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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

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# Linear control ODE - exponential convergence?

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