On the strong convergence of the trajectories of a Tikhonov regularized second order dynamical system with asymptotically vanishing damping

Abstract

This paper deals with a second order dynamical system with vanishing damping that contains a Tikhonov regularization term, in connection to the minimization problem of a convex Fr\'echet differentiable function $g$. We show that for appropriate Tikhonov regularization parameters the value of the objective function in a generated trajectory converges fast to the global minimum of the objective function and a trajectory generated by the dynamical system converges weakly to a minimizer of the objective function. We also obtain the fast convergence of the velocities towards zero and some integral estimates. Nevertheless, our main goal is to extend and improve some recent results obtained in \cite{ABCR} and \cite{AL-nemkoz} concerning the strong convergence of the generated trajectories to an element of minimal norm from the $\argmin$ set of the objective function $g$. Our analysis also reveals that the damping coefficient and the Tikhonov regularization coefficient are strongly correlated.