Convergence of nonlinear circuits towards equilibrium points (EPs) is one of the most basic properties both from a theoretic and a practical viewpoint. It is especially relevant for nonlinear circuits modeling neural networks, since a convergent network with multiple stable EPs is tailor made to implement content addressable memories (CAMs) storing multiple patterns as stable EPs or to solve combinatorial optimization problems in real time. Convergence has been widely investigated in the last few decades. By far, the available convergence results can be applied to circuits without memristors, while the study of convergence in presence of memristors is only in its infancy. In this paper, a class of nonlinear circuits containing memristors, capacitors, passive or active resistors and independent sources, is considered. Active resistors are crucial, since they permit to obtain circuits with multiple stable EPs. A number of basic results on convergence are obtained via the flux-charge analysis method (FCAM) and a fundamental reciprocity principle for the considered class of circuits. The conditions for convergence are robust, i.e., they hold also for perturbations of the circuit parameters and memristor nonlinearities involved. The results are illustrated via selected examples where use is made of the celebrated HP memristor model.

The paper considers a wide class of nonlinear circuits with an ideal memristor, capacitors, inductors and current or voltage sources. A fundamental dynamical property is that each memristor circuit in this class admits a first integral (invariant of motion or preserved quantity), i.e., a function which is constant along the solutions. The first main result is that we can put the state equations of each circuit in a universal form, known as linear-gradient form, given by a state-dependent skew-symmetric matrix times the gradient of the first integral. This is a simple and general form which is both of theoretic and practical interest. First of all, it makes manifest the existence of a first integral. Moreover, it admits an elegant discrete-time (DT) analogue. Indeed, the linear-gradient form, combined with geometric discretization methods and the concept of discrete gradients, yields a DT version of each memristor circuit that exactly preserves the first integral for any discretization time step. This is relevant, since the existence of a first integral is a fragile property that is in general destroyed by typical discretization schemes used in the literature no matter how small the step size is. On one hand, the proposed discretization scheme can be used for constructing numerical algorithms that better approximate the solutions of memristor circuits for small step sizes. On the other hand, for larger step sizes the obtained DT memristor circuits can be of interest by themselves since they are able to easily generate complex dynamics potentially useful for engineering applications (computational chaos). Furthermore, the paper shows that, thanks to the decomposition of the state space in invariant manifolds, the derived DT circuits exhibit extreme multistability, i.e., the coexistence of infinitely many different attractors for fixed set of circuit parameters, memristor nonlinearity and step size.

Nonvolatile memristive devices display nonlinear characteristics suitable for implementing circuits exhibiting oscillations or more complex dynamic behaviors, including chaos. However, the results presented in related works are mostly limited to simulations and employing ideal memristor models whose resistance is governed by a charge-flux relation that is not connected to real devices, thus hindering the realization of such nonlinear oscillators. In this work, we present the framework for the physical implementation of a tunable memristor Chua’s circuit, which is based on a nonvolatile memristive device that provides the nonlinear conductance required by the circuit and the possibility to tune it for the purpose of selecting among different oscillation patterns. We first establish design guidelines to guarantee complex oscillations in the tunable memristor Chua’s circuit. Further, we physically implement the circuit after characterizing and modeling the tunable current-voltage characteristic of a real device. Our circuit successfully generates different oscillation patterns just by programming the nonvolatile memristive device to different states. The devised design guidelines and device modeling were used to extend the experimental work and draw further requirements for device properties for a successful circuit implementation.

