On the Control of Affine Systems with Safety Constraints: Relaxed   In-Block Controllability
  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. That are the cases in which we cannot fully control the affine systems within the interior of a given polytope, representing the intersection of given safety constraints. Instead, we introduce in this paper the notion of relaxed in-block controllability (RIBC), which can be useful for the cases where one can distinguish between soft and hard safety constraints. In particular, we study whether all the states in the interior of a given polytope, formed by the intersection of soft safety constraints, are mutually accessible through the interior of a given bigger polytope, formed by the intersection of hard safety constraints, by applying uniformly bounded control inputs. By exploring the geometry of the problem, we provide necessary conditions for RIBC. We then show when these conditions are also sufficient. Several illustrative examples are also given to clarify the main results. 
Mohamed K. Helwa, Peter E. Caines
cs.SY math.OC