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Some relations between stability and smoothness in discrete-time dynamic models

Many discrete-time dynamic models of current interest are based on functions that, while generally continuous, are nonsmooth; examples include specific multimodels, hinging hyperplane models, and hybrid systems. We consider two models for which we can vary the smoothness and examine its influence on qualitative behavior. In the smooth regime, both models exhibit asymptotic stability for sufficiently small amplitude inputs; in the nonsmooth regime, the simpler model is shown to be BIBO stable but not asymptotically stable, and in both models nonlinear effects become more pronounced as the input amplitude decreases, in marked contrast to the behavior of smooth (i.e., linearizable) systems. Further, in the case of the simpler model the general character of this behavior in the nonsmooth regime cannot be changed with linear proportional feedback.