Passivity theorems for input-to-state stability of forced Lur'e inclusions and equations, and consequent entrainment-type properties

TL;DR

This paper establishes passivity-based input-to-state stability results for forced Lur'e inclusions and equations, enabling analysis of periodic responses and providing a foundation for frequency response in such systems.

Contribution

It introduces passivity theorems for forced Lur'e systems, extending stability analysis and periodic response characterization to a broader class of differential inclusions.

Findings

Input-to-state stability results for Lur'e inclusions

Semi-global incremental stability for forced Lur'e differential equations

Existence and attractivity of periodic responses in forced Lur'e systems

Abstract

A suite of input-to-state stability results are presented for a class of forced differential inclusions, so-called Lur'e inclusions. As a consequence, semi-global incremental input-to-state stability results for systems of forced Lur'e differential equations are derived. The results are in the spirit of the passivity theorem from control theory as both the linear and nonlinear components of the Lur'e inclusion (or equation) are assumed to satisfy passivity-type conditions. These results provide a basis for the analysis of forced Lur'e differential equations subject to (almost) periodic forcing terms and, roughly speaking, ensure the existence and attractivity of (almost) periodic state- and output-responses, comprising another focus of the present work. One ultimate aim of the study is to provide a robust and rigorous theoretical foundation for a well-defined and tractable ``frequency response'' of forced Lur'e systems.