Approximate solution of nonlinear optimal control problems with scale delay function via a composite pseudospectral approach

Abstract

In this paper, a direct discretization method is used to approximate the nonlinear scale delayed optimal control problems. The technique is a pseudospectral collocation approach based on the Legendre-Gauss-Lobatto points. Firstly the domain of interest is divided into several adaptive subintervals, and then the traditional pseudospectral approach is used for each segment. This approach discretizes the optimal control problem with a scale delay function and transforms it into a nonlinear programming problem whose solution can be achieved via the existing solvers. The main advantages of this method are the simplicity of the structure and the ease of its implementation and execution. Moreover, the necessary and sufficient conditions of optimality associated with the scale-delayed control problems are obtained. To do this, a new transformation technique is proposed which transforms the scale delayed control problem into a constant delayed one. These conditions can be used to measure the accuracy of the approximate findings obtained by applying the suggested method. The effectiveness and usefulness of the discretization procedure are demonstrated by the implementation of the proposed method in some experimental examples.

Keywords:

• Optimal control problem
• scale delay systems
• discretization
• composite method
• pseudospectral
• Legendre-Gauss-Lobatto points

Acknowledgments

The author thanks anonymous reviewers for their constructive criticism and helpful guidance which have significantly improved both the quality and the presentation of this paper.

Data availability statement

Data sharing does not apply to this article as no new data were created or analysed in this study. If anyone is interested in our Maple simulation, please contact the author, we are glad to provide the Maple code.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

Sayyed Mohammad Hoseini

Sayyed Mohammad Hoseini received an M.Sc. degree in applied mathematics and a Ph.D. degree in Optimal Control from the Isfahan University of Technology, Iran, in 2007 and 2016, respectively. He is currently an assistant professor at the Ayatollah Boroujerdi University. His research interests include optimal control theory, numerical methods and machine learning.