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This paper deals with a second order dynamical system with vanishing dampingthat contains a Tikhonov regularization term, in connection to the minimizationproblem of a convex Fr\'echet differentiable function $g$.  We show that for appropriate Tikhonov regularization parameters the value ofthe objective function in a generated trajectory converges fast to the globalminimum of the objective function and a trajectory generated by the dynamicalsystem converges weakly to a minimizer of the objective function. We alsoobtain the fast convergence of the velocities towards zero and some integralestimates. Nevertheless, our main goal is to extend and improve some recentresults obtained in \cite{ABCR} and \cite{AL-nemkoz} concerning the strongconvergence of the generated trajectories to an element of minimal norm fromthe $\argmin$ set of the objective function $g$. Our analysis also reveals thatthe damping coefficient and the Tikhonov regularization coefficient arestrongly correlated.

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