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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 LaSalle's 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. Index Terms--Cellular neural networks (CNNs), convergence, differential variational inequalities (DVIs), full-range (FR) model, LaSalle's invariance principle, set-valued derivative.

This correspondence proves a convergence result for the Lotka-Volterra dynamical systems with symmetric interaction parameters between different species. These can be considered as a subclass of the competitive neural networks introduced by Cohen and Grossberg in 1983. The theorem guarantees that each forward trajectory has finite length and converges toward a single equilibrium point, even for those parameters for which there are infinitely many nonisolated equilibrium points. The convergence result in this correspondence, which is proved by means of a new method based on the Lojasiewicz inequality for gradient systems of analytic functions, is stronger than the previous result established by Cohen and Grossberg via LaSalle's invariance principle, which requires, for convergence, the additional assumption that the equilibrium points be isolated. Index Terms--Cohen-Grossberg neural networks, Lojasiewicz inequality, trajectory convergence.

The paper considers the full-range (FR) model of cellular neural networks (CNNs) in the case where the neuron nonlinearities are ideal hard-comparator functions with two vertical straight segments. The dynamics of FR-CNNs, which is described by a differential inclusion, is rigorously analyzed by means of theoretical tools from set-valued analysis and differential inclusions. The fundamental property proved in the paper is that FR-CNNs are equivalent to a special class of differential inclusions termed differential variational inequalities. A sound foundation to the dynamics of FR-CNNs is then given by establishing the existence and uniqueness of the solution starting at a given point, and the existence of equilibrium points. Moreover, a fundamental result on trajectory convergence towards equilibrium points (complete stability) for reciprocal standard CNNs is extended to reciprocal FR-CNNs by using a generalized Lyapunov approach. As a consequence, it is shown that the study of the ideal case with vertical straight segments in the neuron nonlinearities is able to give a clear picture and analytic characterization of the salient features of motion, such as the sliding modes along the boundary of the hypercube defined by the hard-comparator nonlinearities. Finally, it is proved that the solutions of the ideal FR model are the uniform limit as the slope tends to infinity of the solutions of a model where the vertical segments in the nonlinearities are approximated by segments with finite slope. Index Terms--Cellular neural networks (CNNs), differential inclusions, full-range (FR) model, sliding modes, trajectory convergence, variational inequalities.

This paper considers a class of neural networks (NNs) for solving linear programming (LP) problems, convex quadratic programming (QP) problems, and nonconvex QP problems where an indefinite quadratic objective function is subject to a set of affine constraints. The NNs are characterized by constraint neurons modeled by ideal diodes with vertical segments in their characteristic, which enable to implement an exact penalty method. A new method is exploited to address convergence of trajectories, which is based on a nonsmooth Lojasiewicz inequality for the generalized gradient vector field describing the NN dynamics. The method permits to prove that each forward trajectory of the NN has finite length, and as a consequence it converges toward a singleton. Furthermore, by means of a quantitative evaluation of the Lojasiewicz exponent at the equilibrium points, the following results on convergence rate of trajectories are established: 1) for nonconvex QP problems, each trajectory is either exponentially convergent, or convergent in finite time, toward a singleton belonging to the set of constrained critical points; 2) for convex QP problems, the same result as in 1) holds; moreover, the singleton belongs to the set of global minimizers; and 3) for LP problems, each trajectory converges in finite time to a singleton belonging to the set of global minimizers. These results, which improve previous results obtained via the Lyapunov approach, are true independently of the nature of the set of equilibrium points, and in particular they hold even when the NN possesses infinitely many nonisolated equilibrium points. Index Terms--Convergence in finite time, exponential convergence, generalized gradient, Lojasiewicz inequality, neural networks (NNs), programming problems.

