Definition:Liapunov Function
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Definition
Let $x_0$ be an equilibrium point of the system $x' = f \left({x}\right)$.
Then a function $V$ is a Liapunov function of the system on an open set $U$ containing the equilibrium if:
- $V \left({x_0}\right) = 0$
- $V \left({x}\right) > 0$ if $x \in U \setminus \left\{{x_0}\right\}$
- $\nabla V \cdot f \le 0$ for $x \in U$.
What $\nabla$ means in this context.
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If the inequality is strict except at $x_0$, then $V$ is strict.
Source of Name
This entry was named for Aleksandr Lyapunov.
