Robustness of Polynomial Stability of Damped Wave Equations

In this paper we present new results on the preservation of polynomial stability of damped wave equations under addition of perturbing terms. We in particular introduce sufficient conditions for the stability of perturbed two-dimensional wave equations on rectangular domains, a one-dimensional weakly damped Webster’s equation, and a wave equation with an acoustic boundary condition. In the case of Webster’s equation, we use our results to compute explicit numerical bounds that guarantee the polynomial stability of the perturbed equation.

Polynomial stability is a strictly weaker concept than exponential stability, and it can in particular be destroyed under addition of arbitrarily small lower order terms in the partial differential equation. In this paper we employ and refine the general framework introduced in [23,24] to present conditions for preservation of the polynomial stability of the abstract differential equation (1.1) under finite-rank and Hilbert-Schmidt perturbations. Moreover, we study preservation of polynomial stability for selected partial differential equation models, namely, a damped twodimensional wave equation on a rectangular domain, a weakly damped Webster's equation, and a one-dimensional wave equation with a dynamic boundary condition.
As our first main results we present general conditions for the polynomial stability of perturbed second-order systems of the form (1.3) w tt (t) − Lw(t) + D 0 D * 0 w t (t) = B 2 (C 1 w(t) + C 2 w t (t)), t > 0 w(0) = w 0 , w t (0) = w 1 .
Here the operators B 2 ∈ L(Y, X 0 ), C 1 ∈ L(dom (−L) 1/2 , Y ), and C 2 ∈ L(X 0 , Y ) for some Hilbert space Y describe the perturbations to the nominal polynomially stable equation (1.1). As our first main results we adapt and improve the main results in [23,24] to make them more easily verifiable for second-order systems of the form (1.3). Our results show that if the unperturbed equation is polynomially stable so that (1.2) is satisfied with some α ∈ (0, 2], then (1.3) is polynomially stable provided that for exponents β, γ ∈ [0, 1] satisfying β + γ ≥ α the graph norms (−L) β/2 B 2 , (−L) (γ−1)/2 C * 1 , and (−L) γ/2 C * 2 are finite and sufficiently small. Our new results also provide concrete bounds for the required sizes of these graph norms based on lower bounds for the operator D * 0 restricted to the spectral subspaces of L. The results are applicable in the situations where Y is either finite-dimensional or where B 2 , C 1 , and C 2 are Hilbert-Schmidt operators.
We assume the damping coefficient d(·, ·) ≥ 0 is strictly positive on some non-empty open subset of Ω which does not satisfy the Geometric Control Condition (see, e.g., [5]). We apply our abstract results to present conditions for the polynomial stability of perturbed wave equations of the form w tt (t, x, y) − ∆w(t, x, y) + d(x, y)w t (t, x, y) where b k,2 , c k,1 , c k,2 ∈ L 2 (Ω). In particular, our results show that the perturbed wave equation is polynomially stable provided that the coefficient functions b k,2 , c k,1 , and c k,2 have sufficient smoothness properties in the sense that these functions belong to fractional domains of −∆ (the Dirichlet Laplacian on Ω), and the associated fractional graph norms are sufficiently small. We present also analogous results of Hilbert-Schmidt perturbations of (1.4). Finally, we analyse the stability of (1.4) in a situation where the damping term is perturbed in a non-dissipative way with a rank one operator. Our second concrete partial differential equation is a Webster's equation with a weak damping on (0, 1), where a > 0 and d(·) ∈ L 2 (0, 1) is the damping coefficient. In this article we focus on a special case where d(x) = 1 − x. We begin by proving that the Webster's equation is polynomially stable with this particular damping coefficient. Using our abstract results we then present conditions for the preservation of the Webster's equation under addition of a perturbation term. We also present a numerical example where we compute numerical bounds for the coefficient functions in the perturbation to guarantee the preservation of polynomial stability of the Webster's equation.
Finally, as our third partial differential equation we consider a one-dimensional wave equation with a dynamic boundary condition. The polynomial stability of this model was shown in [22,1], and in this paper we present conditions for the preservation of the stability under addition of perturbation terms to the differential equation.
We use the following notation. Given a closed operator A on a Hilbert space X, which will be assumed to be complex, we denote its domain by dom A, its kernel by ker A, and its range by ran A. The spectrum of A is denoted by σ(A), and given λ ∈ ρ(A) := C \ σ(A) we write R(λ, A) for the resolvent operator (λ − A) −1 . The space of bounded linear operators on X is denoted by L(X). Given two functions f, g : (0, ∞) → R + , we write f (t) = O(g(t)) to indicate that f (t) ≤ Cg(t) for some constant C > 0 and for all sufficiently large t > 0.
