We discuss a quadratic criterion optimal control problem for stochastic linear system with delay in both state and control variables. This problem will lead to a kind of generalized forward-backward stochastic differential equations (FBSDEs) with Itô’s stochastic delay equations as forward equations and anticipated backward stochastic differential equations as backward equations. Especially, we present the optimal feedback regulator for the time delay system via a new type of Riccati equations and also apply to a population optimal control problem.
Li Chen. Zhen Wu. Zhiyong Yu. "Delayed Stochastic Linear-Quadratic Control Problem and Related Applications." J. Appl. Math. 2012 1 - 22, 2012. https://doi.org/10.1155/2012/835319