Further Results on Exponentially Robust Stability of Uncertain Connection Weights of Neutral-Type Recurrent Neural Networks

Further results on the robustness of the global exponential stability of recurrent neural network with piecewise constant arguments and neutral terms (NPRNN) subject to uncertain connection weights are presented in this paper. Estimating the upper bounds of the two categories of interference factors and establishing a measuring mechanism for uncertain dual connection weights are the core tasks and challenges. Hence, on the one hand, the new sufficient criteria for the upper bounds of neutral terms and piecewise arguments to guarantee the global exponential stability of NPRNN are provided. On the other hand, the allowed enclosed region of dual connection weights is characterized by a four-variable transcendental equation based on the preceding stable NPRNN. In this way, two interference factors and dual uncertain connection weights are mutually restricted in the model of parameter-uncertainty NPRNN, which leads to a dynamic evolution relationship. Finally, the numerical simulation comparisons with stable and unstable cases are provided to verify the effectiveness of the deduced results.


Introduction
Since recurrent neural networks (RNNs) have the ability of parallel processing, distributed information storage, and associative deep learning, a series of neural networks such as Hopfield neural network, Cohen-Grossberg neural network, cellular neural network, BAM neural network, high-order cellular neural network, and shunt inhibition neural network, which are the typical representatives of RNNs, have attracted extensive attention over the years. Accordingly, with the in-depth study of RNNs, it can be seen that the stability is a forerunner condition for the multifarious practical applications. Hence, the research on the stability of the system is becoming more and more abundant [1][2][3][4][5][6][7][8][9][10][11][12][13][14], such as asymptotic stability [1], exponential stability [2][3][4], multistability [5], synchronization [6], dissipativity [3,7,8], region stability [9], memristorbased dynamic behavior stability [10], and exponential Lagrange stability [11,12]. Additionally, in terms of the widespread application fields such as the visual optimization, image processing, language recognition, associative memory, and other fields, the stability of RNNs has become an indispensable dynamical behavior characteristic which must be further considered.
Exponential stability and robustness, as for the classical dynamic behaviors of neural networks, have been studied extensively in the past few years [2,3,[13][14][15][16][17][18][19][20][21][22]. On the one hand, global exponential stability reveals the superiority that it guarantees the system can be fleetly stable at the equilibrium point with an exponential decay rate, which deservedly leads to a rapid stabilization and dramatically saves the response time. Furthermore, the decay rate value can be intuitively captured [14]. On the other hand, due to the tunable flexibility and broad applicability, the robustness is always endowed with different meanings in disparate practical scenarios. Taking [13] as an example, a natural question is raised and deeply explored: how much the interference intensity can a disturbed system bear to realize the stability again on the basis of its original stability, which implies the exact connotation of the robustness studied in this paper. In addition, the emerging literatures  have also indirectly confirmed that various generalized stability behaviors of the neural networks are inevitably and immensely subject to the category and quantity of disturbances, for instance, time delays [1][2][3][4][5][6][7][8], stochastic disturbances [8,13,15], parameter disturbances [9], piecewise constant arguments [14], neutral terms [22], and Markov switching [12,23]. Hence, the subsequent perturbations will be attached to RNNs to further examine and guarantee the robustness of global exponential stability (RoGES) of RNNs.
Neutral-type RNNs refer to RNNs with the neutral terms appeared in the derivative part, which makes the nonlinear perturbations affect not only the current states but also the states of derivative part. At present, neural networks with neutral terms have been applied in practice, for instance, the electrical interconnect and the electromagnetic interference design in digital computers are used as the specific physical application background for delayed neutral-type differential equations [24]. In combination with the statements mentioned above and some existing literatures, two species of exogenous interferences that diffusely appear in the modern engineering should be taken into consideration for neutraltype RNNs. On the one hand, due to the pervasive restrictions of the switching speed of each node of the neural networks, time delay is such a kind of inevitable interference element that it has been widely applied to various neural networks. And, piecewise constant argument studied here is another form of time delay, which unifies the hysteresis and advance.
