39,305 research outputs found

    The Existence and Application of Unbounded Connected Components

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    Let X be a Banach space and Cn a family of connected subsets of R×X. We prove the existence of unbounded components in superior limit of {Cn}, denoted by lim¯ Cn, which have prescribed shapes. As applications, we investigate the global behavior of the set of positive periodic solutions to nonlinear first-order differential equations with delay, which can be used for modeling physiological processes

    Topology of quadrature domains

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    We address the problem of topology of quadrature domains, namely we give upper bounds on the connectivity of the domain in terms of the number of nodes and their multiplicities in the quadrature identity.Comment: 37 pages, 11 figures in J. Amer. Math. Soc., Published electronically: May 11, 201

    Small gain theorems for large scale systems and construction of ISS Lyapunov functions

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    We consider interconnections of n nonlinear subsystems in the input-to-state stability (ISS) framework. For each subsystem an ISS Lyapunov function is given that treats the other subsystems as independent inputs. A gain matrix is used to encode the mutual dependencies of the systems in the network. Under a small gain assumption on the monotone operator induced by the gain matrix, a locally Lipschitz continuous ISS Lyapunov function is obtained constructively for the entire network by appropriately scaling the individual Lyapunov functions for the subsystems. The results are obtained in a general formulation of ISS, the cases of summation, maximization and separation with respect to external gains are obtained as corollaries.Comment: provisionally accepted by SIAM Journal on Control and Optimizatio
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