Converse Lyapunov Theorem for Input-to-State Stability of Linear Integral Difference Equations
Résumé
This paper investigates necessary and sufficient Lyapunov conditions for Input-to-State Stability (ISS) of Linear Integral Difference Equations in the presence of an additional exogenous signal. Building on recent research in the literature pertaining to necessary conditions for the exponential stability of difference equations, we introduce a quadratic Lyapunov functional that incorporates the derivative of the so-called Lyapunov matrix. We demonstrate that the ISS of the considered class of systems is contingent upon the existence of an ISS Lyapunov functional. The Lyapunov analysis hinges on the properties of the fundamental matrix and the delay Lyapunov matrix.
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