Higher Order Methods for Differential Inclusions

TL;DR

This paper introduces a high-order numerical method for accurately over-approximating the reachable sets of differential inclusions, with rigorous error bounds applicable to all inputs over finite time intervals.

Contribution

It extends existing methods by providing high-order error estimates and formulas based on Lipschitz constants and derivatives, valid for all possible inputs.

Findings

Provides formulas for local error based on Lipschitz constants

Offers uniform error bounds over finite intervals

Extends previous methods with higher-order accuracy

Abstract

We present a numerical method for rigorous over-approximation of a reachable set of differential inclusions. The method gives high-order error bounds for single step approximations and a uniform bound on the error over the finite time interval. We provide formulas for the local error based on Lipschitz constants and bounds on higher-order derivatives. The method is based on a Fliess-like expansion, and extends previous results by providing error estimates which are valid for all possible inputs.