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

  • Farzaneh KheyrinatajEmail author
  • Alireza Nazemi
Methodologies and Application


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.


Müntz–Legendre polynomial Functional link neural network Delay optimal control problems Fractional order Padé approximation Pontryagin minimum principle Error function Optimization scheme 



This study was not funded by any grant.

Compliance with ethical standard

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Mathematical SciencesShahrood University of TechnologyShahroodIran

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