Matrix Analysis Techniques With Applications In The Stability Studies Of Cellular Neural Networks

In this thesis, we study properties of some special matrices, including M-matrices,H-matrices, nonnegative matrices and additively diagonally stable matrices, etc. On thebasis of these matrices, we also study the stability of cellular neural networks with timevaryingdelays. This thesis consists of four parts with six chapters.Two special classes of matrices are studied: additively diagonally stable matricesand M-matrices. On the basis of the previous work, some new properties of additivelydiagonally stable matrices are obtained, which complement the theoretical studies of thiskind of matrices. We propose a linear system based approach for solving some relatedproblems concerning M-matrices: (1) Computing‖A^-1‖_∞; (2) Computing the Skeelcondition number of A; (3) For any given diagonal dominance vector, finding a suitablepositive diagonal matrix D such that AD is a strictly diagonally dominant matrix; (4)Computing A^-1. The proposed linear system can be implemented by application-specificintegrated circuits, which have asynchronous parallel processing ability and can achievehigh computing performance. Some numerical examples and computer simulations aregiven to show the effectiveness of the proposed approach.The global exponential stability of cellular neural networks with time-varying delaysis investigated. Based on the properties of nonnegative matrices and Lyapunov stabilitytheory, a sufficient condition for global exponential stability of neural networks with delaysis obtained, and the degree of exponential stability is estimated. Theoretic analysisshows that the obtained result generalizes and unifies some previous results derived in theliteratures, and the numerical simulations justify the effectiveness of the obtained result.By the obtained result, we introduce the research results on the estimation for spectralradius of nonnegative matrices into the criteria for the stability of neural networks.The global robust stability of interval cellular neural networks with time-varyingdelays is studied. By using nonnegative matrix theory, Halanay inequality and Lyapunov-Razumikhintechnique, some new criteria for global robust stability are presented. Theoreticanalysis shows that the obtained results generalize and include some previous resultsderived in the literatures, and complement the results concerning the robust stability researchof neural networks effectively. Some of the obtained results can be formulated as a semidefinite programming problem, which can bc verified easily in the practical application.Numerical simulations verify the effectiveness of the obtained results.An analytical method based on H-matrices and Schur complements is proposed tostudy the global robust stability of interval ccllular neural networks with multiple timevaryingdelays. By constructing proper Lyapunov functionals, some novel criteria forglobal robust stability arc obtained. Some of the obtained results are linear matrix inequalityconditions, and can be verified by LMI toolbox of MATLAB. Theoretic analysisand numerical examples show that the obtained results include some previous resultsderived in the literatures. Some computer simulations verify the effectiveness of the obtainedresults. Moreover, two results derived in the recent literatures are discussed. Bysome numerical examples, we point out some errors in the proof, and present the modificdversions of these results.