Optimal indirect stability of a weakly damped elastic abstract system of second order equations coupled by velocities

In this paper, by means of the Riesz basis approach, we study the stability of a weakly damped system of two second order evolution equations coupled through the velocities. If the fractional order damping becomes viscous and the waves propagate with equal speeds, we prove exponential stability of the system and, otherwise, we establish an optimal polynomial decay rate. Finally, we provide some illustrative examples.


Introduction
In this paper, we investigate the energy decay rate of the following abstract system of second order evolution equations (1.1) u tt + aAu + A γ u t + αy t = 0, where a > 0, γ ≤ 0, α ∈ R * is the coupling parameter and A is a self-adjoint, coercive operator with a compact resolvent in a separable Hilbert space H and with simple spectrum. The fractional damping term A γ u t is only applied at the first equation and the second equation is indirectly damped through the coupling between the two equations. The fractional order damping of the type A γ , arising from the material property, has been introduced in [16] and, in the cases γ ∈ {0, 1 2 , 1}, is referred to as the so-called viscous damping, square-root (or structural) damping, and Kelvin-Voigt damping respectively. If γ = 1 2 , it was shown in [16] that the semigroup corresponding to the damped elastic model u tt + Au + A γ u t = 0, is analytical, while the subsequent works in [17] and [18] showed that the semigroup is still analytical for 1 2 ≤ γ ≤ 1 but is only of Gevrey class for 0 < γ < 1 2 .
In [29], Liu and Zhang studied the energy decay rate of the weakly damped elastic abstract system described by (1.2) u tt + Au + Bu t = 0, where A is a self-adjoint, positive definite operator on a Hilbert space H. The dissipation operator B is another positive operator satisfying cA γ u ≤ Bu ≤ CA γ u for some constants 0 < c < C. When γ < 0, they proved that the energy of System (1.2) has a polynomial decay rate of type t 1 γ and that this decay rate is in some sense optimal. Regarding System (1.1) when α = 0, it reduces to System (1.2) with B = A γ and a = 1. In this case, we recover the results of [29].
When the coupling acts through displacements, Loreti and Rao studied in [31] the stability of the following abstract system of coupled equations (1.3) u tt + Au + A γ u t + αy = 0, y tt + Ay − αu = 0.
They proved that System (1.3) is not exponentially stable and an optimal polynomial energy decay rate of type t −τ (γ) is obtained where where U ∈ H and A is an unbounded operator on H. We use ·, · and · to denote the inner produc and the induced norm respectively on H. Since the resolvent of A turns out to be compact in H, there exists a non decreasing sequence (µ n ) n≥1 tending to infinity and an orthonormal basis (e n ) n≥1 of H such that, for n ≥ 1, Ae n = µ 2 n e n . We assume the spectrum of A is simple, i.e., the sequence (µ n ) n≥1 is increasing. Our goal in this paper is to establish the optimal stability of System (1.1) using the spectral method for the operator A. For this aim, we study the effect of both the fractional order damping of type A γ and the speeds of the two wave equations on this spectrum and we prove that the latter is made of two branches (λ 1,n ) n∈N and (λ 2,n ) n∈N , whose asymptotics, as n tends to infinity, are given next. Case 1. Assume that γ = 0. If a = 1, i.e., when the two waves propagate with equal speed, we prove that the spectrum of A has an asymptotic expansion, as n tends to infinity, given by λ ± 1,n = ± iµ n − 1 4 + 1 4 1 − 4α 2 + o(1) and λ ± 2,n = ± iµ n − see Lemma 3.2). Note here that if 4α 2 > 1, √ 1 − 4α 2 actually denotes the imaginary number i √ 4α 2 − 1. We then prove that the energy of the system is (uniformly) exponentially stable. If a = 1, i.e., when the waves propagate with different speeds, we show that the spectrum of A has asymptotic expansion, as n tends to infinity, given by see Lemma 4.3). Thus, the real part corresponding to the first branch of eigenvalues is uniformly bounded and the real part corresponding to the second branch of eigenvalues is of magnitude µ −2 n . Therefore, we prove that the total energy decays at the optimal rate 1/t. Case 2. Assume that γ < 0. If a = 1, then the real parts of λ ± 1,n and λ ± 2,n are of magnitude µ 2γ n (see Lemma 4.2) and the total energy decays at the optimal rate t 1 γ . If a = 1, then the real part corresponding to the first branch of eigenvalues is of magnitude µ 2γ n and the real part corresponding to the second branch of eigenvalues is of magnitude µ 2γ−2 n (see Lemma 4.3), yielding a decay rate of the optimal total energy equal to t − 1 1−γ .
