local controllability => global controllability

Question

Consider a general control system $$ \dot x(t) = f(x(t),u(t)), \quad x\in M\; \text{ (smooth manifold)} $$ where $f$ is smooth as derired. I am looking for rigourous statement, when local controllability of such a system at every point of the manifold $M$ implies global controllability on $M$

Answer 1

It will depend on your definitions of controllability right? Maybe a minor result that can get you started is the following.

If we define $$ {\mathbb{A}_x} = \left\{ {f\left( {x,u} \right) | u \in U} \right\} \subseteq {T_x}M $$ Then the system is controllable if ${\mathbb{A}_x} \ni {\mathcal{B}_{\varepsilon ,x}}$ where, $${\mathcal{B}_{\varepsilon ,x}} = \left\{ {y \quad |\quad y \in {T_x}M,\left\| y \right\| \leq \varepsilon } \right\}$$ under an appropriate norm for all $ \varepsilon > 0$ and for all $x$.