I have been struggling with an ostensibly simple problem, that is how to apply perturbation analysis principles on a system of linear differential equations with linear perturbation of the following form:
$$ \dot{x}=\frac{1}{\epsilon} a_0 x + a_1 z + bu\\ \dot{z} = c_0 x + c_1 z $$
where $x,z,u\in\Re$ and $0<\epsilon<<1$. How can I decompose my system and write the solution $x(t),z(t)$ as an asymptotic expansion in $\epsilon$ ?
Update 1: Let us consider the case where $t=\mathcal{O}(\epsilon)$. Then, put $\tau=\frac{t}{\epsilon}=\mathcal{O}(1)$ and the above system becomes:
$$ \frac{d x(\tau)}{d\tau}= a_0 x(\tau) + \epsilon a_1 z(\tau) + \epsilon bu\\ \frac{d z(\tau)}{d\tau} = \epsilon c_0 x + \epsilon c_1 z $$
and let us now set $x(\tau)=x(\frac{t}{\epsilon})=x_0(\tau)+\epsilon x_1(\tau)+\mathcal{O}(\epsilon^2)$ and similarly $z(\tau)=z_0(\tau)+\epsilon z_1(\tau)+\mathcal{O}(\epsilon^2)$. Then we have the following inner system:
$$ \frac{d x_0}{d\tau}=a_0 x_0(\tau)\\ \frac{d x_1}{d\tau}=a_0 x_1(\tau) + a_1 z_0(\tau) + bu(\tau)\\ \frac{d z_0}{d\tau}=0\\ \frac{d z_1}{d\tau}=c_0 x_0(\tau) + c_1 z_0(\tau) $$
and then we may apply matching to both the outer (see @Jon's answer below) and the inner solution.
Question : Can it be considered expedient to consider an asymptotic expansion of the input variable $u$ like $u(t)=\frac{1}{\epsilon}\sum_{i\geq 0}\epsilon^i u_i(t)$?
I would do the following. Just put $\lambda=\frac{1}{\epsilon}$ and now $\lambda\gg 1$. Now you will get
$$ \lambda\dot{x}= a_0 x + \lambda a_1 z + \lambda bu\\ \dot{z} = c_0 x + c_1 z. $$
Then put
$$x(t)=x_0(t)+\frac{1}{\lambda}x_1(t)+\frac{1}{\lambda^2}x_2(t)+\ldots$$
$$z(t)=z_0(t)+\frac{1}{\lambda}z_1(t)+\frac{1}{\lambda^2}z_2(t)+\ldots$$
that gives the first few equations for the perturbation series
$$ \dot{x_0}= a_1 z_0 + bu\\ \dot{z}_0 = c_0 x_0 + c_1 z_0. $$
$$ \dot{x_1}= a_0 x_0 + a_1 z_1\\ \dot{z}_1 = c_0 x_1 + c_1 z_1. $$
and so on.