Editing Lyapunov stability (section)

==Definition for continuous-time systems==
Consider an [[Autonomous system (mathematics)|autonomous]] nonlinear dynamical system

:<math>\dot{x} = f(x(t)), \;\;\;\; x(0) = x_0</math>,

where <math>x(t) \in \mathcal{D} \subseteq \mathbb{R}^n</math> denotes the [[State space representation|system state vector]], <math>\mathcal{D}</math> an open set containing the origin, and <math>f: \mathcal{D} \rightarrow \mathbb{R}^n</math> is a continuous vector field on <math>\mathcal{D}</math>.  Suppose <math>f</math> has an equilibrium at <math>x_e</math>, so that <math> f(x_e)=0 </math>.  Then:

# This equilibrium is said to be '''Lyapunov stable''' if for every <math>\epsilon > 0</math> there exists a <math>\delta > 0</math> such that if <math>\|x(0)-x_e\| < \delta</math> then for every <math>t \geq 0</math> we have <math>\|x(t)-x_e\| < \epsilon</math>.
# The equilibrium of the above system is said to be '''asymptotically stable''' if it is Lyapunov stable and there exists <math>\delta > 0</math> such that if <math>\|x(0)-x_e \|< \delta</math> then <math>\lim_{t \rightarrow \infty} \|x(t)-x_e\| = 0</math>.
# The equilibrium of the above system is said to be '''exponentially stable''' if it is asymptotically stable and there exist <math>\alpha >0, ~\beta >0, ~\delta >0</math> such that if <math>\|x(0)-x_e\| < \delta</math> then <math>\|x(t)-x_e\| \leq \alpha\|x(0)-x_e\|e^{-\beta t}</math> for all <math>t \geq 0</math>.

Conceptually, the meanings of the above terms are the following:
# Lyapunov stability of an equilibrium means that solutions starting "close enough" to the equilibrium (within a distance <math>\delta</math> from it) remain "close enough" forever (within a distance <math>\epsilon</math> from it). Note that this must be true for ''any'' <math>\epsilon</math> that one may want to choose.
# Asymptotic stability means that solutions that start close enough not only remain close enough but also eventually converge to the equilibrium.
# Exponential stability means that solutions not only converge, but in fact converge faster than or at least as fast as a particular known rate <math>\alpha\|x(0)-x_e\|e^{-\beta t}</math>.

The trajectory ''<math>x(t) = \phi(t)</math>'' is (locally) ''attractive'' if

:<math>\|x(t)-\phi(t)\| \rightarrow 0 </math>  as  <math> t \rightarrow \infty</math>

for all trajectories <math>x(t) </math> that start close enough to <math>\phi(t) </math>, and ''globally attractive'' if this property holds for all trajectories.

That is, if ''x'' belongs to the interior of its [[stable manifold]], it is ''asymptotically stable'' if it is both attractive and stable.  (There are examples showing that attractivity does not imply asymptotic stability.<ref>{{cite book |last1=Hahn |first1=Wolfgang |author1-link=Wolfgang Hahn |title=Stability of Motion |date=1967 |publisher=Springer |isbn=978-3-642-50087-9 |pages=191–194, Section 40 |doi=10.1007/978-3-642-50085-5 |url=https://doi.org/10.1007/978-3-642-50085-5}}</ref><ref>{{cite book |last1=Braun |first1=Philipp |last2=Grune |first2=Lars |last3=Kellett |first3=Christopher M. |title=(In-)Stability of Differential Inclusions: Notions, Equivalences, and Lyapunov-like Characterizations |date=2021 |publisher=Springer |isbn=978-3-030-76316-9 |pages=19–20, Example 2.18 |doi=10.1007/978-3-030-76317-6 |s2cid=237964551 |url=https://doi.org/10.1007/978-3-030-76317-6}}</ref><ref>{{cite journal |last1=Vinograd |first1=R. E. |title=The inadequacy of the method of characteristic exponents for the study of nonlinear differential equations |journal=Doklady Akademii Nauk |date=1957 |volume=114 |issue=2 |pages=239–240 |url=http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=dan&paperid=21930&option_lang=eng |language=Russian}}</ref> Such examples are easy to create using [[homoclinic orbit|homoclinic connections]].)

If the [[Jacobian matrix and determinant|Jacobian]] of the dynamical system at an equilibrium happens to be a [[stability matrix]] (i.e., if the real part of each eigenvalue is strictly negative), then the equilibrium is asymptotically stable.

===System of deviations===
Instead of considering stability only near an equilibrium point (a constant solution <math>x(t)=x_e</math>), one can formulate similar definitions of stability near an arbitrary solution <math>x(t) = \phi(t)</math>. However, one can reduce the more general case to that of an equilibrium by a change of variables called a "system of deviations". Define <math>y = x - \phi(t)</math>, obeying the differential equation:

:<math>\dot{y} = f(t, y + \phi(t)) - \dot{\phi}(t) = g(t, y)</math>.

This is no longer an autonomous system, but it has a guaranteed equilibrium point at <math>y=0</math> whose stability is equivalent to the stability of the original solution <math>x(t) = \phi(t)</math>. 

===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.