Stability of an abstract-wave equation with delay and a Kelvin

Abstract

In this paper we consider a stabilization problem for an abstract wave equation with delay and a Kelvin–Voigt damping. We prove an exponential stability result for appropriate damping coefficients. The proof of the main result is based on a frequency-domain approach.

Keywords

internal stabilization Kelvin–Voigt damping abstract wave equation with delay

1. Introduction

Our main goal is to study the internal stabilization of a delayed abstract wave equation with a Kelvin–Voigt damping. More precisely, given a constant time delay $τ > 0$ , we consider the system given by: $\begin{array}{rclr} u^{″} (t) + a {BB}^{*} u^{'} (t) + {BB}^{*} u (t - τ) = 0 in (0, + \infty), & (1.1) \\ u (0) = u_{0}, u^{'} (0) = u_{1}, & (1.2) \\ B^{*} u (t - τ) = f_{0} (t - τ) in (0, τ), & (1.3) \end{array}$ where $a > 0$ is a constant, $B : D (B) \subset H_{1} \to H$ is a linear unbounded operator from a Hilbert space $H_{1}$ into another Hilbert space H equipped with the respective norms $∥ \cdot ∥_{H_{1}}$ , $∥ \cdot ∥_{H}$ and inner products ${(\cdot, \cdot)}_{H_{1}}$ , ${(\cdot, \cdot)}_{H}$ , and $B^{*} : D (B^{*}) \subset H \to H_{1}$ is the adjoint of B. The initial datum $(u_{0}, u_{1}, f_{0})$ belongs to a suitable space.

We suppose that the operator $B^{*}$ satisfies the following coercivity assumption: there exists $C > 0$ such that ${∥ B^{*} v ∥}_{H_{1}} ⩾ C ∥ v ∥_{H} \forall v \in D (B^{*}) .$ (1.4) For shortness we set $V = D (B^{*})$ and we assume that it is closed with the norm $∥ v ∥_{V} : = ∥ B^{*} v ∥_{H_{1}}$ and that it is compactly embedded into H.

Delay effects arise in many applications and practical problems and it is well known that an arbitrarily small delay may destroy the well-posedness of the problem [11,14,17,18] or destabilize a system which is uniformly asymptotically stable in absence of delay (see e.g. [8,9,15,18]). Different strategies were recently developed to restitute either the well-posedness or the stability. In the first case, one idea is to add a non-delay term, see [6,17] for the heat equation. In the second case, we refer to [2–4,10,15,16] for stability results for systems with time delay where a standard feedback compensating the destabilizing delay effect is introduced. Nevertheless recent papers reveal that particular choices of the delay may restitute exponential stability property, see [5,12].

Note that the above system is exponentially stable in absence of time delay, and if $a > 0$ . On the other hand if $a = 0$ and $- {BB}^{*}$ corresponds to the Laplace operator with Dirichlet boundary conditions in a bounded domain of $R^{n}$ , problem (1.1)–(1.3) is not well-posed, see [11,14,17,18]. Therefore in this paper in order to restitute the well-posedness character and its stability we propose to add the Kelvin–Voigt damping term $a {BB}^{*} u^{'}$ . Hence the stabilization of problem (1.1)–(1.3) is performed using a frequency domain approach combined with a precise spectral analysis.

The paper is organized as follows. The second section deals with the well-posedness of the problem while, in the third section, we perform the spectral analysis of the associated operator. In Section 4, we prove the exponential stability of the system (1.1)–(1.3) if $τ ⩽ a$ . In the last section we give an example of an application.

2. Existence results

In this section we will give a well-posedness result for problem (1.1)–(1.3) by using semigroup theory.

Inspired from [15], we introduce the auxiliary variable $z (ρ, t) = B^{*} u (t - τ ρ), ρ \in (0, 1), t > 0 .$ (2.1) Then, problem (1.1)–(1.3) is equivalent to $\begin{array}{rclr} u^{″} (t) + a {BB}^{*} u^{'} (t) + B z (1, t) = 0 in (0, + \infty), & (2.2) \\ τ z_{t} (ρ, t) + z_{ρ} (ρ, t) = 0 in (0, 1) \times (0, + \infty), & (2.3) \\ u (0) = u_{0}, u^{'} (0) = u_{1}, & (2.4) \\ z (ρ, 0) = f_{0} (- ρ τ) in (0, 1), & (2.5) \\ z (0, t) = B^{*} u (t), t > 0 . & (2.6) \end{array}$ If we denote $U : = {(u, u^{'}, z)}^{⊤},$ then $U^{'} : = {(u^{'}, u^{″}, z_{t})}^{⊤} = {(u^{'}, - a {BB}^{*} u^{'} - B z (1, t), - τ^{- 1} z_{ρ})}^{⊤} .$ Therefore, problem (2.2)–(2.6) can be rewritten as $\{\begin{matrix} U^{'} = A U, \\ U (0) = {(u_{0}, u_{1}, f_{0} (- \cdot τ))}^{⊤}, \end{matrix}$ (2.7) where the operator $A$ is defined by $A (\begin{matrix} u \\ v \\ z \end{matrix}) : = (\begin{matrix} v \\ - a {BB}^{*} v - B z (\cdot, 1) \\ - τ^{- 1} z_{ρ} \end{matrix}),$ with domain $\begin{matrix} D (A) : = {{(u, v, z)}^{⊤} \in D (B^{*}) \times D (B^{*}) \times H^{1} (0, 1; H_{1}) : a B^{*} v + z (1) \in D (B), \\ B^{*} u = z (0)}, \end{matrix}$ (2.8) in the Hilbert space $H : = D (B^{*}) \times H \times L^{2} (0, 1; H_{1}),$ (2.9) equipped with the standard inner product ${((u, v, z), (u_{1}, v_{1}, z_{1}))}_{H} = {(B^{*} u, B^{*} u_{1})}_{H_{1}} + {(v, v_{1})}_{H} + ξ \int_{0}^{1} {(z, z_{1})}_{H_{1}} d ρ,$ where $ξ > 0$ is a parameter fixed later on.

