Exponential Stabilization of the Wave Equation by Dirichlet Integral Feedback

Abstract

We consider the problem of boundary feedback stabilization of a vibrating string that is fixed at one end and with control action at the other end. In contrast to previous studies that have required $L^2$-regularity for the initial position and $H^{-1}$-regularity for the initial velocity, in this paper we allow for initial positions with $L^1$-regularity and initial velocities in $W^{-1,1}$ on the space interval. It is well known that for a certain feedback parameter, for sufficiently regular initial states the classical energy of the closed-loop system with Neumann velocity feedback is controlled to zero after a finite time that is equal to the minimal time where exact controllability holds. In this paper, we present a Dirichlet boundary feedback that yields a well-defined closed-loop system in the ($L^1$, $W^{-1,1}$) framework and also has this property. Moreover, for all positive feedback parameters our feedback law leads to exponential decay of a suitably defined $L^1$-energy. For more regular initial states with $(L^2, \,H^{-1})$ regularity, the proposed feedback law leads to exponential decay of an energy that corresponds to this framework. If the initial states are even more regular with $H^1$-regularity of the initial position and $L^2$-regularity of the initial velocity, our feedback law also leads to exponential decay of the classical energy.