On the number of unstable eigenvalues of linear positive systems

This paper investigates the necessary conditions on the number of unstable eigenvalues of the single-input positive systems for stabilizability. For a continuous-time linear positive system, it has been known that if there exists a stabilizing linear time-invariant controller, the number of nonnegative real eigenvalues cannot be greater than one. For the discrete-time systems with nonnegative state constraints, it is demonstrated in this paper that there can exist at most one unstable real eigenvalue as well if every state can be stabilized by a linear time-invariant controller. Regarding the unstable complex eigenvalues, a specific three-dimensional system is studied under the positivity constraint. In addition, the optimal control problem is discussed for the positive double-integrator systems, where only one state needs to be stabilized and the other one is considered in a cost functional.