Definition:Lyapunov Function

Definition

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Let $x_0$ be an equilibrium point of the system of differential equations $x' = \map f x$.

Then a function $V$ is a Lyapunov function of the system $x' = \map f x$ on an open set $U$ containing the equilibrium if and only if:

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

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

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

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Strict Lyapunov Function

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.

Sources

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