We think about the decidability of condition-to-condition reachability in straight line time-invariant control systems over continuous time. We analyse this issue with regards to the allowable control sets, that are assumed is the image within straight line map from the unit hypercube. This naturally models bounded (sometimes known as saturated) controls. Decidability from the form of the reachability condition in which control sets are affine subspaces of $mathbb^n$ is really a fundamental lead to control theory. Our first outcome is decidability in 2 dimensions ($n=2$) when the matrix $A$ satisfies some spectral conditions, and conditional decidablility generally. When the transformation matrix $A$ is diagonal with rational records (or rational multiples of the identical algebraic number) then your reachability issue is decidable. When the transformation matrix $A$ has only real eigenvalues, the reachability issue is conditionally decidable. Time-bounded reachability issue is conditionally decidable, and unconditionally decidable in 2 dimensions. A lot of our decidability answers are conditional for the reason that they depend around the decidability of certain mathematical theories, namely the idea from the reals with exponential ($mathfrak_$) with bounded sine ($mathfrak_crime$). We get yourself a hardness result for any mild generalization from the problem in which the target is straightforward set (hypercube of dimension $n-1$ or hyperplane) rather of the point, and also the control set is really a convex bounded polytope. Within this situation, we reveal that the issue is a minimum of as hard because the emph or even the emph.