A unifying convex analysis and switching system approach to consensus with undirected communication graphs

TL;DR

This paper presents a unified convex analysis and switching system framework to analyze consensus in multi-agent systems with undirected communication graphs, accommodating non-smooth, constrained, and time-varying convex functions.

Contribution

It introduces a novel approach combining convex analysis and switching systems to establish convergence to common minimizers, extending existing consensus results to more general settings.

Findings

Proves convergence of solutions to common convex minimizers.

Derives consensus results as special cases.

Extends analysis to time-varying and set-valued dynamics.

Abstract

Switching between finitely many continuous-time autonomous steepest descent dynamics for convex functions is considered. Convergence of complete solutions to common minimizers of the convex functions, if such minimizers exist, is shown. The convex functions need not be smooth and may be subject to constraints. Since the common minimizers may represent consensus in a multi-agent system modeled by an undirected communication graph, several known results about asymptotic consensus are deduced as special cases. Extensions to time-varying convex functions and to dynamics given by set-valued mappings more general than subdifferentials of convex functions are included.