Asymptotic for the perturbed heavy ball system with vanishing damping term

Mounir Balti; Ramzi May

doi:10.3934/eect.2017010

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2017, Volume 6, Issue 2: 177-186. Doi: 10.3934/eect.2017010

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Asymptotic for the perturbed heavy ball system with vanishing damping term

Mounir Balti^1,2, and
Ramzi May^3, ,

1.
Institut Préparatoire aux Etude Scientifiques et Techniques, Université de Carthage, Bp 51 La Marsa, Tunisia
2.
Faculté des Sciences de Tunis, Laboratoire EDP-LR03ES04, Université de Tunis El Manar Tunis, Tunisia
3.
College of Sciences, Department of Mathematics and Statistics, King Faisal University, P.O. 400 Al Ahsaa 31982, Kingdom of Saudi Arabia

*Corresponding author: Ramzi May
*Corresponding author: Ramzi May

Received: September 2016

Revised: February 2017

Published: May 2017

The authors are grateful to the Deanship of Scientific Research at King Faisal University for financially and morally supporting this work under Project 160052

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Abstract

We investigate the long time behavior of solutions to the differential equation:

$\ddot{x}(t)+\frac{c}{{{\left( 1+t \right)}^{\alpha }}}\dot{x}(t)+\nabla \Phi \left( x(t) \right)=g(t),~t\ge 0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( 1 \right)$

where $c$ is nonnegative constant, $α∈\lbrack0,1[,Φ \ \ {\rm{is \ \ a}}\ \ C^{1}$ convex function defined on a Hilbert space $\mathcal{H}$ and $g∈ L^{1}(0,+∞;\mathcal{H}).$ We obtain sufficient conditions on the source term $g(t)$ ensuring the weak or the strong convergence of any trajectory $x(t)$ of (1) as $t\to ∞$ to a minimizer of the function $Φ$ if one exists.

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Mathematics Subject Classification: Primary: 34A34, 34A40; Secondary: 34D05, 34E10.

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