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

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
Hans Engler
Department of Mathematics, Georgetown University Box 571233
Washington, DC 20057, USA
Sébastien Gadat
Institut de Mathématiques de Toulouse, Université Paul Sabatier
118, Route de Narbonne 31062 Toulouse Cedex 9, France

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