Non-uniform Small-gain Theorems for Systems with Critical and Slow Relaxations

We consider the problem of small-gain analysis of asymptotic behavior in interconnected nonlinear dynamic systems. Mathematical models of these systems are allowed to be uncertain and time-varying. In contrast to standard small-gain theorems that require global asymptotic stability of each interacting component in the absence of inputs, we consider interconnections of systems that can be critically stable and have infinite input-output L8 gains. For this class of systems we derive small-gain conditions specifying state boundedness of the interconnection. The estimates of the domain in which the system’s state remains are also provided. Conditions that follow from the main results of our paper are non-uniform in space. That is they hold generally only for a set of initial conditions in the system’s state space. We show that under some mild continuity restrictions this set has a non-zero volume, hence such bounded yet potentially globally unstable motions are realizable with a non-zero probability. Proposed results can be used for the design and analysis of intermittent, itinerant and metastable dynamics which is the case in the domains of control of chemical kinetics, biological and complex physical systems, and non-linear optimization. The main results are illustrated with simple examples, and relation of our results with the standard small-gain conditions is discussed.