Strong stability of elastic control systems with dissipative saturating feedback

We will consider, with a focus on saturating feedback control laws, two problems associated with damping in a bounded acoustic cavity Omega subset of R-3. Our objective is to verify (compare (Discrete Continuous Dynamical Systems 7 (2001) 319, Math. Control Signals Systems 2 (1989) 265) that these are strongly stable: for every finite-energy solution, the acoustic energy goes to zero as t --> infinity. We will, in each case, formulate the problem in terms of a contraction semigroup, of nonlinear operators on an appropriate Hilbert space and compare this with the corresponding semigroups without saturation-following Avalos and Lasiecka (Semigroup Forum 57 (1998) 278) in using the spectral methods of Arendt and Batty (Trans. Amer. Math. Soc. 8 (1988) 837) to show strong stabilization for those linear semigroups. (C) 2002 Elsevier Science B.V. All rights reserved.