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This paper deals with a second order dynamical system with a Tikhonovregularization term in connection to the minimization problem of a convexFr\'echet differentiable function. The fact that beside the asymptoticallyvanishing damping we also consider an implicit Hessian driven damping in thedynamical system under study allows us, via straightforward explicitdiscretization, to obtain inertial algorithms of gradient type. We show thatthe value of the objective function in a generated trajectory converges rapidlyto the global minimum of the objective function and depending the Tikhonovregularization parameter the generated trajectory converges weakly to aminimizer of the objective function or the generated trajectory convergesstrongly to the element of minimal norm from the $\argmin$ set of the objectivefunction. We also obtain the fast convergence of the velocities towards zeroand some integral estimates. Our analysis reveals that the Tikhonovregularization parameter and the damping parameters are strongly correlated,there is a setting of the parameters that separates the cases when weakconvergence of the trajectories to a minimizer and strong convergence of thetrajectories to the minimal norm minimizer can be obtained.

Solving convex optimization problems via a second order dynamical system with implicit Hessian damping and Tikhonov regularization