# Exponential Stability of Compactly Coupled Wave Equations with Delay Terms in the Boundary Feedbacks

Salah-Eddine Rebiai, Fatima Zohra Sidi Ali

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Salah-Eddine Rebiai, Fatima Zohra Sidi Ali. Exponential Stability of Compactly Coupled Wave Equations with Delay Terms in the Boundary Feedbacks. 26th Conference on System Modeling and Optimization (CSMO), Sep 2013, Klagenfurt, Austria. pp.278-284, 10.1007/978-3-662-45504-3_27 hal-01286436

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# Exponential stability of compactly coupled wave equations with delay terms in the boundary feedbacks

Salah-Eddine Rebiai and Fatima Zohra Sidi Ali

LTM, Department of Mathematics, Faculty of Sciences, University of Batna, 05000 Batna, Algeria

**Abstract.** We consider a linear system of compactly coupled wave equations with Neumann feedback controllers that contain delay terms. First, we prove under some assumptions that the closed-loop system generates a $C_0$-semigroup of contractions on an appropriate Hilbert space. Then, under further assumptions, we show that the closed-loop system is exponentially stable. This result is obtained by introducing a suitable energy function and by using an observability estimate.

**Keywords:** Coupled wave equations, time delays, boundary stabilization

## 1 Introduction

In [1] and [2], Datko et al presented examples of infinite-dimensional second-order systems that become unstable when arbitrary small time delays occur in the damping.

Xu et al established in [9] sufficient conditions that guarantee the exponential stability of the one-dimensional wave equation with a delay term in the boundary feedback. Nicaise and Pignotti [6] extended this result to the multi-dimensional wave equation with a delay term in the boundary or internal feedbacks. The same type of result was obtained by Nicaise and Rebiai [7] for the Schrödinger equation.

Motivated by the references [9], [6], [3] and [5], we investigate in this paper the problem of exponential stability for a linear system of compactly coupled wave equations with delay terms in the boundary feedbacks.

Let $\Omega$ be an open bounded domain of $\mathbb{R}^n$ with a boundary $\Gamma$ of class $C^2$ which consists of two non-empty parts $\Gamma_1$ and $\Gamma_2$ such that $\bar{\Gamma}_1 \cap \bar{\Gamma}_2 = \emptyset$. Furthermore, assume that there exists a real vector field $h \in (C^2(\bar{\Omega}))^n$ such that:
(H.1) The Jacobian matrix $J$ of $h$ satisfies

$$ \int_{\Omega} J(x)\zeta(x)\cdot\zeta(x)d\Omega \geq c \int_{\Omega} |\zeta(x)|^2 d\Omega, $$

for some constant $c > 0$ and for all $\zeta \in L^2(\Omega; \mathbb{R}^n)$,

(H.2) $h(x)\cdot\nu(x) \leq 0$ on $\Gamma_1$,

where $\nu$ is the unit normal on $\Gamma$ pointing towards the exterior of $\Omega$.
Consider the following coupled system of two wave equations with delay
terms in the boundary conditions:

$$
\frac{\partial^2 u(x,t)}{\partial t^2} - \Delta u(x,t) + l(u(x,t) - v(x,t)) = 0 \quad \text{in } \Omega \times (0, +\infty), \quad (1)
$$

$$
\frac{\partial^2 v(x,t)}{\partial t^2} - \Delta v(x,t) + l(v(x,t) - u(x,t)) = 0 \quad \text{in } \Omega \times (0, +\infty), \quad (2)
$$

$$
u(x,0) = u_0(x), \quad \frac{\partial u(x,0)}{\partial t} = u_1(x) \qquad \text{in } \Omega, \tag{3}
$$

$$
v(x, 0) = v_0(x), \quad \frac{\partial v(x, 0)}{\partial t} = v_1(x) \qquad \text{in } \Omega, \tag{4}
$$

$$
u(x,t) = v(x,t) = 0 \qquad \text{on } \Gamma_1 \times (0, +\infty), \tag{5}
$$

$$
\frac{\partial u(x,t)}{\partial \nu} = -\alpha_1 \frac{\partial u(x,t)}{\partial t} - \alpha_2 \frac{\partial u(x,t-\tau)}{\partial t} \quad \text{on } \Gamma_2 \times (0, +\infty), \quad (6)
$$

$$
\frac{\partial v(x,t)}{\partial \nu} = -\beta_1 \frac{\partial v(x,t)}{\partial t} - \beta_2 \frac{\partial v(x,t-\tau)}{\partial t} \quad \text{on } \Gamma_2 \times (0, +\infty), \quad (7)
$$

$$
\frac{\partial u(x, t - \tau)}{\partial t} = g(x, t - \tau) \qquad \text{on } \Gamma_2 \times (0, \tau), \quad (8)
$$

$$
\frac{\partial v(x, t - \tau)}{\partial t} = h(x, t - \tau) \quad \text{on } \Gamma_2 \times (0, \tau). \quad (9)
$$

