Editing Lyapunov stability (section)

==Barbalat's lemma and stability of time-varying systems==
It may be difficult to find a Lyapunov function with a negative definite derivative as required by the Lyapunov stability criterion, however a function <math>V</math> with <math>\dot{V}</math> that is only negative semi-definite may be available. In autonomous systems, [[LaSalle's invariance principle|the invariant set theorem]] can be applied to prove asymptotic stability, but this theorem is not applicable when the dynamics are a function of time.<ref name="Slotine">{{cite book |first=Jean-Jacques E. |last=Slotine |author2=Weiping Li |title=Applied Nonlinear Control |publisher=Prentice Hall |location=NJ |year=1991 }}</ref> 

Instead, Barbalat's lemma allows for Lyapunov-like analysis of these non-autonomous systems. The lemma is motivated by the following observations. Assuming f is a function of time only:

* Having <math>\dot{f}(t) \to 0</math> does not imply that <math>f(t)</math> has a limit at <math>t\to\infty</math>. For example, <math>f(t)=\sin(\ln(t)),\; t>0</math>.
* Having <math>f(t)</math> approaching a limit as <math>t \to \infty</math> does not imply that <math>\dot{f}(t) \to 0</math>. For example, <math>f(t)=\sin\left(t^2\right)/t,\; t>0</math>.
* Having <math>f(t)</math> lower bounded and decreasing (<math>\dot{f}\le 0</math>) implies it converges to a limit. But it does not say whether or not <math>\dot{f}\to 0</math> as <math>t \to \infty</math>.

Barbalat's [[Lemma (mathematics)|Lemma]] says:
:If <math>f(t)</math> has a finite limit as <math>t \to \infty</math> and if <math>\dot{f}</math> is uniformly continuous (a sufficient condition for uniform continuity is that <math>\ddot{f}</math> is bounded), then <math>\dot{f}(t) \to 0</math> as <math>t \to\infty</math>.<ref>I. Barbălat, Systèmes d'équations différentielles d'oscillations non Linéaires, Rev. Math. Pures Appl. 4 (1959) 267–270, p. 269.</ref>

An alternative version is as follows:

:Let <math>p\in [1,\infty)</math> and <math>q\in (1,\infty]</math>. If <math>f \in L^p(0,\infty)</math> and <math>{\dot f}\in L^q(0,\infty)</math>, then <math>f(t)\to 0</math> as <math>t\to \infty.</math> <ref>B. Farkas et al., Variations on Barbălat's Lemma, Amer. Math. Monthly (2016) 128, no. 8, 825-830, DOI: 10.4169/amer.math.monthly.123.8.825, p. 827.</ref>

In the following form the Lemma is true also in the vector valued case:

:Let <math>f(t)</math> be a uniformly continuous function with values in a Banach space <math>E</math> and assume that <math>\textstyle\int_0^t f(\tau)\mathrm {d}\tau</math> has a finite limit as <math>t\to \infty</math>. Then <math>f(t)\to 0</math> as <math>t\to \infty</math>.<ref>B. Farkas et al., Variations on Barbălat's Lemma, Amer. Math. Monthly (2016) 128, no. 8, 825-830, DOI: 10.4169/amer.math.monthly.123.8.825, p. 826.</ref>

The following example is taken from page 125 of Slotine and Li's book ''Applied Nonlinear Control''.<ref name="Slotine" />

Consider a [[Non-autonomous system (mathematics)|non-autonomous system]]
:<math>\dot{e}=-e + g\cdot w(t)</math>
:<math>\dot{g}=-e \cdot w(t).</math>

This is non-autonomous because the input <math>w</math> is a function of time.  Assume that the input <math>w(t)</math> is bounded.

Taking <math>V=e^2+g^2</math> gives <math>\dot{V}=-2e^2 \le 0.</math>

This says that <math>V(t)\leq V(0)</math> by first two conditions and hence <math>e</math> and <math>g</math> are bounded. But it does not say anything about the convergence of <math>e</math> to zero, as <math>\dot{V}</math> is only negative semi-definite (note <math>g</math> can be non-zero when <math>\dot{V}</math>=0) and the dynamics are non-autonomous.

Using Barbalat's lemma:

:<math>\ddot{V}= -4e(-e+g\cdot w)</math>.

This is bounded because <math>e</math>, <math>g</math> and <math>w</math> are bounded. This implies <math>\dot{V} \to 0</math> as <math>t\to\infty</math> and hence <math>e \to 0</math>. This proves that the error converges.