Feedback Stackelberg strategies for the discrete-time mean-field stochastic systems in infinite horizon |
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| Institution: | 1. School of Control Science and Engineering, Shandong University, Jinan 250061, China;2. School of Electrical Engineering and Computer Science, University of Newcastle, NSW 2308, Australia;3. School of Automation, Guangdong University of Technology, Guangzhou, China;4. CIFASIS/CONICET, 2000 Rosario, Argentina;1. School of Control Science and Engineering, Shandong University, Jinan, Shandong, 250061, PR China;2. School of Electrical and Electronic Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798, Singapore;1. Department of Mathematics, School of Science, Tianjin Polytechnic University, Tianjin 300160, China;2. Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;3. School of Mathematical Sciences, University of Adelaide, SA, 5005, Australia;4. Haskayne School of Business, University of Calgary, Calgary, Alberta, Canada;5. Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China;1. School of Mathematics, Shandong University, Jinan 250100, PR China;2. School of Control Science and Engineering, Shandong University, Jinan 250061, PR China;3. Department of Mathematics, Faculty of Science and Technology, University of Macau, Macau, PR China;1. Institute of Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashi-Hiroshima, 739-8527, Japan;2. Graduate School of Business Sciences, The University of Tsukuba, 3-29-1 Bunkyou-ku, Tokyo, 112-0012, Japan |
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| Abstract: | This paper deals with the feedback Stackelberg strategies for the discrete-time mean-field stochastic systems in infinite horizon. The optimal control problem of the follower is first studied. Employing the discrete-time linear quadratic (LQ) mean-field stochastic optimal control theory, the sufficient conditions for the solvability of the optimization of the follower are presented and the optimal control is obtained based on the stabilizing solutions of two coupled generalized algebraic Riccati equations (GAREs). Then, the optimization of the leader is transformed into a constrained optimal control problem. Applying the Karush-Kuhn-Tucker (KKT) conditions, the necessary conditions for the existence and uniqueness of the Stackelberg strategies are derived and the Stackelberg strategies are expressed as linear feedback forms involving the state and its mean based on the solutions of a set of cross-coupled stochastic algebraic equations (CSAEs). An iterative algorithm is put forward to calculate efficiently the solutions of the CSAEs. Finally, an example is solved to show the effectiveness of the proposed algorithm. |
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