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 Ω⊂R3. 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→∞. 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. © 2002 Elsevier Science B.V. All rights reserved.
Systems and Control Letters
Lasiecka, I., & Seidman, T. (2003). Strong stability of elastic control systems with dissipative saturating feedback. Systems and Control Letters, 48 (3-4), 243-252. https://doi.org/10.1016/S0167-6911(02)00269-4