The stability definitions given so far are often called *local*
because they are expressed in terms of a neighborhood of .
*Global* versions can also be defined by extending the
neighborhood to all of . An equilibrium point is *globally
asymptotically stable* if it is Lyapunov stable, and the integral
curve from any converges to as time approaches
infinity. It may be the case that only points in some proper subset
of converge to . The set of all points in that
converge to is often called the *domain of attraction* of
. The funnels of Section
8.5.1 are based on domains of attraction. Also related is
the backward reachable set from Section
14.2.1. In that setting, action trajectories were
considered that lead to in finite time. For the domain of
attraction only asymptotic convergence to is assumed, and the
vector field is given (there are no actions to choose).

Steven M LaValle 2020-08-14