From the above results, we deduce that the maximum decay rate is achieved when γ tends to zero. Therefore, a stronger damping term A γ u t does not necessarily give a better decay rate of the total energy, as it is expected. A good damping term should transmit the damping from one wave to another before the directly damped wave dies out or loses its energy. This effect of a good damping term is interpreted by the real parts of the eigenvalues. This paper is organized as follows. In Section 2, we set the framework of System (1.1) and we establish the characteristic equation satisfied by the eigenvalues of the operator A. Next, in Section 3, relying on the spectrum method, we prove the exponential stability of System (2.1) when a = 1 and γ = 0. In Section 4, we consider the other cases of a and γ. We prove the optimal polynomial energy decay rate of type t −δ(γ) of System (2.1), where δ (γ) = − 1 γ , if a = 1 and γ < 0, 1 1−γ , if a = 1 and γ ≤ 0. Finally, in Section 5, we examine some applications for our study.

Characteristic equation and Riesz basis method
In this paper, we consider the following abstract system of second order evolution equations given by (2.1) u tt + aAu + A γ u t + αy t = 0, where a > 0, γ ≤ 0, α ∈ R * , the operator A is a self-adjoint coercive operator with compact resolvent in a separable Hilbert space H. Let us define the energy space where · denotes the norm in H. We define the linear unbounded operator A in H by Therefore, we can write System (2.1) as an evolution equation of A are uniformly bounded. Then, (e tA ) t≥0 is exponentially stable if and only if its spectral bound s (A), defined as is negative.
If the semigroup fails to be exponentially stable, we search for another type of decay rate such polynomial stability. In that case, the following proposition from [31] provides a useful way to even characterize optimal polynomial stability.
Proposition 2.4. (cf. Theorem 2.1 in [31]). Let (e tA ) t≥0 be a C 0 -semigroup of contractions generated by the operator A on a Hilbert space H. Let (λ k,n ) 1≤k≤K, n≥1 denotes the k-th branch of eigenvalues of A and {e k,n } 1≤k≤K, n≥1 the system of eigenvectors which forms a Riesz basis in H. Assume that for each 1 ≤ k ≤ K there exist a positive sequence (µ k,n ) n≥1 tending to infinity and two positive constants α k ≥ 0, β k > 0 such that and |Im(λ k,n )| ≥ µ k,n ∀n ≥ 1.
Then, for every θ > 0, there exists a constant M > 0 such that, for every u 0 ∈ D(A θ ), one has Moreover, if there exists two constants c 1 > 0, c 2 > 0 such that then (e tA ) t≥0 is polynomially stable with optimal decay rate t −δ , where δ is given in (2.5).
In this work, to check the decay rate, we rely on the Riesz basis method in which we first determine the characteristic equation satisfied by the spectrum. Since the resolvent of A is compact in H, there exists an increasing sequence (µ n ) n≥1 tending to infinity and an orthonormal basis (e n ) n≥1 of H such that (2.6) Ae n = µ 2 n e n ∀ n ≥ 1. In turn, to study the spectrum of System (2.1), let λ be an eigenvalue of the operator A and U = (u, v, y, z) T a corresponding eigenvector. Therefore, we have AU = λU.
