Global Robust Attractive and Invariant Sets of Fuzzy Neural Networks with Delays and Impulses

A class of fuzzy neural networks (FNNs) with time-varying delays and impulses is investigated.With removing some restrictions on the amplification functions, a new differential inequality is established, which improves previouse criteria. Applying this differential inequality, a series of new and useful criteria are obtained to ensure the existence of global robust attracting and invariant sets for FNNs with time-varying delays and impulses. Our main results allow much broader application for fuzzy and impulsive neural networks with or without delays. An example is given to illustrate the effectiveness of our results.


Introduction
The theoretical and applied studies of the current neural networks (CNNs) have been a new focus of studies worldwide because CNNs are widely applied in signal processing, image processing, pattern recognition, psychophysics, speech, perception, robotics, and so on.The scholars have introduced many classes of CNNs models such as Hopfield-type networks [1], bidirectional associative memory networks [2], cellular neural networks [3], recurrent back-propagation networks [4][5][6], optimization-type networks [7][8][9], brain-statein-a-box-(BSB-) type networks [10,11], and Cohen-Grossberg recurrent neural networks (CGCNNs) [12].According to the choice of the variable for CNNs [13], two basic mathematical models of CNNs are commonly adopted: either local field neural network models or static neural network models.The basic model of local field neural network is described as ẋ  () = −  () +  ∑ =1     (  ()) +   ,  = 1, 2, . . ., , (1) where   denotes the activation function of the th neuron;   is the state of the th neuron;   is the external input imposed on the th neuron;   denotes the synaptic connectivity value between the th neuron and the th neuron;  is the number of neurons in the network.
However, in mathematical modeling of real world problems, we will encounter some other inconveniences, for example, the complexity and the uncertainty or vagueness.Fuzzy theory is considered as a more suitable setting for the sake of taking vagueness into consideration.Based on traditional cellular neural networks (CNNs),T.Yang and L.-B.Yang proposed the fuzzy CNNs (FCNNs) [33], which integrate fuzzy logic into the structure of traditional CNNs and maintain local connectedness among cells.Unlike previous CNNs structures, FCNNs have fuzzy logic between its Journal of Applied Mathematics template input and/or output besides the sum of product operation.FCNNs are very useful paradigm for image processing problems, which is a cornerstone in image processing and pattern recognition.In addition, many evolutionary processes in nature are characterized by the fact that their states are subject to sudden changes at certain moments and therefore can be described by impulsive system.Therefore, it is necessary to consider both the fuzzy logic and delay effect on dynamical behaviors of neural networks with impulses.Nevertheless, to the best of our knowledge, there are few published papers considering the global robust domain of attraction for the fuzzy neural network (FNNs).Therefore, in this paper, we will study the global robust attracting set and invariant set of the following fuzzy neural networks (FNNs) with time-varying delays and impulses: where   () The main purpose of this paper is to investigate the global robust attracting and invariant sets of FNNs (2).Different from [34,35], in this paper, we will introduce a new nonlinear differential inequality, which is more effective than the linear differential inequalities for studying the asymptotic behavior of some nonlinear differential equations.Applying this new nonlinear delay differential inequality, sufficient conditions are gained for global robust attracting and invariant sets.
The rest of this paper is organized as follows.In Section 2, we will give some basic definitions and basic results about the attracting domains of FNNs (2).In Section 3, we will obtain the proof of the usefully nonlinear delay differential inequality.In Section 4, our main results will be proved by this delay differential inequality.Finally, an example is given to illustrate the effectiveness of our results in Section 5.
As usual, in the theory of impulsive differential equations, at the points of discontinuity   ,  = 1, 2, . .., we assume that   (  ) ≡   ( −  ) and    (  ) ≡    ( −  ).Inspired by [37], we construct an equivalent theorem between ( 2) and ( 4).Then we establish some lemmas which are necessary in the proof of the main results.
Throughout this paper, we always assume the following.
Consider the following non-impulsive system (4): We have the following lemma, which shows that system (2) and ( 4) is equivalent.
(i) For any given  ∈ Ξ, if for any initial value  ∈  implies that (, , ) ∈  for all  ≥ 0, then  is said to be a robust positive invariant set of system of FNNs (2).

Nonlinear Delay Differential Inequality
In this section, we will establish a new nonlinear delay differential inequality which will play the important role to prove our main results.
From ( 15), we can get In the following, we at first will prove that for any positive constant ,       () We let By applying (11) and ( 21)-( 26), we obtain which contradicts the inequality in (25).Thus the inequality (23) holds.Therefore, letting  → 0, we have (13).The proof is complete.
By the process of proof of Lemma 4, we easily derive the following theorem Theorem 5.Under the conditions of Lemma 4, then

Main Results
In this section, we will state and prove our main results.The following lemma is very useful to prove Theorem 7.
Theorem 8.In addition to ( 1 )-( 5 ), further assume Î = 0. Then FNNs (2) has a zero solution and the zero solution is global robust exponential stability and the exponential convergent rate equals  which is determined by (40).
() is the weight of connection between the th neurons and the th neurons.  (),   , and ]  stand for state, input, and bias of the th neurons, respectively.  () is the transmission delay and   is the activation function.∧ and ∨ denote the fuzzy AND and fuzzy OR operation, respectively.Δ  (  ) is the impulses at moments   , and 0 ≤  1 <  2 < ⋅ ⋅ ⋅ is a strictly increasing sequence such that lim  → ∞   = +∞.  =   ()  () is the impulsive function.
are elements of fuzzy feed-forward template,   () and   () are elements of fuzzy feedback MIN template,   () and   () are elements of fuzzy feedback MAX template, and   () and   () are elements of fuzzy feedforward MIN template and fuzzy feed-forward MAX template, respectively.