Adaptive passivity and synchronization of coupled reaction-diffusion neural networks with multiple state couplings or spatial diffusion couplings

doi:10.1016/j.neucom.2019.10.027

Neurocomputing

Volume 377, 15 February 2020, Pages 168-181

https://doi.org/10.1016/j.neucom.2019.10.027 Get rights and content

Abstract

In this paper, two types of coupled reaction-diffusion neural networks with multiple state couplings or spatial diffusion couplings are presented. By selecting appropriate adaptive control schemes and employing inequality techniques, several passivity conditions for these network models are given. In addition, two sufficient conditions for ensuring the synchronization of the proposed network models are also established by exploiting the output-strictly passivity. Finally, we give two numerical examples to verify the effectiveness of the derived criteria.

Introduction

Recently, the dynamical behaviors of neural networks (NNs) have been widely studied in various fields due to their diversity of applications in optimization, pattern classification, image processing, signal processing, etc. [1], [2], [3], [4]. As is well known, the phenomenon of reaction diffusion is caused by the movement of electrons in a nonuniform electromagnetic field, which is unavoidable [5], [6], [7]. Therefore, it is very important to consider the reaction-diffusion phenomenon in the neural networks. Until now, the dynamical behaviors of reaction-diffusion neural networks (RDNNs) have been studied extensively and deeply [8], [9], [10], [11], [12], [13]. With the help of some inequality techniques and Lyapunov functional method, Wang et al. [8] not only analyzed the passivity of the RDNNs, but also considered the robust passivity under the circumstance that the parameter uncertainties appear in RDNNs. In [9], the robust passivity and passivity criteria were derived for stochastic RDNNs with time-varying delays by employing stochastic analysis and inequality techniques. In [10], the global exponential stability of RDNNs with time-varying delays was researched by exploiting the Wirtinger’s inequality and a diffusion-dependent Lyapunov functional.

More recently, coupled RDNNs (CRDNNs) consisting of several identical or nonidentical RDNNs has attracted much attention in different research areas, such as harmonic oscillation generation, pattern recognition, and chaotic generator design. Therefore, the dynamical behaviors of CRDNNs have been extensively studied, in particular the synchronization [14], [15], [16], [17], [18], [19], [20] and passivity [21], [22], [23], [24], [25]. Wang et al. [14] respectively discussed the pinning synchronization for state coupled and spatial diffusion coupled CRDNNs, and several synchronization criteria for these networks were derived by exploiting several inequality techniques and Lyapunov functional method. In [15], the synchronization problem for CRDNNs with time delays was studied by exploiting the adaptive feedback control method, and the authors presented several sufficient conditions to ensure the synchronization based on the LaSalle invariant principle. In [19], several exponential synchronization criteria were obtained for a class of delayed CRDNNs by using the pinning impulsive control method. In [23], the authors considered the passivity of the CRDNNs with switching topology, several sufficient conditions were given to guarantee the output-strictly passivity and input-strictly passivity by using the Lyapunov functional method and inequality techniques. In [25], Huang et al. proposed a nonlinear CRDNNs, and the pinning passivity and passivity problems of such network model were studied. Unfortunately, in these existing results on the passivity of the CRDNNs, the input and output were required to have the same dimension [21], [22], [23], [24], [25], [26]. As far as we know, passivity of CRDNNs with different input and output dimensions was seldom studied [27], [28]. Wang et al. [27] discussed the passivity for two types of CRDNNs with different dimensions of input and output, and analyzed the internal stability of the CRDNNs. In [28], some passivity criteria for the CRDNNs with undirected and directed topologies were gained based on the designed adaptive laws, and the authors also studied the relationship between the output strict passivity and synchronization.

However, in the above-mentioned works about the dynamical behaviors of CRDNNs are all based upon the single weighted network models [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28]. Actually, many networks in the real life should be modeled by multi-weighted models of complex network, examples are social networks, public traffic roads networks, etc. [29]. The dynamical behaviors for complex networks with multi-weights have been studied by some authors in recent years [30], [31], [32], [33], [34], [35], [36]. In [32], the authors gave some passivity criteria for complex networks with multi-weights by employing Lyapunov functional method and inequality techniques, and designed the pinning control strategy based on nodes and edges for ensuring the passivity, and the relationship between the output-strictly passivity and synchronization was also studied. Qin et al. [33] discussed the $H_{\infty}$ synchronization and synchronization of the complex dynamical networks with multi-weights for switching and fixed topologies, and presented several $H_{\infty}$ synchronization and synchronization criteria by employing Lyapunov functional method and inequality techniques. Regretfully, very few authors have considered the dynamical behaviors about the CRDNNs with multi-weights [37]. In [37], Wang et al. discussed the finite-time synchronization and finite-time passivity problems for CRDNNs with multiple state couplings and multiple delayed state couplings, and several sufficient conditions for ensuring the finite-time passivity or synchronization were given by selecting appropriate controllers. Particularly, no authors discussed the dynamical behaviors for CRDNNs with multiple spatial diffusion couplings. Therefore, it is very meaningful to further research the passivity and synchronization for CRDNNs with multiple state couplings or spatial diffusion couplings.

