Discrete time mean-field stochastic linear-quadratic optimal control problems

• DOI: 10.1016/j.automatica.2013.08.017
• Publication Date: 2013-09-25T10:40:06Z
• Authors: Robert Elliott; Xun Li; Yuan-Hua Ni
• Keywords: mean-field theory; Riccati difference equation; Engineering, Electrical & Electronic; 02 engineering and technology; Automation & Control Systems; Engineering; stochastic linear-quadratic optimal control problem; Optimization and Control (math.OC); Stochastic linear-quadratic optimal control problem; FOS: Mathematics; 0202 electrical engineering, electronic engineering, information engineering; Mean-field theory; Mathematics - Optimization and Control

This paper first presents necessary and sufficient conditions for the solvability of discrete time, mean-field, stochastic linear-quadratic optimal control problems. Then, by introducing several sequences of bounded linear operators, the problem becomes an operator stochastic LQ problem, in which the optimal control is a linear state feedback. Furthermore, from the form of the optimal control, the problem changes to a matrix dynamic optimization problem. Solving this optimization problem, we obtain the optimal feedback gain and thus the optimal control. Finally, by completing the square, the optimality of the above control is validated.