In the setting of
a real Hilbert space ${\cal H}$, we investigate the asymptotic behavior, as time t goes to infinity, of trajectories of second-order evolution
equations
ü(t) + γ$\dot{u}$(t) + ∇ϕ(u(t)) + A(u(t)) = 0,
where ∇ϕ is the gradient operator of a convex
differentiable potential function
ϕ : ${\cal H}\to \R$, A : ${\cal H}\to {\cal H}$ is a maximal monotone operator which is assumed to be
λ-cocoercive, and γ > 0 is a damping parameter.
Potential and non-potential effects are associated respectively to
∇ϕ and A. Under condition λγ2 > 1, it is proved that each trajectory asymptotically weakly
converges to a zero of ∇ϕ + A. This condition, which
only involves the non-potential operator and the damping
parameter, is sharp and consistent with time rescaling. Passing
from weak to strong convergence of the trajectories is obtained by
introducing an asymptotically vanishing Tikhonov-like regularizing
term. As special cases, we recover the asymptotic analysis of the
heavy ball with friction dynamic attached to a convex potential, the
second-order gradient-projection dynamic, and the second-order
dynamic governed by the Yosida approximation of a general maximal
monotone operator. The breadth and flexibility
of the proposed framework is illustrated through applications in the areas of constrained optimization,
dynamical approach to Nash equilibria for noncooperative games, and asymptotic stabilization in the case of a continuum of equilibria.