Touchard–Ritz Method to Solve Variable-Order Fractional Optimal Control Problems

Here, we develop an approximation method to solve a class of variable-order fractional optimal control problems (VO-F-OCPs). This method is designed utilizing the Touchard wavelets, with the help of the Ritz method. To do this, we propose an extra variable-order Caputo (E-VO-C) pseudo-operational matrix, for Touchard wavelet functions. Choosing the best parameters in the Ritz method and the E-VO-C pseudo-operational matrix affects the accuracy of the method directly. The method of obtaining this matrix significantly reduces the error of the present technique. It also produces a sparse matrix, thereby reducing the computational burden. An efficient direct scheme using the expansion of only state variable, and its E-VO-E pseudo-operational matrix is implemented for the considered problem. The proposed technique converts the main problem into a programming one. A brief error estimation is discussed. Three samples are presented to display the effectiveness and precision of the suggested scheme.