Approximation of the reachability set of a nonlinear control system with discrete time

Recieved: 07/10/2024

Accepted: 10/14/2024

Published: 01/31/2025

Keywords: nonlinear dynamics, reachability set, value function, piecewise affine estimates

To cite this article

Tochilin P.A. Approximation of the reachability set of a nonlinear control system with discrete time. // Moscow University Journal. Series 15. Computational Mathematics and Cybernetics. 2025. N 1, p.57-67 https://doi.org/10.55959/MSU/0137–0782–15–2025–49–1–57–67.

This work is licensed under a Сreative Commons Atribiution - NonCommercial 4.0 International (CC BY-NC 4.0)

Tochilin P.A. (Lomonosov Moscow State University (MSU))

Abstract

The paper is devoted to the development of a new method for the approximate construction of the reachability set for a nonlinear control system with discrete time. Pointwise restrictions are imposed on the control parameters. To solve this problem, a technique previously developed and applied for the case of continuous time and differential equations is used. The estimate of the reachability set can be obtained as the level set of a special piecewise affine value function constructed on a grid of simplices in the state space. The paper presents formulas for calculating the coefficients of such a function, which make it possible to analyze the difference between the case with discrete time and the case with continuous time. An example of calculation of piecewise affine value functions and corresponding internal and external estimates of the reachability set is considered.