  In a Hilbert space, we provide a fast dynamic approach to the hierarchical
minimization problem which consists in finding the minimum norm solution of a
convex minimization problem. For this, we study the convergence properties of
the trajectories generated by a damped inertial dynamic with Tikhonov
regularization. When the time goes to infinity, the Tikhonov regularization
parameter is supposed to tend towards zero, not too fast, which is a key
property to make the trajectories strongly converge towards the minimizer of
$f$ of minimum norm. According to the structure of the heavy ball method for
strongly convex functions, the viscous damping coefficient is proportional to
the square root of the Tikhonov regularization parameter. Therefore, it also
converges to zero, which will ensure rapid convergence of values. Precisely,
under a proper tuning of these parameters, based on Lyapunov's analysis, we
show that the trajectories strongly converge towards the minimizer of minimum
norm, and we provide the convergence rate of the values. We show a trade off
between the property of fast convergence of values, and the property of strong
convergence towards the minimum norm solution. This study improves several
previous works where this type of results was obtained under restrictive
hypotheses.
