ABSTRACT

We will establish Riemannian multiplier identities by the Bochner technique for the plate with variable coefficients to derive exact controllability/stabilization. Those multipliers are a geometric version of the classical ones for the plate with constant coefficients. Then we introduce a checkable assumption which guarantees exact controllability/stabilization, which is called escape vector fields for plate. It is different from the one for the wave equation. In particular, Section 3.2 is devoted to study the existence of escape vector fields for plate by curvature and many examples will be given. Finally, in Section 3.5, we will derive a plate model which is defined on a curved surface by the variational principle. In this model only the deformation along its normal is considered while the deformation in the tangential is neglected so it may serve as a curved plate. Stabilization by boundary feedbacks is presented for this model.