e investigation of RNNs subject to piecewise arguments is a breakthrough in literature [14], and the sufficient conditions for the exponential stability of a class of RNNs with piecewise arguments are given by constructing Lyapunov functions. At present, there are also some literatures further exploring the properties of the systems equipped with piecewise arguments [15,25,26]. On the other hand, it can be found that the parameter-intensity of the connection weight matrix is a crucial index affecting the stability of the systems [16-21, 27, 28]. Firstly, since parametric uncertainty was introduced in [16,17,27,28], diverse forms of uncertain connection weights were applied to neural networks. Later, Zhu and Shen [18] visually depicted the boundary of the uncertain connection weights by the enclosed curve graph method and provided the sufficient conditions for RNNs disturbed by uncertain parameters to achieve the global exponential stability. Furthermore, time delays and random perturbations are additionally attached on RNNs in [19][20][21]. At present, uncertain connection weights have been applied to various fields, such as medicine, information transmission, and operations' research and planning [29][30][31][32]. It follows that the intensity of the connection weights is a highly mutable but immensely critical indicator.
By virtue of the existing literatures in the past decades, the stability of neutral-type RNNs with piecewise arguments or other time delays is explored by [22,[33][34][35][36][37] and some relatively mature methods have been widely used, such as the Euler-Maclaurin method, one-leg method, block boundary value method, and multidomain Legendre spectral collocation method. In addition, the investigations of RNNs instead of neutral-type RNNs with uncertain connection weights also have been carried out in [16-21, 27, 28]. However, there is hardly any studies aiming for the RoGES of the recurrent neural network with neutral terms and piecewise constant arguments (NPRNN) with uncertain connection weights. Hence, it is a notable problem that how much the parameter intensity of dual connection weights can a disturbed NPRNN endure to remain stable again on the basis of the original stable NPRNN.
Motivated by the above statements, here we investigate the RoGES of the NPRNN with uncertain connection weights. Naturally, the ultimate goal herein is to quantify the boundary values of the connection weight matrices on the basis of stable NPRNN. Hence, the main contributions are as follows. (1) e upper bounds of piecewise arguments and neutral terms that NPRNN can maintain stability are established. We solve the tricky neutral terms newly appeared in the derivative part of RNN by constructing a Lipschitz condition, and then, we obtain the upper bound of the neutral terms by solving a univariate transcendental equation. Afterwards, by fixing an appropriate value of neutral terms, the upper bound of piecewise arguments is settled by solving several different binary transcendental equations. (2) An enclosed curve about dual connection weights (σ and λ) is acquired by solving the newly established four-element (two disturbances and two uncertain connection weights) transcendental equation on the basis of the preceding stable NPRNN, which is the core significant results of this paper (more details can be seen in Remarks 2 and 3). (3) By virtue of the above two upper bounds of the interference factors and the characterized enclosed curve about the dual-parameter intensity, the simulation results indicate that, as long as one of the interference values exceeds the derived bounds, the preceding stable NPRNN will be unstable, which also intuitively confirmed the validity of deduced results. Based on what has been established above, these four interference elements are mutually restricted and the relationship among these factors established by a four-element transcendental equation is dynamic. e rest of the paper is as follows. e preliminaries and the model descriptions are included in Section 2. en, the feasible threshold values of the piecewise arguments, the neutral terms, and the uncertain connection weights to achieve the RoGES of the parameter-uncertainty NPRNN are discussed in Section 3. Accordingly, the simulation comparisons to verify the validity of the deduced results are shown in Section 4. Finally, a brief conclusion and some feasible prospects for future work are given in Section 5.

Problem Formulation
Based on this paper, denote N as the natural number set and R as the real number set. And, for any constant n, denote n � 1, 2, . . . , n { }. Z + is the positive integer set. R n , R n×m , and R + stand for n-dimensional Euclidean space, n × m real matrix space, and positive real space, respectively. Let ‖ · ‖ be the Euclidean norm, and the operator norm of matrix A is defined as ‖A‖ � sup ‖Ax‖: ‖x‖ � 1 { }. Denote two piecewise constant argument real-value sequences θ i and η i , such that

