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

TL;DR

This paper investigates the conditions under which dissipative infinite-dimensional systems with intermittent damping achieve various forms of asymptotic stability, highlighting the roles of observability, unique continuation, and excitation conditions.

Contribution

It introduces generalized conditions for exponential, weak, and strong asymptotic stability in infinite-dimensional systems with intermittent damping, extending previous results to more general scenarios.

Findings

Exponential stability under generalized observability inequality.

Weak asymptotic stability via a generalized unique continuation principle.

Strong asymptotic stability without persistent excitation conditions.

Abstract

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, Schr\"odinger's equation and, for strong stability, also the special case of finite-dimensional systems.