Fractional Legendre wavelet approach resolving multi-scale optimal control problems involving Caputo-Fabrizio derivative

This article provides an effective numerical approach using the fractional integral operational matrix method for a fractional Legendre wavelet to deal with multi-dimensional fractional optimal control problems. We proposed operational matrices and implemented them to simplify multi-dimensional fractional optimal control problems into a set of equations, utilizing well-known formulas such as the Caputo-Fabrizio operator with a non-singular kernel defined for calculating fractional derivatives and integrals of fractional Legendre wavelets. Finally, the Lagrange multiplier technique is applied, and we get the state and control functions. The convergence analysis and error bounds of the proposed scheme are established. To check the veracity of the presented method, we tested numerical examples using the fractional Legendre wavelet method and obtained the cost function value based on identifying state and control functions.