# Exploration on Robustness of Exponentially Global Stability of Recurrent Neural Networks with Neutral Terms and Generalized Piecewise Constant Arguments

Authors: Wenxiao Si; Tao Xie; Biwen Li
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

With a view to the interference of piecewise constant arguments (PCAs) and neutral terms (NTs) to the original system and the significant applications in the signal transmission process, we explore the robustness of the exponentially global stability (EGS) of recurrent neural network (RNN) with PCAs and NTs (NPRNN). The following challenges arise: what the range of PCAs and the scope of NTs can NPRNN tolerate to be exponentially stable. So we derive two important indicators: maximum interval length of PCAs and the scope of neutral term (NT) compression coefficient here for NPRNN to be exponentially stable. Additionally, we theoretically proved that if the interval length of PCAs and the bound of NT compression coefficient are all lower than the given results herein, the disturbed NPRNN will still remain global exponential stability. Finally, there are two numerical examples to verify the deduced results’ effectiveness here.


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## Body

## 1. Introduction

Investigation and synthesis of recurrent neural networks (RNNs) is an unfailing subject regardless of past and present due to its wide application in image and object recognition, speech recognition, model prediction, automatic control, signal processing, and so forth [1–18]. Various stability types for RNNs have been proposed and deeply explored, for instance, asymptotic stability [1–3], exponential stability [4–6], and multistability [7, 8]. Moreover, some other stability types have been further developed in recent five years, such as synchronization [9, 10], impulsive stochastic stability [11], region stability [12], global Mittag–Leffler stabilization [13, 14], memristor-based dynamic behavior stability [15, 16], antiperiodic stability [17], exponential Lagrange stability [18] and so on. In fact, compared with the general stability, the study of global exponential stability can be convenient for us to know the convergence rate of the system within a fast time interval accurately and intuitively. Therefore, there is still significative work to be done to explore and characterize global exponential stability or other kinds of stability of recurrent neural networks.The neutral-type equation is a general class of utility model originating from the mechanical operation, biological neural, economy management, electric circuit, automation control, and the other scientific fields, which have aroused extremely extensive interests from quite a few investigators [19–30]. In terms of electrical implementations, such as mutual reciprocity, package modelling, and the electromagnetic interference design for multiple digital computers, pretty responses had been obtained in certain studies [19]. Therefore, there are several kinds of reasonable featured methods to research the neutral neural network: the one-step method [20, 21], block boundary value method [22, 23], Euler–Maclaurin method [24], Runge–Kutta method [25], and Legendre multidomain spectral collocation method [26], while there are very few specific ways to study the robustness of the system by giving the supremum of the neutral term compression coefficient affected by other additional distractions at the same time. It may be a challenge to raise some feasible analytical frameworks to investigate the robustness of recurrent neural networks with NTs (NRNNs).Piecewise constant argument (PCA) has a profound influence during the operation process of RNNs. Firstly, the theoretical work on PCA can be traced back to Shah and Joseph [31] and Cooke and Wiener [32]. In quick succession, Akhmet et al. [33] continued to improve this theory and provide evidence to achieve exponentially global stability of RNNs with piecewise constant arguments (PRNNs), appending more integrity with PRNNs at that time. In fact, the method of piecewise argument unifies hysteretic and advance, and can be also applied to other important systems [23, 34, 35], which undoubtedly immensely improved convenience to deal with the impact of time-lag on most systems. Therefore, here we will focus on the discussion of recurrent neural network with piecewise constant arguments and neutral terms (NPRNN) to boost the convenience and completeness of NRNNs.As far as we know, some of the work is about investigating the robustness of hybrid stochastic models, nonlinear models, and recurrent models equipped with NTs without PCAs [27, 28]. Some of the work has been done to explore the robustness of recurrent models with PCA subjects without NTs [14, 33, 36, 37]. Others have also been deeply done for systems with PCAs and NTs [23, 24, 26], but they aim to investigate the stability instead of robustness. But there are very few literatures to explore the robustness as well as the globally exponential stability of NPRNNs through giving the interval length of PCAs and the bound of NT compression coefficient together through appropriate parameter settings of PCAs.Encouraged by these thoughts, this paper will pay attention to the robustness of the globally exponential stability of NPRNN, which can induce faster convergence compared with the general stable system. We will respectively derive the maximum scope of NT compression coefficient and the interval length of PCAs that the system can tolerate to remain stable after interference. First, we may estimate the upper bound of the neutral term compression coefficientk which can be obtained by a linear inequality in one variable. Then, we may select the feasible value of k so that we can successfully determine the supremum of the piecewise argument θ. This has not been fulfilled for the NPRNN model, which is exactly the primary contribution here. Hence, in this paper, the interval length of PCA and the bound of NT compression coefficient for the perturbed NPRNN to be exponentially stable are all provided. Additionally, we will also testify theoretically that if the interval range of the PCAs and the scope of NT compression coefficient of the disturbed system are all lower than the upper bounds given in this paper, then NPRNNs will still remain stable.The composition of the rest part is set out as follows: the establishment of the NPRNN model, symbol description, and several rational assumptions are presented in Section2. The upper bounds of the PCA interval length and NT compression coefficient are all provided in Section 3. Finally, two examples and corresponding figures by MATLAB are performed to confirm the main theorems’ efficiency in Section 4.


