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

$$x_{t+1}=A x_{t}+\xi_{t} u_{t}+\epsilon_{t}$$

for $t=1,2, \ldots, T$, where $T \geqslant 2$ is a fixed integer, $A$ is a real constant, $x_{t}$ is the position of the particle and $u_{t}$ is the control action at time $t$, and $\left(\xi_{t}, \epsilon_{t}\right)_{t=1}^{T}$ is a sequence of independent random vectors with $\mathbb{E} \xi_{t}=\mathbb{E} \epsilon_{t}=0, \operatorname{var}\left(\xi_{t}\right)=V_{\xi}>0, \operatorname{var}\left(\epsilon_{t}\right)=V_{\epsilon}>0$ and $\operatorname{cov}\left(\xi_{t}, \epsilon_{t}\right)=0$.

Find the closed-loop 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

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

where $c>0$ is given. [Note that this function is quadratic in $x$, but linear in $u$.]

Does the closed-loop control depend on $V_{\epsilon}$ or on $V_{\xi}$ ? Deduce the form of the optimal open-loop control.