Consider a linear time-invariant control system given by the differential equation \begin{align*} \dot{x}(t) &= Ax(t) + Bu(t), \;\; x(0) = x_0, \end{align*} where $x\in \mathbb{R}^n$ and $A,B$ are matrices. The standard variation of constant formula gives \begin{align*} x(t) = e^{At}x_0 + \int_{0}^t e^{A(t-\tau)}Bu(\tau)d\tau. \end{align*} One classical question that one asks is what points $x(T)$ one can reach by using a control signal/function $u: [0,T] \mapsto \mathbb R^p$. The answer is of course given by the test of Kalman (checking the rank of the controllability matrix).
Question: If we restrict the input signals to constant signals (!), what can be then said about the reachable points in finite time (!) ?