We will show that $A$ generates a $C_{0}$ semigroup on $H$ by proving that $A - c Id$ is maximal dissipative for an appropriate choice of c in function of ξ, τ and a. Namely we prove the next result.

Lemma 2.1.

If $ξ > \frac{2 τ}{a}$ , then there exists $a^{*} > 0$ such that $A - a_{*}^{- 1} Id$ is maximal dissipative in $H$ .

Proof.

Take $U = {(u, v, z)}^{⊤} \in D (A)$ . Then we have $\begin{array}{rcl} {(A (u, v, z), (u, v, z))}_{H} & = & {(B^{*} v, B^{*} u)}_{H_{1}} - {(B (a B^{*} v + z (1)), v)}_{H} \\ - ξ τ^{- 1} \int_{0}^{1} {(z_{ρ}, z)}_{H_{1}} d ρ . \end{array}$ Hence, we get $\begin{array}{rcl} {(A (u, v, z), (u, v, z))}_{H} & = & {(B^{*} v, B^{*} u)}_{H_{1}} - {(a B^{*} v + z (1), B^{*} v)}_{H_{1}} \\ - \frac{ξ}{2 τ} {∥ z (1) ∥}_{H_{1}}^{2} + \frac{ξ}{2 τ} {∥ z (0) ∥}_{H_{1}}^{2} . \end{array}$ Hence reminding that $z (0) = B^{*} u$ and using Young’s inequality we find that $\begin{matrix} ℜ {(A (u, v, z), (u, v, z))}_{H} \\ ⩽ (ε - a) {∥ B^{*} v ∥}_{H_{1}}^{2} + (\frac{1}{2 ε} - \frac{ξ}{2 τ}) {∥ z (1) ∥}_{H_{1}}^{2} + (\frac{1}{2 ε} + \frac{ξ}{2 τ}) {∥ B^{*} u ∥}_{H_{1}}^{2} . \end{matrix}$ Choosing $ε = \frac{a}{2}$ , we find that $ℜ {(A (u, v, z), (u, v, z))}_{H} ⩽ - \frac{a}{2} {∥ B^{*} v ∥}_{H_{1}}^{2} + (\frac{1}{a} - \frac{ξ}{2 τ}) {∥ z (1) ∥}_{H_{1}}^{2} + (\frac{1}{a} + \frac{ξ}{2 τ}) {∥ B^{*} u ∥}_{H_{1}}^{2} .$ The choice of ξ is equivalent to $\frac{1}{a} - \frac{ξ}{2 τ} < 0$ , and therefore for $a_{*} = {(\frac{1}{a} + \frac{ξ}{2 τ})}^{- 1}$ , $ℜ {(A (u, v, z), (u, v, z))}_{H} ⩽ - \frac{a}{2} {∥ B^{*} v ∥}_{H_{1}}^{2} + (\frac{1}{a} - \frac{ξ}{2 τ}) {∥ z (1) ∥}_{H_{1}}^{2} + a_{*}^{- 1} {∥ B^{*} u ∥}_{H_{1}}^{2} .$ (2.10) As $∥ B^{*} u ∥_{H_{1}}^{2} ⩽ ∥ (u, v, z) ∥_{H}^{2}$ , we get $ℜ {((A - a_{*}^{- 1} Id) (u, v, z), (u, v, z))}_{H} ⩽ - \frac{a}{2} {∥ B^{*} v ∥}_{H_{1}}^{2} + (\frac{1}{a} - \frac{ξ}{2 τ}) {∥ z (1) ∥}_{H_{1}}^{2} ⩽ 0,$ which directly leads to the dissipativeness of $A - a_{*}^{- 1} Id$ .

