The equivalence of several set invariance conditions under saturation

Several equivalent conditions or statements for set invariance were obtained for systems with one saturating actuator in a recent paper. In particular, it was shown that the invariance of an ellipsoid under a saturated linear feedback is equivalent to its controlled invariance and also to the existence of a feedback linear inside the ellipsoid that makes it invariant. In this paper, we attempt to extend the results to systems with multiple saturating actuators. Our analysis reveals that the equivalence holds conditionally for some pairs of the statements and does not hold for some other pairs.