Almost Periodic Solution for Memristive Neural Networks with Time-Varying Delays

Abstract

This paper is concerned with the dynamical stability analysis for almost periodic solution of memristive neural networks with time-varying delays. Under the framework of Filippov solutions, by applying the inequality analysis techniques, the existence and asymptotically almost periodic behavior of solutions are discussed. Based on the differential inclusions theory and Lyapunov functional approach, the stability issues of almost periodic solution are investigated, and a sufficient condition for the existence, uniqueness, and global exponential stability of the almost periodic solution is established. Moreover, as a special case, the condition which ensures the global exponential stability of a unique periodic solution is also presented for the considered memristive neural networks. Two examples are given to illustrate the validity of the theoretical results.

1. Introduction

Memristor (resistor with memory), which was firstly postulated by Chua in [1], is the fourth fundamental electronic component along with the resistor, inductor, and capacitor. On May 1, 2008, the Hewlett-Packard (HP) research team announced their realization of a memristor prototype, with an official publication in Nature [2, 3]. This new circuit element is a two-terminal element, either a charge-controlled memristor or a flux-controlled memristor, and shares many properties of resistors and the same unit of measurement (ohm). Subsequently, memristor has received a great deal of attention from many scientists because of its potential applications in next generation computer and powerful brain-like “neural” computer [4–14].

Recently, various memristor-based networks have been established by means of the memristive circuits, and many applications have been made in science and engineering fields [15–17]; see, for example, Cserey et al. who presented simulation measurements of a memristor crossbar device and designed a PCB memristor package and the appropriate measurement board [17]. It should be pointed that, in many applications, the existing memristor-based networks which many researchers had designed have been found to be computationally restrictive.

Neural networks, such as Hopfield neural networks, Cellular neural networks, Cohen-Grossberg, and bidirectional associative neural networks, are very important nonlinear circuit networks, and, in the past few decades, have been extensively studied due to their potential applications in classification, signal and image processing, parallel computing, associate memories, optimization, cryptography, and so forth. [18–27]. Many results, which deal with the dynamics of various neural networks such as stability, periodic oscillation, bifurcation, and chaos, have been obtained by applying Lyapunov stability theory; see, for example [28–45] and the references therein. Very recently, memristor-based neural networks (memristive neural networks) have been designed by replacing the resistors in the primitive neural networks with memristors in [46–52]. As is well known, the memristor exhibits the feature of pinched hysteresis, which means that a lag occurs between the application and the removal of a field and its subsequent effect, just as the neurons in the human brain have. Because of this feature, the memristive neural networks can remember its past dynamical history, store a continuous set of states, and be “plastic” according to the presynaptic and postsynaptic neuronal activity. In [46], Itoh and Chua designed a memristor cellular automaton and a memristor discrete-time cellular neural network, which can perform a number of applications such as logical operations, image processing operations, complex behaviors, higher brain functions, and RSA algorithm. In [47], Pershin and Di Ventra constructed a simple neural network consisting of three electronic neurons connected by two memristor-emulator synapses and demonstrated experimentally the formation of associative memory in these memristive neural networks. This experimental demonstration opens up new possibilities in the understanding of neural processes using memory devices, an important step forward to reproduce complex learning, adaptive and spontaneous behavior with electronic neural networks.

It is well known that, in the design of practical neural networks, the qualitative analysis of neural network dynamics plays an important role. For example, to solve problems of optimization, neural control, and signal processing, neural networks have to be designed in such a way that, for a given external input, they exhibit only one globally asymptotically/exponentially stable equilibrium point. Hence, in practice applications, it is an essential issue to discuss the stability for the memristive neural networks. In [48], Hu and Wang proposed a piecewise-linear mathematical model of the memristor to characterize the pinched hysteresis feature. Based on this model, the memristor-based recurrent neural network model with time delays was given and two sufficient conditions for the global uniform asymptotic stability of the memristor-based recurrent neural networks were obtained. In [49, 50], Wu et al. investigated the synchronization control issue of a general class of memristor-based recurrent neural networks with time delays, and some sufficient conditions were obtained to guarantee the exponential synchronization of the coupled networks based on drive-response concept. In [51], the dynamic behaviors for a class of memristor-based Hopfield networks were analyzed, and some sufficient conditions were obtained to ensure the essential bound of solutions and global exponential stability. In [52], the stability was considered for the memristor-based recurrent network with bounded activation functions and bounded time-varying delays in the presence of strong external stimuli, and a sufficient condition on the bounds of stimuli was derived for global exponential stability of memristor-based recurrent networks.

It should be noted that, very little attention has been paid to dealing with the periodicity issue, in particular; to the best of our knowledge, the almost periodic dynamics of memristive neural networks with time-varying delays have never been considered in the previous literature, which motivates the work of this paper.

In this paper, our aim is to study the exponential stability of almost periodic solution for memristive neural networks with time-varying delays. By using the concept of Filippov solutions for the differential equations with discontinuous right-hand sides and the inequality analysis techniques, the existence and asymptotically almost periodic behavior of solution will be discussed. Based on the differential inclusions theory, the proof of the existence of the almost periodic solution will be given. By applying Lyapunov functional approach, a sufficient condition will be established to ensure the uniqueness and global exponential stability of the almost periodic solution for the considered memristive neural networks. As a special case, the conditions of the global exponential stability of a unique periodic solution equilibrium point are also presented.

The rest of this paper is organized as follows. In Section 2, the model formulation and some preliminaries are given. In Section 3, the existence and asymptotically almost periodic behavior of solutions are analyzed, the existence of the almost periodic solution is proved, and the uniqueness and global exponential stability of the almost periodic solution are investigated. In Section 4, two numerical examples are presented to demonstrate the validity of the proposed results. Some conclusions are made in Section 5.

Notations. Throughout this paper, 𝑅 denotes the set of real numbers, 𝑅^𝑛 denotes the 𝑛-dimensional Euclidean space, and 𝑅^𝑚×𝑛 denotes the set of all 𝑚×𝑛 real matrices. For any matrix 𝐴, 𝐴^𝑇 denotes the transpose of 𝐴. If 𝐴 is a real symmetric matrix, 𝐴>0(𝐴<0) means that 𝐴 is positive definite (negative definite). Given the column vectors 𝑥=(𝑥₁,…,𝑥_𝑛)^𝑇,  𝑦=(𝑦₁,…,𝑦_𝑛)^𝑇∈𝑅^𝑛, 𝑥^𝑇𝑦=∑^𝑛_𝑖=1𝑥_𝑖𝑦_𝑖, |𝑥|=(|𝑥₁|,…,|𝑥_𝑛|)^𝑇, and ‖𝑥‖=(∑^𝑛_𝑖=1𝑥²_𝑖)^1/2. ‖𝐴‖=𝜆_𝑀(𝐴^𝑇𝐴) represents the norm of 𝐴, where 𝜆_𝑀(𝐴) is the maximum eigenvalue of 𝐴. 𝐶([−𝜏,0];𝑅^𝑛) denotes the family of continuous function 𝜑 from [−𝑟,0] to 𝑅^𝑛 with the norm ‖𝜑‖=sup_{−𝑟≤𝑠≤0}|𝜑(𝑠)|. 𝑥̇(𝑡) denotes the derivative of 𝑥(𝑡). Matrices, if their dimensions are not explicitly stated, are assumed to have compatible dimensions for algebraic operations.

