Consider a $n$-dimentional $p$-input equation: $$\dot{x}=Ax+Bu$$ where $A$ and $B$ are constant $n\times n$ and $n\times p$ real matrices.
By definition, the latter state equation is said to be controllable if for any initial state $x(0)=x_0$ and any final state $x_1$, there exists an input that transfers $x_0$ to $x_1$ in a finite time.
Then, how can I show that if a linear dynamical equation is controllable at $t_0$ then it is controllable at any $t<t_0$?
Recall that $\dot{x} = Ax + Bu$, then $(A,B)$ is controllable with control law (with minimum energy) given by
$$u(t) = -B^{T}e^{A^{T}(t_1-t)}W_c^{-1}(t_1)(e^{At_1}x_0 -x_1), $$ where $$W_c(t) = \int_0^te^{A\tau}BB^{T}e^{A^{T}\tau}d\tau.$$
For $W_c$ nonsingular, then equivalence is $(A,B)$ is controllable.