Coercive quadratic ISS Lyapunov functions for analytic systems

Titelangaben

Mironchenko, Andrii ; Schwenninger, Felix:
Coercive quadratic ISS Lyapunov functions for analytic systems.
In: 2023 62nd IEEE Conference on Decision and Control (CDC) : Proceedings. - Piscataway, NJ, USA : IEEE , 2023 . - S. 4699-4704
ISBN 979-8-3503-0124-3
DOI: https://doi.org/10.1109/CDC49753.2023.10384024

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Abstract

We investigate the relationship between input-to-state stability (ISS) of linear infinite-dimensional systems and existence of coercive ISS Lyapunov functions. We show that input-to-state stability of a linear system does not imply existence of a coercive quadratic ISS Lyapunov function, even if the underlying semigroup is analytic, and the input operator is bounded. However, if in addition the semigroup is similar to a contraction semigroup on a Hilbert space, then a quadratic ISS Lyapunov function always exists.

Next we consider analytic and similar to contraction semi-groups in Hilbert spaces with unbounded input operator B. If B is slightly stronger than 2-admissible, we construct explicitly a coercive $L^2$-ISS Lyapunov function. If the generator of a semigroup is additionally self-adjoint, this Lyapunov function is precisely a square norm in the state space.