# Necessary condition for exponential stability of an LTV system

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!

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