Estabilidade do Sistema Não Linear

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Problem 9.28PP
A first-order nonlinear system is described by the equation jp, = -f{x), where 
^x) is a continuous and differentiable nonlinear function that satisfies the following;
m = o,
f(x) > 0, for X > 0. 
f(x) 0 for X 0
The given L3ra5>nov function is V (x) = —
To prove that the system is stable near the origin.
Step 2 of 2 ^
Find V [x)
r ( x ) = XX
"Whenx > 0 and / ( x ) > 0, we getf^(x) > 0 
Whenx 0 
Whenx = 0 and / ( x ) = 0, we getP^(x) = 0 
Here, for all x ^ 0, T plying Lys^novs stability criterion, we conclude that the system is stable.