Control of the Wave Equation by Time-Dependent Coefficient

¹ CEREMADE, UMR 7534 du CNRS, Université de Paris-Dauphine, 75775 Paris Cedex 16, France; antonin.chambolle@ceremade.dauphine.fr.
² School of Mathematics, University of Minnesota, Minneapolis, MN 55455, U.S.A.; santosa@math.umn.edu.

Received: 14 January 2002
Revised: 15 February 2002

Abstract

We study an initial boundary-value problem for a wave equation with time-dependent sound speed. In the control problem, we wish to determine a sound-speed function which damps the vibration of the system. We consider the case where the sound speed can take on only two values, and propose a simple control law. We show that if the number of modes in the vibration is finite, and none of the eigenfrequencies are repeated, the proposed control law does lead to energy decay. We illustrate the rich behavior of this problem in numerical examples.