Asymptotic for a second-order evolution equation with convex potential andvanishing 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 $\Phi(x(t))-\min\Phi.$

DOI

10.3906/mat-1512-28

Keywords

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

First Page

681

Last Page

685

Recommended Citation

MAY, RAMZI (2017) "Asymptotic for a second-order evolution equation with convex potential andvanishing damping term," Turkish Journal of Mathematics: Vol. 41: No. 3, Article 17. https://doi.org/10.3906/mat-1512-28
Available at: https://journals.tubitak.gov.tr/math/vol41/iss3/17