Global μ-Stability of Impulsive Complex-Valued Neural Networks with Leakage Delay and Mixed Delays

Abstract

The impulsive complex-valued neural networks with three kinds of time delays including leakage delay, discrete delay, and distributed delay are considered. Based on the homeomorphism mapping principle of complex domain, a sufficient condition for the existence and uniqueness of the equilibrium point of the addressed complex-valued neural networks is proposed in terms of linear matrix inequality (LMI). By constructing appropriate Lyapunov-Krasovskii functionals, and employing the free weighting matrix method, several delay-dependent criteria for checking the global 𝜇-stability of the complex-valued neural networks are established in LMIs. As direct applications of these results, several criteria on the exponential stability, power-stability, and log-stability are obtained. Two examples with simulations are provided to demonstrate the effectiveness of the proposed criteria.

1. Introduction

The nonlinear systems are ubiquitous in the real world [1–5]. As one of the most important nonlinear systems, the real-valued neural networks have been extensively studied and developed due to their extensive applications in pattern recognition, associative memory, signal processing, image processing, combinatorial optimization, and other areas [6]. In implementation of neural networks, however, time delays are unavoidably encountered [7]. It has been found that the existence of time delays may lead to instability and oscillation in a neural network [8]. Therefore, dynamics analysis of neural networks with time delays has received much attention. In [6–10], the exponential stability and asymptotic stability of delayed neural networks were investigated; some sufficient conditions for checking stability were given. In [11, 12], authors investigated the synchronization of chaotic neural networks with delay and obtained several criteria for checking the synchronization. In [13], the passivity of uncertain neural networks with both leakage delay and time-varying delay was considered. In [14], authors investigated the state estimation for neural networks with leakage delay and time-varying delays.

However, besides delay effect, impulsive effects are also likely to exist in neural networks [15]. For instance, in implementation of electronic networks, the state of the networks is subject to instantaneous perturbations and experiences abrupt change at certain instants, which may be caused by switching phenomenon, frequency change. or other sudden noise; that is, it exhibits impulsive effects [16–18]. Therefore, it is necessary to consider both impulsive effect and delay effect on dynamical behaviors of neural networks. Some results on impulsive effect have been gained for delayed neural networks; for example, see [15, 16, 19] and the references therein.

Recently, the power-rate global stability of the equilibrium is proposed in [20]. Moreover, in [21], the authors proposed a new concept of global 𝜇-stability to unify the exponential stability, power-rate stability, and log-stability of neural networks. In [22], the authors investigated the global robust 𝜇-stability in the mean square for a class of stochastic neural networks. In [23], the delayed neural systems with impulsion were considered, and the 𝜇-stability criteria were derived by using Lyapunov-Krasovskii functional method. In [24], the multiple 𝜇-stability of delayed neural networks was investigated, and several criteria for the coexistence of equilibrium points and their local 𝜇-stability were derived.

As an extension of real-valued neural networks, complex-valued neural networks with complex-valued state, output, connection weight, and activation function become strongly desired because of their practical applications in physical systems dealing with electromagnetic, light, ultrasonic, and quantum waves [25, 26]. It has been shown that such applications strongly depend on the stability of CVNNs [27]. Therefore, stability analysis of CVNNs has received much attention and various stability conditions have been obtained [28–33]. In [28], authors considered a discrete-time CVNNs and obtained several sufficient conditions for checking global exponential stability of a unique equilibrium. In [29, 30], the discrete-time CVNNs with linear threshold neurons were investigated, and some conditions for the boundedness, global attractivity, and complete stability as well as global exponential stability of the considered neural networks were also derived. In [31], the continuous-time CVNNs with delays were considered, and the boundedness, complete stability, and exponential stability were investigated. In [32, 33], the global stability was investigated for CVNNs on time scales, which is useful to unify the continuous-time and discrete-time CVNNs under the same framework. In [34], the authors investigated the 𝜇-stability of delayed CVNNs and obtained several sufficient conditions to ensure the global 𝜇-stability. To the best of our knowledge, there are no results on 𝜇-stability of impulsive CVNNs with the three kinds of time delays including leakage delay, discrete delay, and distributed delay in the literature, and it remains as an open topic for further investigation.

Motivated by the above discussions, in this paper, we will deal with the problem of 𝜇-stability for CVNNs with leakage delay, discrete delay, and distributed delay under impulsive perturbations. Based on the homeomorphism mapping principle of complex domain, a LMI condition for the existence and uniqueness of the equilibrium point of the addressed CVNNs is proposed. Several delay-dependent criteria for checking the global 𝜇-stability of the CVNNs are obtained by constructing appropriate Lyapunov-Krasovskii functionals and employing the free weighting matrix method. The obtained results can also be applied to several special cases and we can get the exponential stability, power-stability, and log-stability of the CVNNs, correspondingly. Finally, two illustrative examples are provided to show the effectiveness of the proposed criteria.

Notations. The notations are quite standard. Throughout this paper, let ℤ⁺ denote the set of positive integers. Let 𝑖 denote the imaginary unit; that is, 𝑖=√−1. ℂ^𝑛, ℝ^𝑚×𝑛, and ℂ^𝑚×𝑛 denote, respectively, the set of 𝑛-dimensional complex vectors, 𝑚×𝑛 real matrices and complex matrices. The subscripts 𝑇 and ∗ denote matrix transposition and matrix conjugate transpose, respectively. For complex vector 𝑧∈ℂ^𝑛, let |𝑧|=(|𝑧₁|,|𝑧₂|,…,|𝑧_𝑛|)^𝑇 be the module of the vector 𝑧, and let ‖𝑧‖=∑^𝑛_𝑘=1|𝑧_𝑘|² be the norm of the vector 𝑧. 𝐼 denotes the identity matrix with appropriate dimensions. The notation 𝑋≥𝑌 (resp., 𝑋>𝑌) means that 𝑋−𝑌 is positive semidefinite (resp., positive definite). 𝜆_max(𝑃) and 𝜆_min(𝑃) are defined as the largest and the smallest eigenvalue of positive definite matrix 𝑃, respectively. For any 𝑡≥0, 𝑧_𝑡 is defined by 𝑧_𝑡=𝑧(𝑡+𝑠), 𝑧_𝑡−=𝑧(𝑡⁻+𝑠), 𝑠∈[−𝛿,0]. In addition, the notation ⋆ always denotes the conjugate transpose of block in a Hermitian matrix.

2. Problems Formulation and Preliminaries

Consider the following complex-valued neural networks with leakage delay and mixed delays under impulsive perturbations by a nonlinear differential equation of the form 𝑧̇(𝑡)=−𝐷𝑧(𝑡−𝛿)+𝐴𝑓(𝑧(𝑡))+𝐵𝑓𝑧𝑡−𝜏₁+𝐶^𝑡_𝑡−𝜏2𝑓(𝑧(𝑠))𝑑𝑠+ℎ,𝑡≠𝑡_𝑘𝑡,𝑡>0,Δ𝑧_𝑘=𝑧𝑡_𝑘−𝑧𝑡⁻_𝑘=𝐽_𝑘𝑧𝑡⁻_𝑘,𝑧_𝑡⁻_𝑘,𝑘∈ℤ⁺, (1) where the impulse times 𝑡_𝑘 satisfy 0=𝑡₀<𝑡₁<⋯<𝑡_𝑘<⋯ and lim_𝑘→∞𝑡_𝑘=+∞; 𝑧(𝑡) = (𝑧₁(𝑡),𝑧₂(𝑡),…,𝑧_𝑛(𝑡))^𝑇∈ℂ^𝑛 is the state vector of the neural networks; 𝐷=diag{𝑑₁,𝑑₂,…,𝑑_𝑛}∈ℝ^𝑛×𝑛 is the self-feedback connection weight matrix with 𝑑_𝑗>0 (𝑗=1,2,…,𝑛); 𝐴,𝐵, and 𝐶∈ℂ^𝑛×𝑛 are, respectively, the connection weight matrix, the discretely delayed connection weight matrix, and distributively delayed connection weight matrix; 𝑓(𝑧(𝑡)) = (𝑓₁(𝑧₁(𝑡)),𝑓₂(𝑧₂(𝑡)),…,𝑓_𝑛(𝑧_𝑛(𝑡)))^𝑇∈ℂ^𝑛 represents the neuron activation function; ℎ=(ℎ₁,ℎ₂,…,ℎ_𝑛)^𝑇∈ℂ^𝑛 is the external input vector; 𝛿≥0, 𝜏₁≥0, and 𝜏₂≥0 are the leakage time delay, the discrete time delay, and the distributed time delay, respectively; 𝐽_𝑘 is the impulsive function.

