Stability of Almost Periodic Solution for a General Class of Discontinuous Neural Networks with Mixed Time-Varying Delays

Abstract

The global exponential stability issues are considered for almost periodic solution of the neural networks with mixed time-varying delays and discontinuous neuron activations. Some sufficient conditions for the existence, uniqueness, and global exponential stability of almost periodic solution are achieved in terms of certain linear matrix inequalities (LMIs), by applying differential inclusions theory, matrix inequality analysis technique, and generalized Lyapunov functional approach. In addition, the existence and asymptotically almost periodic behavior of the solution of the neural networks are also investigated under the framework of the solution in the sense of Filippov. Two simulation examples are given to illustrate the validity of the theoretical results.

1. Introduction

When applying neural networks to solve many practical problems in optimization, control, and signal processing, neural networks are usually designed to be globally asymptotical or exponentially stable to avoid spurious responses or the problems of local minima. Hence exploring the global stability of the neural networks is of primary importance. In recent years, neural networks with discontinuous activations, as a special class of dynamical systems described by differential equations with discontinuous right-hand sides, have been found useful to address a number of interesting engineering tasks, such as dry friction, impacting machines, switching in electronic circuits, systems oscillating under the effect of an earthquake, and control synthesis of uncertain systems, and therefore have received extensive attention from a lot of scholars so far; see, for example, [1–29], and references therein. In [1], Forti and Nistri firstly dealt with the global asymptotic stability (GAS) and global convergence in finite time of a unique equilibrium point for the neural networks modeled by differential equations with discontinuous right-hand sides, and by using Lyapunov diagonally stable (LDS) matrix and constructing suitable Lyapunov function, several stability conditions were derived. In the subsequent literature, considerable efforts have been devoted to investigate the issue of stability for the neural network systems with discontinuous activation functions. In [2, 3], by applying generalized Lyapunov approach and 𝑀-matrix, Forti et al. discussed the global exponential stability (GES) of the neural networks with discontinuous or non-Lipschitz activation functions. Arguing as in [1, 4], Lu and Chen dealt with GES and GAS of Cohen-Grossberg neural networks with discontinuous activation functions. In [5–9], under the framework of Filippov solutions, by using differential inclusion and Lyapunov approach, a series of results had been obtained for the global stability of the unique equilibrium point of the neural networks with a single constant time-delay and discontinuous activations. In [10], by using differential inclusion and constructing Lyapunov functional, Liu et al. discussed the global stability of the unique equilibrium point for the neural networks with mixed time-varying delays and discontinuous neuron activations, and some stability results were given in terms of certain LMIs. In [11], Nie and Cao studied the existence and global stability of equilibrium point for time-varying delayed competitive neural networks with discontinuous activation functions. In [12, 13], based on Lyapunov stability theory and matrix inequality analysis techniques, the authors analyzed the global robust stability of a unique equilibrium point for the neural networks with time-varying delays and discontinuous neuron activation functions. In [14], Liu and Cao discussed the robust state estimation problems for time-varying delayed neural networks with discontinuous activation functions via differential inclusions, and some criteria had been established to guarantee the existence of robust state estimator. In [15], under the framework of Filippov solutions, by using matrix measure approach, Liu et al. investigated the global dissipativity and quasisynchronization issues for the time-varying delayed neural networks with discontinuous activations and parameter mismatches.

It is well known that any equilibrium point can be regarded as a special case of periodic solution for a neuron system with arbitrary period or zero amplitude. Through the study on periodic solution, more general results can be obtained than those of the study on equilibrium point for a neuron system. Hence to study the global stability of the equilibrium point of the neural networks with discontinuous activation functions, much attention has been paid to deal with the stability of periodic solution for various neural network systems with discontinuous activations; see, for example, [17–28]. In [18–21], under the Filippov inclusion framework, by using Leray-Schauder alternative theorem and Lyapunov approach, the authors presented the conditions of the existence and GES or GAS of a unique periodic solution for Hopfield neural networks or BAM neural networks with discontinuous activation functions. In [22, 23], Wu et al. discussed the existence and GES of the unique periodic solution for the neural networks with discontinuous activation functions under impulsive control. In [25, 26], under the framework of Filippov solutions, by using Lyapunov approach and 𝐻-matrix, the authors presented the stability results of periodic solution for delayed Cohen-Grossberg neural networks with a single constant time-delay and discontinuous activation functions. In [27, 28], the authors explored the periodic dynamical behavior of the neural networks with time-varying delays and discontinuous activation functions. Some conditions were proposed to ensure the existence and GES of the unique periodic solution.

It should be noted that all the results reported in the literature above are concerned with the issue of stability analysis of equilibrium point or periodic solution and neglect the effects of almost periodicity for the neural networks with discontinuous activation functions. As far as we know, the almost periodicity is one of the basic properties for dynamical neural systems and appears to retrace their paths through phase space but not exactly. Almost periodic functions, with a superior spatial structure, can be regarded as a generalization of periodic functions. As pointed out by [29, 30], in practice, almost periodic phenomenon was more common than periodic phenomenon, and almost periodic oscillatory behavior was more accordant with reality. In the past few years, a number of results had been obtained for the global stability of the almost periodic solution of various neural networks with continuous activation functions; see, for instance, [31–39] and references therein. In [31], Ding and Wang presented the results of the existence and stability of 2^𝑁 almost periodic attractors for Cohen-Grossberg-type BAM neural networks with variable coefficients. In [37], Wang et al. investigated the multistability of delayed neural networks and described new attraction basins of almost-periodic solutions. Very recently, under the framework of the theory of Filippov differential inclusions, in [29], Allegretto et al. proved the common asymptotic behavior of almost periodic solution for discontinuous, delayed, and impulsive neural networks. In [30, 40], Lu and Chen and Qin et al. discussed the existence and uniqueness of an almost periodic solution (as well as its global exponential stability) of delayed neural networks with almost periodic coefficients and discontinuous activations. It should be pointed out that, the network model explored in [29, 30, 40] was a class of discontinuous neural networks with a single constant time-delay, and the stability conditions were achieved by using LDS matrix or 𝑀-matrix. Compared with the stability conditions expressed in terms of LMIs, it was obvious that the results obtained in [29, 30, 40] were very conservative. To the authors' knowledge, very few works have been conducted to deal with the stability analysis issues for almost periodic solution of discontinuous neural networks with time-varying delays, particularly to study the global stability of almost periodic solution by applying LMIs analysis technique, which motivates the work of this paper.

In this paper, our aim is to study the delay-dependent exponential stability problem for almost periodic solution of the neural networks with mixed time-varying delays and discontinuous activation functions. Under the framework of Filippov differential inclusions, by applying the nonsmooth Lyapunov stability theory and highly efficient LMI approach, new delay-dependent sufficient conditions are presented to ensure the existence and GES of almost periodic solution in terms of LMIs, which can be solved efficiently by using recently developed convex optimization algorithms [41]. Moreover, the obtained conclusion is applied to prove the existence and stability of periodic solution (or equilibrium point) for the neural networks with mixed time-varying delays and discontinuous activations.

For convenience, some notations are introduced as follows. 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>0(𝐴<0) means that 𝐴 is positive definite (negative definite). 𝐴⁻¹ denotes the inverse of 𝐴. 𝐴^𝑇 denotes the transpose of 𝐴. 𝜆_max(𝐴) and 𝜆_min(𝐴) denote the maximum and minimum eigenvalue of 𝐴, respectively. Given the vectors 𝑥=(𝑥₁,…,𝑥_𝑛)^𝑇, 𝑦=(𝑦₁,…,𝑦_𝑛)^𝑇∈ℝ^𝑛, ‖𝑥‖=(∑^𝑛_𝑖=1𝑥²_𝑖)^1/2, and 𝑥^𝑇𝑦=∑^𝑛_𝑖=1𝑥_𝑖𝑦_𝑖. ‖𝐴‖ denotes the 2-norm of 𝐴; that is, ‖𝐴‖=𝜆(𝐴^𝑇𝐴), where 𝜆(𝐴^𝑇𝐴) denotes the spectral radius of 𝐴^𝑇𝐴. For 𝑟>0, 𝐶([−𝑟,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).

