Equilibrium points and Lyapunov stability

At the very least, it seems that the state should remain fixed at $ {x_{G}}$, if it is reached. A point $ {x_{G}}\in X$ is called an equilibrium point (or fixed point) of the vector field $ f$ if and only if $ f({x_{G}}) = 0$. This does not, however, characterize how trajectories behave in the vicinity of $ {x_{G}}$. Let $ {x_{I}}\in X$ denote some initial state, and let $ x(t)$ refer to the state obtained at time $ t$ after integrating the vector field $ f$ from $ {x_{I}}= x(0)$.

Figure 15.1: Lyapunov stability: (a) Choose any open set $ O_1$ that contains $ {x_{G}}$, and (b) there exists some open set $ O_2$ from which trajectories will not be able to escape $ O_1$. Note that convergence to $ {x_{G}}$ is not required.
...yapunov2.eps,width=2.0in} \\
(a) & & (b)

See Figure 15.1. An equilibrium point $ {x_{G}}\in X$ is called Lyapunov stable if for any open neighborhood15.1 $ O_1$ of $ {x_{G}}$ there exists another open neighborhood $ O_2$ of $ {x_{G}}$ such that $ {x_{I}}\in O_2$ implies that $ x(t) \in O_1$ for all $ t > 0$. If $ X = {\mathbb{R}}^n$, then some intuition can be obtained by using an equivalent definition that is expressed in terms of the Euclidean metric. An equilibrium point $ {x_{G}}\in {\mathbb{R}}^n$ is called Lyapunov stable if, for any $ t > 0$, there exists some $ \delta > 0$ such that $ \Vert{x_{I}}-{x_{G}}\Vert <
\delta$ implies that $ \Vert x(t) - {x_{G}}\Vert < \epsilon$. This means that we can choose a ball around $ {x_{G}}$ 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 $ \delta$. If a single $ \delta$ can be chosen independently of every $ \epsilon$ and $ x$, then the equilibrium point is called uniform Lyapunov stable.

Steven M LaValle 2012-04-20