A smallgain theory for infinite networks via infinitedimensional gain operators
Abstract
In this paper, we develop a new approach to study gain operators built from the interconnection gains of infinite networks of dynamical systems. Our focus is on the construction of paths of strict decay which are used for building Lyapunov functions for the network and thus proving various stability properties, including inputtostate stability. Our approach is based on the study of an augmented gain operator whose fixed points are precisely the points of decay for the original gain operator. We show that plenty of such fixed points exist under a uniform version of the nojointincrease condition. Using these fixed points to construct a path of strict decay, in general, requires specific dynamical properties of associated monotone operators. For particular types of gain operators such as maxtype operators and subadditive operators, these properties follow from uniform global asymptotic stability of the system induced by the original gain operator. This is consistent with former results in the literature which can readily be recovered from our theory.
 Publication:

arXiv eprints
 Pub Date:
 December 2021
 DOI:
 10.48550/arXiv.2112.05809
 arXiv:
 arXiv:2112.05809
 Bibcode:
 2021arXiv211205809K
 Keywords:

 Mathematics  Optimization and Control;
 Mathematics  Dynamical Systems;
 37B25;
 37L15;
 93D05;
 93A15