Asymptotic Stability and Asymptotic Synchronization of Memristive Regulatory-Type Networks

Memristive regulatory-type networks are recently emerging as a potential successor to traditional complementary resistive switch models. Qualitative analysis is useful in designing and synthesizingmemristive regulatory-type networks. In this paper, we propose several succinct criteria to ensure global asymptotic stability and global asymptotic synchronization for a general class ofmemristive regulatory-type networks. The experimental simulations also show the performance of theoretical results.


Introduction
Using memristive devices as synapses is a focus in memristive networks.To extract the benefits of high-efficiency memristive memory, various memristive networks have been reported to date [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18].Unlike conventional two-terminal devices, memristive networks exhibit pinched memristor hysteresis loop characteristics, making them particularly suitable for linear-drift devices [10].Moreover, as the modular compact model for memristors, memristive regulatorytype networks are further broadened to memristive systems that exhibit the phenomenon of closed-form sneak paths, which enable nanoscale geometries with short access latencies.A memristive regulatory-type network contains multiply-threshold synapses, which has been heralded as a new paradigm in large-scale circuits.Compared with some memristive systems, a memristive regulatory-type network has the following advantages: (1) it is more biomimetic in behaviors with simple system structure; (2) it simplifies the structure and complication of circuits and is easy to realize.With these coveted properties, memristive regulatory-type networks have the potential of realizations in module-based nanoscale neuromorphic computing systems.
The underlying physics mechanism of memristor models is extremely complex.In order to explore the characteristics and applications of memristive networks, several attempts in [1][2][3][4][6][7][8][9][10][11][12][13][14][15][16][17][18] have been made, using nonlinear system theory, to develop behavioral models of memristors.An ideal dynamic property is a critical requirement for the development and validation of memristive networks.Evolutional characteristics of memristive networks are an interesting and prosperous research area.However, deploying nonlinear analysis technology in memristive networks is challenging because a memristive network is basically a switched network cluster [13,15].Such switched network cluster thus possesses the synaptic action, in which the synaptic weight can be incrementally ameliorated by adjusting the charge or flux through it.There are two major obstacles to analyze and control the memristive networks, namely, high complexity and switched hybridity [12,13].On the other hand, dynamical analysis for memristive networks can explain carrier dynamics and associated transients.Once the electronic properties of memristive networks are revealed, then the circuit models can be implemented based upon the underlying dynamic nature.By tweaking physical structures and bias conditions, system designer can optimize the circuit performance, and then, numerous potential applications of the memristors have been exploited, such as neuromorphic, digital, and quantum computation.
In spite of having significant progress in the area of nonlinear control systems [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35], memristive regulatorytype networks constituting switched network cluster have received less attention.It has been reasoned that much like neuroevolutionary systems, memristive regulatory-type 2 Advances in Mathematical Physics networks could be responsible for different neuromorphic architectures [36,37].To this end, we focus on the evolution of memristive regulatory-type networks.In this paper, we study global asymptotic stability and global asymptotic synchronization of a class of memristive regulatory-type networks.Based on -matrix theory, we develop less conservative global asymptotic stability results and global asymptotic synchronization results for memristive regulatory-type networks.Such theoretical analysis can significantly help understand and identify system performance, especially in neuromorphic computing era where stability or synchronization is crucial.In fact, dynamic analysis of memristive regulatorytype networks can provide an overview for optimizing the circuit device and enhancing circuit performances.
The rest of this paper is organized as follows.Section 2 introduces model description and preliminaries.Section 3 gives main results.Section 4 discusses two numerical examples to demonstrate the effectiveness of theoretical results.Finally, Section 5 concludes the paper with some remarks.

Model Description and Preliminaries
Consider a general class of memristive regulatory-type networks described by the following delay differential equations: for  = 1, 2, . . ., ,  = 1, 2, . . ., , where   () and   () represent the concentration variations of memristive messenger gene  and affiliated organic compound , respectively,   > 0 and   > 0 denote the degradation rates of memristive messenger gene  and affiliated organic compound , respectively,   ≥ 0 represents the translating rate, nonlinear function   (⋅) is bounded and   (0) = 0, 0 ≤   ≤  and 0 ≤   ≤  ( ≥ 0 is a constant) denote the regulating delay and the translating delay, respectively, and   (  ()) represents regulatory relationship of the network, which is defined as where b and b  are constants.The initial conditions of system (1) are assumed to be where   () and   () are both continuous functions defined on [−, 0].
According to Lyapunov direct method, from Definition 2, as we know, if there exists an appropriate Lyapunov function  which is positive definite and radially unbounded, such that the time-derivative of  along the trajectory of system (1) is negative definite, then the equilibrium point of system (1) is globally asymptotically stable.
then the error dynamics can be expressed by the following form: and we say that the response system can be globally asymptotically synchronized with the drive system if the zero solution of error system is globally asymptotically stable.

