Stability results for some classes of cooperative systems

This paper deals with the constant control problem for homogeneous cooperative and irreducible systems. These systems serve as models for positive systems. A necessary and sufficient condition for global asymptotic stability of the zero solution of this class of systems is known. Adding a constant control allows one to shift, the equilibrium point from zero to a point in the first orthant. We prove that for every nontrivial nonnegative control vector a unique nontrivial equilibrium point is achieved which is globally asymptotically stable if the zero solution of the uncontrolled system is globally, asymptotically stable. Additionally, a stability result for a particular class of Kolmogorov systems is established. We compare our main results to those in the literature.