Given a set 𝐶⊂ℝ^𝑛, 𝐾[𝐶] denotes the closure of the convex hull of 𝐶 and 𝑃_𝑘𝑐(𝐶) denotes the collection of all nonempty, closed, and convex subsets of 𝐶.

Let 𝑉∶ℝ^𝑛→ℝ be a locally Lipschitz continuous function. Clarke’s generalized gradient [42] of 𝑉 at 𝑥 is defined by 𝜕𝑉(𝑥)=𝐾lim_𝑖→∞∇𝑉𝑥_𝑖∶lim_𝑖→∞𝑥_𝑖=𝑥,𝑥_𝑖∈ℝ^𝑛⧵Ω_𝑉∪ℳ, (1) where Ω_𝑉⊂ℝ^𝑛 is the set of Lebesgue measure zero where ∇𝑉 does not exist and ℳ⊂ℝ^𝑛 is an arbitrary set with measure zero.

Let 𝐸⊂𝑅^𝑛. A set-valued map 𝐹∶𝐸↪𝑃_𝑘𝑐(ℝ^𝑛) is said to be measurable, if for all 𝑦∈ℝ^𝑛, ℝ⁺-valued function 𝑥→𝑑(𝑦,𝐹(𝑥))=inf{‖𝑦−𝜐‖,𝜐∈𝐹(𝑥)} is measurable. This definition of measurability is equivalent to saying that Graph(𝐹)=(𝑥,𝜐)∈𝐸×ℝ^𝑛,𝜐∈𝐹(𝑥)∈ℒ(𝐸)×ℬℝ^𝑛 (2) (graph measurability), where ℒ(𝐸) is the Lebesgue 𝜎-field of 𝐸 and ℬ(ℝ^𝑛) is the Borel 𝜎-field of ℝ^𝑛.

Let 𝑌,𝑍 be Hausdorff topological spaces and 𝐺(⋅)∶𝑌↪2^𝑍⧵{∅}. We say that the set-valued map 𝐺(⋅) is upper semicontinuous, if, for all nonempty closed subset 𝐶 of 𝑍, 𝐺⁻¹(𝐶)={𝑦∈𝑌∶𝐺(𝑦)⋂𝐶≠∅} is closed in 𝑌.

The set-valued map 𝐺(⋅) is said to have a closed (convex, compact) image if, for each 𝑥∈𝐸, 𝐺(𝑥) is closed (convex, compact).

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 Filippov solutions are analyzed. Moreover, the proof of the existence of almost periodic solution is given. The global exponential stability is discussed, and a delay-dependent criterion is established in terms of LMIs. In Section 4, two numerical examples are presented to demonstrate the validity of the proposed results. Some conclusions are drawn in Section 5.

2. Model Description and Preliminaries

In this paper, we consider a general class of the neural networks whose dynamics is described by the system of differential equation 𝑥̇_𝑖(𝑡)=−𝑑_𝑖𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗𝑔_𝑗𝑥_𝑗(𝑡+_𝑛𝑗=1)𝑏_𝑖𝑗𝑔_𝑗𝑥_𝑗(𝑡−𝜏(𝑡))+^𝑛_𝑗=1𝑐_𝑖𝑗^𝑡_{(𝑡)𝑡−𝜎}𝑔_𝑗𝑥_𝑗(𝑠)𝑑𝑠+𝐼_𝑖(𝑡),𝑖=1,2,…,𝑛 (3) or equivalently the vector form 𝑥̇(𝑡)=−𝐷𝑥(𝑡)+𝐴𝑔(𝑡)+𝐵𝑔(𝑥(𝑡−𝜏(𝑡)))+𝐶^𝑡_{(𝑡)𝑡−𝜎}𝑔(𝑥(𝑥))𝑑𝑠+𝐼(𝑡), (4) where 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇 is the vector of neuron states at time 𝑡; 𝐷=diag(𝑑₁,𝑑₂,…,𝑑_𝑛) is an 𝑛×𝑛 diagonal matrix, and 𝑑_𝑖>0,𝑖=1,…,𝑛, is the neuron self-inhibition; 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛, 𝐵=(𝑏_𝑖𝑗)_𝑛×𝑛, and 𝐶=(𝑐_𝑖𝑗)_𝑛×𝑛 are real connection weight matrices representing the weighting coefficients of the neurons; 𝑔(𝑥(𝑡))=(𝑔₁(𝑥₁(𝑡)),𝑔₂(𝑥₂(𝑡)),…,𝑔_𝑛(𝑥_𝑛(𝑡)))^𝑇, 𝑔_𝑖, 𝑖=1,…,𝑛 represent the neuron input-output activations; 𝐼(𝑡)=(𝐼₁(𝑡₁),𝐼₂(𝑡₂),…,𝐼_𝑛(𝑡_𝑛))^𝑇 is a real vector function representing the external inputs of the neuron at time 𝑡; the function 𝜏(𝑡) and 𝜎(𝑡) denote the discrete and distributed time-varying delays, respectively, satisfying 0⩽𝜏(𝑡)⩽𝜏_𝑀,𝜏̇(𝑡)⩽𝜏_𝐷(𝑡)<1,0⩽𝜎⩽𝜎_𝑀,𝜎̇(𝑡)⩽𝜎_𝐷<1. (5) The activation function 𝑔(𝑥) satisfies the assumption that

(𝐴₁): (1)𝑔_𝑖, 𝑖=1,…,𝑛, is piecewise continuous; that is, 𝑔_𝑖 is continuous in ℝ except a countable set of jump discontinuous points and, in every compact set of ℝ, has only a finite number of jump discontinuous points;

(2)𝑔_𝑖, 𝑖=1,2,…,𝑛, is nondecreasing.

Under the assumption (𝐴₁), 𝑔(𝑥) is undefined at the points where 𝑔(𝑥) is discontinuous, and 𝐾[𝑔(𝑥)]=(𝐾[𝑔₁(𝑥₁)],…,𝐾[𝑔_𝑛(𝑥_𝑛)])^𝑇, where 𝐾[𝑔_𝑖(𝑥_𝑖)]=[𝑔_𝑖(𝑥⁻_𝑖),𝑔_𝑖(𝑥⁺_𝑖)], 𝑖=1,…,𝑛. The system (4) is a differential equation with discontinuous right-hand side. For the system (4), we adopt the following definition of the solution in the sense of Filippov [43].

Definition 1. A function 𝑥(𝑡)∶[−𝜄,𝑇)→ℝ^𝑛, 𝑇∈(0,+∞], is a solution of the system (4) on [−𝜄,𝑇) if(1)𝑥(𝑡) is continuous on [−𝜄,𝑇) and absolutely continuous on [0,𝑇);(2)𝑥(𝑡) satisfies 𝑥̇(𝑡)∈−𝐷𝑥(𝑡)+𝐴𝐾𝑔(𝑥(𝑡))+𝐵𝐾𝑔(𝑥(𝑡−𝜏(𝑡)))+𝐶^𝑡_{(𝑡)𝑡−𝜎}𝐾𝑔(𝑥(𝑠(𝑡),[),))𝑑𝑠+𝐼fora.a.𝑡∈0,𝑇 (6)where 𝜄=max{𝜏_𝑀,𝜎_𝑀}.