This paper introduces a general class of neural networks with arbitrary constant delays in the neuron interconnections, and neuron activations belonging to the set of discontinuous monotone increasing and (possibly) unbounded functions. The discontinuities in the activations are an ideal model of the situation where the gain of the neuron amplifiers is very high and tends to infinity, while the delay accounts for the finite switching speed of the neuron amplifiers, or the finite signal propagation speed. It is known that the delay in combination with high-gain nonlinearities is a particularly harmful source of potential instability. The goal of this paper is to single out a subclass of the considered discontinuous neural networks for which stability is instead insensitive to the presence of a delay. More precisely, conditions are given under which there is a unique equilibrium point of the neural network, which is globally exponentially stable for the states, with a known convergence rate. The conditions are easily testable and independent of the delay. Moreover, global convergence in finite time of the state and output is investigated. In doing so, new interesting dynamical phenomena are highlighted with respect to the case without delay, which make the study of convergence in finite time significantly more difficult. The obtained results extend previous work on global stability of delayed neural networks with Lipschitz continuous neuron activations, and neural networks with discontinuous neuron activations but without delays. Index Terms--Convergence in finite time, delayed neural networks, discontinuous dynamical systems, global exponential stability.

In 1988 Kennedy and Chua introduced the dynamical canonical nonlinear programming circuit (NPC) to solve in real time nonlinear programming problems where the objective function and the constraints are smooth (twice continuously differentiable) functions. In this paper, a generalized circuit is introduced (G-NPC), which is aimed at solving in real time a much wider class of nonsmooth nonlinear programming problems where the objective function and the constraints are assumed to satisfy only the weak condition of being regular functions. G-NPC, which derives from a natural extension of NPC, has a neural-like architecture and also features the presence of constraint neurons modeled by ideal diodes with infinite slope in the conducting region. By using the Clarke's generalized gradient of the involved functions, G-NPC is shown to obey a gradient system of differential inclusions, and its dynamical behavior and optimization capabilities, both for convex and nonconvex problems, are rigorously analyzed in the framework of nonsmooth analysis and the theory of differential inclusions. In the special important case of linear and quadratic programming problems, salient dynamical features of G-NPC, namely the presence of sliding modes, trajectory convergence infinite lime, and the ability to compute the exact optimal solution of the problem being modeled, are uncovered and explained in the developed analytical framework. Index Terms--Convergence in finite time, gradient inclusions, neural networks, nonlinear programming, nonsmooth nonconvex optimization, sliding modes.

In a recent work, a new method has been introduced to analyze complete stability of the standard symmetric cellular neural networks (CNNs), which are characterized by local interconnections and neuron activations modeled by a three-segment piecewise-linear (PWL) function. By complete stability it is meant that each trajectory of the neural network converges toward an equilibrium point. The goal of this paper is to extend that method in order to address complete stability of the much wider class of symmetric neural networks with an additive interconnecting structure where the neuron activations are general PWL functions with an arbitrary number of straight segments. The main result obtained is that complete stability holds for any choice of the parameters within the class of symmetric additive neural networks with PWL neuron activations, i.e., such a class of neural networks enjoys the important property of absolute stability of global pattern formation. It is worth pointing out that complete stability is proved for generic situations where the neural network has finitely many (isolated) equilibrium points, as well as for degenerate situations where there are infinite (nonisolated) equilibrium points. The extension in this paper is of practical importance since it includes neural networks useful to solve significant signal processing tasks (e.g., neural networks with multilevel neuron activations). It is of theoretical interest too, due to the possibility of approximating any continuous function (e.g., a sigmoidal function), using PWL functions. The results in this paper confirm the advantages of the method of Forti and Tesi, with respect to LaSalle approach, to address complete stability of PWL neural networks. Index Terms--Neural networks, piecewise-linear (PWL) functions, stability.

It is known that additive neural networks with a symmetric interconnection matrix are completely stable, i.e., each trajectory converges toward some equilibrium point. This paper addresses the fundamental question of robustness of complete stability of additive neural networks with respect to small perturbations of the nominal symmetric interconnections. It is shown that in the general case, complete stability is not robust. More precisely, the paper considers a class of neural networks, and gives a necessary and sufficient condition for the existence of Hopf bifurcations (HBs) at the equilibrium point at the origin, arbitrarily close to symmetry. Such HBs originate stable limit cycles and hence cause the loss of complete stability. Furthermore, the paper highlights situations where the HBs are particularly critical, in the sense that the amplitude of the limit cycles is very sensitive to errors due to tolerances in the electronic implementation of the neuron interconnections. It is shown that sensitivity is crucially dependent on the neuron nonlinearity, and it is also significantly influenced by the features of the interconnection matrix and the network dimension. Finally, limitations of the obtained results are discussed and hints for future work are given. Index Terms--Complete stability, Hopf bifurcations, limit cycles, neural networks, robustness.

