Uniform Stabilizability of a Full Von Karman System with Nonlinear Boundary Feedback

Abstract

Full von Karman system accounting for in-plane accelerations 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. It is known [J. Puel and M. Tucsnak, SIAM J. Control Optim., 33 (1995), pp. 255--273] that in the case of linear dissipation and star-shaped domains, boundary velocity feedback with the tangential derivatives of horizontal displacements leads to the exponential decay rates for the energy of the resulting closed loop system. The main goal of the paper is to derive the uniform energy decay rates valid for the model without the above-mentioned restrictions. In particular, it is shown that simple, monotone nonlinear feedback (without the tangential derivatives of the horizontal displacements) 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 recently in [V. Isakov, J. Differential Equations, 97 (1997), pp. 134--147] for the dynamical systems of elasticity which are overdetermined on the boundary.