2. Robustness of stability for generalized wave equations 2.1. Polynomial stability of strongly continuous semigroups. The secondorder differential equation (1.1) with a negative and boundedly invertible operator L : dom L ⊂ X 0 → X 0 and D 0 ∈ L(U, X 0 ) can be represented as a first-order abstract Cauchy problem with state u(t) = (w(t), w t (t)) ⊤ as with the initial condition u(0) = (w 0 , w 1 ) ⊤ . We choose the state space of this linear system as H = dom (−L) 1/2 × X 0 . The space H is a Hilbert space with inner product defined by Here A 0 is a skew-adjoint operator and A generates a strongly continuous semigroup T (t) on H by the Lumer-Phillips theorem [ H) and C := (C 1 , C 2 ) ∈ L(H, Y ), the perturbed system can be represented as an abstract Cauchy problem du dt = (A + BC)u. The following theorem presented in [24] provides general conditions for the preservation of the polynomial stability of the semigroup T A+BC (t) generated by A + BC.
The following theorem introduces a concrete bound κ > 0 for the norms of the perturbations in Theorem 2.2 for A = A 0 − DD * with a skew-adjoint operator A 0 in the important special case β, γ ≥ 0 are chosen so that β + γ = ⌈α⌉ (here ⌈α⌉ ∈ N denotes the ceiling of α > 0). The first part of the result is a special case of [13,Thm. 3.5] with a proof which has been modified in a trivial manner to yield an explicit constant M R > 0. Theorem 2.3. Let X and U be Hilbert spaces, and assume A = A 0 − DD * where A 0 : dom A 0 ⊂ X → X is skew-adjoint and D ∈ L(U, X). Let P (a,b) ∈ L(X) be the spectral projection of A 0 corresponding to the interval (ia, ib) ⊂ iR. Assume there exist η 0 , δ 0 > 0 and functions η : R → (0, η 0 ] and δ : R → (0, δ 0 ] such that ∀x ∈ ran P (s−δ(s),s+δ(s)) .
If there exists M 0 > 0 such that η(s) −2 δ(s) −2 ≤ M 0 (1 + |s| α ) for all s ∈ R, then for β, γ ≥ 0 with β + γ = ⌈α⌉ in Theorem 2.2 it is possible to choose any κ > 0 such that where M C > 0 is defined with an arbitrary s 0 > 0 by Proof. Assume that the functions η and δ satisfy the assumptions of the theorem. Let y ∈ X and s ∈ R be arbitrary and write x = R(is, A)y ∈ dom A. We then have (is − A 0 + DD * )x = y, and thus Denote P s := P (s−δ(s),s+δ(s)) for brevity and write X = X s ⊕ ⊥ X ∞ where X s = P s X and X ∞ = (I − P s )X. If we write x = x 0 + x ∞ and y = y 0 + y ∞ according to this decomposition, then By assumption we have If we denote q(s) = 1 + 2η(s) −2 D 2 , we can use D * x 2 ≤ x y and the Young's inequality to estimate This estimate implies This completes the first part of the proof.
Assume now that there exists M 0 > 0 such that η(s) −2 δ(s) −2 ≤ M 0 (1 + |s| α ) for all s ∈ R and denote n α = ⌈α⌉ ∈ N. Then R(is, A) ≤ M 0 M R (1 + |s| α ) for all s ∈ R, and Theorem 2.2 implies that if β, γ ≥ 0 are such that β + γ = n α , then the constant κ > 0 is required to satisfy κ < (sup λ∈C+ R(λ, A)A −nα ) −1/2 . The approach in the proof of [8,Lem. 5.3] can be used to show that This implies that κ > 0 in Theorem 2.2 can be chosen to have any value κ < In order to show this, let s 0 > 0 be arbitrary and fixed. For any s ∈ R with |s| ≤ s 0 we have On the other hand, if |s| ≥ s 0 , then using the resolvent identity R(is, Combining the above two estimates shows that sup s∈R R(is, A)A −β−γ ≤ M C for the constant M C > 0 in the statement of the theorem, and thus the proof is complete.