Let us go on with the maximality, namely let us show that $λ I - A$ is surjective for a fixed $λ > 0$ . Given ${(f, g, h)}^{⊤} \in H$ , we look for a solution $U = {(u, v, z)}^{⊤} \in D (A)$ of $(λ I - A) (\begin{matrix} u \\ v \\ z \end{matrix}) = (\begin{matrix} f \\ g \\ h \end{matrix}),$ (2.11) that is, verifying $\{\begin{matrix} λ u - v = f, \\ λ v + B (a B^{*} v + z (1)) = g, \\ λ z + τ^{- 1} z_{ρ} = h . \end{matrix}$ (2.12) Suppose that we have found u with the appropriate regularity. Then, $v = λ u - f$ (2.13) and we can determine z. Indeed, by (2.8), $z (0) = B^{*} u,$ (2.14) and, from (2.12), $λ z (ρ) + τ^{- 1} z_{ρ} (ρ) = h (ρ) for ρ \in (0, 1) .$ (2.15) Then, by (2.14) and (2.15), we obtain $z (ρ) = B^{*} u e^{- λ ρ τ} + τ e^{- λ ρ τ} \int_{0}^{ρ} h (σ) e^{λ σ τ} d σ .$ (2.16) In particular, we have $z (1) = B^{*} u e^{- λ τ} + z_{0},$ (2.17) with $z_{0} \in H_{1}$ defined by $z_{0} = τ e^{- λ τ} \int_{0}^{1} h (σ) e^{λ σ τ} d σ .$ (2.18) This expression in (2.12) shows that the function u verifies formally $λ^{2} u + B (a B^{*} (λ u - f) + B^{*} u e^{- λ τ} + z_{0}) = g + λ f,$ that is, $λ^{2} u + (λ a + e^{- λ τ}) {BB}^{*} u = g + λ f + B (a B^{*} f) - B z_{0} .$ (2.19) Problem (2.19) can be reformulated as ${(λ^{2} u + (λ a + e^{- λ τ}) {BB}^{*} u, w)}_{H} = {(g + λ f + B (a B^{*} f) - B z_{0}, w)}_{H} \forall w \in V .$ (2.20) Using the definition of the adjoint of B, we get $\begin{array}{rclr} λ^{2} {(u, w)}_{H} + (λ a + e^{- λ τ}) {(B^{*} u, B^{*} w)}_{H_{1}} = {(g + λ f, w)}_{H} + {(a B^{*} f - z_{0}, B^{*} w)}_{H_{1}} \forall w \in V . & (2.21) \end{array}$ As the left-hand side of (2.21) is coercive on $D (B^{*})$ , the Lax–Milgram lemma guarantees the existence and uniqueness of a solution $u \in V$ of (2.21). Once u is obtained we define v by (2.13) that belongs to V and z by (2.16) that belongs to $H^{1} (0, 1; H_{1})$ . Hence we can set $r = a B^{*} v + z (1)$ , it belongs to $H_{1}$ but owing to (2.21), it fulfils $λ {(v, w)}_{H} + {(r, B^{*} w)}_{H_{1}} = {(g, w)}_{H} \forall w \in D (B^{*}),$ or equivalently ${(r, B^{*} w)}_{H_{1}} = {(g - λ v, w)}_{H} \forall w \in D (B^{*}) .$ As $g - λ v \in H$ , this implies that r belongs to $D (B)$ with $B r = g - λ v .$ This shows that the triple $U = (u, v, z)$ belongs to $D (A)$ and satisfies (2.11), hence $λ I - A$ is surjective for every $λ > 0$ . □

We have then the following result. Proposition 2.2.

The system (1.1)–(1.3) is well posed. More precisely, for every $(u_{0}, u_{1}, f_{0}) \in H$ , there exists a unique solution $(u, v, z) \in C (0, + \infty, H)$ of (2.7). Moreover, if $(u_{0}, u_{1}, f_{0}) \in D (A)$ then $(u, v, z) \in C (0, + \infty, D (A)) \cap C^{1} (0, + \infty, H)$ with $v = u^{'}$ and u is indeed a solution of (1.1)–(1.3) .

3. The spectral analysis

As $D (B^{*})$ is compactly embedded into H, the operator ${BB}^{*} : D ({BB}^{*}) \subset H \to H$ has a compact resolvent. Hence let ${(λ_{k})}_{k \in N^{*}}$ be the set of eigenvalues of ${BB}^{*}$ repeated according to their multiplicity (that are positive real numbers and are such that $λ_{k} \to + \infty$ as $k \to + \infty$ ) and denote by ${(φ_{k})}_{k \in N^{*}}$ the corresponding eigenvectors that form an orthonormal basis of H (in particular for all $k \in N^{*}$ , ${BB}^{*} φ_{k} = λ_{k} φ_{k}$ ).

3.1. The discrete spectrum

We have the following lemma.

Lemma 3.1.
If $τ ⩽ a$ , then any eigenvalue λ of $A$ satisfies $ℜ λ < 0$ .
Proof.
Let $λ \in C$ and $U = {(u, v, z)}^{⊤} \in D (A)$ be such that $(λ I - A) (\begin{matrix} u \\ v \\ z \end{matrix}) = 0,$ or equivalently $\{\begin{matrix} v = λ u, \\ - B (a B^{} v + z (\cdot, 1)) = λ v, \\ - τ^{- 1} z_{ρ} = λ z . \end{matrix}$ (3.1) By (2.14), we find that $z (ρ) = λ^{- 1} B^{} v e^{- λ ρ τ} .$ (3.2) Using this property in (3.1), we find that $u \in D (B^{})$ is solution of $λ^{2} u + (a λ + e^{- λ τ}) {BB}^{} u = 0 .$

Hence a non-trivial solution exists if and only if there exists $k \in N^{}$ such that $\frac{λ^{2}}{a λ + e^{- λ τ}} = - λ_{k} .$ (3.3)

