Further results on dissipativity and stability analysis of Markov jump generalized neural networks with time-varying interval delays☆
Introduction
In recent years, many achievements have been made in the research on neural networks, and they have been used in many fields, such as robotics, pattern recognition, optimum combination, multi-agent systems and associative memory [1], [2], [3], [4], [5], [6], [7]. As pointed out in [8], neural networks often have information latching phenomenon in their applications. In this regard, one can solve this issue by extracting finite state representation from a trained network. Neural networks, in other words, often switch from one state to another, and through research, Markov chains could generate the switching of neural networks among different patterns [9], [10], [11]. That is the reason why the investigation of Markov jump neural networks is becoming a research hotspot. For more details, one can refer to the relevant literature [12], [13], [14], [15].
In addition, in order to define the dynamical evolution law of neural networks, we can use the neural states or local field states, and the corresponding neural network models could be divided into two kinds, static [16] and local field neural networks [1]. The above two kinds could transform each other under certain assumptions, but such assumptions are not always suitable for most applications [17]. Naturally, how to develop a new unified neural networks model, which may cover the local field neural network models or the static neural network models as a special case, is an interesting question. Fortunately, researchers have recently established such a model referred to the generalized neural networks (GNNs) [17], [18], [19].
In the study of GNNs, it has to mention that time delays could cause the networks under consideration to oscillate and unstable in the response of neurons in view of some existing researches [20], [21], [22], [23], [24], [25], [26], [27], [28]. In this regard, it is necessary and reasonable to investigate GNNs subject to time delays. As a consequence, the last decades have witnessed an increasing interest in the dynamic behavior of GNNs subject to time delays, where stability is always one of the most concerned problems by researchers come from many subjects such as mathematics, physics, computer science and so on. In this case, the key is to determine available stability criteria for the GNNs taken into account. Roughly speaking, the stability criteria of GNNs are divided into two kinds, delay-dependent stability criteria and delay-independent ones. In particular, the delay-dependent ones of the greatest concern, due to the fact that when the time delays are relatively small or it varies within an interval, will be less conservative than delay-independent ones. Until now, there are many effective delay-dependent stability criteria for different neural networks with time delays [29], [30]. Very recently, for Markov jump GNNs, the authors in [17] developed some new delay-dependent stability criteria by utilizing Wirtinger-based inequality technique. It should be remarkable that such criteria presented in [17] have room for improvement. For instance, some auxiliary function-based integral inequalities could provide much tighter bounds than the Wirtinger-based inequality used in [17]. In addition, mode-dependent matrices in the Lyapunov–Krasovskii functional (LKF) were not fully considered in [17]. More mode-dependent matrices should be introduced in the choice of LKF [31], and then one may make the better use of special switching structure of Markov jump GNNs. Furthermore, the performance of neural networks corresponds to input and output relations that plays an important role in many applications [32]. Naturally, this becomes a driving force for investigating robust performance of Markov jump GNNs with time delays, which was overlooked in [17]. In this regard, how to improve the criteria presented in [17] is a significant question in the study of Markov jump GNNs with time delays, which is the main motivation of this work.
Summing up the above discussions, we address the problems of dissipativity and stability analysis for Markov jump GNNs with time delays in this paper. Some novel dissipativity and stability criteria with less conservatism are presented. Compared with those in [17], the potential advantages of the presented method are reflected in the following three aspects: (1). A more general dissipative property index inequality is introduced for Markov jump GNNs with time delays, which is conducive to developing more general performance analysis conditions. By means of this, not only the conditions with H∞ performance [1], passivity performance, dissipativity performance, but also, with performance [13] are established in a unified framework; (2). In order to make the better use of special switching structure of Markov jump GNNs, an appropriate LKF, which contains mode-dependent matrices, is utilized; (3). As an effective way to reduce the conservatism of delay-dependent criteria, some auxiliary function-based integral inequalities are employed, and the less conservatism of the presented criteria is explained by three comparative examples used in [17].
Section snippets
Problem formulation
In what follows, a right continuous Markov chain {λt} needs to be introduced in a fixed probabilistic space as pointed out in [17]. More specifically, such a Markov chain has its values in a finite set and stands for its transition probability matrix, which follows the principle as: where h > 0, lim λij ≥ 0 is the transition rate from i to j if j ≠ i while
In the work, let us pay attention to the
Main results
In this section, we will show our main results for the Markov jump GNNs (1). A theorem and a corollary will be presented. In particular, in order to facilitate and simplify the calculation, the following formulas are first defined.
Numerical examples
Three examples are employed in this section. Based on these, the superiority and reduced conservatism of the presented criteria for the considered Markov jump generalized delayed neural networks may be explained.
Example 1: In this case, we pay attention to Markov jump generalized delayed neural networks in (4) with system mode where the parameters are borrowed in [17], [42] where
Then, one
Conclusions
In this work, we have studied the extended dissipativity analysis of Markov jump generalized neural networks (MJGNNs) with interval time-varying delays by introducing a more general dissipative property index inequality. A novel mode-dependent Lyapunov–Krasovskii functional has been selected and some auxiliary function-based integral inequalities have been utilized to reduce the conservatism of the presented criteria. In the end of the work, three compared examples have been provided to show
Closure statement
No potential conflict of interest was reported by the authors.
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This work was supported by the National Natural Science Foundation of China under Grants 61304066, 61573008, 61473178, 61703004, National Natural Science Foundation of Anhui Province under Grants 1708085MF165, 1808085QA18, SDUST Research Fund under Grant 2014TDJH102.