## 2. Preliminaries and Notations

Letℕ, ℤ+, ℝ+ be the natural number set, the positive integer set, and the nonnegative real-number set, respectively. For n∈ℕ, denote r=r1,r2,…,rnT to be a general vector in ℝn (n -D real-vector space); the norm of r is expressed as r=∑j=1nrj, where T is the transpose of a vector. Denote N¯=1,2,…,n. θq, ϖq are two real-valued sequences such that θq≤ϖq≤θq+1, θq⟶∞ when k⟶∞, q∈ℕ.In the first step, we aim at a kind of RNN model with the effect of neutral terms and generalized piecewise constant arguments (NPRNN), which is written as(1)dyit−Giyit=−aiyit+∑j=1nbijfjyjt+∑j=1ncijgjyjδt+Iidt,t≥t0≥0,yit0=yi0∈ℝ,i∈N¯,where n represents the neurocyte numbers in system (1). δt=ϖq for q∈ℕ and θq≤t<θq+1. Giyit:ℝ⟶ℝ is the neutral-type function for yit∈ℝ of the i th neurocyte, where yit is a function which stands for the state trajectory of given time t of the neurocyte i. fjyjt and gjyjδt represent a class of output (activation functions) at time t and δt of the jth neurocyte, respectively. ai means the self-inhibition of the ith neurocyte, ai>0. bij and cij are the intensities of the neurocyte j on the neurocyte i, corresponding to time t and time δt, respectively. Ii refers to the extra interference on the ith neurocyte.Remark 1.
Obviously, system (1) is a class of a hybrid system. That is, if we consider the performance expression of (1) within the interval θq,θq+1, q∈ℕ. System (1) is shown as an advanced system when θq≤t<ϖq, and system (1) is shown as a hysteretic system when ϖq≤t<θq+1. Therefore, system (1) is a mixed system which unifies the advanced and hysteretic systems.Remark 2.
Additionally, compared with (1), in case of δt=t and without neutral terms, the NPRNN (1) degenerates into the RNN:(2)yi.t=−aiyit+∑j=1nbijfjyjt+∑j=1ncijgjyjt+Ii,t≥t0≥0,yit0=yi0∈ℝ,i∈N¯.
Apparently, (2) has a zero solution, state yt,t0,y0 exists and is unique for any given initial values t0 and y0 [37]. What follows is the exponentially global stability definition of RNN (2).Definition 1.
Denote thatyˇ is the equilibrium of model (2). State yt can achieve exponentially global stability (EGS) if yt−yˇ<α¯yt−yˇexp−β¯t−t0 exists for all t≥t0, where α¯, β¯>0. In particular, if yˇ=0 is the equilibrium of model (2), state yt can achieve EGS if yt<αytexp−βt−t0 is valid, t≥t0, where α, β>0.
To derive the main results, here are some of the assumptions needed:(A1)fi,gi are all continuous activation functions, fi,gi∈ℝ for each i∈N¯, and fi0=0, gi0=0.(A2) There are Lipschitz constantsLi1 and Li2>0, ζi and ηi∈ℝ, i∈N¯, such that(3)fiζi−fiηi≤Li1ζi−ηi,giζi−giηi≤Li2ζi−ηi.(A3) There are Lipschitz constantski, and for any variable ζi, ηi∈ℝ, i∈N¯, we assume(4)Giζi−Giηi≤kiζi−ηi,ki∈0,1.Letk=maxi∈N¯ki, ζ and η∈ℝn; the above inequality can be simply expressed as(5)Gζ−Gη≤kζ−η,k∈0,1.(A4) There is aθ>0 such that θq+1−θq≤θ, q∈ℕ.(A5)m2θ+m1θm2θ+1expm1θ<1.(A6)supt∈t0+T,t0+2Trt≥1.(A7)αexp−βT+4m2T+2m2TK+kM2exp2Tm1+m2<1.Remark 3.
For notational brevity, let(6)m1=max1≤i≤nai+Li1∑j=1nbji,m2=max1≤i≤nLi2∑j=1ncji,K=K11−ω1,K1=2kM11+m1θexpm1θ,ρ=11−ω1,ω1=m2θ+m1θ1+m2θexpm1θ,M1=supt∈θq,θq+1rt,M2=2supt∈t0,t0+2Trt,ν1=2m2T1+ρ,ν2=2m2TK,ν3=m1+m2,T=lnαβ,where m1, m2, ν1, ν2, ν3, M1, K1, K, T>0; 0<k<1; and M2≥2.


