A particle follows a discrete-time trajectory on $\mathbb{R}$ given by

$$x_{t+1}=\left(A x_{t}+u_{t}\right) \xi_{t}+\epsilon_{t}$$

for $t=1,2, \ldots, T$. Here $T \geqslant 2$ is a fixed integer, $A$ is a real constant, $x_{t}$ and $u_{t}$ are the position of the particle and control action at time $t$, respectively, and $\left(\xi_{t}, \epsilon_{t}\right)_{t=1}^{T}$ is a sequence of independent random vectors with

$$\mathbb{E} \xi_{t}=\mathbb{E} \epsilon_{t}=0, \operatorname{var}\left(\xi_{t}\right)=V_{\xi}>0, \operatorname{var}\left(\epsilon_{t}\right)=V_{\epsilon}>0 \text { and } \operatorname{cov}\left(\xi_{t}, \epsilon_{t}\right)=0$$

Find the optimal control, i.e. the control action $u_{t}$, defined as a function of $\left(x_{1}, \ldots, x_{t} ; u_{1}, \ldots, u_{t-1}\right)$, that minimizes

$$\sum_{t=1}^{T} x_{t}^{2}+c \sum_{t=1}^{T-1} u_{t}^{2}$$

where $c>0$ is given.

On which of $V_{\epsilon}$ and $V_{\xi}$ does the optimal control depend?

Find the limiting form of the optimal control as $T \rightarrow \infty$, and the minimal average cost per unit time.