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

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.Comment: This is a preprint of a paper whose final and definite form is with 'Journal of Vibration and Control', available from [http://journals.sagepub.com/home/jvc]. Submitted 02-June-2018; Revised 03-Sept-2018; Accepted 12-Oct-201

A new framework for solving fractional optimal control problems using fractional pseudospectral methods

The main purpose of this work is to provide new fractional pseudospectral methods for solving fractional optimal control problems (FOCPs). We develop differential and integral fractional pseudospectral methods and prove the equivalence between them from the distinctive perspective of Caputo fractional Birkhoff interpolation. As a result, the present work establishes a new unified framework for solving fractional optimal control problems using fractional pseudospectral methods, which can be viewed as an extension of existing frameworks. Furthermore, we provide exact, efficient, and stable approaches to compute the associated fractional pseudospectral differentiation/integration matrices even at millions of Jacobi-type points. Numerical results on two benchmark FOCPs including a fractional bang–bang problem demonstrate the performance of the proposed methods.MOE (Min. of Education, S’pore