Wavelets method for solving fractional optimal control problems

In this paper, an efficient and accurate computational method based on the Legendre wavelets (LWs) is proposed for solving a class of fractional optimal control problems (FOCPs). In the proposed method, the FOCP under consideration is reduced to a system of nonlinear algebraic equations which can be simply solved. To this end, the fractional derivative of the state variable and the control variable are expanded by the LWs with unknown coefficients. Then, the operational matrix of the Riemann–Liouville fractional integration with some properties of the LWs are employed to achieve a nonlinear algebraic equation, in place of the performance index and a linear system of algebraic equations, in place of the dynamical system in terms of the unknown coefficients. Finally, the method of constrained extrema, which consists of adjoining the constraint equations derived from the given dynamical system to the performance index by a set of undetermined Lagrange multipliers is applied. As a result, the necessary conditions of optimality are derived as a system of algebraic equations in the unknown coefficients of the state variable, control variable and Lagrange multipliers. Furthermore, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient.