Asymptotic for a second-order evolution equation with convex potential and vanishing damping term

Authors: RAMZI MAY

Abstract: In this short note, we recover by a different method the new result due to Attouch, Chbani, Peyrouqet, and Redont concerning the weak convergence as $t\rightarrow+\infty$ of solutions $x(t)$ to the second-order differential equation $x^{\prime\prime}(t)+\frac{K}{t}x^{\prime}(t)+\nabla\Phi(x(t))=0,$ where $K>3$ and $\Phi$\ is a smooth convex function defined on a Hilbert space $\mathcal{H}.$ Moreover, we improve their result on the rate of convergence of $\Phi(x(t))-\min\Phi.$

Keywords: Dynamical systems, asymptotically small dissipation, asymptotic behavior, energy function, convex function, convex optimization

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