Dirichlet boundary stabilization of the wave equation with damping feedback of finite range

Abstract

A “closed loop” system consisting of the wave equation with a feedback acting in the Dirichlet boundary condition in the form of a nonlocal, one-dimensional range operator, defined on the “velocity” vector (damping) is considered. Beside the well-posedness question (generation of a feedback C 0 -semigroup), the boundary feedback stabilization problem is studied: while asymptotic decay to zero in the uniform operator norm can never occur for the considered class of closed loop systems, however checkable sufficient conditions on the vectors in the boundary conditions are provided that guarantee asymptotic decay to zero in the strong norm of appropriate Sobolev spaces. These conditions include, but are not limited to, the standard case of dissipativity of the feedback system. The starting point of this approach is a functional analytic model for the study of nonsmooth boundary input hyperbolic equations recently developed (I. Lasiecka and R. Triggiani, Appl. Math. Optim. 7 (1) (1981), 35–93).