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Łojasiewicz inequality and exponential convergence of the full‐range model of CNNs
Author(s) -
Di Marco Mauro,
Forti Mauro,
Grazzini Massimo,
Pancioni Luca
Publication year - 2012
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.717
Subject(s) - convergence (economics) , exponent , mathematics , degenerate energy levels , range (aeronautics) , equilibrium point , exponential function , ideal (ethics) , nonlinear system , mathematical analysis , matrix (chemical analysis) , interconnection , point (geometry) , physics , computer science , telecommunications , geometry , law , economics , quantum mechanics , composite material , differential equation , economic growth , linguistics , philosophy , materials science , political science
SUMMARY This paper considers the Full‐range (FR) model of Cellular Neural Networks (CNNs) in the case where the signal range is delimited by an ideal hard‐limiter nonlinearity with two vertical segments in the i − v characteristic. A Łojasiewicz inequality around any equilibrium point, for a FRCNN with a symmetric interconnection matrix, is proved. It is also shown that the Łojasiewicz exponent is equal to . The main consequence is that any forward solution of a symmetric FRCNN has finite length and is exponentially convergent toward an equilibrium point, even in degenerate situations where the FRCNN possesses non‐isolated equilibrium points. The obtained results are shown to improve the previous results in literature on convergence or almost convergence of symmetric FRCNNs. Copyright © 2010 John Wiley & Sons, Ltd.