Equilibria in Motion: Stability, Tracking, and Convergence

Abstract

We study the stability, tracking, and convergence of nonautonomous systems with moving nonisolated equilibrium sets. We develop a Lyapunov framework based on coupled dissipation channels to analyze the evolution of trajectories relative to a moving equilibrium family whose motion is quantified by an equilibrium speed through localized Attouch--Wets estimates. Under suitable dissipation and energy--distance comparison conditions, we establish Lyapunov stability, quantitative tracking estimates, asymptotic tracking under integrable equilibrium motion, and input-to-state stability (ISS) estimates relative to the moving equilibrium family. We further show that, under an appropriate nonescape condition, integrable equilibrium speed guarantees the existence of a limiting equilibrium geometry in the Attouch--Wets sense, allowing convergence to the moving equilibrium family to be transferred to convergence relative to the limiting equilibrium set. Explicit convergence-rate estimates are also derived. The theoretical results are illustrated by a dynamic resource allocation model with time-varying demand.

Revised and corrected version. Some statements and arguments have been corrected and strengthened; the main scope and conclusions of the paper remain unchanged