Complexity
Consider a NPRNN as where ) are continuous vector-value activation functions, and v(t) and v(c(t)) are the current state and piecewise argument state, respectively. For the case that G(v(t)) � 0 and c(t) � t, NPRNN (1) becomes the following RNN model: (2) holds, and the state r(t, t 0 , r 0 ) of (2) can achieve globally exponential stability. (1) can achieve globally exponential stability.
To deduce the main results, some needed assumptions throughout the paper are given: e activation functions f(·), g(·) ∈ R n , and there are Lipschitz constants l 1 and l 2 > 0 such that hold for any ς, ϱ ∈ R n , where f(·) and g(·) are endowed with the initial values f(0) � 0 and g(0) � 0.
(H4) Assume that Remark 1. For the sake of convenience, some symbol descriptions are listed as follows: Complexity 3

Main Results
On the basis of NPRNN (1), in this paper, we mainly explore NPRNN with uncertain connection weights (called parameter-uncertainty NPRNN throughout the text): where σ and λ ∈ R are the extra interference intensity of connection weight matrices A and B, corresponding to the state u(t) and activation function f(u(t)). en, the following auxiliary Lemma 1 aims to clarify the relationship between piecewise argument state u(c(t)) and current state u(t). (2), and (H1)-(H4) hold; then, the following inequality, holds, where Proof. Fix k ∈ N, and for Applying the norm inequality on both sides of (11) and in accordance with (H1), we obtain In terms of (H2) and the norm inequality, we derive Complexity According to Gronwall-Bellman lemma, it follows that In combination with (H1)-(H3) and (16), similarly, we obtain where ξ � L + l 2 θ‖C‖ + (1 + |σ|)‖A‖ + (1 + |λ|)l 1 ‖B‖ erefore, by (18) and (H4), for η k � c(t) and k ∈ N, we can obtain where (b) Let θ 3 be the upper bound of piecewise constant arguments, where θ 3 is given by where θ 1 and θ 2 are the upper bounds satisfying (H4) and (H5), respectively, T > (ln α)/β > 0.
Furthermore, if the real selected values L < L and θ < θ 3 hold by (22) and (23), then the allowed intensity of (σ, λ) to achieve the RoGES of parameter-uncertainty NPRNN (8) should be in the inner of the following enclosed curve: where and the other parameters are the same as Lemma 1. Besides, the boundary value of |σ| is expressed as |σ| sup , and |λ| is expressed as |λ| sup hereinbelow.

Complexity
Besides, since z(Γ(θ))/zθ > 0 and z(F(Γ(θ)), L)/zθ > 0, the upper bound of neutral terms can be given by Finally, if we select the exact L and θ which stabilize NPRNN (1), the intensity boundary of dual parameters σ and λ to stabilize the new parameter-uncertainty NPRNN (8) will be obtained by the following transcendental equation in accordance with the above L and θ: and some corresponding symbolic descriptions of (40) are shown in Remark 1.
Proof. Similarly, if we set λ � 0 and σ � 0 for eorem 1, respectively, the sufficient conditions to ensure the globally 8 Complexity exponential stability of (44) and (45) can be given promptly.
□ Remark 2. It is not easy to directly handle (24) (i.e., (40)), a transcendental equation with four variables in eorem 1. Hence, we adopt a special method: fix θ and L which can maintain the globally exponential stability of NPRNN (1), and then, in combination with some known parameters α, β, l 1 , l 2 , and T, (24) becomes an implicit transcendental equation with only two variables |σ| and |λ|. Transcendental equation (24) is a key step to obtain the enclosed curve such as the one in Figure 1, which is also the core work of this paper.