Physically, $u$ and $v$ may represent the displacements of two vibratings objects
measured from their equilibrium positions, the coupling terms $\pm l(u-v)$ are the
distributed springs linking the two vibrating objects. $l, \alpha_1, \alpha_2, \beta_1, \beta_2$ are positive
constants, $\tau$ is the time delay, $u_0, u_1, v_0, v_1, g$ and $h$ are the initial data.

It is well known that in the absence of delay (i.e. $\alpha_2 = \beta_2 = 0$), the solution of
(1)-(9) with $\alpha_1$ and $\beta_1$ positive, decays exponentially to zero in the energy space
$H_{\Gamma_1}^1(\Omega) \times L^2(\Omega) \times H_{\Gamma_1}^1(\Omega) \times L^2(\Omega)$ (see [5] and [3]).

The purpose of this paper is to investigate the uniform exponential stability of
system (1)–(9) in the case where all the boundary damping coefficients $\alpha_1, \alpha_2, \beta_1$
and $\beta_2$ are positive. To this end, assume as in [6] that

$$
\alpha_1 > \alpha_2, \beta_1 > \beta_2 \tag{10}
$$

and define the energy of a solution of (1) – (9) by

$$
\begin{equation}
\begin{split}
E(t) = \frac{1}{2} \int_{\Omega} [ & |\nabla u(x,t)|^2 + \left| \frac{\partial u(x,t)}{\partial t} \right|^2 + |\nabla v(x,t)|^2 + \left| \frac{\partial v(x,t)}{\partial t} \right|^2 + \\
& l |u(x,t) - v(x,t)|^2 ] dx + \frac{1}{2} \int_{\Gamma_2} \int_0^1 [\mu \left| \frac{\partial u(x,t-\tau\rho)}{\partial t} \right|^2 + \\
& \qquad \xi \left| \frac{\partial v(x,t-\tau\rho)}{\partial t} \right|^2 ] d\rho \, d\Gamma
\end{split}
\tag{11}
\end{equation}
$$

where

$$
\tau \alpha_2 < \mu < \tau(2\alpha_1 - \alpha_2) \tag{12}
$$
and

$$ \tau\beta_2 < \xi < \tau(2\beta_1 - \beta_2) \qquad (13) $$

We show that if {$\Omega, \Gamma_1, \Gamma_2$} satisfies (H.1) and (H.2), then there is an exponential decay rate for $E(t)$. The proof of this result is based on Carleman estimates for a system of coupled nonconservative hyperbolic systems established by Lasieka and Triggiani in [4] and on compactness-uniqueness arguments. The main result of this paper can be stated as follows.

**Theorem 1.** Assume (H1), (H.2), (10), (12) and (13). Then there exist constants $M \ge 1$ and $\omega > 0$ such that

$$ E(t) \le Me^{-\omega t}E(0). $$

Theorem 1 is proved in Section 3. In Section 2, we study the well-posedness of system (1) – (9) using semigroup theory.

## 2 Well-posedness of system (1) – (9)

Inspired from [6] and [7], we introduce the auxiliary variables

$$ y(x, \rho, t) = \frac{\partial u(x, t - \tau \rho)}{\partial t} $$

$$ z(x, \rho, t) = \frac{\partial v(x, t - \tau \rho)}{\partial t} $$

With these new unknowns, system (1) – (9) is equivalent to

$$ \frac{\partial^2 u(x, t)}{\partial t^2} - \Delta u(x, t) + l(u(x, t) - v(x, t)) = 0 \quad \text{in } \Omega \times (0, +\infty), \qquad (14) $$

$$ \frac{\partial y(x, \rho, t)}{\partial t} + \frac{1}{\tau} \frac{\partial y(x, \rho, t)}{\partial \rho} = 0 \quad \text{on } \Gamma_2 \times (0, 1) \times (0, +\infty), \qquad (15) $$