Equivalently, we have the following system Similar to the analysis done in [31], we will see in Proposition 3.6 and in Proposition 4.8 that, for every n ≥ 1, there exists (B n , C n ) = (0, 0) such that the eigenvector U of A is of the form (2.8) u = B n e n , v = λB n e n , y = C n e n , z = λC n e n .
Inserting (2.8) in (2.7) and using (2.6), we obtain (2.9) aµ 2 n + λ 2 + λµ 2γ n B n e n + αλC n e n = 0, −αλB n e n + µ 2 n + λ 2 C n e n = 0, which has a non-trivial solution (B n , C n ) = (0, 0) if and only if λ is a solution of the equation (2.10) aµ 4 n + λ (a + 1) λ + µ 2γ n µ 2 n + λ 2 λ 2 + λµ 2γ n + α 2 = 0, that we refer to as the characteristic equation associated with the eigenvalue µ 2 n of A. The four roots of this equation are eigenvalues of A and called the eigenvalues of A corresponding to µ n of A. We also have the following result. Lemma 2.5. Let λ n = λ ± j,n , j = 1, 2 be one of the fourth eigenvalues of A corresponding to µ n . Then, there exists two positive constants m, M , such that, for n large enough, Proof. Set Z n = λn µn . Then, from (2.10), one has that Z n is one of the four roots of the polynomial f n of degree four given by Let g be the the polynomial of degree four given by g(Z) = Z 4 + (a + 1)Z 2 + a, which has exactly four non zero roots. Since the coefficients of f n converge to those of g as n tends to infinity, one gets the result.

Exponential stability
In this Section, we consider the case where a = 1 and γ = 0. Our main result is the following theorem. For the proof of Theorem 3.1, we first need to study the asymptotic behaviour of the spectrum of A and, in that direction, we have the following Lemma.
Lemma 3.2. Assume that a = 1 and γ = 0. Then, for n ≥ 1 large enough, the four eigenvalues of A corresponding to the eigenvalue µ 2 n of A and denoted λ ± 1,n , λ ± 2,n , satisfy the following asymptotic expansions Case 2. If α 2 > 1 4 , then Proof. We divide the proof into two cases.
We next provide the form of the eigenvectors and root vectors of A. We start with the following Lemma.
Lemma 3.3. If a = 1 and γ = 0, then the eigenvectors of A take the following form.
We now search for the asymptotic behaviour of the eigenvectors of A. From Lemma 3.2, we remark that if α 2 = 1 4 , we have double eigenvalues. In this case, we look for the corresponding root vectors. Lemma 3.4. If a = 1 and γ = 0, then the eigenvectors e ± 1,n , e ± 2,n of A satisfy the following asymptotic expansion.
Case 2. If α 2 = 1 4 , we suppose that α = 1 2 since the analysis follows similarly if α = − 1 2 , then the eigenvectors e ± n of A satisfy the following asymptotic expansion Case 3. If α 2 > 1 4 , then we have Proof. When a = 1 and γ = 0, we must now subdivide the proof into three cases.
Let now E ± 1,n , E ± 2,n be linearly independent eigenvectors of the decoupled system (corresponding to α = 0). Then one has Moreover, for every n ≥ 1, define From what precedes, one gets the following corollary.
In the sequel, our aim is to prove the following proposition. To prove Proposition 3.6, we first recall Lemma 3.1 in [31].  Assume furthermore that there exist a Riesz basis {f n,i } 1≤i≤In (I n ≤ +∞) in each X n and positive constants c, C independent of n such that, for every x n = In i=1 α n,i f n,i , one has Then the sequence forms a Riesz basis in H.
Proof of Proposition 3.6. First, we prove that {W n } n≥1 , defined in (3.29), is a Riesz basis of subspaces of H. Let U = (u, v, y, z) ⊤ ∈ H.
