Massimo Grazzini

Limit Set Dichotomy and Multistability for a Class of Cooperative Neural Networks With Delays

Recent papers have pointed out the interest to study convergence in the presence of multiple equilibrium points (EPs) (multistability) for neural networks (NNs) with nonsymmetric cooperative (nonnegative) interconnections and neuron activations modeled by piecewise linear (PL) functions. One basic difficulty is that the semiflows generated by such NNs are monotone but, due to the horizontal segments in the PL functions, are not eventually strongly monotone (ESM). This notwithstanding, it has been shown that there are subclasses of irreducible interconnection matrices for which the semiflows, although they are not ESM, enjoy convergence properties similar to those of ESM semiflows. The results obtained so far concern the case of cooperative NNs without delays. The goal of this paper is to extend some of the existing results to the relevant case of NNs with delays. More specifically, this paper considers a class of NNs with PL neuron activations, concentrated delays, and a nonsymmetric cooperative interconnection matrix A and delay interconnection matrix Aτ. The main result is that when A+Aτ satisfies a full interconnection condition, then the generated semiflows, which are monotone but not ESM, satisfy a limit set dichotomy analogous to that valid for ESM semiflows. It follows that there is an open and dense set of initial conditions, in the state space of continuous functions on a compact interval, for which the solutions converge toward an EP. The result holds in the general case where the NNs possess multiple EPs, i.e., is a result on multistability, and is valid for any constant value of the delays.

Lyapunov Method and Convergence of the Full-Range Model of CNNs

This paper develops a Lyapunov approach for studying convergence and stability of a class of differential inclusions termed differential variational inequalities (DVIs). The DVIs describe the dynamics of a general system evolving in a compact convex subset of the state space. In particular, they include the dynamics of the full-range (FR) model of cellular neural networks (CNNs), which is characterized by hard-limiter nonlinearities with vertical segments in the i-v characteristic. The approach is based on the following two main tools: 1) a set-valued derivative, which enables to compute the evolution of a Lyapunov function along the solutions of the DVIs without involving integrations, and 2) an extended version of LaSalles invariance principle, which permits to study the limiting behavior of the solutions with respect to the invariant sets of the DVIs. Then, this paper establishes conditions for convergence (complete stability) of DVIs in the presence of multiple equilibrium points (EPs), global asymptotic stability (GAS), and global exponential stability (GES) of the unique EP. These conditions are applied to investigate convergence, GAS, and GES for FR-CNNs and some extended classes of FR-CNNs. It is shown that, by means of the techniques developed in this paper, the analysis of convergence and stability of FR-CNNs is no more difficult than that of the standard (S)-CNNs. In addition, there are significant cases, such as the symmetric FR-CNNs and the nonsymmetric FR-CNNs with a Lyapunov diagonally stable matrix, where the proof of convergence or global stability is much simpler than that of the S-CNNs.

Limit Set Dichotomy and Convergence of Cooperative Piecewise Linear Neural Networks

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 t → +∞. 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.

Necessary and sufficient condition for multistability of neural networks evolving on a closed hypercube.

The paper considers nonsmooth neural networks described by a class of differential inclusions termed differential variational inequalities (DVIs). The DVIs include the relevant class of neural networks, introduced by Li, Michel and Porod, described by linear systems evolving in a closed hypercube of R(n). The main result in the paper is a necessary and sufficient condition for multistability of DVIs with nonsymmetric and cooperative (nonnegative) interconnections between neurons. The condition is easily checkable and provides a sharp bound between DVIs that can store multiple patterns, as asymptotically stable equilibria, and those for which this is not possible. Numerical examples and simulations are presented to confirm and illustrate the theoretic findings.

