Construction of a new family of orthogonal functions and its utilization in optimal control systems governed by delay fractional integro-differential equations

The contributions of this investigation are highlighted below. First, a novel family of orthogonal functions is constructed based on the fractional Taylor functions. Also, an effective spectral collocation method is offered and new theoretical convergence results are acquired. Furthermore, by utilizing the Caputo derivative operator, a new operator is constructed. Second, a challenging family of control systems is investigated whose state equation is a nonlinear fractional integro-differential equation consisting of delays. This category of control problems has not been investigated in the literature yet. Five challenging problems have been discussed to establish the fruitfulness and success of the constructed numerical scheme. Simulation results indicate that the proposed basis is capable of producing an exact solution for those optimal control problems having non-smooth behavior.