Consider the deterministic dynamical system

$$\dot{x}_{t}=A x_{t}+B u_{t}$$

where $A$ and $B$ are constant matrices, $x_{t} \in \mathbb{R}^{n}$, and $u_{t}$ is the control variable, $u_{t} \in \mathbb{R}^{m}$. What does it mean to say that the system is controllable?

Let $y_{t}=e^{-t A} x_{t}-x_{0}$. Show that if $V_{t}$ is the set of possible values for $y_{t}$ as the control $\left\{u_{s}: 0 \leq x \leq t\right\}$ is allowed to vary, then $V_{t}$ is a vector space.

Show that each of the following three conditions is equivalent to controllability of the system.

(i) The set $\left\{v \in \mathbb{R}^{n}: v^{\top} y_{t}=0\right.$ for all $\left.y_{t} \in V_{t}\right\}=\{0\}$.

(ii) The matrix $H(t)=\int_{0}^{t} e^{-s A} B B^{\top} e^{-s A^{\top}} d s$ is (strictly) positive definite.

(iii) The matrix $M_{n}=\left[\begin{array}{lllll}B & A B & A^{2} B & \cdots & A^{n-1} B\end{array}\right]$ has rank $n$.

Consider the scalar system

$$\sum_{j=0}^{n} a_{j}\left(\frac{d}{d t}\right)^{n-j} \xi_{t}=u_{t}$$

where $a_{0}=1$. Show that this system is controllable.