In the analysis of complex-valued neural networks, it is usually assumed that the activation functions are differentiable [31]. However, in this paper, we adopt the following assumption on the activation functions in which the differentiability is not be required:(H1)the neuron activation functions 𝑓_𝑗 are continuous and satisfy 𝑓_𝑗𝑧₁−𝑓_𝑗𝑧₂≤𝛾_𝑗𝑧₁−𝑧₂, (2) for any 𝑧₁, 𝑧₂∈ℂ, 𝑗=1,2,…,𝑛, where 𝛾_𝑗 is a constant. Moreover, we define Γ=diag{𝛾²₁,𝛾²₂,…,𝛾²_𝑛}.

Remark 1. Note that the assumptions on activation functions are weaker than those generally used in the literature. Namely, the boundedness and differentiability of the activation functions 𝑓_𝑗 are not required in this paper.

Next we introduce some definitions and lemmas to be used in the stability analysis.

Definition 2. Let 𝑧 be an equilibrium point of system (1). Suppose that 𝑧(𝑡) is an arbitrary solution of system (1); 𝜇(𝑡) is a positive continuous function and satisfies 𝜇(𝑡)→∞ as 𝑡→∞. If there is a positive constant 𝑀 such that ‖𝑧(𝑡)−𝑧‖≤𝑀𝜇(𝑡),𝑡≥0, (3) then the equilibrium point 𝑧 is said to be 𝜇-stable.

In particular, if taking 𝜇(𝑡) in Definition 2 to exponential function, power function, and logarithmic function, we can get the definitions of exponential stability, power-stability, and log-stability, correspondingly.

Definition 3. Let 𝑧 be an equilibrium point of system (1). Suppose that 𝑧(𝑡) is an arbitrary solution of system (1). If there are two positive constants 𝜀 and 𝑀 such that ‖𝑧(𝑡)−𝑧‖≤𝑀𝑒^𝜀𝑡,𝑡≥0, (4) then the equilibrium point 𝑧 is said to be exponentially stable.

Definition 4. Let 𝑧 be an equilibrium point of system (1). Suppose that 𝑧(𝑡) is an arbitrary solution of system (1). If there are two positive constants 𝜀 and 𝑀 such that ‖𝑧(𝑡)−𝑧‖≤𝑀𝑡^𝜀,𝑡≥0, (5) then the equilibrium point 𝑧 is said to be power-stable.

Definition 5. Let 𝑧 be an equilibrium point of system (1). Suppose that 𝑧(𝑡) is an arbitrary solution of system (1). If there are two positive constants 𝜀 and 𝑀 such that ‖𝑧(𝑡)−𝑧‖≤𝑀ln(𝜀𝑡+1),𝑡>0, (6) then the equilibrium point 𝑧 is said to be log-stable.

Lemma 6. If 𝐻(𝑧)∶ℂ^𝑛→ℂ^𝑛 is a continuous map and satisfies the following conditions: (i)𝐻(𝑧) is injective on ℂ^𝑛,(ii)lim_{‖𝑧‖→∞}‖𝐻(𝑧)‖=∞, then 𝐻(𝑧) is a homeomorphism of ℂ^𝑛 onto itself.

Proof. Let 𝑧=𝑥+𝑖𝑦 and 𝛼=(𝑥^𝑇,𝑦^𝑇)^𝑇, where 𝑥,𝑦∈ℝ^𝑛. Define a homeomorphism ℐ∶ℂ^𝑛→ℝ^2𝑛 by ℐ(𝑧)=𝛼. (7) Obviously, ℐ is invertible. Let ℒ=ℐ∘𝐻∘ℐ⁻¹. Then ℒ is injective on ℝ^2𝑛, since 𝐻 and ℐ are injective. In addition, ‖ℒ(𝛼)‖→∞ when ‖𝛼‖→∞, since lim_{‖𝑧‖→∞}‖ℐ(𝑧)‖=∞ and lim_{‖𝑧‖→∞}‖𝐻(𝑧)‖=∞. Therefore, ℒ is a homeomorphism of ℝ^2𝑛 onto itself. Then 𝐻=ℐ⁻¹∘ℒ∘ℐ is a homeomorphism of ℂ^𝑛 onto itself. The proof is completed.

Lemma 7. For any 𝑎,𝑏∈ℂ^𝑛, if 𝑃∈ℂ^𝑛×𝑛 is a positive definite Hermitian matrix, then 𝑎^∗𝑏+𝑏^∗𝑎≤𝑎^∗𝑃𝑎+𝑏^∗𝑃⁻¹𝑏.

Proof. Since 𝑃 is a positive definite Hermitian matrix, there exists an invertible matrix 𝑄∈ℂ^𝑛×𝑛, such that 𝑃=𝑄^∗𝑄. For any 𝑎,𝑏∈ℂ^𝑛, it follows from Cauchy inequality that 𝑎^∗𝑏+𝑏^∗𝑎=(𝑄𝑎)^∗𝑄^∗−1𝑏+𝑄^∗−1𝑏^∗())𝑄𝑎=2Re(𝑄𝑎^∗𝑄^∗−1𝑏≤2(𝑄𝑎)^∗𝑄^∗−1𝑏≤2‖𝑄𝑎‖𝑄^∗−1𝑏≤(𝑄𝑎)^∗(𝑄𝑎)+𝑄^∗−1𝑏^∗𝑄^∗−1𝑏=𝑎^∗𝑃𝑎+𝑏^∗𝑃⁻¹𝑏. (8) The proof is completed.

Lemma 8 (see [33]). For any constant matrix 𝑊∈ℂ^𝑛×𝑛 and 𝑊>0, a scalar function 𝜔∶[𝑎,𝑏]→ℂ^𝑛 with scalars 𝑎<𝑏 such that the integration concerned are well defined, then (𝑏−𝑎)^𝑏_𝑎𝜔^∗(𝑠)(𝑠)≤𝑊𝜔d𝑠_𝑏𝑎𝜔(𝑠)d𝑠^∗𝑊^𝑏_𝑎𝜔(𝑠)d𝑠. (9)

Lemma 9. A given Hermitian matrix 𝑆𝑆=¹¹𝑆₁₂𝑆₂₁𝑆₂₂<0, (10) where 𝑆^∗₁₁=𝑆₁₁, 𝑆^∗₁₂=𝑆₂₁ and 𝑆^∗₂₂=𝑆₂₂, is equivalent to any one of the following conditions: (1)𝑆₂₂<0 and 𝑆₁₁−𝑆₁₂𝑆⁻¹₂₂𝑆₂₁<0,(2)𝑆₁₁<0 and 𝑆₂₂−𝑆₂₁𝑆⁻¹₁₁𝑆₁₂<0.

Proof. (1) Note that 𝑆𝑆=¹¹𝑆₁₂𝑆₂₁𝑆₂₂=𝐼𝑆₁₂𝑆⁻¹₂₂𝑆0𝐼¹¹−𝑆₁₂𝑆⁻¹₂₂𝑆₂₁00𝑆₂₂𝐼𝑆₁₂𝑆⁻¹₂₂0𝐼^∗. (11) Therefore, 𝑆<0⇔𝑆₂₂<0 and 𝑆₁₁−𝑆₁₂𝑆⁻¹₂₂𝑆₂₁<0.
(2) Note that 𝑆𝑆=¹¹𝑆₁₂𝑆₂₁𝑆₂₂=𝑆𝐼0₂₁𝑆⁻¹₁₁𝐼𝑆₁₁00𝑆₂₂−𝑆₂₁𝑆⁻¹₁₁𝑆₁₂𝑆𝐼0₂₁𝑆⁻¹₁₁𝐼^∗. (12) Therefore, 𝑆<0⇔𝑆₁₁<0 and 𝑆₂₂−𝑆₂₁𝑆⁻¹₁₁𝑆₁₂<0. The proof is completed.

3. Existence and Uniqueness of Equilibrium Point

Now we study the existence and uniqueness of the equilibrium point of system (1). As usual, we denote an equilibrium point of the system (1) by the constant complex vector 𝑧∈ℂ^𝑛, where 𝑧 satisfies −𝐷𝑧+𝐴𝑓(𝑧)+𝐵𝑓(𝑧)+𝜏₂(𝑧)𝐽𝐶𝑓+ℎ=0,_𝑘(𝑧,𝑧)=0,𝑘∈ℤ⁺. (13) In this paper, it is assumed that the constant complex vector 𝑧 satisfies 𝐽_𝑘(𝑧,𝑧)=0 for 𝑘∈ℤ⁺, if 𝑧 satisfies −𝐷𝑧+𝐴𝑓(𝑧)+𝐵𝑓(𝑧)+𝜏₂𝐶𝑓(𝑧)+ℎ=0. Hence, to prove the existence and uniqueness of a solution of (13), it suffices to show that the following map ℋ∶ℂ^𝑛→ℂ^𝑛 has a unique zero point: ℋ(𝑧)=−𝐷𝑧+𝐴+𝐵+𝜏₂𝐶𝑓(𝑧)+ℎ. (14) In the following, we will give some conditions for checking that ℋ is a homeomorphism on ℂ^𝑛, that is, for assuring the existence and uniqueness of the equilibrium point of system (1).

