Robust boundary tracking for reachable sets of nonlinear differential inclusions

TL;DR

This paper introduces a boundary tracking method for reachable sets of nonlinear differential inclusions that reduces computational cost and enhances robustness by focusing on boundaries rather than entire sets.

Contribution

It proposes a novel boundary tracking approach for the Euler scheme in differential inclusions, improving efficiency and robustness compared to traditional methods.

Findings

Significant reduction in computational cost.

Method is robust against topology changes.

Numerical examples demonstrate efficiency gains.

Abstract

The Euler scheme is up to date the most important numerical method for ordinary differential inclusions, because the use of the available higher-order methods is prohibited by their enormous complexity after spatial discretization. Therefore, it makes sense to reassess the Euler scheme and optimize its performance. In the present paper, a considerable reduction of the computational cost is achieved by setting up a numerical method that computes the boundaries instead of the complete reachable sets of the fully discretized Euler scheme from lower-dimensional data only. Rigorous proofs for the propriety of this method are given, and numerical examples illustrate the gain of computational efficiency as well as the robustness of the scheme against changes of topology of the reachable sets.