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June  2017, 6(2): 177-186. doi: 10.3934/eect.2017010

## Asymptotic for the perturbed heavy ball system with vanishing damping term

 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

Received  September 2016 Revised  February 2017 Published  April 2017

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

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.
Citation: Mounir Balti, Ramzi May. Asymptotic for the perturbed heavy ball system with vanishing damping term. Evolution Equations & Control Theory, 2017, 6 (2) : 177-186. doi: 10.3934/eect.2017010
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