborhood of a portion of the boundary were obtained by the method of Carleman estimates. Stability results for layered beam systems closely related to (1) subject to internal damping proportional to rate-of-shear in one or more layers have been studied in several papers; [2], [3], [8], [27]. In particular, the approach used in [8] was successfully applied in the dissertation [1] to obtain uniform exponential stability results the system (1) with rate-of shear damping included in the compliant layers.
Concerning boundary feedback stabilization of layered beam models, spectral methods (based on the Riesz basis property) are applied in [28] to prove exponential stability results for a laminated beam model in [13]. A similar approach is used in [29] for the Mead-Markus model described in [5]. There are also several results concerning the boundary feedback stabilization of a single Rayleigh beam equation e.g., [23], where a uniform exponential decay result is obtained for a Rayleigh beam by means of a compact perturbation argument and [6] where the Riesz basis approach is used to obtain a similar stabilization result. Some related uniform stabilization results for the Kirchhoff plate are proved in [16].
Our main result is the following.
In the above, A, ε(t), and H are defined in (5), (6), and (7) respectively. Our methodology in this paper is a combination of techniques used in [23], [26], and [28]. The decoupled system (i.e., (1), with G E ≡ 0) consists of a Rayleigh beam equation and (m + 1) wave equations. We prove that the decoupled system has a Riesz basis of eigenfunctions and obtain explicit asymptotic estimates on the eigenvalues. In particular, the eigenvalues of the decoupled system asymptotically lie along a finite number of vertical lines in the left half plane. We are able to prove that the family of eigenfunctions and generalized eigenfunctions of the decoupled system form a Riesz basis and consequently (see [25]), the spectrum determined growth condition holds. This allows us to prove the exponential stability of the decoupled closed-loop system (see Theorem 3.8). We mention that the exponential stabilization for the portion of the uncoupled system corresponding to the wave equations is well-known results, e.g., [4], [14], [15], [17]. Furthermore a number of results are known for stabilization of the Rayleigh beam, e.g. [6], [16], [23], however none of these results are applicable to the Rayleigh beam with clamped-hinged boundary conditions which we consider. Therefore we include a detailed proof of the exponential stability for the Rayleigh beam. Next, we prove that the system (1)-(3) has a compact resolvent and is a compact perturbation of the decoupled system. Therefore, exponential stability of (1)-(3) follows from a perturbation theorem due to Triggiani [26] once it is shown that the semigroup generated by A is strongly stable (see Theorem 4.2). Proving the strong stability involves use of dissipativity of the semigroup together with a nontrivial unique continuation argument that is proved in [20] in application to the associated boundary control problem.
Our paper is organized as the following. In Section 2, we give a semigroup formulation of (1)-(3). We prove that the semigroup is a C 0 −semigroup of contractions on an appropriate Hilbert space. In Section 3, we first prove that the generalized functions corresponding to the single Rayleigh beam equation with the feedback applied to the moment forms a Riesz basis. Then, we show that the decoupled system, i.e. G E ≡ 0 in (1), has the Riesz basis property. Finally, we show that the decoupled system is exponentially stable. In Section 4, we prove that the system (1) is a compact perturbation of the decoupled system. Finally, we prove our main stabilization result in Theorem 1.1. .
Let u, v Ω = Ω u · v dx where u and v may be scalar or vector valued. Define the bilinear forms a and c by The natural energy of the beam is given by where a(·), c(·) are the quadratic forms that agree with a(· ; ·), c(· ; ·) on the diagonal. We define the Hilbert space H by with the energy inner product where

Characterization of the domain of
In the above, Hence, using the definition of a in (5), we find that the following identity holds: We use (9) as a basis for the variational definition of the space D(A). More precisely, (9) holds}.
Note that the operator A : D(A) ⊂ H → H is densely defined.
Proof. By an easy calculation we have the following with the boundary conditions has only the trivial solution.
