The speaker will be presenting some of the results from his DPhil thesis (2017, University of Oxford). In particular, the talk will be centered on the problem of determining point-to-point reachability for discrete linear time-invariant dynamical systems, when the set of controls is either a convex polyhedron or a finite union of convex polyhedra. The speaker will present a proof that the latter case is undecidable, by encoding Hilbert's Tenth Problem; time permitting, a proof of hardness of the former case will also be presented. The talk will be conducted in a self-contained way, and a background on elementary linear algebra should suffice to understand its contents.