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.
