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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.

This paper considers a class of nonsymmetric cooperative neural networks (NNs) where the neurons are fully interconnected and the neuron activations are modeled by piecewise linear (PL) functions. The solution semiflow generated by cooperative PLNNs is monotone but, due to the horizontal segments in the neuron activations, is not eventually strongly monotone (ESM). The main result in this paper is that it is possible to prove a peculiar form of the LIMIT SET DICHOTOMY for this class of cooperative PLNNs. Such a form is slightly weaker than the standard form valid for ESM semiflows, but this notwithstanding it permits to establish a result on convergence analogous to that valid for ESM semiflows. Namely, for almost every choice of the initial conditions, each solution of a fully interconnected cooperative PLNN converges toward an equilibrium point, depending on the initial conditions, as + . From a methodological viewpoint, this paper extends some basic techniques and tools valid for ESM semiflows, in order that they can be applied to the monotone semiflows generated by the considered class of cooperative PLNNs.Index Terms-Convergence, cooperative neural networks, dynamical systems, limit set dichotomy, monotone and eventually strongly monotone semiflows.

SUMMARYThis paper compares the dynamical behaviour of the standard (S) cellular neural networks (CNNs) and the full-range (FR) CNNs, when the two CNN models are characterized by the same set of parameters (interconnections and inputs). The FR-CNNs are assumed to be characterized by ideal hard-limiter nonlinearities with two vertical segments in the i-v characteristic. The main result is that some basic conditions ensuring global exponential stability (GES) of the unique equilibrium point of S-CNNs, with or without delay, continue to ensure the same property for FR-CNNs for the same set of parameters. The significance of this result is discussed with respect to the results in a paper by Corinto and Gilli addressing the similarity of the qualitative behaviour of S-CNNs and FR-CNNs. FR-CNNs are analysed in this paper from a rigorous mathematical viewpoint by means of theoretical tools from set-valued analysis and differential inclusions. In particular, GES is investigated via an extended Lyapunov approach that is applicable to the differential inclusion describing the dynamics of FR-CNNs.