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

Falk Hante (1) , Mario Sigalotti (2, 3) , Marius Tucsnak (4, 5)

1 IWR - Interdisciplinary Center for Scientific Computing
2 GECO - Geometric Control Design
3 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
4 CORIDA - Robust control of infinite dimensional systems and applications
5 IECN - Institut Élie Cartan de Nancy

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, Schrödinger's equation and, for strong stability, also the special case of finite-dimensional systems.

Mots clés: maximal dissipative operator; Intermittent damping; asymptotic behavior; persistent excitation

Domaines: Optimisation et contrôle [math.OC]

Falk Hante, Mario Sigalotti, Marius Tucsnak. On conditions for asymptotic stability of dissipative infinite-dimensional systems with intermittent damping. Journal of Differential Equations, 2012, 252 (10), pp.5569-5593. ⟨10.1016/j.jde.2012.01.037⟩. ⟨inria-00616474v2⟩