A uniqueness theorem for linear control systems with coinciding reachable sets

Two multivariable linear control systems are considered with control $u(t)$ satisfying the inequality $\| {u(t)} \|_p \leqq 1$, $1 \leqq p \leqq \infty $, and with coinciding reachable sets. Under certain conditions (of which $p \ne 2$ seems to be the most remarkable), it is shown that the control systems have equal system matrices and equal control matrices up to the signs and the ordering of the columns. The proof depends on a theorem of Banach on rotations.