Then T is a densely defined, self-adjoint, positive definite, and unbounded operator on (L 2 (Ω)) m , and therefore (h E T + P G E ) −1 exists and is a bounded operator defined on all of (L 2 (Ω)) m where Then J extends to a continuous and self-adjoint operator on (L 2 (Ω)) m , and To show this, let s = Jz = −(h E T + P G E ) −1 Tz so that s ∈ Dom(T) and Tz = −h E Ts + P G E s. Then By applying (h E T + P G E ) −1 to both sides of (20), we get (19).  (17) yields But since J is non-positive, the operator KD 4 is a positive operator (using (18)). This implies that u = 0. Therefore v O = 0 by (17).
We have the following theorem for the well-posedness of the Cauchy problem (1)-(3). 3. Uniform stabilization of the decoupled system, i.e. G E ≡ 0. In this section, we prove the exponential stability of the decoupled system: with initial and boundary conditions 3.1. Semigroup formulation. Let Then the semigroup corresponding to (21) is given by Define the bilinear forms a d and c d by where a d (·), c d (·) are the quadratic forms that agree with a(·; ·), c(·; ·) on the diagonal. The corresponding energy inner product on H is given by  (9). The proof of the Theorem 2.4 remains valid when G E ≡ 0, and hence Theorem 3.1 follows. Now we find the adjoint operator A * d which is needed in the proof of Lemma 3.5. Lemma 3.2. The infinitesimal generator A d satisfies Then, U 1 and U 2 satisfy the following boundary conditions A calculation using (24) shows that Now let λ = is 0 . Then the solution of (25) is where θ 0 (s 0 ) = By using the first three boundary conditions u(0) = u ′ (0) = 0, and u ′′ (L) + is 0 γ 0 u ′ (L) = 0 for (26) we get By using the last boundary condition u ′ (L) = 0, we obtain the characteristic equation that s 0 satisfies Since we have θ 0 ξ 0 = s 2 K , by (27), we find that Multiplying (29) (30) and (31) yields cos θ 0 L + iγs 0 θ 0 sinh ξ 0 L = 0, as s 0 → ∞ (or θ 0 → ∞). (32) The following theorem characterizes the eigenvalues of (25).
Theorem 3.3. Assume (4). The eigenvalues {λ ± 0,n } of (25) for sufficiently large n consist of complex conjugate pairs λ − 0,n and λ + 0,n with asymptotic form λ + 0,n = i K α σ 0,n + O( 1 n ) as n → ∞ where Proof. First, note that {σ 0,n } n∈Z+ are the solutions of (33) when the right hand side of the equation is zero. We claim that {σ 0,n + O( 1 n )} solve (33) for all sufficiently large n ∈ Z + . Without loss of generality, we only consider the case By a simple calculation one easily obtains the following for sufficiently large n : where we have used (29), (33) and (34). Therefore, by Rouché's theorem, f (θ 0 ) + g(θ 0 ) has a unique zero in the ball B n for sufficiently large n. That is, there exists a unique solution of the equation (33) in B n . This proves our claim and hence λ + 0,n = i K α σ 0,n + O( 1 n ) as n → ∞ by (30).
Case I. Let γ 0 > α K . By (30) we have When n is odd, it is clear that the complex number  has a nonzero (but constant) real part. Therefore ζ n = O(1) = 0.
Proof. We first prove that the eigenfunctions {(e * n , λ * 0,n e * n ) T , n ∈ Z} of the adjoint eigenvalue problem (see (38) below), are biorthogonal to the eigenfunctions {(e n , λ 0,n e n ) T , n ∈ Z} of (25). By using Lemma 3.2, we consider the following adjoint eigenvalue problem: This is exactly the same boundary value problem as (25). Therefore λ * 0,n = λ 0,n . The only difference is the expression of the eigenfunctions of (38) given by {(e * n , λ * 0,n e * n ) T , n ∈ Z} = {(e n , −λ 0,n e n , ) T , n ∈ Z}. It is possible to check that is uniformly bounded (from (35) each term has a uniform asymptotic bound). Hence a uniform bound for (39) exists. Therefore, {(e n , λ 0,n e n ) T , n ∈ Z} is ω−linearly independent in H 2 # (Ω) × H 1 0 (Ω). This proves the first part of Lemma 3.5.
