A Sharp Geometric Condition for the Boundary Exponential Stabilizability of a Square Plate by Moment Feedbacks only

Abstract

We consider a boundary stabilization problem for the plate equation in a square. The feedback law gives the bending moment on a part of the boundary as function of the velocity field of the plate. The main result of the paper asserts that the obtained closed loop system is exponentially stable if and only if the controlled part of the boundary contains a vertical and a horizontal part of non-zero length (the geometric optics condition introduced by Bardos, Lebeau and Rauch in [2] for the wave equation is thus not necessary in this case). The proof of the main result uses the methodology introduced in Ammari and Tucsnak [1], where the exponential stability for the closed loop problem is reduced to an observability estimate for the corresponding uncontrolled system combined to a boundedness property of the transfer function of the associated open loop system. The second essential ingredient of the proof is an observability inequality recently proved by Ramdani, Takahashi, Tenenbaum and Tucsnak [7]