Theorem 10. Under condition (H1), the system (1) has a unique equilibrium point, if there exist two complex matrices 𝑈₁,𝑈₂ and a real positive diagonal matrix 𝑅, such that the following LMI holds: Π₁₁Π₁₂⋆Π₂₂<0, (15) where Π₁₁=Γ^∗𝑅Γ−𝑈₁𝐷−𝐷𝑈^∗₁, Π₁₂=𝑈₁(𝐴+𝐵+𝜏₂𝐶)−𝐷𝑈^∗₂, Π₂₂=𝑈₂(𝐴+𝐵+𝜏₂𝐶) + (𝐴^∗+𝐵^∗+𝜏₂𝐶^∗)𝑈^∗₂−𝑅.

Proof. In the following, we will prove that ℋ(𝑧) is a homeomorphism of ℂ^𝑛 onto itself.
First, we prove that ℋ(𝑧) is an injective map on ℂ^𝑛. Suppose that there exist 𝑧,𝑧^′∈ℂ^𝑛 with 𝑧≠𝑧^′, such that ℋ(𝑧)=ℋ(𝑧^′). Then −𝐷𝑧−𝑧^′+𝐴+𝐵+𝜏₂𝐶𝑓(𝑧)−𝑓𝑧^′=0. (16) Multiply both sides above by (𝑧−𝑧^′)^∗𝑈₁+(𝑓(𝑧)−𝑓(𝑧^′))^∗𝑈₂; we get 0=−𝑧−𝑧^′∗𝑈₁𝐷𝑧−𝑧^′+𝑧−𝑧^′∗𝑈₁𝐴+𝐵+𝜏₂𝐶𝑓(𝑧)−𝑓𝑧^′−𝑓(𝑧)−𝑓𝑧^′∗𝑈₂𝐷𝑧−𝑧^′+𝑓(𝑧)−𝑓𝑧^′∗𝑈₂𝐴+𝐵+𝜏₂𝐶𝑓(𝑧)−𝑓𝑧^′. (17) Taking the conjugate transpose of (17), we get that 0=−𝑧−𝑧^′∗𝐷𝑈^∗₁𝑧−𝑧^′+𝑓(𝑧)−𝑓𝑧^′∗𝐴^∗+𝐵^∗+𝜏₂𝐶^∗𝑈^∗₁𝑧−𝑧^′−𝑧−𝑧^′∗𝐷𝑈^∗₂𝑓(𝑧)−𝑓𝑧^′+𝑓(𝑧)−𝑓𝑧^′∗×𝐴^∗+𝐵^∗+𝜏₂𝐶^∗𝑈^∗₂𝑓(𝑧)−𝑓𝑧^′. (18) Let Σ₁=𝑅−𝑈₂(𝐴+𝐵+𝜏₂𝐶)−(𝐴^∗+𝐵^∗+𝜏₂𝐶^∗)𝑈^∗₂, Σ₂=−𝑈₁𝐷−𝐷𝑈^∗₁ + [𝑈₁(𝐴+𝐵+𝜏₂𝐶) − 𝐷𝑈^∗₂]Σ₁⁻¹[(𝐴^∗+𝐵^∗+𝜏₂𝐶^∗)𝑈^∗₁−𝑈₂𝐷]. Then Σ₁ is a positive definite Hermitian matrix from Lemma 9 and LMI (15). Summing (17) and (18), applying Lemma 7, we have 0=−𝑧−𝑧^′∗𝑈₁𝐷+𝐷𝑈^∗₁𝑧−𝑧^′+𝑧−𝑧^′∗𝑈₁𝐴+𝐵+𝜏₂𝐶−𝐷𝑈^∗₂𝑓(𝑧)−𝑓𝑧^′+𝑓(𝑧)−𝑓𝑧^′∗×𝐴^∗+𝐵^∗+𝜏₂𝐶^∗𝑈^∗₁−𝑈₂𝐷𝑧−𝑧^′+𝑓(𝑧)−𝑓𝑧^′∗×𝑈₂𝐴+𝐵+𝜏₂𝐶+𝐴^∗+𝐵^∗+𝜏₂𝐶^∗𝑈^∗₂×𝑓(𝑧)−𝑓𝑧^′≤−𝑧−𝑧^′∗𝑈₁𝐷+𝐷𝑈^∗₁𝑧−𝑧^′+𝑧−𝑧^′∗𝑈₁𝐴+𝐵+𝜏₂𝐶−𝐷𝑈^∗₂×Σ₁⁻¹𝐴^∗+𝐵^∗+𝜏₂𝐶^∗𝑈^∗₁−𝑈₂𝐷𝑧−𝑧^′+𝑓(𝑧)−𝑓𝑧^′∗Σ₁𝑓(𝑧)−𝑓𝑧^′+𝑓(𝑧)−𝑓𝑧^′∗×𝑈₂𝐴+𝐵+𝜏₂𝐶+𝐴^∗+𝐵^∗+𝜏₂𝐶^∗𝑈^∗₂×𝑓(𝑧)−𝑓𝑧^′=𝑧−𝑧^′∗Σ₂𝑧−𝑧^′+𝑓(𝑧)−𝑓𝑧^′∗𝑅𝑓(𝑧)−𝑓𝑧^′. (19) Since 𝑅 is a positive diagonal matrix, from condition (H1), we can get 𝑓(𝑧)−𝑓𝑧^′∗𝑅𝑓(𝑧)−𝑓𝑧^′≤𝑧−𝑧^′∗Γ^∗𝑅Γ𝑧−𝑧^′. (20) It follows from (19) and (20) that 0≤𝑧−𝑧^′∗Σ₃𝑧−𝑧^′, (21) where Σ₃=Γ^∗𝑅Γ+Σ₂=Γ^∗𝑅Γ−𝑈₁𝐷−𝐷𝑈^∗₁+[𝑈₁(𝐴+𝐵+𝜏₂𝐶)−𝐷𝑈^∗₂]Σ₁⁻¹[(𝐴^∗+𝐵^∗+𝜏₂𝐶^∗)𝑈^∗₁−𝑈₂𝐷]. From Lemma 9 and LMI (15), we can know Σ₃<0. Then 𝑧−𝑧^′=0 from (21). Therefore, ℋ(𝑧) is an injective map on ℂ^𝑛.
Second, we prove ‖ℋ(𝑧)‖→∞ as ‖𝑧‖→∞. Let ℋ(𝑧)=ℋ(𝑧)−ℋ(0). By Lemma 7, we have 𝑧^∗𝑈₁+𝑓(𝑧)−𝑓(0)^∗𝑈₂ℋ(𝑧)+ℋ(𝑧)^∗𝑧^∗𝑈₁+𝑓(𝑧)−𝑓(0)^∗𝑈₂^∗=−𝑧^∗𝑈₁𝐷+𝐷𝑈^∗₁𝑧+𝑧^∗𝑈₁𝐴+𝐵+𝜏₂𝐶−𝐷𝑈^∗₂𝑓(𝑧)(0)+𝑓(𝑧)(0)−𝑓−𝑓^∗𝐴^∗+𝐵^∗+𝜏₂𝐶^∗𝑈^∗₁−𝑈₂𝐷𝑧+𝑓(𝑧)−𝑓(0)^∗×𝑈₂𝐴+𝐵+𝜏₂𝐶+𝐴^∗+𝐵^∗+𝜏₂𝐶^∗𝑈^∗₂×𝑓(𝑧)−𝑓(0)≤−𝑧^∗𝑈₁𝐷+𝐷𝑈^∗₁𝑧+𝑧^∗𝑈₁𝐴+𝐵+𝜏₂𝐶−𝐷𝑈^∗₂×Σ₁⁻¹𝐴^∗+𝐵^∗+𝜏₂𝐶^∗𝑈^∗₁−𝑈₂𝐷𝑧+𝑓(𝑧)−𝑓(0)^∗Σ₁𝑓(𝑧)(0)+𝑓(𝑧)(0)−𝑓−𝑓^∗×𝑈₂𝐴+𝐵+𝜏₂𝐶+𝐴^∗+𝐵^∗+𝜏₂𝐶^∗𝑈^∗₂×𝑓(𝑧)−𝑓(0)=𝑧^∗Σ₂𝑧+𝑓(𝑧)−𝑓(0)^∗𝑅𝑓(𝑧)−𝑓(0). (22) It follows from condition (H1) that 𝑓(𝑧)−𝑓(0)^∗𝑅𝑓(𝑧)−𝑓(0)≤𝑧^∗Γ^∗𝑅Γ𝑧. (23) Then from (22), (23), and Σ₃=Γ^∗𝑅Γ+Σ₂<0, we obtain 𝑧^∗𝑈₁+𝑓(𝑧)−𝑓(0)^∗𝑈₂ℋ(𝑧)+ℋ(𝑧)^∗𝑧^∗𝑈₁+𝑓(𝑧)−𝑓(0)^∗𝑈₂^∗≤𝑧^∗Σ₃𝑧≤−𝜌_min−Σ₃‖𝑧‖², (24) which can imply that 𝜆_min−Σ₃‖𝑧‖²≤2𝑧^∗𝑈₁+𝑓(𝑧)−𝑓(0)^∗𝑈₂ℋ(𝑧𝑈)≤2₁+Γ^∗𝑈₂‖𝑧‖ℋ(𝑧). (25) When 𝑧≠0, we have ℋ(𝑧≥𝜆)^min−Σ₃‖𝑧‖2𝑈₁+Γ^∗𝑈₂. (26) Therefore, ‖ℋ(𝑧)‖→∞ as ‖𝑧‖→∞ which implies ‖ℋ(𝑧)‖→∞ as ‖𝑧‖→∞. From Lemma 6, we know that ℋ(𝑧) is a homeomorphism of ℂ^𝑛. Thus, the system (1) has a unique equilibrium point.

