Boundary Stabilization of the Wave Equation with Time-Varying and Nonlinear Feedback

Abstract

We study the stabilization of the wave equation with variable coefficients in a bounded domain and a time-varying and nonlinear term. By the Riemannian geometry methods and a suitable assumption of nonlinearity and the time-varying term, we obtain the uniform decay of the energy of the system.

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Jian-Sheng Tian 1, 2 and Wei Wang 1, 2 and Fei Xue 1, 2 and Pei-Yong Cong 1, 2

Academic Editor:Yoshinori Hayafuji

1, College of Computer Science, Beijing University of Technology, Beijing 100124, China
2, State Engineering Laboratory of Information System Classified Protection Key Technologies, Beijing University of Technology, Beijing 100124, China

Received 24 April 2014; Accepted 7 June 2014; 29 June 2014

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

There are many results concerning the boundary stabilization of classical wave equations. See [1-6] for linear cases and [7-12] for nonlinear ones. The stability of the wave equation with variable coefficients has attracted much attention. See [13-23], and many others. In [20], by the methods in [11, 24], the authors study the stability of the wave equation with nonlinear term and time-varying term. However, under the condition the nonlinear term has upper bound and the time-varying term has lower bound, the stability of the wave equation was not studied in [20]. In this paper, our purpose is to study the stability of the wave equation under the condition the nonlinear term has upper bound and the time-varying term has lower bound.

Let Ω be a bounded domain in ^{R n} with smooth boundary Γ . It is assumed that Γ consists of two parts _{Γ 1} and _{Γ 2} ( Γ = _{Γ 1} ∪ _{Γ 2} ) with _{Γ 2} ...0; ∅ , _{Γ ¯ 1} ∩ _{Γ ¯ 2} = ∅ . Define [figure omitted; refer to PDF] where div ... is the divergence operator of the standard metric of ^{R n} ; A ( x ) = ( _{a i j} ( x ) ) is symmetric, positively definite matrices for each x ∈ ^{R n} and _{a i j} ( x ) are smooth functions on ^{R n} .

We consider the stabilization of the wave equations with variable coefficients and time-varying delay in the dissipative boundary feedback: [figure omitted; refer to PDF] _{g 1} ∈ C ( R ) and there exists a positive constant _{c 1} such that [figure omitted; refer to PDF] and [varphi] ( t ) ∈ C ( [ 0 , + ∞ ) ) satisfies [figure omitted; refer to PDF] where _{[varphi] 0} is a positive constant and F ( t ) = _{max ... 0 ...4; ρ ...4; t} [varphi] ( ρ ) .

∂ u / ∂ _{ν A} is the conormal derivative [figure omitted; refer to PDF] where Y9; · , · YA; denotes the standard metric of the Euclidean space ^{R n} and ν ( x ) is the outside unit normal vector for each x ∈ Γ . Moreover, the initial data ( _{u 0} , _{u 1} ) belongs to a suitable space.

Define the energy of the system (2) by [figure omitted; refer to PDF]

We define [figure omitted; refer to PDF] as a Riemannian metric on ^{R n} and consider the couple ( ^{R n} , g ) as a Riemannian manifold with an inner product: [figure omitted; refer to PDF]

Let _{D g} denote the Levi-Civita connection of the metric g . For the variable coefficients, the main assumptions are as follows.

Assumption A.

There exists a vector field H on Ω ¯ and a constant _{ρ 0} > 0 such that [figure omitted; refer to PDF] Moreover we assume that [figure omitted; refer to PDF] where δ is a positive constant.

Assumption (10) was introduced by [13] as a checkable assumption for the exact controllability of the wave equation with variable coefficients. For examples on the condition, see [13, 14].

Based on Assumption (10), Assumption A was given by [19] to study the stabilization of the wave equation with variable coefficients and boundary condition of memory type.

Define [figure omitted; refer to PDF] To obtain the stabilization of the system (2), we assume the system (2) is well-posed such that [figure omitted; refer to PDF]

The main result of this paper is stated as follows.

