Robust Dissipativity Analysis of Hopfield-Type Complex-Valued Neural Networks with Time-Varying Delays and Linear Fractional Uncertainties

We study the robust dissipativity issue with respect to the Hopfield-type of complex-valued neural network (HTCVNN) models incorporated with time-varying delays and linear fractional uncertainties. To avoid the computational issues in the complex domain, we divide the original complex-valued system into two real-valued systems. We devise an appropriate Lyapunov-Krasovskii functional (LKF) equipped with general integral terms to facilitate the analysis. By exploiting the multiple integral inequality method, the sufficient conditions for the dissipativity of HTCVNN models are obtained via the linear matrix inequalities (LMIs). The MATLAB software package is used to solve the LMIs effectively. We devise a number of numerical models and their empirical results positively ascertain the obtained results.


Introduction
Nowadays, many investigations related to the dynamical properties with respect to a variety of complex-valued neural network (CVNN) models have been published in the literature. In the engineering science domain, the applications of CVNN models have been reported by many researchers, e.g., for sonic wave, electromagnetic wave, light wave, quantum devices, image processing as well as signal processing. In regard to both the mathematical analysis and practical application, CVNN models have been widely studied, and many effective methods on various dynamical analysis of CVNN models are available [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. Mainly, the Hopfield-type of neural network (HTNN) models has been considered a key development owing to their adaptive mathematical model capability, along with many powerful methods concerning the stability of HTNN models [1,13,[16][17][18].
Time delays naturally occur in almost every dynamical system, which could cause the unstable behaviors of the resulting system [19][20][21][22][23][24][25]. Because of these characteristics, the stability of delayed NN
(A 3 ): The time-varying delay function is r(t), which satisfies 0 ≤ r(t) ≤ r,ṙ(t) ≤ µ, where µ and r are known constants. It should be remembered that the uncertainties associated with the weight coefficients of neurons are unavoidable in network models. Therefore, the parameter uncertainties cannot be overlooked when analyzing the stability of NN models. Therefore, the UHTCVNN model can be described by (5) are the parameter uncertainties, and they satisfy: where the known constant real matrices are G, H 1 , H 2 , H 3 and J ; the time-varying uncertain matrix is F (t), which satisfies (8) and (9), it is confirmed that (I − J F (t)) is invertible. Given J = 0, the following norm-bounded parametric uncertainty form ∆ T (t)∆(t) = F T (t)F (t) ≤ I can be obtained from the linear fractional uncertainty of the form (6).

Remark 1. From the inequalities
For a comprehensive analysis, let The real and imaginary parts of the HTCVNN model in (5) are From (10), an equivalent form of the model is which is equivalent to Then, we can express the model in (12) in an equivalent form of From (6)-(8), the parameter uncertainties D , Ã satisfy: According to (A 2 ), it is straightforward to obtain where L = diag{l 1 , ..., l n }. The real and imaginary parts of Equation (17) are Given the model in (13), the initial condition is

Remark 2.
It should be noted that, if we let D = Ã = 0, the NN model in (13) becomes The real-valued NN model in (20) is an equivalent form of the original model in (2). In addition, the model in (5) is an equivalent form of the real-valued NN model in (13).
A number of key lemmas and definitions used to derive the main results are explained. Definition 1 ([31]). Given the existence of a compact set S ⊆ C n , whereby ∀p 0 ∈ C n , ∃T > 0, when t ≥ t 0 + T, p(t, t 0 , p 0 ) ⊆ S in which the solution of (5) from the initial state and time of p 0 is denoted by p(t, t 0 , p 0 ), the CVNN model in (5) is said to be globally dissipative. In this case, S is called a globally attractive set. A set S is called positive invariant if ∀p 0 ∈ S implies p(t, t 0 , p 0 ) ⊆ S for t ≥ t 0 .
Similar to the publications in [31,34,35,37], the energy supply function for the NN model in (5) is defined as where Q ≤ 0, and Q, S, R ∈ C n×n . In addition, Definition 2 ( [31,37]). Subject to zero initial state, and given any T ≥ 0 and scalar α > 0, under zero initial state, the CVNN model in (5) is said to be strictly (Q, S, R)-dissipative. The following inequality holds with respect to any non-zero input u ∈ L 2 [0, ∞).
For the model in (5), we can express the relation (22) in an equivalent dissipativity performance index, as follows: Remark 3. We notice from the available publications that a number of definitions for strictly (Q, S, R)-dissipativity [27,30], global exponential disspativity and global dissipativity [31,33] are provided in the Euclidean space R n . These definitions have been extended in recent publication [31,[34][35][36][37] to the complex plane C n .
At the same time, the energy supply function for the NN model in (13) can be defined as follows: whereQ,S,R ∈ R n×n , and Definition 3. Given scalar α > 0 and T ≥ 0, and subject to zero initial condition, the NN model in (13) is said to be strictly (Q,S,R)-dissipative. The following inequality holds for any nonzero inputũ ∈ L 2 [0, ∞).
Consider the NN model in (13), by dividing into the real and imaginary parts, we can write the Inequality (25)