Remark 11. It should be noted that Theorem 10 is independent of leakage time delay and initial conditions. So, the time delays in the leakage terms do not affect the existence and uniqueness of the equilibrium point.

4. Global 𝜇-Stability Results

In the preceding section, we have shown the existence and uniqueness of the equilibrium point for system (1). In this section, we will further investigate the global 𝜇-stability of the unique equilibrium point. For this purpose, the impulsive function 𝐽_𝑘 which is viewed as a perturbation of the equilibrium point 𝑧 of system (1) without impulses is defined by 𝐽_𝑘𝑧𝑡⁻_𝑘,𝑧_𝑡⁻_𝑘=𝐸_𝑘𝑧𝑡⁻_𝑘−𝑧−𝐷^𝑡𝑘_{𝑡𝑘−𝛿}(𝑧(𝑠)−𝑧)d𝑠, (27) where 𝑘∈ℤ⁺, 𝐸_𝑘∈ℂ^𝑛×𝑛. It is obvious that 𝐽_𝑘(𝑧,𝑧)=0.

Remark 12. The type of impulse such as (27) describes the fact that the instantaneous perturbations are not only related to the state of neurons at impulse times 𝑡_𝑘 but also related to the state of neurons in recent history, which reflects more realistic dynamics [19].

Theorem 13. Under the conditions of Theorem 10, the equilibrium point of system (1) is globally 𝜇-stable, if there exist six positive definite Hermitian matrices 𝑃₁, 𝑃₂, 𝑃₃,𝑃₄, 𝑃₅, and 𝑃₆, two real positive diagonal matrices 𝑅₁ and 𝑅₂, three complex matrices 𝑄₁, 𝑄₂, and 𝑄₃, a positive differential function 𝜇(𝑡), and three constants 𝛼≥0, 𝛽>0, and 𝑇>0 such that 𝜇̇(𝑡)0≤𝜇(𝑡)𝜇()≤𝛼,min𝑡−𝛿,𝜇𝑡−𝜏₁,𝜇𝑡−𝜏₂𝜇(𝑡)[),≥𝛽,∀𝑡∈𝑇,∞ (28) and the following LMIs hold:Ω₁₁Ω₁₂𝑃₁𝐷𝛽²𝜏₁𝑃₅Ω₁₅000⋆Ω₂₂𝑄^∗₁𝐷𝑄₃−𝑃₁𝐷−𝑄^∗₁𝐴−𝑄^∗₁𝐵−𝑄^∗₁𝐶⋆⋆Ω₃₃𝐷𝑄₃−𝐷𝑃₁𝐷−𝑄^∗₂𝐴−𝑄^∗₂𝐵−𝑄^∗₂𝐶⋆⋆⋆Ω₄₄0−𝑄^∗₃𝐴−𝑄^∗₃𝐵−𝑄^∗₃𝐶𝛽⋆⋆⋆⋆−²𝛿𝑃₃000⋆⋆⋆⋆⋆Ω₆₆00⋆⋆⋆⋆⋆⋆−𝑅₂0𝛽⋆⋆⋆⋆⋆⋆⋆−²𝜏₂𝑃₆<0, (29)𝑃₁𝐼+𝐸_𝑘𝑃₁⋆𝑃₁>0, (30) where Ω₁₁=2𝛼𝑃₁−𝐷𝑃₁−𝑃₁𝐷+2𝛼𝑃₂+𝛿𝑃₃+𝑃₄−(𝛽²/𝜏₁)𝑃₅+𝑅₁Γ, Ω₁₂=𝑃₁+𝑃₂, Ω₁₅=𝐷𝑃₁𝐷 − 2𝛼𝑃₁𝐷, Ω₂₂=𝑄^∗₁+𝑄₁+𝜏₁𝑃₅, Ω₃₃=𝑄^∗₂𝐷+𝐷𝑄₂, Ω₄₄=−𝛽²𝑃₄−(𝛽²/𝜏₁)𝑃₅+𝑅₂Γ, and Ω₆₆=𝜏₂𝑃₆−𝑅₁.

