Convergence and Stability of a Catching-Up Algorithm for Differential Inclusions with Maximal Monotone Operators

TL;DR

This paper analyzes a catching-up algorithm for differential inclusions with maximal monotone operators, proving existence, stability, convergence, and feasibility properties, supported by explicit examples.

Contribution

It introduces a novel convergence and stability analysis for a catching-up algorithm applied to differential inclusions with maximal monotone operators, including discretization and feasibility results.

Findings

Existence of solutions under mild assumptions.

Convergence of discrete schemes to continuous solutions.

Explicit error bounds and stability estimates.

Abstract

We study a catching-up algorithm for a class of differential inclusions driven by maximal monotone operators with continuous perturbations. Using a decomposition of the monotone operator into the closed convex hull of its single-valued part and the normal cone to a closed convex set, we establish existence of solutions and derive global energy bounds under a mild tangent dissipativity assumption. Under an additional local Lipschitz assumption on the perturbation, we also obtain uniqueness and stability with respect to the initial data. We then analyze a time-discretized catching-up scheme with variable step sizes and approximate projections. On every finite horizon, we prove convergence of the discrete trajectories to solutions of the continuous problem. A discrete velocity decomposition together with a discrete energy inequality yields uniform boundedness of the iterates, quantitative stability estimates, and explicit error bounds. We also establish asymptotic feasibility of the predictor step in an $L^2$ sense, as well as a Ces\`aro-type averaged feasibility property, showing that the constraint violations generated by the free step vanish as the discretization is refined. Finally, we illustrate the theory on explicit examples, including a fully explicit one--dimensional test case and a multidimensional constrained dry-friction system.