The uniform stabilization of a full von Karman system with nonlinear boundary feedback

Abstract

A full von Karman system accounting for in-plane acceleration and describing the transient deformations of a thin, elastic plate subject to edge loading is considered. The energy dissipation is introduced via the nonlinear velocity feedback acting on a part of the edge of the plate. This paper aims to derive the uniform energy decay rates valid for the model without the above mentioned restrictions. In particular, it is shown that a simple, monotone nonlinear feedback (without the tangential derivatives of the horizontal displacement) provides the uniform decay rates for the energy, in the absence of geometric hypotheses imposed on the controlled part of the boundary. This is accomplished by establishing, among other things, sharp regularity results valid for the boundary traces of solutions corresponding to this nonlinear model and by employing a Holmgren type uniqueness result proved by Isakov (1997) for the dynamical systems of elasticity which are overdetermined on the boundary.