$$ \frac{\partial^2 v(x, t)}{\partial t^2} - \Delta v(x, t) + l(v(x, t) - u(x, t)) = 0 \quad \text{in } \Omega \times (0, +\infty), \qquad (16) $$

$$ \frac{\partial z(x, \rho, t)}{\partial t} + \frac{1}{\tau} \frac{\partial z(x, \rho, t)}{\partial \rho} = 0 \quad \text{on } \Gamma_2 \times (0, 1) \times (0, +\infty), \qquad (17) $$

$$ u(x, t) = v(x, t) = 0 \quad \text{on } \Gamma_1 \times (0, +\infty), \qquad (18) $$

$$ \frac{\partial u(x, t)}{\partial \nu} = -\alpha_1 \frac{\partial u(x, t)}{\partial t} - \alpha_2 y(x, 1, t) \quad \text{on } \Gamma_2 \times (0, +\infty), \qquad (19) $$

$$ \frac{\partial v(x, t)}{\partial \nu} = -\beta_1 \frac{\partial u(x, t)}{\partial t} - \beta_2 z(x, 1, t) \quad \text{on } \Gamma_2 \times (0, +\infty), \qquad (20) $$

$$ y(x, 0, t) = \frac{\partial u(x, t)}{\partial t}, z(x, 0, t) = \frac{\partial v(x, t)}{\partial t} \quad \text{on } \Gamma_2 \times (0, +\infty), \qquad (21) $$

$$ u(x, 0) = u_0(x), \quad \frac{\partial u(x, 0)}{\partial t} = u_1(x) \quad \text{in } \Omega, \qquad (22) $$

$$ v(x, 0) = v_0(x), \quad \frac{\partial v(x, 0)}{\partial t} = v_1(x) \quad \text{in } \Omega, \qquad (23) $$

$$ y(x, \rho, 0) = g(x, -\tau \rho), \quad z(x, \rho, 0) = h(x, -\tau \rho) \quad \text{on } \Gamma_2 \times (0, 1). \qquad (24) $$
Denote by $\mathcal{H}$ the Hilbert space

$$ \mathcal{H} = H_{\Gamma_1}^1(\Omega) \times L^2(\Omega) \times L^2(\Gamma_2 \times L^2(0,1)) \times H_{\Gamma_1}^1(\Omega) \times L^2(\Omega) \times L^2(\Gamma_2 \times L^2(0,1)) $$

where

$$ H_{\Gamma_1}^1(\Omega) = \{u \in H^1(\Omega) : u = 0 \text{ on } \Gamma_1\} $$

We equip $\mathcal{H}$ with the inner product

$$ \left\langle \begin{pmatrix} \zeta \\ \eta \\ \theta \\ \phi \\ \chi \\ \psi \end{pmatrix} ; \begin{pmatrix} \tilde{\zeta} \\ \tilde{\eta} \\ \tilde{\theta} \\ \tilde{\phi} \\ \tilde{\chi} \\ \tilde{\psi} \end{pmatrix} \right\rangle = \int_{\Omega} (\nabla \zeta(x) \cdot \nabla \tilde{\zeta}(x) + \eta(x) \tilde{\eta}(x)) dx + \\ \mu \int_{\Gamma_2} \int_0^1 \theta(x, \rho) \tilde{\theta}(x, \rho) d\rho d\Gamma + \int_{\Omega} (\nabla \phi(x) \cdot \nabla \tilde{\phi}(x) + \chi(x) \tilde{\chi}(x)) dx + \\ \xi \int_{\Gamma_2} \int_0^1 \psi(x, \rho) \tilde{\psi}(x, \rho) d\rho d\Gamma + l \int_{\Omega} (\zeta(x) - \phi(x))(\tilde{\zeta}(x) - \tilde{\phi}(x)) dx $$