## 3. Main Results

In what follows, we aim to give Lemma1 and Theorem 1 to study the robustness of EGS of the NPRNN system.At first, we expect to simplify system (1). Actually, we may denote yjt−yˇj as rjt and yjδt−yˇj as rjδt. Then, system (1) can be rewritten as a class of NPRNN model (7) as(7)drit−Girit=−airit+∑j=1nbijφjrjt+∑j=1ncijψjrjδtdt,t≥t0≥0,rit0=ri0=yi0∈ℝ,i∈N¯,where φjrjt=fjrjt+yˇj−fjyˇj and ψjzjδt=gjrjδt+yˇj−gjyˇj. Generally, we denote φj0=0 and ψj0=0, j∈N¯. In this way, φj⋅ and ψj⋅ also satisfy (A2), which are all Lipschitzian with Lj1 and Lj2, respectively.Next, Lemma1 is given, which directly shows the relationship between the generalized PCA state rδt and the state rt at present, laying a benign foundation for the proof of the theorem later.Lemma 1.
Define thatrt is the current state of (1), rδt is the state with the generalized PCA of (1), where t∈ℝ+. If assumptions (A1)–(A5) hold, the following inequation exists:(8)rδt≤ρrt+K,where(9)ρ=1−ω1−1,ω1=m2θ+m1θ1+m2θexpm1θ,K=K11−ω1,K1=2kM11+m1θexpm1θ,m2=max1≤i≤nLi2∑j=1ncji,m1=max1≤i≤nai+Li1∑j=1nbji,M1=supt∈θq,θq+1rt.Proof.
For a generalized PCA functionδt=ϖq, define a set ϕ=t|t∈ℝ+,θq≤t<θq+1, let t∈ϕ and q∈ℕ, and then we have(10)rit−riϖq+Giriϖq−Girit≤∫ϖqt−airis+∑j=1nbijφjrjs+∑j=1ncijψjrjϖqds.
In combination with (A2), it follows that(11)rt−rϖq−Grϖq−Grt≤∑i=1n∫ϖktairis+∑j=1nLj1bijrjs+∑j=1nLj2cijrjϖqds.
Subsequently, forδt=ϖq and t∈θq,θq+1, it follows that(12)rt≤Grt−Grϖq+rϖq+∑i=1n∫ϖqtairis+∑j=1nLj1bijrjs+∑j=1nLj2cijrjϖqds≤Grt−Grδt+rϖq+Grδt−Grϖq+∑i=1n∫ϖqtairis+∑j=1nLj1bijrjs+∑j=1nLj2cijrjϖqds≤krt−rϖq+rϖq+∑i=1n∫ϖqtairis+∑j=1nLj1bijrjs+∑j=1nLj2cijrjϖqds≤krt−rϖq+rϖq+∫ϖqt∑i=1nairis+Li1∑j=1nbjiris+Li2∑j=1ncjiriϖqds≤2ksupt∈θq,θq+1rt+rϖq+∫ϖqt∑i=1nairis+Li1∑j=1nbjiris+Li2∑j=1ncjiriϖqds≤2kM1+rϖq+∫ϖqt∑i=1nai+Li1∑j=1nbjiris+∑i=1n∑j=1nLi2cjiriϖqds≤2kM1+1+∫ϖqt∑i=1nLi2∑j=1ncjiriϖqds+∫ϖqt∑i=1nai+Li1∑j=1nbjirisds≤2kM1+1+max1≤i≤nLi2∑j=1ncji∫ϖqt∑i=1nriϖqds+max1≤i≤nai+Li1∑j=1nbji×∫ϖqt∑i=1nrisds≤2kM1+1+m2θrϖq+∫ϖqtm1rsds,where M1=supt∈θq,θq+1rt. Besides, m1 and m2 are defined in Remark 3.
Applying the Gronwall–Bellman Lemma to (12), we can derive(13)rt≤1+m2θrϖq+2kM1expm1θ.
Otherwise, forθq≤t<θq+1, similarly, we get(14)rϖq≤2kM1+rt+∫ϖqt∑i=1nai+Li1bjiris+∑i=1n∑j=1nLi2cjiriϖqds≤2kM1+rt+m2θrϖq+∫ϖqtm1rsds≤2kM1+rt+m2θrϖq+m1θexpm1θ1+m2θrϖq+2kM1≤rt+m2θ+m1θexpm1θ1+m2θrϖq+2kM11+m1θexpm1θ≤rt+ω1rϖq+K1,where ω1=m2θ+m1θ1+m2θexpm1θ, K1=2kM11+m1θexpm1θ and m1, m2, and M1 are defined in Remark 3.
Consequently, by combining the above equation with similar terms, we can obtain(15)1−ω1rϖq≤rt+K1.