Remark 2.4. In the case where −L has a complete set of orthonormal eigenvectors −Lφ n = µ n φ n with 0 < µ 1 ≤ µ 2 ≤ · · · , the operator A 0 in (2.1) has eigenvalues λ n = sign (n)i √ µ n and a complete set of orthonormal eigenvectors {ψ n } n∈Z\{0} such that In this situation for every s ∈ R the spectral subspace ran P (s−δ(s),s+δ(s)) of A 0 consists of linear combinations of the eigenvectors ψ n with every n ∈ Z \ {0} for which s − δ(s) < sign (n) √ µ n < s + δ(s). The functions η : R → (0, η 0 ] and δ : R → (0, δ 0 ] in Theorem 2.3 should then be chosen so that is chosen in such a way that δ(−s) = δ(s) and every interval (i(s − δ(s)), i(s + δ(s))) contains at most one eigenvalue λ n = sign (n)i √ µ n , then η : R → (0, η 0 ] in Theorem 2.3 can be chosen to be an even function satisfying Remark 2.5. The second part of the proof of Theorem 2.3 can be extended in a straightforward manner to the more general case where β, γ ≥ 0 are any exponents satisfying β +γ ≥ α. Indeed, if we denote n βγ = ⌈β +γ⌉, the moment inequality [17, Thm. II. 5.34] with θ = (n βγ − β − γ)/n βγ and a constant M β,γ > 0 can first be used estimate and R(is, A)A −n βγ can be estimated using the resolvent identity similarly as before. However, in this case the constant M C in the bound for κ > 0 has a more complicated formula. Throughout this section C * 1 denotes the adjoint of C 1 as an operator C 1 : dom C 1 ⊂ X 0 → Y . Theorem 2.6. Assume that the strongly continuous semigroup T (t) generated by then the semigroup generated by A + BC is polynomially stable with the same α.
For proving this result we use the following theorem from [19].
Proof of Theorem 2.6. Let 0 ≤ β, γ ≤ 1 be such that β + γ ≥ α. Our aim is to show that if B 2 , C 1 , and C 2 satisfy the given assumptions, then B = (0, B 2 ) ⊤ and C = (C 1 , C 2 ) satisfy (2.4) with the same κ > 0. The stability of the semigroup generated by A + BC then follows directly from Theorem 2.2. To this end let κ > 0 be as in Theorem 2.2 and suppose that (2.7) hold. Define

closed and maximally accretive operators and dom
By Theorem 2.2 the semigroup generated by A + BC is polynomially stable with α.
If the operator L is diagonalizable [29, Sec. 2.6], then for θ ∈ R the spaces dom (−L) θ and the graph norms of (−L) θ have the forms where µ k are the eigenvalues of −L and φ k are the corresponding orthonormal eigenvectors. With these definitions the space H −θ (L) is the dual of H θ (L) with respect to the pivot space X 0 [29, Sec. 2.9].
then the semigroup generated by A + BC is polynomially stable with the same α.
Here K θ = e
We suppose that the set ω = {d(x, y) > 0} contains an open, nonempty subset and does not satisfy Geometric Control Condition (GCC) (see a definition of GCC for example in [5,Sec. 1]). It was shown in [18] that for such damping the Schrödinger group is observable, i.e., the pair (D * 0 , i(−∆)) is exactly observable [29, Def. 6.1.1] (see also [12]). In this case the damped wave equation (1.4)  Our assumptions together with the results in [18] and [13,Prop. 3.9] also imply that the condition (2.5) is satisfied for some functions η : R → (0, η 0 ] and δ : R → (0, δ 0 ] satisfying η(s) −2 δ(s) −2 ≤ M 0 (1 + s 2 ) for all s ∈ R. Because of this, Theorem 2.3 could in principle be used to derive numerical values for κ > 0 for particular damping functions d(·, ·). In practice, however, finding suitable concrete functions η and δ can be challenging, and in the case of the two-dimensional wave equation this is an important topic for further research.
Remark 3.1. In some cases of damping functions the estimate for the exponent of polynomial stability can be improved. For example, in [28] the exponent of polynomial stability for the damping function was shown to be α = 3/2. Moreover, additional differentiability assumptions on d(·, ·) improve the rate of polynomial decay, as shown in [11,5,14].
3.1. Rank one perturbations. We begin by considering perturbed wave equations of the form with b 2 , c 2 ∈ L 2 (Ω) and c 1 ∈ H −1/2 (∆). The following theorem presents sufficient conditions for the polynomial stability of (3.2).

3.3.
