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

Authors: Wenxiao Si; Tao Xie; Biwen Li
Journal: Complexity
 (2021)

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## Abstract

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.


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## Body

## 1. 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 [1–14], such as asymptotic stability [1], exponential stability [2–4], multistability [5], synchronization [6], dissipativity [3, 7, 8], region stability [9], memristor-based 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, 13–22]. 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 [1–23] 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 [1–8], 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 neutral-type 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. The 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 [19–21]. At present, uncertain connection weights have been applied to various fields, such as medicine, information transmission, and operations’ research and planning [29–32]. 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, 33–37] 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) The 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.The rest of the paper is as follows. The preliminaries and the model descriptions are included in Section2. Then, 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.


## 2. Problem Formulation

Based on this paper, denoteN 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. ℝn, ℝn×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=supAx:x=1. Denote two piecewise constant argument real-value sequences θi and ηi, such that θi<ηi<θi+1, θi⟶∞ when i⟶∞ and i∈ℕ.Consider a NPRNN as(1)dvt−Gvt=−Avt+Bfvt+Cgvγtdt,vt0=v0∈ℝn,where vt=v1t,…,vnt∈ℝn represents the state vector of the neurons, Gvt:ℝn⟶ℝn is the neutral-type function of vt∈ℝn, A=diaga1,…,an is a self-feedback connection weight matrix, B=bijn×n and C=cijn×n are the connection weight matrices of fvt and gvγt, respectively, where A,B,C∈ℝn×n, fvt and gvγt are continuous vector-value activation functions, and vt and vγt are the current state and piecewise argument state, respectively.For the case thatGvt=0 and γt=t, NPRNN (1) becomes the following RNN model:(2)r˙t=−Art+Bfrt+Cgrt,rt0=r0∈ℝn.Definition 1.
For anyt0∈ℝ+ and r0∈ℝn, if there are α,β>0, such that rt,t0,r0≤αr0exp−βt−t0 holds, and the state rt,t0,r0 of (2) can achieve globally exponential stability.Definition 2.
For anyt0∈ℝ+and v0∈ℝn, if there are α˜,β˜>0, such that vt,t0,v0≤α˜v0exp−β˜t−t0 holds, and the state vt,t0,v0 of (1) can achieve globally exponential stability.To deduce the main results, some needed assumptions throughout the paper are given:(H1) The activation functionsf⋅, g⋅∈ℝn, and there are Lipschitz constants l1 and l2>0 such that(3)fς−fϱ≤l1ς−ϱ,gς−gϱ≤l2ς−ϱ,hold for anyς, ϱ∈ℝn, where f⋅ and g⋅ are endowed with the initial values f0=0 and g0=0.(H2) Assume that neutral-type functionG⋅ satisfies a Lipschitz condition:(4)Gς−Gϱ≤Lς−ϱ,whereς, ϱ∈ℝn.(H3) For a real-value sequenceθk, there is a θ>0, which makes θk+1−θk≤θ true for any k∈N.(H4) Assume that(5)L+l2θC+θA+l1B1+L+l2θC1−L×expθA+l1B1−L<1.(H5) Assume that(6)αexp−βT−θ+Lαexp−βθ+L1−L+2αl2Cβ1−L2×exp2TA+l1B1−L+2T3−Ll2C1−L2<1.Remark 1.
For the sake of convenience, some symbol descriptions are listed as follows:(7)ϖ1=1+L+l2θC1−L,ϖ2=1+σA+1+λl1B1−L,ϖ3=l2C1+Γ+σA+λl1Bα/β1−L+Lαexp−βθ+L1−L,ϖ4=1+σA+l11+λB+l2C2+Γ1−L,ξ=L+l2θC+1+σA+1+λl1B×θϖ1expϖ2θ,Γ=1+L1−ξ,ξ<1,T>lnαβ>0.


