| 王晔.随机线性二次最优控制: 从离散到 连续时间模型[J].数学年刊A辑,2018,(4):429~448 |
| 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 |
| Received:September 01, 2016 Revised:November 30, 2017 |
| DOI：10.16205/j.cnki.cama.2018.0035 |
| 中文关键词: Stochastic linear quadratic optimal control, Indefinitestochastic LQ control, Riccati equation, Numerical method |
| 英文关键词:Stochastic linear quadratic optimal control, Indefinitestochastic LQ control, Riccati equation, Numerical method |
| 基金项目: |
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| 中文摘要: |
| 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. |
| 英文摘要: |
| 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. |
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