LIMIT SET DICHOTOMY AND CONVERGENCE OF SEMIFLOWS DEFINED BY COOPERATIVE STANDARD CNNs

Abstract

The paper analyzes some fundamental properties of the solution semiflow of nonsymmetric cooperative standard (S) cellular neural networks (CNNs) with a typical three-segment piecewise-linear (pwl) neuron activation. Two relevant subclasses of SCNNs, corresponding to one-dimensional circular SCNNs with two-sided or single-sided positive interconnections between nearest neighboring neurons only, are considered. For these subclasses it is shown that the associated solution semiflow satisfies the fundamental properties of the CONVERGENCE CRITERION, the NONORDERING OF LIMIT SETS and the LIMIT SET DICHOTOMY, and that this is true although the semiflow is not eventually strongly monotone. As a consequence such CNNs are almost convergent, i.e., almost all solutions converge toward an equilibrium point as time tends to infinity. To the authors' knowledge the paper is the first rigorous investigation on the geometry of limit sets and convergence properties of cooperative SCNNs with a pwl neuron activation. All available convergence results in the literature indeed concern a modified cooperative CNN model where the original pwl activation of the SCNN model is replaced by a continuously differentiable strictly increasing sigmoid function. The main results in the paper are established by conducting a deep analysis of the properties of the omega-limit sets of the solution semiflow defined by the considered subclasses of SCNNs. In doing so the paper exploits and extends some mathematical tools for monotone systems in order that they can be applied to pwl vector fields that govern the dynamics of SCNNs. By using some transformations and referring to specific examples it is also shown that the treatment in the paper can be extended to other subclasses of SCNNs.

Limit Set Dichotomy and Convergence of Cooperative Piecewise Linear Neural Networks

“…In fact the other existing convergence results in the literature concern cooperative NNs where the PL activation is replaced by a strictly increasing sigmoid activation. It is believed that there are other subclasses of irreducible matrices for which the semiflow generated by a cooperative PLNN still enjoys convergence properties as those established in this paper and in [35]. Probably we should proceed on a case by case basis in order to discover such subclasses and investigate their dynamical properties by ad hoc extensions of the mathematical machinery of monotone systems.…”

“…It has been shown that, although the semiflow is not ESM, it is possible to prove a form of the LIMIT SET DICHOTOMY and results on convergence analogous to those valid for ESM semiflows. To the authors' knowledge, the results here obtained and those in a recent paper [35] for a class of sparsely interconnected irreducible matrices, are the only available results on convergence of PLNNs. In fact the other existing convergence results in the literature concern cooperative NNs where the PL activation is replaced by a strictly increasing sigmoid activation.…”

“…2) To the authors' knowledge convergence of cooperative PLNNs have been investigated so far only in [35]. There, a class of one-dimensional circular standard cellular NNs defined by the cloning template , and a PL activation as in (2), is considered.…”

“…On the other hand, in this paper we are dealing with fully interconnected matrices, which are otherwise unstructured. The techniques used in [35] for addressing convergence exploit the very peculiar structure of the interconnection matrix (5) and they are different from those here employed for studying convergence in the completely interconnected case.…”

Exponential synchronization of discontinuous neural networks with time-varying mixed delays via state feedback and impulsive control

“…Although neural networks with piecewise-linear neuron activation or continuously differentiable strictly increasing sigmoid activation have important applications in signal and image processing (Di Marco et al 2010Kamel and Xia 2009;Cao and Wan 2014;Zhang et al 2014), it is reported that neural networks with discontinuous (or non-Lipschitz) neuron activations are an ideal model for neuron amplifiers with very high gain . For example, the sigmoidal neuron activations of the classical Hopfield network with high-gain amplifiers would approach a discontinuous hard-comparator function (Li et al 1989).…”