## 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):(8)dut−Gut=−1+σAut+1+λBfut+Cguγtdt,ut0=u0∈ℝn,where σ and λ∈R are the extra interference intensity of connection weight matrices A and B, corresponding to the state ut and activation function fut.Then, the following auxiliary Lemma1 aims to clarify the relationship between piecewise argument state uγt and current state ut.Lemma 1.
Denote thatut=u1t,…,untT is a solution of (2), and (H1)-(H4) hold; then, the following inequality,(9)uγt≤Γut,holds, where(10)ϖ1=1+L+l2θC1−L,ϖ2=1+σA+1+λl1B1−L,Γ=1+L1−ξ,ξ<1,ξ=L+l2θC+1+σA+1+λl1B×θϖ1expϖ2θ.Proof.
Fixk∈N, and for γt=ηk and t∈θk,θk+1, t∈ℝ+, we have(11)ut−Gut−uηk−Guηk=∫ηkt−1+σAus+1+λBfus+Cguηkds.
Applying the norm inequality on both sides of (11) and in accordance with (H1), we obtain(12)ut−uηk−Guηk−Gut≤1+σA∫ηktusds+1+λB×∫ηktfusds+C∫ηktguηkds≤1+σA∫ηktusds+1+λB×∫ηktl1usds+C∫ηktl2uηkds≤1+σA∫ηktusds+1+λl1B×∫ηktusds+l2C∫ηktuηkds.
In terms of (H2) and the norm inequality, we derive(13)ut≤uηk+Luηk+Lut+1+σA+1+λl1B∫ηktusds+l2C∫ηktuηkds≤Lut+1+L+l2θCuηk+1+σA+1+λ|l1B∫ηktusds.
Directly, forL∈0,1, we have(14)ut≤1+L+l2θC1−Luηk+1+σA+1+λl1B1−L∫ηktusds≤ϖ1uηk+ϖ2∫ηktusds,where(15)ϖ1=1+L+l2θC1−L,ϖ2=1+σA+1+λl1B1−L.
According to Gronwall–Bellman lemma, it follows that(16)ut≤ϖ1expϖ2θuηk.
Moreover, fort∈θk,θk+1, we obtain(17)Guηk−Gut=uηk−ut+∫ηkt−1+σAus+1+λBfus+Cguηkds.
In combination with (H1)–(H3) and (16), similarly, we obtain(18)uηk≤ut+Luηk+Lut+1+σA+1+λl1B∫ηktusds+l2C∫ηktuηkds≤1+Lut+Luηk+1+σA+1+λl1Bθϖ1expϖ2θuηk+l2C∫ηktuηkds≤1+Lut+L+l2θC+1+σA+1+λl1Bθϖ1expϖ2θuηk≤1+Lut+ξuηk,where(19)ξ=L+l2θC+1+σA+1+λl1B×θϖ1expϖ2θ.
Therefore, by (18) and (H4), for ηk=γt and k∈N, we can obtain(20)uγt≤1+L1−ξut≕Γut,where(21)Γ=1+L1−ξ,ξ<1.Theorem 1.
If (H1)–(H5) hold, (2) achieves globally exponential stability. Then, the conditions for achieving the RoGES of NPRNN can be given by the following (a) and (b):(a)
LetL¯ be the upper bound of neutral term compressibility coefficient; then, L<L¯ and L¯∈0,1, and L¯ is given by the following transcendental equation:(22)αexp−βT+L¯α+L¯1−L¯β+2αl2Cβ1−L¯2×exp2T1−L¯A+l1B+3−L¯l2C1−L¯2<1.(b)
Letθ3 be the upper bound of piecewise constant arguments, where θ3 is given by(23)θ<θ3=minT2,θ1,θ2,where θ1 and θ2 are the upper bounds satisfying (H4) and (H5), respectively, T>lnα/β>0.Furthermore, if the real selected valuesL<L¯ and θ<θ3 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:(24)αexp−βT−θ+ϖ3exp2Tϖ4=1,where(25)ϖ3=Lαexp−βθ+L1−L+l2C1+Γ+σA+λl1Bα/β1−L,ϖ4=1+σA+l11+λB+l2C2+Γ1−L,and the other parameters are the same as Lemma 1. Besides, the boundary value of σ is expressed as σsup, and λ is expressed as λsup hereinbelow.Proof.
From (2) and (8), for t≥t0, we obtain(26)ut−rt=Gut−Gut0+∫t0t−Aus−rs+Bfus−frs+Cguγs−gus+gus−grs−σAus−rs+rs+λBfus−frs+frsds.
Taking the absolute value of (26) for t≤t0+2T, we obtain(27)ut−rt≤Gut−Gut0+A+l1B+l2C+σA+λl1B∫t0tus−rsds+C∫t0tguγs−gusds+σA+λl1B∫t0trsds≤Lut+Lut0+1+σA+l11+λB+l2C∫t0tus−rsds+Cl2∫t0tuγs+usds+σA+λl1B∫t0trsds≤Lut−rt+Lrt+Lut0+1+σA+l11+λB+l2C∫t0tus−rsds+Cl21+Γ∫t0tus−rs+rsds+σA+λl1B∫t0trsds.
In accordance with Definition 1 and the exponential stability of (2), L∈0,1, when t∈t0−θ,t0+θ, further we derive(28)ut−rt≤L1−Lαu0exp−βθ+L1−Lu0+1+σA+l11+λB+l2C2+Γ1−L×∫t0tus−rsds+l2C1+Γ+σA+λl1B1−L×∫t0tαr0exp−βs−t0ds≤Lαexp−βθ+L1−L+l2C1+Γ+σA+λl1Bα/β1−Lu0+1+σA+l11+λB+l2C2+Γ1−L×∫t0tus−rsds≤ϖ3u0+ϖ4∫t0tus−rsds,where(29)ϖ3=Lαexp−βθ+L1−L+l2C1+Γ+σA+λl1Bα/β1−L,ϖ4=1+σA+l11+λB+l2C2+Γ1−L.
