ArticleOriginal scientific text
Title
On the nonlinear stabilization of the wave equation
Authors 1
Affiliations
- Institut de recherche mathématique avancée, Université Louis Pasteur et CNRS, 7, rue René Descartes, 67084 Strasbourg Cedex, France
Abstract
We obtain a precise decay estimate of the energy of the solutions to the initial boundary value problem for the wave equation with nonlinear internal and boundary feedbacks. We show that a judicious choice of the feedbacks leads to fast energy decay.
Keywords
wave equation, nonlinear damping, integral inequality
Bibliography
- J. B. Ball, On the asymptotic behavior of generalized processes with applications to nonlinear evolution equations, J. Differential Equations 27 (1978), 224-265.
- F. Conrad and M. Pierre, Stabilization of second order evolution equations by unbounded nonlinear feedback, Ann. Inst. Henri Poincaré 11 (1994), 485-515.
- A. Guesmia, Stabilisation frontière non linéaire d'un système isotropique d'élasticité, submitted.
- V. Komornik, Exact Controllability and Stabilization, the Multiplier Method, Masson-Wiley, Paris, 1994.
- V. Komornik, Decay estimates for the wave equation with internal damping, in: Proc. Conf. Control Theory, Vorau 1993, Internat. Ser. Numer Anal. 118, Birkhäuser, Basel, 1994, 253-266.
- V. Komornik, On the nonlinear boundary stabilization of the wave equation, Chinese Ann. Math. Ser. B 14 (1993), 153-164.
- V. Komornik, Rapid boundary stabilization of the wave equation, SIAM J. Control Optim. 29 (1991), 197-208.
- V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. Pures Appl. 69 (1990), 33-54.
- M. Nakao, On the decay of solutions of some nonlinear dissipative wave equations in higher dimensions, Math. Z. 193 (1986), 227-234.
- M. Nakao, Energy decay for the wave equation with a nonlinear weak dissipation, Differential Integral Equations 8 (1995), 681-688.
- E. Zuazua, Uniform stabilization of the wave equation by nonlinear boundary feedback, SIAM J. Control Optim. 28 (1990), 446-477.