Hence (e n , λ 0,n e n ) T , n ∈ Z forms a Riesz basis in H 2 # (Ω) × H 1 0 (Ω). The following theorem can be obtained by the same procedure.
Then, the eigenfunctions of (41) (e k,n , λ k,n e k,n ) T , k = 1, 3, . . . , 2m + 1, n ∈ Z corresponding to the branches of eigenvalues {∪{λ k,n }, k = 1, 3, . . . , 2m+1, n ∈ Z} forms a Riesz basis in H 1 * (Ω) (m+1) × (L 2 (Ω)) (m+1) where e k,n = (0, . . . , e k,n , . . . , 0) T , e k,n = θ −1 k,n sin θ k,n x, λ k,n = i E k ρ k θ k,n for all n ∈ Z, and Theorem 3.8. Assume (4). Then the semigroup generated by A d is exponentially stable on H, i.e., ∃M > 0 such that Proof. The Riesz basis property for the Rayleigh beam equation (Theorem 3.6) together with Riesz basis property for the system of wave equations (Theorem 3.7) imply that the eigenfunctions {(e n , e k,n , λ k,n e n , λ k,n e k,n ) T , n ∈ Z} of the operator A d form a Riesz basis in H. Hence, as is well known, the growth bound for the associated semigroup is determined by spectrum of the generator. We know from Theorems 3.3 and 3.7 that the eigenvalues {λ k,n , k = 0, 1, 3, . . . , 2m + 1, n ∈ Z} of A d have the expressions λ 0,n = i K α σ 0,n +O( 1 n ) as n → ∞ and λ k,n = i E k ρ k θ k,n for all n ∈ Z + . Furthermore since A d is dissipative, all eigenvalues have non-positive real parts. Hence, if we show that there are no eigenvalues on the imaginary axis, then the theorem is proved. For the wave equations, this is trivial to show, and is well-known. For the boundary conditions we have, for the Rayleigh beam, the possibility of imaginary eigenvalues lead to the following overdetermined eigensystem This system was shown in [19] to have only the trivial solution. Therefore {e A d t } t≥0 is an exponentially stable semigroup on H, and (42) holds. 4. Uniform stabilization of the coupled system. In this section, we show that one boundary feedback for each equation is enough to obtain the uniform stabilization of the multilayer RN beam. First, we will consider the decomposition A = A d + B of the semigroup generator of the original problem (5) where A d is the semigroup generator of the decoupled system and it is defined by (23), and the operator B : H → H is the coupling between the layers defined as the following where φ E = h −1 E Bu + N u ′ When (u, u, v, v) T ∈ H, we have u ∈ H 2 # (Ω) and u ∈ (H 1 * (Ω)) (m+1) , and therefore φ E ∈ (H 1 * (Ω)) (m+1) . Since L : H 2 (Ω) ∩ H 1 0 (Ω) → L 2 (Ω) is an isomorphism, the last terms in (43) satisfy which are compactly embeddded in H 1 0 (Ω) and (L 2 (Ω)) (m+1) , respectively. Hence the operator B is compact in H. Proof. We know that our system (1) with initial and overdetermined boundary conditions u(0) = u ′ (0) = u(L) = u ′ (L) = u ′′ (L) = 0 u(0) = u(L) = u ′ (L) = 0 has only the trivial solution, i.e. u = 0, u = 0. This same overdetermined system came up in proving observability for the corresponding boundary control problem in [20], where the uniqueness of the zero solution was proved using a multiplier type argument. Now we prove our main theorem for the exponential stability of the solutions (1)-(3): Proof of Theorem 1.1. We know that A = A d + B. The semigroup {e (A d +B)t } t≥0 is strongly stable on H by Theorem 4.2 and the operator B is a compact in H by Lemma 4.1. Therefore, since the semigroup generated by (A d + B) − B is uniformly exponentially stable in H then the semigroup {e (A d +B)t } t≥0 = {e At } t≥0 is uniformly exponentially stable in H by e.g., the perturbation theorem of Triggiani [26].