2. Model Description and Preliminaries

The KCL equation of the 𝑖th subsystem of a general class of neural networks with time-varying delays can be written as 𝑥̇_𝑖(𝑡)1=−ℂ_𝑖^𝑛_𝑗=11ℝ_𝑖𝑗+1𝔽_𝑖𝑗×sgn_𝑖𝑗+1ℛ_𝑖𝑥_𝑖(𝑡)+1ℂ_𝑖^𝑛_𝑗=1𝑔_𝑗𝑥_𝑗(𝑡)ℝ_𝑖𝑗×sgn_𝑖𝑗+1ℂ_𝑖^𝑛_𝑗=1𝑔_𝑗𝑥_𝑗𝑡−𝜏_𝑗(𝑡)𝔽_𝑖𝑗×sgn_𝑖𝑗+𝕀_𝑖ℂ_𝑖,𝑡≥0,𝑖=1,2,…,𝑛, (1) where 𝑥_𝑖(𝑡) is the voltage of the capacitor ℂ_𝑖; ℝ_𝑖𝑗 denotes the resistor between the feedback function 𝑔_𝑖(𝑥_𝑖(𝑡)) and 𝑥_𝑖(𝑡); 𝔽_𝑖𝑗 denotes the resistor between the feedback function 𝑔_𝑖(𝑥_𝑖(𝑡−𝜏_𝑖(𝑡))) and 𝑥_𝑖(𝑡);𝜏_𝑖(𝑡) corresponds to the transmission delay; ℛ_𝑖 represents the parallel resistor corresponding to the capacitor ℂ_𝑖; 𝕀_𝑖 is the external input or bias; sgn_𝑖𝑗=1,𝑖≠𝑗,−1,𝑖=𝑗. (2) Let 𝑑_𝑖=−(1/ℂ_𝑖)[∑^𝑛_𝑗=1(1/ℝ_𝑖𝑗+1/𝔽_𝑖𝑗)×sgn_𝑖𝑗+1/ℛ_𝑖], 𝑎_𝑖𝑗=sgn_𝑖𝑗/ℂ_𝑖ℝ_𝑖𝑗,  𝑏̃_𝑖𝑗=sgn_𝑖𝑗/ℂ_𝑖𝔽_𝑖𝑗, and 𝐼_𝑖=𝕀_𝑖/ℂ_𝑖, then (1) can be rewritten as 𝑥̇_𝑖(𝑡)=−𝑑_𝑖𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗𝑔_𝑗𝑥_𝑗(𝑡+_𝑛𝑗=1)𝑏̃_𝑖𝑗𝑔_𝑗𝑥_𝑗𝑡−𝜏_𝑗(𝑡)+𝐼_𝑖,𝑡≥0,𝑖=1,2,…,𝑛. (3) By replacing the resistors ℝ_𝑖𝑗 and 𝔽_𝑖𝑗 in the primitive neural networks (1) or (3) with memristors whose memductances are 𝕎_𝑖𝑗 and 𝕄_𝑖𝑗, respectively, then memristive neural networks with time-varying delays can be designed as 𝑥̇_𝑖(𝑡)=−𝑑_𝑖𝑥_𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗𝑥_𝑖(𝑡)𝑔_𝑗𝑥_𝑗(𝑡+_𝑛𝑗=1)𝑏_𝑖𝑗𝑥_𝑖(𝑡)𝑔_𝑗𝑥_𝑗𝑡−𝜏_𝑗(𝑡)+𝐼_𝑖,𝑡≥0,𝑖=1,2,…,𝑛, (4) where 𝑑_𝑖(𝑥_𝑖(𝑡))=−(1/ℂ_𝑖)[∑^𝑛_𝑗=1(𝕎_𝑖𝑗+𝕄_𝑖𝑗)×sgn_𝑖𝑗+1/ℛ_𝑖], 𝑎_𝑖𝑗(𝑥_𝑖(𝑡))=(𝕎_𝑖𝑗/ℂ_𝑖)×sgn_𝑖𝑗, 𝑏_𝑖𝑗(𝑥_𝑖(𝑡))=(𝕄_𝑖𝑗/ℂ_𝑖)×sgn_𝑖𝑗, and 𝐼_𝑖=𝕀_𝑖/ℂ_𝑖.

Combining the typical current-voltage characteristics of memristor (see Figure  1 in [48]), similarly to discussion in [49, 50], the coefficient parameters of the system (4) 𝑑_𝑖(𝑥_𝑖(𝑡)), 𝑎_𝑖𝑗(𝑥_𝑖(𝑡)), and 𝑏_𝑖𝑗(𝑥_𝑖(𝑡)) can be modeled as 𝑑_𝑖𝑥_𝑖(𝑡)=𝑑_𝑖,𝑥_𝑖(𝑡)<𝑇_𝑖,𝑑̌_𝑖,𝑥_𝑖(𝑡)>𝑇_𝑖,𝑎_𝑖𝑗𝑥_𝑖(𝑡)=𝑎_𝑖𝑗,𝑥_𝑖(𝑡)<𝑇_𝑖,𝑎̌_𝑖𝑗,𝑥_𝑖(𝑡)>𝑇_𝑖,𝑏_𝑖𝑗𝑥_𝑖(𝑡)=𝑏̂_𝑖𝑗,𝑥_𝑖(𝑡)<𝑇_𝑖,𝑏̌_𝑖𝑗,𝑥_𝑖(𝑡)>𝑇_𝑖, (5) where switching jumps 𝑇_𝑖>0, 𝑑_𝑖>0, 𝑑̌_𝑖>0, 𝑎_𝑖𝑗, 𝑎̌_𝑖𝑗, 𝑏̂_𝑖𝑗, and 𝑏̌_𝑖𝑗, 𝑖,𝑗=1,2,…,𝑛, are constant numbers.

The initial value associated with the system (4) is 𝑥_𝑖(𝑡)=𝜑_𝑖(𝑡)∈𝐶([−𝜏,0];𝑅),𝑖=1,2,…,𝑛.

Let 𝑑_𝑖=max{𝑑_𝑖,𝑑̆_𝑖}, 𝑑_𝑖=min{𝑑_𝑖,𝑑̆_𝑖}, 𝑎_𝑖𝑗=max{𝑎_𝑖𝑗,𝑎̆_𝑖𝑗}, 𝑎_𝑖𝑗=min{𝑎_𝑖𝑗,𝑎̆_𝑖𝑗}, 𝑏_𝑖𝑗=max{𝑏̂_𝑖𝑗,𝑏̆_𝑖𝑗}, and 𝑏_𝑖𝑗=min{𝑏̂_𝑖𝑗,𝑏̆_𝑖𝑗}. Notice that the system (4) is a differential equation with discontinuous right-hand sides, and based on the theory of differential inclusions [53], if 𝑥_𝑖(𝑡) is a solution of (4) in the sense of Filippov [54], then 𝑥̇_𝑖(𝑡)∈−𝑑_𝑖,𝑑_𝑖𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗,𝑎_𝑖𝑗𝑔_𝑗𝑥_𝑗(𝑡+_𝑛𝑗=1)𝑏_𝑖𝑗,𝑏_𝑖𝑗𝑔_𝑗𝑥_𝑗𝑡−𝜏_𝑗(𝑡)+𝐼_𝑖(𝑡),𝑡≥0,𝑖=1,2,…,𝑛. (6)