Define in $\mathcal{H}$ a linear operator $\mathcal{A}$ by

$$ D(\mathcal{A}) = \{(\zeta, \eta, \theta, \phi, \chi, \psi)^T \in H^2(\Omega) \times H_{\Gamma_1}^1(\Omega) \times L^2(\Gamma_2 \times H^1(0,1)) \times H^2(\Omega) \times H_{\Gamma_1}^1(\Omega) \times L^2(\Gamma_2 \times H^1(0,1)); \frac{\partial \zeta}{\partial \nu} = -\alpha_1 \eta - \alpha_2 \theta(\cdot, 1), \\ \eta = \theta(\cdot, 0) \text{ on } \Gamma_2; \frac{\partial \phi}{\partial \nu} = -\beta_1 \chi - \beta_2 \psi(\cdot, 1), \chi = \psi(\cdot, 0) \text{ on } \Gamma_2 \} \quad (25) $$

$$ A(\zeta, \eta, \theta, \phi, \chi, \psi)^T = (\eta, \Delta\zeta + l\phi - l\zeta, -\tau^{-1}\frac{\partial\theta}{\partial\rho}, \chi, \Delta\phi - l\phi + l\zeta, -\tau^{-1}\frac{\partial\psi}{\partial\rho})^T \quad (26) $$

Then we can rewrite (14) – (24) as an abstract Cauchy problem in $\mathcal{H}$

$$ \left\{ \begin{array}{l} \frac{d}{dt} W(t) = \mathcal{A}W(t) \\ W(0) = W_0 \end{array} \right. \quad (27) $$

where

$$ W(t) = (u(x,t), \frac{\partial u(x,t)}{\partial t}, y(x,\rho,t), v(x,t), \frac{\partial v(x,t)}{\partial t}, z(x,\rho,t))^T, $$ 
and $W_0 = (u_0, u_1, g(\cdot, -\tau), v_0, v_1, h(\cdot, -\tau))^T$

We verify that $\mathcal{A}$ is dissipative and that $\lambda I - \mathcal{A}$ is onto for a fixed $\lambda > 0$. Thus, by the Lumer-Phillips Theorem (see for instance [8]) $\mathcal{A}$ generates a strongly continuous semigroup on $\mathcal{H}$ and consequently we have
**Proposition 1.** For every $W_0 \in \mathcal{H}$, problem (27) has a unique solution $W$ whose regularity depends on the initial datum $W_0$ as follows:

$$ W(\cdot) \in C([0, +\infty); \mathcal{H}) \text{ if } W_0 \in \mathcal{H}, \\ W(\cdot) \in C^1([0, +\infty); \mathcal{H}) \cap C([0, +\infty); D(\mathcal{A})) \text{ if } W_0 \in D(\mathcal{A}). $$

### 3 Proof of Theorem 1

We prove Theorem 1 for smooth initial data. The general case follows by a standard density argument.

We proceed in several steps.

**Step 1.**

Differentiating $E(t)$ with respect to time, we obtain

$$ \frac{d}{dt} E(t) \leq -k \int_{\Gamma_2} \left\{ \left| \frac{\partial u(x, t)}{\partial t} \right|^2 + \left| \frac{\partial u(x, t - \tau)}{\partial t} \right|^2 + \left| \frac{\partial v(x, t)}{\partial t} \right|^2 + \left| \frac{\partial v(x, t - \tau)}{\partial t} \right|^2 \right\} d\Gamma \quad (28) $$

where

$$ k = \min\{\alpha_1 - \frac{\alpha_2}{2} - \frac{\mu}{2\tau}, \frac{\mu}{2\tau} - \frac{\alpha_2}{2}, \beta_1 - \frac{\beta_2}{2} - \frac{\xi}{2\tau}, \frac{\xi}{2\tau} - \frac{\beta_2}{2}\} $$

**Step 2.**

We rewrite

$$ E(t) = \mathcal{E}(t) + E_d(t) $$

where

$$ \mathcal{E}(t) = \frac{1}{2} \int_{\Omega} \left\{ \left| \nabla u(x, t) \right|^2 + \left| \frac{\partial u(x, t)}{\partial t} \right|^2 + \left| \nabla v(x, t) \right|^2 + \left| \frac{\partial v(x, t)}{\partial t} \right|^2 + l \left| u(x, t) - v(x, t) \right|^2 \right\} dx $$

and

$$ E_d(t) = \frac{1}{2} \int_{\Gamma_2} \int_0^1 \left\{ \mu \left| \frac{\partial u(x, t - \tau \rho)}{\partial t} \right|^2 + \xi \left| \frac{\partial v(x, t - \tau \rho)}{\partial t} \right|^2 \right\} d\rho d\Gamma $$