Protection against the transients originated by the connection or disconnection of loads in the power mains is of fundamental importance for the electric energy quality. The knowledge of the parameters of the disturbances and of the characteristics of the mains lines is essential for the mitigation of the unwanted phenomena and for the design of protection measures. This paper offers a lumped-constant analysis of the influence of the network loads on the transient waveshapes, and on the impedance seen when looking into the mains. The complex problem involving two loads and two line sections is reduced to a simplified model permitting analytical treatment with the consequent possibility of prediction and of extraction of the important features. The simplifying assumptions are 1) the line losses are low, and 2) the distance between the loads involved in the transient is short with respect to the total line length. The approximation limits are pointed out and the experimental validation demonstrates excellent agreement. Only the more significant case involving capacitive loading of the line is worked out in detail. Other cases of interest also involving resistive-inductive loading have been treated and are here delineated in summary.

We give a condition which is necessary and sufficient for the injectivity (i.e., for the global invertibility) of vector fields defining a class of piece-wise-linear neural networks which include the Cellular Neural Networks as a special case. It is shown that this is the sharpest obtainable condition for injectivity, since it enables one to ascertain such property for each specific nonlinear piece-wise-linear function modeling the neuron activations. This result establishes an exact bound between neural circuits possessing a unique equilibrium point (which are tailor made, e.g., for solving global optimization problems) and those possessing multiple equilibrium points (which are suitable, e.g., for implementing a Content Addressable Memory or a Cellular Neural Network for image processing). We also prove conceptually similar results on injectivity in case of continuously differentiable neuron activations. The proof of the main results exploits topological concepts from degree theory, such as the concept of homotopy of odd vector fields.

In this paper, we present new conditions ensuring existence, uniqueness, and Global Asymptotic Stability (GAS) of the equilibrium point for a large class of neural networks. The results are applicable to both symmetric and nonsymmetric interconnection matrices and allow for the consideration of all continuous nondecreasing neuron activation functions. Such functions may be unbounded (but not necessarily surjective), may have infinite intervals with zero slope as in a piece-wise-linear model, or both. The conditions on GAS rely on the concept of Lyapunov Diagonally Stable (or Lyapunov Diagonally Semi-Stable) matrices and are proved by employing a class of Lyapunov functions of the generalized Lur'e-Postnikov type. Several classes of interconnection matrices of applicative interest are shown to satisfy our conditions for GAS. In particular, the results are applied to analyze GAS for the class of neural circuits introduced in [10] for solving linear and quadratic programming problems. In this application, the principal result here obtained is that the networks in [10] are GAS also when the constraint amplifiers are dynamical, as it happens in any practical implementation.

A sufficient condition is proved guaranteeing that a class of neural circuits that includes the Hopfield model as a special case is globally convergent towards a unique stable equilibrium. The condition only requires symmetry and negative semi-definiteness of the neuron connection matrix T and is extremely simple to check and apply in practice. The consequences of the above result are discussed in the context of neural circuits for optimization of quadratic cost functions.

In this paper we discuss the design of a cellular neural network (CNN) to solve a class of optimization problems of importance for communication networks. The CNN optimization capabilities are exploited to implement an efficient cell scheduling algorithm in a fast packet switching fabric. The neural-based switching fabric maximizes the cell throughput and, at the same time, it is able to meet a variety of quality of service (QoS) requirements by optimizing a suitable function of the switching delay and priority of the cells. We also show that the CNN approach has advantages with respect to that based on Hopfield neural networks (HNN's) to solve the considered class of optimization problems. In particular, we exploit existing techniques to design CNN's with a prescribed set of stable binary equilibrium points as a basic tool to suppress spurious responses and, hence to optimize the neural switching fabric performance. Index Terms--Cellular neural networks, communication net. works, optimization problems.