  In a Hilbertian framework, for the minimization of a general convex
differentiable function $f$, we introduce new inertial dynamics and algorithms
that generate trajectories and iterates that converge fastly towards the
minimizer of $f$ with minimum norm. Our study is based on the non-autonomous
version of the Polyak heavy ball method, which, at time $t$, is associated with
the strongly convex function obtained by adding to $f$ a Tikhonov
regularization term with vanishing coefficient $\epsilon(t)$. In this dynamic,
the damping coefficient is proportional to the square root of the Tikhonov
regularization parameter $\epsilon(t)$. By adjusting the speed of convergence
of $\epsilon(t)$ towards zero, we will obtain both rapid convergence towards
the infimal value of $f$, and the strong convergence of the trajectories
towards the element of minimum norm of the set of minimizers of $f$. In
particular, we obtain an improved version of the dynamic of Su-Boyd-Cand\`es
for the accelerated gradient method of Nesterov. This study naturally leads to
corresponding first-order algorithms obtained by temporal discretization. In
the case of a proper lower semicontinuous and convex function $f$, we study the
proximal algorithms in detail, and show that they benefit from similar
properties.