By the assumption (𝐴₁)(1), it is easy to check that 𝜙: (𝑥,𝑡)↪−𝐷𝑥(𝑡)+𝐴𝐾[𝑔(𝑥(𝑡))]+𝐵𝐾[𝑔(𝑥(𝑡−𝜏(𝑡)))]+𝐶∫^𝑡_{𝑡−𝜎(𝑡)}𝐾[𝑔(𝑥(𝑠))]𝑑𝑠+𝐼(𝑡) is an upper semicontinuous set-valued map with nonempty, compact, and convex values. Hence 𝜙(𝑥,𝑡) is measurable [44]. By the measurable selection theorem, if 𝑥(⋅) is a solution of the system (4), then there exists a measurable function 𝛾=(𝛾₁,𝛾₂,…,𝛾_𝑛)^𝑇∶[−𝜄,𝑇)→ℝ^𝑛 such that 𝛾(𝑡)∈𝐾[𝑔(𝑥(𝑡))] and 𝑥̇(𝑡)=−𝐷𝑥(𝑡)+𝐴𝛾(𝑡)+𝐵𝛾(𝑡−𝜏(𝑡))+𝐶^𝑡_{(𝑡)𝑡−𝜎}𝛾(𝑠)𝑑𝑠+𝐼(𝑡), (7) for a.a. 𝑡∈[0,𝑇).

The function 𝛾(𝑡) in (7) is called an output solution associated with the state variable 𝑥(𝑡) and represents the vector of neural network outputs.

Definition 2. For any continuous function 𝜙(𝑠)∶[−𝜄,0]→ℝ^𝑛 and any measurable selection 𝜓∶[−𝜄,0]→ℝ^𝑛, such that 𝜓(𝑠)∈𝐾[𝑔(𝜙(𝑠))] for a.a. 𝑠∈[−𝜄,0], an absolutely continuous function 𝑥(𝑡)=𝑥(𝑡,𝜙,𝜓) associated with a measurable function 𝛾(𝑡) is said to be a solution of the initial value problem (IVP) for the system (4) on [0,𝑇) (𝑇 might be ∞) with the initial value (𝜙(𝑠),𝜓(𝑠)), 𝑠∈[−𝜄,0], if 𝑥̇(𝑡)(𝑡)(𝑡)((𝑡=−𝐷𝑥+𝐴𝛾+𝐵𝛾𝑡−𝜏))+𝐶^𝑡_{(𝑡)𝑡−𝜎}𝛾(𝑠)(𝑡),[),𝑥(𝑠)(𝑠)[],𝛾(𝑠)(𝑠)[].𝑑𝑠+𝐼fora.a.𝑡∈0,𝑇=𝜙,∀𝑠∈−𝜄,0=𝜓,fora.a.𝑠∈−𝜄,0 (8)

Definition 3 (see [45]). A continuous function 𝑥(𝑡)∶ℝ→ℝ^𝑛 is said to be almost periodic on ℝ, if for any scalar 𝜀>0 there exist scalars 𝑙=𝑙(𝜀)>0 and 𝜔=𝜔(𝜀) in any interval with the length of 𝑙, such that ‖𝑥(𝑡+𝜔)−𝑥(𝑡)‖<𝜀 for all 𝑡∈ℝ.

Definition 4. 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 𝑥𝑡,𝜙,𝜓−𝑥^∗(𝑡)≤𝜂𝑒^−𝛿𝑡,𝑡≥0, (9) where 𝑥(𝑡,𝜙,𝜓) is the solution of the system (4) with the initial value (𝜙(𝑠),𝜓(𝑠)), 𝑠∈[−𝜄,0], and 𝛿 is called as the exponential convergence rate.

To obtain the main results of this paper, the following lemmas will be needed.

Lemma 5 (chain rule [42]). If 𝑉(𝑥)∶ℝ^𝑛→ℝ is C-regular and 𝑥(𝑡)∶[0,+∞)→ℝ^𝑛 is absolutely continuous on any compact interval of [0,+∞) then 𝑥(𝑡) and 𝑉(𝑥(𝑡))∶[0,+∞)→ℝ are differentials for a.a. 𝑡∈[0,+∞), and 𝑉̇(𝑥(𝑡))=⟨𝜍,𝑥̇(𝑡)⟩,∀𝜍∈𝜕𝑉(𝑥). (10)

Lemma 6 (Jensen’s inequality). For any constant matrix 𝐴>0, any scalars 𝑎 and 𝑏 with 𝑏>𝑎, and a vector function 𝑥(𝑡)∶[𝑎,𝑏]→ℝ^𝑛 such that the integrals concerned as well defined, ^𝑏_𝑎𝑥(𝑠)𝑑𝑠^𝑇𝐴^𝑏_𝑎𝑥(𝑠)𝑑𝑠⩽(𝑏−𝑎)^𝑏_𝑎𝑥^𝑇(𝑠)𝐴𝑥(𝑠)𝑑𝑠. (11)

Lemma 7. Let scalar 𝜈>0. Given any 𝑥,𝑦∈ℝ^𝑛 and 𝐴∈ℝ^𝑛×𝑛, 2𝑥^𝑇1𝐴𝑦⩽𝜈𝑥^𝑇𝐴𝐴^𝑇𝑥+𝜈𝑦^𝑇𝑦. (12)

Before proceeding to the main results, we further make the following assumptions.

(𝐴₂): 𝐼_𝑖(𝑡), 𝜏(𝑡), and 𝜎(𝑡) are continuous functions and possess almost periodic property; that is, for any 𝜀>0, there exist 𝑙=𝑙(𝜀)>0 and 𝜔=𝜔(𝜀) in any interval with the length of 𝑙, such that 𝐼_𝑖(𝑡+𝜔)−𝐼_𝑖(𝑡|𝜏()(𝑡|𝜎()(𝑡)<𝜀,𝑡+𝜔−𝜏)|<𝜀,𝑡+𝜔−𝜎)|<𝜀. (13)

(𝐴₃): For any 𝜂_𝑖∈𝐾[𝑔_𝑖(𝑥_𝑖)], 𝜁_𝑖∈𝐾[𝑔_𝑖(𝑦_𝑖)], 𝜁_𝑖≠𝜂_𝑖, there exists constant 𝑒_𝑖>0, such that 𝜂_𝑖−𝜁_𝑖𝑥_𝑖−𝑦_𝑖⩽𝑒_𝑖,𝑖=1,2,…,𝑛. (14)

(𝐴₄): There exist a positive definite diagonal matrix 𝑃 and two positive definite symmetric matrices 𝑄, 𝐻, such that 𝑁=−𝑃𝐴−𝐴^𝑇𝑃−𝑄−𝜎_𝑀𝐻−𝑃𝐵−𝑃𝐶−𝐵^𝑇𝑃1−𝜏_𝐷𝑄0−𝐶^𝑇𝑃01−𝜎_𝐷𝜎_𝑀𝐻>0. (15)

3. Main Results

In this section, the main results concerned with the stability of the almost periodic solution are addressed for a general class of neural networks (4).

Firstly, we give the proof of the existence of the solution and discuss the asymptotically almost periodic behavior of the solution for the system (4).

Proposition 8. If the assumption (𝐴₄) holds, then there exist a positive constant 𝛿<𝑑=min{𝑑₁,𝑑₂,…,𝑑_𝑛}, a positive diagonal matrix 𝑃, and two positive definite symmetric matrices 𝑄 and 𝐻 such that 𝑅=−2𝐷+2𝛿𝐸_𝑛𝐴𝐴𝐵𝐶_𝑇Γ𝑃𝐵𝐵𝑃𝐶_𝑇𝐵^𝑇𝑃−1−𝜏_𝐷𝐶𝑄0_𝑇𝐶^𝑇𝑃0−1−𝜎_𝐷𝜎_𝑀𝐻<0, (16) where Γ=𝑃𝐴+𝐴^𝑇𝑃+𝑒^𝛿𝜏𝑀𝑄+𝑒^𝛿𝜎𝑀𝜎_𝑀𝐻+𝛿𝐸_𝑛 and 𝐸_𝑛 is the 𝑛-dimensional identity matrix.

