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

Using the Generalize Laguerre Operational Matrix for The Caputo Fractional Derivatives to Solve Some Classes of Fractional Optimal Control Problems

Abstract This paper aims to develop the new numerical solution method of the fractional optimal control problems (FOCPs) based on the generalized Laguerre polynomials. The generalize Laguerre operational matrix for the Caputo fractional derivatives has been derived. The operational matrix was used together with the properties of the generalized Laguerre orthonormal polynomials to reduce FOCPs to the system of algebraic equations by the direct method. Five examples are included to demonstrate efficient and accurate numerical algorithms for some classes of FOCPs is considered.</jats:p

An Indirect Spectral Collocation Method Based on Shifted Jacobi Functions for Solving Some Class of Fractional Optimal Control Problems

Abstract A new approximation formula of the Riemann-Liouville fractional derivatives is derived based on shifted classical Jacobi polynomial in spectral approximations. This formula is presented to approximate indirect solution of fractional optimal control problems (FOCPs) with a fractional differential equation as the dynamic constrain. The properties of new formula allows us to use spectral collocation method to reduce FOCPs by indirect method to a system of liner/nonlinear algebraic equations. Four test examples are presented to examine the applicability and validity of a newly purposed method.</jats:p