Neumann Boundary Stabilization of One-Dimensional Linear Parabolic Systems With Input Delay

This paper addresses the boundary stabilization problem of one-dimensional linear parabolic systems with input delay. Especially, the case of Neumann boundary control is studied. The purpose is to derive the stabilizing controller of predictor type in a Hilbert space, by using a backstepping method combined with the semigroup theory. The use of the semigroup theory makes the proof of continuity of the inverse transformation easy. Also, it is shown that the implementation of the abstract controller is feasible by using a finite number of eigenvalues and eigenfunctions of the system operator.

I. Introduction

Since early times, the control problem of the system with input delay has been investigated by many researchers (see, e.g., the monograph [9] and references therein). The sphere of the control objects treated in [9] ranges from lumped parameter systems to distributed parameter systems, and from linear systems to nonlinear systems. Especially, when one considers the thermal control problem, large time lag exists in the feedback loop, since it is generally difficult to regulate heat quickly. So, it is very important to study the control problem of parabolic systems with input delay.