Proof. Let 𝑈=𝐴𝐵𝐶, 𝜆₁=𝜆_min(𝑁), and 𝜆₂=𝜆_max((1/2)𝑈^𝑇𝐷⁻¹𝑈). Choose a scalar 𝜃>𝜆₂/𝜆₁; then 1𝜃𝑁−2𝑈^𝑇𝐷⁻¹𝑈>0. (17) By Schur's complements, (17) is equivalent to𝑅₁=𝐴−2𝐷𝐴𝐵𝐶_𝑇𝜃𝑃𝐴+𝐴^𝑇𝑃+𝑄+𝜎_𝑀𝐻𝐵𝜃𝑃𝐵𝜃𝑃𝐶_𝑇𝜃𝐵^𝑇𝑃−1−𝜏_𝐷𝐶𝜃𝑄0_𝑇𝜃𝐶^𝑇𝑃0−1−𝜎_𝐷𝜎_𝑀𝜃𝐻<0. (18)Thus, we can choose a sufficiently small positive constant 𝛿<𝑑, such that −2𝐷+2𝛿𝐸_𝑛𝐴𝐴𝐵𝐶_𝑇Γ^′𝐵𝜃𝑃𝐵𝜃𝑃𝐶_𝑇𝜃𝐵^𝑇𝑃−1−𝜏_𝐷𝐶𝜃𝑄0_𝑇𝜃𝐶^𝑇𝑃0−1−𝜎_𝐷𝜎_𝑀𝜃𝐻<0, (19) where Γ^′=𝜃(𝑃𝐴+𝐴^𝑇𝑃+𝑒^𝛿𝜏𝑀𝑄+𝑒^𝛿𝜎𝑀𝜎_𝑀𝐻)+𝛿𝐸_𝑛. Let 𝑃=𝜃𝑃, 𝑄=𝜃𝑄, and 𝐻=𝜃𝐻; then Proposition 8 is proved.

Theorem 9. If the assumptions (𝐴₁), (𝐴₂), and (𝐴₄) are satisfied, then the neural network system (4) has a solution of IVP on [0,+∞) for any initial value (𝜙(𝑠),𝜓(𝑠)), 𝑠∈[−𝜄,0].

Proof. For any initial value (𝜙(𝑠),𝜓(𝑠)), 𝑠∈[−𝜄,0], similar to the proof of Lemma 1 in [2], under the assumptions (𝐴₁)(1), the system (4) has a local solution 𝑥(𝑡) associated with a measurable function 𝛾(𝑡) with initial value (𝜙(𝑠),𝜓(𝑠)), 𝑠∈[−𝜄,0] on [0,𝑇), where 𝑇∈(0,+∞) or 𝑇=+∞ and [0,𝑇) is the maximal right-side existence interval of the local solution.
Consider the following Lyapunov functional candidate: 𝑉(𝑡)=𝑒^𝛿𝑡𝑥^𝑇(𝑡)𝑥(𝑡)+2^𝑛_𝑖=1𝑒^𝛿𝑡𝑝_𝑖^𝑥𝑖₀^(𝑡)𝑔_𝑖(𝑠)+𝑑𝑠_{𝑡(𝑡)𝑡−𝜏}𝑒^{𝛿(𝑠+𝜏𝑀)}𝛾^𝑇(𝑠)𝑄𝛾(𝑠)𝑑𝑠+⁰_{(𝑡)−𝜎}^𝑡_𝑡+𝜃𝑒^{𝛿(𝑠+𝜎𝑀)}𝛾^𝑇(𝑠)𝐻𝛾(𝑠)𝑑𝑠𝑑𝜃, (20) where 𝛿, 𝑃, 𝑄, and 𝐻 are defined as in Proposition 8. By Lemma 5, by calculating the time derivative of 𝑉(𝑡) along the local solution of the system (4) on [0,𝑇), it yields 𝑉̇(𝑡)=𝛿𝑒^𝛿𝑡𝑥^𝑇(𝑡)𝑥(𝑡)+2𝑒^𝛿𝑡𝑥^𝑇(𝑡)×−𝐷𝑥(𝑡)+𝐴𝛾(𝑡)+𝐵𝛾(𝑡−𝜏(𝑡))+𝐶^𝑡_{(𝑡)𝑡−𝜎}𝛾(𝑠)𝑑𝑠+𝐼(𝑡)+2𝑒^𝛿𝑡𝛾^𝑇(𝑡)𝑃−𝐷𝑥(𝑡)+𝐴𝛾(𝑡)+𝐵𝛾(𝑡−𝜏(𝑡))+𝐶^𝑡_{(𝑡)𝑡−𝜎}𝛾(𝑠)𝑑𝑠+𝐼(𝑡)+2𝛿𝑒^𝑛𝛿𝑡_𝑖=1𝑝_𝑖^𝑥𝑖₀^(𝑡)𝑔_𝑖(𝑠)𝑑𝑠+𝑒^{𝛿(𝑡+𝜏𝑀)}𝛾^𝑇(𝑡)(𝑡)−(𝜏̇(𝑡𝑒𝑄𝛾1−))^{𝛿(𝑡+𝜏𝑀−𝜏(𝑡))}𝛾^𝑇(𝑡−𝜏(𝑡))((𝑡(𝑡)𝑒𝑄𝛾𝑡−𝜏))+𝜎^{𝛿(𝑡+𝜎𝑀)}𝛾^𝑇(𝑡)(𝑡)−(𝜎̇(𝑡×𝐻𝛾1−))_{𝑡(𝑡)𝑡−𝜎}𝑒^{𝛿(𝑠+𝜎𝑀)}𝛾^𝑇(𝑠)𝐻𝛾(𝑠)𝑑𝑠. (21) Without loss of generality, we can suppose 0∈𝐾[𝑔(0)]. In fact, if this is not the case, set 𝐺(𝑥)=𝑔(𝑥)−𝜂, 𝜂∈𝐾[𝑔(0)]; then the system (7) could be equivalently changed as 𝑥̇(𝑡)=−𝐷𝑥(𝑡)+𝐴𝛾(𝑡)+𝐵𝛾(𝑡−𝜏(𝑡))+𝐶^𝑡_{(𝑡)𝑡−𝜎}𝛾(𝑠)𝑑𝑠+𝐼(𝑡), (22) where 𝛾(𝑡)=𝛾(𝑡)−𝜂∈𝐾[𝐺(𝑥(𝑡))], for a.a. 𝑡∈[0,𝑇), and 𝐼(𝑡)=(𝐴+𝐵+𝜎(𝑡)𝐶)𝜂+𝐼(𝑡). It is obvious that 0∈𝐾[𝐺(0)]. By 0<𝛿<𝑑, 0∈𝐾[𝑔_𝑖(0)], and the assumption (𝐴₁)(2), we have 𝛿^𝑥𝑖₀^(𝑡)𝑔_𝑖(𝑠)𝑑𝑠⩽𝛿𝑥_𝑖(𝑡)𝛾_𝑖(𝑡)⩽𝑑_𝑖𝑥_𝑖(𝑡)𝛾_𝑖(𝑡). (23) Using Lemmas 6 and 7, we can obtain that 𝑉̇(𝑡)⩽𝑒^𝛿𝑡2𝑥^𝑇(𝑡)𝛿𝐸_𝑛−𝐷𝑥(𝑡)+2𝑥^𝑇(𝑡)𝐴𝛾(𝑡)+2𝑥^𝑇(𝑡)𝐵𝛾(𝑡−𝜏(𝑡))+2𝑥^𝑇(𝑡)𝐶^𝑡_{(𝑡)𝑡−𝜎}𝛾(𝑠)𝑑𝑠+2𝛾^𝑇(𝑡)𝑃𝐴𝛾(𝑡)+2𝛾^𝑇(𝑡)𝑃𝐵𝛾(𝑡−𝜏(𝑡))+2𝛾^𝑇(𝑡)𝑃𝐶^𝑡_{(𝑡)𝑡−𝜎}𝛾(𝑠)𝑑𝑠+𝛿𝛾^𝑇(𝑡)𝛾(𝑡)+𝑒^𝛿𝜏𝑀𝛾^𝑇(𝑡)(𝑡)−𝑄𝛾1−𝜏_𝐷𝛾^𝑇(𝑡−𝜏(𝑡))𝑄𝛾(𝑡−𝜏(𝑡))+𝜎_𝑀𝑒^𝛿𝜎𝑀𝛾^𝑇(𝑡)(𝑡)−𝐻𝛾1−𝜎_𝐷𝜎_𝑀^𝑡_{(𝑡)𝑡−𝜎}𝛾^𝑇(𝑠)𝑑𝑠𝐻^𝑡_{(𝑡)𝑡−𝜎}𝛾(𝑠)+𝑒𝑑𝑠_𝛿𝑡𝛿𝐼^𝑇(𝑡)𝐼(𝑡)+𝐼^𝑇(𝑡)𝑃𝑃𝐼(𝑡)=𝑒^𝛿𝑡𝑧^𝑇𝑒𝑅𝑧+^𝛿𝑡𝛿𝐼^𝑇(𝑡)𝐼(𝑡)+𝐼^𝑇(𝑡)𝑃(𝑡)⩽𝑒𝑃𝐼_𝛿𝑡𝛿‖𝐼(𝑡)‖²+𝐼^𝑇(𝑡)𝑃𝑃𝐼(𝑡), (24) where 𝑧=𝑥^𝑇(𝑡)𝛾^𝑇(𝑡)𝛾^𝑇(𝑡−𝜏(𝑡))∫^𝑡_{(𝑡)𝑡−𝜎}𝛾^𝑇(𝑠)𝑑𝑠^𝑇. (25)𝑅 is defined as in Proposition 8. By the assumption (𝐴₂), 𝐼(𝑡) is bounded for 𝑡⩾0. Hence there exists a constant 𝑀>0 such that 𝑒0<^𝛿𝑡𝛿‖𝐼(𝑡)‖²+𝐼^𝑇(𝑡)𝑃𝑃𝐼(𝑡)<𝑀,𝑡⩾0. (26) This implies that 𝑉̇(𝑡)⩽𝑀𝑒^𝛿𝑡,∀𝑡∈[0,𝑇). (27) Integrating both sides of (27) from 0 to 𝑡, 𝑡∈[0,𝑇), it follows that 𝑉(𝑡)⩽𝑉(0)+^𝑡₀𝑀𝑒^𝛿𝑠(0)+𝑀𝑑𝑠=𝑉𝛿𝑒^𝛿𝑡−1. (28) In view of the definition of 𝑉(𝑡) in (17) and the fact that all the terms in 𝑉(𝑡) are not negative, we have 𝑉(𝑡)⩾𝑒^𝛿𝑡𝑥^𝑇(𝑡)𝑥(𝑡),𝑡∈[0,𝑇). (29) By combining (28) and (29), it is easy to obtain ‖𝑥(𝑡)‖²⩽𝑒^−𝛿𝑡𝑉(0)+𝑀𝛿1−𝑒^−𝛿𝑡,𝑡∈[0,𝑇). (30) Therefore, lim_𝑡→𝑇⁻‖𝑥(𝑡)‖<+∞. By viability theorem in differential inclusions theory [44], one yields 𝑇=+∞. That is, the system (4) has a solution of IVP on [0,+∞) for any initial value. The proof is completed.

