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.

Also known as

The term Lyapunov function is sometimes seen spelled as Liapunov function.

Also see

• Results about Lyapunov functions can be found here.

Source of Name

This entry was named for Aleksandr Mikhailovich Lyapunov.