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M ‐matrices and global convergence of discontinuous neural networks
Author(s) -
Forti Mauro
Publication year - 2006
Publication title -
international journal of circuit theory and applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.364
H-Index - 52
eISSN - 1097-007X
pISSN - 0098-9886
DOI - 10.1002/cta.381
Subject(s) - uniqueness , lemma (botany) , artificial neural network , convergence (economics) , mathematics , equilibrium point , class (philosophy) , differential equation , lyapunov function , topology (electrical circuits) , computer science , mathematical analysis , combinatorics , nonlinear system , physics , ecology , poaceae , quantum mechanics , machine learning , artificial intelligence , economics , biology , economic growth
The paper considers a general class of neural networks possessing discontinuous neuron activations and neuron interconnection matrices belonging to the class of M ‐matrices or H ‐matrices. A number of results are established on global exponential convergence of the state and output solutions towards a unique equilibrium point. Moreover, by exploiting the presence of sliding modes, conditions are given under which convergence in finite time is guaranteed. In all cases, the exponential convergence rate, or the finite convergence time, can be quantitatively estimated on the basis of the parameters defining the neural network. As a by‐product, it is proved that the considered neural networks, although they are described by a system of differential equations with discontinuous right‐hand side, enjoy the property of uniqueness of the solution starting at a given initial condition. The results are proved by a generalized Lyapunov‐like approach and by using tools from the theory of differential equations with discontinuous right‐hand side. At the core of the approach is a basic lemma, which holds under the assumption of M ‐matrices or H ‐matrices, and enables to study the limiting behaviour of a suitably defined distance between any pair of solutions to the neural network. Copyright © 2006 John Wiley & Sons, Ltd.