# 随机控制

## 离散时间系统

### 例子

${\displaystyle {\text{E}}_{1}\sum _{t=1}^{S}[y_{t}^{T}Qy_{t}+u_{t}^{T}Ru_{t}]}$

${\displaystyle y_{t}=A_{t}y_{t-1}+B_{t}u_{t},}$

${\displaystyle u_{t}^{*}=-[{\text{E}}(B^{T}X_{t}B+R)]^{-1}{\text{E}}(B^{T}X_{t}A)y_{t-1},}$

${\displaystyle X_{t-1}=Q+{\text{E}}[A^{T}X_{t}A]-{\text{E}}[A^{T}X_{t}B][{\text{E}}(B^{T}X_{t}B+R)]^{-1}{\text{E}}(B^{T}X_{t}A),\,}$

X的稳态特征若存在，会和S延伸到无限大的的无限时间问题相关。可以用重复迭代动态方程中的X，一直到收敛为止来计算，此时的动态方程中的X就不用有关时间的下标了。

## 参考文献

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8. ^ Fleming, W.; Rishel, R. Deterministic and Stochastic Optimal Control. 1975. ISBN 0-387-90155-8.
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