#### Equilibrium points and Lyapunov stability

At the very least, it seems that the state should remain fixed at , if it is reached. A point is called an equilibrium point (or fixed point) of the vector field if and only if . This does not, however, characterize how trajectories behave in the vicinity of . Let denote some initial state, and let refer to the state obtained at time after integrating the vector field from .

See Figure 15.1. An equilibrium point is called Lyapunov stable if for any open neighborhood15.1 of there exists another open neighborhood of such that implies that for all . If , then some intuition can be obtained by using an equivalent definition that is expressed in terms of the Euclidean metric. An equilibrium point is called Lyapunov stable if, for any , there exists some such that implies that . This means that we can choose a ball around with a radius as small as desired, and all future states will be trapped within this ball, as long as they start within a potentially smaller ball of radius . If a single can be chosen independently of every and , then the equilibrium point is called uniform Lyapunov stable.

Steven M LaValle 2012-04-20