Is T completely arbitrary in the definition of reachability?

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Q Is T completely arbitrary in the definition of reachability, or is it limited by the system? For instance, if the system is a car, x0 = LA, xf = NYC, does the fact that I can't get there from here in 1 second mean that it isn't reachable?

A If a system is continuous time and linear, then the property has to be veryfied for any time T. Indeed, when you check using the reachability matrix

\([B\quad AB\quad A^2B\quad... \quad A^{n-1}B]\)

there is no time dependence in this.

On the other hand, the notion of reachability for discrete time systems (you have not seen this, and won't probably see it) requires that a desired state \(\bar x\) be reached within \(n-1\) steps, where \(n\) is always the order (dimension) of the system.

For a nonlinear system, things are more complicated: one has to talk about reachable regions in a certain finite or desired time. So unfortunately your car is a highly constrained nonlinear system, therefore NY is in your reachability region, but you takes a while to get there...