A trajectory of the autonomous ODE \(\dot{x} = f(x)\) is a globally defined map \(\varphi : \mathbb {R} \to \mathbb {R}^{n}\) satisfying

A point \(x_{\mathrm{eq}} \in \mathbb {R}^{n}\) is an equilibrium of \(\dot{x} = f(x)\) when \(f(x_{\mathrm{eq}}) = 0\).

A class \(\mathcal{K}\) function on \([0,a)\) is a continuous strictly increasing map \(\alpha : [0,a) \to [0,b)\) with \(\alpha (0) = 0\). The structure records both \(\alpha \) and its inverse \(\alpha ^{-1} : [0,b) \to [0,a)\).

A class \(\mathcal{K}_{\infty }\) function is a continuous strictly increasing map \(\alpha : [0,\infty ) \to [0,\infty )\) with \(\alpha (0) = 0\) and \(\alpha (r) \to \infty \) as \(r \to \infty \).

The equilibrium \(x_{\mathrm{eq}}\) is Lyapunov stable when

The equilibrium \(x_{\mathrm{eq}}\) is locally asymptotically stable (LAS) when it is Lyapunov stable (definition 30) and there exists \(c {\gt} 0\) such that every trajectory \(\varphi \) with \(\| \varphi (0) - x_{\mathrm{eq}}\| {\lt} c\) satisfies \(\varphi (t) \to x_{\mathrm{eq}}\) as \(t \to \infty \).

The equilibrium \(x_{\mathrm{eq}}\) is globally asymptotically stable (GAS) when it is Lyapunov stable (definition 30) and every trajectory \(\varphi \) satisfies \(\varphi (t) \to x_{\mathrm{eq}}\) as \(t \to \infty \).

The sublevel set of \(V : \mathbb {R}^{n} \to \mathbb {R}\) at level \(c \in \mathbb {R}\) is

A set \(S \subseteq \mathbb {R}^{n}\) is positively invariant for \(\dot{x} = f(x)\) when every trajectory \(\varphi \) starting in \(S\) remains in \(S\) for all future time:

A function \(V : \mathbb {R}^{n} \to \mathbb {R}\) is a local Lyapunov function on an open domain \(D \ni x_{\mathrm{eq}}\) when \(V\) is smooth, \(V(x_{\mathrm{eq}}) = 0\), \(V {\gt} 0\) on \(D \setminus \{ x_{\mathrm{eq}}\} \), and the Lie derivative satisfies \(\dot{V}(x) = \nabla V(x) \cdot f(x) \le 0\) for all \(x \in D\).

A strict local Lyapunov function on \(D\) strengthens definition 36: the Lie derivative satisfies \(\dot{V}(x) {\lt} 0\) for all \(x \in D \setminus \{ x_{\mathrm{eq}}\} \), and there exists \(c {\gt} 0\) such that the compact sublevel set \(\Omega _{c}(V) \subseteq D\) (definition 33).

A global strict Lyapunov function for \(\dot{x} = f(x)\) at \(x_{\mathrm{eq}}\) is a \(C^{1}\) map \(V : \mathbb {R}^{n} \to \mathbb {R}\) with \(V(x_{\mathrm{eq}}) = 0\), \(V {\gt} 0\) everywhere else, \(\dot{V}(x) {\lt} 0\) on \(\mathbb {R}^{n} \setminus \{ x_{\mathrm{eq}}\} \), and all sublevel sets \(\Omega _{c}(V)\) compact (coercivity).

The classical GAS Lyapunov certificate: a \(C^{1}\) map \(V : \mathbb {R}^{n} \to \mathbb {R}\) with \(V(x_{\mathrm{eq}}) = 0\), \(V {\gt} 0\) elsewhere, \(\dot{V} {\lt} 0\) on \(\mathbb {R}^{n} \setminus \{ x_{\mathrm{eq}}\} \), and radially unbounded (\(V(x) \to \infty \) as \(\| x\| \to \infty \)). Implies definition 38 via theorem 35.

A class \(\mathcal{KL}\) function on \([0,a) \times [0,\infty )\) is continuous, class \(\mathcal{K}\) in the first argument, and for each fixed \(r {\gt} 0\) is strictly decreasing and tends to \(0\) as \(s \to \infty \). It arises as the bound \(\| x(t)\| \le \beta (\| x_0\| , t)\) in asymptotic stability estimates.