The differential inclusion system (6) can be transformed into the vector form as 𝑥̇(𝑡)∈−𝐷,𝐷𝑥(𝑡)+𝐴,𝐴𝑔(𝑥(𝑡+𝐵)),𝐵𝑔(𝑥(𝑡−𝜏(𝑡)))+𝐼(𝑡),𝑡≥0, (7) where 𝐷=diag(𝑑₁,𝑑₂,…,𝑑_𝑛), 𝐷=diag(𝑑₁,𝑑₂,…,𝑑_𝑛), 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛, 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛, 𝐵=(𝑏_𝑖𝑗)_𝑛×𝑛, and 𝐵=(𝑏_𝑖𝑗)_𝑛×𝑛. Or equivalently, there exist measurable functions 𝒟(𝑡)∈[𝐷,𝐷], 𝒜(𝑡)∈[𝐴,𝐴], and ℬ(𝑡)∈[𝐵,𝐵], such that 𝑥̇(𝑡)(𝑡)𝑥(𝑡)(𝑡)𝑔(𝑥(𝑡(𝑡)𝑔(𝑥((𝑡(𝑡)=−𝒟+𝒜))+ℬ𝑡−𝜏)))+𝐼,𝑡≥0. (8)

To obtain the main results of this paper, some definitions and lemmas are introduced as follows.

Definition 1 (see [55]). A continuous function 𝑥(𝑡)∶𝑅→𝑅^𝑛 is said to be almost periodic on 𝑅, if, for any 𝜀>0, there exists a scalar 𝑙=𝑙(𝜀)>0 and, for any interval with length 𝑙, there exists a scalar 𝜔=𝜔(𝜀)>0 in this interval, such that ‖𝑥(𝑡+𝜔)−𝑥(𝑡)‖<𝜀 for all 𝑡∈𝑅.

Definition 2. The almost periodic solution 𝑥^∗(𝑡) of the system (4) is said to be globally exponentially stable, if there exist scalars 𝜂>0 and 𝛿>0, such that 𝑥(𝑡)−𝑥^∗(𝑡)≤𝜂𝑒^−𝛿𝑡,𝑡≥0, (9) where 𝑥(𝑡) is the solution of the system (4) with the initial value 𝑥(𝑡)=𝜑(𝑡)∈𝐶([−𝜏,0];𝑅^𝑛). 𝛿 is called as the exponential convergence rate.

Definition 3 (see [55]). The solution 𝑥(𝑡) of the system (4) with the initial value 𝑥(𝑡)=𝜑(𝑡)∈𝐶([−𝜏,0];𝑅^𝑛) is said to be asymptotically almost periodic, if, for any 𝜀>0, there exist scalars 𝑇>0, 𝑙=𝑙(𝜀)>0, and 𝜔=𝜔(𝜀)>0 in any interval with length 𝑙, such that ‖𝑥(𝑡+𝜔)−𝑥(𝑡)‖<𝜀, for all 𝑡≥𝑇.

Lemma 4 (see [29]). For any 𝑄(𝑡)∈[𝑄,𝑄], the following inequality holds: ‖𝑄(𝑡)‖≤𝑄^∗+𝑄_∗, (10) where 𝑄^∗=(𝑄+𝑄)/2,  𝑄_∗=(𝑄−𝑄)/2.

Lemma 5 (see [45]). Let scalar 𝜀>0,𝑥,𝑦∈𝑅^𝑛, and𝐴∈𝑅^𝑛×𝑛, then 𝑥^𝑇1𝐴𝑦≤𝑥2𝜀^𝑇𝐴𝐴^𝑇𝜀𝑥+2𝑦^𝑇𝑦. (11)

Throughout this paper, the following assumptions are made on (4):(𝐴₁)𝐼_𝑖(𝑡) is an almost periodic function. (𝐴₂)𝑔_𝑖∶𝑅→𝑅 is a nondecreasing continuous function. (𝐴₃)𝜏_𝑖(𝑡) is an almost periodic function, and 0<𝜏_𝑖(𝑡)<𝜏, 𝜏̇_𝑖(𝑡)≤𝜇<1, 𝜏 and 𝜇 are constants.

3. Main Results

In this section, the main results concerned with the existence, uniqueness, and global exponential stability of the almost periodic solution are addressed for the memristive neural network in (4).

Theorem 6. Under the assumptions (𝐴₁)−(𝐴₃), if there exists a diagonal matrix 𝑃=diag(𝑝₁,𝑝₂,…,𝑝_𝑛)>0 such that 1𝐵1−𝜇𝑈+𝒫^∗+𝐵_∗²𝑈−𝑆<0, (12) where 𝒫=max𝑝_𝑖, 𝑈 is the identity matrix, 𝑆=(𝑠_𝑖𝑗)_𝑛×𝑛,𝑠_𝑖𝑖=−2𝑝_𝑖𝑎_𝑖𝑖, 𝑠_𝑖𝑗=−max{|𝑝_𝑖𝑎_𝑖𝑗+𝑝_𝑗𝑎_𝑗𝑖|,|𝑝_𝑖𝑎_𝑖𝑗+𝑝_𝑗𝑎_𝑗𝑖|},  𝑖≠𝑗, 𝐵^∗=(𝐵+𝐵)/2,  and 𝐵_∗=(𝐵−𝐵)/2. then
(1) For any initial value 𝑥(𝑡)=𝜑(𝑡)∈𝐶([−𝜏,0];𝑅^𝑛), there exists a solution of the memristive neural network (4) on [0,+∞), and this solution is asymptotically almost periodic.
(2) The memristive neural network (4) has a unique almost periodic solution which is globally exponentially stable.

