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

and Applied Analysis 3 Definition 2. Let ?̂? 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 constantM such that ‖z (t) − ?̂?‖ ≤ M μ (t) , t ≥ 0, (3) then the equilibrium point ?̂? 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 ?̂? 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 andM such that ‖z (t) − ?̂?‖ ≤ M e , t ≥ 0, (4) then the equilibrium point ?̂? is said to be exponentially stable. Definition 4. Let ?̂? 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 andM such that ‖z (t) − ?̂?‖ ≤ M t , t ≥ 0, (5) then the equilibrium point ?̂? is said to be power-stable. Definition 5. Let ?̂? 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 andM such that ‖z (t) − ?̂?‖ ≤ M ln (εt + 1) , t > 0, (6) then the equilibrium point ?̂? is said to be log-stable. Lemma 6. If H(z) : Cn → Cn is a continuous map and satisfies the following conditions: (i) H(z) is injective on C, (ii) lim ‖z‖→∞ ‖H(z)‖ = ∞, thenH(z) is a homeomorphism of Cn onto itself. Proof. Let z = x + iy and α = (xT, y)T, where x, y ∈ R. Define a homeomorphismI : Cn → R2n by I (z) = α. (7) Obviously, I is invertible. Let L = I ∘ H ∘ I. Then L is injective on R, 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 R2n onto itself. Then H = I−1 ∘L ∘I is a homeomorphism of Cn onto itself. The proof iscompleted. Lemma 7. For any a, b ∈ C, if P ∈ Cn×n is a positive definite Hermitian matrix, then a∗b + b∗a ≤ a∗Pa + bPb. Proof. Since P is a positive definite Hermitian matrix, there exists an invertible matrix Q ∈ C, such that P = QQ. For any a, b ∈ C, it follows from Cauchy inequality that


Introduction
The nonlinear systems are ubiquitous in the real world [1][2][3][4][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][7][8][9][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 timevarying 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][17][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 -stability to unify the exponential stability, power-rate stability, and log-stability of neural networks.In [22], the authors investigated the global robust -stability in the mean square for a class of stochastic neural networks.In [23], the delayed neural systems with impulsion were considered, and the -stability criteria were derived by using Lyapunov-Krasovskii functional method.In [24], the multiple -stability of delayed neural networks was investigated, and several criteria for the coexistence of equilibrium points and their local -stability were derived.
As an extension of real-valued neural networks, complexvalued 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][29][30][31][32][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 discretetime CVNNs under the same framework.In [34], the authors investigated the -stability of delayed CVNNs and obtained several sufficient conditions to ensure the global -stability.To the best of our knowledge, there are no results on 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 -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 -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  denote the imaginary unit; that is,  = √ −1.C  , R × , and C × denote, respectively, the set of -dimensional complex vectors,  ×  real matrices and complex matrices.The subscripts  and * denote matrix transposition and matrix conjugate transpose, respectively.For complex vector  ∈ C  , let || = (| 1 |, | 2 |, . . ., |  |)  be the module of the vector , and let ‖‖ = √∑  =1 |  | 2 be the norm of the vector . denotes the identity matrix with appropriate dimensions.The notation  ≥  (resp.,  > ) means that  −  is positive semidefinite (resp., positive definite). max () and  min () are defined as the largest and the smallest eigenvalue of positive definite matrix , respectively.For any  ≥ 0,   is defined by   = (+),   − = ( − +),  ∈ [−, 0].In addition, the notation ⋆ always denotes the conjugate transpose of block in a Hermitian matrix.

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 where the impulse times 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   are continuous and satisfy for any  1 ,  2 ∈ C,  = 1, 2, . . ., , where   is a constant.Moreover, we define Γ = diag{ 2 1 ,  2 2 , . . .,  2  }.
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   are not required in this paper.
Next we introduce some definitions and lemmas to be used in the stability analysis.Definition 2. Let ẑ be an equilibrium point of system (1).Suppose that () is an arbitrary solution of system (1); () is a positive continuous function and satisfies () → ∞ as  → ∞.If there is a positive constant  such that then the equilibrium point ẑ is said to be -stable.
In particular, if taking () 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 ẑ be an equilibrium point of system (1).Suppose that () is an arbitrary solution of system (1).If there are two positive constants  and  such that then the equilibrium point ẑ is said to be exponentially stable.
Definition 4. Let ẑ be an equilibrium point of system (1).Suppose that () is an arbitrary solution of system (1).If there are two positive constants  and  such that then the equilibrium point ẑ is said to be power-stable.
Definition 5. Let ẑ be an equilibrium point of system (1).Suppose that () is an arbitrary solution of system (1).If there are two positive constants  and  such that then the equilibrium point ẑ is said to be log-stable.Lemma 6.If () : C  → C  is a continuous map and satisfies the following conditions: Proof.Let  =  +  and  = (  ,   )  , where ,  ∈ R  .Define a homeomorphism I : Obviously, Then L is injective on R 2 , since  and I are injective.
In addition, Proof.Since  is a positive definite Hermitian matrix, there exists an invertible matrix  ∈ C × , such that  =  * .For any ,  ∈ C  , it follows from Cauchy inequality that The proof is completed.

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 ẑ ∈ C  , where ẑ satisfies In this paper, it is assumed that the constant complex vector Hence, to prove the existence and uniqueness of a solution of ( 13), it suffices to show that the following map H : C  → C  has a unique zero point: In the following, we will give some conditions for checking that H is a homeomorphism on C  , 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  1 ,  2 and a real positive diagonal matrix , such that the following LMI holds: where Proof.In the following, we will prove that H() is a homeomorphism of C  onto itself.First, we prove that H() is an injective map on C  .Suppose that there exist ,   ∈ C  with  ̸ =   , such that
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.

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 -stability of the unique equilibrium point.For this purpose, the impulsive function   which is viewed as a perturbation of the equilibrium point ẑ of system (1) without impulses is defined by where  ∈ Z + ,   ∈ C × .It is obvious that   (ẑ, ẑ) = 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   but also related to the state of neurons in recent history, which reflects more realistic dynamics [19].
Remark 17.It is noted that LMIs (15), (29), and (30) are complex-valued, which cannot be directly handled via MAT-LAB LMI Toolbox.However, the authors in [33] give the result that a complex Hermitian matrix  satisfies  < 0 if and only if 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.

Numerical Examples
The following two illustrative examples will demonstrate the effectiveness and superiority of our results.
Remark 18.If we take leakage delay  ≥ 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  ≥ 0.37.From the simulations, it is easy to check that the unique equilibrium point of system (58) with  = 0.37 is not stable (see Figure 3); this implies that the effect of leakage delay on the dynamics of CVNNs cannot be ignored.

Conclusion
In this paper, the -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 -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 logstability have been obtained.Two numerical examples are given to illustrate the effectiveness of the proposed theoretical results.