Consider the continuous linear time-invariant system
$$
\begin{array}{l}
    \dot{\mathbf{x}}(t) = A \mathbf{x}(t) + B \mathbf{u}(t)\\
    \mathbf{y}(t) = C \mathbf{x}(t) + D \mathbf{u}(t) 
\end{array}
$$
where
$\mathbf{x}$ is the $n \times 1$ state vector,
$\mathbf{y}$ is the $m \times 1$ output vector,
$\mathbf{u}$ is the $r \times 1$ input (or control) vector,
$A$ is the $n \times n$ state matrix,
$B$ is the $n \times r$ input matrix,
$C$ is the $m \times n$ output matrix,
$D$ is the $m \times r$ feedthrough (or feedforward) matrix. 

For this system, we say it is controllable iff the controllability matrix
$$
    R = \begin{bmatrix}B & AB & A^{2}B & ...& A^{n-1}B\end{bmatrix} 
$$
is full rank, i.e. for any $\mathbf{x}_0,\mathbf{x}_1\in {\mathbb R}^n$ and $T>0$, there exist a control $\mathbf{u}$, such that for the system
$$
\begin{array}{l}
    \dot{\mathbf{x}}(t) = A \mathbf{x}(t) + B \mathbf{u}(t)\\
    \mathbf{x}(0) = \mathbf{x}_0 
\end{array}
$$
we have $\mathbf{x}(T)=\mathbf{x}_1$. 

Beside this, I am looking for a condition on the speed of reaching vs boundedness of $\mathbf{u}$, i.e. for a given speed value $V$ and a bound $u_\max$, what is the condition on $A$ and $B$, such that for any given $\mathbf{x}_0,\mathbf{x}_1\in {\mathbb R}^n$, there exist a control input $\mathbf{u}$ and a final time $T$, where $||\mathbf{u}||_\infty\le u_\max$ and $ T\le \frac{||\mathbf{x}_1-\mathbf{x}_0||}{V}$,
such that for the system
$$
\begin{array}{l}
    \dot{\mathbf{x}}(t) = A \mathbf{x}(t) + B \mathbf{u}(t)\\
    \mathbf{x}(0) = \mathbf{x}_0 
\end{array}
$$
we have $\mathbf{x}(T)=\mathbf{x}_1$. 

A special case: Let $A = 0$ and $B =  \frac{V}{u_\max} I_n$. So the resulting system is $$\dot{\mathbf{x}}(t) = \frac{V}{u_\max}\mathbf{u}(t).$$
This system satisfies the requested property.  The question is a generalization for systems with similar property.

It seems to be a standard question which might be found in the texts, so in the case of any references please mention the reference, either books or papers.