I have a question regarding Shankar Sastry's text "Nonlinear Systems, Analysis, Stability and Control", Page 199, Spring-mass system with damper
The equations that describe this system is,
$\dot x_1 = x_2$
$\dot x_2 = -f(x_2) - g(x_1)$,
where, $f$ and $g$ are assumed to satisfy, $\sigma f(\sigma) \geq 0, \forall \sigma \in [-\sigma_0, \sigma_0]$.
The author wishes to show that the system converges to an equilibrium state.
So he proposes a Lyapunov function,
$v(x_1, x_2) = x_2^2/2 + \int_0^{x_1} g(\sigma) d\sigma$
It is easy to show (and I agree),
$\dot v(x_1, x_2) = -x_2 f(x_2) \leq 0$ for $x_2 \in [-\sigma_0, \sigma_0]$.
Now here's the "God move" by the author,
choose $c = \min(v(-\sigma_0, 0), v(\sigma_0, 0))$,
Then the author claims: $\dot v \leq 0$ for $x \in \Omega_c = \{(x_1, x_2): v(x_1, x_2) \leq c\}$.
As a consequence of LaSalle's principle, the trajectory enters the largest invariant set in $\Omega_c \cap \{(x_1, x_2): \dot v = 0\} = \Omega_c \cap \{x_1, 0\}$.
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My question is why is it that by choosing $c = \min(v(-\sigma_0, 0), v(\sigma_0, 0))$, we can conclude that $\dot v \leq 0$ for $x$ in $\Omega_c$. I have been staring at this for a while now and could not really understand how this works.