Proof. We should prove this theorem in four steps.
Step  1. In this step, we will prove the existence of the solution, that is, prove that the system (4) has a global solution for any initial value 𝑥(𝑡)=𝜑(𝑡)∈𝐶([−𝜏,0];𝑅^𝑛).
Similar to the proof of Lemma  1 in [37], under the assumptions of Theorem 6, it is easy to obtain the existence of the local solution of (4) with initial value 𝑥(𝑡)=𝜑(𝑡)∈𝐶([−𝜏,0];𝑅^𝑛) on [0,𝑇(𝜑)), where 𝑇(𝜑)∈(0,+∞) or 𝑇(𝜑)=+∞, and [0,𝑇(𝜑)) is the maximal right-side existence interval of the local solution.
Due to 𝑑_𝑖>0, 𝑖=1,2,…,𝑛, by (12) we can choose constants 𝛿>0 and 𝜈>0, such that 0<𝛿<𝑑_𝑖,  𝑖=1,2,…,𝑛, (𝛿+3𝜈)𝑈−2𝐷<0, and 𝑒^𝛿𝑡𝐵1−𝜇𝑈+𝒫^∗+𝐵_∗²𝑈−𝑆<0. (13)
Without loss of generality, we suppose that 𝑔(0)=0. Let scalars𝛼>0,  𝛽>0. Consider a Lyapunov functional defined by 𝑉(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡)+𝑉₃(𝑡),𝑡≥0, (14) where 𝑉₁(𝑡)=𝑒^𝛿𝑡𝑥^𝑇(𝑡)𝑥(𝑡),𝑉₂=2𝛼𝑒^𝑛𝛿𝑡_𝑖=1𝑝_𝑖^𝑥𝑖₀^(𝑡)𝑔_𝑖(𝜃)𝑉𝑑𝜃,₃=𝛼+𝛽1−𝜇^𝑡_{𝑡−𝜏(𝑡)}𝑔^𝑇(𝑥(𝜃))𝑔(𝑥(𝜃))𝑒^{𝛿(𝜃+𝜏)}𝑑𝜃. (15) Calculate the time derivative of 𝑉(𝑡) along the local solution of (4) on [0,𝑇(𝜑)). By (8) and Lemma 5, we have 𝑉̇₁(𝑡)=𝛿𝑒^𝛿𝑡𝑥^𝑇(𝑡)𝑥(𝑡)+2𝑒^𝛿𝑡𝑥^𝑇(𝑡)𝑥̇(𝑡)=𝛿𝑒^𝛿𝑡𝑥^𝑇(𝑡)𝑥(𝑡)−2𝑒^𝛿𝑡𝑥^𝑇(𝑡)𝒟(𝑡)𝑥(𝑡)+2𝑒^𝛿𝑡𝑥^𝑇(𝑡)𝒜(𝑡)𝑔(𝑥(𝑡))+2𝑒^𝛿𝑡𝑥^𝑇(𝑡)ℬ(𝑡)𝑔(𝑥(𝑡−𝜏(𝑡)))+2𝑒^𝛿𝑡𝑥^𝑇(𝑡)𝐼(𝑡)≤𝑒^𝛿𝑡𝑥^𝑇(𝑡)(𝛿𝑈−2𝒟(𝑡))𝑥(𝑡)+𝜈𝑒^𝛿𝑡𝑥^𝑇(𝑡)𝑥(𝑡)+𝑒^𝛿𝑡𝜈𝑔^𝑇(𝑥(𝑡))𝒜(𝑡)^𝑇𝒜(𝑡)𝑔(𝑥(𝑡))+𝜈𝑒^𝛿𝑡𝑥^𝑇(𝑡)𝑥(𝑡)+𝑒^𝛿𝑡𝜈𝑔^𝑇(𝑥(𝑡−𝜏(𝑡)))ℬ(𝑡)^𝑇ℬ(𝑡)𝑔(𝑥(𝑡−𝜏(𝑡)))+𝜈𝑒^𝛿𝑡𝑥^𝑇(𝑡)𝑥(𝑡)+𝑒^𝛿𝑡𝜈𝐼^𝑇(𝑡)𝐼(𝑡)≤𝑒^𝛿𝑡𝑥^𝑇(𝑡)(𝑡𝑥(𝑡)+𝑒)((𝛿+3𝜈𝑈−2𝒟))_𝛿𝑡𝜈𝑔^𝑇(𝑥(𝑡))‖𝒜(𝑡)‖²𝑔(𝑥(𝑡+𝑒))_𝛿𝑡𝜈𝑔^𝑇(𝑥(𝑡−𝜏(𝑡)))‖ℬ(𝑡)‖²𝑔(𝑥((𝑡+𝑒𝑡−𝜏)))_𝛿𝑡𝜈𝐼^𝑇(𝑡)𝐼(𝑡)≤𝑒^𝛿𝑡𝜈𝑔^𝑇(𝑥(𝑡))‖𝒜(𝑡)‖²𝑔(𝑥(𝑡+𝑒))_𝛿𝑡𝜈𝑔^𝑇(𝑥(𝑡−𝜏(𝑡)))‖ℬ(𝑡)‖²𝑔(𝑥((𝑡+𝑒𝑡−𝜏)))_𝛿𝑡𝜈𝐼^𝑇(𝑡)𝐼(𝑡). (16) By the assumptions (𝐴₂) and (𝐴₃), for any constant 𝛾>0, we have 𝑉̇₂(𝑡)=2𝛼𝛿𝑒^𝑛𝛿𝑡_𝑖=1𝑝_𝑖^𝑥𝑖₀^(𝑡)𝑔_𝑖(𝜃)𝑑𝜃+2𝛼𝑒^𝑛𝛿𝑡_𝑖=1𝑝_𝑖𝑔_𝑖𝑥_𝑖(𝑡)𝑥̇_𝑖(𝑡)≤2𝛼𝛿𝑒^𝑛𝛿𝑡_𝑖=1𝑝_𝑖𝑔_𝑖𝑥_𝑖(𝑡)𝑥_𝑖(𝑡)+2𝛼𝑒^𝑛𝛿𝑡_𝑖=1𝑝_𝑖𝑔_𝑖𝑥_𝑖(𝑡)𝑥̇_𝑖(𝑡)=2𝛼𝛿𝑒^𝛿𝑡𝑔^𝑇(𝑥(𝑡))𝑃𝑥(𝑡)+2𝛼𝑒^𝛿𝑡𝑔^𝑇(𝑥(𝑡))𝑃𝑥̇(𝑡)=2𝛼𝛿𝑒^𝛿𝑡𝑔^𝑇(𝑥(𝑡))𝑃𝑥(𝑡)−2𝛼𝑒^𝛿𝑡𝑔^𝑇(𝑥(𝑡))𝑃𝒟(𝑡)𝑥(𝑡)+2𝛼𝑒^𝛿𝑡𝑔^𝑇(𝑥(𝑡))𝑃𝒜(𝑡)𝑔(𝑥(𝑡))+2𝛼𝑒^𝛿𝑡𝑔^𝑇(𝑥(𝑡))𝑃ℬ(𝑡)𝑔(𝑥(𝑡−𝜏(𝑡)))+2𝛼𝑒^𝛿𝑡𝑔^𝑇(𝑥(𝑡))𝑃𝐼(𝑡)≤2𝛼𝑒^𝛿𝑡𝑔^𝑇(𝑥(𝑡))𝑃(𝛿𝐼−𝒟(𝑡))(𝑡)𝑥(𝑡)+𝛼𝑒^𝛿𝑡𝑔^𝑇(𝑥(𝑡))𝑃𝒜(𝑡)+𝒜(𝑡)^𝑇𝑃𝑔(𝑥(𝑡))+𝛼𝑒^𝛿𝑡𝑔^𝑇(𝑥(𝑡))𝑃ℬ(𝑡)(𝑃ℬ(𝑡))^𝑇𝑔(𝑥(𝑡))+𝛼𝑒^𝛿𝑡𝑔^𝑇(𝑥(𝑡−𝜏(𝑡)))𝑔(𝑥(𝑡−𝜏(𝑡)))+𝛾𝛼𝑒^𝛿𝑡𝑔^𝑇(𝑥(𝑡))𝑔(𝑥(𝑡))+𝛼𝒫²𝑒^𝛿𝑡𝛾𝐼^𝑇(𝑡)𝐼(𝑡)≤𝛼𝑒^𝛿𝑡𝑔^𝑇(𝑥(𝑡))𝑃𝒜(𝑡)+𝒜(𝑡)^𝑇𝑃+𝑃ℬ(𝑡)(𝑃ℬ(𝑡))^𝑇+𝛾𝑈𝑔(𝑥(𝑡))+𝛼𝑒^𝛿𝑡𝑔^𝑇(𝑥((𝑡𝑔(𝑥((𝑡+𝑡−𝜏)))𝑡−𝜏)))𝛼𝒫²𝑒^𝛿𝑡𝛾𝐼^𝑇(𝑡)𝐼(𝑡).𝑉̇₃=𝛼+𝛽𝑒1−𝜇^{𝛿(𝑡+𝜏)}𝑔^𝑇(𝑥(𝑡))𝑔(𝑥(𝑡))−𝑒^{𝛿(𝑡−𝜏(𝑡)+𝜏)}×(1−𝜏̇(𝑡))𝑔^𝑇(𝑥((𝑡(𝑥((𝑡≤𝑡−𝜏)))×𝑔𝑡−𝜏)))𝛼+𝛽𝑔1−𝜇^𝑇(𝑥(𝑡))𝑔(𝑥(𝑡))𝑒^{𝛿(𝑡+𝜏)}−𝛼+𝛽𝑔^𝑇(𝑥(𝑡−𝜏(𝑡)))𝑔(𝑥(𝑡−𝜏(𝑡)))𝑒^𝛿𝑡. (17) From (16) and (17) and by Lemma 4, one yields 𝑉̇(𝑡)≤𝛼𝑒^𝛿𝑡𝑔^𝑇(𝑥(𝑡×(𝑡)(𝑡)))𝑃𝒜+𝒜^𝑇𝑃+𝑃𝐵(𝑡)(𝑃𝐵(𝑡))^𝑇+‖𝒜(𝑡𝛾+)‖²+𝑒𝛼𝜈^𝛿𝜏×𝛽1−𝜇1+𝛼𝑈𝑔(𝑥(𝑡))+𝑒^𝛿𝑡𝑔^𝑇(𝑥((𝑡‖ℬ(𝑡𝑡−𝜏))))‖²𝜈(𝑥((𝑡−𝛽×𝑔𝑡−𝜏)))+𝑒^𝛿𝑡1𝜈+𝛼𝒫²𝛾‖𝐼(𝑡)‖²≤𝛼𝑒^𝛿𝑡𝑔^𝑇(𝑥(𝑡×𝒫))₂𝐵^∗+𝐵_∗²+𝐴𝑈−𝑆𝛾+^∗+𝐴^∗2+𝑒𝛼𝜈_𝛿𝜏𝛽1−𝜇1+𝛼𝑈𝑔(𝑥(𝑡))+𝑒^𝛿𝑡𝑔^𝑇(𝑥((𝑡×𝐵𝑡−𝜏)))_∗+𝐵^∗2𝜈−𝛽𝑔(𝑥(𝑡−𝜏(𝑡)))+𝑒^𝛿𝑡1𝜈+𝛼𝒫²𝛾‖𝐼(𝑡)‖², (18) where 𝐴^∗=(𝐴+𝐴)/2, 𝐴_∗=(𝐴−𝐴)/2. By (13), we can choose suitable constants 𝛾>0,𝛼>0, 𝜈>0, and 𝛽>0, such that 𝐵_∗+𝐵^∗2𝜈𝒫−𝛽<0,₂𝐵^∗+𝐵_∗²+𝐴𝑈−𝑆𝛾+^∗+𝐴^∗2+𝑒𝛼𝜈^𝛿𝜏𝛽1−𝜇1+𝛼𝑈<0. (19) This implies that 𝑉̇(𝑡)≤𝑒^𝛿𝑡1𝜈+𝛼𝒫²𝛾‖𝐼(𝑡)‖². (20) Moreover, by the assumption (𝐴₁), we can obtain that 𝐼(𝑡) is a bounded function. Hence, there exist a constant 𝑀>0, such that 10<𝜈+𝛼𝒫²𝛾‖𝐼(𝑡)‖²<𝑀,𝑡≥0. (21) By (20) and (21), it follows that 𝑉̇(𝑡)≤𝑀𝑒^𝛿𝑡,𝑡∈0,𝑇𝜑. (22) From the definition of 𝑉(𝑡) and (22), we have 𝑒^𝛿𝑡‖𝑥(𝑡)‖²≤𝑉(𝑡)≤𝑉(0)+^𝑡₀𝑉̇(𝑠)(0)+𝑑𝑠≤𝑉^𝑡₀𝑀𝑒^𝛿𝑠(0)+𝑀𝑑𝑠=𝑉𝛿𝑒^𝛿𝑡−1,𝑡∈0,𝑇𝜑. (23) Thus, ‖𝑥(𝑡)‖²≤𝑒^−𝛿𝑡𝑉(0)+𝑀𝛿1−𝑒^−𝛿𝑡(0)+𝑀≤𝑉𝛿,𝑡∈0,𝑇𝜑. (24) This shows that the local solution 𝑥(𝑡) of (4) is bounded on [0,𝑇(𝜑)) and hence is defined on [0,+∞). That is, the system (4) has a global solution for any initial value 𝑥(𝑡)=𝜑(𝑡)∈𝐶([−𝜏,0];𝑅^𝑛).
Step  2. In this step, the global solution 𝑥(𝑡) of the system (4) will be proved to be asymptotically almost periodic.
