Lyapunov function: Difference between revisions

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===Stable equilibrium===

If <math>V</math> is a Lyapunov function, then the equilibrium is [[Lyapunov stable]]. The converse is also true, and was proved by [[José Luis Massera]].

===Globally asymptotically stable equilibrium===

If the Lyapunov-candidate-function <math>V</math> is globally positive definite, [[Radially unbounded function|radially unbounded]], the equilibrium isolated and the time derivative of the Lyapunov-candidate-function is globally negative definite:

:<math>\dot{V}(x) < 0 \quad \forall x \in \R ^n\setminus\{0\},</math>

then the equilibrium is proven to be [[Stability theory|globally asymptotically stable]].

The Lyapunov-candidate function <math>V(x)</math> is radially unbounded if

:<math>\| x \| \to \infty \Rightarrow V(x) \to \infty. </math>

(This is also referred to as norm-coercivity.)

The converse is also true,<ref name=Massera1949>{{Citation

| author = Massera, José Luis

| year = 1949

| title = On Liapounoff's conditions of stability

| journal = Annals of Mathematics |series=Second Series

| volume = 50

| issue = 3

| pages = 705–721

| doi = 10.2307/1969558

| mr = 0035354

| jstor = 1969558

}}</ref> and was proved by [[José Luis Massera]] (see also [[Massera's lemma]]).

==Example==