  We consider the decidability of state-to-state reachability in linear
time-invariant control systems over discrete time. We analyse this problem with
respect to the allowable control sets, which in general are assumed to be
defined by boolean combinations of linear inequalities. Decidability of the
version of the reachability problem in which control sets are affine subspaces
of $\mathbb{R}^n$ is a fundamental result in control theory. Our first result
is that reachability is undecidable if the set of controls is a finite union of
affine subspaces. We also consider versions of the reachability problem in
which (i)~the set of controls consists of a single affine subspace together
with the origin and (ii)~the set of controls is a convex polytope. In these two
cases we respectively show that the reachability problem is as hard as Skolem's
Problem and the Positivity Problem for linear recurrence sequences (whose
decidability has been open for several decades). Our main contribution is to
show decidability of a version of the reachability problem in which control
sets are convex polytopes, under certain spectral assumptions on the transition
matrix.