Let 𝑦(𝑡)=𝑥(𝑡+𝜔)−𝑥(𝑡), then 𝑦̇(𝑡)(𝑡)𝑦(𝑡)(𝑡)𝑔𝑦(𝑡)(𝑡)𝑔𝑦((𝑡+𝐼̃(),=−𝒟+𝒜+ℬ𝑡−𝜏))𝜔,𝑡 (25) where 𝑔𝑦(𝑡)(𝑥((𝑥(𝑡,𝑔𝑦((𝑡(𝑥((𝑡(𝑥((𝑡,𝐼̃()()(𝑡).=𝑔𝑡+𝜔))−𝑔))𝑡−𝜏))=𝑔𝑡+𝜔−𝜏)))−𝑔𝑡−𝜏)))𝜔,𝑡=𝐼𝑡+𝜔−𝐼 (26) Similar to 𝑉(𝑡), define Lyapunov functional 𝑊(𝑡) as 𝑊(𝑡)=𝑒^𝛿𝑡𝑦^𝑇(𝑡)𝑦(𝑡)+2𝛼𝑒^𝑛𝛿𝑡_𝑖=1𝑝_𝑖^𝑦𝑖₀^(𝑡)𝑔_𝑖(𝜃)+𝑑𝜃𝛼+𝛽1−𝜇^𝑡_{(𝑡)𝑡−𝜏}𝑔^𝑇𝑦(𝜃)𝑔𝑦(𝜃)𝑒^{𝛿(𝜃+𝜏)}𝑑𝜃. (27) Calculate the derivative of 𝑊(𝑡) along the solution of the system (25). Arguing as in Step  1, we can choose the appropriate positive constants 𝜈, 𝛾, 𝛼, and 𝛿, such that 𝑊̇(𝑡)≤𝑒^𝛿𝑡1𝜈+𝛼𝒫²𝛾𝐼̃(𝜔,𝑡)²,𝑡>0. (28) By the assumption (𝐴₁), 𝐼_𝑖(𝑡) is an almost periodic function. Thus, by Definition 1, for any 𝜀>0, there exists 𝑙=𝑙(𝜀)>0, and for any interval with length 𝑙, there exists a scalar 𝜔 in this interval, such that 1𝜈+𝛼𝒫²𝛾𝐼̃(𝜔,𝑡)²≤12𝜀²𝛿,𝑡>0. (29) It follows from (28) and (29) that 𝑊̇(𝑡)≤(1/2)𝜀²𝛿𝑒^𝛿𝑡, 𝑡>0, which implies 𝑒^𝛿𝑡𝑦(𝑡)²≤𝑊(𝑡)≤𝑊(0)+^𝑡₀12𝛿𝜀²𝑒^𝛿𝑠(0)+1𝑑𝑠=𝑊2𝜀²𝑒^𝛿𝑡‖𝑥()(𝑡≤𝑒−1,𝑡>0,𝑡+𝜔−𝑥)‖_−𝛿𝑡𝑊(0)+12𝜀²1−𝑒^{−𝛿𝑡1/2},𝑡>0. (30) Therefore, there exists a constant 𝑇>0; when 𝑡>𝑇, we have ‖𝑥(𝑡+𝜔)−𝑥(𝑡)‖≤𝜀. (31) This shows that the solution of the system (4) is asymptotically almost periodic.
Step  3. In this step, we will prove that the system (4) has an almost periodic solution.
Let 𝑥(𝑡) be the solution of the system (4) with the initial value 𝑥(𝑡)=𝜑(𝑡)∈𝐶([−𝜏,0];𝑅^𝑛), then 𝑥(𝑡) satisfies (8). Take a sequence {𝑡_𝑘}, 𝑡₁<𝑡₂<⋯<𝑡_𝑘<𝑡_𝑘+1<⋯, lim_𝑘→+∞𝑡_𝑘=+∞. It is easy to derive that the function sequence {𝑥(𝑡+𝑡_𝑘)} is equicontinuous and uniformly bounded. Hence, by using Arzela-Ascoli theorem and diagonal selection principle, a subsequence of {𝑡_𝑘} (still denoted by {𝑡_𝑘}) can be selected, such that {𝑥(𝑡+𝑡_𝑘)} uniformly converges to a continuous function 𝑥^∗(𝑡) on any compact set of 𝑅.
By applying Lebesgue's dominated convergence theorem on (8), we can obtain that 𝑥^∗(𝑡+ℎ)−𝑥^∗(𝑡)=lim_𝑘→+∞𝑥𝑡+𝑡_𝑘+ℎ−𝑥𝑡+𝑡_𝑘=lim_{𝑘→+∞𝑡}^𝑡+ℎ−𝒟(𝑡)𝑥𝑡_𝑘(𝑡)𝑔𝑥𝑡+𝜃+𝒜_𝑘(𝑡)𝑔𝑥𝑡+𝜃+ℬ_𝑘+𝜃−𝜏𝑡_𝑘+𝜃+𝐼𝑡_𝑘+𝜃𝑑𝜃=_𝑡^𝑡+ℎ−𝒟(𝑡)𝑥^∗(𝜃)+𝒜(𝑡)𝑔𝑥^∗(𝜃)+ℬ(𝑡)𝑔𝑥^∗(𝜃−𝜏(𝜃))+𝐼(𝜃)𝑑𝜃 (32) for any 𝑡∈[−𝜏,+∞) and ℎ∈𝑅. This implies that 𝑥^∗(𝑡) is a solution of (4). By the result obtained in Step  2, 𝑥(𝑡) is asymptotically almost periodic. That is, for any 𝜀>0, there exist 𝑇>0, 𝑙=𝑙(𝜀), and for any interval with length 𝑙, there exists a scalar 𝜔 in this interval, such that ‖𝑥(𝑡+𝜔)−𝑥(𝑡)‖≤𝜀 for all 𝑡≥𝑇. Thus, there exists a constant 𝐾>0; for all 𝑡∈[−𝜏,+∞) and 𝑘>𝐾, we can get that 𝑥𝑡+𝑡_𝑘+𝜔−𝑥𝑡+𝑡_𝑘≤𝜀. (33) Let 𝑘→+∞ in (33), it follows that ‖𝑥^∗(𝑡+𝜔)−𝑥^∗(𝑡)‖≤𝜀 for all 𝑡∈[−𝜏,+∞). This shows that 𝑥^∗(𝑡) is the almost periodic solution of (4). The proof of the existence of the almost periodic solution has been completed.
Step  4. In this step, we will prove that the uniqueness and global exponential stability of the almost periodic solution for the system (4).
Let 𝑥(𝑡) be any solution of (4), and let 𝑥^∗(𝑡) be an almost periodic solution of (4). Set 𝑧(𝑡)=𝑥(𝑡)−𝑥^∗(𝑡), then 𝑧̇(𝑡)(𝑡)𝑧(𝑡)(𝑡)𝑔(𝑧(𝑡(𝑡)𝑔(𝑧((𝑡,=−𝒟+𝒜))+ℬ𝑡−𝜏))) (34) where 𝑔(𝑧(𝑡))=𝑔(𝑥(𝑡))−𝑔𝑥^∗(𝑡),𝑔(𝑧(𝑡−𝜏(𝑡)))=𝑔(𝑥(𝑡−𝜏(𝑡)))−𝑔𝑥^∗(𝑡−𝜏(𝑡)). (35) Similar to 𝑉(𝑡), define Lyapunov functional 𝐿(𝑡) as 𝐿(𝑡)=𝐿₁(𝑡)+𝐿₂(𝑡)+𝐿₃(𝑡), (36) where 𝐿₁(𝑡)=𝑒^𝛿𝑡𝑧^𝑇(𝑡)𝑧(𝑡),𝐿₂(𝑡)=2𝛼𝑒^𝑛𝛿𝑡_𝑖=1𝑝_𝑖^𝑧𝑖₀^(𝑡)𝑔_𝑖(𝜃)𝐿𝑑𝜃,₃(𝑡)=𝛼+𝛽^𝑡_{(𝑡)𝑡−𝜏}𝑔^𝑇(𝑧(𝜃))𝑔(𝑧(𝜃))𝑒^{𝛿(𝜃+𝜏)}𝑑𝜃. (37) Arguing as in Step  1, we have 𝐿̇₁(𝑡)≤𝑒^𝛿𝑡𝜈𝑔^𝑇(𝑧(𝑡))‖𝒜(𝑡)‖²𝑔(𝑧(𝑡+𝑒))_𝛿𝑡𝜈𝑔^𝑇(𝑧(𝑡−𝜏(𝑡)))‖ℬ(𝑡)‖²𝑔(𝑧((𝑡,𝐿̇𝑡−𝜏)))₂(𝑡)≤𝛼𝑒^𝛿𝑡𝑔(𝑧(𝑡))𝑃𝒜(𝑡)+𝒜(𝑡)^𝑇𝑃+𝑃ℬ(𝑡)(𝑃ℬ(𝑡))^𝑇𝑔(𝑧(𝑡))+𝛼𝑒^𝛿𝑡𝑔^𝑇(𝑧((𝑡𝑔(𝑧((𝑡,𝐿̇𝑡−𝜏)))𝑡−𝜏)))₃(𝑡)≤𝛼+𝛽𝑔(𝑧(𝑡𝑔(𝑧(𝑡𝑒1−𝜇))))^{𝛿(𝑡+𝜏)}−𝛼+𝛽𝑔^𝑇𝑧(𝑡−𝜏(𝑡))𝑔(𝑧(𝑡−𝜏(𝑡)))𝑒^𝛿𝑡. (38) Thus 𝐿̇(𝑡)≤𝛼𝑒^𝛿𝑡𝑔^𝑇(𝑧(𝑡))𝑃𝒜(𝑡)+𝒜(𝑡)𝑃+𝑃ℬ(𝑡)(𝑃ℬ(𝑡))^𝑇+‖𝒜(𝑡)‖²+𝑒𝛼𝜈^𝛿𝜏𝛽1−𝜇1+𝛼𝑈×𝑔(𝑧(𝑡))+𝑒^𝛿𝑡𝑔(𝑧((𝑡‖ℬ(𝑡𝑡−𝜏))))‖²𝜈−𝛽𝑔(𝑧(𝑡−𝜏(𝑡)))≤𝛼𝑒^𝛿𝑡𝑔^𝑇(𝑥(𝑡×𝒫))₂𝐵^∗+𝐵_∗²+𝐴𝑈−𝑆_∗+𝐴^∗2+𝑒𝛼𝜈^𝛿𝜏𝛽1−𝜇1+𝛼𝑈×𝑔(𝑥(𝑡))+𝑒^𝛿𝑡𝑔^𝑇(𝑥((𝑡𝐵𝑡−𝜏)))^∗+𝐵^∗2𝜈(𝑥((𝑡.−𝛽×𝑔𝑡−𝜏))) (39) By (13), we can choose appropriate constants 𝛾>0,  𝛼>0, 𝜈>0, and 𝛽>0, such that 𝒫²𝐵^∗+𝐵_∗²+𝐴𝑈−𝑆_∗+𝐴^∗2+𝑒𝛼𝜈^𝛿𝜏𝛽1−𝜇1+𝛼𝐵𝑈<0,_∗+𝐵^∗2𝜈−𝛽<0. (40) This implies that 𝐿̇(𝑡)≤0. Therefore, combined with the definition of 𝐿(𝑡), it follows that ‖𝑧(𝑡)‖²=𝑥(𝑡)−𝑥^∗(𝑡)²≤𝑒^−𝛿𝑡𝐿(𝑡)≤𝐿(0)𝑒^−𝛿𝑡. (41) This shows that the almost periodic solution 𝑥^∗(𝑡) of the system (4) is globally exponentially stable. Consequently, the periodic solution 𝑥^∗(𝑡) is unique. This completes the proof of Theorem 6.

