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

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.