A space-time pseudospectral discretization method for solving diffusion optimal control problems with two-sided fractional derivatives

TL;DR

This paper introduces a novel numerical method combining pseudospectral discretization techniques to efficiently solve two-sided space-fractional diffusion optimal control problems, transforming them into convex quadratic programs.

Contribution

The paper presents a new approach for discretizing two-sided fractional derivatives and solving fractional optimal control problems efficiently with demonstrated accuracy.

Findings

Method achieves high accuracy in test examples

Low CPU time required for solutions

Effective handling of fractional derivatives and constraints

Abstract

We propose a direct numerical method for the solution of an optimal control problem governed by a two-side space-fractional diffusion equation. The presented method contains two main steps. In the first step, the space variable is discretized by using the Jacobi-Gauss pseudospectral discretization and, in this way, the original problem is transformed into a classical integer-order optimal control problem. The main challenge, which we faced in this step, is to derive the left and right fractional differentiation matrices. In this respect, novel techniques for derivation of these matrices are presented. In the second step, the Legendre-Gauss-Radau pseudospectral method is employed. With these two steps, the original problem is converted into a convex quadratic optimization problem, which can be solved efficiently by available methods. Our approach can be easily implemented and extended to cover fractional optimal control problems with state constraints. Five test examples are provided to demonstrate the efficiency and validity of the presented method. The results show that our method reaches the solutions with good accuracy and a low CPU time.