Proof. Under the condition of Theorem 10, system (1) has a unique equilibrium point 𝑧. Then we shift the equilibrium point of (1) to the origin by the translation 𝑧(𝑡)=𝑧(𝑡)−𝑧 and obtain ̇𝑧(𝑡)=−𝐷𝑧(𝑡−𝛿)+𝐴𝑔(𝑧(𝑡))+𝐵𝑔𝑧𝑡−𝜏₁+𝐶^𝑡_𝑡−𝜏2𝑔(𝑧(𝑠))d𝑠,𝑡≠𝑡_𝑘Δ𝑧𝑡,𝑡>0,_𝑘=𝑧𝑡_𝑘−𝑧𝑡⁻_𝑘=𝐸_𝑘𝑧𝑡⁻_𝑘−𝐷^𝑡𝑘_{𝑡𝑘−𝛿}𝑧(𝑠)d𝑠,𝑘∈ℤ⁺. (31) Consider the following Lyapunov-Krasovskii functional candidate: 𝑉(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡)+𝑉₃(𝑡)+𝑉₄(𝑡), (32) where 𝑉₁(𝑡)=𝜇²(𝑡)𝑧(𝑡)−𝐷^𝑡_𝑡−𝛿𝑧(𝑠)d𝑠^∗×𝑃₁𝑧(𝑡)−𝐷^𝑡_𝑡−𝛿𝑧(𝑠),𝑉d𝑠₂(𝑡)=𝜇²(𝑡)𝑧^∗(𝑡)𝑃₂𝑧(𝑡)+^𝛿₀^𝑡_𝑡−𝑢𝜇²(𝑠)𝑧^∗(𝑠)𝑃₃𝑧(𝑠)𝑉d𝑠d𝑢,₃(𝑡)=^𝑡_𝑡−𝜏1𝜇²(𝑠)𝑧^∗(𝑠)𝑃₄𝑧(𝑠)d𝑠+^𝜏1₀^𝑡_𝑡−𝑢𝜇²(𝑠)̇𝑧^∗(𝑠)𝑃₅̇𝑧(𝑠)𝑉d𝑠d𝑢,₄(𝑡)=^𝜏2₀^𝑡_𝑡−𝑢𝜇²(𝑠)𝑔^∗(𝑧(𝑠))𝑃₆𝑔(𝑧(𝑠))d𝑠d𝑢. (33) Calculating the upper right derivative of 𝑉 along the solution of (31), applying Lemma 8, we get 𝐷⁺𝑉₁(𝑡)=2𝜇(𝑡)𝜇̇(𝑡)𝑧(𝑡)−𝐷^𝑡_𝑡−𝛿𝑧(𝑠)d𝑠^∗×𝑃₁𝑧(𝑡)−𝐷^𝑡_𝑡−𝛿𝑧(𝑠)d𝑠+𝜇²(𝑡̇)𝑧(𝑡)−𝐷𝑧(𝑡)+𝐷𝑧(𝑡−𝛿)^∗×𝑃₁𝑧(𝑡)−𝐷^𝑡_𝑡−𝛿𝑧(𝑠)d𝑠+𝜇²(𝑡)𝑧(𝑡)−𝐷^𝑡_𝑡−𝛿𝑧(𝑠)d𝑠^∗×𝑃₁̇𝑧(𝑡)−𝐷𝑧(𝑡)+𝐷𝑧(𝑡−𝛿)≤𝜇²(𝑡)2𝛼𝑧^∗(𝑡)𝑃₁𝑧(𝑡)−2𝛼𝑧^∗(𝑡)𝑃₁𝐷^𝑡_𝑡−𝛿𝑧(𝑠)d𝑠−2𝛼^𝑡_𝑡−𝛿𝑧^∗(𝑠)d𝑠𝐷𝑃₁𝑧(𝑡)d𝑠+2𝛼^𝑡_𝑡−𝛿𝑧^∗(𝑠)d𝑠𝐷𝑃₁𝐷^𝑡_𝑡−𝛿𝑧(𝑠)+̇𝑧d𝑠_∗(𝑡)𝑃₁𝑧(𝑡)−̇𝑧^∗(𝑡)𝑃₁𝐷^𝑡_𝑡−𝛿𝑧(𝑠)−𝑧d𝑠_∗(𝑡)𝐷𝑃₁𝑧(𝑡)+𝑧^∗(𝑡)𝐷𝑃₁𝐷^𝑡_𝑡−𝛿𝑧(𝑠)+𝑧d𝑠_∗(𝑡−𝛿)𝐷𝑃₁𝑧(𝑡)−𝑧^∗(𝑡−𝛿)𝐷𝑃₁𝐷^𝑡_𝑡−𝛿𝑧(𝑠)d𝑠+𝑧^∗(𝑡)𝑃₁̇𝑧(𝑡)−^𝑡_𝑡−𝛿𝑧^∗(𝑠)d𝑠𝐷𝑃₁̇𝑧(𝑡)−𝑧^∗(𝑡)𝑃₁𝐷𝑧(𝑡)+^𝑡_𝑡−𝛿𝑧^∗(𝑠)d𝑠𝐷𝑃₁𝐷𝑧(𝑡)+𝑧^∗(𝑡)𝑃₁𝐷𝑧()−𝑡−𝛿_{𝑡𝑡−𝛿}𝑧^∗(𝑠)d𝑠𝐷𝑃₁𝐷𝑧(𝑡−𝛿)=𝜇²(𝑡)𝑧^∗(𝑡)2𝛼𝑃₁−𝐷𝑃₁−𝑃₁𝐷𝑧(𝑡)+𝑧^∗(𝑡)𝐷𝑃₁𝐷−2𝛼𝑃₁𝐷^𝑡_𝑡−𝛿𝑧(𝑠)+d𝑠_{𝑡𝑡−𝛿}𝑧^∗(𝑠)d𝑠𝐷𝑃₁𝐷−2𝛼𝐷𝑃₁𝑧(𝑡)+2𝛼^𝑡_𝑡−𝛿𝑧^∗(𝑠)d𝑠𝐷𝑃₁𝐷^𝑡_𝑡−𝛿𝑧(𝑠)+̇𝑧d𝑠_∗(𝑡)𝑃₁𝑧(𝑡)+𝑧^∗(𝑡)𝑃₁̇𝑧(𝑡)−̇𝑧^∗(𝑡)𝑃₁𝐷^𝑡_𝑡−𝛿𝑧(𝑠)d𝑠−^𝑡_𝑡−𝛿𝑧^∗(𝑠)d𝑠𝐷𝑃₁̇𝑧(𝑡)+𝑧^∗(𝑡−𝛿)𝐷𝑃₁𝑧(𝑡)+𝑧^∗(𝑡)𝑃₁𝐷𝑧()−𝑧𝑡−𝛿_∗(𝑡−𝛿)𝐷𝑃₁𝐷^𝑡_𝑡−𝛿𝑧(𝑠)−d𝑠_{𝑡𝑡−𝛿}𝑧^∗(𝑠)d𝑠𝐷𝑃₁𝐷𝑧(𝑡−𝛿),𝐷⁺𝑉₂(𝑡)=2𝜇(𝑡)𝜇̇(𝑡)𝑧^∗(𝑡)𝑃₂𝑧(𝑡)+𝜇²(𝑡)̇𝑧^∗(𝑡)𝑃₂𝑧(𝑡)+𝜇²(𝑡)𝑧^∗(𝑡)𝑃₂̇𝑧(𝑡)+𝛿𝜇²(𝑡)𝑧^∗(𝑡)𝑃₃𝑧(𝑡)−^𝛿₀𝜇²(𝑡−𝑢)𝑧^∗(𝑡−𝑢)𝑃₃𝑧(𝑡−𝑢)d𝑢≤2𝛼𝜇²(𝑡)𝑧^∗(𝑡)𝑃₂𝑧(𝑡)+𝜇²(𝑡)̇𝑧^∗(𝑡)𝑃₂𝑧(𝑡)+𝜇²(𝑡)𝑧^∗(𝑡)𝑃₂̇𝑧(𝑡)+𝛿𝜇²(𝑡)𝑧^∗(𝑡)𝑃₃𝑧(𝑡)−^𝑡_𝑡−𝛿𝜇²(𝑠)𝑧^∗(𝑠)𝑃₃𝑧(𝑠)d𝑠≤2𝛼𝜇²(𝑡)𝑧^∗(𝑡)𝑃₂𝑧(𝑡)+𝜇²(𝑡)̇𝑧^∗(𝑡)𝑃₂𝑧(𝑡)+𝜇²(𝑡)𝑧^∗(𝑡)𝑃₂̇𝑧(𝑡)+𝛿𝜇²(𝑡)𝑧^∗(𝑡)𝑃₃𝑧(𝑡)−𝛽²𝜇²(𝑡)^𝑡_𝑡−𝛿𝑧^∗(𝑠)𝑃₃𝑧(𝑠)d𝑠≤𝜇²(𝑡)𝑧^∗(𝑡)2𝛼𝑃₂+𝛿𝑃₃𝑧(𝑡)+𝑧^∗(𝑡)𝑃₂̇𝑧(𝑡)+̇𝑧^∗(𝑡)𝑃₂𝑧(𝑡)−𝛽²𝛿^𝑡_𝑡−𝛿𝑧^∗(𝑠)d𝑠𝑃₃^𝑡_𝑡−𝛿𝑧(𝑠)d𝑠,𝐷⁺𝑉₃(𝑡)=𝜇²(𝑡)𝑧^∗(𝑡)𝑃₄𝑧(𝑡)−𝜇²𝑡−𝜏₁𝑧^∗𝑡−𝜏₁𝑃₄𝑧𝑡−𝜏₁+𝜏₁𝜇²(𝑡)̇𝑧^∗(𝑡)𝑃₅̇𝑧(𝑡)−^𝜏1₀𝜇²()̇𝑡−𝑢𝑧^∗(𝑡−𝑢)𝑃₅̇𝑧(𝑡−𝑢)d𝑢=𝜇²(𝑡)𝑧^∗(𝑡)𝑃₄𝑧(𝑡)−𝜇²𝑡−𝜏₁𝑧^∗𝑡−𝜏₁𝑃₄𝑧𝑡−𝜏₁+𝜏₁𝜇²(𝑡)̇𝑧^∗(𝑡)𝑃₅̇𝑧(𝑡)−^𝑡_𝑡−𝜏1𝜇²(𝑠)̇𝑧^∗(𝑠)𝑃₅̇𝑧(𝑠)d𝑠≤𝜇²(𝑡)𝑧^∗(𝑡)𝑃₄𝑧(𝑡)−𝛽²𝑧^∗𝑡−𝜏₁𝑃₄𝑧𝑡−𝜏₁+𝜏₁̇𝑧^∗(𝑡)𝑃₅̇𝑧(𝑡)−𝛽²𝜏₁^𝑡_𝑡−𝜏1̇𝑧^∗(𝑠)d𝑠𝑃₅^𝑡_𝑡−𝜏1̇𝑧(𝑠)d𝑠=𝜇²(𝑡)𝑧^∗(𝑡)𝑃₄𝑧(𝑡)−𝛽²𝑧^∗𝑡−𝜏₁𝑃₄𝑧𝑡−𝜏₁+𝜏₁̇𝑧^∗(𝑡)𝑃₅̇𝑧(𝑡)−𝛽²𝜏₁𝑧^∗(𝑡)−𝑧^∗𝑡−𝜏₁×𝑃₅𝑧(𝑡)−𝑧𝑡−𝜏₁,𝐷⁺𝑉₄(𝑡)=𝜏₂𝜇²(𝑡)𝑔^∗(𝑧(𝑡))𝑃₆𝑔(𝑧(𝑡−))_𝜏20𝜇²(𝑡−𝑢)𝑔^∗(𝑧(𝑡−𝑢))𝑃₆𝑔(𝑧(𝑡−𝑢))d𝑢=𝜏₂𝜇²(𝑡)𝑔^∗(𝑧(𝑡))𝑃₆𝑔(𝑧(𝑡−))_{𝑡𝑡−𝜏2}𝜇²(𝑠)𝑔^∗(𝑧(𝑠))𝑃₆𝑔(𝑧(𝑠))d𝑠≤𝜏₂𝜇²(𝑡)𝑔^∗(𝑧(𝑡))𝑃₆𝑔(𝑧(𝑡))−𝛽²𝜇²(𝑡)^𝑡_𝑡−𝜏2𝑔^∗(𝑧(𝑠))𝑃₆𝑔(𝑧(𝑠))d𝑠≤𝜇²(𝑡)𝜏₂𝑔^∗(𝑧(𝑡))𝑃₆𝑔(𝑧(𝑡−𝛽))₂𝜏₂^𝑡_𝑡−𝜏2𝑔^∗(𝑧(𝑠))d𝑠𝑃₆^𝑡_𝑡−𝜏2𝑔(𝑧(𝑠))d𝑠. (34)(35)(36)(37) Combining (34), (35), (36), and (37), we can deduce that 𝐷⁺𝑉(𝑡)≤𝜇²(𝑡)𝑧^∗(𝑡)2𝛼𝑃₁−𝐷𝑃₁−𝑃₁𝐷+2𝛼𝑃₂+𝛿𝑃₃+𝑃₄−𝛽²𝜏₁𝑃₅𝑧(𝑡)+𝑧^∗(𝑡)𝑃₁+𝑃₂̇𝑧(𝑡)+̇𝑧^∗(𝑡)𝑃₁+𝑃₂𝑧(𝑡)+𝑧^∗(𝑡)𝑃₁𝐷𝑧()+𝑧𝑡−𝛿_∗(𝑡−𝛿)𝐷𝑃₁𝑧(𝑡)+𝛽²𝜏₁𝑧^∗(𝑡)𝑃₅𝑧𝑡−𝜏₁+𝛽²𝜏₁𝑧^∗𝑡−𝜏₁𝑃₅𝑧(𝑡)+𝑧^∗(𝑡)𝐷𝑃₁𝐷−2𝛼𝑃₁𝐷^𝑡_𝑡−𝛿𝑧(𝑠)+d𝑠_{𝑡𝑡−𝛿}𝑧^∗(𝑠)d𝑠𝐷𝑃₁𝐷−2𝛼𝐷𝑃₁𝑧(𝑡)−̇𝑧^∗(𝑡)𝑃₁𝐷^𝑡_𝑡−𝛿𝑧(𝑠)−d𝑠_{𝑡𝑡−𝛿}𝑧^∗(𝑠)d𝑠𝐷𝑃₁̇𝑧(𝑡)−𝑧^∗(𝑡−𝛿)𝐷𝑃₁𝐷^𝑡_𝑡−𝛿𝑧(𝑠)−d𝑠_{𝑡𝑡−𝛿}𝑧^∗(𝑠)d𝑠𝐷𝑃₁𝐷𝑧()−𝛽𝑡−𝛿₂𝛿^𝑡_𝑡−𝛿𝑧^∗(𝑠)d𝑠𝑃₃^𝑡_𝑡−𝛿𝑧(𝑠)−𝑧d𝑠_∗𝑡−𝜏₁𝛽²𝑃₄+𝛽²𝜏₁𝑃₅𝑧𝑡−𝜏₁+𝜏₂𝑔^∗(𝑧(𝑡))𝑃₆𝑔(𝑧(𝑡−𝛽))₂𝜏₂^𝑡_𝑡−𝜏2𝑔^∗(𝑧(𝑠))d𝑠𝑃₆^𝑡_𝑡−𝜏2𝑔(𝑧(𝑠))d𝑠. (38) From assumption (H1), we have 𝑔_𝑗𝑧_𝑗(𝑡)≤𝛾_𝑗𝑧_𝑗(𝑡) (39) for 𝑗=1,2,…,𝑛. Let 𝑅₁=diag(𝑟₁,𝑟₂,…,𝑟_𝑛)>0. It follows from (39) that 𝑟_𝑗𝜇²(𝑡)𝑔^∗_𝑗𝑧_𝑗(𝑡)𝑔_𝑗𝑧_𝑗(𝑡)−𝑟_𝑗𝜇²(𝑡)𝛾²_𝑗𝑧^∗_𝑗(𝑡)𝑧_𝑗(𝑡)≤0 (40) for 𝑗=1,2,…,𝑛. Hence 𝜇²(𝑡)𝑔^∗(𝑧(𝑡))𝑅₁𝑔(𝑧(𝑡))−𝜇²(𝑡)𝑧^∗(𝑡)𝑅₁Γ𝑧(𝑡)≤0. (41) Also, we can get that 𝜇²(𝑡)𝑔^∗𝑧𝑡−𝜏₁𝑅₂𝑔𝑧𝑡−𝜏₁−𝜇²(𝑡)𝑧^∗𝑡−𝜏₁𝑅₂Γ𝑧𝑡−𝜏₁≤0. (42) From (31), we have that 0=𝜇²(𝑡)𝑄₁̇𝑧(𝑡)+𝑄₂𝑧(𝑡−𝛿)+𝑄₃𝑧𝑡−𝜏₁^∗×̇𝑧(𝑡)+𝐷𝑧(𝑡−𝛿)−𝐴𝑔(𝑧(𝑡))−𝐵𝑔𝑧𝑡−𝜏₁−𝐶^𝑡_𝑡−𝜏2𝑔(𝑧(𝑠))d𝑠+𝜇²(𝑡̇)𝑧(𝑡)+𝐷𝑧(𝑡−𝛿)−𝐴𝑔(𝑧(𝑡))−𝐵𝑔(𝑧(𝑡−𝜏₁))−𝐶^𝑡_𝑡−𝜏2𝑔(𝑧(𝑠))d𝑠^∗×𝑄₁̇𝑧(𝑡)+𝑄₂𝑧(𝑡−𝛿)+𝑄₃𝑧𝑡−𝜏₁. (43) It follows from (38), (41), (42), and (43) that 𝐷⁺𝑉(𝑡)≤𝜇²(𝑡)𝑤^∗(𝑡)Ω𝑤(𝑡), (44) where 𝑤(𝑡)=(𝑧^∗̇(𝑡),𝑧^∗(𝑡),𝑧^∗(𝑡−𝛿),𝑧(𝑡−𝜏₁),∫^𝑡_𝑡−𝛿𝑧^∗(𝑠)d𝑠,𝑔^∗(𝑧(𝑡)),𝑔^∗(𝑧(𝑡−𝜏₁)),∫^𝑡_𝑡−𝜏2𝑔^∗(𝑧(𝑠))d𝑠)^∗ andΩΩ=¹¹Ω₁₂𝑃₁𝐷𝛽²𝜏₁𝑃₅Ω₁₅000⋆Ω₂₂𝑄^∗₁𝐷𝑄₃−𝑃₁𝐷−𝑄^∗₁𝐴−𝑄^∗₁𝐵−𝑄^∗₁𝐶⋆⋆Ω₃₃𝐷𝑄₃−𝐷𝑃₁𝐷−𝑄^∗₂𝐴−𝑄^∗₂𝐵−𝑄^∗₂𝐶⋆⋆⋆Ω₄₄0−𝑄^∗₃𝐴−𝑄^∗₃𝐵−𝑄^∗₃𝐶𝛽⋆⋆⋆⋆−²𝛿𝑃₃000⋆⋆⋆⋆⋆Ω₆₆00⋆⋆⋆⋆⋆⋆−𝑅₂0𝛽⋆⋆⋆⋆⋆⋆⋆−²𝜏₂𝑃₆ (45)with Ω₁₁=2𝛼P₁−𝐷𝑃₁−𝑃₁𝐷+2𝛼𝑃₂+𝛿𝑃₃+𝑃₄−(𝛽²/𝜏₁)𝑃₅+𝑅₁Γ, Ω₁₂=𝑃₁+𝑃₂, Ω₁₅=𝐷𝑃₁𝐷−2𝛼𝑃₁𝐷, Ω₂₂=𝑄^∗₁+𝑄₁+𝜏₁𝑃₅, Ω₃₃=𝑄^∗₂𝐷+𝐷𝑄₂, Ω₄₄=−𝛽²𝑃₄−(𝛽²/𝜏₁)𝑃₅+𝑅₂Γ, Ω₆₆=𝜏₂𝑃₆−𝑅₁. Thus, we get from (29) and (44) that 𝐷⁺𝑉(𝑡)≤0,𝑡∈𝑡_𝑘−1,𝑡_𝑘∩[𝑇,∞),𝑘∈ℤ⁺. (46) In addition, we note that 𝑃₁𝐼+𝐸_𝑘𝑃₁⋆𝑃₁⟺>0𝐼00𝑃₁⁻¹𝑃₁𝐼+𝐸_𝑘𝑃₁⋆𝑃₁𝐼00𝑃₁⁻¹⟺𝑃>0₁𝐼+𝐸_𝑘⋆𝑃₁⁻¹>0⟺𝑃₁−𝐼+𝐸_𝑘^∗𝑃₁𝐼+𝐸_𝑘>0 (47) in which the last equivalent relation is obtained by Lemma 9. Thus, it yields 𝑉₁𝑡_𝑘=𝜇²𝑡_𝑘𝑧𝑡_𝑘−𝐷^𝑡𝑘_{𝑡𝑘−𝛿}𝑧(𝑠)d𝑠^∗×𝑃₁𝑧𝑡_𝑘−𝐷^𝑡𝑘_{𝑡𝑘−𝛿}𝑧(𝑠)d𝑠=𝜇²𝑡⁻_𝑘𝑧𝑡⁻_𝑘−𝐷^𝑡𝑘_{𝑡𝑘−𝛿}𝑧(𝑠)d𝑠^∗×𝐼+𝐸_𝑘^∗𝑃₁𝐼+𝐸_𝑘×𝑧𝑡_𝑘−𝐷^𝑡𝑘_{𝑡𝑘−𝛿}𝑧(𝑠)d𝑠≤𝜇²𝑡⁻_𝑘𝑧𝑡⁻_𝑘−𝐷^𝑡𝑘_{𝑡𝑘−𝛿}𝑧(𝑠)d𝑠^∗×𝑃₁𝑧𝑡_𝑘−𝐷^𝑡𝑘_{𝑡𝑘−𝛿}𝑧(𝑠)d𝑠=𝑉₁𝑡⁻_𝑘. (48) Hence, we can deduce that 𝑉𝑡_𝑘≤𝑉𝑡⁻_𝑘,𝑘∈ℤ⁺. (49) From (46) and (49), we know that 𝑉 is monotonically nonincreasing for 𝑡∈[𝑇,∞), which implies that 𝑉(𝑡)≤𝑉(𝑇),𝑡≥𝑇. (50) From the definition of 𝑉(𝑡) in (32), we obtain that 𝜇²(𝑡)𝜆_min𝑃₂‖𝑧(𝑡)‖²≤𝑉(𝑡)≤𝑉₀<∞,𝑡≥0, (51) where 𝑉₀=max_{0≤𝑠≤𝑇}𝑉(𝑠). It implies that ‖𝑧(𝑡≤𝑀)‖𝜇(𝑡),𝑡≥0, (52) where 𝑀=𝑉₀/𝜆_min(𝑃₂). The proof is completed.

