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07 February 2025
 
  » arxiv » 1609.0135

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Asymptotic for the perturbed heavy ball system with vanishing damping term
Mounir Balti ; Ramzi May ;
Date 1 Sep 2016
AbstractWe investigate the long time behavior of solutions to the differential equation $ddot{x}(t)+frac{c}{left( t+1 ight) ^{alpha}}dot{x}(t)+ abla Phileft( x(t) ight) =g(t),~tgeq0, $ where $c$ is nonnegative constant, $alphainlbrack0,1[,$ $Phi$ is a $C^{1}$ convex function on a Hilbert space $mathcal{H}$ and $gin L^{1} (0,+infty;mathcal{H}).$ We obtain sufficient conditions on the source term $g(t)$ ensuring the weak or the strong convergence of any trajectory $x(t)$ as $t ightarrow+infty$ to a minimizer of the function $Phi$ if one exists.
Source arXiv, 1609.0135
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