As a follow-up for the question: What is the difference between Reachability and Controllability
According to the definitions I have found, the problems of Controllability and Reachability consist in the following:
Controllability problem consists in finding the set of initial states $x$($t_0$) that are controllable to a given final state $x$($t_1$). Reachability problem consists in finding the set of final states $x$($t_1$) that can be reachable from a given initial state $x$($t_0$). The two give the sense of being notions "inverse" to each other:
In addition, Steve Brunton in the 15th video of his Control Bootcamp, mentions that Controllability is about if we can steer our system anywhere with a given input $u(t)$ . I (being more of a beginner) would believe that this looks closer to what Reachability is all about, rather than Controllability. Besides that, proving a system as controllable or not through the rank of the controllability-matrix, is independent of any input, and only depends on $A$ and $B$ matrices. In other words, whether a system is or is not controllable seems to be something like a property linked to the placement of the actuators of the controller (which of course has to be linked to the dynamics of the system). If this property is satisfied, then we can steer our system through different inputs $u(t)$ towards different final states. In conclusion, if a state is Reachable, then it is implied that the direction leading to this state is already controllable, while by itself Controllability does not imply Reachability, since it is also necessary for an appropriate input $u(t)$ to "reach" somewhere.
My problem is that it's difficult for me to combine all the previous along with the figure, since Controllability shouldn't be something quantitative based on initial and final states and input, but rather something qualitative which "paves the way" for Reachability. This dependence on initial-final states and inputs I sense that it's appropriate for Reachability. In a few words, what is the difference between controllability and reachability and how does it depend on whether we talk about LTI or LTV or nonlinear systems?
