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

Authors : Ramzi MAY

Pages : 681-685

View : 9 | Download : 8

Publication Date : 0000-00-00

Article Type : Research Paper

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 $xinsert ignore into journalissuearticles values(t);$ to the second-order differential equation $x^{\prime\prime}insert ignore into journalissuearticles values(t);+\frac{K}{t}x^{\prime}insert ignore into journalissuearticles values(t);+\nabla\Phiinsert ignore into journalissuearticles values(xinsert ignore into journalissuearticles values(t););=0,$ 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 $\Phiinsert ignore into journalissuearticles values(xinsert ignore into journalissuearticles values(t););-\min\Phi.$
Keywords : Dynamical systems, asymptotically small dissipation, asymptotic behavior, energy function, convex function, convex optimization