Liapunov function: Difference between revisions

Revision as of 20:03, 22 August 2005

Revision as of 20:05, 22 August 2005

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Suppose we are given an [[autonomous system]] of [[differential equation|first order differential equation]]s.

<math> \frac{dx}{dt}=F(x,y)\quad\frac{dy}{dt}=G(x,y)</math>

dx/dt=F(x,y) dy/dt=G(x,y)

Let the origin be an [[isolated critical point]] of the above system.

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A [[function]] V(x,y) that is of class C¹ and satisfies V(0,0)=0 is called a '''Liapunov function''' if every [[open ball]] B<sub>d<\math><\sub>(0,0) contains at least one [[point]] where V>0. If there happens to exist d^* such that the function '''.'''{V}, given by

'''.'''<math> \displaystyle \dot{V}(x,y)=V_{x}(x,y)F(x,y)+V_{y}(x,y)G(x,y) $</math>

is [[positive definite]] in <math> B_{\delta}^{*}(0,0) </math>, then the [[origin]] is an [[unstable critical point]] of the system.