The article considers a large class of delayed neural networks (NNs) with extended memristors obeying the Stanford model. This is a widely used and popular model that accurately describes the switching dynamics of real nonvolatile memristor devices implemented in nanotechnology. The article studies via the Lyapunov method complete stability (CS), i.e., convergence of trajectories in the presence of multiple equilibrium points (EPs), for delayed NNs with Stanford memristors. The obtained conditions for CS are robust with respect to variations of the interconnections and they hold for any value of the concentrated delay. Moreover, they can be checked either numerically, via a linear matrix inequality (LMI), or analytically, via the concept of Lyapunov diagonally stable (LDS) matrices. The conditions ensure that at the end of the transient capacitor voltages and NN power vanish. In turn, this leads to advantages in terms of power consumption. This notwithstanding, the nonvolatile memristors can retain the result of computation in accordance with the in-memory computing principle. The results are verified and illustrated via numerical simulations. From a methodological viewpoint, the article faces new challenges to prove CS since due to the presence of nonvolatile memristors the NNs possess a continuum of nonisolated EPs. Also, for physical reasons, the memristor state variables are constrained to lie in some given intervals so that the dynamics of the NNs need to be modeled via a class of differential inclusions named differential variational inequalities.

Controlling multistability, i.e., designing control laws for switching among different attractors, is an emerging issue in the area of memristor circuits. The paper considers the Chua’s memristor circuit which is known to display infinitely many attractors, each one contained in an invariant manifold of the circuit state space. The problem of switching among these attractors via pulse-programmed feedforward control laws, which are implementable via a unique current/voltage source, is investigated. In particular, it is shown that if the shape of the voltage source in series to the inductor is suitably designed, then it is possible to switch in finite time from one attractor to another.

This article introduces a new class of memristor neural networks (NNs) for solving, in real-time, quadratic programming (QP) and linear programming (LP) problems. The networks, which are called memristor programming NNs (MPNNs), use a set of filamentary-type memristors with sharp memristance transitions for constraint satisfaction and an additional set of memristors with smooth memristance transitions for memorizing the result of a computation. The nonlinear dynamics and global optimization capabilities of MPNNs for QP and LP problems are thoroughly investigated via a recently introduced technique called the flux-charge analysis method. One main feature of MPNNs is that the processing is performed in the flux-charge domain rather than in the conventional voltage-current domain. This enables exploiting the unconventional features of memristors to obtain advantages over the traditional NNs for QP and LP problems operating in the voltage-current domain. One advantage is that operating in the flux-charge domain allows for reduced power consumption, since in an MPNN, voltages, currents, and, hence, power vanish when the quick analog transient is over. Moreover, an MPNN works in accordance with the fundamental principle of in-memory computing, that is, the nonlinearity of the memristor is used in the dynamic computation, but the same memristor is also used to memorize in a nonvolatile way the result of a computation.

Memristors are more and more seen as the basic elements for the development of new unconventional computational schemes. One of the appealing property of memristor circuits is multistability, i.e., the coexistence in the state space of a rich variety of different attractors. This paper considers the problem of controlling multistability for a circuit with a charge-controlled memristor which displays infinite stable equilibrium points and limit cycles. Specifically, it is shown how voltage and current sources can be pulse programmed in order to steer the circuit dynamics from one stable equilibrium point to a different stable equilibrium point within a given finite time interval.

The paper considers a relevant class of networks containing memristors and (possibly) nonlinear capacitors and inductors. The goal is to unfold the nonlinear dynamics of these networks by highlighting some main features that are potentially useful for real-time signal processing and in-memory computing. In particular, an analytic treatment is provided for dynamic phenomena as the presence of invariant manifolds, the coexistence of different regimes, complex dynamics and attractors and the phenomenon of bifurcations without parameters, i.e., bifurcations due to changing the initial conditions of the state variables for a fixed set of circuit parameters. The paper also addresses the issue of how to design pulse independent voltage or current sources to steer the network dynamics through different manifolds and attractors. Two relevant examples are worked out in details, namely, a variant of Chua's circuit with a memristor and a nonlinear capacitor and a relaxation oscillator with a memristor and a nonlinear inductor. In the latter example, the paper also studies the effect on manifolds and coexisting dynamics when real memristive devices are accounted for using a class of extended memristor models. The analysis is conducted by means of a recently developed technique named flux-charge analysis method (FCAM). Numerical simulations are presented to confirm the theoretic findings.

