$k$-Contraction in a Generalized Lurie System

TL;DR

This paper establishes a sufficient condition for $k$-contraction in generalized Lurie systems, extending contraction theory to analyze stability and convergence in nonlinear feedback systems, with applications to biochemical control circuits.

Contribution

It introduces a new sufficient condition for $k$-contraction in generalized Lurie systems, broadening the understanding of stability in nonlinear feedback dynamics.

Findings

For $k=1$, the condition reduces to standard contraction criteria.

For $k=2$, solutions converge to an equilibrium, not necessarily unique.

Application to biochemical control circuits demonstrates the theoretical results.

Abstract

We derive a sufficient condition for $k$-contraction in a generalized Lurie system~(GLS), that is, the feedback connection of a nonlinear dynamical system and a memoryless nonlinear function. For $k=1$, this reduces to a sufficient condition for standard contraction. For $k=2$, this condition implies that every bounded solution of the GLS converges to an equilibrium, which is not necessarily unique. We demonstrate the theoretical results by analyzing $k$-contraction in a biochemical control circuit with nonlinear dissipation terms.