Theorem 1.

Let Assumption A holds true. Then there exist positive constants C , _{C 2} , such that [figure omitted; refer to PDF]

2. Basic Inequality of the System

In this section we work on Ω with two metrics at the same time, the standard dot metric Y9; · , · YA; and the Riemannian metric g = _{Y9; · , · YA; g} given by (8).

If f ∈ ^{C 1} ( ^{R n} ) , we define the gradient _{∇ g} f of f in the Riemannian metric g , via the Riesz representation theorem, by [figure omitted; refer to PDF] where X is any vector field on ( ^{R n} , g ) . The following lemma provides further relations between the two metrics; see [13] in Lemma 2.1.

Lemma 2.

Let x = ( _{x 1} , ... , _{x n} ) be the natural coordinate system in ^{R n} . Let f , h be functions and let H , X be vector fields. Then

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

: where ∇ f is the gradient of f in the standard metric;

(c) [figure omitted; refer to PDF]

: where the matrix A ( x ) is given in formula (1).

To prove Theorem 1, we still need several lemmas further. Define [figure omitted; refer to PDF] Then, we have [figure omitted; refer to PDF]

Lemma 3.

Let ( u ) be the solution of system (2). Then there exists a constant _{C 1} such that [figure omitted; refer to PDF] where T ...5; 0 . The assertion (22) implies that E ( t ) is decreasing.

Proof.

Differentiating (7), we obtain [figure omitted; refer to PDF] Then the inequality (22) holds true.

3. Proofs of Theorem 1

From Proposition 2.1 in [13], we have the following identities.

Lemma 4.

Suppose that u ( x , t ) solves equation _{u t t} + A u = 0 , ( x , t ) ∈ Ω × ( 0 , + ∞ ) and that H is a vector field defined on Ω ¯ . Then, for T ...5; 0 , [figure omitted; refer to PDF]

Moreover, assume that P ∈ ^{C 1} ( Ω ¯ ) . Then [figure omitted; refer to PDF]

Lemma 5.

Suppose that all assumptions in Theorem 1 hold true. Let u be the solution of the system (2). Then there exist positive constants C , _{T 0} for which [figure omitted; refer to PDF] where T ...5; _{T 0} .

Proof.

We let θ be a positive constant satisfying [figure omitted; refer to PDF] Set [figure omitted; refer to PDF] Substituting the identity (25) into the identity (24), we obtain [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Decompose _{Π Γ} as [figure omitted; refer to PDF] Since _{u | Γ 2} = 0 , we obtain _{∇ Γ u | Γ 2} = 0 ; that is, [figure omitted; refer to PDF] Similarly, we have [figure omitted; refer to PDF] Using the formulas (32) and (33) in the formula (30) on the portion _{Γ 2} , with (12), we obtain [figure omitted; refer to PDF] From (12), we have [figure omitted; refer to PDF]

Substituting the formulas (34) and (35) into the formula (29), with (27), we obtain [figure omitted; refer to PDF]

It follows from (22) that [figure omitted; refer to PDF]

Substituting the formulas (22) and (37) into the formula (36), the inequality (26) holds.

Proof of Theorem 1.

Since E ( t ) is decreasing, with (4) and (26), for sufficiently large T , we have [figure omitted; refer to PDF] Note that E ( t ) is decreasing; the estimate (15) holds.

4. Application of the System (2)

Nonlinear feedback describes a property of a physical system; that is, the response by the physical system to an applied force is nonlinear in its effect. One of the applications of the system (2) is in sound waves, where the system (2) describes the reflection of sound in heterogeneous materials at surfaces of some materials with nonlinearity of interest in engineering practice. Theorem 1 indicates that the energy of the sound waves with the reflection of sound at surfaces in heterogeneous materials at surfaces of some materials with nonlinearity is uniform decay under a suitable assumption of the nonlinearity.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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