By virtue of Gronwall inequality, fort0+θ≤t≤t0+2T, we have(30)ut−rt≤ϖ3u0expϖ4t−t0≤ϖ3exp2Tϖ4supt∈t0−θ,t0+θut.
Note thatθ<T/2, when t0−θ+T≤t0−θ+2T, by the exponential stability of (2), we further obtain(31)ut≤rt+ut−rt≤αexp−βT−θ+ϖ3exp2Tϖ4×supt0−θ≤t0−θ+Tut≕C˜supt0−θ≤t0−θ+Tut,where C˜=αexp−βT−θ+ϖ3exp2Tϖ4. Then, if we define(32)FΓθ,L=αexp−βT−θ+ϖ3exp2Tϖ4,obviously, there is(33)FΓθ,L≤1.
Firstly, by (H4), denoteθ1 as the unique maximal positive solution satisfying ξθ|σ,λ=0<1, that is,(34)L+l2θC+θA+l1B1+L+l2θC1−LexpθA+l1B1−L<1,so(35)Γθ=1+L1−ξθ∈1+L1−L,∞,L∈0,1,holds for any θ∈0,θ1. According to (32) and (H5), it derives(36)F1+L1−L,L|λ,σ=0=αexp−βT−θ+Lαexp−βθ+L1−L+2αl2Cβ1−L2×exp2TA+l1B1−L+2T3−Ll2C1−L2<1,(37)F∞,L>1.
Thus, by the monotonicity ofFΓθ,L, there exists some Γ^θ∈1+l/1−L,∞, which makes FΓθ,L|λ,σ=0=1 true. Then, if we denote θ2 as the upper bound satisfying (36) and select(38)θ3=minT2,θ1,θ2,hence, the upper bound of the piecewise arguments unaffected by the intensity of uncertain parameters is given.
Besides, since∂Γθ/∂θ>0 and ∂FΓθ,L/∂θ>0, the upper bound of neutral terms can be given by(39)F1+L1−L,L|θ,λ,σ=0=αexp−βT+Lα+L1−Lβ+2αl2Cβ1−L2×exp2T1−LA+l1B+3−Ll2C1−L2<1.
Finally, if we select the exactL 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:(40)αexp−βT−θ+ϖ3exp2Tϖ4=1,and some corresponding symbolic descriptions of (40) are shown in Remark 1.
Accordingly, selectingΛ=−lnC˜/T, where Λ≥0, and from (31), we can obtain(41)supt0−θ+T≤t≤t0−θ+2Tut,t0,u0≤exp−ΛTsupt0−θ≤t≤t0−θ+Tut,t0,u0.
Subsequently, for any positive integerq∈n¯, when t≥t0−θ+q−1T, by the continuity of the solutions of the system [13], invoking (41), further we have(42)supt0−θ+qT≤t≤t0−θ+q+1Tut;t0,u0=supt0−θ+q−1T+T≤t≤t0−θ+qT+Tut;t0−θ+q−1T,ut0−θ+q−1T;t0,u0≤exp−ΛTsupt0−θ+q−1T≤t≤t0−θ+qTut;t0,u0≤exp−ΛqTsupt0−θ≤t≤t0−θ+Tut;t0,u0=Cexp−ΛqT,where C=supt0−θ≤t≤t0−θ+Tut,t0,u0. Hence, when t0−θ+qT≤t≤t0−θ+q+1T, for any integer q∈n¯, we have(43)ut≤Cexp−ΛqT≤CexpΛ−t−t0+T−θ≤CexpΛTexp−Λt−t0.
In this way, (8) achieves globally exponential stability.Corollary 1.
Obviously, the allowed intensity of (σ, λ) to achieve the RoGES of parameter-uncertainty NPRNN (8) is derived in Theorem 1. In fact, Theorem 1 is also applicable for the following models (44) and (45):(44)dut−Gut=−1+σAut+Bfut+Cguγtdt,ut0=u0∈ℝn,(45)dut−Gut=−Aut+1+λBfut+Cgyγtdt,ut0=u0∈ℝn,corresponding to the cases λ=0 and σ=0 for parameter-uncertainty NPRNN (8), respectively.Proof.
Similarly, if we setλ=0 and σ=0 for Theorem 1, respectively, the sufficient conditions to ensure the globally 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 Theorem 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, l1, l2, 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.Figure 1
Stable enclosed curve ofσ and λ in system (48).Remark 3.
The order of the calculation in this paper is as follows. At first, some parametersα, β, l1, l2, 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 (θ3) is obtained by (23). Furthermore, in accordance with the appropriately selected values of L and θ (i.e., L<L¯ and θ<θ3) and equation (24), we can obtain the bounds of σ and λ which can guarantee the RoGES of parameter-uncertainty NPRNN (8) by MATLAB. The robustness of the system means that if the original NPRNN (1) is stable, the perturbed parameter-uncertainty 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).