Main Results
In this section, we will first give two lemmas, which play important role in the analysis and synthesis of memristive regulatory-type networks (1).
Using standard arguments as Lemmas 1 and 2 in [15], Lemmas 4 and 5 of this paper can be proved, respectively.

Global Asymptotic
where According to Lemma   Proof.Since matrix W is a nonsingular -matrix, by the matrix theory, it follows that W  is a nonsingular -matrix.
Based on the fact that W  is a nonsingular -matrix, then there exists an ( + )-dimensional vector  > 0 such that W   > 0; that is, Choose and then we get Calculating the upper right Dini derivative of ((), ()) along the trajectory of system (12)  Proof.The proof is a direct result of Theorem 6.
Proof.Since matrix W is a nonsingular -matrix, by the matrix theory, it follows that W  is a nonsingular -matrix.
Based on the fact that W  is a nonsingular -matrix, then there exists an ( + )-dimensional vector  > 0 such that W   > 0; that is, + > 0, for  = 1, 2, . . ., , Choose and then we get Consider the following positive definite and radially unbounded Lyapunov function: Calculating the upper right Dini derivative of ((), E()) along the trajectory of system (28) yields By Lyapunov global asymptotic stability theory, we can conclude that system (28) is globally asymptotically stable.Thus, the response system (24) can be globally asymptotically synchronized with the drive system (1).The proof is completed.
Next we extend Theorem 9 to other possible cases.
Corollary 10.The zero solution of system ( 28) is globally asymptotically stable; that is, the response system (24) can be globally asymptotically synchronized with the drive system (1), if b   , for  = 1, 2, . . ., . (37) Proof.Select the ( + )-dimensional unit vector as  in the proof of Theorem 9, from (37), it follows that (32) hold.Therefore, the conclusion of Corollary 10 is obvious.
Corollary 11.When  = , the zero solution of system ( 28) is globally asymptotically stable; that is, the response system (24) can be globally asymptotically synchronized with the drive system (1), if the matrix is a nonsingular -matrix, where A = diag(

Illustrative Examples
In this section, we discuss two numerical examples to illustrate the theoretical results.

Conclusion
Memristive network can achieve more expedient goal-finding behavior in spiking networks via memristive connections, which has aroused considerable interest by electronics researchers.The practical applications of memristive network popularizes real-time processing and recognition of natural signals.It is of great significance to investigate its nonlinear dynamics.In this paper, we study global asymptotic stability and global asymptotic synchronization for memristive regulatory-type networks, based on the -matrix theory and Lyapunov stability theory.These criteria, which can be directly derived from the system parameters, are easily verified.The theoretical results developed in this paper may be applied to the synthesis of memristive regulatory-type networks.

Figure 2 :
Figure 2: Phase portraits of system (39) in the three-dimensional space.
1 + K 1 ,  2 + K 2 , ...,   + K  ), B = ( b   ) × , C = diag( 1 + H 1 ,  2 + H 2 , ...,   + H  ), and D = (  ) × .Proof.The proof is a direct result of Theorem 9.Remark 12. Theorem 9 and Corollaries 10 and 11 show the feasibility of linear feedback scheme for designing a perfect control in memristive regulatory-type networks, and the sufficient conditions only depend on some system parameters, which are easy to be checked.Remark 13.Compared with many other control strategies, linear feedback scheme is more suitable for implementation in memristive regulatory-type networks.For one thing, transient states are quite prevalent in memristive regulatory-type networks; that is, state-dependent jump abruptly spikes up or down with uncertainty.For another thing, linear feedback scheme itself is relatively cheaper and simpler to operate.It is more reasonable and implementable for linear feedback scheme only carried out at finite gain and bandwidth.