Stabilization of solutions for a class of parabolic integro-differential equations

Abstract

Abstract : Integro-differential equations arise in the description of feed-back control systems, where the control variables are derived from filtered observations of the state or where the control mechanism possesses inertia. The author studies a model equation for a distributed control system (e.g., the state varies over some space-like domain) which contains also some diffusion effects and give conditions under which the state will tend to some limit, as time goes to infinity, regardless of the initial situation. The limit is shown to satisfy an elliptic differential equation. Convergence rates are also given; these show the slowing-down effect of a slow control mechanism on the convergence of the state variable. The problem under study can also be viewed as a natural extension of a type of reaction-diffusion equation that has received wide attention in the literature. (Author)