Domains of attraction

The stability definitions given so far are often called local because they are expressed in terms of a neighborhood of ${x_{G}}$. Global versions can also be defined by extending the neighborhood to all of $X$. An equilibrium point is globally asymptotically stable if it is Lyapunov stable, and the integral curve from any $x_0 \in X$ converges to ${x_{G}}$ as time approaches infinity. It may be the case that only points in some proper subset of $X$ converge to ${x_{G}}$. The set of all points in $X$ that converge to ${x_{G}}$ is often called the domain of attraction of ${x_{G}}$. 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 ${x_{G}}$ in finite time. For the domain of attraction only asymptotic convergence to ${x_{G}}$ is assumed, and the vector field is given (there are no actions to choose).