Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions


In this paper we eliminate altogether geometrical conditions that were assumed (even) with control action on the entire boundary in prior literature: (i) strict convexity of our paper [LT4] on uniform stabilization of the wave equation in the (optimal) state space L2(Ω)×H-1(Ω) with L2(Σ) Dirichlet feedback control, as well as (ii) "star-shaped" conditions in papers [C1], [La1], and [Tr1] on uniform stabilization and [Lio1] and [LT5] on exact controllability in the energy space H1(Ω)×L2(Ω) of the wave equation with L2(Σ)-Neumann feedback control. Key to the present improvements is a pseudodifferential analysis which permits us to express certain boundary traces of the solution in terms of other traces modulo lower-order interior terms. See Lemma 3.1 for the Dirichlet case and Lemma 7.2 for the Neumann case. © 1992 Springer-Verlag New York Inc.

Publication Title

Applied Mathematics & Optimization