Discrete time mean-field stochastic linear-quadratic optimal control problems |
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| Authors: | Robert Elliott Xun Li Yuan-Hua Ni |
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| Affiliation: | 1. School of Mathematical Sciences, University of Adelaide, SA 5005, Australia;2. Haskayne School of Business, University of Calgary, Calgary, Alberta, Canada;3. Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China;4. Department of Mathematics, School of Science, Tianjin Polytechnic University, Tianjin 300160, China;5. Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China |
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| Abstract: | This paper firstly presents necessary and sufficient conditions for the solvability of discrete time, mean-field, stochastic linear-quadratic optimal control problems. Secondly, the optimal control within a class of linear feedback controls is investigated using a matrix dynamical optimization method. Thirdly, by introducing several sequences of bounded linear operators, the problem is formulated as an operator stochastic linear-quadratic optimal control problem. By the kernel-range decomposition representation of the expectation operator and its pseudo-inverse, the optimal control is derived using solutions to two algebraic Riccati difference equations. Finally, by completing the square, the two Riccati equations and the optimal control are also obtained. |
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| Keywords: | Stochastic linear-quadratic optimal control problem Mean-field theory Riccati difference equation |
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