# Lyapunov Function

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## Definition

Consider a differentiable vectorfield \(f:X \rightarrow X\ ,\) \(x \mapsto f(x)\ ,\) \(X \subset \mathbb{R}^n.\) A differentiable function \(V:U \rightarrow \mathbb{R}\ ,\) defined on an open subset \(U \subset X\) is called a **Lyapunov function** for \(f\) on \(U\) if the inequality\[ \overset{\circ}{V}(x) := \nabla V(x)^T f(x) \, \leq 0 \] is satisfied for all \(x \in U\ .\)

\(\overset{\circ}{V}\) defined as above is called the orbital differential of \(V\) at \(x\ .\)

In other words, a Lypunov function is decreasing along the orbits of points in \(U\) that are introduced by the flow corresponding to the vectorfield \(f\ .\)