Circuits containing memelements (memory elements) are suitable for the design of new unconventional computational schemes. The coexistence of a rich variety of different attractors is one of the appealing property of these circuits, which has stimulated the so-called “multistability control” problem. This paper considers the multistability control problem for a circuit with a charge-controlled memristor. It is shown how pulse control inputs can be generated via an external current generator in order to drive the system dynamics from an attractor to another one in a given finite time interval.

Nonlinear dynamic memory elements, as memristors, memcapacitors, and meminductors (also known as mem-elements), are of paramount importance in conceiving the neural networks, mem-computing machines, and reservoir computing systems with advanced computational primitives. This paper aims to develop a systematic methodology for analyzing complex dynamics in nonlinear networks with such emerging nanoscale mem-elements. The technique extends the flux-charge analysis method (FCAM) for nonlinear circuits with memristors to a broader class of nonlinear networks N containing also memcapacitors and meminductors. After deriving the constitutive relation and equivalent circuit in the flux-charge domain of each two-terminal element in N , this paper focuses on relevant subclasses of N for which a state equation description can be obtained. On this basis, salient features of the dynamics are highlighted and studied analytically: 1) the presence of invariant manifolds in the autonomous networks; 2) the coexistence of infinitely many different reduced-order dynamics on manifolds; and 3) the presence of bifurcations due to changing the initial conditions for a fixed set of parameters (also known as bifurcations without parameters). Analytic formulas are also given to design nonautonomous networks subject to pulses that drive trajectories through different manifolds and nonlinear reduced-order dynamics. The results, in this paper, provide a method for a comprehensive understanding of complex dynamical features and computational capabilities in nonlinear networks with mem-elements, which is fundamental for a holistic approach in neuromorphic systems with such emerging nanoscale devices.

Recent works have introduced an effective technique to analyze nonlinear dynamics of a class LM of circuits containing ideal flux- or charge-controlled memristors and linear lossless elements (i.e. ideal capacitors and inductors). The technique, named Flux-Charge Analysis Method (FCAM), is based on analyzing the circuits in the flux-charge domain instead of the traditional voltage-current domain. Goal of this paper is to extend the FCAM to a larger class N of circuits containing also nonlinear capacitors and inductors. Nonlinear circuits with memristors and nonlinear lossless elements are widely used to several real nanoscale devices including the well-known Josephson junction. After deriving the constitutive relation in the flux-charge domain of each two-terminal element in N, the work focuses on a relevant subclass of N for which a state equation description can be obtained. State Equations (SE) formulation provides the fundamental basis for studying the chief features of the nonlinear dynamics: presence of invariant manifolds in autonomous circuits; coexistence of infinitely many different reduced-order dynamics on the manifolds; bifurcations due to changing of initial conditions for a fixed set of parameters, a.k.a. bifurcations without parameters.

Recent papers in the literature introduced a class of neural networks (NNs) with memristors, named dynamic-memristor (DM) NNs, such that the analog processing takes place in the charge-flux domain, instead of the typical current-voltage domain as it happens for Hopfield NNs and standard cellular NNs. One key advantage is that, when a steady state is reached, all currents, voltages, and power of a DM-NN drop off, whereas the memristors act as nonvolatile memories that store the processing result. Previous work in the literature addressed multistability of DM-NNs, i.e., convergence of solutions in the presence of multiple asymptotically stable equilibrium points (EPs). The goal of this paper is to study a basically different dynamical property of DM-NNs, namely, to thoroughly investigate the fundamental issue of global asymptotic stability (GAS) of the unique EP of a DM-NN in the general case of nonsymmetric neuron interconnections. A basic result on GAS of DM-NNs is established using Lyapunov method and the concept of Lyapunov diagonally stable matrices. On this basis, some relevant classes of nonsymmetric DM-NNs enjoying the property of GAS are highlighted.

