On conditions for asymptotic stability of dissipative infinite-dimensional systems with intermittent damping

• Published in: Journal of Differential Equations Vol. 252; no. 10; pp. 5569 - 5593
• Main Authors: Hante, Falk M., Sigalotti, Mario, Tucsnak, Marius
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15-05-2012; Elsevier
• Subjects: Asymptotic behavior; Intermittent damping; Mathematics; Maximal dissipative operator; Optimization and Control; Persistent excitation; Maximal dissipative operator; Intermittent damping; Persistent excitation; Asymptotic behavior; maximal dissipative operator; asymptotic behavior; persistent excitation
• ISSN: 0022-0396; 1090-2732
• DOI: 10.1016/j.jde.2012.01.037

We study the asymptotic stability of a dissipative evolution in a Hilbert space subject to intermittent damping. We observe that, even if the intermittence satisfies a persistent excitation condition, if the Hilbert space is infinite-dimensional then the system needs not being asymptotically stable (not even in the weak sense). Exponential stability is recovered under a generalized observability inequality, allowing for time-domains that are not intervals. Weak asymptotic stability is obtained under a similarly generalized unique continuation principle. Finally, strong asymptotic stability is proved for intermittences that do not necessarily satisfy some persistent excitation condition, evaluating their total contribution to the decay of the trajectories of the damped system. Our results are discussed using the example of the wave equation, Schrödingerʼs equation and, for strong stability, also the special case of finite-dimensional systems.