Explicit construction of a boundary feedback law to stabilize a class of parabolic equations

Abstract

In this paper, a problem of boundary feedback stabilization for a class of parabolic equations is considered. The problem is reformulated to convert a parabolic equation into a well known one by using an integral transformation with a kernel required to satisfy an appropriate partial differential equation (PDE). The well-posedness of the kernel PDE and the smoothness of the solution are studied. By varying a parameter ${\lambda}>0$ in the kernel PDE, the solution is exploited to explicitly construct a boundary feedback law such that the solution of the closed loop system decays exponentially at the desired rate of ${\lambda}.$ Moreover, a control which, simultaneously, stabilizes the output function and minimizes an appropriate cost functional is derived.