Lyapunov function

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Definition

Consider a differentiable vectorfield $f:X \rightarrow X$ , $x \mapsto f(x)$ , defined on a complete metric space $X.$ A differentiable function $V:U \rightarrow \mathbb{R}$ , defined on an open subset $U$ 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 $f$ .

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Categories: Differential Equations

Lyapunov function

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