In this paper, we respectively study the passivity and synchronization of CRDNNs with multiple state couplings or spatial diffusion couplings. The main contributions of this paper are as follows. First, several passivity criteria for the multiple state coupled CRDNNs are obtained by employing some inequality techniques and the adaptive state feedback controller. Second, the synchronization problem of CRDNNs with multiple state couplings are studied based on the output-strictly passivity. Third, the adaptive passivity and synchronization problems for CRDNNs with multiple spatial diffusion couplings are also discussed.

Notations: $R^{ϵ} ∋ Ω = {a = {(a_{1}, a_{2}, \dots, a_{ϵ})}^{T} | | a_{h} | < c_{h}, h = 1, 2, \dots, ϵ}$ . γ_m( · ) and γ_M( · ) denote the minimum and the maximum eigenvalue of the corresponding matrix. $D = {1, 2, \dots, Z}$ represents node set, $B \subseteq D \times D$ represents undirected edge set. $Z_{p} = {q \in D$ : $(p, q) \in B$ }.

Section snippets

Definitions

Definition 2.1

(see [27])

If there exists a non-negative function W and a constant matrix $H \in R^{β \times ζ}$ such that $\int_{t_{0}}^{t_{r}} \int_{Ω} y^{T} (a, t) H u (a, t) d a d t ⩾ W (t_{r}) - W (t_{0})$ for any $t_{r}, t_{0} \in R^{+}$ and t_r ≥ t₀, the system with input $u (a, t) \in R^{ζ}$ and output $y (a, t) \in R^{β}$ is passive.

Definition 2.2

(see [27])

If there exists a non-negative function W and two constant matrices $R^{ζ \times ζ} ∋ Q_{1} > 0$ and $H \in R^{β \times ζ}$ such that $\begin{matrix} \int_{t_{0}}^{t_{r}} \int_{Ω} y^{T} (a, t) H u (a, t) d a d t \\ ⩾ W (t_{r}) - W (t_{0}) + \int_{t_{0}}^{t_{r}} \int_{Ω} u^{T} (a, t) Q_{1} u (a, t) d a d t \end{matrix}$ for any $t_{r}, t_{0} \in R^{+}$ and t_r ≥ t₀, the system is input-strictly passive.

Definition 2.3

(see [27])

If there exists a non-negative function W and two constant

Network model

The network model considered in this section is described by: $\begin{matrix} \frac{\partial χ_{p} (a, t)}{\partial t} & = & M ▵ χ_{p} (a, t) - A χ_{p} (a, t) + E f (χ_{p} (a, t)) + J + K u_{p} (a, t) \\ + \sum_{s = 1}^{d} \sum_{q = 1}^{Z} b_{s} B_{p q}^{s} Γ^{s} χ_{q} (a, t) + v_{p} (a, t), p = 1, 2, \dots, Z, \end{matrix}$ where $χ_{p} (a, t) = {(χ_{p 1} (a, t), χ_{p 2} (a, t), \dots, χ_{p z} (a, t))}^{T} \in R^{z}$ is the state vector of node p; $u_{p} (a, t) \in R^{η}$ is the external input of node p; $v_{p} (a, t) \in R^{z}$ is the control input of node p; $0 < M = diag (m_{1}, m_{2}, \dots, m_{z}) \in R^{z \times z}$ ; $0 < A = diag (Λ_{1}, Λ_{2}, \dots, Λ_{z}) \in R^{z \times z}$ ; $▵ = \sum_{h = 1}^{ϵ} (\partial^{2} / \partial a_{h}^{2})$ ; $f (χ_{p} (a, t)) = {(f_{1} (χ_{p 1} (a, t)), f_{2} (χ_{p 2} (a, t)), \dots, f_{z} (χ_{p z} (a, t)))}^{T} \in R^{z}$ ; $J = {(J_{1}, J_{2}, \dots, J_{z})}^{T} \in R^{z}$ ; $E \in R^{z \times z}$ and $K \in R^{z \times η}$ are

Network model

In this section, the network (15) is connected. The boundary value and initial value for network (15) are given as follows: $\begin{matrix} χ_{p} (a, t) & = & 0, (a, t) \in \partial Ω \times [0, + \infty) \end{matrix}$

