Stabilization of Abstract Parabolic Equations

Abstract

In this chapter, we present a technique to design asymptotically exponentially stabilizing boundary proportional-type feedback controllers for nonlinear parabolic-like equations, namely equations for which their linear parts are generated by analytic $C_0$-semigroups. In what follows, we will simply refer to them as parabolic equations, in concordance with the title of this book. The feedback law’s main features are that it is expressed in an explicit simple form and has a finite-dimensional structure involving only the eigenfunctions of the linear operator obtained from the linearized equation. As we will see, these features will enable us to obtain the first results to appear in the literature regarding the stabilization of different equations, such as the stochastic heat equation, the Chan–Hilliard equations, and for boundary stabilization to nonsteady states for parabolic-type equations.