Equilibria in Motion: Stability, Tracking, and Convergence

Abstract

{We study nonautonomous systems whose equilibrium sets are nonisolated and move with time. Their motion is measured on each ball by a localized Attouch--Wets distance, which yields an equilibrium speed that may depend on the radius. The distance from a trajectory to the current equilibrium set is estimated in two ways: through a composite Lyapunov function compared with a power of this distance, and, without any Lyapunov function, through a decay condition on the frozen vector fields. In the first setting, we prove practical stability, and Lyapunov stability when the dissipation has no residual, a case that forces the equilibrium sets to be nested. In the second, we obtain, in the region where the decay condition holds, an explicit tracking estimate driven by the equilibrium speed and the inputs, and we prove asymptotic tracking whenever their averages over unit time intervals vanish, a condition that cannot be weakened among conditions on the speed alone. We also derive input-to-state stability bounds and explicit convergence rates. When the equilibrium speeds are integrable, the equilibrium sets converge in the Attouch--Wets sense to a limiting set at an explicit rate. If, moreover, the velocity is bounded by a multiple of the distance to the current equilibrium set, every trajectory remaining in the region considered has finite length and converges to a point of the limiting set; for a fixed equilibrium set, this yields semistability. The results are illustrated on a scalar example and on resource-allocation models with time-varying demand and with rotating weights.

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