Second order dynamical systems associated to variational inequalities

Abstract

We investigate the asymptotic convergence of the trajectories generated by the second order dynamical system $\ddot x(t) + \gamma\dot x(t) + \nabla \phi(x(t))+\beta(t)\nabla \psi(x(t))=0$, where $\phi,\psi:{\cal H}\rightarrow \R$ are convex and smooth functions defined on a real Hilbert space ${\cal H}$, $\gamma>0$ and $\beta$ is a function of time which controls the penalty term. We show weak convergence of the trajectories to a minimizer of the function $\phi$ over the (nonempty) set of minima of $\psi$ as well as convergence for the objective function values along the trajectories, provided a condition expressed via the Fenchel conjugate of $\psi$ is fulfilled. When the function $\phi$ is assumed to be strongly convex, we can even show strong convergence of the trajectories. The results can be seen as the second order counterparts of the ones given by Attouch and Czarnecki (Journal of Differential Equations 248(6), 1315--1344, 2010) for first order dynamical systems associated to constrained variational inequalities. At the same time we give a positive answer to an open problem posed in \cite{att-cza-16} by the same authors.