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 $$\ddot{x}(t)+\text{g}(t)\dot{x}(t)+x(t)-J_{\lambda (t)A}(x(t)-\lambda (t)D(x(t))-\lambda (t)\beta (t)B(x(t)))=0,$$ where $A:\mathcal{H}\text{toto}\mathcal{H}$ is a maximal monotone operator, $J_{\lambda (t)A}:\mathcal{H}\text{To}\mathcal{H}$ is the resolvent operator of $\lambda (t)A$ and $D,B:\mathcal{H}\to \mathcal{H}$ are cocoercive operators, and $\lambda ,\beta :[0,+\mathrm{\infty })\to (0,+\mathrm{\infty })$, and $\gamma :[0,+\mathrm{\infty })\to (0,+\mathrm{\infty })$ 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 $A+D+N_C,$ where $C=\zer B$and$N_C$denotesthenormalconeoperatorof$C$. 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 $A+D+N_C$, provided that $A$ is a strongly monotone operator.