• Views 319
• Citations 0
• ePub 11
• PDF 232
`Mathematical Problems in EngineeringVolume 2014 (2014), Article ID 176583, 5 pageshttp://dx.doi.org/10.1155/2014/176583`
Research Article

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

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

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

Copyright © 2014 Jian-Sheng Tian et al. 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.

#### 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.

#### 1. Introduction

There are many results concerning the boundary stabilization of classical wave equations. See [16] for linear cases and [712] for nonlinear ones. The stability of the wave equation with variable coefficients has attracted much attention. See [1323], 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 be a bounded domain in with smooth boundary . It is assumed that consists of two parts and with . Define where is the divergence operator of the standard metric of ; is symmetric, positively definite matrices for each and are smooth functions on .

We consider the stabilization of the wave equations with variable coefficients and time-varying delay in the dissipative boundary feedback: and there exists a positive constant such that and satisfies where is a positive constant and .

is the conormal derivative where denotes the standard metric of the Euclidean space and is the outside unit normal vector for each . Moreover, the initial data belongs to a suitable space.

Define the energy of the system (2) by

We define as a Riemannian metric on and consider the couple as a Riemannian manifold with an inner product:

Let denote the Levi-Civita connection of the metric . For the variable coefficients, the main assumptions are as follows.

Assumption A. There exists a vector field on and a constant such that Moreover we assume that where 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 To obtain the stabilization of the system (2), we assume the system (2) is well-posed such that

The main result of this paper is stated as follows.

Theorem 1. Let Assumption A holds true. Then there exist positive constants , such that

#### 2. Basic Inequality of the System

In this section we work on with two metrics at the same time, the standard dot metric and the Riemannian metric given by (8).

If , we define the gradient of in the Riemannian metric , via the Riesz representation theorem, by where is any vector field on . The following lemma provides further relations between the two metrics; see [13] in Lemma 2.1.

Lemma 2. Let be the natural coordinate system in . Let , be functions and let , be vector fields. Then(a)(b)where is the gradient of in the standard metric;(c)where the matrix is given in formula (1).

To prove Theorem 1, we still need several lemmas further. Define Then, we have

Lemma 3. Let be the solution of system (2). Then there exists a constant such that where . The assertion (22) implies that is decreasing.

Proof. Differentiating (7), we obtain 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 solves equation , and that is a vector field defined on . Then, for ,
Moreover, assume that . Then

Lemma 5. Suppose that all assumptions in Theorem 1 hold true. Let be the solution of the system (2). Then there exist positive constants for which where .

Proof. We let be a positive constant satisfying Set Substituting the identity (25) into the identity (24), we obtain where
Decompose as Since , we obtain ; that is, Similarly, we have Using the formulas (32) and (33) in the formula (30) on the portion , with (12), we obtain From (12), we have
Substituting the formulas (34) and (35) into the formula (29), with (27), we obtain
It follows from (22) that
Substituting the formulas (22) and (37) into the formula (36), the inequality (26) holds.

Proof of Theorem 1. Since is decreasing, with (4) and (26), for sufficiently large , we have Note that 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.

