Asymptotics for a gradient system with memory term

Abstract:

Given a Hilbert space $H$ and a function $\Phi :H\to \mathbb {R}$ of class $\mathcal {C}^1$, we investigate the asymptotic behavior of the trajectories associated to the following dynamical system: \[ (\mathcal {S})\qquad \qquad \qquad \dot x(t) +\frac {1}{k(t)} \int _{t_0}^t h(s) \nabla \Phi (x(s)) ds=0, \qquad t\geq t_0,\qquad \qquad \quad \]where $h$, $k: [t_0,+\infty )\to \mathbb {R}_+^*$ are continuous maps. When $k(t) \sim \int _{t_0}^t h(s) ds$ as $t\to +\infty$, this equation can be interpreted as an averaged gradient system. We define a natural energy function $E$ associated to system $(\mathcal {S})$ and we give conditions which ensure that $E(t)$ decreases to $\inf \Phi$ as $t\to +\infty$. When $\Phi$ is convex and has a set of non-isolated minima, we show that the trajectories of $(\mathcal {S})$ cannot converge if the average process does not “privilege” the recent past. Special attention is devoted to the particular case $h(t)=t^\alpha$, $k(t)=t^\beta$, which is fully treated.