This condition implies that λ does not belong to $Σ : = {λ \in C : a λ + e^{- λ τ} = 0},$ (3.4) and that $e^{- λ τ} + \frac{λ^{2}}{λ_{k}} + a λ = 0 .$ (3.5) Writing $λ = x + i y$ , with $x, y \in R$ , we see that this identity is equivalent to $\begin{array}{rclr} e^{- τ x} cos (τ y) + \frac{x^{2} - y^{2}}{λ_{k}} + a x = 0, & (3.6) \\ - e^{- τ x} sin (τ y) + \frac{2 x y}{λ_{k}} + a y = 0 . & (3.7) \end{array}$

The second equation is equivalent to $e^{τ x} (\frac{2 x}{λ_{k}} + a) y = sin (τ y) .$

Hence if $y \neq 0$ , we will get $\frac{e^{τ x}}{τ} (\frac{2 x}{λ_{k}} + a) = \frac{sin (τ y)}{τ y} .$ As the modulus of the right-hand side is ⩽1, we obtain $| \frac{e^{τ x}}{τ} (\frac{2 x}{λ_{k}} + a) | ⩽ 1,$ or equivalently $| \frac{2 x}{λ_{k}} + a | ⩽ τ e^{- τ x} .$

Therefore if $x ⩾ 0$ , we find that $\frac{2 x}{λ_{k}} + a ⩽ τ e^{- τ x} ⩽ τ,$ which implies that $\frac{2 x}{λ_{k}} ⩽ τ - a .$ For $τ < a$ , we arrive to a contradiction. For $τ = a$ , the sole possibility is $x = 0$ and by (3.7), we find that $sin (τ y) = τ y,$ which yields $y = 0$ and again we obtain a contradiction.

If $y = 0$ , we see that (3.7) always holds and (3.6) is equivalent to $e^{- τ x} = - x (\frac{x}{λ_{k}} + a) .$ This equation has no non-negative solutions x since for $x ⩾ 0$ , the left-hand side is positive while the right-hand side is non-positive, hence again if a solution x exists, it has to be negative.

The proof of the lemma is complete. □

If $a < τ$ , we now show that there exist some pairs of $(a, τ)$ for which the system (1.1)–(1.3) becomes unstable. Hence the condition $τ ⩽ a$ is optimal for the stability of this system. Lemma 3.2.
There exist pairs of* $(a, τ)$ such that $0 < a < τ$ and for which the associated operator $A$ has a pure imaginary eigenvalue.
Proof.
We look for a purely imaginary eigenvalue $i y$ of $A$ , hence system (3.6)–(3.7) reduces to $\begin{array}{rclr} cos (τ y) = \frac{y^{2}}{λ_{k}}, & (3.8) \\ sin (τ y) = a y . & (3.9) \end{array}$ Such a solution exists if $\frac{y^{4}}{λ_{k}^{2}} + a^{2} y^{2} = 1 .$ (3.10) One solution of this equation is $y_{k} = {(\frac{- a^{2} λ_{k}^{2} + \sqrt{a^{4} λ_{k}^{4} + 4 λ_{k}^{2}}}{2})}^{1 / 2} .$ We now take any $τ \in (0, \frac{π}{2 y_{k}})$ and $a = \frac{sin (τ y_{k})}{y_{k}}$ . Then (3.9) automatically holds, while (3.8) is valid owing to (3.10) (as $cos (τ y_{k}) > 0$ ). Finally $a < τ$ because $\frac{a}{τ} = \frac{sin (τ y_{k})}{τ y_{k}} < 1 .$ Therefore with such a choice of a and τ, the operator $A$ has a purely imaginary eigenvalue equal to $i y_{k}$ . □

3.2. The continuous spectrum

Inspired from Section 3 of [1], by using a Fredholm alternative technique, we perform the spectral analysis of the operator $A$ .

Recall that an operator T from a Hilbert space X into itself is called singular if there exists a sequence $u_{n} \in D (T)$ with no convergent subsequence such that $∥ u_{n} ∥_{X} = 1$ and $T u_{n} \to 0$ in X, see [19]. According to Theorem 1.14 of [19] T is singular if and only if its kernel is infinite dimensional or its range is not closed. Let Σ be the set defined in (3.4). The following results hold theorem.

Theorem 3.3.

If $λ \in Σ$ , then $λ I - A$ is singular.

If $λ \notin Σ$ , then $λ I - A$ is a Fredholm operator of index zero.

Proof.
For the proof of point 1, let us fix $λ \in Σ$ and for all $k \in N^{}$ set $U_{k} = {(u_{k}, λ u_{k}, B^{} u_{k} e^{- λ τ \cdot})}^{⊤},$ with $u_{k} = \frac{1}{\sqrt{λ_{k}}} φ_{k}$ . Then $U_{k}$ belongs to $D (A)$ and easy calculations yield (due to the assumption $λ \in Σ$ ) $(λ I - A) U_{k} = λ^{2} {(0, u_{k}, 0)}^{⊤} .$ Therefore we deduce that ${∥ (λ I - A) U_{k} ∥}_{H} \to 0 as k \to \infty .$ Moreover, due to the property $∥ B^{} u_{k} ∥_{H_{1}} = 1$ , there exist positive constants c, C, such that $c ⩽ ∥ U_{k} ∥_{H} ⩽ C \forall k \in N^{} .$

This shows that $λ I - A$ is singular.

