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This is a homework problem from my adaptive control course:

Consider the IVP $\dot x(t) = -u(t)^2x(t)$ with $x(0) = x_0$. Suppose the system is exponentially stable. Show that there exist some $\epsilon, T > 0$ such that $$ \int_t^{t + T} u(\tau)^2d\tau \geqslant \epsilon T $$ for any $t > 0$.

Below is my attempt:

Knowing that the system is exponentially stable, there are some $m, \alpha > 0$ and $t_0\geqslant 0$ such that for any $t > t_0$, $$ \Vert x(t)\Vert = \exp\left( -\int_{t_0}^t u(\tau)^2d\tau \right)\Vert x_0\Vert \leqslant m\exp[-\alpha(t - t_0)]\Vert x_0\Vert $$ $$ \implies \int_{t_0}^t u(\tau)^2d\tau\geqslant \alpha(t - t_0)-\log m $$

But after this step I am not sure how to proceed to extract $\int_t^{t + t_0}u^2d\tau$ and bound the RHS below. Any hint or help will be greatly appreciated!

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  • $\begingroup$ the integral in your attempt probably starts at $t_0$ instead of $0$, right? $\endgroup$
    – Trb2
    Feb 19 at 22:16
  • $\begingroup$ @Trb2 I am not sure because the system starts at $t = 0$ and I just followed the definition of exponential stability... $\endgroup$
    – ArGenya
    Feb 19 at 22:18
  • $\begingroup$ @Trb2 Just checked my textbook and yes, you are right, the integral should start at $t_0$. Thanks for pointing this out! $\endgroup$
    – ArGenya
    Feb 19 at 22:34
  • $\begingroup$ You are pretty much done. The definition of exponential stability is for all initial times $t_0$ and final times $t$. $\endgroup$
    – KBS
    Feb 20 at 11:59
  • $\begingroup$ The condition must hold for any $t_0$ and $t$. So, select $t_0 = t$ and $t=t+T$. $\endgroup$
    – obareey
    Mar 1 at 7:30

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