A recent work [1] introduced a flux-charge analysis method (FCAM) to study the nonlinear dynamics and bifurcations of a large class of memristor circuits. FCAM relies on the use of Kirchhoff Flux and Charge Laws and constitutive relations of circuits elements in the flux-charge domain. In [1], the saddle-node bifurcations of equilibrium points in the simplest memristor circuit composed of an ideal flux-controlled memristor and a capacitor, were studied. This paper is devoted to analyze via FCAM more complex bifurcations, such as Hopf bifurcations and period-doubling bifurcations originating complex attractors, in higher-order memristor circuits. It is shown analytically and quantitatively how these bifurcations can be induced by varying the initial conditions of dynamic circuit elements in the voltage-current domain while assuming that circuits parameters are held fixed. Such bifurcations are known in the literature as bifurcations without parameters.

Recent work has considered a class of cellular neural networks (CNNs) where each cell contains an ideal capacitor and an ideal flux-controlled memristor. One main feature is that during the analog computation the memristor is assumed to be a dynamic element, hence each cell is second-order with state variables given by the capacitor voltage and the memristor flux. Such CNNs, named dynamic memristor (DM)-CNNs, were proved to be convergent when a symmetry condition for the cell interconnections is satisfied. The goal of this paper is to investigate convergence and multistability of DM-CNNs in the general case of nonsymmetric interconnections. The main result is that convergence holds when there are (possibly) nonsymmetric, non-negative interconnections between cells and an irreducibility assumption is satisfied. This result appears to be similar to the classic convergence result for standard (S)-CNNs with positive cell-linking templates. Yet, due to the presence of DMs, a DM-CNN displays some basically different and peculiar dynamical properties with respect to S-CNNs. One key difference is that the DM-CNN processing is based on the time evolution of memristor fluxes instead of capacitor voltages as it happens for S-CNNs. Moreover, when a steady state is reached, all voltages and currents, and hence power consumption of a DM-CNN vanish. This notwithstanding the memristors are able to store in a nonvolatile way the result of the processing. Voltages, currents and power instead do not vanish when an S-CNN reaches a steady state.

This paper considers a class of nonsmooth neural networks with discontinuous hard-limiter (signum) neuron activations for solving time-dependent (TD) systems of algebraic linear equations (ALEs). The networks are defined by the subdifferential with respect to the state variables of an energy function given by the L1norm of the error between the state and the TD-ALE solution. It is shown that when the penalty parameter exceeds a quantitatively estimated threshold the networks are able to reach in finite time, and exactly track thereafter, the target solution of the TD-ALE. Furthermore, this paper discusses the tightness of the estimated threshold and also points out key differences in the role played by this threshold with respect to networks for solving time-invariant ALEs. It is also shown that these convergence results are robust with respect to small perturbations of the neuron interconnection matrices. The dynamics of the proposed networks are rigorously studied by using tools from nonsmooth analysis, the concept of subdifferential of convex functions, and that of solutions in the sense of Filippov of dynamical systems with discontinuous nonlinearities.

This paper introduces a nonsmooth (NS) neural network that is able to operate in a time-dependent (TD) context and is potentially useful for solving some classes of NS-TD problems. The proposed network is named nonsmooth time-dependent network (NTN) and is an extension to a TD setting of a previous NS neural network for programming problems. Suppose C(t), t ≥ 0, is a nonempty TD convex feasibility set defined by TD inequality constraints. The constraints are in general NS (nondifferentiable) functions of the state variables and time. NTN is described by the subdifferential with respect to the state variables of an NS-TD barrier function and a vector field corresponding to the unconstrained dynamics. This paper shows that for suitable values of the penalty parameter, the NTN dynamics displays two main phases. In the first phase, any solution of NTN not starting in C(0) at t=0 is able to reach the moving set C(·) in finite time th , whereas in the second phase, the solution tracks the moving set, i.e., it stays within C(t) for all subsequent times t ≥ t(h). NTN is thus able to find an exact feasible solution in finite time and also to provide an exact feasible solution for subsequent times. This new and peculiar dynamics displayed by NTN is potentially useful for addressing some significant TD signal processing tasks. As an illustration, this paper discusses a number of examples where NTN is applied to the solution of NS-TD convex feasibility problems.

