Finite-time input-to-state stability for infinite-dimensional systems

TL;DR

This paper extends finite-time input-to-state stability concepts to infinite-dimensional systems, providing Lyapunov-based criteria and verifying stability for certain PDEs with disturbances through theoretical analysis and simulations.

Contribution

It introduces an FTISS Lyapunov theorem for infinite-dimensional systems and offers sufficient conditions for its existence, advancing stability analysis in complex systems.

Findings

FTISS Lyapunov theorem established for infinite-dimensional systems

Sufficient conditions for FTISS Lyapunov functional existence provided

Numerical simulations confirm theoretical stability results

Abstract

In this paper, we extend the notion of finite-time input-to-state stability (FTISS) for finite-dimensional systems to infinite-dimensional systems. More specifically, we first prove an FTISS Lyapunov theorem for a class of infinite-dimensional systems, namely, the existence of an FTISS Lyapunov functional (FTISS-LF) implies the FTISS of the system, and then, provide a sufficient condition for ensuring the existence of an FTISS-LF for a class of abstract infinite-dimensional systems under the framework of compact semigroup theory and Hilbert spaces. As an application of the FTISS Lyapunov theorem, we verify the FTISS for a class of parabolic PDEs involving sublinear terms and distributed in-domain disturbances. Since the nonlinear terms of the corresponding abstract system are not Lipschitz continuous, the well-posedness is proved based on the application of compact semigroup theory and the FTISS is assessed by using the Lyapunov method with the aid of an interpolation inequality. Numerical simulations are conducted to confirm the theoretical results.