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

TL;DR

This paper analyzes the long-term behavior of a class of second-order dynamical systems with inertia and multiscale features in Hilbert spaces, showing weak convergence to hierarchical minimizers under specific conditions.

Contribution

It introduces new convergence results for gradient-like systems with inertia and multiscale aspects, extending understanding of their asymptotic behavior.

Findings

Trajectories converge weakly in Hilbert space.

Limit points solve hierarchical minimization problems.

Conditions on damping and multiscale parameters ensure convergence.

Abstract

In a Hilbert space $\mathcal H$, we study the asymptotic behaviour, as time variable $t$ goes to $+\infty$, of nonautonomous gradient-like dynamical systems involving inertia and multiscale features. Given $\mathcal H$ a general Hilbert space, $\Phi: \mathcal H \rightarrow \mathbb R$ and $\Psi: \mathcal H \rightarrow \mathbb R$ two convex differentiable functions, $\gamma$ a positive damping parameter, and $\epsilon (t)$ a function of $t$ which tends to zero as $t$ goes to $+\infty$, we consider the second-order differential equation $$\ddot{x}(t) + \gamma \dot{x}(t) + \nabla \Phi (x(t)) + \epsilon (t) \nabla \Psi (x(t)) = 0. $$ 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 $\mathcal H$. 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 $ \int_{0}^{+\infty}\epsilon (t) dt = + \infty $ and that, for every $p$ belonging to a convex cone $\mathcal C$ depending on the data $\Phi$ and $\Psi$ $$ \int_{0}^{+\infty} \left[\Phi^* \left(\epsilon (t)p\right) -\sigma_C \left(\epsilon (t)p\right)\right]dt < + \infty $$ where $\Phi^*$ is the Fenchel conjugate of $\Phi$, and $\sigma_C $ is the support function of $C$. An application is given to coupled oscillators.