This note addresses two problems: the characterization of positivity of autonomous descriptor systems and the problem of finding conditions for their stability. For these fundamental issues, we provide necessary and sufficient conditions complemented with a numerically reliable computational approach. The given conditions can be checked by means of linear programming (LP). Also, we derive a necessary and sufficient condition for stability that is connected to the underlying eigenstructure of the system.

The aim of this technical note is to provide an efficient control design technique for discrete-time positive periodic systems. In particular, stability, positivity and periodic invariance of such systems are studied. Moreover, the concept of periodic invariance with respect to a collection of boxes is introduced and investigated with connection to stability. It is shown how such concept can be used for deriving a stabilizing state-feedback control that maintains the positivity of the closed-loop system and respects states and control signals constraints. In addition, all the proposed results can be efficiently solved in terms of linear programming.