Exponential stabilization of solutions for the 1-d transmission wave equation with boundary feedback

Abstract

The purpose of this work is to study the exponential decay of the energy for

the one-dimensional transmission wave equation with a boundary velocity feedback.

Thanks to the perturbed energy method developed by some authors in several contexts, and

under certain conditions, we prove that the feedback controller exponentially stabilizes the

equilibrium to zero of the system below, i.e. the feedback leads to faster energy decay.

Keywords

• Boundary feedback,decay rate of energy,exponential stabilization,perturbed energy

Kaynakça

1. [1] K. Ammari, Derichlet boundary stabilization of the wave equation , Asymptot. Anal.30 (2002) 117-130.
2. [2] G. Chen, Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain . J. Math. Pures Appl. 58, 249-273 (1979)
3. [3] G. Chen, Control and stabilization for the wave equation in a bounded domain . SIAM J. Control Optim. 17, 66-81 (1979).
4. [4] G. Chen, Control and stabilization for the wave equation, part III: Domain with moving boundary. SIAM J. Control Optim.19, 123-138 (1981).
5. [5] C. Deng,Y. Liu, W. Jiang, F. Huang, Exponential decay rate for a wave equation with Dirichlet boundary control, Applied Mathematics letters, 20 (2007) 861-865.
6. [6] L.C. Evans, Partial Differential Equations, Vol. 19, American Mathematical Society, 1997.
7. [7] I. Lasiecka& R. Trigiani, Uniform exponential energy decay of the wave equation in a bounded region with feedback control in the Dirichlet boundary conditions, J. Differential Equations. 66 (1987) 340-390.
8. [8] J.L. Lions, Controlabilite exacte perturbation et stabilisation de systemes distribues, Tome 1, Controlabilite exacte. Masson, Paris (1988).

9. [9] J.L. Lions, Controlabilit´e exacte perturbation et stabilisation de systemes distribues, Tome 2, Perturbation. Masson, Paris (1988).
10. [10] W. Liu, Stabilization and controllability for the transmission wave equation. IEEE Transcation on Automatic Control 46, 1900-1907 (2001).
11. [11] W. Liu, E. Zuazua, Decay rates for dissipative wave equations. Ricerche di Matimatica. 48, 61-75 (1999).
12. [12] W. Liu, E. Zuazua, Uniform stabilization of the higher dimensional system of thermoelastisity with boundary feedback. Quartyely Appl. Math. 59, 269-314 (2001).
13. [13] M. Nakao, Energy decay for the wave equation with nonlinear weak dissipation. Differential Integral Equation, 8, 681-688 (1995).
14. [14] J. Rauch& M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domains. India J. Math. 24, 79-83 (1974)
15. [15] B. Straughan, The Energy Method, Stability, and Nonlinear Convection, Springer Science Business Media, New York, 2004.
16. [16] E. Zuazua, Uniform stabilization of the wave equation by nonlinear boundary feedback. SIAM J. Control and optim. 28 (1990) 466-478.
17. [17] E. Zuazua, Exponential decay for the semi-linear wave equation with locally distributed damping. Commun. in Partial Differential Equations 15, 205-235 (1990)