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On the long time behavior of second order differential equations with asymptotically small dissipation

Authors: Alexandre Cabot, Hans Engler and Sébastien Gadat
Journal: Trans. Amer. Math. Soc. 361 (2009), 5983-6017
MSC (2000): Primary 34G20, 34A12, 34D05
Published electronically: June 9, 2009
MathSciNet review: 2529922
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Abstract: We investigate the asymptotic properties as $ t\to \infty$ of the following differential equation in the Hilbert space $ H$:

$\displaystyle (\mathcal{S})\qquad\qquad\qquad\ddot{x}(t)+a(t)\dot{x}(t)+ \nabla G(x(t))=0, \quad t\geq 0,\qquad\qquad\qquad\qquad\quad$

where the map $ a:\mathbb{R}_+\to \mathbb{R}_+$ is nonincreasing and the potential $ G:H\to \mathbb{R}$ is of class $ \mathcal{C}^1$. If the coefficient $ a(t)$ is constant and positive, we recover the so-called ``Heavy Ball with Friction'' system. On the other hand, when $ a(t)=1/(t+1)$ we obtain the trajectories associated to some averaged gradient system. Our analysis is mainly based on the existence of some suitable energy function. When the function $ G$ is convex, the condition $ \int_0^\infty a(t) dt =\infty$ guarantees that the energy function converges toward its minimum. The more stringent condition $ \int_0^{\infty} e^{-\int_0^t a(s) ds}dt<\infty$ is necessary to obtain the convergence of the trajectories of $ (\mathcal{S})$ toward some minimum point of $ G$. In the one-dimensional setting, a precise description of the convergence of solutions is given for a general nonconvex function $ G$. We show that in this case the set of initial conditions for which solutions converge to a local minimum is open and dense.

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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

Hans Engler
Affiliation: Department of Mathematics, Georgetown University, Box 571233, Washington, DC 20057

Sébastien Gadat
Affiliation: Laboratoire de Statistique et Probabilités, Université Paul Sabatier, 31062 Toulouse Cedex 9, France

Keywords: Differential equation, dissipative dynamical system, vanishing damping, averaged gradient system, asymptotic behavior, Bessel equation
Received by editor(s): October 22, 2007
Published electronically: June 9, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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