Chapter2. lyapunov theorem

Given by: Dr. Hamid Sadeghian

Lyapunov theorem

Asymp. stability with n.s.d function for non-autonomous sys.

• For autonomous systems, by LaSalle lemma it is possible to show asymptotic stability for s.p.d Lyapunov functions.
• For non-autonomous systems, use Barbalat lemma.

If the differentiable function $f(t)$ has a finite limit as $t\rightarrow\infty$, and if $\dot{f}$ is uniformly continous, then $\dot{f}(t)\rightarrow 0$ as $t\rightarrow\infty$

uniformly continuity: a function $g(t)$ is said to be uniformly continous on $[0,\infty)$ if

$$\forall R>0,\ \exists \eta(R)>0, \ \forall t_1\ge0,\ \forall t\ge0, \ |t-t_1|<\eta \Rightarrow |g(t)-g(t_1)|<R$$

, which means as long as the input change is sufficiently small, the output change will necessarily be small(globally). For differentiable function to be uniformly continous is that its derivative be bounded.