Liapunov function: Difference between revisions

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#REDIRECT [[Lyapunov function]]

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>

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

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