Definition:Lyapunov Function/Strict

Definition
Let $V$ be a Lyapunov function of a system of differential equations $x' = \map f x$.


 * $(1): \quad \map V {x_0} = 0$


 * $(2): \quad \map V x > 0$ if $x \in U \setminus \set {x_0}$


 * $(3): \quad \nabla V \cdot f \le 0$ for $x \in U$.

where $x_0$ is an equilibrium point of $\mathbf x' = \map f {\mathbf x}$.

If the inequality $(3)$ is strict except at $x_0$:
 * $(3): \quad \nabla V \cdot f > 0$ for $x \in U$.

then $V$ is a strict Lyapunov function.