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

作者：Attouch Hedy; Czarnecki Marc Olivier

来源：Journal of Differential Equations, 2017, 262(3): 2745-2770.

DOI：10.1016/j.jde.2016.11.009

• Asymptotic control
• Convex optimization
• Hierarchical minimization
• Inertial gradient dynamics
• Lyapunov analysis
• Weakly coupled wave equations

In a Hilbert space H, we study the asymptotic behavior, as time variable t goes to +infinity, of nonautonomous gradient-like dynpmical systems involving inertia and multiscale features. Given Phi : H -> R and Psi : H -> 111 two convex differentiable functions, gamma a positive damping parameter, and epsilon(t) a function oft which tends to zero as t goes to +infinity, we consider the second-order differential equation
sic(t) + y(x) over dot (t) + del Phi(x(t)) + epsilon(t)del Psi(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 epsilon(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 Psi over the set C of minimizers of Phi. As key assumptions, we suppose that integral(+infin

• 出版日期2017-2-5