  In this work we investigate dynamical systems designed to approach the
solution sets of inclusion problems involving the sum of two maximally monotone
operators. Our aim is to design methods which guarantee strong convergence of
trajectories towards the minimum norm solution of the underlying monotone
inclusion problem. To that end, we investigate in detail the asymptotic
behavior of dynamical systems perturbed by a Tikhonov regularization where
either the maximally monotone operators themselves, or the vector field of the
dynamical system is regularized. In both cases we prove strong convergence of
the trajectories towards minimum norm solutions to an underlying monotone
inclusion problem, and we illustrate numerically qualitative differences
between these two complementary regularization strategies. The so-constructed
dynamical systems are either of Krasnoselskii-Mann, of forward-backward type or
of forward-backward-forward type, and with the help of injected regularization
we demonstrate seminal results on the strong convergence of Hilbert space
valued evolutions designed to solve monotone inclusion and equilibrium
problems.
