Quasi-invariant and attractive sets of inertial neural networks with time-varying and infinite distributed delays
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Quasi-invariant and attractive sets of inertial neural networks with time-varying and infinite distributed delays Qian Tang1,2 · Jigui Jian1,2 Received: 4 June 2019 / Revised: 17 March 2020 / Accepted: 6 May 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract This paper aims at analyzing the quasi-invariant and attractive sets for a class of inertial neural networks with time-varying and infinite distributed delays. By utilizing the properties of nonnegative matrix, a new bidirectional-like delay integral inequality is developed. Some sufficient conditions are obtained for the existence of the quasi-invariant and attractive sets of the discussed system according to the bidirectional-like integral inequality. Besides, the framework of the quasi-invariant and attractive sets for the concerned system is provided. Finally, one example is analyzed to clarify the validity of our results. Keywords Inertial neural network · Infinite distributed delay · Quasi-invariant set · Globally attractive set · Delay integral inequality Mathematics Subject Classification 34A34 · 34D05 · 34D23 · 37L25
1 Introduction Over the years, the dynamical characteristic analysis of artificial neural networks has received considerable attentions for their extensive applications in pattern recognition, signal processing, associative memory, and other areas. At the same time, the dynamical behaviors such as stability, dissipativity, convergence, oscillation, and periodicity have been investigated for various neural networks (Song et al. 2017; Zhang and Yu 2016; Velmurugan et al. 2017; Xu and Li 2017; Manivannan et al. 2018; Velmurugan et al. 2016; Jian and Wan 2015; Li et al. 2017; Zhang and Zeng 2018a, b). In reality, time delay is an inherent character of signal trans-
Communicated by Leonardo Tomazeli Duarte.
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Jigui Jian [email protected] Qian Tang [email protected]
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College of Science, China Three Gorges University, Yichang 443002, Hubei, China
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Three Gorges Mathematical Research Center, China Three Gorges University, Hubei Yichang 443002, China 0123456789().: V,-vol
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mission between neurons, which may lead to oscillations, instability and poor performances of neural networks. Generally, the time delays considered in neural networks can be categorized as constant delays (Zhang and Yu 2016; Velmurugan et al. 2016, 2017), time-varying delays (Zhang and Zeng 2018a, b), and distributed delays (Song et al. 2017; Manivannan et al. 2018; Jian and Wan 2015). Therefore, for the need of applications, the dynamical characteristic analysis of neural networks with delays has turned into a hot research topic. It is noteworthy that lots of former studies mainly concentrated on neural networks with only the first derivative of the states, yet it is also of great significance to recommend the second-order derivative of states of inertial neural networks (INNs). Wheeler and Schieve (Wheeler and Schieve 1997) first proposed a second-order INN model and discussed its sta
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