So, forδt=ϖq and ω1<1 for (A5), we can get(16)rδt≤1−ω1−1⋅rt+1−ω1−1⋅K1≕ρrt+K,where ρ=1/1−ω1 and K=K1/1−ω1. In this way, (8) is valid for any t≥0. And the proof is over.Remark 4.
Different from the previous recurrent neural network with piecewise arguments [37], Lemma 1 has one more term than the previous lemma in [37] due to the presence of neutral terms. It is reasonable, which shows the impact of neutral terms on PRNN.Theorem 1.
If (A1)–(A7) exist and RNN (2) achieves exponentially global stability. And if(17)k<2M1+M2−11−αexp−βTexp−2m1+m2T−4m2T,θ<θ4=minθ3,minsupθ1,θ2,whereθ3>0, θ3is written as the only solution to the following transcendental equation:(18)αexp−βT+ν1+ν2+kM2exp2ν3T=1.θ1 and θ2satisfyA5andA7, respectively; then, NPRNN (7) can be globally exponentially stable, where(19)ν1=2m2T1+ρ,ν2=2m2TK,ν3=m1+m2,M2=2supt0≤t≤t0+2Trt.
Besides,m1, m2, T, M1, ρ, ω1, K, andK1here are the same as those in Lemma 1.Proof.
For convenience, denoteyt to be shorthand for yt,t0,y0 and rt to be shorthand for rt,t0,r0, which are the states of systems (2) and (7), respectively. Thus from (2) and (7), we have(20)yit−rit+Girit−Giri0=∫t0t−aiyis−ris+∑j=1nbijfjyjs−fjrjs+∑j=1ncijgjyjs−gjrjδsds.
Utilize (A2) and norm inequation for (20), i∈N¯; that is(21)yt−rt−Grt−Gr0≤∑i=1n∫t0taiyis−ris+∑j=1nLj1bijyjs−rjs+∑j=1nLj2cijyjs−rjδsds≤∑i=1n∫t0taiyis−ris+∑j=1nLj1bijyjs−rjs+∑j=1nLj2cijyjs−rjs+rjs−rjδsds≤∫t0t∑i=1naiyis−ris+∑i=1nLi1∑j=1nbji+Li2∑j=1ncjiyis−ris+∑i=1n∑j=1nLi2cjiris−riδsds.
Based on Lemma1 and Remark 3, for t0≤t≤t0+2T, we can derive(22)yt−rt≤∫t0tmaxi∈N¯ai+Li1∑j=1nbji+Li2∑j=1ncjiys−rsds+∫t0tmaxi∈N¯Li2∑j=1ncjirs−rδsds+Grt−Gr0≤m1+m2∫t0tys−rsds+m2∫t0trδs+rsds+2ksupt0≤t≤t0+2Trt≤m1+m2∫t0tys−rsds+2m2T1+ρrt+2m2TK+kM2≤ν1rt+kM2+ν2+ν3∫t0tys−rsds,where(23)ν1=2m2T1+ρ,ν2=2m2TK,ν3=m1+m2,M2=2supt0≤t≤t0+2Trt.
Applying the Gronwall inequation and (A6), fort0+T≤t≤t0+2T, we derive(24)yt−rt≤ν2+kM2+ν1rtexp2ν3T≤ν2+kM2supt∈t0+T,t0+2Trt⋅exp2ν3T+ν1supt∈t0+T,t0+2Trt⋅exp2ν3T≤ν1+ν2+kM2exp2ν3Tsupt∈t0+T,t0+2Trt,for ν2+kM2exp2ν3T>0 in Remark 3.
Subsequently, by Definition1 and (24), for t0+T≤t≤t0+2T, we further have(25)rt≤yt−rt+yt≤ν1+ν2+kM2exp2ν3Tsupt∈t0+T,t0+2Trt+yt≤ν1+ν2+kM2exp2ν3T+αexp−βt−t0supt∈t0+T,t0+2Trt≤ν1+ν2+kM2exp2ν3T+αexp−βTsupt∈t0+T,t0+2Trt=C^supt∈t0+T,t0+2Trt,with C^=αexp−βT+ν1+ν2+kM2exp2ν3T.
Denote(26)Fρ,k=αexp−βT+ν1+ν2+kM2exp2ν3T=αexp−βT+2m2T1+ρ+2m2TK+kM2exp2m1+m2T,since Fρ,k is strictly and continuously monotonous in terms of variable ρ on the interval 1,+∞, from Theorem 1 and (A7), we have(27)F1,k=αexp−βT+4m2T+2m2TK+kM2exp2m1+m2T<1,and(28)F∞,k>1.