For all $λ \in C$ , introduce the (linear and continuous) mapping $A_{λ}$ from V into its dual by ${⟨ A_{λ} v, w ⟩}_{V^{'} - V} = λ^{2} {(v, w)}_{H} + (a λ + e^{- λ τ}) {(B^{} v, B^{} w)}_{H_{1}} \forall v, w \in D (B^{}) .$ Then from the proof of Lemma 2.1, we know that for $λ > 0$ , $A_{λ}$ is an isomorphism.

Now for $λ \in C ∖ Σ$ , we can introduce the operator $B_{λ} = {(a λ + e^{- λ τ})}^{- 1} A_{λ} .$ Hence for $λ \in C ∖ Σ$ , $A_{λ}$ is a Fredholm operator of index 0 if and only if $B_{λ}$ is a Fredholm operator of index 0. Furthermore for $λ, μ \in C ∖ Σ$ as $B_{λ} - B_{μ}$ is a multiple of the identity operator, due to the compact embedding of V into $V^{'}$ , and as $B_{μ}$ is an isomorphism for $μ > 0$ , we finally deduce that $A_{λ}$ is a Fredholm operator of index 0 for all $λ \in C ∖ Σ$ .

Now we readily check that, for any $λ \in C ∖ Σ$ , we have the equivalence $u \in ker A_{λ} ⟺ {(u, λ u, B^{} u e^{- λ τ \cdot})}^{⊤} \in ker (λ I - A) .$ (3.11)

This equivalence implies that $dim ker (λ I - A) = dim ker A_{λ} \forall λ \in C ∖ Σ .$ (3.12)

For the range property for all $λ \in C ∖ Σ$ introduce the inner product ${(u, z)}_{λ, V} : = {({(u, λ u, B^{} u e^{- λ τ \cdot})}^{⊤}, {(z, λ z, B^{} z e^{- λ τ \cdot})}^{⊤})}_{H},$ on V whose associated norm is equivalent to the standard one.

Denote by ${y^{(i)}}_{i = 1}^{N}$ an orthonormal basis of $ker A_{λ}$ for this new inner product (for shortness the dependence of λ is dropped), i.e. ${(y^{(i)}, y^{(j)})}_{λ, V} = δ_{i j} \forall i, j = 1, \dots, N .$ Finally, for all $i = 1, \dots, N$ , we set $Z^{(i)} = {(y^{(i)}, λ y^{(i)}, B^{} y^{(i)} e^{- λ τ \cdot})}^{⊤},$ the element of $ker (λ I - A)$ associated with $y^{(i)}$ that are orthonormal with respect to the inner product of $H$ .

Let us now show that for all $λ \in C ∖ Σ$ , the range $R (λ I - A)$ of $λ I - A$ is closed. Indeed, let us consider a sequence $U_{n} = {(u_{n}, v_{n}, z_{n})}^{⊤} \in D (A)$ such that $(λ I - A) U_{n} = F_{n} = {(f_{n}, g_{n}, h_{n})}^{⊤} \to F = {(f, g, h)}^{⊤} in H .$ (3.13) Without loss of generality we can assume that ${(U_{n}, Z^{(i)})}_{H} = - α_{n, i} \forall i = 1, \dots, N,$ (3.14) where $α_{n, i} : = {({(0, f_{n}, - τ e^{- λ τ \cdot} \int_{0}^{\cdot} h_{n} (σ) e^{λ σ τ} d σ)}^{⊤}, Z^{(i)})}_{H} .$ Indeed, if this is not the case, we can consider ${\tilde{U}}_{n} = U_{n} - \sum_{i = 1}^{N} β_{i} Z^{(i)}$ that still belongs to $D (A)$ and satisfies $(λ I - A) {\tilde{U}}_{n} = F_{n},$ as well as ${({\tilde{U}}_{n}, Z^{(i)})}_{H} = - α_{n, i} \forall i = 1, \dots, N,$ by setting $β_{i} = {(U_{n}, Z^{(i)})}_{H} + α_{n, i} \forall i = 1, \dots, N .$

Note that the condition (3.14) is equivalent to ${(u_{n}, y^{(i)})}_{λ, V} = 0 \forall i = 1, \dots, N .$ In other words, $u_{n} \in {(ker A_{λ})}^{⊥_{λ, V}},$ (3.15) where $⊥_{λ, V}$ means that the orthogonality is taken with respect to the inner product ${(\cdot, \cdot)}_{λ, V}$ .

