Solving Fractional Optimal Control-Affine Problems via Fractional-Order Hybrid Jacobi Functions

This paper proposes and analyzes an applicable approach for numerically computing the solution of fractional optimal control-affine problems. The fractional derivative in the problem is considered in the sense of Caputo. The approach is based on a fractional-order hybrid of block-pulse functions and Jacobi polynomials. ‎First‎, ‎the corresponding Riemann-Liouville fractional integral operator of the introduced basis functions is calculated‎. ‎ Then, an approximation of the fractional derivative of the unknown state function is obtained by considering an approximation in terms of these basis functions‎. ‎ Next, ‎using the dynamical system and applying the fractional integral operator‎, ‎an approximation of the unknown control function is obtained based on the given approximations of the state function and its derivatives‎. ‎ Subsequently‎, ‎all the given approximations are substituted into the performance index‎. ‎Finally‎, ‎the optimality conditions transform the problem into a system of algebraic equations‎. ‎An error upper bound of the approximation of a function based on the fractional hybrid functions is provided‎. ‎The method is applied to several numerical examples‎, and ‎the experimental results confirm the efficiency and capability of the method.  Furthermore, they demonstrate a good agreement between the approximate and exact solutions‎. ‎