Asymptotic behavior of gradient-like dynamical systems involving inertia and multiscale aspects

In a Hilbert space \ H, we study the asymptotic behaviour, as time t goes to +\∞, of nonautonomous gradient-like dynamical involving inertia and multiscale features. Given \ H a general Hilbert space, \Φ: \ H \→\ R and \Ψ: \ H \→ \ R two convex functions, \γ a positive damping parameter, and \ε(t) a function of t which tends to zero as t goes to +\∞, we the second-order differential equation ̈(t) + \γ̇(t) + \∇ \Φ (x(t)) + \ε (t) \∇ \Ψ (x(t)) = 0. This models the emergence of various collective behaviors in game theory, as as the asymptotic control of coupled nonlinear oscillators. Assuming that\ε(t) tends to zero moderately slowly as t goes to infinity, we show the trajectories converge weakly in \ H. The limiting equilibria solutions of the hierarchical minimization problem which consists in \Ψ over the set C of minimizers of \Φ. As key assumptions, suppose that \∫₀+\∞\ε (t) dt = + \∞ and that, for p belonging to a convex cone \ C depending on the data \Φ \Ψ \∫₀+\∞ \[\Φ^* \(\ε (t)p\)-\σC \(\ε (t)p\)\]dt < + \∞ where \Φ^* is Fenchel conjugate of \Φ, and \σC is the support function of. An application is given to coupled oscillators.