Returning to (3.13), the arguments of the proof of Lemma 2.1 imply that $A_{λ} u_{n} = L_{F_{n}} in V^{'},$ where $L_{F}$ is defined by $L_{F} (w) : = {(g, w)}_{H} - τ e^{- λ τ} {(\int_{0}^{1} h (σ) e^{λ σ τ} d σ, B^{} w)}_{H_{1}} + {(λ f + a B^{} f, w)}_{H_{1}},$ when $F = {(f, g, h)}^{⊤}$ . But it is easy to check that $L_{F_{n}} \to L_{F} in V^{'} .$ Moreover, as $λ \in C ∖ Σ$ , $A_{λ}$ is an isomorphism from ${(ker A_{λ})}^{⊥_{λ, V}}$ into $R (A_{λ})$ , hence by (3.15) we deduce that there exists a positive constant $C (λ)$ such that $∥ u_{n} - u_{m} ∥_{V} ⩽ C (λ) ∥ L_{F_{n}} - L_{F_{m}} ∥_{V^{'}} \forall n, m \in N .$ Hence, ${(u_{n})}_{n}$ is a Cauchy sequence in V, and therefore there exists $u \in V$ such that $u_{n} \to u in V,$ as well as $A_{λ} u = L_{F} in V^{'} .$ Then defining v by (2.13) and z by (2.16), we deduce that $U : = {(u, v, z)}^{⊤}$ belongs to $D (A)$ and $(λ I - A) U = F .$ In other words, F belongs to $R (λ I - A)$ . The closedness of $R (λ I - A)$ is thus proved.

At this stage, for any $λ \in C ∖ Σ$ , we show that $codim R (A_{λ}) = codim R (λ I - A),$ (3.16) where for $W \subset H$ , $codim W$ is the dimension of the orthogonal in $H$ of W, while for $W^{'} \subset V^{'}$ , $codim W^{'}$ is the dimension of the annihilator $A : = {v \in V : {⟨ v, w ⟩}_{V - V^{'}} = 0 \forall w \in W^{'}},$ of $W^{'}$ in V.

Indeed, let us set $N = codim R (A_{λ})$ , then there exist N elements $φ_{i} \in V$ , $i = 1, \dots, N$ , such that $f \in R (A_{λ}) ⟺ f \in V^{'} and {⟨ f, φ_{i} ⟩}_{V^{'} - V} = 0 \forall i = 1, \dots, N .$ Consequently, for $F \in H$ , if $L_{F}$ (that belongs to $V^{'}$ ) satisfies $L_{F} (φ_{i}) = 0 \forall i = 1, \dots, N,$ (3.17) then there exists a solution $u \in V$ of $A_{λ} u = L_{F} in V^{'},$ and the arguments of the proof of Lemma 2.1 imply that F is in $R (λ I - A)$ . Hence, the N conditions on $F \in H$ from (3.17) allow to show that it belongs to $R (λ I - A)$ , and therefore $codim R (λ I - A) ⩽ N = codim R (A_{λ}) .$ (3.18) This shows that $λ I - A$ is a Fredholm operator.

Conversely, set $M = codim R (λ I - A)$ , then there exist M elements $Ψ_{i} = (u_{i}, v_{i}, z_{i}) \in H$ , $i = 1, \dots, M$ , such that $F \in R (λ I - A) ⟺ F \in H and {(F, Ψ_{i})}_{H} = 0 \forall i = 1, \dots, M .$ Then, for any $g \in H$ , if ${(g, v_{i})}_{H} = {({(0, g, 0)}^{⊤}, Ψ_{i})}_{H} = 0 \forall i = 1, \dots, M,$ (3.19) there exists $U = {(u, v, z)}^{⊤} \in D (A)$ such that $(λ I - A) U = (0, g, 0),$ which implies that $A_{λ} u = g .$ This shows that $R (A_{λ}) \supset H_{0},$ where $H_{0} : = {g \in H satisfying (3.19)}$ . This inclusion implies that (here ⊥ means the annihilator of the set in V) $R {(A_{λ})}^{⊥} \subset H_{0}^{⊥} .$ Therefore $\begin{array}{rcl} R {(A_{λ})}^{⊥} & \subset & {v \in V : {⟨ v, g ⟩}_{V - V^{'}} = 0 \forall g \in H_{0}} \\ = & {v \in V : {(v, g)}_{H} = 0 \forall g \in H_{0}} \\ \subset & Span {v_{i}}_{i = 1}^{M} \cap V . \end{array}$ Hence, $codim R (A_{λ}) ⩽ M = codim R (λ I - A) .$ (3.20)

The inequalities (3.18) and (3.20) imply (3.16). □
Lemma 3.4.
If $τ ⩽ a$ , then* $Σ \subset {λ \in C : ℜ λ < 0} .$
Proof.
Let $λ = x + i y \in Σ$ , with $x, y \in R$ we deduce that $\begin{array}{rcl} a x + e^{- τ x} cos (τ y) = 0, \\ a y - e^{- τ x} sin (τ y) = 0 . \end{array}$ This corresponds to the system (3.6)–(3.7) with $k = \infty$ , hence the arguments as in the proof of Lemma 3.1 yield the result. □
Corollary 3.5.
It holds $σ (A) = σ_{p p} (A) \cup Σ,$ and therefore if $τ ⩽ a$ $σ (A) \subset {λ \in C : ℜ λ < 0} .$
Proof.
By Theorem 3.3, $C ∖ Σ \subset σ_{p p} (A) \cup ρ (A) .$ The first assertion directly follows.

The second assertion follows from Lemmas 3.1 and 3.4. □

4. Asymptotic behavior

In this section, we show that if $τ ⩽ a$ and $ξ > \frac{2 τ}{a}$ , the semigroup $e^{t A}$ decays to the null steady state with an exponential decay rate. To obtain this, our technique is based on a frequency domain approach and combines a contradiction argument to carry out a special analysis of the resolvent.

