Fractional power series neural network for solving delay fractional optimal control problems

Abstract

ABSTRACTIn this paper, we develop a numerical method for solving the delay optimal control problems of fractional-order. The fractional derivatives are considered in the Caputo sense. The process begins with the assumption that the problem is first transformed into an equivalent problem with a fractional dynamical system without delay, using a Padé approximation. We then try to approximate the solution of the Hamiltonian conditions based on the Pontryagin minimum principle. The main feature is to implement nonlinear polynomial expansions in a neural network adaptive structure. The transfer functions of the employed neural network follow a fractional power series. The proposed technique does not use sigmoid or hyperbolic tangent nonlinear transfer functions commonly adopted in conventional neural networks at the output. Instead, linear transfer functions are employed which lead to explicit fractional power series formulae for the fractional optimal control problem. To do this, we use trial solutions for the states, Lagrange multipliers and control functions where these trial solutions are constructed by fractional power series neural network model. We then minimise the error function using an unconstrained optimisation scheme where weight parameters (or coefficients of the series) and biases associated with all neurons are unknown. Some numerical examples are given to illustrate the effectiveness of the proposed scheme.