Non-coercive Lyapunov functions for infinite-dimensional systems

Titelangaben

Mironchenko, Andrii ; Wirth, Fabian:
Non-coercive Lyapunov functions for infinite-dimensional systems.
In: Journal of Differential Equations. Bd. 266 (2019) Heft 11 . - S. 7038-7072.
ISSN 1090-2732
DOI: https://doi.org/10.1016/j.jde.2018.11.026

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Abstract

We show that the existence of a non-coercive Lyapunov function is sufficient for uniform global asymptotic stability (UGAS) of infinite-dimensional systems with external disturbances provided the speed of decay is measured in terms of the norm of the state and an additional mild assumption is satisfied. For evolution equations in Banach spaces with Lipschitz continuous nonlinearities these additional assumptions become especially simple. The results encompass some recent results on linear switched systems on Banach spaces. Finally, we derive new non-coercive converse Lyapunov theorems and give some examples showing the necessity of our assumptions.