Theorem 10. Suppose that the assumptions (𝐴₁)--(𝐴₄) are satisfied; then any solution of the neural network system (4) is asymptotically almost periodic.

Proof. Let 𝑥(𝑡) be a solution of IVP of the system (4) associated with a measurable function 𝛾(𝑡) with initial value (𝜙(𝑠),𝜓(𝑠)), 𝑠∈[−𝜄,0]. Set 𝑦(𝑡)=𝑥(𝑡+𝜔)−𝑥(𝑡), we have 𝑦(̇𝑡)=−𝐷𝑥(𝑡+𝜔)+𝐴𝛾(𝑡+𝜔)+𝐵𝛾(𝑡+𝜔−𝜏(𝑡+𝜔))+𝐶₍₎^𝑡+𝜔_{𝑡+𝜔−𝜎𝑡+𝜔}𝛾(𝑠)()−(𝑡)(𝑡)((𝑡𝑑𝑠+𝐼𝑡+𝜔−𝐷𝑥+𝐴𝛾+𝐵𝛾𝑡−𝜏))+𝐶^𝑡_{(𝑡)𝑡−𝜎}𝛾(𝑠)(𝑡(𝑡)𝛾()(𝑡𝛾((𝑡((𝑡𝑑𝑠+𝐼)=−𝐷𝑦+𝐴𝑡+𝜔−𝛾)+𝐵𝑡+𝜔−𝜏))−𝛾𝑡−𝜏))+𝐶^𝑡_{(𝑡)𝑡−𝜎}𝛾(𝑠+𝜔)−𝛾(𝑠)𝑑𝑠+𝜌(𝜔,𝑡), (31) where 𝜌(𝜔,𝑡)=𝐵𝛾(𝑡+𝜔−𝜏(𝑡+𝜔))−𝛾(𝑡+𝜔−𝜏(𝑡))+𝐶₍₎^{𝑡−𝜎(𝑡)}_{𝑡−𝜎𝑡+𝜔}𝛾(𝑠+𝜔)𝑑𝑠+𝐼(𝑡+𝜔)−𝐼(𝑡). (32) Consider a Lyapunov functional candidate as 𝑊(𝑡)=𝑒^𝛿𝑡𝑦^𝑇(𝑡)𝑦(𝑡)+2^𝑛_𝑖=1𝑒^𝛿𝑡𝑝_𝑖^𝑦𝑖₀^(𝑡)𝑔_𝑖(𝑠)+𝑑𝑠_{𝑡(𝑡)𝑡−𝜏}𝑒^{𝛿(𝑠+𝜏𝑀)}𝛾(𝑠+𝜔)−𝛾(𝑠)^𝑇×𝑄𝛾()(𝑠)+𝑠+𝜔−𝛾𝑑𝑠_{0(𝑡)−𝜎𝑡𝑡+𝜃}𝑒^{𝛿(𝑠+𝜎𝑀)}𝛾(𝑠+𝜔)−𝛾(𝑠)^𝑇×𝐻𝛾(𝑠+𝜔)−𝛾(𝑠)𝑑𝑠𝑑𝜃. (33) Calculating the time derivative of 𝑊(𝑡) along trajectories of the system (31), similar to the proof of Theorem 9, we can get 𝑊̇(𝑡)⩽𝑒^𝛿𝑡𝛿𝜌(𝜔,𝑡)²+𝜌^𝑇(𝜔,𝑡)𝑃𝑃𝜌(𝜔,𝑡). (34) From the proof of Theorem 9, we can get that 𝑥(𝑡) is bounded. Consequently, 𝛾(𝑡) is also bounded. By the assumption (𝐴₃), there exist positive constants 𝛼 and 𝛽, such that 𝜌(‖𝐵𝜏()(𝑡‖𝐶𝜎()(𝑡+‖𝐼()(𝑡.𝜔,𝑡)⩽𝛼‖|𝑡+𝜔−𝜏)|+𝛽‖|𝑡+𝜔−𝜎)|𝑡+𝜔−𝐼)‖ (35) Therefore, by using the assumption (𝐴₂), it is easy to obtain that, for any 𝜀>0, there exist 𝑙=𝑙(𝜀) and 𝜔=𝜔(𝜀) in any interval with the length of 𝑙, such that 1𝛿𝜌(𝜔,𝑡)²+𝜌^𝑇(𝜔,𝑡)𝑃(⩽1𝑃𝜌𝜔,𝑡)2𝛿𝜀²,𝑡⩾0. (36) This implies 𝑊̇(𝑡)⩽12𝜀²𝛿𝑒^𝛿𝑡. (37) By combining (33) and (37), we have 𝑦(𝑡)²⩽𝑒^−𝛿𝑡𝑊(𝑡)⩽𝑒^−𝛿𝑡𝑊(0)+12𝑒^{−𝛿𝑡𝑡}₀𝜀²𝛿𝑒^𝛿𝑠𝑑𝑠=𝑒^−𝛿𝑡𝑊(0)+12𝜀²1−𝑒^−𝛿𝑡. (38) Therefore, there exists 𝑇>0, such that, for any 𝑡>𝑇,‖𝑦(𝑡)‖<(1/√2)𝜀<𝜀; that is, ‖𝑥(𝑡+𝜔)−𝑥(𝑡)‖<𝜀. This shows that any solution of the system (4) is asymptotically almost periodic. The proof is completed.

