Fastest stabilization time of a system of n integrators.

Question

Given a set $n$ integrators of the form $$\begin{align} \dot x_i &= x_{i+1}\\ \dot x_n &= u \end{align}$$ for $i = 1, 2, \ldots, n-1$, with nonzero initial conditions $x_0$, is it possible to determine the fastest time the system could be stabilized to origin, or even in a ball around it, while having the states remain inside a chosen closed set? And if yes, is an analytic solution/method possible, or would the results have to be numerical? By searching through optimal control bibliography, I've started leaning towards the latter. Keep in mind that the input is not constrained.