Catching-Up Algorithm for Differential Inclusions

Convergence, stability, and quantitative error bounds for discretizations of differential inclusions driven by maximal monotone operators.

The catching-up algorithm (Moreau’s algorithme de rattrapage) is a time-discretization scheme that constructs piecewise-constant approximate solutions of differential inclusions by projecting onto admissible sets at each step. It is the foundational numerical tool for sweeping processes and projected dynamical systems.

Current work (with Hassan Saoud, Gulf University)

We study a catching-up scheme for differential inclusions driven by maximal monotone operators with continuous perturbations, decomposing the operator into its single-valued part and the normal cone to a closed convex set. The work establishes:

  • Existence and global energy bounds under a mild tangent dissipativity assumption.
  • Uniqueness and stability with respect to initial data under local Lipschitz perturbations.
  • Quantitative error bounds and uniform boundedness of iterates for the variable-step-size scheme with approximate projections.
  • Asymptotic feasibility of the predictor step in an $L^2$ sense and a Cesàro-averaged feasibility property.
  • Explicit examples: a one-dimensional test case and a multidimensional dry-friction system.

📄 Status: submitted to Applied Mathematics and Optimization (2026).

  • Catching-up with errors for sweeping processes driven by a fixed set.
  • Connections to projected dynamical systems and complementarity problems.

References