Feedback and positive feedback stabilizability and holdability

Consider the following problem. Given a discrete-time linear system, find, if possible, linear state-feedback control laws such that under these laws all trajectories originating in the non-negative orthant of the state space remain non-negative while asymptotically deteriorating to the origin. This problem is called feedback stabilizability-holdability problem (FSH). If, in addition, the requirement of nonnegativity is imposed on controls, the problem is a positive feedback stabilizabilityholdability (PFSH}. The feedback stabilizability-holdability problem has been studied for continuous-time linear systems with scalar controls in [1, 2], where conditions requiring that all principal minors of the closed-loop system matrix are positive or, equivalently, the closed-loop system matrix is a non-singular M-matrix have been obtained. Related results are given in [3] for discrete-time periodic linear systems. Stabilization of positive linear systems by state-feedback is considered in [5] but the author does not take into account the restriction on non-negativity of controls in the closed loop. This restriction is essential because the control variables in many real-life systems represent quantities, which do not have meaning unless being non-negative. In all the aforementioned works the requirement on positivity (non-negativity) of controls in the closed loop is not considered. The geometry of the problems and the related computational aspects are not at all exposed either. At the same time the requirement on positivity of controls is important since it imposes additional restrictions on the closed-loop system dynamics. The computational aspects of the problem are also quite important for the design of stable and holdable systems.