An improved Lyapunov functional with application to stability of Cohen–Grossberg neural networks of neutral-type with multiple delays
Introduction
During the last few decades, the analysis of the dynamics of some special models of neural networks known as Hopfield neural networks, cellular neural networks and Cohen–Grossberg neural networks has been given quite much attention due to the important applications of these neural network models in various practical engineering problems related to image and signal processing, control theory applications, parallelly computing systems, associative memory applications, fault diagnosis, pattern recognitions and optimization related problems (Chua and Yang, 1988, Cohen and Grossberg, 1983, Guez et al., 1998, Hopfield, 1982, Kosko, 1988, Wang et al., 2011). If neural networks are employed to deal with the dynamics of these types of engineering problems, then, dynamical behaviors of employed neural networks must meet the required criteria for the intended applications. In some practical problems such as optimization related problems, the designed neural network needs possessing some unique equilibrium points to the corresponding independent inputs. In addition to establishing the uniqueness of the equilibrium point, this unique equilibrium point also needs to be globally asymptotically stable. A critical issue is that if a neural system is realized by using VLSI technology to solve these targeted critical issues, the finite switching speeds of electronic components within the circuit and signal transmission times of the neurons in the network will bring about some indispensable time delays. These time delays may cause the designed neural network to change the required dynamical properties to the undesired dynamical properties. For instance, a stable neural network without time delays may exhibit an unstable dynamical behavior when time delays are introduced into neurons states. Due to the existence of such undesired dynamics, it becomes critical to address some required stability characteristics of these types of neural systems taking into account the existence of such delayed parameters. In recent years, a huge number of scientific articles have carried out stability analysis of neural networks having mathematical representations which possess the time delay parameters in states of neurons (Chen et al., 2018, Ma et al., 2018, Manivannan et al., 2018, Song and Chen, 2018, Song et al., 2018, Wang et al., 2018, Xie and Zhu, 2015, Xie et al., 2018, Zhu and Cao, 2010, Zhu and Cao, 2011, Zhu and Cao, 2012, Zhu et al., 2015, Zhu et al., 2018). However, introducing time delays into states of neurons may not always be adequate to establish the desired exact behaviors of neural reaction processes because of the existence of different complex dynamics of neuronal activities. In order to avoid the unexpected complex dynamical behaviors of time delayed neural network models, some suitable delay parameters are also introduced into derivatives of states of neurons. These new delays are defined as neutral delays and neural networks involving neutral delays are known as neutral-type neural networks. These classes of neural systems proved to be employed in some specific important applications in the areas of population ecology, distributed networks involving lossless transmission lines, propagation and diffusion models (Chen et al., 2016, Kolmanovskii and Nosov, 1986, Kuang, 1993, Zhang et al., 2016).
If a neutral-type neural network is implemented with neurons with being time delays and being neutral delays occurring during the transmission of related information from neuron to neuron (), then, depending on the number of delay parameters within the system, the mathematical model of such a neural network may include three different forms of delay parameters. If and for all , then, this neural network will have a single time delay and a single neutral delay parameter. If and for all , then, this neural network will possess discrete time delay parameters and discrete neutral delay parameters. If and are assumed to have different values for all , then, this neural network will have multiple time delay parameters and the multiple neutral delay parameters. If a neutral-type neural network possesses single or discrete delays, then the mathematical model of this neural system will have the property of being represented by the equations of vector–matrix form. Thus, it will be possible to analyze the stability of this neural network model by using linear matrix inequality (LMI) approach and some additional suitable mathematical techniques and methods. Noting the fact that a neutral-type neural network involving multiple delays does not have the property of being represented by the equations of vector and matrix forms, mathematical description of such a system is given by some additionally complicated nonlinear differential equations in which case dynamical analysis of these neural systems becomes an important task to overcome. Thus, most of the past and existing research results have focused on the stability of neutral-type neural networks having the single-discrete delays.