Theorem 4.1.
If $ξ > \frac{2 τ}{a}$ and $τ ⩽ a$ , then there exist constants $C, ω > 0$ such that the semigroup $e^{t A}$ satisfies the following estimate ${∥ e^{t A} ∥}_{L (H)} ⩽ C e^{- ω t} \forall t > 0 .$ (4.1)
Proof.
We will employ the following frequency domain theorem for uniform stability from [13, Theorem 8.1.4] of a $C_{0}$ semigroup on a Hilbert space. Lemma 4.2.
A $C_{0}$ semigroup $e^{t L}$ on a Hilbert space $H$ satisfies ${∥ e^{t L} ∥}_{L (H)} ⩽ C e^{- ω t},$ for some constant $C > 0$ and for $ω > 0$ if and only if $ℜ λ < 0 \forall λ \in σ (L),$ (4.2) and $sup_{ℜ λ ⩾ 0} {∥ {(λ I - L)}^{- 1} ∥}_{L (H)} < \infty,$ (4.3) where $σ (L)$ denotes the spectrum of the operator $L$ .

According to Corollary 3.5 the spectrum of $A$ is fully included into $ℜ λ < 0$ , which clearly implies (4.2). Then the proof of Theorem 4.1 is based on the following lemma that shows that (4.3) holds with $L = A$ . Lemma 4.3.
The resolvent operator of $A$ satisfies condition $sup_{ℜ λ ⩾ 0} {∥ {(λ I - A)}^{- 1} ∥}_{L (H)} < \infty .$ (4.4)
Proof.
Suppose that condition (4.4) is false. By the Banach–Steinhaus theorem (see [7]), there exists a sequence of complex numbers $λ_{n}$ such that $ℜ λ_{n} ⩾ 0$ , $| λ_{n} | \to + \infty$ and a sequence of vectors $Z_{n} = {(u_{n}, v_{n}, z_{n})}^{t} \in D (A)$ with $∥ Z_{n} ∥_{H} = 1$ (4.5) such that ${∥ (λ_{n} I - A) Z_{n} ∥}_{H} \to 0 as n \to \infty,$ (4.6) i.e., $\begin{array}{rclr} λ_{n} u_{n} - v_{n} \equiv f_{n} \to 0 in D (B^{}), & (4.7) \\ λ_{n} v_{n} + a B (B^{} v_{n} + z_{n} (1)) \equiv g_{n} \to 0 in H, & (4.8) \\ λ_{n} z_{n} + τ^{- 1} \partial_{ρ} z_{n} \equiv h_{n} \to 0 in L^{2} ((0, 1); H_{1}) . & (4.9) \end{array}$

Our goal is to derive from (4.6) that $∥ Z_{n} ∥_{H}$ converges to zero, that furnishes a contradiction.

We notice that from (2.10) and (4.7) we have $\begin{array}{rcl} {∥ (λ_{n} I - A) Z_{n} ∥}_{H} & ⩾ & | ℜ {((λ_{n} I - A) Z_{n}, Z_{n})}_{H} | \\ ⩾ & ℜ λ_{n} - a_{}^{- 1} {∥ B^{} u_{n} ∥}_{H_{1}}^{2} + (\frac{ξ}{2 τ} - \frac{1}{a}) {∥ z_{n} (1) ∥}_{H_{1}}^{2} + \frac{a}{2} {∥ B^{} v_{n} ∥}_{H_{1}}^{2} \\ = & ℜ λ_{n} - a_{}^{- 1} {∥ \frac{B^{} v_{n} + B^{} f_{n}}{λ_{n}} ∥}_{H_{1}}^{2} + (\frac{ξ}{2 τ} - \frac{1}{a}) {∥ z_{n} (1) ∥}_{H_{1}}^{2} + \frac{a}{2} {∥ B^{} v_{n} ∥}_{H_{1}}^{2} . \end{array}$ Hence using the inequality ${∥ B^{} v_{n} + B^{} f_{n} ∥}_{H_{1}}^{2} ⩽ 2 {∥ B^{} v_{n} ∥}_{H_{1}}^{2} + 2 {∥ B^{} f_{n} ∥}_{H_{1}}^{2},$ we obtain that $\begin{array}{rcl} {∥ (λ_{n} I - A) Z_{n} ∥}_{H} & ⩾ & ℜ λ_{n} - 2 a_{}^{- 1} | λ_{n} |^{- 2} {∥ B^{} f_{n} ∥}_{H_{1}}^{2} + (\frac{ξ}{2 τ} - \frac{1}{a}) {∥ z_{n} (1) ∥}_{H_{1}}^{2} \\ + (\frac{a}{2} - 2 a_{}^{- 1} | λ_{n} |^{- 2}) {∥ B^{} v_{n} ∥}_{H_{1}}^{2} . \end{array}$ Hence for n large enough, say $n ⩾ n^{}$ , we can suppose that $\frac{a}{2} - 2 a_{}^{- 1} | λ_{n} |^{- 2} ⩾ \frac{a}{4}$ and therefore for all $n ⩾ n^{}$ , we get ${∥ (λ_{n} I - A) Z_{n} ∥}_{H} ⩾ ℜ λ_{n} - 2 a_{}^{- 1} | λ_{n} |^{- 2} {∥ B^{} f_{n} ∥}_{H_{1}}^{2} + (\frac{ξ}{2 τ} - \frac{1}{a}) {∥ z_{n} (1) ∥}_{H_{1}}^{2} + \frac{a}{4} {∥ B^{} v_{n} ∥}_{H_{1}}^{2} .$ By this estimate, (4.6) and (4.7), we deduce that $z_{n} (1) \to 0, B^{} v_{n} \to 0 in H_{1} as n \to \infty,$ (4.10) and in particular, from the coercivity (1.4), that $v_{n} \to 0 in H as n \to \infty .$ This implies according to (4.7) that $u_{n} = \frac{1}{λ_{n}} v_{n} + \frac{1}{λ_{n}} f_{n} \to 0 in D (B^{}) as n \to \infty,$ (4.11) as well as $z_{n} (0) = B^{} u_{n} \to 0 in H_{1} as n \to \infty .$ (4.12) By integration of the identity (4.9), we have $z_{n} (ρ) = z_{n} (0) e^{- τ λ_{n} ρ} + τ \int_{0}^{ρ} e^{- τ λ_{n} (ρ - γ)} h_{n} (γ) d γ .$ (4.13) Hence recalling that $ℜ λ_{n} ⩾ 0$ $\int_{0}^{1} {∥ z_{n} (ρ) ∥}_{H_{1}}^{2} d ρ ⩽ 2 {∥ z_{n} (0) ∥}_{H_{1}}^{2} + 2 τ^{2} \int_{0}^{1} \int_{0}^{ρ} {∥ h_{n} (γ) ∥}_{H_{1}}^{2} d γ ρ d ρ \to 0 as n \to \infty .$ All together we have shown that $∥ Z_{n} ∥_{H}$ converges to zero, that clearly contradicts $∥ Z_{n} ∥_{H} = 1$ . □

