Uniform stabilization of a full von Karman system with nonlinear boundary feedback


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 feedbacks acting on a part of the edge of the plate. It is known 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 a 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 [6] for the dynamical systems of elasticity which are overdetermined on the boundary.

Publication Title

Proceedings of the IEEE Conference on Decision and Control

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