Analysis of global exponential stability and periodic solutions of neural networks with time-varying delays

Multi-periodicity of switched neural networks with time delays and periodic external inputs under stochastic disturbances

This paper presents new theoretical results on the multi-periodicity of recurrent neural networks with time delays evoked by periodic inputs under stochastic disturbances and state-dependent switching. Based on the geometric properties of activation function and switching threshold, the neuronal state space is partitioned into 5ⁿ regions in which 3ⁿ ones are shown to be positively invariant with probability one. Furthermore, by using Itô's formula, Lyapunov functional method, and the contraction mapping theorem, two criteria are proposed to ascertain the existence and mean-square exponential stability of a periodic orbit in every positive invariant set. As a result, the number of mean-square exponentially stable periodic orbits increases to 3ⁿ from 2ⁿ in a neural network without switching. Two illustrative examples are elaborated to substantiate the efficacy and characteristics of the theoretical results.

A sparse deep belief network with efficient fuzzy learning framework

Deep belief network (DBN) is one of the most feasible ways to realize deep learning (DL) technique, and it has been attracting more and more attentions in nonlinear system modeling. However, DBN cannot provide satisfactory results in learning speed, modeling accuracy and robustness, which is mainly caused by dense representation and gradient diffusion. To address these problems and promote DBN's development in cross-models, we propose a Sparse Deep Belief Network with Fuzzy Neural Network (SDBFNN) for nonlinear system modeling. In this novel framework, the sparse DBN is considered as a pre-training technique to realize fast weight-initialization and to obtain feature vectors. It can balance the dense representation to improve its robustness. A fuzzy neural network is developed for supervised modeling so as to eliminate the gradient diffusion. Its input happens to be the obtained feature vector. As a novel cross-model, SDBFNN combines the advantages of both pre-training technique and fuzzy neural network to improve modeling capability. Its convergence is also analyzed as well. A benchmark problem and a practical problem in wastewater treatment are conducted to demonstrate the superiority of SDBFNN. The extensive experimental results show that SDBFNN achieves better performance than the existing methods in learning speed, modeling accuracy and robustness.

Synchronization of neural networks with stochastic perturbation via aperiodically intermittent control

In this paper, the synchronization problem for neural networks with stochastic perturbation is studied with intermittent control via adaptive aperiodicity. Under the framework of stochastic theory and Lyapunov stability method, we develop some techniques of intermittent control with adaptive aperiodicity to achieve the synchronization of a class of neural networks, modeled by stochastic systems. Some effective sufficient conditions are established for the realization of synchronization of the underlying network. Numerical simulations of two examples are provided to illustrate the theoretical results obtained in the paper.

Dynamical behaviors for discontinuous and delayed neural networks in the framework of Filippov differential inclusions

This paper is concerned with the periodic dynamics of a class of delayed neural networks with discontinuous neural activation functions. Under the Filippov framework, the cone expansion and compression fixed point theorems of set-valued maps are successfully employed to derive the existence of the ω-periodic positive solution. However, before the discussion of the periodicity, there still remains a fundamental issue about viability to be solved due to the presence of general mixed time-delays involving both time-varying delays and distributed delays. This difficulty can be overcome by a transformation and the continuation theorem. Then, for the discontinuous and delayed neural network system with time-periodic coefficients, the uniqueness and global exponential stability of the periodic state solution are proved by using non-smooth analysis theory with generalized Lyapunov approach. Furthermore, the global convergence in measure of the periodic output is also investigated. The obtained results are a very good extension and improvement of previous works on discontinuous dynamical neuron systems with a broad range of neuron activations dropping the assumption of boundedness or monotonicity. Finally, numerical simulations are provided to illustrate the effectiveness of the theoretical results.

Estimator design for discrete-time switched neural networks with asynchronous switching and time-varying delay

This brief deals with the estimator design problem for discrete-time switched neural networks with time-varying delay. One main problem is the asynchronous-mode switching between the neuron state and the estimator. Our goal is to design a mode-dependent estimator for the switched neural networks under average dwell time switching such that the estimation error system is exponentially stable with a prescribed l2 gain (in the H∞ sense) from the noise signal to the estimation error. A new Lyapunov functional is constructed that may increase during the mismatched switchings. New results on the stability and l2 gain analysis are then obtained. The admissible estimator gains are computed by solving a set of linear matrix inequalities. The relations among the switching law, the maximal delay upper bound, and the optimal H∞ disturbance attenuation level are established. The effectiveness of the proposed design method is finally illustrated by a numerical example.

Robust stability of stochastic delayed additive neural networks with Markovian switching

This paper is concerned with the problem of robust stability for stochastic interval delayed additive neural networks (SIDANN) with Markovian switching. The time delay is assumed to be time-varying. In such neural networks, the features of stochastic systems, interval systems, time-varying delay systems and Markovian switching are taken into account. The mathematical model of this kind of neural networks is first proposed. Secondly, the global exponential stability in the mean square is studied for the SIDANN with Markovian switching. Based on the Lyapunov method, several stability conditions are presented, which can be expressed in terms of linear matrix inequalities. As a subsequent result, the stochastic interval additive neural networks with time-varying delay are also discussed. A sufficient condition is given to determine its stability. Finally, two simulation examples are provided to illustrate the effectiveness of the results developed.