# Linear control ODE - exponential convergence?

 P: 1 Hello, I'm having hard times with the following simple linear ODE coming from a control problem: $$u(t)' \leq \alpha(t) - u(t)\,,\quad u(0) = u_0 > 0$$ with a given smooth α(t) satisfying $$0 \leq \alpha(t) \leq u(t) \quad\mbox{for all } t\geq 0.$$ My intuition is that $$\lim_{t\to\infty} u(t) - \alpha(t) = 0,$$ and that the convergence is exponential, i.e., $$|u(t) - \alpha(t)| = u(t) - \alpha(t) \leq c_1 e^{-c_2 t}.$$ For instance, if α was a constant, then the exponential convergence clearly holds (just solve the related ODE and use a "maximum principle"). Do you see a simple proof for time-dependent α (could not prove neither of the "statements" - probably I'm missing something very elementary); or is my intuition wrong? Many thanks, Peter