Integral fractional pseudospectral methods for solving fractional optimal control problems

The main purpose of this work is to provide a unified framework and develop integral fractional pseudospectral methods for solving fractional optimal control problems. As a generalization of conventional pseudospectral integration matrices, fractional pseudospectral integration matrices (FPIMs) and their efficient and stable computation are the key to our new approach. In order to achieve this goal, we take a special and smart way to compute FPIMs. The essential idea is to transform the fractional integral of Lagrange interpolating polynomials through a change of variables into their Jacobi-weighted integral which can be calculated exactly using the Jacobi-Gauss quadrature. This, together with the stable barycentric representation of Lagrange interpolating polynomials and the explicit barycentric weights for the Gauss-, flipped Radau-, and Radau-type points corresponding to the Jacobi polynomials, leads to an exact, efficient, and stable scheme to compute FPIMs even at millions of Jacobi-type points. Numerical results on two benchmark optimal control problems demonstrate the performance of the proposed pseudospectral methods.