Lyapunov function

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In mathematics, especially in stability theory, Lyapunov functions, named after Aleksandr Mikhailovich Lyapunov, are functions which can be used to proof the stability or instability of fixed points in dynamical systems and autonomous differential equations.

In general there is no method to construct a Lyapunov function and the inability to find a Lyapunov function is inconclusive with respect to stability or instability. For physical systems conservation laws can often be used to construct a Lyapunov function.

1 Definition
2 Lyapunov stability criteria
3 See also
4 References
5 External Links

[edit] Definition

Let

$f : D \subset \mathbb{R}^n \to \mathbb{R}^n$

be a vector field in Rⁿ which is two times continuously differentiable with

f (p) = 0

for some p in D. Given an open neighbourhood U of p contained in D a scalar function

$V : U \to \mathbb{R}$

is called Lyapunov function for f and p if

V is continuous on U and continuously differentiable on U \ {p}
V(p) = 0 and V(x) > 0 for all x in U \ {p}, that is V is a positive-definite function
gradV(x) o f(x) ≤ 0 for all x in U \ {p}, in other words the derivative of V in direction f(x) is decreasing.

If additionally the stronger condition gradV(x) o f(x) < 0 is satisfied, we call V a strict Lyapunov function.

[edit] Lyapunov stability criteria

Main article: Lyapunov stability

The existence of a Lyapunov function for a fixed point of a autonomous differential equation provides sufficient condition for the stability of the point.

Given a vector field in Rⁿ which is two times continuously differentiable

$f : D \subset \mathbb{R}^n \to \mathbb{R}^n$

with the associated autonomous differential equation

$\mathbf{x}^' = f(\mathbf{x})$

and a fixed point p in D, that is

$f(\mathbf{p}) = 0.$

The fixed point p is stable if there exists a Lyapunov function on an open neighbourhood U of p. The point is asymptotically stable if there exists a strong Lyapunov function on an open neighbourhood U of p.

[edit] See also

[edit] References

Eric W. Weisstein, Lyapunov Function at MathWorld.
This article incorporates material from Liapunov function on PlanetMath, which is licensed under the GFDL.