A novel high accurate numerical approach for the time-delay optimal control problems with delay on both state and control variables

Mehrnoosh Hedayati; Hojjat Ahsani Tehrani; Alireza Fakharzadeh Jahromi; Mohammad Hadi Noori Skandari; Dumitru Baleanu; Mehrnoosh Hedayati; Hojjat Ahsani Tehrani; Alireza Fakharzadeh Jahromi; Mohammad Hadi Noori Skandari; Dumitru Baleanu

doi:10.3934/math.2022545

AIMS Mathematics

2022, Volume 7, Issue 6: 9789-9808. doi: 10.3934/math.2022545

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A novel high accurate numerical approach for the time-delay optimal control problems with delay on both state and control variables

1.
Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran
2.
Faculty of Mathematics, Shiraz University of Technology, P.O. Box 71555-313, Shiraz, Iran
3.
Fars Elites Foundation, P.O. Box 71966-98893, Shiraz, Iran
4.
Department of Mathematics, Cankaya University, Ankara, Turkey
5.
Institute of Space Sciences, Magurele-Bucharest, Romania
6.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

Received: 10 December 2021 Revised: 10 February 2022 Accepted: 22 February 2022 Published: 17 March 2022
MSC : 65M70, 49J15, 90C30

In this study, we intend to present a numerical method with highly accurate to solve the time-delay optimal control problems with delay on both the state and control variables. These problems can be seen in many sciences such as medicine, biology, chemistry, engineering, etc. Most of the methods used to work out time delay optimal control problems have high complexity and cost of computing. We extend a direct Legendre-Gauss-Lobatto spectral collocation method for numerically solving the issues mentioned above, which have some difficulties with other methods. The simple structure, convergence, and high accuracy of our approach are the advantages that distinguish it from different processes. At first, by replacing the delay functions of state and control variables in the dynamical method, we propose an equivalent system. Then discretizing the problem at the collocation points, we achieve a nonlinear programming problem. We can solve this discrete problem to obtain the approximate solutions for the main problem. Moreover, we prove the gained approximate solutions convergent to the exact optimal solutions when the number of collocation points increases. Finally, we show the capability and the superiority of the presented method by solving some numeral examples and comparing the results with those of others.
- time delay optimal control problem,
- Legendre spectral method,
- nonlinear programming,
- Legendre-Gauss-Lobatto points,
- convergence analysis
Citation: Mehrnoosh Hedayati, Hojjat Ahsani Tehrani, Alireza Fakharzadeh Jahromi, Mohammad Hadi Noori Skandari, Dumitru Baleanu. A novel high accurate numerical approach for the time-delay optimal control problems with delay on both state and control variables[J]. AIMS Mathematics, 2022, 7(6): 9789-9808. doi: 10.3934/math.2022545

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Abstract

In this study, we intend to present a numerical method with highly accurate to solve the time-delay optimal control problems with delay on both the state and control variables. These problems can be seen in many sciences such as medicine, biology, chemistry, engineering, etc. Most of the methods used to work out time delay optimal control problems have high complexity and cost of computing. We extend a direct Legendre-Gauss-Lobatto spectral collocation method for numerically solving the issues mentioned above, which have some difficulties with other methods. The simple structure, convergence, and high accuracy of our approach are the advantages that distinguish it from different processes. At first, by replacing the delay functions of state and control variables in the dynamical method, we propose an equivalent system. Then discretizing the problem at the collocation points, we achieve a nonlinear programming problem. We can solve this discrete problem to obtain the approximate solutions for the main problem. Moreover, we prove the gained approximate solutions convergent to the exact optimal solutions when the number of collocation points increases. Finally, we show the capability and the superiority of the presented method by solving some numeral examples and comparing the results with those of others.

References

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