Asymptotic behavior of second-order dissipative evolution equations combining potential with non-potential effects

TL;DR

This paper analyzes the long-term behavior of second-order dissipative evolution equations that combine potential and non-potential effects, providing conditions for convergence to equilibria with applications in optimization and game theory.

Contribution

It introduces a sharp convergence condition involving damping and cocoercivity, extending understanding of asymptotic behavior in complex dynamical systems.

Findings

Established a precise condition for convergence to equilibria.

Applied results to optimization, fixed points, and Nash equilibria.

Demonstrated stabilization in systems with multiple equilibria.

Abstract

We study the asymptotic convergence properties, as the time variable goes to infinity, of trajectories of second-order dissipative evolution equations combining potential with non-potential effects. We exhibit a sharp condition, involving the damping parameter and the cocoercive coefficient of the non-potential operator, which guarantees convergence to equilibria of the trajectories. Applications are given to constrained optimization, fixed point problems, dynamical approach to Nash equilibria, and asymptotic stabilization in the case of a continuum of equilibria.