$E_d(t)$ can be rewritten via a change of variable as

$$ E_d(t) = \frac{1}{2\tau} \int_t^{t+\tau} \int_{\Gamma_2} \left\{ \mu \left| \frac{\partial u(x, s-\tau)}{\partial t} \right|^2 + \xi \left| \frac{\partial v(x, s-\tau)}{\partial t} \right|^2 \right\} d\Gamma ds \quad (29) $$

From (29), we obtain (here and throughout the rest of the paper $C$ is some positive constant different at different occurences)

$$ E_d(t) \leq C \int_0^T \int_{\Gamma_2} \left\{ \left| \frac{\partial u(x, s-\tau)}{\partial t} \right|^2 + \left| \frac{\partial v(x, s-\tau)}{\partial t} \right|^2 \right\} d\Gamma ds \quad (30) $$

for $0 \le t + \tau \le T$ and $T$ large enough.
### Step 3.

From Poincaré inequality and Proposition 3.5 of [4], we have for $T$ sufficiently large and for any $\epsilon > 0$

$$ \mathcal{E}(0) \le C \int_0^T \int_{\Gamma_2} \left\{ \left| \frac{\partial u(x,t)}{\partial \nu} \right|^2 + \left| \frac{\partial u(x,t)}{\partial t} \right|^2 + \left| \frac{\partial v(x,t)}{\partial \nu} \right|^2 + \left| \frac{\partial v(x,t)}{\partial t} \right|^2 \right\} d\Gamma dt + C \left\{ \|u\|_{L^2(0,T;H^{1/2+\epsilon}(\Omega))}^2 + \|v\|_{L^2(0,T;H^{1/2+\epsilon}(\Omega))}^2 \right\} \quad (31) $$

Inserting the boundary conditions (6) and (7) into (31), we obtain

$$ \mathcal{E}(0) \le C \int_0^T \int_{\Gamma_2} \left\{ \left| \frac{\partial u(x,t)}{\partial t} \right|^2 + \left| \frac{\partial u(x, t-\tau)}{\partial t} \right|^2 + \left| \frac{\partial v(x,t)}{\partial t} \right|^2 + \left| \frac{\partial v(x, t-\tau)}{\partial t} \right|^2 \right\} d\Gamma dt + C \left\{ \|u\|_{L^2(0,T;H^{1/2+\epsilon}(\Omega))}^2 + \|v\|_{L^2(0,T;H^{1/2+\epsilon}(\Omega))}^2 \right\} \quad (32) $$

### Step 4.

Estimate (30) together with (32) yields

$$ E(0) \le C \int_0^T \int_{\Gamma_2} \left\{ \left| \frac{\partial u(x,t)}{\partial t} \right|^2 + \left| \frac{\partial u(x, t-\tau)}{\partial t} \right|^2 + \left| \frac{\partial v(x,t)}{\partial t} \right|^2 + \left| \frac{\partial v(x, t-\tau)}{\partial t} \right|^2 \right\} d\Gamma dt + C \left\{ \|u\|_{L^2(0,T;H^{1/2+\epsilon}(\Omega))}^2 + \|v\|_{L^2(0,T;H^{1/2+\epsilon}(\Omega))}^2 \right\} \quad (33) $$

### Step 5.

We drop the lower order terms on the right-hand side of (33) by a compactness-uniqueness argument to obtain

$$ E(0) \le C \int_0^T \int_{\Gamma_2} \left\{ \left| \frac{\partial u(x,t)}{\partial t} \right|^2 + \left| \frac{\partial u(x, t-\tau)}{\partial t} \right|^2 + \left| \frac{\partial v(x,t)}{\partial t} \right|^2 + \left| \frac{\partial v(x, t-\tau)}{\partial t} \right|^2 \right\} d\Gamma dt \quad (34) $$

### Step 6.

From (28), we have

$$ E(T)-E(0) \le -k \int_0^T \int_{\Gamma_2} \left\{ \left| \frac{\partial u(x,t)}{\partial t} \right|^2 + \left| \frac{\partial u(x, t-\tau)}{\partial t} \right|^2 + \left| \frac{\partial v(x,t)}{\partial t} \right|^2 + \left| \frac{\partial v(x, t-\tau)}{\partial t} \right|^2 \right\} d\Gamma dt $$

which together with (34) leads to

$$ E(T) \le \frac{Ck^{-1}}{1+Ck^{-1}} E(0) \quad (35) $$

The desired conclusion follows now from (35).

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