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.