## 4. Illustrative Example

Example 1.
For a two-state RNN,(46)r˙1tr˙2t=−1001r1tr2t+0.09000.09tanhr1ttanhr2t+0.10.10.10.1sinr1t2sinr2t2.Equation (46) can achieve globally exponential stability with α=1 and β=0.6. According to Theorem 1, let T=0.5>ln1/0.6=0. From (H1), let l1=1 and l2=0.5. Besides, other parameters A=1, B=0.09, and C=0.02 can also be listed; then, the stable state trajectory can be seen in Figure 2.Figure 2
Stable state trajectories ofr1t and r2t in (46).When the neutral terms and piecewise arguments are attached to RNN (46), (46) can be written as NPRNN (47):(47)ddtv1t+Lsinv1tv2t+Lsinv2t=−1001v1tv2t+0.09000.09tanhv1ttanhv2t+0.10.10.10.1sinv1γt2sinv2γt2,where θk=k/10andηk=2k+1/20, k∈N. The piecewise constant argument function γt=ηk when t∈θk,θk+1, k∈N.Lastly, if we consider additional dual-parameter perturbations for (47), then parameter-uncertainty NPRNN (48) is formulated as(48)ddtu1t+Lsinu1tu2t+Lsinu2t=−1+σ1001u1tu2t+1+λ0.09000.09×tanhu1ttanhu2t+0.10.10.10.1sinu1γt2sinu2γt2.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,(49)exp−0.3+2L1−L×0.6+0.020.6×1−L2×exp1.09×1−L+0.01×3−L1−L2=1,then the upper bound of L is obtained: L¯=0.0231.Next, if we selectL=0.01<L¯=0.0231 and substitute L, α, β, l1, l2, and T into (H4) and (H5), we get θ1=0.5053 and θ2=0.2004, respectively. So, the upper bound of θ is given by the following θ3:(50)θ<θ3=minT2,θ1,θ2=min0.25,0.5053,0.2004=0.2004.According to Theorem1, the states v1t and v2t of (47) will be stable with L=0.01 and θ=0.1, which is shown in Figure 3.Figure 3
Stability ofv1t and v2t in (47) with L=0.01 and θ=0.1.On the other hand, we will explain the intensity of connection weightsσ and λ that parameter-uncertainty NPRNN (48) can tolerate based on the stable NPRNN (47) shown in Figure 3.Therefore, we fix parametersL=0.01 and θ=0.1 so as to satisfy the stable conditions in Theorem 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:(51)exp−0.24+0.01×exp−0.06+0.010.99+0.01×1+1.01/1−ξ+σ+0.09λ×1/0.60.99×exp1+σ+0.091+λ+0.01×2+1.01/1−ξ0.99=1,where(52)ξ=0.011+1+σ+0.091+λ×0.1×1.0110.99exp0.1×1+σ+0.091+λ0.99.According to the above implicit transcendental equation, the stable region ofσ 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). Therefore, 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).(i)
Ifσ,λ=0.001,0.01, where σ=0.001<σsup=0.0071809 and λ=0.01<λsup=0.0797873, according to Figure 1, the stable state trajectories of u1t and u2t are shown in Figure 4.Otherwise, some unstable cases should be provided to illustrate. If the connection weightσ exceeds the boundary value σsup or λ exceeds the boundary value λsup in Figure 1, parameter-uncertainty NPRNN (48) will be unstable even based on stable NPRNN (47), which can be illustrated in the following:(ii)
Ifσ,λ=0,18, where λ=18>λsup=0.0797873, the unstable state trajectories are shown in Figure 5.(iii)
Ifσ,λ=−1.6,0, where σ=1.6>σsup=0.0071809, the unstable state trajectories are shown in Figure 6.(iv)
Ifσ,λ=0.5,20, where σ=0.5>σsup=0.0071809 and λ=20>λsup=0.0797873, the unstable state trajectories are shown in Figure 7.(v)
