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Diagonal stability of a class of cyclic systems and its connection with the secant criterion. (English) Zbl 1132.39002
Authors’ abstract: We consider a class of systems with a cyclic interconnection structure that arises, among other examples, in dynamic models for certain biochemical reactions. We first show that a “secant” criterion for local stability, derived earlier in the literature, is in fact a necessary and sufficient condition for diagonal stability of the corresponding class of matrices. We then revisit a recent generalization of this criterion to output strictly passive systems, and recover the same stability condition using our diagonal stability result as a tool for constructing a Lyapunov function. Using this procedure for Lyapunov construction we exhibit classes of cyclic systems with sector nonlinearities and characterize their global stability properties.

##### MSC:
 39A11 Stability of difference equations (MSC2000) 93C55 Discrete-time control systems 93D25 Input-output approaches to stability of control systems 92E20 Classical flows, reactions, etc. 39A12 Discrete version of topics in analysis
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