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!