Müntz–Legendre neural network construction for solving delay optimal control problems of fractional order with equality and inequality constraints

In this paper, an artificial intelligence approach using neural networks is described to solve a class of delay optimal control problems of fractional order with equality and inequality constraints. In the proposed method, a functional link neural network based on the Müntz–Legendre polynomial is developed. The problem is first transformed into an equivalent problem with a fractional dynamical system without delay, using a Padé approximation. According to the Pontryagin’s minimum principle for optimal control problems of fractional order and by constructing an error function, the authors then define an unconstrained minimization problem. The authors use trial solutions for the states, Lagrange multipliers and control functions where these trial solutions are constructed by a single-layer Müntz–Legendre neural network model. The authors then exploit an unconstrained optimization scheme for adjusting the network parameters (weights and bias) and to minimize the computed error function. Some numerical examples are given to illustrate the effectiveness of the proposed method.