Hence, on the one hand, there is a uniqueρ˜∈1,+∞ which makes Fρ˜,k=1 true. From assumption (A5), let(29)Γθ=m2θ+m1θ1+m2θexpm1θ<1,such that θ1θ1>0 satisfy inequation (29), and surely θ1 satisfy (A5). Obviously,(30)ρθ=11−Γθ∈1,+∞holds towards any θ∈0,θ1. Hence, there is a unique θ˜∈0,θ1, which makes it possible for ρ=ρ˜. Then, if we assume that θ2 satisfies (27), θ3 is the unique positive solution of (26).
Let(31)θ4=minθ3,minsupθ1,θ2,by virtue of the monotonicity of F⋅ and ρ⋅, we know that 0<Fρ,k=C^<1 holds when 0<θ<θ4, and we obtain the interval length of PCA so far.
On the other hand, surely, we know that there is somek˜<k such that F1,k˜<1 from (27). Let θ=0 in (27); then the bound to k can be described as(32)k<2M1+M2−11−αexp−βTexp−2m1+m2T−4m2T,
Denoteξ=−lnC^/T; according to (25), we can obtain(33)supt0+T≤t≤t0+2Trt≤exp−ξTsupt0≤t≤t0+Trt.
Consequently, for any positive integerl∈ℤ+, when t≥t0+l−1T, by the uniqueness of solution [27], we can express the solutions of the NPRNN in the following flow modality:(34)rt≡rt;t0,r0=rt;t0+l−1T,rt0+l−1T;t0,r0.
Taking (33) and (34) into account, it follows that(35)supt0+lT≤t≤t0+l+1Trt=supt0+l−1T+T≤t≤t0+l−1T+2Trt;t0+l−1T,rt0+l−1T;t0,r0≤exp−ξTsupt0+l−1T≤t≤t0+lTrt≤exp−lξTsupt∈t0,t0+Trt=Cexp−lξT,where C=supt∈t0,t0+Trt. Hence, in the time interval t0+lT,t0+l+1T, l∈ℤ+, we obtain(36)zt≤Cexp−lξT≤Cexpξ−t+t0+T≤CexpξTexp−ξt−t0.
In this way, the NPRNN model (7) achieves exponentially global stability.Remark 5.
Few studies have been finished to comprehensively consider the influence of robustness and exponentially global stability on NPRNN model by giving the upper bounds of PCAs and NTs. In this paper, from Lemma1, by the method of norm scaling inequality technique, the relationship between the generalized PCA state rδt and the state rt at present is clarified. From Theorem 1, by utilizing Gronwall Lemma, norm scaling inequality technique, and the scientific method of controlling variables, the interval length of PCAs and the maximum scope of NT compression coefficient of NPRNN (7) are derived intuitively.Remark 6.
The interval length of PCAs and the maximum scope of NT compression coefficient can be estimated by software tool MATLAB. The order of calculation is as follows: firstly, we choose suitableT=lnα/β>0. Furthermore, since α, β, Li1 and Li2 are known and other parameters can be computed in terms of Remark 3, i∈N¯. Thus, the bound of k can be calculated easily by (32). Additionally, we select the appropriate k and substitute it into transcendental equation (18) with other parameters together so that the length of the argument interval can be acquired. Finally, a comprehensive selection of values that satisfy both k and θ inequality conditions in Theorem 1 can ensure the exponential stability of NPRNN (7).


