Second order dynamical systems with penalty terms associated to monotone inclusions

In this paper, we investigate in a Hilbert space setting a second-order dynamical system of the form (formula presented) are step size, penalization and, respectively, damping functions, all depending on time. We show the existence and uniqueness of strong global solutions in the framework of the Cauchy-Lipschitz-Picard Theorem and prove ergodic asymptotic convergence for the generated trajectories to a zero of the operator A + D + NC, where C = zerB and NC denotes the normal cone operator of C. To this end, we use Lyapunov analysis combined with the celebrated Opial Lemma in its ergodic continuous version. Furthermore, we show strong convergence for trajectories to the unique zero of A + D + NC, provided that A is a strongly monotone operator.