Lojasiewicz inequality and exponential convergence of the full-range model of CNNs

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 1/2. 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.