Linear positive systems may have a reachable subset from the origin that is either polyhedral or nonpolyhedral

Reference:
Y. Zeinaly, J.H. van Schuppen, and B. De Schutter, "Linear positive systems may have a reachable subset from the origin that is either polyhedral or nonpolyhedral," SIAM Journal on Matrix Analysis and Applications, vol. 41, no. 1, pp. 297-307, 2020.

Abstract:
Positive systems with positive inputs and positive outputs are used in several branches of engineering, biochemistry, and economics. Both control theory and system theory require the concept of reachability of a time-invariant discrete-time linear positive system. The subset of the state set that is reachable from the origin is therefore of interest. The reachable subset is in general a cone in the positive vector space of the positive real numbers. It is established in this paper that the reachable subset can be either a polyhedral or a nonpolyhedral cone. For a single-input case, a characterization is provided of when the infinite-time and the finite-time reachable subset are polyhedral. An example is provided for which the reachable subset is nonpolyhedral. Finally, for the case of polyhedral reachable subset(s), a method is provided to verify if a target set can be reached from the origin using positive inputs.