Hilbert-Schmidt perturbations. Now we consider a more general case of perturbations of the wave equation w tt (t, x, y) − ∆w(t, x, y) + d(x, y)w t (t, x, y) where the functions b k,2 , c k,2 ∈ L 2 (Ω) and c k,1 ∈ H −1/2 (∆) of the perturbation are assumed to satisfy The stability of this perturbed wave equation can be studied using Corollary 2.8 for Hilbert-Schmidt perturbations.
3.4. Wave equation with "almost dissipative" damping. Finally, we consider the two-dimensional damped wave equation with a perturbed damping term, namely with b 2 , c ∈ L 2 (Ω). We also make an additional assumption that d ∈ C 2 (Ω). The structure of the perturbed semigroup generator is now A := A 0 − (D + B)D * = A − BD * , where D = (0, d(·)) ⊤ and B = (0, b 2 ·, c L 2 ) ⊤ . Because of this structure, the damping in the wave equation (3.7) can be thought to be "almost dissipative".
Since we assumed that the damping coefficient is smooth, i.e. d(·) ∈ C 2 (Ω), it is possible to characterise the higher order domain dom A 2 and the stability of (3.7) can be studied using Theorem 2.2 with the parameters β = 2 and γ = 0, as shown in the following theorem. Theorem 3.7. Assume that damped wave equation (3.7) is polynomially stable with α ≤ 2 in the case where b 2 = 0. There exists κ > 0 such that if b 2 ∈ dom ∆ and c ∈ L 2 (Ω) satisfy is polynomially stable with the same α. For such perturbations there exists M T > 0 such that the solutions of (3.7) corresponding to initial conditions w 0 ∈ H 2 (Ω) ∩ H 1 0 (Ω) and w 1 ∈ H 1 0 (Ω) satisfy Proof. Let κ > 0 be as in Theorem 2.2 and suppose the assumptions on b 2 and c are satisfied. We define B := (0, b 2 ) ⊤ and C := (0, ·, √ dc ). It is clear that B C = BD * . Our aim is to verify that the conditions of Theorem 2.2 are satisfied for the perturbed operator for A − B C with parameters β = 2 and γ = 0. We have The norm of A 2 B can be estimated by where the last estimate is completed as in the proof of Theorem 3.2. Moreover, (3.10) Thus (−A) 2 B < κ. We also have that (−A * ) 0 C * = √ dc L 2 < κ. The polynomial stability of the semigroup generated by A − B C = A − BD * follows from Theorem 2.2 and then wave equation (3.7) is polynomial stable with α.

Perturbations of Webster's equations
In this section we show the polynomial stability of weakly damped Webster's equation and use Theorems 2.6 and 2.3 to derive sufficient conditions for the preservation of the stability under addition of perturbing terms. We begin by considering an undamped Webster's equation on Ω = (0, 1) which has the form We consider r(x) = e ax , where a ≥ 0. Then Webster's equation takes the form w tt (t, x) = w xx (t, x) + aw x (t, x).
We denote by L 2 a (0, 1) the Hilbert space L 2 (0, 1) with the inner product Let us define the operator L = d 2 dx 2 + a d dx from L 2 a (0, 1) to L 2 a (0, 1) with dom L = {h ∈ L 2 a (0, 1) : h, h ′ are absolutely continuous , h ′′ ∈ L 2 a (0, 1) and h(0) = h(1) = 0}. In the next lemma we state some properties of L. 1) is a negative self-adjoint operator with a bounded inverse. The eigenvalues and eigenvectors of L are respectively, for n ∈ N.
Now we can consider an auxiliary operator L : It is well known that L is a negative self-adjoint operator with a bounded inverse. Hence L is also a negative self-adjoint operator with a bounded inverse. The eigenvalues and the eigenvectors of the operator L are µ n = − a 2 4 − π 2 n 2 , ϕ n (x) = sin(πnx), n ∈ N.
Since L = V * LV , the operators L and L have the same eigenvalues and the eigenvectors of L are given by the formula ϕ n (x) = (V ϕ n )(x) = e − ax 2 sin(πnx).
