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

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 (n...

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Hauptverfasser: Hante, Falk Michael (VerfasserIn) , Sigalotti, Mario (VerfasserIn) , Tucsnak, Marius (VerfasserIn)
Dokumenttyp: Article (Journal)
Sprache:Englisch
Veröffentlicht: 16 February 2012
In: Journal of differential equations
Year: 2012, Jahrgang: 252, Heft: 10, Pages: 5569-5593
ISSN:1090-2732
DOI:10.1016/j.jde.2012.01.037
Online-Zugang:Verlag, Volltext: http://dx.doi.org/10.1016/j.jde.2012.01.037
Verlag, Volltext: http://linkinghub.elsevier.com/retrieve/pii/S0022039612000617
Volltext
Verfasserangaben:Falk M. Hante, Mario Sigalotti, Marius Tucsnak

MARC

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520 |a 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. 
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