Asymptotic for the perturbed heavy ball system with vanishing damping term

Abstract

We investigate the long time behavior of solutions to the differential equation $\ddot{x}(t)+\frac{c}{{\left(t+1\right)}^{\alpha }}\dot{x}(t)+\nabla \Phi \left(x(t)\right)=g(t),t\ge 0,$ where $c$ is nonnegative constant, $\alpha \in [0,1[,$ $\Phi$ is a $C^1$ convex function on a Hilbert space $\mathcal{H}$ and $g\in L^1(0,+\infty ;\mathcal{H}).$ We obtain sufficient conditions on the source term $g(t)$ ensuring the weak or the strong convergence of any trajectory $x(t)$ as $t\to +\infty$ to a minimizer of the function $\Phi$ if one exists.

arXiv comment12 pages