consider this linear, non-autonomic system:
$x_1 ̇=-x_1-f(t)(x_2-x_3 )$,
$\ x_2 ̇= -x_2+x_1$,
$x_3 ̇=-x_3-x_1$
where $f(t)$ is continuously differentiable and satisfies $0≤f'(t)≤f(t)≤k$ for all $0≤t $
I showed that $x=0$ is uniformly asymptotically stable which lead to that it is exponentially stable by using this lyaponove function
$V(x) = V^*(x_1,x_2,x_3) + f(t)V^*(0,x_2,x_3)$
$V^*(x_1,x_2,x_3) = \frac12(x_1^2+x_2^2+x_3^2)$
we can show that:
$\frac12||x||^2≤V(x)≤\frac12(1+k)||x||^2$
$V'(x)≤-\frac 1{10} ||x||^2$
I need to find $C,\alpha>0$ (at least one of them) in terms of k such that
$||x(t)||≤Ce^{-\alpha t}||x(0)||$
but Im having trouble to find them.
can someone show me how to find $C$ or $\alpha$ in terms of k?
reminder: $||x(t)||^2 = (x_1^2+x_2^2+x_3^2)$
thanks!