EDIT:
In one of Steve Brunton's video lectures which I really like, he designs this energy ellipsoid.
The description about it (as I perceive it), is that: even if a system is not globally controllable, some controllable directions can be found using the Singular Value Decomposition (SVD) of the Controllability Matrix, and be ordered from more to less controllable ones. The main axes of the ellipsoid are the singular vectors and the ellipsoid represents "how far the system can reach along the controllable direction, in comparison to other less controllable directions, given the same input energy".
Assuming though a controllable 3-dimension system, given a specific $B$,where following the SVD, we can order the controllable directions from more controllable to less controllable, e.g.:
[$ξ_1$ $ξ_2$ $ξ_3$], where $ξ_1$ is the most controllable while the $ξ_3$ is the least controllable.
What confuses me:
On the one hand, for a linear transformation R (reachability), R: U→X, where U is the domain and contains the inputs, and $X$ is the codomain and contains the final states, if I translate correctly the information from the energy ellipsoid, one $u(t)$ can be mapped in more than one value of ξ. How is this supposed to happen?
On the other hand, it does make sense that with the same input there can be more possible "final destinations". In that case however, both $ξ_1$ and $ξ_2$, as well as $ξ_3$, are all calculated by the same integral: $$\int_0^t e^{A(t-τ)}Bu(τ)dτ$$ Shouldn't there be a factor to distinguish between the different vectors?
I feel like I'm missing something very obvious here. Any advice?