Second order dynamical systems with penalty terms associated to monotone inclusions

Abstract

In this paper we investigate in a Hilbert space setting a second order dynamical system of the form $$\ddot{x}(t)+\g(t)\dot{x}(t)+x(t)-J_{λ(t) A}\big(x(t)-λ(t) D(x(t))-λ(t)β(t)B(x(t))\big)=0,$$ where $A:{\mathcal H}\toto{\mathcal H}$ is a maximal monotone operator, $J_{λ(t) A}:{\mathcal H}\To{\mathcal H}$ is the resolvent operator of $\lambda (t)A$ and $D,B:\mathcal{H}\to \mathcal{H}$ are cocoercive operators, and $\lambda ,\beta :[0,+\infty )\to (0,+\infty )$, and $\gamma :[0,+\infty )\to (0,+\infty )$ 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 $N_C$ 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 $A+D+N_C$, provided that $A$ is a strongly monotone operator.