Now we consider weakly damped Webster's equation where the damping coefficient is d ∈ L 2 a (0, 1). This equation is of the form (1.1) on X 0 = L 2 a (0, 1) with L defined above and with a rank one operator D 0 = d(·) ∈ L(C, L 2 a (0, 1)) and D * 0 = ·, d L 2 a . The polynomial stability of the weakly damped Webster's equation can be analyzed using [25,Thm. 6.3]. The following result in particular shows that (4.1) is polynomially stable for the particular choice of damping d(x) = 1 − x. Proof. We can write with dom A 0 = dom A and D ∈ L(C, X). We will use [25, Thm. 6.3] to show that R(is, A) ≤ M (1 + s 2 ) for some M > 0. To this end, we need to estimate the quantities |D * ψ n | from below, where ψ n are the normalized eigenvectors of A 0 . Since L has eigenvalues µ n = − a 2 4 − π 2 n 2 with the corresponding eigenvectors ϕ n (x) = e − ax 2 sin(πnx) for n ∈ N, the eigenvectors ψ n and the corresponding eigenvalues λ n of A 0 are given by λ n = sign (n)i a 2 4 + π 2 n 2 , and ψ n (x) = 1 λ n ϕ |n| (x) λ n ϕ |n| (x) , n ∈ Z\{0}.
For any n ∈ Z \ {0} we thus have for some constant c > 0 and for all sufficiently large |n|. By [25,Thm. 6.3] we have R(is, A) = O(s 2 ) for |s| large, and thus [10,Thm. 2.4] implies that the semigroup generated by A = A 0 − DD * is polynomially stable with α = 2. We consider the weakly damped Webster's equation with additional perturbing terms of the form where b 2 , c 2 ∈ L 2 (0, 1) and c 1 ∈ H −1/2 (L). The following theorem presents conditions for the polynomial stability of the perturbed Webster's equation (4.2). The spaces H θ (L) and the corresponding norms are defined as in (2.8). The above perturbations correspond to rank one perturbation operators in the abstract wave equation. Addition of multiple perturbation terms can be treated similarly as in the case of the two-dimensional wave equation in Section 3.
then the perturbed Webster's equation (4.2) is polynomially stable with the same α.
In the next step we calculate M R . To this end, we need also D which is We can now use Matlab to compute M R = 5.451. Now we will find the constant M 0 > 0. A direct estimate using (x + y) 2 ≤ 2(x 2 + y 2 ) and δ 0 < 1 shows that To compute M C > 0, we also need an estimate for A −1 . We have If we take s 0 = 2.8 in the formula for M C , we obtain M C = 17.0664. This way, we finally see that κ > 0 in Theorem 2.2 can take any value such that κ < 1 √ 2MC = 0.1712. Now we are able to give explicit upper bounds for the norms of b 2 , c 1 , and c 2 for the preserving of polynomial stability. From Corollary 4.5 we have that b 2 , c 2 ∈ H 1/2 (L), c 1 ∈ L 2 a (0, 1) satisfy

Wave equation with an acoustic boundary condition
In this section we consider a one-dimensional wave equation with an "acoustic boundary condition" on the interval Ω = (0, 1), with k, d > 0 [1, Sec. 6.1]. The spectral properties and polynomial stability of differential equations of this form (also on multidimensional spatial domains) have been studied in detail in [9,22,1,2]. In particular, it was shown in [22,Thm. 1.3] that the energy of the classical solutions of (5.1) decays at a rational rate, and the optimality of this decay rate was proved in [1, Sec. 6.1]. This model is not of the form (1.1), but the preservation of its polynomial stability can be studied using Theorem 2.2. Equation (5.1) can be formulated as an abstract Cauchy problem with state u(t) = (w x (t, ·), w t (t, ·), a(t), a t (t)) ⊤ on the Hilbert space H = L 2 (0, 1) × L 2 (0, 1) × C 2 with inner product defined as In this situation the semigroup generator is defined as with domain dom A = (u 1 (·), u 2 (·), u 3 , u 4 ) ⊤ ∈ (H 1 (0, 1)) 2 × C 2 : u 2 (0) = 0, u 2 (1) = u 4 .
The operator A generates a contraction semigroup on H, and it was shown in [22, Thm. 1.3] (see also [1,Sec. 6.1], [26,Sec. 4]) that this semigroup is polynomially stable with α = 2. In the context of the wave equation (5.1) this means that there exists a constant M T > 0 such that for all initial conditions w 0 , w 1 , a 0 , a 1 such that (w ′ 0 , w 1 , a 0 , a 1 ) ⊤ ∈ dom A the solutions of (5.1) satisfy w x (t, ·) 2 L 2 + w t (t, ·) 2 for all t > 0.
Similarly, applying Theorem 2.2 with β = 2 and γ = 0 we obtain the following alternative conditions for the polynomial stability of (5.3).