Quasi-Solutions, Vector Lyapunov Functions and Monotone Method

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Quasi-Solutions, Vector Lyapunov Functions and Monotone Method

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Title: Quasi-Solutions, Vector Lyapunov Functions and Monotone Method
Author: Leela, S.; Lakshmikantham, V.; Oguztoreli, M. N.
Abstract: It is now well known that the method of vector Lyapunov functions provides an effective tool to investigate the properties of large scale interconnected dynamical and control systems. [3,4,5,11-15]. Several Lyapunov functions result in a natural way in the study of such systems by the decomposition and aggregation method [1-4,12-14]. However an unpleasant fact in this approach is the requirement of quasi-monotone property on the comparison system since comparison systems with a desired property like stability exist without satisfying quasi-monotone property. Also in the study of comparison theorems and extremal solutions for differential systems one usually imposes this quasi-monotone property. To avoid this difficulty two ideas are suggested recently, namely to use an appropriate cone other than [see pdf for notation] and to exploit the new notion of quasi-solutions [6,9,10], In [8] the idea of quasi-solutions is developed to some extent. In this paper we obtain further results on quasi-solutions. We introduce the notion of coupled quasi-solutions in addition to quasi-solutions. We show how the idea of quasi-solutions leads to isolated subsystems, obtain error estimates between solutions and quasi-solutions and develop monotone iterative techniques to obtain coupled maximal and minimal quasi-solutions.
Date: 1980-02

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