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Numerical solution of delay fractional optimal control problems with free terminal time

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Abstract

This paper considers a class of delay fractional optimal control problems with free terminal time. The fractional derivatives in this class of problems are described in the Caputo sense and they can be of different orders. We first show that for this class of problems, the well-known time-scaling transformation for mapping the free time horizon into a fixed time interval yields a new fractional-order system with variable time-delay. Then, we propose an explicit numerical scheme for solving the resulting fractional time-delay system, which gives rise to a discrete-time optimal control problem. Furthermore, we derive gradient formulas of the cost and constraint functions with respect to decision variables. On this basis, a gradient-based optimization approach is developed to solve the resulting discrete-time optimal control problem. Finally, an example problems is solved to demonstrate the effectiveness of our proposed solution approach.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos. 12271307 and 11771008), the Australian Research Council (No. DP190103361), the China Scholarship Council (No. 201902575002), and the Natural Science Foundation of Shandong Province, China (Nos. ZR2017MA005 and ZR2019MA031).

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Authors and Affiliations

  1. School of Mathematics and Information Science, Shandong Technology and Business University, Yantai, 264005, China

    Chongyang Liu & Zhaohua Gong

  2. School of Electrical Engineering, Computing, and Mathematical Sciences, Curtin University, Perth, 6845, Australia

    Chongyang Liu & Song Wang

  3. School of Mathematical Sciences, Sunway University, 47500, Kuala Lumpur, Malaysia

    Kok Lay Teo

  4. Coordinated Innovation Center for Computable Modeling in Management Science, Tianjin University of Finance and Economics, Tianjin, 300222, China

    Kok Lay Teo

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  1. Chongyang Liu
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  2. Zhaohua Gong
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Correspondence to Chongyang Liu.

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Liu, C., Gong, Z., Wang, S. et al. Numerical solution of delay fractional optimal control problems with free terminal time. Optim Lett 17, 1359–1378 (2023). https://doi.org/10.1007/s11590-022-01926-1

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