Feedback stabilization of semilinear heat equations

V Barbu,G Wang

doi:10.1155/s1085337503212100

Abstract

This paper is concerned with the internal and boundary stabilization of the steady‐state solutions to quasilinear heat equations via internal linear feedback controllers provided by an LQ control problem associated with the linearized equation.

Highlights

Let Ω ∈ Rn be an open and bounded subset with smooth boundary ∂Ω, let ω ⊂ Ω be an open subset, and let m be the characteristic function of ω
It turns out that if f = f (x, r) is locally Lipschitz continuous in r, any sufficiently smooth steady-state solution ye is locally controllable in any finite time T and so, in particular, it is stabilizable
By defining an appropriate infinite horizon LQ problem with unbounded cost functional, we find a linear selfadjoint and positive operator P, which is the solution to an algebraic Riccati equation associated with the LQ problem such that the feedback law (1.4) makes ye locally exponentially stable

Summary

Introduction

Let Ω ∈ Rn be an open and bounded subset with smooth boundary ∂Ω, let ω ⊂ Ω be an open subset, and let m be the characteristic function of ω. It turns out (see [1, 3]) that if f = f (x, r) is locally Lipschitz continuous in r, any sufficiently smooth steady-state solution ye is locally controllable in any finite time T and so, in particular, it is stabilizable. By defining an appropriate infinite horizon LQ problem with unbounded cost functional, we find a linear selfadjoint and positive operator P, which is the solution to an algebraic Riccati equation associated with the LQ problem such that the feedback law (1.4) makes ye locally exponentially stable. We obtain a boundary feedback controller which locally exponentially stabilizes (1.2) Such a result was obtained (see [2]) for 3-dimensional Navier-Stokes equations.

Assumptions and the main results

Boundary feedback stabilization of semilinear equation