# Accessibility in control theory: are these 2 different definitions equivalent?

So I have been studying control theory, and I have found different authors use different definitions for the concept of accessibility. In particular, I find that with one definition it is trivial to prove that accessibility is a necessary condition for controllability, and with the other one...not so much.

Let me introduce some concepts first. We'll be dealing with a differential equation on a smooth $$n$$-manifold $$M$$ of the form $$$$\label{controlsys} \dot{x} = f(x,u(t)),$$$$ where $$u(t)$$ is a time-dependent map from the nonnegative reals $$\mathbb{R}^+$$ to a constraint set $$\Omega \subset \mathbb{R}^m$$; $$u$$ is the control.

Define the reachable sets: Given $$x_0 \in M$$, we define $$R(x_0,t)$$ to be the set of all $$x \in M$$ for which there exists an admissible control $$u$$ such that there is a trajectory of the control system with $$x(0)=x_0, x(t) = x$$. The reachable set from $$x_0$$ at time $$T$$ is defined to be $$$$R_T(x_0) = \bigcup_{0 \leq t \leq T} R(x_0,t).$$$$

Now here come the two definitions of accessibility I have found.

1. The control system on $$M$$ is said to be accessible from $$p \in M$$ if for every $$T>0$$, the reachable set $$R_T(p)$$ contains a nonempty open set.
2. The control system on $$M$$ is said to be accessible from $$p \in M$$ if for some $$T>0$$, the reachable set $$R_T(p)$$ contains a nonempty open set.

Since my definition of controllability is simply: for all $$p \in M$$ there exists a $$T>0$$ such that $$R_T(p) = M$$ , it is clear that controllability implies accessibility from every point $$p$$ according to the second definition.

However, having seen how many papers use either one of the two definitions of accessibility above nonchalantly, I believe (1) and (2) are equivalent. Intuitively it kinda makes sense (it'd be very weird for $$R_T(p)$$ to go from having an empty interior to a nonempty interior all of a sudden as we vary $$T$$--and accessibility after time $$T$$ implies accessibility for all later times, of course), but I'm stuck trying to prove it. I also couldn't find any authoritative references discussing the distinction, which I thought was weird.

They're not equivalent. Take $$\ M=\mathbb{R}^2\$$, $$\ \Omega=[0,2]^2\$$ and $$\ f\$$ to be the function defined by $$f(x,u)=\cases{(1,0)&if \ x_1<1\\ x+((u_1x_1-1,u_2x_2-1))&if \ x_1\ge1\ . }$$ For this control system $$R_T((0,1))=\cases{ [0,T]\times\{1\}\ &for \ 01\ , }$$ which contains an open subset of $$\ M\$$ for $$\ T>1\$$ but not for $$\ 0. This system is therefor accessible from $$\ (0,1)\$$ according to the second definition but not according to the first.
• I think there's something wrong with this-- it seems you have computed $R(0,T)$ (that is, the points we can reach in the manifold at exactly time $T$) instead of $R_T(0)$ (that is, the points we can reach in the manifold for times $t \leq T$). In fact, reachable sets should form a chain under inclusion for increasing T, and the proposed formula doesn't satisfy that. Playing around with this control system gives me reachable sets with non-empty interior right from the get-go. Jun 5, 2021 at 22:12
• Yes, you're right. I misread the definition of $\ R\$. I've now modified my counterxample by adding an extra dimension to the state space. As long as the differential equation has a unique solution for any (sufficiently regular) $\ u(t)\$, I don't think there exists a counterxample with a one-dimensional state space. Thank you for picking up the error. Jun 5, 2021 at 23:21
• Yup, probably because these $R_T(x_0)$ sets are path-connected, which implies they are intervals on $\mathbb{R}$. This is a sweet counterexample extending that 1D intuition you had. Thank you! Jun 6, 2021 at 3:02