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Proceedings of the American Mathematical Society 137 (2009) 3013-3024

Asymptotics for a gradient system with memory term

Alexandre Cabot 1

(2009)

Given a Hilbert space $H$ and a function $\Phi:H\to\R$ of class $\cC^1$, we investigate the asymptotic behavior of the trajectories associated to the following dynamical system $$\dot x(t) +\frac{1}{k(t)}\, \int_{t_0}^t h(s)\, \nabla \Phi(x(s))\, ds=0, \qquad t\geq t_0,\leqno (S)$$ where $h$, $k: [t_0,+\infty)\to \R_+^*$ 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 $(S)$ 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 $(S)$ cannot converge if the average process does not ''privilege'' the recent past. A special attention is devoted to the particular case $h(t)=t^\alpha$, $k(t)=t^\beta$, which is fully treated.

1 :	Institut de Mathématiques et de Modélisation de Montpellier (I3M)
	CNRS : UMR5149 – Université Montpellier II - Sciences et techniques

Domaine

:

Mathématiques/Analyse classique Mathématiques/Optimisation et contrôle

Mots Clés : Differential equation – dissipative dynamical system – averaged gradient system – memory effect – Bessel equation

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Contributeur : Alexandre Cabot <>

Soumis le : Mercredi 22 Octobre 2008, 16:21:09

Dernière modification le : Mardi 19 Février 2013, 12:09:51