  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.
