Lyapunov function

Definition

Let $U$ be an open subset of $\mathbb{R}^n$, let $f : U \to \mathbb{R}^n$ be locally Lipschitz, and let $x^* \in U$ be an equilibrium point of $x' = f(x)$. Let $W \subseteq U$ be an open neighbourhood of $x^*$ and let $V : W \to \mathbb{R}$ be continuously differentiable. The orbital derivative of $V$ is the function $\dot{V}(x) = \nabla V(x) \cdot f(x)$ on $W$. The function $V$ is a Lyapunov function for $x^*$ if $V(x^*) = 0$, $V(x) > 0$ for each $x \in W \setminus \{x^*\}$, and $\dot{V}(x) \le 0$ for each $x \in W$. It is a strict Lyapunov function if also $\dot{V}(x) < 0$ for each $x \in W \setminus \{x^*\}$.

Remarks

For each solution $\gamma$ in $W$, $\frac{d}{dt} V(\gamma(t)) = \dot{V}(\gamma(t))$. So $V$ does not increase along solutions. For $x' = -x$ on $\mathbb{R}$, the function $V(x) = x^2$ is a strict Lyapunov function for $0$, since $\dot{V}(x) = -2x^2$. In mechanics, the total energy is often a Lyapunov function. See Lyapunov's stability theorem.

Explanations

L1 human

A Lyapunov function is like a height above the equilibrium, which is the lowest point. If the height never increases along the motion, the state cannot climb away from the equilibrium.