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Asymptotics for a gradient system with memory term

Author: Alexandre Cabot
Journal: Proc. Amer. Math. Soc. 137 (2009), 3013-3024
MSC (2000): Primary 34G20, 34A12, 34D05
Published electronically: May 4, 2009
MathSciNet review: 2506460
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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:

$\displaystyle (\mathcal{S})\qquad\qquad\qquad\dot x(t) +\frac{1}{k(t)} \int_{t_0}^t h(s) \nabla \Phi(x(s)) ds=0, \qquad t\geq t_0,\qquad\qquad\quad$

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.

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Additional Information

Alexandre Cabot
Affiliation: Département de Mathématiques, Université Montpellier II, CC 051, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France

Keywords: Differential equation, dissipative dynamical system, averaged gradient system, memory effect, Bessel equation
Received by editor(s): October 22, 2008
Published electronically: May 4, 2009
Communicated by: Walter Craig
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.