Given a controllable discrete-time linear system
$x(k+1) = A x(k) + B u(k)$
the input sequence leading from state $x_0$ to $x_f$ is given by $C^{-1} (x_f - A^n x_0)$ where $C$ is the controllability matrix.
Suppose we add linear state/input constraints to the system, i.e.,
$W \left( \begin{matrix} x(k) \\ u(k) \end{matrix} \right) \leq b ~~ \forall k$
Given a state pair $x_0$, $x_f$, is there a result by which we can determine whether a feasible input sequence exists to go from $x_0$ to $x_f$ and if so can the input sequence be determined independent of its length?
Note that the bounded version of this question, namely whether an input sequence of length $N$ exists that leads from $x_0$ to $x_f$ can be easily expressed as a linear program.