  In this manuscript we would like to address the classical optimization
problem of minimizing a proper, convex and lower semicontinuous function via
the second order in time dynamics, combining viscous and Hessian-driven damping
with a Tikhonov regularization technique. In our analysis we heavily exploit
the Moreau envelope of the objective function and its properties as well as
Tikhonov properties, which we extend to a nonsmooth case. We introduce the
setting, which at the same time guarantees the fast convergence of the function
(and Moreau envelope) values and strong convergence of the trajectories of the
system to a minimal norm solution -- the element of the minimal norm of all the
minimizers of the objective. Moreover, we deduce the precise rates of
convergence of the values for the particular choice of parameter function.
Various numerical examples are also included as an illustration of the
theoretical results.
