On the Decay Rates of Homogeneous Positive Systems of Any Degree With Time-Varying Delays

This technical note studies the stability problem of homogeneous positive systems of any degree with time-varying delays. Delay-independent conditions are derived for asymptotic and finite-time stability. Estimates on the decay rates, which reveal how the system delays affect the rates of convergence, are obtained. More precisely, this technical note features three contributions. First, we derive a necessary and sufficient condition for global polynomial stability of continuous-time homogeneous cooperative systems with time-varying delays when the degree of homogeneity is greater than one. Second, we characterize finite-time stability of continuous-time homogeneous cooperative delay-free systems of degree smaller than one. Finally, for discrete-time positive systems with time-varying delays, a local exponential stability criterion is established when the vector fields are order-preserving and homogeneous of degree greater than one. An illustrative example is given to show the effectiveness of our results.