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→Locally asymptotically stable equilibrium: The equilibrium x has to be isolated for applying this method - elsewhise ∀ε>0 ∃x' ∈ B(x,ε): x' is an equilibrium, which means a solution starting in x' won't converge to x, thus preventing asymptotic stability. Tags: Mobile edit Mobile web edit |
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===Locally asymptotically stable equilibrium===
If the equilibrium is isolated, the Lyapunov-candidate-function <math>V</math> is locally positive definite and the time derivative of the Lyapunov-candidate-function is locally negative definite:
:<math>\dot{V}(x) < 0 \quad \forall x \in \mathcal{B}\setminus\{0\}</math>
for some neighborhood <math>\mathcal{B}</math> of origin then the equilibrium is proven to be locally asymptotically stable.
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