Boundary Observability, Controllability, and Stabilization of Linear Elastodynamic Systems

Abstract

In 1988 Lions obtained observability and exact controllability results for linear homogeneous isotropic elastodynamic systems [ SIAM Rev., 30 (1988), pp. 1--68]. Applying some new identities we extend his theorems to nonisotropic systems. In 1991 Lagnese obtained uniform stabilizability results for two-dimensional linear homogeneous isotropic systems by applying a somewhat artificial feedback [Nonlinear Anal., 16 (1991), pp. 35--54]. Then he asked whether analogous results hold for a natural and physically implementable boundary feedback. Using some new identities and applying a method introduced in 1987 by Zuazua and the second author [ J. Math. Pures. Appl., 69 (1990), pp. 33--54], we give an affirmative answer to this question in all dimensions and also for nonisotropic systems. Moreover, we obtain good decay estimates. Finally, applying a recent general method of uniform stabilization, we construct boundary feedbacks leading to arbitrarily large energy decay rates.