Discontinuous neural networks

The paper introduces a nonsmooth (NS) neural network which 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 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 a NS-TD barrier function and a vector field corresponding to the unconstrained dynamics. The 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 t h , 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 the paper discusses a number of examples where NTN is applied to the solution of NS-TD convex feasibility problems.

The paper considers a class of additive neural networks where the neuron activations are modeled by discontinuous functions or by continuous non-Lipschitz functions. Some tools are developed which enable us to apply a Lyapunov-like approach to differential equations with discontinuous right-hand side modeling the neural network dynamics. The tools include a chain rule for computing the time derivative along the neural network solutions of a nondifferentiable Lyapunov function, and a comparison principle for this time derivative, which yields conditions for exponential convergence or convergence in finite time. By means of the Lyapunov-like approach, a general result is proved on global exponential convergence toward a unique equilibrium point of the neural network solutions. Moreover, new results on global convergence in finite time are established, which are applicable to neuron activations with jump discontinuities, or neuron activations modeled by means of continuous (non-Lipschitz) Hölder functions.

The paper introduces a nonsmooth (NS) neural network which 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 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 a NS-TD barrier function and a vector field corresponding to the unconstrained dynamics. The 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 t h , 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 the paper discusses a number of examples where NTN is applied to the solution of NS-TD convex feasibility problems.

In this paper, we introduce the notion of generalized pseu-dolinearity for nondifferentiable and nonconvex but locally Lipschitz functions defined on a Banach space. We present some characterizations of generalized pseudolinear functions. The characterizations of the solution set of a convex and nondifferentiable but generalized pseu-dolinear program are obtained. The results of this paper extend various results for pseudolinear functions, pseudoinvex functions and pseudo-linear functions, and also for pseudoinvex programs, pseudolinear programs and pseudolinear programs.

The paper considers a class of additive neural networks where the neuron activations are modeled by discontinuous functions or by continuous non-Lipschitz functions. Some tools are developed which enable us to apply a Lyapunov-like approach to differential equations with discontinuous right-hand side modeling the neural network dynamics. The tools include a chain rule for computing the time derivative along the neural network solutions of a nondifferentiable Lyapunov function, and a comparison principle for this time derivative, which yields conditions for exponential convergence or convergence in finite time. By means of the Lyapunov-like approach, a general result is proved on global exponential convergence toward a unique equilibrium point of the neural network solutions. Moreover, new results on global convergence in finite time are established, which are applicable to neuron activations with jump discontinuities, or neuron activations modeled by means of continuous (non-Lipschitz) Hölder functions.

We consider the Full-Range (FR) model of Cellular described by a specific class of differential inclusions named Neural Networks (CNNs) in the ideal case where the neuron nondifferential variational inequalities. Accordingly, the paper linearities are hard-comparator functions with two unbounded gives a sound theoretical foundation to the dynamics of FRvertical segments. The dynamics of FR-CNNs is rigorously givsbasodd stheo fondtiont te dynicseoeFR analyzed by using theoretical tools from set-valued analysis and CN by addresin the d nitin existence, d u qens differential inclusions. The fundamental property proved in the of the solution of FR-CNNs, and the existence of at least paper is that FR-CNNs are equivalent to a special class of one stationary solution (an equilibrium point). The analysis differential inclusions named differential variational inequalities. is valid in the general case where the neuron interconnection On this basis, a sound foundation to the dynamics of FR-CNNs matrix T is symmetric or non-symmetric. Moreover under the is given, by establishing results on the existence and uniqueness matrxTis osymmetric r n symt Moreove, uNde hey of the solution starting at a given point, and on the existence hypothess of symmetrilc T, ltis shown that FR-CNNs obey of equilibrium points. Moreover, some fundamental results on a gradient differential inclusion, and a fundamental result on trajectory convergence towards equilibrium points (complete trajectory convergence towards equilibrium points (complete stability) for reciprocal standard CNNs are extended to reciprocal stability) is obtained via a generalized Lyapunov approach. FR-CNNs by using a generalized Lyapunov approach. The results obtained in the paper can be considered as a

The paper introduces a general class of neural networks where the neuron activations are modeled by discontinuous functions. The neural networks have an additive interconnecting structure and they include as particular cases the Hopfield neural networks (HNNs), and the standard cellular neural networks (CNNs), in the limiting situation where the HNNs and CNNs possess neurons with infinite gain. Conditions are derived which ensure the existence of a unique equilibrium point, and a unique output equilibrium point, which are globally attractive for the state and the output trajectories of the neural network, respectively. These conditions, which are applicable to general nonsymmetric neural networks, are based on the concept of Lyapunov diagonally-stable neuron interconnection matrices, and they can be thought of as a generalization to the discontinuous case of previous results established for neural networks possessing smooth neuron activations. Moreover, by suitably exploiting the presence of sliding modes, entirely new conditions are obtained which ensure global convergence in finite time, where the convergence time can be easily estimated on the basis of the relevant neural-network parameters. The analysis in the paper employs results from the theory of differential equations with discontinuous right-hand side as introduced by Filippov. In particular, global convergence is addressed by using a Lyapunov-like approach based on the concept of monotone trajectories of a differential inclusion.

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.

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.

The paper introduces a nonsmooth (NS) neural network which 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 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 a NS-TD barrier function and a vector field corresponding to the unconstrained dynamics. The 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 t h , 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 the paper discusses a number of examples where NTN is applied to the solution of NS-TD convex feasibility problems.