Numerical examples

Example 5.1

The CRDNNs with multiple state couplings is given as follows: $\begin{matrix} \frac{\partial χ_{p} (a, t)}{\partial t} & = & M ▵ χ_{p} (a, t) - A χ_{p} (a, t) + E f (χ_{p} (a, t)) + J \\ + 2 \sum_{q = 1}^{5} B_{p q}^{1} Γ^{1} χ_{q} (a, t) + K u_{p} (a, t) \\ + 3 \sum_{q = 1}^{5} B_{p q}^{2} Γ^{2} χ_{q} (a, t) + v_{p} (a, t) \\ + 4 \sum_{q = 1}^{5} B_{p q}^{3} Γ^{3} χ_{q} (a, t), \end{matrix}$ where $p = 1, 2, \dots, 5, f_{k} (ξ) = \frac{| ξ + 1 | - | ξ - 1 |}{4}, k = 1, 2, 3, Γ^{1} = diag (0.5, 0.6, 0.5), Γ^{2} = diag (0.7, 0.4, 0.8), Γ^{3} = diag (0.5, 0.7, 0.8), Ω = {a ∣ - 0.5 < a < 0.5}, J = {(0.6, 0.4, 0.8)}^{T}, M = diag (0.3, 0.5, 0.2), A = diag (0.7, 0.9, 0.8) .$ $\begin{matrix} E = (\begin{matrix} 0.4 & 0.7 & 0.5 \\ 0.5 & 0.5 & 0.6 \\ 0.2 & 0.3 & 0.7 \end{matrix}), K = (\begin{matrix} 0.4 & 0.3 \\ 0.4 & 0.4 \\ 0.5 & 0.3 \end{matrix}), \end{matrix}$ $\begin{matrix} B^{1} = (\begin{matrix} - 0.4 & 0.1 & 0 & 0.2 & 0.1 \\ 0.1 & - 0.4 & 0.1 & 0.1 & 0.1 \\ 0 & 0.1 & - 0.6 & 0 & 0.5 \\ 0.2 & 0.1 & 0 & - 0.6 & 0.3 \\ 0.1 & 0.1 & 0.5 & 0.3 & - 1 \end{matrix}), \\ B^{2} = (\begin{matrix} - \end{matrix} \end{matrix}$

Conclusion

In this paper, the CRDNNs with multiple state couplings or spatial diffusion couplings have been studied. By choosing appropriate adaptive state feedback controllers and making use of some inequality techniques, we have given several passivity conditions for these network models. Furthermore, two synchronization criteria for these networks also have been established by exploiting the obtained output strictly passivity results. Finally, two numeral examples have been provided to verify the

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.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grant 61773285, in part by the Natural Science Foundation of Tianjin, China, under Grant 19JCYBJC18700, and in part by the Program for Innovative Research Team in University of Tianjin (No. TD13-5032).

Lu Wang received the B.E. degree in Software Engineering from Tiangong University, Tianjin, China, in 2018. She is currently pursuing the M.E. degree in Computer Technology with the School of Computer Science and Technology, Tiangong University, Tianjin, China. Her current research interests include stability, passivity, synchronization and neural networks.

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It is worth noticing that most results above for CRDNNs only consider the state information, while spatial information is a great resource for synchronization. Thus, it is very necessary to develop dynamical properties of CRDNNs with hybrid coupling (CRDNNHC), and some synchronization results have been reported recently [15,22,35,40]. [35] discussed adaptive synchronization and passivity problems for CRDNNHC by exploiting its state information and spatial information.
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Third, several passivity and synchronization criteria for CRDNNMSDCs are obtained by taking advantage of the PD method. In recent years, some authors have investigated the dynamical behaviors for CRDNNs by using the state feedback controllers [24–26]. Considering that the derivative control can enhance stability and speed up the system response, it is also beneficial to utilize the PD controller to study the dynamical behaviors of CRDNNs, but the result about this topic has not yet been reported.
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Jin-Liang Wang received the Ph.D. degree in control theory and control engineering from the School of Automation Science and Electrical Engineering, Beihang University, Beijing, China, in January 2014.

In January 2014, he joined the School of Computer Science and Technology, Tiangong University, Tianjin, China, as an Associate Professor, where he has been promoted to a Professor since March 2018. In 2014, he was a Program Aid with Texas A & M University at Qatar, Doha, Qatar, for two months. From June 2015 to July 2015 and from July 2016 to August 2016, he was a Postdoctoral Research Associate with Texas A & M University at Qatar. From June 2017 to September 2017, he was an Associate Research Scientist in Texas A & M University at Qatar. He has authored two books entitled Analysis and control of coupled neural networks with reaction-diffusion terms (Springer, 2017) and Analysis and control of output synchronization for complex dynamical networks (Springer, 2018). His current research interests include passivity, synchronization, cooperative control, complex networks, coupled neural networks, coupled reaction-diffusion neural networks, and multiagent systems.

Dr. Wang currently serves as an Associate Editor for the Neurocomputing and IEEE Access, was a Managing Guest Editor for the Special Issue of Dynamical behaviors of coupled neural networks with reaction-diffusion terms: analysis, control and applications in Neurocomputing.

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Adaptive passivity and synchronization of coupled reaction-diffusion neural networks with multiple state couplings or spatial diffusion couplings

Abstract

Introduction

Section snippets

Definitions

(see [27])

(see [27])

(see [27])

Network model

Network model

Numerical examples

Conclusion

Declaration of Competing Interest

Acknowledgments

Appl. Math. Comput.

Commun. Nonlinear Sci. Numer. Simul.

Automatica

J. Frankl. Inst.

Neurocomputing

Neurocomputing

Neurocomputing

Commun. Nonlinear Sci. Numer. Simul.

IEEE Trans. Cybern.

J. Frankl. Inst.

Neurocomputing

Neurocomputing

Neural Process Lett.

J. Frankl. Inst.

J. Frankl. Inst.

IEEE Trans. Neural Netw. Sci. Eng.

Phys. A

Complexity

IEEE Trans. Cybern.