Approximation of Fractional Order Conflict-Controlled Systems

TL;DR

This paper introduces a numerical approximation method for nonlinear conflict-controlled systems governed by fractional differential equations, enabling the application of functional-differential system results to fractional control problems.

Contribution

It proposes a finite-difference approximation of fractional systems using Grünwald-Letnikov formulas and a mutual aiming procedure to ensure system proximity.

Findings

Approximation method effectively models fractional conflict-controlled systems.

Numerical simulations demonstrate the accuracy of the approximation.

The approach facilitates solving control problems in fractional systems.

Abstract

We consider a conflict-controlled dynamical system described by a nonlinear ordinary fractional differential equation with the Caputo derivative of an order $\alpha \in (0, 1).$ Basing on the finite-difference Gr\"{u}nwald-Letnikov formulas, we propose an approximation of the considered system by a system described by a functional-differential equation of a retarded type. A mutual aiming procedure between the initial conflict-controlled system and the approximating system is given that guarantees the desired proximity between their motions. This procedure allows to apply, via the approximating system, the results obtained for functional-differential systems for solving control problems in fractional order systems. Examples are considered, results of numerical simulations are presented.