Relaxation of Euler-Type Discrete-Time Control System

This paper investigates what is the Hausdorff distance between the set of Euler curves of a Lipschitz continuous differential inclusion and the set of Euler curves for the corresponding convexified differential inclusion. It is known that this distance can be estimated by O(root h), where h is the Euler discretization step. It has been conjectured that, in fact, an estimation O(h) holds. The paper presents results in favor of the conjecture, which cover most of the practically relevant cases. However, the conjecture remains unproven, in general.