Boundedness of the input $$
\dot x(t)=Ax(t)+Bu(t),\quad t\in[0,T]\\
x(0)=x_0\\
x(t)\text{ is bounded on }t\in[0,T]\\
A\text{ and }B\text{ are given constants},\quad B\neq 0
$$
My question:

Is $u(t)$ also bounded on $t\in[0,T]$?

Thanks in advance.

Let $x(0)=0$ and $x(t)=t^2\cos (t^{-2})$ for $t\ne 0$.The function $x(t)$ is differentiable at every $t$ but $x'(t)$ is unbounded on $[0,T]$ for any $T>0 $ . (Consider $t=1/\sqrt {2\pi n}$ for $n\in N.$) Note that $x'(t)$ is not continuous at $t=0$. 