Notice that periodic function can be regarded as special almost periodic function. Hence, when 𝐼_𝑖(𝑡) is a periodic external input in the system (4), we can get the following corollary.

Corollary 7. Suppose that the assumptions (𝐴₂)-(𝐴₃) hold. If there exists a diagonal matrix 𝑃=diag(𝑝₁,𝑝₂,…,𝑝_𝑛)>0 such that 1𝐵1−𝜇𝑈+𝒫^∗+𝐵_∗²𝑈−𝑆<0, (42) where 𝒫=max𝑝_𝑖, 𝑈 is the identity matrix, and 𝑆=(𝑠_𝑖𝑗)_𝑛×𝑛,𝑠_𝑖𝑖=−2𝑝_𝑖𝑎_𝑖𝑖, 𝑠_𝑖𝑗=−max{|𝑝_𝑖𝑎_𝑖𝑗+𝑝_𝑗𝑎_𝑗𝑖|,|𝑝_𝑖𝑎_𝑖𝑗+𝑝_𝑗𝑎_𝑗𝑖|},  and 𝑖≠𝑗, then one has the following.(1)For any initial value 𝑥(𝑡)=𝜑(𝑡)∈𝐶([−𝜏,0];𝑅^𝑛), there exists a solution of the memristive neural network (4) on [0,+∞), and this solution is asymptotically periodic.(2)The memristive neural network (4) has a unique periodic solution which is globally exponentially stable.

