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.

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随机线性二次控制问题的最优反馈特征

控制理论中的基本问题之一是设计反馈控制。众所周知，在确定性线性二次控制问题的研究中引入Riccati方程的目的正是为了构建所需的反馈。迄今为止，随机环境中的相同问题只是部分被很好地理解。在本文中，我们建立了具有随机系数的随机线性二次控制问题的最优反馈控制的存在与相应倒向随机Riccati方程在适当意义上的可解性之间的等价性。我们还给出了一个反例，表明不存在可解决的随机线性二次控制问题的反馈控制。这是随机环境中的新现象，