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In a Hilbert space, we provide a fast dynamic approach to the hierarchicalminimization problem which consists in finding the minimum norm solution of aconvex minimization problem. For this, we study the convergence properties ofthe trajectories generated by a damped inertial dynamic with Tikhonovregularization. When the time goes to infinity, the Tikhonov regularizationparameter is supposed to tend towards zero, not too fast, which is a keyproperty to make the trajectories strongly converge towards the minimizer of$f$ of minimum norm. According to the structure of the heavy ball method forstrongly convex functions, the viscous damping coefficient is proportional tothe square root of the Tikhonov regularization parameter. Therefore, it alsoconverges to zero, which will ensure rapid convergence of values. Precisely,under a proper tuning of these parameters, based on Lyapunov's analysis, weshow that the trajectories strongly converge towards the minimizer of minimumnorm, and we provide the convergence rate of the values. We show a trade offbetween the property of fast convergence of values, and the property of strongconvergence towards the minimum norm solution. This study improves severalprevious works where this type of results was obtained under restrictivehypotheses.

Damped inertial dynamics with vanishing Tikhonov regularization: strong asymptotic convergence towards the minimum norm solution