Remark 11. (1) If discrete time-varying delay reduces to constant delay (i.e., 𝜏(𝑡)=𝜏), then 𝜌(𝜔,𝑡)=𝐶∫^{−𝜎(𝑡)}_{−𝜎(𝑡+𝜔)}𝛾(𝑠+𝑡+𝜔)𝑑𝑠+𝐼(𝑡+𝜔)−𝐼(𝑡). Without making the assumption (𝐴₃), from the proof above, it is easy to see that Theorem 10 can be also obtained. (2) If both 𝜏(𝑡) and 𝜎(𝑡) reduce to constant delays (i.e., 𝜏(𝑡)=𝜏, 𝜎(𝑡)=𝜎), then 𝜌(𝜔,𝑡)=𝐼(𝑡+𝜔)−𝐼(𝑡). In this case, Theorem 10 is also true.

Next, we will discuss the existence, uniqueness, and global exponential stability of the almost periodic solutions for the system (4).

Theorem 12. Suppose that the assumptions (𝐴₁)--(𝐴₄) hold; then the neural network system (4) has a unique almost periodic solution which is globally exponentially stable.

Proof. Firstly, we prove the existence of the almost periodic solution for the system (4).
Under the assumptions of Theorem 12, for any initial value (𝜙(𝑠),𝜓(𝑠)), 𝑠∈[−𝜄,0], the neural network (4) has a solution which is asymptotically almost periodic. Let 𝑥(𝑡) be any solution of the system (4) associated with a measurable function 𝛾(𝑡) with the initial value (𝜙(𝑠),𝜓(𝑠)), 𝑠∈[−𝜄,0]. Then 𝑥̇(𝑡)=−𝐷𝑥(𝑡)+𝐴𝛾(𝑡)+𝐵𝛾(𝑡−𝜏(𝑡))+𝐶^𝑡_{(𝑡)𝑡−𝜎}𝛾(𝑠)𝑑𝑠+𝐼(𝑡), (39) for a.a. 𝑡∈[−𝜄,+∞).
By using (36), we can pick a sequence {𝑡_𝑘} satisfying lim_𝑘→+∞𝑡_𝑘=+∞ and ‖𝜌(𝑡_𝑘,𝑡)‖<1/𝑘, for all 𝑡⩾0, where 𝜌(𝜔,𝑡) is defined in (32). In addition, the sequence {𝑥(𝑡+𝑡_𝑘)} is equicontinuous and uniformly bounded. By the Arzela-Ascoli theorem and diagonal selection principle, we can select a subsequence of {𝑡_𝑘} (still denoted by {𝑡_𝑘}), such that {𝑥(𝑡+𝑡_𝑘)} uniformly converges into an absolute continuous function 𝑥^∗(𝑡) on any compact set of ℝ.
On the other hand, since 𝛾(𝑡+𝑡_𝑘)∈𝐾[𝑔(𝑥(𝑡+𝑡_𝑘))] and 𝐾[𝑔(𝑥(𝑡+𝑡_𝑘))] are bounded by the boundedness of 𝑥(𝑡), the sequence {𝛾(𝑡+𝑡_𝑘)} is bounded. Hence we can also select a sub-sequence of 𝑡_𝑘 (still denoted by {𝑡_𝑘}), such that {𝛾(𝑡+𝑡_𝑘)} converges into a measurable function 𝛾^∗(𝑡) for any 𝑡∈[−𝜄,+∞). According to the fact that(i)𝐾[𝑔(⋅)] is an upper semicontinuous set-valued map,(ii)for 𝑡∈[−𝜄,+∞), 𝑥(𝑡+𝑡_𝑘)→𝑥^∗(𝑡) as 𝑘→+∞,we can get that, for any 𝜖>0, there exists 𝑁>0, such that 𝐾[𝑔(𝑥(𝑡+𝑡_𝑘))]⊆𝐾[𝑔(𝑥^∗(𝑡))]+𝜖ℬ for 𝑘>𝑁 and 𝑡∈[−𝜄,+∞), where ℬ is an 𝑛-dimensional unit ball. Hence the fact that 𝛾(𝑡+𝑡_𝑘)∈𝐾[𝑔(𝑥(𝑡+𝑡_𝑘))] implies that 𝛾(𝑡+𝑡_𝑘)∈𝐾[𝑔(𝑥^∗(𝑡))]+𝜖ℬ. On the other hand, since 𝐾[𝑔(𝑥^∗(𝑡))]+𝜖ℬ is a compact subset of ℝ^𝑛, we have 𝛾^∗(𝑡)=lim_𝑘→+∞𝛾(𝑡+𝑡_𝑘)∈𝐾[𝑔(𝑥^∗(𝑡))]+𝜖ℬ. Noting the arbitrariness of 𝜖, it follows that 𝛾^∗(𝑡)∈𝐾[𝑔(𝑥^∗(𝑡))] for a.a. 𝑡∈[−𝜄,+∞).
By Lebesgue's dominated convergence theorem, we can obtain 𝑥^∗(𝑡+ℎ)−𝑥^∗(𝑡)=lim_𝑘→+∞𝑥𝑡+𝑡_𝑘+ℎ−𝑥𝑡+𝑡_𝑘=lim_{𝑘→+∞𝑡}^𝑡+ℎ−𝐷𝑥𝑡_𝑘+𝜃+𝐴𝛾𝑡_𝑘𝑡+𝜃+𝐵𝛾_𝑘+𝜃−𝜏𝑡_𝑘+𝜃+𝐶^𝑡𝑘_𝑡^+𝜃_{𝑘𝑡+𝜃−𝜎𝑘+𝜃}𝛾(𝑠)𝑑𝑠+𝐼𝑡_𝑘+𝜃𝑑𝜃=lim_{𝑘→+∞𝑡}^𝑡+ℎ−𝐷𝑥𝑡_𝑘+𝜃+𝐴𝛾𝑡_𝑘𝑡+𝜃+𝐵𝛾_𝑘+𝜃−𝜏(𝜃)+𝐶^𝜃_{(𝜃)𝜃−𝜎}𝛾𝑡_𝑘(𝜃)𝑡+𝑠𝑑𝑠+𝐼+𝜌_𝑘=,𝜃𝑑𝜃_𝑡𝑡+ℎ−𝐷𝑥^∗(𝜃)+𝐴𝛾^∗(𝜃)+𝐵𝛾^∗(𝜃−𝜏(𝜃))+𝐶^𝜃_{(𝜃)𝜃−𝜎}𝛾^∗(𝑠)𝑑𝑠+𝐼(𝜃)𝑑𝜃, (40) for any 𝑡∈[−𝜄,+∞) and ℎ∈ℝ. This implies that 𝑥^∗(𝑡) is a solution of the system (4).
Notice that 𝑥(𝑡) is asymptotically almost periodic; then, for any 𝜀>0, there exist 𝑇>0, 𝑙=𝑙(𝜀), and 𝜔=𝜔(𝜀) in any interval with the length of 𝑙, such that ‖𝑥(𝑡+𝜔)−𝑥(𝑡)‖<𝜀, for all 𝑡>𝑇. Therefore, there exists a constant 𝑁>0, when 𝑘>𝑁 and ‖𝑥(𝑡+𝑡_𝑘+𝜔)−𝑥(𝑡+𝑡_𝑘)‖<𝜀, for any 𝑡∈[−𝜄,+∞). Let 𝑘→+∞; it follows that ‖𝑥^∗(𝑡+𝜔)−𝑥^∗(𝑡)‖<𝜀, for any 𝑡∈[−𝜄,+∞). This shows that 𝑥^∗(𝑡) is a almost periodic solution of the system (4).
Consider the changes of variables 𝑧(𝑡)=𝑥(𝑡)−𝑥^∗(𝑡), which transforms (4) into the differential equation 𝑧(̇𝑡)=−𝐷𝑧(𝑡)+𝐴𝛾(𝑡)+𝐵𝛾(𝑡−𝜏(𝑡))+𝐶^𝑡_{(𝑡)𝑡−𝜎}𝛾(𝑠)𝑑𝑠, (41) where 𝛾(𝑡)∈𝐾[𝐺(𝑧(𝑡))] is measurable, 𝐺(𝑧(𝑡))=(𝐺₁(𝑧₁(𝑡)),𝐺₂(𝑧₂(𝑡)),…,𝐺_𝑛(𝑧_𝑛(𝑡)))^𝑇 and 𝐺_𝑖(𝑧_𝑖(𝑡))=𝑔_𝑖(𝑧_𝑖(𝑡)+𝑥^∗_𝑖(𝑡))−𝛾^∗_𝑖(𝑡)(𝑖=1,2,…,𝑛) and 𝛾^∗(𝑡) is the output solution associated with 𝑥^∗(𝑡).
Similar to 𝑉(𝑡) in (17), define a Lyapunov functional candidate 𝐿(𝑡) as 𝐿(𝑡)=𝑒^𝛿𝑡𝑧^𝑇(𝑡)𝑧(𝑡)+2^𝑛_𝑖=1𝑒^𝛿𝑡𝑝_𝑖^𝑧𝑖₀^(𝑡)𝐺(𝑠)+𝑑𝑠_{𝑡(𝑡)𝑡−𝜏}𝑒^{𝛿(𝑠+𝜏𝑀)}𝛾^𝑇(𝑠)𝑄𝛾(𝑠)𝑑𝑠+⁰_{(𝑡)−𝜎}^𝑡_𝑡+𝜃𝑒^{𝛿(𝑠+𝜎𝑀)}𝛾^𝑇(𝑠)𝐻𝛾(𝑠)𝑑𝑠𝑑𝜃. (42) Calculate the derivative of 𝐿(𝑡) along the solution of the system (41). Arguing as in the proof of Theorem 9, we have 𝐿̇(𝑡)⩽𝑒^𝛿𝑡𝜐^𝑇𝑅𝜐−𝛿𝑒^𝛿𝑡𝑧^𝑇(𝑡)𝑧(𝑡)−𝛿𝑒^𝛿𝑡𝛾^𝑇(𝑡)𝛾(𝑡)<0, (43) where 𝜐=𝑧^𝑇(𝑡)𝛾^𝑇(𝑡)𝛾^𝑇(𝑡−𝜏(𝑡))∫^𝑡_{(𝑡)𝑡−𝜎}𝛾^𝑇(𝑠)^𝑇. (44) Combining (42) and (43) gives ‖𝑧(𝑡)‖⩽𝐿(𝑡)𝑒^{−(𝛿/2)𝑡}⩽𝐿(0)𝑒^{−(𝛿/2)𝑡}. (45) This implies that the almost periodic solution 𝑥^∗(𝑡) of the system (4) is globally exponentially stable. Consequently, the almost periodic solution of the system (4) is unique. The proof is completed.

