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

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 [

The underlying physics mechanism of memristor models is extremely complex. In order to explore the characteristics and applications of memristive networks, several attempts in [

In spite of having significant progress in the area of nonlinear control systems [

The rest of this paper is organized as follows. Section

Consider a general class of memristive regulatory-type networks described by the following delay differential equations: for

The initial conditions of system (

In addition, we also assume that the nonlinear function

In this paper, solutions of all the systems considered in the following are in Filippov’s sense.

When considering memristive regulatory-type networks (

Obviously, for

By the theory of differential inclusions, from (

A solution

The constant vectors

The equilibrium point of system (

According to Lyapunov direct method, from Definition

For drive system

In this section, we will first give two lemmas, which play important role in the analysis and synthesis of memristive regulatory-type networks (

In system (

For system (

Using standard arguments as Lemmas

According to Lemma

According to Lemma

From (

The equilibrium points

Since matrix

Choose

Consider the following positive definite and radially unbounded Lyapunov function:

Calculating the upper right Dini derivative of

By Lyapunov global asymptotic stability theory, we can conclude system (

Next we extend Theorem

The equilibrium points

Select the

When

The proof is a direct result of Theorem

Let (

Next, the linear feedback scheme is used to achieve synchronization between drive memristive regulatory-type networks (

Let

To apply the theories of set-valued maps and differential inclusions, (

According to Lemma

From (

The zero solution of system (

Since matrix

Choose

Consider the following positive definite and radially unbounded Lyapunov function:

Calculating the upper right Dini derivative of

By Lyapunov global asymptotic stability theory, we can conclude that system (

Next we extend Theorem

The zero solution of system (

Select the

When

The proof is a direct result of Theorem

Theorem

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.

The asymptotic synchronization strategy contains more general synchronization behaviors. Through the node cluster, asymptotic synchronization in each group can achieve complete synchronization.

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

Consider the following memristive regulatory-type networks:

Obviously, we can calculate that

The simulation results of system (

Transient behaviors of system (

Phase portraits of system (

Consider the following memristive regulatory-type networks:

Let (

The controllers

The error dynamics

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

The author declares that there is no conflict of interests regarding the publication of this paper.

The work is supported by the Research Project of Hubei Provincial Department of Education of China under Grant T201412.