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

Abstract

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