Asymptotic behavior of gradient-like dynamical systems involving inertia and multiscale aspects
Hédy Attouch, Marc-Olivier Czarnecki · arXiv (Cornell University) · 2016
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 $$\\ddot{x}(t) + \\gamma \\dot{x}(t) + \ abla \\Phi (x(t)) + \\epsilon (t) \ abla \\Psi (x(t)) = 0. $$ 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.