The two hypotheses of Lemma 4.2 are proved, then (4.1) holds. The proof of Theorem 4.1 is then finished. □

5. Application to the stabilization of the wave equation with delay and a Kelvin–Voigt damping

We study the internal stabilization of a delayed wave equation. More precisely, we consider the system given by: $\begin{array}{rclr} u_{t t} (x, t) - a Δ u_{t} (x, t) - Δ u (x, t - τ) = 0 in Ω \times (0, + \infty), & (5.1) \\ u = 0 on \partial Ω \times (0, + \infty), & (5.2) \\ u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x) in Ω, & (5.3) \\ \nabla u (x, t - τ) = f_{0} (t - τ) in Ω \times (0, τ), & (5.4) \end{array}$ where Ω is a smooth open bounded domain of $R^{n}$ and $a, τ > 0$ are constants.

This problem enters in our abstract framework with $H = L^{2} (Ω)$ , $B = - div : D (B) = H^{1} {(Ω)}^{n} \to L^{2} (Ω)$ , $B^{*} = \nabla : D (B^{*}) = H_{0}^{1} (Ω) \to H_{1} : = L^{2} {(Ω)}^{n}$ , the assumption (1.4) being satisfied owing to Poincaré’s inequality. The operator $A$ is then given by $A (\begin{matrix} u \\ v \\ z \end{matrix}) : = (\begin{matrix} v \\ a Δ v + div z (\cdot, 1) \\ - τ^{- 1} z_{ρ} \end{matrix}),$ with domain $\begin{matrix} D (A) : = {{(u, v, z)}^{⊤} \in H_{0}^{1} (Ω) \times H_{0}^{1} (Ω) \times L^{2} (Ω; H^{1} (0, 1)) : a \nabla v + z (\cdot, 1) \in H^{1} (Ω), \\ \nabla u = z (\cdot, 0) in Ω}, \end{matrix}$ (5.5) in the Hilbert space $H : = H_{0}^{1} (Ω) \times L^{2} (Ω) \times L^{2} (Ω \times (0, 1)) .$ (5.6)

According to Lemma 3.5 and Theorem 4.1 we have: Corollary 5.1.

If $τ ⩽ a$ , the system (5.1)–(5.4) is exponentially stable in $H$ , namely for $ξ > \frac{2 τ}{a}$ , the energy $E (t) = \frac{1}{2} (\int_{Ω} ({| \nabla u (x, t) |}^{2} + {| u_{t} (x, t) |}^{2}) d x + ξ \int_{Ω} \int_{0}^{1} {| \nabla u (x, t - τ ρ) |}^{2} d x d ρ),$ satisfies $E (t) ⩽ M e^{- ω t} E (0) \forall t > 0$ for some positive constants M and ω.

6. Conclusion

By a careful spectral analysis combined with a frequency domain approach, we have shown that the system (1.1)–(1.3) is exponentially stable if $τ ⩽ a$ and that this condition is optimal. But from the general form of (1.1), we can only consider interior Kelvin–Voigt dampings. Hence an interesting perspective is to consider the wave equation with dynamical Ventcel boundary conditions with a delayed term and a Kelvin–Voigt damping.

Footnotes

Acknowledgements

The authors are grateful to the referee for her/his thorough and careful reading of the paper, as well as for helpful suggestions and comments.

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Stability of an abstract-wave equation with delay and a Kelvin–Voigt damping

Abstract

Keywords

1. Introduction

3.1. The discrete spectrum

Footnotes

Acknowledgements

References