Remark 13. Under the assumptions (𝐴₁) and (𝐴₂), if the assumption (𝐴₄) changes as the assumption (𝒜₄), there exists a positive diagonal matrix 𝑃, such that the coefficient matrices 𝐴, 𝐵, and 𝐶 of the system (4) satisfy the following LMI: 𝒩=−𝑃𝐴−𝐴^𝑇𝑃−𝐵^𝑇𝐵−𝜎_𝑀𝐶^𝑇𝐶−𝑃−𝑃−𝑃1−𝜏_𝐷𝐸_𝑛0−𝑃01−𝜎_𝐷𝜎_𝑀𝐸_𝑛>0, (46)
then, the conclusion of Theorem 9 is still true. In fact, similar with the proof of Proposition 8, we can choose positive constants 𝜃 and 𝛿<𝑑, such that ℛ=−2𝐷+2𝛿𝐸_𝑛𝐴𝐸_𝑛𝐸_𝑛𝐴^𝑇𝐸Σ𝜃𝑃𝜃𝑃_𝑛𝜃𝑃1−𝜏_𝐷𝜃𝐸_𝑛0𝐸_𝑛𝜃𝑃01−𝜎_𝐷𝜎_𝑀𝜃𝐸_𝑛<0, (47) where Σ=𝜃(𝑃𝐴+𝐴^𝑇𝑃+𝑒^𝛿𝜏𝑀𝐵^𝑇𝐵+𝜎_𝑀𝑒^𝛿𝜎𝑀𝐶^𝑇𝐶+𝛿𝐸_𝑛).
Consider the following Lyapunov functional candidate: 𝒱(𝑡)=𝑒^𝛿𝑡𝑥^𝑇(𝑡)𝑥(𝑡)+2𝜃^𝑛_𝑖=1𝑒^𝛿𝑡𝑝_𝑖^𝑥𝑖₀^(𝑡)𝑔_𝑖(𝑠)𝑑𝑠+𝜃^𝑡_{(𝑡)𝑡−𝜏}𝑒^{𝛿(𝑠+𝜏𝑀)}𝛾^𝑇(𝑠)𝐵^𝑇𝐵𝛾(𝑠)𝑑𝑠+𝜃⁰_{(𝑡)−𝜎}^𝑡_𝑡+𝜌𝑒^{𝛿(𝑠+𝜎𝑀)}𝛾^𝑇(𝑠)𝐶^𝑇𝐶𝛾(𝑠)𝑑𝑠𝑑𝜌. (48) Under the assumptions (𝐴₁) and (𝐴₂), calculating the time derivative of 𝒱(𝑡) along the local solution of the system (4), we can obtain 𝒱̇(𝑡)⩽𝑒^𝛿𝑡𝑢^𝑇ℛ𝑢+𝑒^𝛿𝑡𝛿𝐼^𝑇(𝑡)𝐼(𝑡)+𝜃𝐼^𝑇(𝑡)𝑃𝑃𝐼(𝑡), (49) where 𝑢=𝑥^𝑇(𝑡)𝛾^𝑇(𝑡)𝛾^𝑇(𝑡−𝜏(𝑡))𝐵^𝑇∫^𝑡_{(𝑡)𝑡−𝜎}𝛾^𝑇(𝑠)𝑑𝑠𝐶^𝑇𝑇. (50) This implies that the conclusion of Theorem 9 is still true.
In addition, under the case above, if the assumption (𝐴₃) is satisfied, then it is easy to get that the conclusions of Theorems 10 and 12 are also correct.

Notice that periodic function can be regarded as special almost periodic function. Hence based on Theorems 9 and 12, we can obtain the following corollary.