Corollary 14. Under the conditions of Theorem 10, the equilibrium point of system (1) is globally exponentially stable, if there exist six positive definite Hermitian matrices 𝑃₁, 𝑃₂, 𝑃₃, 𝑃₄, 𝑃₅, and 𝑃₆, two real positive diagonal matrices 𝑅₁ and 𝑅₂, three complex matrices 𝑄₁, 𝑄₂, and 𝑄₃, and a positive constant 𝜀 such that conditions (29) and (30) in Theorem 13 are satisfied, where 𝛼=𝜀, 𝛽=𝑒^{−𝜀max{𝛿,𝜏1,𝜏2}}.

Proof. Let 𝜇(𝑡)=𝑒^𝜀𝑡 (𝑡≥0); then 𝜇̇(𝑡)/𝜇(𝑡)=𝜀, min{𝜇(𝑡−𝛿),𝜇(𝑡−𝜏₁),𝜇(𝑡−𝜏₂)}/𝜇(𝑡) = min{𝑒^−𝜀𝛿,𝑒^−𝜀𝜏1,𝑒^−𝜀𝜏2}=𝑒^{−𝜀max{𝛿,𝜏1,𝜏2}}. Take 𝛼=𝜀, 𝛽=𝑒^{−𝜀max{𝛿,𝜏1,𝜏2}}, and 𝑇=0. Then it is obvious that condition (28) in Theorem 13 is satisfied. The proof is completed.

Corollary 15. Under the conditions of Theorem 10, the equilibrium point of system (1) is globally power-stable, if there exist six positive definite Hermitian matrices 𝑃₁, 𝑃₂, 𝑃₃, 𝑃₄, 𝑃₅, and 𝑃₆, two real positive diagonal matrices 𝑅₁ and 𝑅₂, three complex matrices 𝑄₁, 𝑄₂, and 𝑄₃, and a positive constant 𝜀 such that conditions (29) and (30) in Theorem 13 are satisfied, where 𝛼=𝜀/2, 𝛽=2^−𝜀.

Proof. Let 𝜇(𝑡)=𝑡^𝜀. For all 𝑡≥2max{1,𝛿,𝜏₁,𝜏₂}, we have 𝜇̇(𝑡)/𝜇(𝑡)=𝜀/𝑡≤𝜀/2, min{𝜇(𝑡−𝛿),𝜇(𝑡−𝜏₁),𝜇(𝑡−𝜏₂)}/𝜇(𝑡) = (1−(max{𝛿,𝜏₁,𝜏₂}/𝑡))^𝜀≥2^−𝜀. Take 𝛼=𝜀/2, 𝛽=2^−𝜀, and 𝑇=2max{1,𝛿,𝜏₁,𝜏₂}. Then it is obvious that condition (28) in Theorem 13 is satisfied. The proof is completed.

Corollary 16. Under the conditions of Theorem 10, the equilibrium point of system (1) is globally log-stable, if there exist six positive definite Hermitian matrices 𝑃₁, 𝑃₂, 𝑃₃,𝑃₄, 𝑃₅, and 𝑃₆, two real positive diagonal matrices 𝑅₁ and 𝑅₂, three complex matrices 𝑄₁, 𝑄₂, and 𝑄₃, and a positive constant 𝜀 such that conditions (29) and (30) in Theorem 13 are satisfied, where 𝛼=𝜀/𝑒, 𝛽=1/ln(𝑒+𝜀max{𝛿,𝜏₁,𝜏₂}).

Proof. Let 𝜇(𝑡)=ln(𝜀𝑡+1). For all 𝑡≥((𝑒−1)/𝜀)+max{𝛿,𝜏₁,𝜏₂}, we have 𝜇̇(𝑡)/𝜇(𝑡) = 𝜀/(𝜀𝑡+1)ln(𝜀𝑡+1)≤𝜀/𝑒, min{𝜇(𝑡−𝛿),𝜇(𝑡−𝜏₁),𝜇(𝑡−𝜏₂)}/𝜇(𝑡) = (𝑙𝑛[𝜀(𝑡−𝑚𝑎𝑥{𝛿,𝜏_1,𝜏_2})+1])0𝑥02044(𝑙𝑛(𝜀𝑡+1)) ≥ 1/ln(𝑒+𝜀max{𝛿,𝜏₁,𝜏₂}). Take 𝛼=𝜀/𝑒, 𝛽=1/ln(𝑒+𝜀max{𝛿,𝜏₁,𝜏₂}), and 𝑇=((𝑒−1)/𝜀)+max{𝛿,𝜏₁,𝜏₂}. Then it is obvious that condition (28) in Theorem 13 is satisfied. The proof is completed.

Remark 17. It is noted that LMIs (15), (29), and (30) are complex-valued, which cannot be directly handled via MATLAB LMI Toolbox. However, the authors in [33] give the result that a complex Hermitian matrix 𝐿 satisfies 𝐿<0 if and only if (𝐿)(𝐿)(𝐿)(𝐿)ReIm−ImRe<0. (53) Therefore, applying the result, the complex-valued LMIs (15), (29), and (30) can be turned into real-valued LMIs, which can be checked numerically using LMI toolbox in MATLAB.

5. Numerical Examples

The following two illustrative examples will demonstrate the effectiveness and superiority of our results.

