Seventh Mississippi State - UAB Conference on Differential Equations and
Computational Simulations.
Electronic Journal of Differential Equations,
Conference 17 (2009), pp. 33-38.

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

Authors: Alexandre Cabot (Univ. Montpellier II, Cedex 5, France)
         Hans Engler     (Georgetown Univ., Washington, DC, USA)
	 Sebastien Gadat (Univ. Paul Sabatier, Cedex 9, France)
 
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 $\mathop{\rm argmin} G$, in the case where  $G$ is 
 convex and $\mathop{\rm argmin} G$  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: 4G20, 34A12, 34D05.
Key Words: Differential equation; dissipative dynamical system;
           vanishing damping; asymptotic behavior.