Fractional Chebyshev functional link neural network‐optimization method for solving delay fractional optimal control problems with Atangana‐Baleanu derivative

• Published in: Optimal control applications & methods Vol. 41; no. 3; pp. 808 - 832
• Main Authors: Kheyrinataj, Farzaneh, Nazemi, Alireza
• Format: Journal Article
• Language: English
• Published: Glasgow Wiley Subscription Services, Inc 01.05.2020
• Subjects: Algorithms; Atangana‐Baleanu derivative; Boundary value problems; Chebyshev approximation; Delay; delay fractional optimal control problems; Error functions; fractional Chebyshev polynomials; functional link neural network; Neural networks; Nonlinear control; Optimal control; Optimization; Padé approximation
• ISSN: 0143-2087; 1099-1514
• DOI: 10.1002/oca.2572

Summary In this article, we propose a higher order neural network, namely the functional link neural network (FLNN), for the model of linear and nonlinear delay fractional optimal control problems (DFOCPs) with mixed control‐state constraints. We consider DFOCPs using a new fractional derivative with nonlocal and nonsingular kernel that was recently proposed by Atangana and Baleanu. The derivative possesses more important characteristics that are very useful in modelling. In the proposed method, a fractional Chebyshev FLNN is developed. At the first step, the delay problem is transformed to a nondelay problem, using a Padé approximation. The necessary optimality condition is stated in a form of fractional two‐point boundary value problem. By applying the fractional integration by parts and by constructing an error function, we then define an unconstrained minimization problem. In the optimization problem, trial solutions for state, co‐state and control functions are utilized where these trial solutions are constructed by using single‐layer fractional Chebyshev neural network model. We then minimize the error function using an unconstrained optimization scheme based on the gradient descent algorithm for updating the network parameters (weights and bias) associated with all neurons. To show the effectiveness of the proposed neural network, some numerical results are provided.