Equilibria in Motion: Stability, Tracking, and Convergence

Abstract: We study the stability, tracking, and convergence of nonautonomous systems with time-varying nonisolated equilibrium sets. A Lyapunov framework based on coupled dissipation channels is developed to analyze the evolution of trajectories relative to a moving equilibrium family whose variation is quantified by an equilibrium speed measured through local Hausdorff estimates. Under suitable dissipation and energy--distance comparison conditions, we establish Lyapunov stability, quantitative tracking bounds, asymptotic tracking under integrable equilibrium drift, and an input-to-state stability estimate relative to the moving equilibrium family. We further show that integrable equilibrium speed implies the existence of a limiting equilibrium geometry obtained through local Hausdorff convergence of the equilibrium sets and that convergence to the moving equilibrium family can be transferred to convergence relative to the limiting equilibrium set. Quantitative convergence estimates are also derived. The theory is illustrated by a dynamic resource allocation model with time-varying demand.