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We consider affine systems defined on polytopes and study the cases where thesystems are not in-block controllable with respect to the given polytopes. Thatare the cases in which we cannot fully control the affine systems within theinterior of a given polytope, representing the intersection of given safetyconstraints. Instead, we introduce in this paper the notion of relaxed in-blockcontrollability (RIBC), which can be useful for the cases where one candistinguish between soft and hard safety constraints. In particular, we studywhether all the states in the interior of a given polytope, formed by theintersection of soft safety constraints, are mutually accessible through theinterior of a given bigger polytope, formed by the intersection of hard safetyconstraints, by applying uniformly bounded control inputs. By exploring thegeometry of the problem, we provide necessary conditions for RIBC. We then showwhen these conditions are also sufficient. Several illustrative examples arealso given to clarify the main results.

On the Control of Affine Systems with Safety Constraints: Relaxed In-Block Controllability