Non-Fragile H∞ Synchronization for Markov Jump Singularly Perturbed Coupled Neural Networks Subject to Double-Layer Switching Regulation

This work explores the $H_{\infty }$ synchronization issue for singularly perturbed coupled neural networks (SPCNNs) affected by both nonlinear constraints and gain uncertainties, in which a novel double-layer switching regulation containing Markov chain and persistent dwell-time switching regulation (PDTSR) is used. The first layer of switching regulation is the Markov chain to characterize the switching stochastic properties of the systems suffering from random component failures and sudden environmental disturbances. Meanwhile, PDTSR, as the second-layer switching regulation, is used to depict the variations in the transition probability of the aforementioned Markov chain. For systems under double-layer switching regulation, the purpose of the addressed issue is to design a mode-dependent synchronization controller for the network with the desired controller gains calculated by solving convex optimization problems. As such, new sufficient conditions are established to ensure that the synchronization error systems are mean-square exponentially stable with a specified level of the $H_{\infty }$ performance. Eventually, the solvability and validity of the proposed control scheme are illustrated through a numerical simulation.

I. Introduction

As a special kind of complex dynamical networks, coupled neural networks (CNNs) have received high concern from researchers than neural networks because neural networks may encounter difficulties in modeling increasingly intricate actual systems, and then fruitful results focused on CNNs have been achieved and further reported [1]–[4]. To name just a few, an optimized synchronization controller method of CNNs with the nonuniform coupling strengths was proposed in [5]; the synchronization problem was studied in [6] for CNNs containing state coupling and spatial diffusion; Poznyak et al. [7], Wang et al. [8], Tan and Wang [9], and Diesmann et al. [10] reflected the significance of the stability of CNNs. From these results, it is easy to observe that though synchronization, as a fundamental dynamical behavior, has been widely explored, it is only one of the emphases for research in CNNs. The situations about the switching characteristics on the structures or parameters in response to an intricate environment should also be taken into account [11]–[15]. In this regard, the work [16] adopted a kind of spontaneous switching which relied on the stochastic process (e.g., Markov chain) to describe the variations in system parameters due perhaps to the switching phenomenon in CNNs. Besides that, considerable research efforts have been done regarding Markov jump CNNs [17]–[20]. Nonetheless, most results developed above all considered under the assumption that the transition probability of the Markov chain is time-invariant, which is obviously limiting. Subsequently, enlightened by the nonhomogeneous Markov chain in [21], persistent dwell-time switching regulation (PDTSR), which is deemed as a general switching regulation, is used in this work to regulate the frequency of multiple sets of transition probabilities in Markov chain. To the authors’ knowledge, compared with the regular cooperation between dwell-time switching regulation (DTSR)/average dwell-time switching regulation (ADTSR) and Markov chain [14], few works use double-layer switching regulation which contains PDTSR and Markov chain on CNNs to further address synchronization problem. Therefore, carrying out this work will be a small step in the novel attempt.