On general relations between null-controllable and controlled invariant sets for linear constrained systems

We prove some general relations between null-controllable and controlled invariant sets for linear systems with input and state constraints.We show that the closure of the largest null-controllable set is identical to the largest controlled invariant set. In order to prove this claim, we demonstrate that the interior of every controlled invariant set is null-controllable in the linear case. While some of these properties appear to be obvious, formal proofs are missing to the best of the authors´ knowledge. To highlight the importance of careful proofs, we show that these properties are specific to linear systems and generally do not hold in the nonlinear case.