  We consider the decidability of state-to-state reachability in linear
time-invariant control systems over continuous time. We analyse this problem
with respect to the allowable control sets, which are assumed to be the image
under a linear map of the unit hypercube. This naturally models bounded
(sometimes called saturated) controls. 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
decidability in two dimensions ($n=2$) if the matrix $A$ satisfies some
spectral conditions, and conditional decidablility in general. If the
transformation matrix $A$ is diagonal with rational entries (or rational
multiples of the same algebraic number) then the reachability problem is
decidable. If the transformation matrix $A$ only has real eigenvalues, the
reachability problem is conditionally decidable. The time-bounded reachability
problem is conditionally decidable, and unconditionally decidable in two
dimensions. Some of our decidability results are conditional in that they rely
on the decidability of certain mathematical theories, namely the theory of the
reals with exponential ($\mathfrak{R}_{\exp}$) and with bounded sine
($\mathfrak{R}_{\exp,\sin}$). We also obtain a hardness result for a mild
generalization of the problem where the target is simple set (hypercube of
dimension $n-1$ or hyperplane) instead of a point, and the control set is a
convex bounded polytope. In this case, we show that the problem is at least as
hard as the \emph{Continuous Positivity problem} or the \emph{Nontangential
Continuous Positivity problem}.
