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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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
Trans. Amer. Math. Soc. 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.
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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