The reachable set of a linear system under control constraints

We investigate the structure of the reachable set for a discrete time linear control system Xk+1 =A xk +B Uk when the control inputs (Uk)k=0,1 required to lie in a given set U. We first study the two-dimensional case when U=[-M, M] and give explicit formulas for the equations of the sides, the lengths of the sides, the nature of the comers, etc. Then we generalize these results to higher dimensional cases when the control set U is an arbitrary polyhedron.