Feedback stabilization of some unstable elliptic-parabolic systems

Abstract

This work presents feedback designs of exponentially stabilizing boundary controllers for a one-dimensional system coupling elliptic and parabolic equations. The control acts on one Dirichlet boundary condition of the parabolic equation. First, we use a backstepping approach and deal with the case where the parabolic solution can be fully measured. We prove that the classical backstepping design for the heat equation allows to get the stabilization in some unstable cases. Then, we assume that we can only measure the Neumann boundary condition of the elliptic equation at the left end-point. By using an observer we get the desired feedback law. Finally splitting the system in the unstable and stable part, we design a feedback controller using the Pole Placement Method and the Artstein Transformation even in the case where the control has time-delay. Using a Lyapunov function we show the stability of the entire system. Moreover, as consequence of an intermediate result we prove the null controllability of the system via Moments method.