随机线性二次最优控制: 从离散到 连续时间模型
Stochastic Linear Quadratic Optimal Control Problem: From Discrete to Continuous Time

在一般情形下, 分析了离散时间~LQ~问题与连续时间情形两者之间的自然联系. 首先回顾了连续时间和离散时间随机~LQ~问题及对应~Riccati~微分/差分方程的相关结论. 接下来在假设~Riccati~微分方程有解的前提下,~证明了离散化步长足够小时, Riccati~差分方程有解.~然后针对连续和离散时间模型,~采用配对问题最优控制的反馈形式, 分别构造了一个辅助反馈控制,~并证明该控制可驱使对应模型的性能指标逼近于配对问题的值函数, 以此得到了关于两个模型之间联系的初步结论. 最后藉由前述结论以及控制问题的特性, 揭晓了连续时间和离散时间模型之间的自然联系, 并给出了~Riccati~差分方程和微分方程的解之间的误差估计. 由此联系,~可构造相应离散系统和~LQ~问题,~以适当的阶估计连续时间~LQ~问题的解, 抑或为离散时间模型构造一个近似最优控制.~无论哪种思路, 都旨在降低直接求解原问题的难度和复杂性.
This paper deals with the continuous-time stochastic LQ problem involving state and control dependent noises and its discrete-time counterparts. Given the unique solvability of the continuous-time LQ problem, it is shown that time-discrete LQ problems admit solutions in cases where the step-size is sufficiently small. Moreover, the author reveals the natural connections between them and makes it possible to approximate the original continuous-time LQ problem with a proper order by a sequence of discrete-time ones. Besides, based on the optimal control of the continuous (discrete)-time LQ problem, optimal controls for the associated discrete (continuous)-time LQ problem and demonstrate their asymptotic optimality are constructed.