#### References

1. J. E. Lagnese, “Note on boundary stabilization of wave equations,” SIAM Journal on Control and Optimization, vol. 26, no. 5, pp. 1250–1256, 1988.
2. D. L. Russell, “Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions,” SIAM Review, vol. 20, no. 4, pp. 639–739, 1978.
3. R. Triggiani, “Wave equation on a bounded domain with boundary dissipation: an operator approach,” Journal of Mathematical Analysis and Applications, vol. 137, no. 2, pp. 438–461, 1989.
4. Y. You, “Energy decay and exact controllability for the Petrovsky equation in a bounded domain,” Advances in Applied Mathematics, vol. 11, no. 3, pp. 372–388, 1990.
5. M. Aassila, M. M. Cavalcanti, and V. N. D. Cavalcanti, “Existence and uniform decay of the wave equation with nonlinear boundary damping and boundary memory source term,” Calculus of Variations and Partial Differential Equations, vol. 15, no. 2, pp. 155–180, 2002.
6. M. M. Cavalcanti, V. N. Domingos Cavalcanti, and P. Martinez, “Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term,” Journal of Differential Equations, vol. 203, no. 1, pp. 119–158, 2004.
7. F. Conrad and B. Rao, “Decay of solutions of the wave equation in a star-shaped domain with nonlinear boundary feedback,” Asymptotic Analysis, vol. 7, no. 3, pp. 159–177, 1993.
8. V. Komornik, “On the nonlinear boundary stabilization of the wave equation,” Chinese Annals of Mathematics B, vol. 14, no. 2, pp. 153–164, 1993.
9. V. Komornik, Exact Controllability and Stabilization: The Multiplier Method, John Wiley & Sons, Chichester, UK, 1994.
10. V. Komornik and E. Zuazua, “A direct method for the boundary stabilization of the wave equation,” Journal de Mathématiques Pures et Appliquées, vol. 69, no. 1, pp. 33–54, 1990.
11. I. Lasiecka and D. Tataru, “Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping,” Differential and Integral Equations, vol. 6, no. 3, pp. 507–533, 1993.
12. E. Zuazua, “Uniform stabilization of the wave equation by nonlinear boundary feedback,” SIAM Journal on Control and Optimization, vol. 28, no. 2, pp. 466–477, 1990.
13. P. F. Yao, “On the observability inequalities for exact controllability of wave equations with variable coefficients,” SIAM Journal on Control and Optimization, vol. 37, no. 5, pp. 1568–1599, 1999.
14. P.-F. Yao, Modeling and Control in Vibrational and Structural Dynamics: A Differential Geometric Approach, Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series, CRC Press, Boca Raton, Fla, USA, 2011.
15. I. Lasiecka, R. Triggiani, and P. F. Yao, “Inverse/observability estimates for second-order hyperbolic equations with variable coefficients,” Journal of Mathematical Analysis and Applications, vol. 235, no. 1, pp. 13–57, 1999.
16. Z. H. Ning and Q. X. Yan, “Stabilization of the wave equation with variable coefficients and a delay in dissipative boundary feedback,” Journal of Mathematical Analysis and Applications, vol. 367, no. 1, pp. 167–173, 2010.
17. Z. H. Ning, C. X. Shen, and X. P. Zhao, “Stabilization of the wave equation with variable coefficients and a delay in dissipative internal feedback,” Journal of Mathematical Analysis and Applications, vol. 405, no. 1, pp. 148–155, 2013.
18. Z. H. Ning, C. X. Shen, X. Zhao, H. Li, C. Lin, and Y. M. Zhang, “Nonlinear boundary stabilization of the wave equations with variable coefficients and time dependent delay,” Applied Mathematics and Computation, vol. 232, pp. 511–520, 2014.
19. S. Nicaise and C. Pignotti, “Stabilization of the wave equation with variable coefficients and boundary condition of memory type,” Asymptotic Analysis, vol. 50, no. 1-2, pp. 31–67, 2006.
20. B. Gong and X. Zhao, “Boundary stabilization of a semilinear wave equation with variable coefficients under the time-varying and nonlinear feedback,” Abstract and Applied Analysis, vol. 2014, Article ID 728760, 6 pages, 2014.
21. H. Li, C. S. Lin, S. P. Wang, and Y. M. Zhang, “Stabilization of the wave equation with boundary time-varying delay,” Advances in Mathematical Physics, vol. 2014, Article ID 735341, 6 pages, 2014.
22. B.-Z. Guo and Z.-C. Shao, “On exponential stability of a semilinear wave equation with variable coefficients under the nonlinear boundary feedback,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, pp. 5961–5978, 2009.
23. Z. H. Ning, C. X. Shen, and X. P. Zhao, “stabilization of the wave equation with variable coefficients and a internal memory type,” Nonlinear Analysis: Real World Applications. In press.
24. M. M. Cavalcanti, V. N. D. Cavalcanti, and I. Lasiecka, “Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping—source interaction,” Journal of Differential Equations, vol. 236, no. 2, pp. 407–459, 2007.