Q:

state variable response of non-homogenuous ODE

The following one-dimensional ODE is considered:
$\dot x(t) = -a \, x(t) + u(t),\quad x(0)=x_0,$
with $a>0$.
The solution is given as
$x(t) = x_0\exp(-a\,t) + \exp(-a\,t)\int_0^\tau\exp(a\,\tau)u(\tau)\,d\tau$.
I want analyse the behavior of $x$ for $a\to\infty$. Intuitively it is clear that for large $a$ and if $u(t)$ does not vary too fast, the solution $x(t) \approx \frac{1}{a}u(t)$. The larger $a$ is, the better $x\,a$ approaches $u(t)$. In simulation this can be easily verified.
Can soemone give me a hint how to analyse this observation mathematically? More clearly how to relate the solution $x(t)$ with $a$ and the variation of $u(t)$?
I tried to consider fast and slow time-scales, however, so far I stucked in the deviations.

A:

In the explicit form of the solution
$$
x(t) = x_0 e^{-at} + e^{-at} \int_0^t e^{a\tau} u(\tau)\,d\tau
$$
make the substitution $\tau = t - s$ to get
$$
x(t) = x_0 e^{-at} + \int_0^t e^{-as}u(t-s)\,ds
$$
then apply Watson's lemma to approximate the integral.

