# Numerical Solution of Fractional Control Problems via Fractional Differential Transformation

## TL;DR

This paper introduces a new numerical algorithm combining fractional differential transformation and step methods to efficiently approximate solutions of linear fractional control problems with delays, validated on a two-dimensional example.

## Contribution

A novel numerical method using fractional differential transformation and step methods for solving fractional control problems with delays.

## Key findings

- The algorithm provides reliable and efficient approximations.
- Exact solutions are obtained for initial intervals, confirming accuracy.
- The method converges to classical solutions when fractional order approaches 1.

## Abstract

In the paper we deal with linear fractional control problems with constant delays in the state. Single-order systems with fractional derivative in Caputo sense of orders between 0 and 1 are considered. The aim is to introduce a new algorithm convenient for numerical approximation of a solution of the studied problem. The method consists of the fractional differential transformation in combination with general methods of steps. The original system is transformed to a system of recurrence relations. Approximation of the solution is given in the form of truncated fractional power series. The choice of order of the fractional power series is discussed and the order is determined in relation to the order of the system. An application on a two-dimensional fractional system is shown. Exact solution is found for the first two intervals of the method of steps. The result for Caputo derivative of order 1 coincides with the solution of first-order system with classical derivative. We conclude that the algorithm is applicable, efficient and gives reliable results.