Consider a Linear Time System with the admissble control set $$U = \left\{ u: R \rightarrow R^m \;|\;\text{u is integrable in any finite interval} \right\} $$.
Show that, if starting on $x_0=0$ we can reach a state $x_1$ in time $T_1$, then we can reach this state in any time $T \geq T_1$. That is, if there is a trajetory linking $0$ to $x_1$ in time $T_1$, there is a trajetory linking those states in any time greater than $T_1$
My attempt: We have that
$$x_1 = x(T_1) = \int_0^{T_1} e^{sA} B u(s) ds$$ where $$x'(t)=Ax(t)+Bu(t), \; x_0 = 0 \text{ and } u \in U$$
I need to show that for any $T\geq T_1, \; \exists \bar{u} \in U$ such that $$x_1 = x(T) = \int_0^Te^{sA}B\bar{u}(s)ds$$
I had the following ideas:
1. $$\bar{u}(s)=\left\{\begin{matrix} u(s), s \in [0, T_1] \\ 0, s\in (T_1, T] \end{matrix}\right.$$
2. The gramian of controlabillity is defined as $$Q_T = \int_0^Te^{sA}BB^te^{sA^t} ds $$ and the state $x_1$ is reachable iff $\exists \eta$ such that $x_1 = Q_T \eta$, ie, $x_1 \in \text{Im }Q_T$ In this case, we know that $$x_1 = Q_{T_1}\eta = \int_0^{T_1} e^{sA}BB^te^{sA^t}\eta\; ds$$ So I thought that $$\bar{u}(s)=\left\{\begin{matrix} B^te^{sA^t}\eta, s \in [0, T] \\ -B^te^{sA^t}\eta, s\in (T_1, T] \end{matrix}\right.$$
I would like to know if both approaches are correct and, if not, what is wrong with them.
Thanks in advance!