  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.
