In this paper, a linear quadratic optimal control problem driven by a stochastic differential delay system is investigated, where both state delay and control delay can appear in the state equation, especially in the diffusion term. Three kinds of solvability for the delayed control problem are proposed—open-loop solvability, the closed-loop representation of open-loop optimal control, and closed-loop solvability—and their necessary and sufficient conditions are obtained. The delayed control problem is transformed into an infinite dimensional optimal control problem without delay but with a new control operator. Some novel auxiliary equations are constructed to overcome the difficulties caused by the new control operator, because state delay and control delay coexist, and some stochastic analysis tools are lacking in the study of the above three kinds of solvability. The open-loop solvability is ensured by the solvability of a constrained forward-backward stochastic evolution system and a convexity condition, or by the solvability of an anticipated-backward stochastic differential delay system and a convexity condition; the closed-loop representation of the open-loop optimal control is given via a coupled matrix-valued Riccati equation; the closed-loop solvability is ensured by the solvability of an operator-valued Riccati equation or a coupled matrix-valued Riccati equation.

Publication:

SIAM JOURNAL ON CONTROL AND OPTIMIZATION

http://dx.doi.org/10.1137/23M1625482

Author:

WEIJUN MENG

School of Mathematics & Statistics, Nanjing University of Science and Technology, Nanjing210094, China

mengwj@mail.sdu.edu.cn

JINGTAO SHI

School of Mathematics, Shandong University, Jinan 250100, China

shijingtao@sdu.edu.cn

JI-FENG ZHANG

Corresponding author

School of Automation and Electrical Engineering, Zhongyuan Univer-sity of Technology, Zhengzhou 450007; State Key Laboratory of Mathematical Sciences, Academy ofMathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190; and school of Math-ematical Sciences, University of Chinese Academy of Sciences, Beijing 100149, China

jif@iss.ac.cn

YANLONG ZHAO

State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190; and school of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100149, China

ylzhao@amss.ac.cn