Editing Lyapunov stability (section)

===Lyapunov's second method for stability===
Lyapunov, in his original 1892 work, proposed two [[convergence proof techniques|methods for demonstrating stability]].<ref name=lyapunov/> The first method developed the solution in a series which was then proved convergent within limits. The second method, which is now referred to as the '''Lyapunov stability criterion''' or the Direct Method, makes use of a ''Lyapunov function V(x)'' which has an analogy to the potential function of classical dynamics. It is introduced as follows for a system <math> \dot{x} = f(x)</math> having a point of equilibrium at <math>x=0</math>. Consider a function <math>V : \mathbb{R}^n \rightarrow \mathbb{R} </math> such that
* <math>V(x)=0</math>  if and only if <math>x=0</math>
* <math>V(x)>0</math>  if and only if <math>x \ne 0</math>
* <math> \dot{V}(x) = \frac{d}{dt}V(x) = \sum_{i=1}^n\frac{\partial V}{\partial x_i}f_i(x) = \nabla V \cdot f(x) \le 0</math> for all values of <math>x\ne 0</math> . Note: for asymptotic stability, <math> \dot{V}(x)<0 </math> for <math>x \ne 0</math> is required.
Then ''V(x)'' is called a [[Lyapunov function]] and the system is stable in the sense of Lyapunov. (Note that <math>V(0)=0</math> is required; otherwise for example <math>V(x) = 1/(1+|x|)</math> would "prove" that <math>\dot x(t) = x</math> is locally stable.) An additional condition called "properness" or "radial unboundedness" is required in order to conclude global stability. Global asymptotic stability (GAS) follows similarly.

It is easier to visualize this method of analysis by thinking of a physical system (e.g. vibrating spring and mass) and considering the [[energy]] of such a system. If the system loses energy over time and the energy is never restored then eventually the system must grind to a stop and reach some final resting state. This final state is called the [[attractor]]. However, finding a function that gives the precise energy of a physical system can be difficult, and for abstract mathematical systems, economic systems or biological systems, the concept of energy may not be applicable.

Lyapunov's realization was that stability can be proven without requiring knowledge of the true physical energy, provided a [[Lyapunov function]] can be found to satisfy the above constraints.