Relaxed in-block controllability of affine systems on polytopes

We consider affine systems defined on polytopes and study the cases where the systems are not in-block controllable with respect to the given polytopes. Instead, we introduce in this paper the notion of relaxed in-block controllability (RIBC). In particular, we study whether all the states in the interior of a given polytope are mutually accessible through the interior of a given bigger polytope by applying uniformly bounded control inputs. By exploring the geometry of the problem, we provide necessary conditions for RIBC. Then, we present where these conditions are also sufficient. Several illustrative examples are also given to clarify the main results.