When 𝐼_𝑖(𝑡) is a constant external input 𝐼_𝑖, the system (4) changes as 𝑥̇_𝑖(𝑡)=−𝑑_𝑖𝑥_𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗𝑥_𝑖(𝑡)𝑔_𝑗𝑥_𝑗(𝑡+_𝑛𝑗=1)𝑏_𝑖𝑗𝑥_𝑖(𝑡)𝑔_𝑗𝑥_𝑗(𝑡−𝜏(𝑡))+𝐼_𝑖,𝑖=1,2,…,𝑛. (43) Since a constant can be also regarded as a special almost periodic function, by applying Theorem 6 on the neural network (43), we have the following.

Corollary 8. Suppose that the assumptions (𝐴₂)-(𝐴₃) hold. If there exists a diagonal matrix 𝑃=diag(𝑝₁,𝑝₂,…,𝑝_𝑛)>0 such that 1𝐵1−𝜇𝑈+𝒫^∗+𝐵_∗²𝑈−𝑆<0, (44) where 𝒫=max𝑝_𝑖, 𝑈 is the identity matrix, and 𝑆=(𝑠_𝑖𝑗)_𝑛×𝑛,𝑠_𝑖𝑖=−2𝑝_𝑖𝑎_𝑖𝑖, 𝑠_𝑖𝑗=−max{|𝑝_𝑖𝑎_𝑖𝑗+𝑝_𝑗𝑎_𝑗𝑖|,|𝑝_𝑖𝑎_𝑖𝑗+𝑝_𝑗𝑎_𝑗𝑖|}, and 𝑖≠𝑗, then one has the following. (1)For any initial value 𝑥(𝑡)=𝜑(𝑡)∈𝐶([−𝜏,0];𝑅^𝑛), there exists a solution of the memristive neural network (43) on [0,+∞).(2)The memristive neural network (43) has a unique equilibrium point which is globally exponentially stable.

4. Illustrative Examples

In this section, two examples will be given to illustrate the effectiveness of the results obtained in this paper.

Example 1. Consider the second-order memristive neural network with time-varying delays in (4) described by 𝑥̇₁(𝑡)𝑥̇₂(𝑡)=𝑑₁₁𝑥₁00𝑑₂₂𝑥₂𝑥₁(𝑡)𝑥₂(𝑡)+𝑎₁₁𝑥₁𝑎₁₂𝑥₁𝑎₂₁𝑥₂𝑎₂₂𝑥₂𝑥³₁(𝑡)𝑥³₂(𝑡)+𝑏₁₁𝑥₁𝑏₁₂𝑥₁𝑏₂₁𝑥₂𝑏₂₂𝑥₂𝑥³₁((𝑥𝑡−0.5sin𝑡+1))₃₂((+√𝑡−0.5sin𝑡+1))2−4sin𝑡−3−4cos,2𝑡 (45) where 𝑑₁₁𝑥₁=0.8,𝑥₁(𝑡𝑥)<1.5,1,₁(𝑡𝑑)>1.5,₂₂𝑥₂=2,𝑥₂(𝑡𝑥)<1.5,2.4,₂(𝑡𝑎)>1.5,₁₁𝑥₁=−150,𝑥₁(𝑡𝑥)<1.5,−145,₁(𝑡𝑎)>1.5,₁₂𝑥₁=0,𝑥₁(𝑡𝑥)<1.5,1,₁(𝑡𝑎)>1.5,₂₁𝑥₂=0,𝑥₂(𝑡𝑥)<1.5,2,₂(𝑡𝑎)>1.5,₂₂𝑥₂=−162,𝑥₂(𝑡𝑥)<1.5,−160,₂(𝑡𝑏)>1.5,₁₁𝑥₁=−1,𝑥₁(𝑡𝑥)<1.5,3,₁(𝑡𝑏)>1.5,₁₂𝑥₁=0,𝑥₁(𝑡𝑥)<1.5,4,₁(𝑡𝑏)>1.5,₂₁𝑥₂=1,𝑥₂(𝑡𝑥)<1.5,2,₂(𝑡𝑏)>1.5,₂₂𝑥₂=2,𝑥₂(𝑡𝑥)<1.5,3,₂(𝑡)>1.5. (46) It is obvious that 𝜏=1, 𝜇=0.5. Choose the positive definite diagonal matrix 𝑃=diag(1,0.1), then 𝑆=^290.0−1.2_−1.2320.0. It is easy to check that 1𝐵1−𝜇𝑈+𝒫^∗+𝐵_∗²=𝑈−𝑆−226.71571.20001.2000−256.7157<0. (47)
All conditions of Theorem 6 hold; hence the memristive neural network in this example has a unique almost periodic solution which is globally exponentially stable.
Figure 1 displays the state trajectory of the network with initial condition (𝜑₁(𝑡),𝜑₂(𝑡))^𝑇=(−0.5𝑡²,0.5𝑡)^𝑇,𝑡∈[−1,0). It can be seen that this trajectory converges to the unique almost periodic solution of the network. This is in accordance with the conclusion of Theorem 6.