Example 1. Consider the following two-neuron CVNNs with leakage, discrete, and distributed delays, described by 𝑧̇(𝑡)=−𝐷𝑧(𝑡−𝛿)+𝐴𝑓(𝑧(𝑡))+𝐵𝑓𝑧𝑡−𝜏₁+𝐶^𝑡_𝑡−𝜏2𝑓(𝑧(𝑠))d𝑠+ℎ,𝑡≠𝑡_𝑘𝑡,𝑡>0,Δ𝑧_𝑘=𝑧𝑡_𝑘−𝑧𝑡⁻_𝑘=𝐸_𝑘𝑧𝑡⁻_𝑘−𝐷^𝑡𝑘_{𝑡𝑘−𝛿}𝑧(𝑠)d𝑠,𝑘∈ℤ⁺, (54) where 𝑓₁(𝑢)=𝑓₂(𝑢)=(1/20)(|𝑢+1|−|𝑢−1|), 𝛿=0.1, 𝜏₁=0.2, 𝜏₂=0.2, ℎ=(0,0)^𝑇, 𝑡_𝑘=0.4𝑘, 𝑘∈ℤ⁺, and the parameter matrices 𝐷, 𝐴, 𝐵, 𝐶, and 𝐸_𝑘 are given as follows: ,,,𝐸𝐷=3003,𝐴=0.5+𝑖−0.2−0.3𝑖0.2+0.5𝑖1−0.5𝑖𝐵=0.4+0.3𝑖−0.1−𝑖−1.2+𝑖−0.4−0.3𝑖𝐶=0.1−0.5𝑖0.6+0.4𝑖0.2−0.5𝑖0.2_𝑘=−0.3+0.2𝑖00.1−0.6𝑖−0.2+0.2𝑖,𝑘∈ℤ⁺. (55) It can be verified that the activation functions 𝑓₁ and 𝑓₂ satisfy condition (H1), and Γ=diag{0.01,0.01}. Then LMI in Theorem 10 has the following feasible solution via the MATLAB LMI toolbox: 𝑈₁=,𝑈19.0818+6.7535𝑖−9.2220+2.2044𝑖8.0728+0.4398𝑖19.0186+9.3571𝑖₂=,.4.7472−0.2821𝑖−6.9954−6.3515𝑖1.8858+0.8953𝑖4.5109+1.3211𝑖𝑅=161.011700129.7777 (56) Also there exists a constant 𝜀=0.1, and by employing MATLAB LMI Toolbox, we can find the solutions to the LMIs in Corollary 14 as follows: 𝑃₁=,𝑃12.4935−0.6208−3.5072𝑖−0.6208+3.5072𝑖38.9645₂=,𝑃16.5671−1.5742−1.2335𝑖−1.5742+1.2335𝑖27.6891₃=,𝑃43.4484−1.3614−5.3324𝑖−1.3614+5.3324𝑖87.1860₄=,𝑃12.8707−0.3721−2.5769𝑖−0.3721+2.5769𝑖30.8832₅=,𝑃18.0542−0.3996+0.2524𝑖−0.3996−0.2524𝑖16.1543₆=,𝑄49.8004−0.2065+3.8010𝑖−0.2065−3.8010𝑖48.9421₁=,𝑄−33.7168+0.0480𝑖3.0570−2.0568𝑖2.2459+1.1090𝑖−39.3508+0.0752𝑖₂=,𝑄−66.6664−1.5581𝑖6.8518−15.0791𝑖1.2207+10.1224𝑖−73.9430+1.7006𝑖₃=,𝑅−4.2664+0.1047𝑖0.4653−0.9119𝑖0.3881+1.2719𝑖0.9145−0.0078𝑖₁=,𝑅260.795600276.7577₂=.293.913300256.1810 (57) Hence, from Theorem 10 and Corollary 14, the unique equilibrium (0,0)^𝑇 of system (54) is globally exponentially stable. Figure 1 depicts the real and imaginary parts of states of the considered system (54), where the initial condition is 𝑧₁(𝑡)=0.5−0.4𝑖,𝑧₂(𝑡)=−0.4+0.5𝑖.

(a)

(b)

Figure 1 

State trajectories of system (54).

Example 2. Consider the following two-neuron CVNNs with leakage, discrete, and distributed delays, described by 𝑧̇(𝑡)=−𝐷𝑧(𝑡−𝛿)+𝐴𝑓(𝑧(𝑡))+𝐵𝑓𝑧𝑡−𝜏₁+𝐶^𝑡_𝑡−𝜏2𝑓(𝑧(𝑠))d𝑠+ℎ,𝑡≠𝑡_𝑘𝑡,𝑡>0,Δ𝑧_𝑘=𝑧𝑡_𝑘−𝑧𝑡⁻_𝑘=𝐸_𝑘𝑧𝑡⁻_𝑘−𝐷^𝑡𝑘_{𝑡𝑘−𝛿}𝑧(𝑠)d𝑠−𝛾,𝑘∈ℤ⁺, (58) where 𝑓₁(𝑢)=𝑓₂(𝑢)=(1/4)[max{0,Re(𝑢)}+𝑖max{0,Im(𝑢)}], 𝛿=0.2, 𝜏₁=0.6, 𝜏₂=1, ℎ=(1+𝑖,−1+𝑖)^𝑇, 𝛾=(0.0295+0.0623𝑖,−0.0604+0.0397𝑖)^𝑇, 𝑡_𝑘=0.8𝑘, 𝑘∈ℤ⁺, and the parameter matrices 𝐷, 𝐴, 𝐵, 𝐶, and 𝐸_𝑘 are given as follows: ,,𝐸𝐷=4004,𝐴=−1+3𝑖−1+2𝑖−2−2𝑖1+0.5𝑖𝐵=0.8+𝑖𝑖−1+0.5𝑖1+0.5𝑖,𝐶=1+𝑖1−2𝑖−1+𝑖0.5+𝑖_𝑘=−10.5𝑖0.5𝑖−0.5,𝑘∈ℤ⁺. (59) It can be verified that the activation functions 𝑓₁ and 𝑓₂ satisfy condition (H1), and Γ=diag{1/16,1/16}. Then LMI in Theorem 10 has the following feasible solution via the MATLAB LMI toolbox: 𝑈₁=,𝑈2.9099−5.3413𝑖2.7078+7.2154𝑖−2.5180+7.0990𝑖2.6469+0.8628𝑖₂=,.5.4524+4.8583𝑖−11.7495+2.6580𝑖−0.5273−7.2479𝑖−0.8154−1.1665𝑖𝑅=79.40220037.6964 (60) Also there exists a constant 𝜀=0.1, and by employing MATLAB LMI Toolbox, we can find the solutions to LMIs in Corollary 15 as follows: 𝑃₁=,𝑃1.65370.0741+0.0987𝑖0.0741−0.0987𝑖2.5236₂=,𝑃0.19840.0640−0.0118𝑖0.0640+0.0118𝑖0.2985₃=,𝑃21.56590.7498+1.4712𝑖0.7498−1.4712𝑖32.7091₄=,𝑃0.68520.0304+0.0669𝑖0.0304−0.0669𝑖1.1632₅=,𝑃0.24190.0941−0.1415𝑖0.0941+0.1415𝑖0.5347₆=,𝑄7.7723−0.2071−0.7161𝑖−0.2071+0.7161𝑖10.5317₁=,𝑄−0.7898+0.0341𝑖−0.1269−0.5099𝑖−0.1170+0.5513𝑖−1.2707−0.0430𝑖₂=,𝑄−2.5965+0.0654𝑖−0.5192−2.1874𝑖−0.5396+2.4027𝑖−4.2140−0.0583𝑖₃=,𝑅−0.0376+0.0030𝑖−0.0029−0.0779𝑖−0.0046+0.0854𝑖−0.0596−0.0047𝑖₁=,𝑅19.98800032.1768₂=.6.82680014.7991 (61) Hence, from Theorem 10 and Corollary 15, system (58) has a unique equilibrium, which is globally exponentially power-stable. Figure 2 depicts the real and imaginary parts of states of the considered system (58), where the initial condition is 𝑧₁(𝑡)=0.5−0.2𝑖,𝑧₂(𝑡)=−0.5+0.6𝑖.

(a)

(b)

Figure 2 

State trajectories of system (58) with 𝛿=0.2.

Remark 18. If we take leakage delay 𝛿≥0.37 in system (58), one may check that LMIs in Corollary 15 do not have a feasible solution via MATLAB. In other words, our results cannot guarantee the stability of system (58) with 𝛿≥0.37. From the simulations, it is easy to check that the unique equilibrium point of system (58) with 𝛿=0.37 is not stable (see Figure 3); this implies that the effect of leakage delay on the dynamics of CVNNs cannot be ignored.

(a)

(b)

Figure 3 

State trajectories of system (58) with 𝛿=0.37.

6. Conclusion

In this paper, the 𝜇-stability of impulsive CVNNs with leakage delay, discrete delay, and distributed delay has been investigated. Several sufficient conditions to ensure the existence, uniqueness, and global 𝜇-stability of the equilibrium point of the considered neural networks have been established in LMIs by applying the homeomorphism mapping principle of complex domain, constructing appropriate Lyapunov-Krasovskii functionals, and employing the free weighting matrix method. As direct applications of these results, several criteria on the exponential stability, power-stability and log-stability have been obtained. Two numerical examples are given to illustrate the effectiveness of the proposed theoretical results.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants 61273021 and 11172247 and in part by the Natural Science Foundation Project of CQ cstc2013jjB40008.