I have this problem to solve:
Use Barbalat’s Lemma to show that $\lim_{t\to \infty} x_1(t) = 0$ for the system:
$\dot x_1= − x_1 + x_1 x_2 $
$\dot x_2= − \gamma x_1^2$ ,
where $\gamma > 0$. Can we you anything about $x_2(t)$ based on this analysis?
First, I defined the following function:
$V(x_1, x_2)=\frac{1}{2} (x_1^2+x_2^2)$
This is clearly lower bounded. I then proceeded to check if $\dot V\leq0$. I get:
$\dot V(x_1, x_2)=x_1 \dot x_1+x_2 \dot x_2=-x_1^2-x_1^2x_2(\gamma-1)$
Here's where I can't proceed any further. The sign of $\dot V$ depends on the quantity $(\gamma-1)$, which I'm not told if it's always positive... I only know that $\gamma >0$. What do you guys suggest?
This system is a degenerate one; it has infinitely many connected poles due to degenerate linearized system.
It seems to have an attractor, but it doesn't. More precisely, if $|x_2-1|$>1 then $x_2$ starts to decrease because of increasing in $x_1$ until it traps in the region $|x_2-1|$<1 which can be easily proved by your Lyapunov candidate. the term $\gamma-1$ can be vanished by using some multipliers like $(1-x_2)x_1$ for $x_1>0$; and $(x_2-1)x_1$ for $x_1<0$. look at Multiplier Method for Lyapunov function.