15.1.3 Controllability

Now suppose that a system ${\dot x}= f(x,u)$ is given on a smooth manifold $X$ as defined throughout Chapter 13 and used extensively in Chapter 14. The system can be considered as a parameterized family of vector fields in which $u$ is the parameter. For stability, it was assumed that this parameter was fixed by a feedback plan to obtain some ${\dot x}= f(x)$. This section addresses controllability, which indicates whether one state is reachable from another via the existence of an action trajectory ${\tilde{u}}$. It may be helpful to review the reachable set definitions from Section 14.2.1.