(a)

(b)

Figure 1 

The state trajectory of the network with the initial condition (𝜑1(𝑡),𝜑2(𝑡))𝑇=(−0.5𝑡2,0.5𝑡)𝑇,𝑡∈[−1,0).

Example 2. Consider the third-order memristive neural network with time-varying delays in (4) described by 𝑥̇₁(𝑡)𝑥̇₂(𝑡)𝑥̇₃(𝑡)=𝑑₁₁𝑥₁000𝑑₂₂𝑥₂000𝑑₃₃𝑥₃𝑥₁(𝑡)𝑥₂(𝑡)𝑥₃(𝑡)+𝑎₁₁𝑥₁𝑎₁₂𝑥₁𝑎₁₃𝑥₁𝑎₂₁𝑥₂𝑎₂₂𝑥₂𝑎₂₃𝑥₂𝑎₃₁𝑥₃𝑎₃₂𝑥₃𝑎₃₃𝑥₃𝑥³₁(𝑡)𝑥³₂(𝑡)𝑥³₃(𝑡)+𝑏₁₁𝑥₁𝑏₁₂𝑥₁𝑏₁₃𝑥₁𝑏₂₁𝑥₂𝑏₂₂𝑥₂𝑏₂₃𝑥₂𝑏₃₁𝑥₃𝑏₃₂𝑥₃𝑏₃₃𝑥₃×𝑥³₁((𝑥𝑡−0.5cos𝑡+1))₃₂((𝑥𝑡−0.5cos𝑡+1))₃₃((+√𝑡−0.5cos𝑡+1))2+2sin√2𝑡−3−3cos√3𝑡sin,6𝑡 (48) where 𝑑₁₁𝑥₁=0.1,𝑥₁(𝑡𝑥)<1,0.3,₁(𝑡𝑑)>1,₂₂𝑥₂=1,𝑥₂(𝑡𝑥)<1,1.9,₂(𝑡𝑑)>1,₃₃𝑥₃=3,𝑥₃(𝑡𝑥)<1,3.3,₃(𝑡𝑎)>1,₁₁𝑥₁=−50,𝑥₁(𝑡𝑥)<1,−49.3,₁(𝑡𝑎)>1,₁₂𝑥₁=−8,𝑥₁(𝑡𝑥)<1,−7.6,₁(𝑡𝑎)>1,₁₃𝑥₁=−16,𝑥₁(𝑡𝑥)<1,−15.4,₁(𝑡𝑎)>1,₂₁𝑥₂=−1,𝑥₂(𝑡𝑥)<1,−0.7,₂(𝑡𝑎)>1,₂₂𝑥₂=−160,𝑥₂(𝑡𝑥)<1,−159.5,₂(𝑡𝑎)>1,₂₃𝑥₂=2,𝑥₂(𝑡𝑥)<1,2.3,₂(𝑡𝑎)>1,₃₁𝑥₃=−3,𝑥₃(𝑡𝑥)<1,−2.4,₃(𝑡𝑎)>1,₃₂𝑥₃=−4,𝑥₃(𝑡𝑥)<1,−3.6,₃(𝑡𝑎)>1,₃₃𝑥₃=−120,𝑥₃(𝑡𝑥)<1,−119.7,₃(𝑡𝑏)>1,₁₁𝑥₁=−1,𝑥₁(𝑡𝑥)<1,−0.8,₁(𝑡𝑏)>1,₁₂𝑥₁=0,𝑥₁(𝑡𝑥)<1,0.1,₁(𝑡𝑏)>1,₁₃𝑥₁=1,𝑥₁(𝑡𝑥)<1,1.2,₁(𝑡𝑏)>1,₂₁𝑥₂=−0.5,𝑥₂(𝑡𝑥)<1,−0.4,₂(𝑡𝑏)>1,₂₂𝑥₂=−0.3,𝑥₂(𝑡𝑥)<1,−0.2,₂(𝑡𝑏)>1,₂₃𝑥₂=0.7,𝑥₂(𝑡𝑥)<1,0.8,₂(𝑡𝑏)>1,₃₁𝑥₃=0,𝑥₃(𝑡𝑥)<1,0.1,₃(𝑡𝑏)>1,₃₂𝑥₃=0,𝑥₃(𝑡𝑥)<1,0.2,₃(𝑡𝑏)>1,₃₃𝑥₃=−1,𝑥₃(𝑡𝑥)<1,−0.6,₃(𝑡)>1. (49) It is obvious that 𝜏=1,𝜇=0.5. Choose the positive definite diagonal matrix 𝑃=diag(1,1,0.1), then 𝑆=^{98.60−9.00−16.30−9.00319.00−1.94}_{−16.30−1.94239.40}. It is easy to check that 1𝐵1−𝜇𝑈+𝒫^∗+𝐵_∗²=𝑈−𝑆−91.72539.000016.30009.0000−312.12531.940016.30001.9400−232.5253<0. (50)
All conditions of Theorem 6 hold; hence the memristive neural network in this example has a unique almost periodic solution which is globally exponentially stable.
Figure 2 displays the state trajectory of the network with initial condition (𝜑₁(𝑡),𝜑₂(𝑡),𝜑₃(𝑡))^𝑇=(0.3sin𝑡,0.2sin𝑡,0.4cos𝑡)^𝑇,𝑡∈[−1,0). It can be seen that this trajectory converges to the unique almost periodic solution of the network. This is in accordance with the conclusion of Theorem 6.

(a)

(b)

Figure 2 

The state trajectory of the network with the initial condition (𝜑1(𝑡),𝜑2(𝑡),𝜑3(𝑡))𝑇=(0.3sin𝑡,0.2sin𝑡,0.4cos𝑡)𝑇,𝑡∈[−1,0).

5. Conclusion

In this paper, the exponential stability issue of the almost periodic solution for memristive neural networks with time-varying delays has been investigated. A sufficient condition has been obtained to ensure the existence, uniqueness, and global exponential stability of the almost periodic solution. As special cases, when the external input is a periodic or constant function in the network, the conditions which ensure the global exponential stability of a unique periodic solution or equilibrium point have been established for the considered memristive neural networks with time-varying delay. Two illustrative examples have been also given to demonstrate the effectiveness and validity of the proposed results in this paper.

In [56], the distributed filtering issue have been studied for a class of time-varying systems over sensor networks with quantization errors and successive packet dropouts. In [57], authors considered the the exponential stabilization of a class of stochastic system with Markovian jump parameters and mode-dependent mixed time delays. In [58], authors discussed the fuzzy-model-based robust fault detection with stochastic mixed time delays and successive packet dropouts. However, the issues of distributed filtering, stochastic stabilization, and robust fault detection have not been investigated for memristive neural networks in the existing literature. These will be the topic of our research on memristive neural networks with mode-dependent mixed time delays and Markovian jump parameters in future.

Acknowledgments

This work was supported by the Natural Science Foundation of Hebei Province of China (A2011203103) and the Hebei Province Education Foundation of China (2009157).