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 \begin{equation}\label{controlsys} \dot{x} = f(x,u(t)), \end{equation} 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 \begin{equation} R_T(x_0) = \bigcup_{0 \leq t \leq T} R(x_0,t). \end{equation}
Now here come the two definitions of accessibility I have found.
- 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.
- 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.
Can you please help me prove these 2 definitions are equivalent? Thanks!
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 $$ R_T((0,1))=\cases{ [0,T]\times\{1\}\ &for $\ 0<T\le1$\\ \big[0,\frac{2+e^{3(T-1)}}{3}\big]\times\big[1,\frac{2+e^{3(T-1)}}{3}\big]&for $\ T>1\ $, } $$ which contains an open subset of $\ M\ $ for $\ T>1\ $ but not for $\ 0<T\le0\ $. This system is therefor accessible from $\ (0,1)\ $ according to the second definition but not according to the first.