The Forward Euler Scheme for Nonconvex Lipschitz Differential Inclusions Converges with Rate One

TL;DR

This paper proves that the Forward Euler scheme for nonconvex Lipschitz differential inclusions converges with rate one in the sense of solution paths, extending previous results on set convergence and improving error constants for specific cases.

Contribution

It strengthens previous convergence results from reachable sets to solution paths and provides improved error constants for set-valued functions with few smooth components.

Findings

Convergence rate of one for the Forward Euler scheme on differential inclusions.

Extension of convergence from reachable sets to solution paths.

Improved error constants for set-valued functions with few smooth components.

Abstract

In a previous paper it was shown that the Forward Euler method applied to differential inclusions where the right-hand side is a Lipschitz continuous set-valued function with uniformly bounded, compact values, converges with rate one. The convergence, which was there in the sense of reachable sets, is in this paper strengthened to the sense of convergence of solution paths. An improvement of the error constant is given for the case when the set-valued function consists of a small number of smooth ordinary functions.