Globally Stabilizing Finite-Dimensional Damping Control for Dissipative PDE with Bounded Inputs

Abstract

The main aim of this paper is to address the global asymptotic stabilization (GAS) of a class of dissipative partial differential equations (pdes) via finite-dimensional admissible (regular and bounded) damping (output) controls. To this end, we use the inertial manifold theory to derive infinite-dimensional dissipative control systems given by interconnection of a finite-dimensional control system on the inertial manifold plus an infinite-dimensional zero-input system on the complement. We show that the GAS of such systems is reduced to the finite-dimensional one. Then, we prove that the finite-dimensional systems which are zero-input point-dissipative (have global attractors K) and those which are B-strictly passive (passivity relative to bounded sets) are connected. Finally, we use the control Lyapunov functions (CLF) theory to design admissible feedback damping controls for the GAS of B-strictly passive systems.