Barbalat's lemma for Stability Analysis Good day,
We have: 
Lyapunov-Like Lemma: If a scaler function V(t, x) satisfies the
following conditions:


*

*$V(t,x)$ is lower bounded

*$\dot{V}(t,x)$ is negative semi-definite

*$\dot{V}(t,x)$ is uniformly continuous in time
then $\dot{V}(t,x) \to 0$ as $t \to \infty $.


Now if we have the following system:
$\dot{e} = -e + \theta w(t) \\
 \dot{\theta} = -e w(t)$
and assume that $w(t)$ is a bounded function, then we can select the following Lyapunov function:
$V(x,t) = e^2 + \theta^2$
Taking the time derivative:
$\dot{V}(x,t) = -2e^2 \leq 0$
Taking the time derivative again:
$\ddot{V}(x,t) = -4e(-e+\theta w)$
Now $\ddot{V}(x,t)$ satisfies condition (3) when $e$ and $\theta$ are bounded, but how can I be sure that these two variables are indeed bounded? Should I perform some other test, or...?

Since $\dot{V} = -2e^2 \leq 0$, from the Lyapunov stability theory, one concludes that the system states $(e,\theta)$ are bounded. 
Observe above that $\dot{V}$ is a function of only one state ($e$); if it were a function of the two states $(e,\theta)$ and $\dot{V} < 0$ (except in $e=\theta = 0$, in that case $V=0$), one would conclude asymptotic stability. 
