Mean-field stochastic linear quadratic optimal control problems: closed-loop solvability

Xun Li; Jingrui Sun; Jiongmin Yong

doi:10.1186/s41546-016-0002-3

Article Contents

2016, Volume 1: 2. Doi: 10.1186/s41546-016-0002-3

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Mean-field stochastic linear quadratic optimal control problems: closed-loop solvability

1 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China;
2 Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA

Jiongmin Yong, malixun@polyu.edu.hk

Received: February 23, 2016

Revised: June 09, 2016

Published: September 2020

supported by Hong Kong RGC under grants 519913, 15209614 and 15224215. Jingrui Sun was partially supported by the National Natural Science Foundation of China (11401556) and the Fundamental Research Funds for the Central Universities (WK 2040000012). Jiongmin Yong was partially supported by NSF DMS-1406776.

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Abstract

An optimal control problem is studied for a linear mean-field stochastic differential equation with a quadratic cost functional. The coefficients and the weighting matrices in the cost functional are all assumed to be deterministic. Closedloop strategies are introduced, which require to be independent of initial states; and such a nature makes it very useful and convenient in applications. In this paper, the existence of an optimal closed-loop strategy for the system (also called the closedloop solvability of the problem) is characterized by the existence of a regular solution to the coupled two (generalized) Riccati equations, together with some constraints on the adapted solution to a linear backward stochastic differential equation and a linear terminal value problem of an ordinary differential equation.

Keywords:

Mean-field stochastic differential equation,
Linear quadratic optimal control,
Riccati equation,
Regular solution,
Closed-loop solvability.

Mathematics Subject Classification: 49N10;49N35;93E20.

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