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 A(τ). The main result is that when A+A(τ) 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.

The paper considers a class of nonsymmetric cooperative standard cellular neural networks (SCNNs), which are defined by a cell-linking template, and are characterized by neuron activations modeled by a typical three-segment pwl function. The paper establishes conditions ensuring that the monotone solution semiflow associated to the considered class of SCNNs satisfies the LIMIT SET DICHOTOMY and is convergent toward equilibrium points. The conditions, which involve only static aspects of the equilibrium point configuration of the SCNNs, are easier to verify with respect to those in previous results in the literature. By means of a standard numerical program for locating the equilibrium points of pwl SCNNs, parameter ranges for which the conditions are verified, and the cooperative SCNNs are convergent, are established.

The paper considers a class of delayed standard (S) cellular neural networks (CNNs) with non-negative interconnections between distinct neurons and a typical three-segment pwl neuron activation. It is also assumed that such cooperative SCNNs satisfy an irreducibility condition on the interconnection and delayed interconnection matrix. By means of a counterexample it is shown that the solution semiflow associated to such SCNNs in the general case does not satisfy the fundamental property of the omega-limit set dichotomy and is not eventually strongly monotone. The consequences of this result are discussed in the context of the existing methods for addressing convergence of monotone semiflows defined by delayed cooperative dynamical systems.

The paper considers a general neural network model with impulses at a given sequence of instants, discontinuous neuron activations, delays, and time-varying data and inputs. It is shown that when the neuron interconnections satisfy an M-matrix condition, or a dominance condition, then the state solutions and the output solutions display a common asymptotic behavior as time t--> +infinity. It is also shown, via a new technique based on prolonging the solutions of the delayed neural network to -infinity, that it is possible to select a unique special solution that is globally exponentially stable and can be considered as the unique global attractor for the network. Finally, this paper shows that for almost periodic data and inputs the selected solution is almost periodic; moreover, it is robust with respect to a large class of perturbations of the data. Analogous results also hold for periodic data and inputs. A by-product of the analysis is that a sequence of almost periodic impulses is able to induce in the generic case (nonstationary) almost periodic solutions in an otherwise globally convergent nonimpulsive neural network. To the authors' knowledge the results in this paper are the only available results on global exponential stability of the unique periodic or almost periodic solution for a general neural network model combining three main features, i.e., impulses, discontinuous neuron activations and delays. The results in this paper are compared with several results in the literature dealing with periodicity or almost periodicity of some subclasses of the neural network model here considered and some hints for future work are given.

The paper proposes an alternate definition of set-valued derivative, with respect to that in a previous paper, for computing the evolution of a (candidate) Lyapunov function along the solutions of a class of differential variational inequalities (DVIs). The class of DVIs is of interest in that it includes as a special case the dynamics of full-range (FR) cellular neural networks (CNNs). The usefulness of the new definition is discussed in the context of a generalized Lyapunov method for addressing stability and convergence of solutions of DVIs and FR-CNNs.

The paper considers a class of Full-Range (FR) cellular neural networks (CNNs) characterized by a finite constant delay in the neuron interconnections and intervalized interconnection parameters. A theorem is proved which ensures global robust stability (GRS), i.e., global stability of the equilibrium point for any FR-CNN whose parameters belong to given intervals. The theorem extends to FR-CNNs a result on GRS for standard (S) CNNs obtained in a recent paper by Shen and Zhang. The significance of the result in this paper is discussed in relation to the results in a paper by Corinto and Gilli, which addresses the equivalence of the dynamical behavior of FR-CNNs and S-CNNs, when they are defined by the same set of parameters.

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 Łojasiewicz 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.

In a series of papers published in the seventies, Grossberg has developed a geometric approach for analyzing the global dynamical behavior and convergence properties of a class of competitive dynamical systems. In this paper, Grossberg approach is extended to competitive standard cellular neural networks (CNNs), and it is used to investigate convergence of classes of non-symmetric competitive CNNs under the hypothesis that they induce a globally consistent decision scheme.