Main Results
The dissipativity analysis of the HTCVNN model in (13) is yet to be fully studied in this literature. To overcome this issue, we derive some sufficient conditions with respect to dissiaptivity pertaining to the NN model in (13). For clarity, we use the following notations:

Dissipativity Analysis
By employing the Lyapunov stability theory and integral inequality approach, some sufficient conditions are derived. The aim is to make sure the (Q,S,R)-α dissipativity of the CVHNN model in (20) in terms of LMIs, as in Theorem 1.

Theorem 1.
Based on Assumption (A 2 ), we can divide the activation function into both the real and imaginary parts. The NN model in (20) is strictly (Q,S,R)-α dissipative subject to scalars µ > 0 and r > 0, and with the existence of scalars 0 < 1 , 0 < 2 , 0 < α and matrices 0 < P, 0 < Q, 0 < R n (n = 1, 2, ...m) whereby the following LMI holds:0 Proof. Given the NN model in (20), the following Lyapunov function candidate is considered We can obtain the time-derivative of V (t), i.e.
We can estimate the terms in (30) through Lemma 1, i.e., Moreover, from (18), it follows that which is equivalent toV where0 is defined in (27), whileξ t is defined in the main results. From (26), we can obtain It can be deducted from (27) that It can be concluded that (25) holds, subject to zero initial condition. This implies the NN model in (20) is strictly (Q,S,R) − α− dissipative in accordance with Definition 3. The proof is completed.
Based on Theorem 1, the (Q,S,R) − α− dissipative criteria with respect to the UHTCVNN model in (13) is given to Theorem 2 along with linear fractional uncertainties.

Corollary 1.
Based on Assumption (A 2 ), we can divide the activation function into two: real and imaginary parts. The NN model in (20) withũ(t) = 0 is global asymptotic stable subject to scalars r > 0 and µ > 0 and with the existence of scalars 0 < 1 , 0 < 2 and matrices 0 < P, 0 < Q, 0 < R n (n = 1, 2, ..., m) in such a way that the following LMI holds: where

Remark 6.
Unlike some existing studies on dissipativity analysis of the CVNN models [31][32][33][34][35][36][37], we derive the sufficient conditions to safeguard the dissipativity of HTCVNN models. An equivalent real-valued model is formulated from the original model. Moreover, the obtained dissipativity conditions (27) and (39) are expressed in LMIs. The feasible solutions can be obtained using the MATLAB software package.

Remark 7.
In this paper, we construct an appropriate LKF candidate along with multiple integral terms, such    Furthermore, to solve this term, we define,ỹ n t = Remark 8. With respect to Assumption (A 2 ), the presented dissipativity and stability results in this paper are invalid in the situation when we cannot convert the complex-valued activation function g y (·), y = 1, ..., n into its real and imaginary part.

Numerical Examples
We assess the usefulness of the results using a number of numerical examples. (20) is considered, i.e.,

Example 2. The UHTCVNN model in
Assume thatg x (ẽ) =g R x (x, y) + ig I x (x, y), x = 1, 2. By simple calculation, we havẽ  Figure 9 depict the time responses with respect to both the real and imaginary parts pertaining to the model in (13), in whichũ(t) = sin(0.02 t)e −0.005 t , t ≥ 0. Based on the same initial conditions and withũ(t) = 0, the time responses with respect to both the real and imaginary parts of the model in (13) are shown in Figure 10.

Example 3. The HTCVNN model in
). and L = Take r(t) = 0.6 + 0.2sin t which satisfies r = 0.8 and µ = 0.3, and with the above parameters, we can use the MATLAB software package, the LMI (41) is true with n = 1, 2, 3. Based on 20 randomly generated initial values, Figures 11-14 depict the time responses with respect to both the imaginary and real parts pertaining to the model in (20). From the illustrations, we can confirm that the equilibrium point of the model in (20) is global asymptotic stable.

Conclusions
An investigation on the robust dissipativity with respect to the HTCVNN models with linear fractional uncertainties and time-varying delays was conducted. To facilitate the analysis, we devised an appropriate LKF with general integral terms and employed the multiple integral inequality method to yield the sufficient conditions of dissipativity with respect to the HTCVNN models in the form of LMIs. The MATLAB software package was used to solve the LMIs effectively. We also illustrated the feasibility of the results through several numerical models and their simulation results. Note that the features of HTCVNNs are closely connected with other CVNN models. As a result, we intend to extend the obtained results to study various dynamical behaviours of different fractional-order CVNN models in our future research.