I'm having hard times with the following simple linear ODE coming from a control problem:

$\displaystyle u(t)' \leq \alpha(t) - u(t)\,,\quad u(0) = u_0 > 0$

with a given smooth $\displaystyle \alpha(t)$ satisfying

$\displaystyle 0 \leq \alpha(t) \leq u(t)$ for all $\displaystyle t\geq 0$.

My intuition is that $\displaystyle \lim_{t\to\infty} u(t) - \alpha(t) = 0$, and that the convergence is exponential, i.e., $\displaystyle |u(t) - \alpha(t)| = u(t) - \alpha(t) \leq c_1 e^{-c_2 t}$.
For instance, if $\displaystyle \alpha$ was a constant, then the exponential convergence clearly holds.
Do you see a simple proof for time-dependent $\displaystyle \alpha$ (I'm getting grey, but could not prove it), or is my intuition wrong?

Many thanks, Peter