Lyapunov function

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In the theory of dynamical systems, and control theory, Lyapunov functions, named after Aleksandr Mikhailovich Lyapunov, are a family of functions that can be used to demonstrate the stability or instability of some state points of a system.

The demonstration of stability or instability requires finding a Lyapunov function for that system. There is no direct method to obtain a Lyapunov function but there may be tricks to simplify the task. The inability to find a Lyapunov function is inconclusive with respect to stability or instability.

Definition

Given an autonomous system of two first order differential equations:

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

Let the origin (0,0) be an isolated critical point of the above system.

A function V(x,y) that is of class C^{{1}} and satisfies V(0,0)=0 is called a Lyapunov function if every open ball B_{\delta }(0,0) contains at least one point where V>0. If there happens to exist \delta ^{*} such that the function {\dot  {V}}, given by

{\dot  {V}}(x,y)=V_{{x}}(x,y)F(x,y)+V_{{y}}(x,y)G(x,y)

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

See also

References

This article incorporates material from Liapunov function on PlanetMath, which is licensed under the GFDL.Template:Mathapplied-stub

External Links

  • Example of determining the stability of the equilibrium solution of a system of ODEs with a Lyapunov function