The paper considers a large class of additive neural networks where the neuron activations are modeled by discontinuous functions or by non-Lipschitz functions. A result is established guaranteeing that the state solutions and output solutions of the neural network are globally convergent in finite time toward a unique equilibrium point. The obtained result, which generalizes previous results on convergence in finite time in the literature, is of interest for designing neural networks aimed at solving global optimization problems in real time

We consider the full-range (FR) model of cellular neural networks (CNNs) in the ideal case where the neuron nonlinearities are hard-comparator functions with two unbounded vertical segments. The dynamics of FR-CNNs is rigorously analyzed by using 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 named differential variational inequalities. On this basis, a sound foundation to the dynamics of FR-CNNs is given, by establishing results on the existence and uniqueness of the solution starting at a given point, and on the existence of equilibrium points. Moreover, some fundamental results on trajectory convergence towards equilibrium points (complete stability) for reciprocal standard CNNs are extended to reciprocal FR-CNNs by using a generalized Lyapunov approach

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.

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.

In the last decade, code division multiple access (CDMA) has gained even more importance due to its capabilities of wider band occupancy without any time constraints. A lot of the recent implemented systems for wireless communications, as Universal Mobile Telecommunication System (UMTS) or IEEE 802.11b wireless local area network (WLAN), use the CDMA approach to allow the simultaneous access of multiple users. One of the main drawbacks of CDMA systems is the so called multiple access interference (MAI). In the literature, several multiuser receivers were developed. Among them, receivers that perform mean square error minimization are very attractive for their very simple implementation. On the other hand, neural networks have gained recently an increasing importance due to their capabilities in solving many engineering problems, involving minimization of errors or some other cost functionals. In this paper, an advanced MMSE receiver based on the use of neural networks is proposed, where at every bit time neural network achieves the optimum values for the coefficient set of receiving filter, thus minimizing the error rate.

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.

The issue of the loss of complete stability for a class of cellular neural networks (CNNs) is analyzed. It is shown that there are CNNs in this class for which a Hopf bifurcation is present, even if the interconnection matrix is arbitrarily close to some symmetric matrix. This shows that, in the general case, complete stability is not robust with respect to perturbations of nominal symmetric interconnection matrices.

We propose an implementation in terms of a cellular neural network (CNN) of the packet selection discipline in an input queueing fast packet switching (FPS) fabric. A neural network is designed which is devoid of spurious (suboptimal) responses and is guaranteed to take optimal decisions for switching packets. Such a neural network permits to lower the buffer memory requirements and to achieve throughput-mean switching delay performance close to the optimum (output) queueing alternative.

This paper discusses the design of a neural network for solving some classes of combinatorial optimization problems in real time. By means of a suitable design procedure which is not based on energy arguments, it is guaranteed that the network is devoid of spurious responses. An important application is considered to a typical optimization problem arising in the telecommunications field. More specifically, we show how the neural network can be used to take decisions for switching packets and improve switching performance in a fast packet switching fabric with input buffers.

The aim of this paper is to discuss the role of Absolute Stability (ABST) in the design of neural optimization solvers and to find necessary and sufficient conditions for ABST for some classes of neural networks of applicative interest. By ABST it is meant that there is a unique equilibrium point attracting all trajectories of motion and that this property is valid for all neuron activation functions belonging to a specified class of nonlinear mappings and for all constant neural network inputs. ABST neural networks are best suited for solving optimization problems being devoid of spurious suboptimal responses for every choice of the activation function and of the input vector. A necessary and sufficient condition for ABST has been found for symmetric neural networks of the Hopfield type. In this paper, we show that the concept of ABST can be applied also to special classes of nonsymmetric Hopfield neural networks and to neural models different from the Hopfield one. It is shown in particular that necessary and sufficient conditions for ABST can be found for two interesting classes of nonsymmetric networks, namely, cooperative Hopfield-type networks and composite neural networks with variable and constraint neurons used for solving linear and quadratic programming problems in real time.>