Lyapunov function

From Scholarpedia
Jump to: navigation, search

    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 \(f\ .\)

    Personal tools
    Namespaces

    Variants
    Actions
    Navigation
    Focal areas
    Activity
    Tools