On damped second-order gradient systems

TL;DR

This paper analyzes damped second-order gradient systems, establishing their quasi-gradient nature, exploring their asymptotic behavior, convergence properties, and rates, with applications to definable, real-analytic, and convex functions.

Contribution

It introduces a novel perspective by viewing these systems as quasi-gradient systems and provides new convergence and rate results using KL inequality and desingularizing functions.

Findings

Any desingularizing function for the potential also applies to the total energy.

Trajectories either converge or their norm diverges to infinity.

Convergence is established for various cases including definable, real-analytic, and convex functions.

Abstract

Using small deformations of the total energy, as introduced in [31], we establish that damped second order gradient systems $$u^{\prime\prime}(t)+\gamma u^\prime(t)+\nabla G(u(t))=0,$$may be viewed as quasi-gradient systems. In order to study the asymptotic behavior of these systems, we prove that any (nontrivial) desingularizing function appearing in KL inequality satisfies $\varphi(s)\ge c\sqrt s$ whenever the original function is definable and $C^2.$ Variants to this result are given. These facts are used in turn to prove that a desingularizing function of the potential $G$ also desingularizes the total energy and its deformed versions. Our approach brings forward several results interesting for their own sake: we provide an asymptotic alternative for quasi-gradient systems, either a trajectory converges, or its norm tends to infinity. The convergence rates are also analyzed by an original method based on a one-dimensional worst-case gradient system.We conclude by establishing the convergence of solutions of damped second order systems in various cases including the definable case. The real-analytic case is recovered and some results concerning convex functions are also derived.