1. Second-order dynamical systems with penalty terms associated to monotone inclusions
- Author
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Szilárd László, Radu Ioan Boţ, and Ernö Robert Csetnek
- Subjects
021103 operations research ,Dynamical systems theory ,Computer Science::Information Retrieval ,Applied Mathematics ,010102 general mathematics ,Astrophysics::Instrumentation and Methods for Astrophysics ,0211 other engineering and technologies ,Hilbert space ,Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) ,02 engineering and technology ,01 natural sciences ,Combinatorics ,symbols.namesake ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Monotone polygon ,symbols ,Computer Science::General Literature ,Order (group theory) ,0101 mathematics ,Dynamical system (definition) ,ComputingMilieux_MISCELLANEOUS ,Analysis ,Mathematics - Abstract
In this paper, we investigate in a Hilbert space setting a second-order dynamical system of the form [Formula: see text] where [Formula: see text]image[Formula: see text] is a maximal monotone operator, [Formula: see text] is the resolvent operator of [Formula: see text] and [Formula: see text] are cocoercive operators, and [Formula: see text], and [Formula: see text] 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 [Formula: see text] where [Formula: see text] and [Formula: see text] denotes the normal cone operator of [Formula: see text]. 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 [Formula: see text], provided that [Formula: see text] is a strongly monotone operator.
- Published
- 2018