GRAPH-THEORETIC STABILITY CONDITIONS FOR METZLER MATRICES AND MONOTONE SYSTEMS

This paper studies the graph-theoretic conditions for stability of positive monotone systems. Using concepts from input-to-state stability and network small-gain theory, we first establish necessary and sufficient conditions for the stability of linear positive systems described by Metzler matrices. Specifically, we derive two sets of stability conditions based on two forms of input-to state stability gains for Metzler systems, namely max-interconnection gains and sum-interconnection gains. Based on the max-interconnection gains, we show that the cyclic small-gain theorem becomes necessary and sufficient for the stability of Metzler systems; based on the sum-interconnection gains, we obtain novel graph-theoretic conditions for the stability of Metzler systems. All these conditions highlight the role of cycles in the interconnection graph and unveil how the structural properties of the graph affect stability. Finally, we extend our results to the nonlinear monotone system and obtain similar sufficient conditions for global asymptotic stability.

Automated summary

This study investigates the stability of positive monotone systems using graph-theoretic conditions. Necessary and sufficient conditions for the stability of linear positive systems described by Metzler matrices are established, with novel graph-theoretic conditions obtained for the stability of Metzler systems. The results are also extended to nonlinear monotone systems with similar sufficient conditions for global asymptotic stability.