# A numerical approach for solving fractional optimal control problems using modified hat functions

## TL;DR

This paper presents a novel numerical method using modified hat functions to solve fractional optimal control problems, simplifying computations and avoiding numerical integration, with proven error bounds and applicability to constrained problems.

## Contribution

The paper introduces a new numerical approach based on modified hat functions for fractional optimal control problems, including error analysis and extension to inequality constraints.

## Key findings

- Effective in solving fractional optimal control problems
- Reduces to solving nonlinear algebraic equations
- Demonstrates high accuracy and ease of implementation

## Abstract

We introduce a numerical method, based on modified hat functions, for solving a class of fractional optimal control problems. In our scheme, the control and the fractional derivative of the state function are considered as linear combinations of the modified hat functions. The fractional derivative is considered in the Caputo sense while the Riemann-Liouville integral operator is used to give approximations for the state function and some of its derivatives. To this aim, we use the fractional order integration operational matrix of the modified hat functions and some properties of the Caputo derivative and Riemann-Liouville integral operators. Using results of the considered basis functions, solving the fractional optimal control problem is reduced to the solution of a system of nonlinear algebraic equations. An error bound is proved for the approximate optimal value of the performance index obtained by the proposed method. The method is then generalized for solving a class of fractional optimal control problems with inequality constraints. The most important advantages of our method are easy implementation, simple operations, and elimination of numerical integration. Some illustrative examples are considered to demonstrate the effectiveness and accuracy of the proposed technique.