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

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- by Alexandre Cabot, Hans Engler and Sébastien Gadat PDF
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**361**(2009), 5983-6017 Request permission

## Abstract:

We investigate the asymptotic properties as $t\to \infty$ of the following differential equation in the Hilbert space $H$: \begin{equation*}(\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 \end{equation*} 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.## References

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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
- Email: acabot@math.univ-montp2.fr
**Hans Engler**- Affiliation: Department of Mathematics, Georgetown University, Box 571233, Washington, DC 20057
- MR Author ID: 63565
- Email: engler@georgetown.edu
**Sébastien Gadat**- Affiliation: Laboratoire de Statistique et Probabilités, Université Paul Sabatier, 31062 Toulouse Cedex 9, France
- Email: Sebastien.Gadat@math.ups-tlse.fr
- Received by editor(s): October 22, 2007
- Published electronically: June 9, 2009
- © Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**361**(2009), 5983-6017 - MSC (2000): Primary 34G20, 34A12, 34D05
- DOI: https://doi.org/10.1090/S0002-9947-09-04785-0
- MathSciNet review: 2529922