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A Brief Review on Mean Field Optimal Control Problem from a Linear Quadratic Perspective

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Abstract

This paper reviews the mean field social (MFS) optimal control problem for multi-agent dynamic systems and the mean-field-type (MFT) optimal control problem for single-agent dynamic systems within the linear quadratic (LQ) framework. For the MFS control problem, this review discusses the existing conclusions on optimization in dynamic systems affected by both additive and multiplicative noises. In exploring MFT optimization, the authors first revisit researches associated with single-player systems constrained by these dynamics. The authors then extend the proposed review to scenarios that include multiple players engaged in Nash games, Stackelberg games, and cooperative Pareto games. Finally, the paper concludes by emphasizing future research on intelligent algorithms for mean field optimization, particularly using reinforcement learning method to design strategies for models with unknown parameters.

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Authors and Affiliations

  1. College of New Energy, China University of Petroleum (East China), Qingdao, 266580, China

    Xiushan Jiang

  2. Department of Mathematics, City University of Hong Kong, Hong Kong, 518057, China

    Daniel W. C. Ho

  3. College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao, 266590, China

    Weihai Zhang

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  1. Xiushan Jiang
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Correspondence to Daniel W. C. Ho.

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This research was supported by the National Natural Science Foundation of China under Grant Nos. 62103442, 12326343, 62373229, the Research Grants Council of the Hong Kong Special Administrative Region, China under Grant Nos. CityU 11213023, 11205724, the Natural Science Foundation of Shandong Province under Grant No. ZR2021QF080, the Taishan Scholar Project of Shandong Province under Grant No. tsqn202408110, the Fundamental Research Foundation of the Central Universities under Grant No. 23CX06024A, and the Outstanding Youth Innovation Team in Shandong Higher Education Institutions under Grant No. 2023KJ061.

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Jiang, X., Ho, D.W.C. & Zhang, W. A Brief Review on Mean Field Optimal Control Problem from a Linear Quadratic Perspective. J Syst Sci Complex 38, 390–420 (2025). https://doi.org/10.1007/s11424-025-4501-0

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