Verify the controllability of the system $(A,B)$, for
$$A=\begin{pmatrix}1&0\\1&-1\end{pmatrix}, \; B=\begin{pmatrix}1\\1\end{pmatrix}$$
Find a control $u \in L([0,1];R)$ such that $(0,0)\xrightarrow{u}(1,e^2)$.
My attempt My doubt is about the second part. First, I've tried to suppose that the system starts on $(0,0)$. In this case, we have a solution of the form
$$x(t)=\int_0^t e^{(t-s)A}Bu(s)ds$$
I've calculated the exponential matrix as $$e^{(t-s)A} = \begin{pmatrix} e^{(t-s)} & 0 \\ \sinh (t-s) & e^{-(t-s)}\end{pmatrix}$$
Substituting this and multiplying by $B$, I couldn't see a good form for the integrals so that a function $u$ looked obvious. Probably there is a better way of doing it.
Thanks in advance!
If $W(0,T)=\int_0^Te^{sA}BB^te^{sA^t}ds$ is the gremian of controllability, since the system is controllable the control
$$u(s)=B^te^{(T-s)A^t}W(0,T)^{-1}x(T)$$ for $x(T)=(1,e^2)$ do what we want.