We discuss stability analysis for uncertain stochastic neural networks (SNNs) with time delay in this letter. By constructing a suitable Lyapunov-Krasovskii functional (LKF) and utilizing Wirtinger inequalities for estimating the integral inequalities, the delay-dependent stochastic stability conditions are derived in terms of linear matrix inequalities (LMIs). We discuss the parameter uncertainties in terms of norm-bounded conditions in the given interval with constant delay. The derived conditions ensure that the global, asymptotic stability of the states for the proposed SNNs. We verify the effectiveness and applicability of the proposed criteria with numerical examples.
The significance of neural networks (NNs) cannot be limited to being a class of mathematical models and information processing systems. Their application is far-reaching in many areas, among them automatic control, signal processing, pattern recognition, and quadric recognition (Haykin, 2007). The stability of NNs has been discussed by many researchers (Anbuvithya, Mathiyalagan, Sakthivel, & Prakash, 2016; Cichocki & Unbehauen, 1993; Lakshmanan, Prakash, Rakkiyappan, & Joo, 2020; Liu, Zeng, & Wang, 2017; Liu, Wang, & Liu, 2006; Li, Zheng, & Lin, 2011; Lv et al., 2017; Zhang, Liu, & Zhou, 2012; Wong & Selvi, 1998; Zheng, Zhang, & Wang, 2009). However, in many practical NNs, time delays are unavoidable, and they lead to NN instability, oscillation, and poor performance. Due to this, stability investigation of NNs with time delays has become an important area for research, and many levant reports have been published (Chen & Rong, 2003; Chen & Wu, 2009; Chen, Sun, Liu, & Rees, 2010; Fu & Li, 2011; Lakshmanan et al., 2018; Li, Wang, Yang, Zhang, & Wang, 2008; Li & Chen, 2009; Qiu, Cui, & Wu, 2009; Shao, Huang, & Zhou, 2009; Yu, Zhang, & Quan, 2015; Zhang, Cao, Wu, Chen, & Alsaadi, 2018; Zhang & Quan, 2015). A global exponential stability condition and inequality based on a linear matrix inequality (LMI) that forms an global exponential stability condition for inertial Cohen-Grossberg NNs with time delays is discussed in Yu et al. (2015). Projective synchronization of fractional-order NNs with multiple time delays was studied in Zhang et al. (2018), Zhang and Yu (2016), and Zhang and Quan (2015). Zhang and Quan (2015) sought to obtain sufficient LMI-based conditions for the existence and global exponential stability of inertial bidirectional associative memory NNs with time delays. Therefore, it becomes imperative to include the factor of time delays in the dynamical analysis of NNs.
Stochastic disturbance generally is affected by network models. Thus, when the stability of NNs is analyzed, stochastic disturbance becomes unavoidable. This happens due to the common factor that synaptic transmission is a noisy process, and the neurons' connection weights rely on certain values of resistance and capacitance where there are uncertainties. In this regard, a great deal of work has been conducted on stability analysis for delayed SNNs and robust stability for uncertain stochastic neural networks (SNNs). As a result, scientific results have been published in relation to the stability of NNs with stochastic disturbance (Balasubramaniam & Lakshmanan, 2011; Blythe, Mao, & Liao, 2001; Chen & Wu, 2009; Liao & Mao, 1996; Mao, 1997; Muralisankar, Manivannan, & Balasubramaniam, 2015; Xia, Yu, Li, & Zheng, 2012; Zhao, Gao, & Mou, 2008; Zhu & Cao, 2010a, 2010b, 2014). Stability analysis for NNs by using specific stochastic inputs was discussed in Blythe et al. (2001) and Liao and Mao (1996). For Markovian jump impulsive stochastic Cohen-Grossberg NNs with mixed time delays, Zhu and Cao (2010b) used the Lyapunov-Krasovskii functional (LKF) method for structuring a novel robust exponential stability criterion and known or unknown parameters to be achieved. Zhu and Cao (2014) investigated the stability of stochastic delayed recurrent NNs with the use of an augmented LKF method. This leads to the need for increased attention to the issue of stability investigation for SNNs with time delays.
