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Stability of Interconnections of ISS Systems

Interconnections of several nonlinear systems with inputs are considered in this paper. Each system is assumed to be input-to-state stable (ISS). However an interconnection of such systems is not stable in general. We provide a stability condition for interconnections of such systems. Some interpretations of this condition are given. Moreover we show how an ISS-Lyapunov function for such a network can be constructed explicitly if this condition is satisfied. Further we give the idea of numerical verification of the small gain condition for networks. Nonlinear systems appear frequently in applications and their stability properties are important for design and performance of such systems. We consider several nonlinear stable systems and refer to the question, whether an interconnection of these systems is also stable. First consider one nonlinear system _