Remark 3.
e order of the calculation in this paper is as follows. At first, some parameters α, β, l 1 , l 2 , and T are given in advance, and the upper bound of the neutral terms (L) can be derived by (22). Next, we fix an exact L < L which can ensure the exponential stability of NPRNN (1), and then, the supremum of the piecewise arguments (θ 3 ) is obtained by (23). Furthermore, in accordance with the appropriately selected values of L and θ (i.e., L < L and θ < θ 3 ) and equation (24), we can obtain the bounds of σ and λ which can guarantee the RoGES of parameter-uncertainty NPRNN (8) by MATLAB. e robustness of the system means that if the original NPRNN (1) is stable, the perturbed parameteruncertainty NPRNN (8) can still remain stable as long as the selected values of σ and λ are included inside the closed curve given by equation (24).
When the neutral terms and piecewise arguments are attached to RNN (46), (46) can be written as NPRNN (47): where θ k � k/10 Lastly, if we consider additional dual-parameter perturbations for (47), then parameter-uncertainty NPRNN (48) is formulated as d dt In order to make the process clearer, the following explanations will be divided into two parts to illustrate the RoGES of system (48). On the one hand, we will explain how much the interference intensity of neutral terms and piecewise arguments the system (47) can tolerate to be stable again based on the stable RNN (46) depicted in Figure 2.
Firstly, calculating the following equation by MATLAB, then the upper bound of L is obtained: L � 0.0231. Next, if we select L � 0.01 < L � 0.0231 and substitute L, α, β, l 1 , l 2 , and T into (H4) and (H5), we get θ 1 � 0.5053 and θ 2 � 0.2004, respectively. So, the upper bound of θ is given by the following θ 3 : According to eorem 1, the states v 1 (t) and v 2 (t) of (47) will be stable with L � 0.01 and θ � 0.1, which is shown in Figure 3.
On the other hand, we will explain the intensity of connection weights σ and λ that parameter-uncertainty NPRNN (48) can tolerate based on the stable NPRNN (47) shown in Figure 3. erefore, we fix parameters L � 0.01 and θ � 0.1 so as to satisfy the stable conditions in eorem 1 and be consistent with the parameter setting in Figure 3. Subsequently, the stable region with (σ, λ) can be solved by a transcendental equation by MATLAB: According to the above implicit transcendental equation, the stable region of σ and λ is depicted in Figure 1, that is, if the intensity range of (σ, λ) is in the inner of the enclosed curve in Figure 1, then parameter-uncertainty NPRNN (48) will be stable again based on the stable NPRNN (47). erefore, some stable and unstable cases are given to verify the RoGES of system (48), such as the stable case (i) and unstable cases (ii)-(v).

Remark 4.
Remark 5. Figures 1-8 systematically and intuitively prove the robustness of the system (48). From eorem 1, the supremum of neutral terms L is derived by (22), and the supremum of piecewise arguments θ 3 is derived by (23), and the boundary of uncertain-parameter intensity is characterized by (24). Figures 1-8 are obtained as follows.

Conclusion
is paper further explored the robustness of global exponential stability of NPRNN with uncertain dual connection weight matrices. Firstly, a foundational lemma is given. e relationship between the piecewise argument state and the current state is obtained by virtue of Gronwall inequality and the Lipchitz conditions. In addition, a main theorem is given. e upper bounds of the two interference factors and the coupling restrictions of uncertain dual connection weights are deduced by solving a four-element transcendental equation based on stable NPRNN. Finally, the stable and unstable cases of parameter-uncertainty NPRNN are analyzed by a systematic comparison of several numerical simulations, and it is also visually explained that if the two categories of interferences (L and θ) and dual uncertain connection weights (σ and λ) are lower than the deduced values (L, θ 3 , |σ| sup , and |λ| sup ) in eorem 1, respectively, the disturbed parameter-uncertainty NPRNN based on stable NPRNN can be stabilized again.
Additionally, there is a prospect for future work. On the one hand, future work may focus on the other diverse dynamical stability analysis of NPRNN with connection weight matrices, such as passivity and dissipativity [3,8], multistability [5], and asynchronous analysis [6,23]. On the other hand, we may also pay attention to attaching more exogenous interferences on the system established in this paper, such as mixed time-varying delays [3], stochastic disturbances [8], and Markov switching [23]. Furthermore, the uncertain-parameter NPRNN can be extended to more multidimensional spaces, such as fractional-order systems [1] and high-order systems [4].

Data Availability
No data were used to support this study. Using the above L and θ, then |σ| sup and |λ| sup are obtained by (3.11).

Conflicts of Interest
Stable Figure 1 Stable Figure 2 Stable Figure 4 Unstable Figure 5 Unstable Figure 6 e enclosed curve in Figure 3 Unstable  14 Complexity