Corollary 14. Suppose that 𝐼(𝑡), 𝜏(𝑡), and 𝜎(𝑡) are periodic functions; if the assumptions (𝐴₁), (𝐴₃), and (𝐴₄) (or (𝒜₄)) are satisfied, then(1)the neural network system (4) has a solution of IVP on [0,+∞) for any initial value (𝜙(𝑠),𝜓(𝑠)), 𝑠∈[−𝜄,0];(2)the neural network system (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(𝑡)=−𝐷𝑥(𝑡)+𝐴𝑔(𝑡)+𝐵𝑔(𝑥(𝑡−𝜏(𝑡)))+𝐶^𝑡_{(𝑡)𝑡−𝜎}𝑔(𝑥(𝑥))𝑑𝑠+𝐼. (51) Since a constant function can be also regarded as a special almost periodic function, by applying Theorems 9 and 12 on the neural network system (51), we can obtain the following corollary.

Corollary 15. If the assumptions (𝐴₁), (𝐴₃) and (𝐴₄) (or (𝒜₄)) are satisfied, then(1)the neural network system (51) has a solution of IVP on [0,+∞) for any initial value (𝜙(𝑠),𝜓(𝑠)), 𝑠∈[−𝜄,0];(2)the neural network system (51) has a unique equilibrium point which is globally exponentially stable.

4. Illustrative Examples

Example 1. Consider the second-order neural network (4) with the following system parameters: ,.𝐷=2002,𝐴=−3−0.50.5−3𝐵=0.85−0.350.350.45,𝐶=0.7−0.450.250.65 (52) Set 𝑔₁(𝑠)=[𝑠+0.5], 𝑔₂(𝑠)=[𝑠], 𝜏(𝑡)=0.8−0.2sin𝑡, and 𝜎(𝑡)=0.5(1+cos𝑡). It is easy to check that assumptions (𝐴₁)--(𝐴₃) hold, and 𝜏_𝑀=1,𝜏_𝐷=0.2, 𝜎_𝑀=1, and 𝜎_𝐷=0.5.
Solving the LMI in (15) by using appropriate LMI solver in the Matlab, the feasible positive definite diagonal matrix 𝑃 and positive definite matrices 𝑄, 𝐻 could be as ,.𝑃=0.7900000.8540,𝑄=1.6282−0.0081−0.00811.7253𝐻=1.7451−0.0106−0.01061.8725 (53) The assumption (𝐴₄) is also satisfied.
When the external input of the network 𝐼(𝑡)=(2sin𝑡,−2cos𝑡)^𝑇 is a periodic function, by Corollary 14, this neural network has a unique periodic solution which is globally exponentially stable.
Figures 1 and 2 display the state trajectories of this neural network with initial values 𝜙(𝑡)=(sin2𝑡,cos2𝑡)^𝑇, 𝑡∈[−1,0] when 𝐼(𝑡)=(2sin𝑡,−2cos𝑡)^𝑇. It can be seen that these trajectories converge into the unique periodic solution of the network. This is in accordance with the conclusion of Corollary 14.
When the external input of the network 𝐼(𝑡)=(1,−3)^𝑇 is constant, by Corollary 15, this neural network has a unique equilibrium point which is globally exponentially stable.
Figure 3 displays the state trajectories of this neural network with state initial value 𝜙(𝑡)=(sin2𝑡,cos2𝑡)^𝑇, 𝑡∈[−1,0], when 𝐼(𝑡)=(1,−3)^𝑇. It can be seen that these trajectories converge into the unique equilibrium point of the network. This is in accordance with the conclusion of Corollary 15.

Figure 1 

Time-domain behavior of the state variables 𝑥1 and 𝑥2 when 𝐼(𝑡)=(2sin𝑡,−2cos𝑡)𝑇.

Figure 2 

Phase plane behavior of the state variables 𝑥1 and 𝑥2 when 𝐼(𝑡)=(2sin𝑡,−2cos𝑡)𝑇.

Figure 3 

Time-domain behavior of the state variables 𝑥1 and 𝑥2 when 𝐼(𝑡)=(1,−3)𝑇.

Example 2. Consider the third-order neural network (4) with the following system parameters: ,.𝐷=200020002,𝐴=−3.000.530.130.15−3.000.250.300.25−3.00𝐵=0.80−0.650.060.41−0.820.350.270.340.60,𝐶=−0.670.140.04−0.300.690.040.05−0.250.75 (54) Set 𝑔₁(𝑠)=[𝑠], 𝑔₂(𝑠)=𝑔₃(𝑠)=[𝑠+0.5], 𝜏(𝑡)=0.8−0.2sin𝑡, and 𝜎(𝑡)=0.5(1+cos𝑡). It is easy to check that assumptions (𝐴₁)--(𝐴₃) hold, and 𝜏_𝑀=1,𝜏_𝐷=0.2, 𝜎_𝑀=1, and 𝜎_𝐷=0.5.
By solving the LMI in (15) by using appropriate LMI solver in the Matlab, the feasible positive definite diagonal matrix 𝑃 and positive definite matrices 𝑄,𝐻 could be as ,,.𝑃=22.850400023.524700023.3745𝑄=41.7777−10.5914−0.2800−10.591443.1320−2.8559−0.2800−2.855941.0031𝐻=46.2777−11.2084−4.7838−11.208447.0601−3.6590−4.7838−3.659048.8991 (55) The assumption (𝐴₄) is also satisfied.
When the external input of the network 𝐼(𝑡)=(2sin𝑡,−cos𝑡,−5sin2𝑡)^𝑇 is a periodic function, by Corollary 14, this neural network has a unique periodic solution which is globally exponentially stable.
Figures 4 and 5 display the state trajectories of this neural network with state initial value 𝜙(𝑡)=(−(1/2)𝑒^2𝑡,𝑒^𝑡,−(1/5)sin𝑡)^𝑇, 𝑡∈[−1,0], when 𝐼(𝑡)=(2sin𝑡,−cos𝑡,−5sin2𝑡)^𝑇. It can be seen that these trajectories converge into the unique periodic of the network. This is in accordance with the conclusion of Corollary 14.
When the external input of the network 𝐼(𝑡)=(−3,−4,−5)^𝑇 is constant, by Corollary 15, this neural network has a unique equilibrium point which is globally exponentially stable.
Figure 6 displays the state trajectories of this neural network with initial values 𝜙(𝑡)=(−(1/2)𝑒^2𝑡,𝑒^𝑡,−(1/5)sin𝑡)^𝑇, 𝑡∈[−1,0], when 𝐼(𝑡)=(−3,−4,−5)^𝑇. It can be seen that these trajectories converge into the unique equilibrium point of the network. This is in accordance with the conclusion of Corollary 15.

Figure 4 

Time-domain behavior of the state variables 𝑥1, 𝑥2, and 𝑥3 when 𝐼(𝑡)=(2sin𝑡,−cos𝑡,−5sin2𝑡)𝑇.

Figure 5 

Phase plane behavior of the state variables 𝑥1, 𝑥2, and 𝑥3 when 𝐼(𝑡)=(2sin𝑡,−cos𝑡,−5sin2𝑡)𝑇.

Figure 6 

Time-domain behavior of the state variables 𝑥1, 𝑥2, and 𝑥3 when 𝐼(𝑡)=(−3,−4,−5)𝑇.

5. Conclusion

In this paper, the almost periodic oscillation issue has been investigated for the neural networks with mixed time-varying delays and discontinuous activation functions. Some sufficient conditions which ensure the existence, uniqueness, and global exponential stability of almost periodic solution have been obtained in terms of LMIs, which is easy to be checked and applied in practice. Two numerical examples have been given to illustrate the effectiveness of the present results.

In [2], Forti et al. conjectured that all solutions of the delayed neural networks with discontinuous neuron activations and periodic inputs converge into an asymptotically stable limit cycle. In this paper, under the assumptions (𝐴₁)--(𝐴₄), the obtained results confirm that Forti's conjecture is true for the neural networks with mixed time-varying delays and discontinuous activation functions. In the near future, the stability issues of almost periodic solution for stochastic/impulsive neural networks with discontinuous activations will be expected to be solved.

Acknowledgments

This work was supported by the National Science and Technology Major Project of China (2011ZX05020-006) and the Natural Science Foundation of Hebei Province of China (A2011203103).