In Samidurai, Marshal, and Balachandran (2010), stability analysis of neutral-type neural networks having discrete delays has been carried out by establishing some adequate Lyapunov functionals together with utilizing LMI techniques. In Song, Long, Zhao, Liu, and Alsaadi (2020) and Muralisankar, Manivannan, and Balasubramaniam (2015), by using the proper Lyapunov functionals and the integral inequality approaches, a variety of sufficient stability conditions of LMI forms have been given for neutral-type neural systems defined by the single time and neutral delays. In Lee, Kwon, and Park (2010), the Lyapunov functional approach and the general delay partitioning techniques have been combined with the generalizing the Jensen inequality to derive some new stability criteria have been given in LMIs forms for the neural network models with the single time and neutral delays. In Shi et al. (2015), the semimartingale convergence theorem and Lyapunov functional approach have been used to present some stability results in LMI forms for neural systems with the single time and neutral delays. In Faydasicok and Arik (2013), an alternative LMI result on stability of neutral neural network having the single time and neutral delays has been given by exploiting the Homeomorphism mapping theorem and Lyapunov functional techniques. In Zhang, Liu, and Yang (2012), some stability results for neutral-type neural systems whose model involves the single time and neutral delays have been derived by utilizing the delay partitioning approaches and the appropriate Lyapunov functionals. In Dharani, Rakkiyappan, and Cao (2015), by making the use of the adequate Lyapunov functionals possessing triple and quadruple integral terms, various sets of stability conditions in LMI forms have been obtained for the neural network models with the single time and neutral delays. In Shi, Zhong, Zhu, Liu and Zeng (2015), using the Wirtinger-based integral inequality with employing convex combination techniques and proper Lyapunov functionals, various sufficient stability conditions in LMI forms have been given for neutral-type neural system including the single time and neutral delays. In Zhang, Wang, Li, and Fei (2018), by utilizing Lyapunov functional technique and Leibniz–Newton formula, alternative stability conditions in LMI forms have been derived for neutral-type neural system having discrete delays. On the other hand, in Liu (2013), by using Lyapunov stability theorems, stochastic analysis theory with adequate LMI techniques, various useful stability conditions in LMI forms have been obtained for neutral-type neural network models having discrete delays. In Huang, Du, and Kang (2013), by utilizing various Lyapunov functionals and descriptor transformation approach, new alternative global stability conditions which are represented by LMI forms have been obtained for neutral-type neural systems whose mathematical models have discrete delays. In addition to the stability results given in Dharani et al., 2015, Faydasicok and Arik, 2013, Lee et al., 2010, Liu, 2013, Muralisankar et al., 2015, Samidurai et al., 2010, Shi, Zhong et al., 2015, Shi, Zhu et al., 2015, Song et al., 2020, Zhang et al., 2012, Zhang et al., 2018 and Huang et al. (2013) in LMIs forms, some recent papers have also proposed the different sets of easily checkable algebraic stability results on neutral-type neural networks including the single or discrete delays (Akca et al., 2015, Arik, 2014, Cheng et al., 2008, Lien et al., 2008, Samli and Arik, 2009, Yang et al., 2015).
Taking into account the fact that the investigation of stability into neutral-type neural systems whose mathematical models have multiple delays is not an easy task to achieve due to complexity of mathematical model of such neural systems, up to date, only a few research papers addressing this problem have been published (Arik, 2020, Faydasicok, 2020a, Faydasicok, 2020b, Ozcan, 2019). Therefore, this paper will carry out a further investigation into stability issue of these neural networks. By using a modified and improved version of the recently introduced Lyapunov functional together with employing basic properties of nonsingular M-matrices, some alternative results on neutral-type neural networks whose mathematical models possess multiple delays are presented.
Section snippets
Neutral-type neural systems and preliminaries
In this research article, we will analyze a Cohen–Grossberg neural network of neutral-type with multiple time and neutral delays, whose mathematical model is determined by the nonlinear equations given below: in which a constant value represents the numbers of neurons, the variable represents the state of th neuron, and are some nonlinear functions. The constant
Stability analysis of neutral-type neural systems
In this section, an improved alternative criterion establishing global asymptotic stability of neutral-type neural system (1) is presented. First, neutral system (1) is being transformed into a corresponding neutral system. Let be the state vector of neural network model (1). Suppose that denotes the equilibrium points of neutral-type neural system (1). Consider that represents a transformed state vector such that
Comparisons and a numerical example
In the current section, by giving a constructive numerical example, we will proceed further to compare the stability conditions of Theorem 1 with the existing stability criteria derived for system (2). In order to make a proper comparison, these previous results are restated:
Theorem 2 For neural system (2), suppose that hold. Let the network parameters satisfy the conditions Ozcan, 2019
Conclusions
This paper has addressed the stability issue for the neutral-type Cohen–Grossberg neural networks involving multiple time delays in states of the neurons and multiple neutral delays in time derivative of states of neurons. By using a modified and improved version of a previously introduced Lyapunov functional, some new sufficient criteria have been derived for global asymptotic stability of Cohen–Grossberg neural networks of neutral-type with multiple delays. Our proposed new stability
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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