However, there are also inevitable uncertainties in modeling NNs due to errors in modeling and fluctuating parameters at the time of execution, resulting in instability and poor performance. There have also been many interesting results recently (Chen & Qin, 2010; Deng, Hua, Liu, Peng, & Fei, 2011; Hua, Liu, Deng, & Fei, 2010; Huang & Cao, 2007; Li, Chen, Zhou, & Fang, 2008; Wang, Shu, Fang, & Liu, 2006; Wu, Su, Chu, & Zhou, 2009; Zhang, Shi, & Qiu, 2007; Zhang, Shi, Qiu, & Yang, 2008) on the stability of uncertain SNNs with delay. Chen and Qin (2010), Hua et al. (2010), Huang and Cao (2007), Li et al. (2008), and Zhang et al. (2008) investigated uncertain SNNs with robust stability and time-varying delays in terms of LKF and stochastic analysis approaches. The robust stability in terms of stochastic Hopfield NNs with time delays was examined by using the LKF functional and conducting stochastic analysis by, Wang, Shu, Fang, and Liu (2006) and Zhang et al. (2007). Deng et al. (2011) studied delay-dependent exponential stability of uncertain where SNNs with mixed delays, based on the LKF method. Wu, Su, Chu, and Zhou (2009) discussed some novel delay-dependent conditions, sufficient to ensure the global exponential stability of discrete, recurrent NNs with time-varying delays. Thus, it is evident that many researchers have contributed to the analysis of the stability of time-delayed NNs. A number of methods have been developed to minimize the conservatism of stability criteria: the multiple integral approach (Fang & Park, 2013), model transformation (Kwon & Park, 2004), free-weighting matrix techniques (He, Liu, Rees, & Wu, 2007; Liu, Wu, Martin, & Tang, 2007), park inequality (Park, 1999), the convex combination technique (Park & Ko, 2007), and reciprocally convex optimization (Park, Ko, & Jeong, 2011). Most important, since estimating a lower bound of the quadratic integral term such as is one of the major research topics on time-delay systems, Jensen's inequality has been used widely as a key lemma in obtaining delay-dependent stability criteria. The Wirtinger-based integral inequality, introduced recently in Seuret and Gouaisbaut (2013), also reduced the conservatism of Jensen's inequality, and its advantage was reflected in the comparisons of delay bounds for numerous systems, such as systems with constant, known, and time-varying delay. However, some new LKFs were not considered, and use of the Wirtinger-based integral inequality was concentrated only in Seuret and Gouaisbaut (2013). Therefore, further improvement on the reduction of conservatism in stability analysis for a system with time delays can be achieved, the motivation behind the research we present in this letter.
This letter discusses robust stability analysis for SNNs with time delay. We also consider parameter uncertainties in the system matrices of delayed SNNs. Based on suitable LKF, we derive the delay stability conditions in line with LMIs.
This letter focuses on the following points:
Parameter uncertainties and stochastic disturbance are taken into account.
Integral terms are estimated based on Wirtinger's integral inequalities. With appropriate LKF and stochastic stability theory, the delay-dependent stability conditions are attained to ensure the global asymptotic stability of the proposed system. We have employed well-known software to identify the effectiveness of the intended LMIs. Finally, we provide a number of figures to check the effectiveness of our intended method.
2 Problem Formulation and Preliminaries
We make following assumptions throughout this letter.
The structure of the parameter uncertainty as in equations 2.6 and 2.7 was extensively exploited in the analysis of robust control and filtering of uncertain systems (Wang, Xie, & De Souza, 1992; Wang & Qiao, 2002). Many practical systems have unknown parameters that can either be modeled exactly or overbound by equation 2.7.
The following lemmas are useful in deriving the stability results for SNNs, equation 2.5:
(Yue, Tian, Zhang, & Peng, 2009). Let and be real matrices of appropriate dimensions with , . Then for any scalar satisfying , we have
3 Main Results
In this section, we derive a delay-dependent stochastic stability condition based on suitable LKF and LMI approaches.
To show that our major results are sufficiently general to cover certain cases that have been discussed in the literature, we give a few corollaries.
4 Numerical Examples
4.1 Example 1
4.2 Example 2
This letter has discussed robust, asymptotic stability analysis for uncertain, stochastic-delayed NNs. In theorem 6, by constructing a suitable LKF and utilizing Wirtinger-based inequality, we derived the sufficient condition for asymptotic stability of the system with time delay. An LMI approach has been proposed to check the mean square stability of stochastic uncertain neural networks, which can be tested easily using Matlab's LMI toolbox. We provided examples to illustrate the effectiveness of our main results.