Convergence of a Class of Cooperative Standard Cellular Neural Network Arrays

This paper considers a nonsymmetric standard (S) cellular neural network (CNN) array with cooperative (nonnegative) interconnections between neurons and a typical three-segment piecewise-linear (PL) neuron activation. The CNN is defined by a one-dimensional cell-linking (irreducible) cloning template with nearest-neighbor interconnections and has periodic boundary conditions. The flow generated by the SCNN is monotone but, due to the squashing effect of the horizontal segments in the PL activations, is not eventually strongly monotone (ESM). A new method for addressing convergence of the cooperative SCNN array is developed, which is based on the two main tools: (1) the concept of a frozen saddle, i.e., an unstable saddle-type equilibrium point (EP) enjoying certain dynamical properties that hold also for an asymptotically stable EP (a sink); (2) the analysis of the order relations satisfied by the sinks and saddle-type EPs. The analysis permits to show a fundamental result according to which any pair of ordered EPs of the SCNN contains at least a sink or a frozen saddle. On this basis it is shown that the flow generated by the SCNN enjoys a LIMIT SET DICHOTOMY and convergence properties analogous to those valid for ESM flows. Such results hold in the case where the SCNN displays either a local diffusion or a global propagation behavior.

Global Robust Stability Criteria for Interval Delayed Full-Range Cellular Neural Networks

This brief considers a class of delayed full-range (FR) cellular neural networks (CNNs) with uncertain interconnections between neurons modeled by means of intervalized matrices. Using mathematical tools from the theory of differential inclusions, a fundamental result on global robust stability of standard (S) CNNs is extended to prove global robust exponential stability for the corresponding class (same interconnection weights and inputs) of FR-CNNs. The result is of theoretical interest since, in general, the equivalence between the dynamical behavior of FR-CNNs and S-CNNs is not guaranteed.

Robustness of convergence in finite time for linear programming neural networks

A recent work has introduced a class of neural networks for solving linear programming problems, where all trajectories converge toward the global optimal solution in finite time. In this paper, it is shown that global convergence in finite time is robust with respect to tolerances in the electronic implementation, and an estimate of the allowed perturbations preserving convergence is obtained. Copyright

FOURTH-ORDER NEARLY-SYMMETRIC CNNS EXHIBITING COMPLEX DYNAMICS

In this paper, the possible presence of complex dynamics in nearly-symmetric standard Cellular Neural Networks (CNNs), is investigated. A one-parameter family of fourth-order CNNs is presented, which exhibits a cascade of period-doubling bifurcations leading to the birth of a complex attractor, close to some nominal symmetric CNN. Different from previous work on this topic, the bifurcations and complex dynamics are obtained for small relative errors with respect to the nominal interconnections. The fourth-order CNNs have negative (inhibitory) interconnections between distinct neurons, and are designed by a variant of a technique proposed by Smale to embed a given dynamical system within a competitive dynamical system of larger order.

The dichotomy of omega-limit sets fails for cooperative standard CNNs

The paper investigates some basic aspects of the solution semiflow associated to a class of cooperative standard (S) cellular neural networks (CNNs) with a typical three-segment pwl neuron activation. It is assumed that the SCNN neuron interconnection matrix is irreducible. By means of two counter-examples the following basic facts are shown: 1) in general the semiflow associated to the SCNN is not eventually strongly monotone; 2) in the general case also the fundamental property of the omega-limit set dichotomy fails. The consequences of these results are discussed in the context of the existing methods for addressing convergence of cooperative dynamical systems.

Convergent Dynamics of Nonreciprocal Differential Variational Inequalities Modeling Neural Networks

The paper addresses convergence of solutions for a class of differential inclusions termed differential variational inequalities (DVIs). Each DVI describes the dynamics of a neural network (NN) evolving in a closed hypercube of \BBR n and defined by a continuously differentiable, cooperative and (possibly) nonreciprocal vector field f. The main result in the paper is that under a new condition on f, which is called strong Kamke-Müller (SKM) condition, the solution semiflow generated by the DVI is strongly order preserving (SOP) and hence it satisfies a Limit Set Dichotomy and enjoys generic convergence properties. A characterization of the SKM condition is given in terms of the interconnection properties of the Jacobian matrix of f. In the case where f is an affine, or a linear, vector field the considered DVIs include two relevant classes of NNs, namely, the linear systems operating on a closed hypercube, also known as linear systems in saturated mode (LSSMs), and the full-range (FR) model of cellular neural networks (CNNs). By applying the results to LSSMs it is obtained that any cooperative LSSM with a (possibly) nonsymmetric and fully interconnected matrix is generically convergent. Analogous results hold for FRCNNs. All the obtained convergence results hold in the general case where the DVIs, and the LSSMs and FRCNNs, possess multiple equilibrium points.