Since (e n ) n≥1 is a Hilbert basis of H, then v n e n y = n≥1 y n e n z = n≥1 z n e n . Hence, (u n e n , v n e n y n e n , z n e n ) T form a Riesz sequence of subspaces in H. Next, we divide the proof into three cases: 0 < α 2 < 1 4 , α 2 = 1 4 and If α 2 > 1 4 . Since the argument is entirely similar for the three cases, we only provide one of them.
If α 2 = 1 4 , then from Corollary 3.5, we remark that L n has a constant leading term which is invertible. This, together with the fact that L n is invertible for every n ≥ 1, implies condition (3.37). Moreover, the condition (3.38) is satisfied since {E + 1,n , E − 1,n , E + 2,n , E − 2,n } forms a Hilbert basis in the subspace W n . Then applying Proposition 3.7, we obtain that the system of eigenvectors e + 1,n , e − 1,n , e + 2,n , e − 2,n n≥1 forms a Riesz basis in H. Thus, the proof is complete.
Proof of Theorem 3.1. From Lemma 3.2, the large eigenvalues λ ± k,n k=1,2, n≥n0 of A satisfy the following and Im λ ± k,n ≥ µ n .
In addition to that, from Proposition 3.6, the system of eigenvectors and root vectors of A forms a Riesz basis of H. Then, applying Proposition 2.3, we get that System (2.1) is exponentially stable. Thus, the proof is complete.
Before we study the asymptotic behavior of the eigenvalues in case a = 1 and γ ≤ 0, we prove the following lemma. We now study the asymptotic behavior of the eigenvalues in the case when a = 1 and γ ≤ 0. We prove the following lemma.
For the proof of Lemma 4.3, we need the following lemmas.
Next, when −2γ ≥ 1, we try to replace the powers of λ n in (4.20) and in (4.21) with powers of µ n as shown in the following lemma.
Our next goal is to prove (4.55). From (4.56), we get Therefore, we have From (4.63) we can find a real number F 2 (N + 1) depending on a and α such that Inserting (4.64) in (4.21), we get (4.55).
We now study the asymptotic behavior of the eigenvectors in the different cases a = 1 and γ < 0 or a = 1 and γ ≤ 0. We prove the following lemma.
Similar to Proposition 3.6, we can prove the following proposition. Therefore, by Proposition 2.4, we get (4.1) where δ (γ) = 1 1 − γ . Furthermore, from Proposition 4.8, the system of eigenvectors of A forms a Riesz basis in H. Then, applying Proposition 2.4, we get the optimal polynomial energy decay rate given in (4.1). Thus, the proof is complete.

Examples
Let Ω ⊂ R N be a bounded open set with a smooth boundary Γ. where γ ≤ 0, a > 0, and α is a real number. We define the operator A in L 2 (Ω) by A = −∆ with D (A) = H 2 (Ω) ∩ H 1 0 (Ω) . We easily get that A is a densely defined, closed, self-adjoint and coercive operator with compact resolvent in L 2 (Ω). We also assume that the spectrum of A is simple. Note that this assumption is generic (in the Baire sense) with respect to the domain Ω, according to [23]. Therefore: When a = 1 and γ = 0, applying Theorem 3.1, we obtain an exponential energy decay rate given by e tA u 0 H ≤ M e −ǫt u 0 H , t > 0, u 0 ∈ H.
When a = 1 and γ < 0 or when a = 0 and γ ≤ 0, applying Theorem 4.1, we obtain an optimal polynomial energy decay rate of the form where γ ≤ 0, a > 0 and α is a real number. We define the operator A in L 2 (Ω) by A = ∆ 2 with D (A) = H 4 (Ω) ∩ H 2 0 (Ω) . Here, A is a densely defined, closed, self-adjoint and coercive operator with compact resolvent in L 2 (Ω) and we furthermore assume that the spectrum of A is simple, assumption which holds generically with respect to the domain Ω, according to [23]. Then: