Feedback control for an abstract parabolic equation

Abstract

Abstract : In their previous paper, the authors considered the problem of the exponential stabilization of the heat equation with Neumann boundary conditions in the smooth and bounded domain OMEGA in R(exp n), where partial derivative/partial derivative nu denotes the differentiation in the direction normal to the boundary partial derivative OMEGA. The solutions to this problem, although stable, are not asymptotically stable, because spatially homogeneous functions do not decay. A practical way to induce asymptotic stability is to introduce a feedback mechanism that observes the temperature at a patch of the boundary partial derivative OMEGA and causes an appropriate exchange of heat through the boundary or through an interior subset of OMEGA. A feedback mechanism of the latter type may be modeled by introducing a term on the right hand side of the equation. The functions phi and sigma, prescribed on partial derivative OMEGA and OMEGA, respectively, are indicators of the sites where the observation and the feedback take place. The coefficient epsilon is the of the feedback. Feedback mechanisms of the former type may also be treated by combining the method of section 6 of [6], and the estimates developed in this paper. More generally, one may consider a more complex diffusion process governed by a general (autonomous) linear elliptic operator ALPHA, and several feedback mechanisms of the type above that act simultaneously and independently. The method introduced in [6] for the stability analysis extends to this and more general situations. In this paper, the authors study an abstract evolution equation that includes the problem in [6] and the problem corresponding to equation (1.3) as special cases. Their main result is that if certain hypotheses are satisfied, and if epsilon is sufficiently small, then the system is exponentially asymptotically stable. The smallness of the feedback gain factor epsilon is essential for stability.