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

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.