Characterization of optimal feedback for stochastic linear quadratic control problems

Qi Lü; Tianxiao Wang; Xu Zhang

doi:10.1186/s41546-017-0022-7

Article Contents

2017, Volume 2: 11. Doi: 10.1186/s41546-017-0022-7

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Characterization of optimal feedback for stochastic linear quadratic control problems

School of Mathematics, Sichuan University, Chengdu 610064, Sichuan Province, China

Received: June 20, 2016

Revised: September 2017

Published: September 2020

This work is supported by the NSF of China under grants 11471231, 11221101, 11231007, 11301298 and 11401404, the PCSIRT under grant IRT 16R53 and the Chang Jiang Scholars Program from Chinese Education Ministry, the Fundamental Research Funds for the Central Universities in China under grant 2015SCU04A02, and the NSFC-CNRS Joint Research Project under grant 11711530142.

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Abstract

One of the fundamental issues in Control Theory is to design feedback controls. It is well-known that, the purpose of introducing Riccati equations in the study of deterministic linear quadratic control problems is exactly to construct the desired feedbacks. To date, the same problem in the stochastic setting is only partially well-understood. In this paper, we establish the equivalence between the existence of optimal feedback controls for the stochastic linear quadratic control problems with random coefficients and the solvability of the corresponding backward stochastic Riccati equations in a suitable sense. We also give a counterexample showing the nonexistence of feedback controls to a solvable stochastic linear quadratic control problem. This is a new phenomenon in the stochastic setting, significantly different from its deterministic counterpart.

Keywords:

Stochastic linear quadratic problem,
Feedback control,
Backward stochastic Riccati equation,
Backward stochastic differential equation.

Mathematics Subject Classification: 93E20;93B52;93C05;60H10.

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