Ifσ,λ=−0.5,20, similarly, the unstable state trajectories are shown in Figure 8.Figure 4
Stability ofu1t and u2t in (48) with σ,λ=0.001,0.01.Figure 5
Instability ofu1t and u2t in (48) with σ,λ=0,18.Figure 6
Instability ofu1t and u2t in (48) with σ,λ=−1.6,0.Figure 7
Instability ofu1t and u2t in (48) with σ,λ=0.5,20.Figure 8
Instability ofu1t and u2t in (48) with σ,λ=−0.5,20.Remark 4.
Figure1 is an enclosed curve about σ and λ with the boundary value σsup and λsup. Figure 1 is depicted by (24); more details of the calculation process of (24) can be seen in Remark 2. According to Remark 2, we transform (24) from a four-variable equation to a two-variable equation only related to σ and λ. By virtue of computer software MATLAB, an enclosed region of σ and λ, and the boundary values σsup=0.0071809 and λsup=0.0797873 are all directly exhibited in Figure 1 to demonstrate the RoGES of parameter-uncertainty NPRNN (48). If we select σ<σsup and λ<λsup, system (48) will be stable again based on stable Figure 3. Or else, if σ>σsup or λ>λsup holds, system (48) will lose its stability even on the basis of stable Figure 3.Remark 5.
Figures1–8 systematically and intuitively prove the robustness of the system (48). From Theorem 1, the supremum of neutral terms L¯ is derived by (22), and the supremum of piecewise arguments θ3 is derived by (23), and the boundary of uncertain-parameter intensity is characterized by (24). Figures 1–8 are obtained as follows. (1) Fix some necessary parameters α, β, l1, l2, and T; an initial globally exponentially stable RNN (46) is given by Figure 2. (2) According to L¯ and θ3 settled by (22) and (23), here we select L=0.01<L¯=0.0231 and θ=0.1<θ3=0.2004; then, the NPRNN (47) can achieve exponential stability and the stable trajectories are shown in Figure 3. (3) Figure 1 is an stable enclosed curve about σ and λ depicted by (24) (more details for Figure 1 is shown in Remark 4). (4) If connection weights σ=0.001<σsup=0.0071809 and λ=0.01<λsup=0.0797873, the parameter-uncertainty NPRNN (48) will be stable again, and the stable trajectories are shown in Figure 4. (5) However, as long as either σ>σsup or λ>λsup is true, system (48) will be unstable, that is, the final-time state value does not converge to the same equilibrium point, which can be seen in Figures 5–8. More importantly, Figures 5–8 are essential and favourable reference frames for demonstrating the robustness of the parameter-uncertainty NPRNN. Furthermore, a brief calculation flowchart of Figures 1–8 is provided in Figure 9.Figure 9
The calculation process of Figures1–8 in Example 1.


## 5. Conclusion

This paper further explored the robustness of global exponential stability of NPRNN with uncertain dual connection weight matrices. Firstly, a foundational lemma is given. The 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. The 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¯, θ3, σsup, and λsup) in Theorem 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].



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