Approximation Methods for Solving Fractional Optimal Control Problems Using Legendre Polynomials

Abstract

This paper develops a high-accuracy Legendre spectral method for solving a broad class of fractional optimal control problems governed by Caputo derivatives of order $0<\alpha<1$. The state and control variables are approximated by truncated Legendre expansions, and a new operational matrix for the Caputo derivative is constructed explicitly in the Legendre basis. By enforcing the fractional dynamics at Gauss--Legendre collocation nodes, the continuous optimal control problem is transformed into a finite-dimensional nonlinear optimization problem that preserves the structure of the original system. We establish well-posedness of the fractional state equation, prove existence
and uniqueness of optimal controls, and derive a fractional Pontryagin maximum principle consistent with modern formulations of fractional variational theory. A detailed convergence analysis of the proposed spectral discretization is presented, showing algebraic rates for weakly regular data and spectral (exponential) convergence for analytic solutions. Several numerical experiments illustrate the efficiency and accuracy of the method. For smooth and analytic problems the convergence is essentially exponential, while for weakly singular solutions the method retains favorable algebraic decay. These results demonstrate the robustness of the Legendre spectral framework and its potential as an effective tool for the numerical solution of fractional optimal control problems.