Solution of Fractional Optimal Control Problems with Noise Function Using the Bernstein Functions

This paper presents a numerical solution of a class of fractional optimal control problems (FOCPs) in a bounded domain having a noise function by the spectral Ritz method‎. ‎The Bernstein polynomials with the fractional operational matrix are applied to approximate the unknown functions‎. ‎By substituting these estimated functions into the cost functional‎, ‎an unconstrained nonlinear optimization problem is achieved‎. In order to solve this optimization problem‎, ‎the Matlab software and its optimization toolbox are used‎. ‎In the considered FOCP‎, ‎the performance index is expressed as a function of both state and control functions‎. ‎The method is robust enough because of its computational consistency in the presence of the noise function‎. ‎Moreover‎, ‎the proposed scheme has a good pliability satisfying the given initial and boundary conditions‎. ‎At last‎, ‎some test problems are investigated to confirm the efficiency and applicability of the new method.