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.