Asymmetric iISS and ISS Properties Yielding Stability of Interconnected Systems and An Application to A Circadian Model

This paper pursues effective utilization of input- to-state stability (ISS) and integral input-to-state stability (iISS) properties for establishing stability of nonlinear interconnected systems. The purpose is to develop sharp and simple stability criteria readily applicable to differential equations of practical models. This paper focuses on networks of one-dimensional systems, and shows how we can effectively solve state-dependent scaling problems leading to global asymptotic stability, iISS and ISS properties using asymmetric dissipation. The previous formulation of state-dependent scaling problems is relaxed by allowing dissipation asymmetric with respect to equilibria, and employing a sort of balancing property, which are often furnished by models of biological process on any level. As an illustration, the proposed approach is applied to a model of circadian rhythm in cells. A lower bound of transcription rate with which circadian oscillations are maintained is derived.