Construction of the Minimum Time Function for Linear Systems Via Higher-Order Set-Valued Methods

TL;DR

This paper introduces a higher-order set-valued numerical method for approximating the minimum time function in linear control systems, providing error estimates and demonstrating improved convergence in various regularity scenarios.

Contribution

It develops a novel higher-order discretization approach for reachable sets in linear control problems, with theoretical error bounds and practical numerical validation.

Findings

Higher-order methods improve convergence rates for smooth problems.

The approach handles Holder continuous and discontinuous minimum time functions.

Numerical examples demonstrate the method's effectiveness and error estimates.

Abstract

The paper is devoted to introducing an approach to compute the approximate minimum time function of control problems which is based on reachable set approximation and uses arithmetic operations for convex compact sets. In particular, in this paper the theoretical justification of the proposed approach is restricted to a class of linear control systems. The error estimate of the fully discrete reachable set is provided by employing the Hausdorff distance to the continuous-time reachable set. The detailed procedure solving the corresponding discrete set-valued problem is described. Under standard assumptions, by means of convex analysis and knowledge of the regularity of the true minimum time function, we estimate the error of its approximation. Higher-order discretization of the reachable set of the linear control problem can balance missing regularity (e.g., Holder continuity) of the minimum time function for smoother problems. To illustrate the error estimates and to demonstrate differences to other numerical approaches we provide a collection of numerical examples which either allow higher order of convergence with respect to time discretization or where the continuity of the minimum time function cannot be sufficiently granted, i.e., we study cases in which the minimum time function is Holder continuous or even discontinuous.