Direct transcription methods based on fractional integral approximation formulas for solving nonlinear fractional optimal control problems

• Published in: Communications in nonlinear science & numerical simulation 2019-02, Vol.67, p.334-350
• Main Authors: Salati, Abubakar Bello, Shamsi, Mostafa, Torres, Delfim F. M.
• Format: Article
• Language: English
• Subjects: Approximation; Control systems; Direct numerical solution; Fractional integration matrix; Fractional optimal control; Fractions; Grünwald–Letnikov; HIV; Human immunodeficiency virus; Integrals; Nonlinear control; Nonlinear programming; Nonlinear systems; Numerical analysis; Optimal control; Performance indices; Reliability analysis; Trapezoidal and Simpson fractional integral formulas
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2018.05.011

•A general class of fractional optimal control problems was considered, which contains the bang-bang, free final time and path constraints problems.•Fractional integration matrices of Grünwald–Letnikov, trapezoidal and Simpson’s formulas are derived.•Using the fractional integration matrix, the fractional optimal control problems are reduced to a finite-dimensional optimization problem.•In order to improve the speed and accuracy, the gradient of objective function and the Jacobian of constraints are supplied to the optimization solver.•Numerical simulations are provided to illustrate the presented method on different types of fractional optimal control problems. This paper presents three direct methods based on Grünwald–Letnikov, trapezoidal and Simpson fractional integral formulas to solve fractional optimal control problems (FOCPs). At first, the fractional integral form of FOCP is considered, then the fractional integral is approximated by Grünwald–Letnikov, trapezoidal and Simpson formulas in a matrix approach. Thereafter, the performance index is approximated either by trapezoidal or Simpson quadrature. As a result, FOCPs are reduced to nonlinear programming problems, which can be solved by many well-developed algorithms. To improve the efficiency of the presented method, the gradient of the objective function and the Jacobian of constraints are prepared in closed forms. It is pointed out that the implementation of the methods is simple and, due to the fact that there is no need to derive necessary conditions, the methods can be simply and quickly used to solve a wide class of FOCPs. The efficiency and reliability of the presented methods are assessed by ample numerical tests involving a free final time with path constraint FOCP, a bang-bang FOCP and an optimal control of a fractional-order HIV-immune system.