Seventh Mississippi State - UAB Conference on Differential Equations and Computational Simulations. Electron. J. Diff. Eqns., Conference 17 (2009), pp. 33-38

Second-order differential equations with asymptotically small dissipation and piecewise flat potentials

Alexandre Cabot, Hans Engler, Sebastien Gadat

Abstract:
We investigate the asymptotic properties as $t\to \infty$ of the differential equation
$$ \ddot{x}(t)+a(t)\dot{x}(t)+ \nabla G(x(t))=0, \quad t\geq 0 $$
where $x(\cdot)$ is $\mathbb{R}$-valued, the map $a:\mathbb{R}_+\to \mathbb{R}_+$ is non increasing, and $G:\mathbb{R} \to \mathbb{R}$ is a potential with locally Lipschitz continuous derivative. We identify conditions on the function $a(\cdot)$ that guarantee or exclude the convergence of solutions of this problem to points in $\hbox{\rm argmin} G$, in the case where $G$ is convex and $\hbox{\rm argmin} G$ is an interval. The condition
$$ \int_0^{\infty} e^{-\int_0^t a(s)\, ds}dt<\infty $$
is known to be necessary for convergence of trajectories. We give a slightly stronger condition that is sufficient.

Published April 15, 2009.
Math Subject Classifications: 34G20, 34A12, 34D05.
Key Words: Differential equation; dissipative dynamical system; vanishing damping; asymptotic behavior.

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Alexandre Cabot
Département de Mathématiques, Université Montpellier II, CC 051
Place Eugène Bataillon, 34095 Montpellier Cedex 5, France
email: acabot@math.univ-montp2.fr
Hans Engler
Department of Mathematics, Georgetown University Box 571233
Washington, DC 20057, USA
email: engler@georgetown.edu
Sébastien Gadat
Institut de Mathématiques de Toulouse, Université Paul Sabatier
118, Route de Narbonne 31062 Toulouse Cedex 9, France
email: Sebastien.Gadat@math.ups-tlse.fr

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