## 4. Numerical Examples

For the sake of exploring the robustness of the EGS of NPRNN, we derive two important indicators: the supremum of the interval length of PCAθ and the feasible upper range of NT compression coefficient k in this paper. Thus, there are two elucidative examples to testify the efficiency for the criteria in Theorem 1.Example 1.
The following bistate RNN is considered:(37)y˙1ty˙2t=−200−1y1ty2t+0.020.010.010.02siny1t20siny2t20.
By calculation, we easily get that RNN (37) can achieve exponentially global stability when we set suitable parameters α=1, β=0.5; one can see Figures 1 and 2.
Next, in the presence of the neutral terms and piecewise constant arguments, (37) becomes(38)ddtr1t−ksinr1tr2t−ksinr2t=−200−1r1tr2t+0.020.010.010.02sinr1t20sinr2t20+−0.0010.0020.002−0.001sinr1δtsinr2δt.
Fix two consequences:θq=q/9, ϖq=2q+1/18, q∈ℕ. Define activation functions fjrjt and gjrjt here as sinrjt/20 and sinrjδt, respectively, so we have L11=L21=1/20, L12=L22=1, where j=1,2. Then by calculation, we can obtain m1=2.0015, m2=0.001, ν3=2.0025.
According to Theorem1 and (32), let T=0.3>lnα/β=0, and then we have(39)k<0.25×1−exp−0.15×exp−1.2015−0.0012=0.0102.
From Theorem1, if we set(40)f1θ=m2θ+m1θ1+m2θexpm1θ=1,the unique positive solution of (40) θ1˜=0.2833 is the supremum of (A5). Solving (41)f2θ=αexp−βT+4m2T+2m2TK+kM2exp2m1+m2T=1,the unique positive solution of (41) θ2˜=0.2832 is the supremum of (A7). So we get(42)minθ1˜,θ2˜=0.2832.
Letθ3θ3>0 be the supremum of equation (18); then we can obtain θ3=0.2829 by MATLAB. So(43)θ<θ4=minθ3,minθ1˜,θ2˜=min0.2829,0.2832=0.2829.
Due to the monotonicity off1θ and f2θ in (40) and (41), it is easy to get that (A5) and (A7) hold when θ=1/9, k=0.01.
So, in the case that if the real selected values are all lower than the derived values, that is, the criteria in Theorem1 and (A1)–(A7) are all satisfied, NPRNN (7) will be exponentially globally stable if we take θ=1/9=0.1111<θ4=0.2829 and k=0.01<0.0102. Then the simulations of the NPRNN model with the interval length of the argument θ=1/9 and the bound to neutral term compression coefficient k=0.01 of state r1t and r2t are given in Figures 3 and 4. That is to say, if the interval length of PCA θ and the neutral term compression coefficient k are all lower than the results given in Theorem 1, transient behavior will remain exponentially stable.Figure 1
Stable behavior ofy1t of RNN (37).Figure 2
Stable behavior ofy2t of RNN (37).Figure 3
The EGS ofr1t with θ=1/9, k=0.01 in (38).Figure 4
The EGS ofr2t with θ=1/9, k=0.01 in (38).Example 2.
The following one-dimensional dynamical system is considered:(44)y˙t=−3.1yt+0.1fyt.
Letf⋅=tanh⋅, according to many existing criteria, the neurodynamic system (44) can be stable when we take α=1, β=3. Figure 5 illustrates the convergence stability of RNN (44).
In what follows, if we add neutral terms and piecewise constant arguments to system (44), we can get the NPRNN (45) as(45)ddtrt−ksinrt=−3.1rt+0.088tanhrt+0.002tanhrδt.
Similarly, if we fix two consequencesθq=q/10 and ϖq=2q+1/20, according to the characteristics of (45), we can easily get L11=L12=L21=L22=1. Then the parameters that we need by easy calculation are listed as m1=3.188, m2=0.002, ν3=3.19. By Theorem 1, let T=0.25>lnα/β=0. Substituting them into (32), we get(46)k<0.25×1−exp−0.75×exp−1.595−0.002=0.0263.
Suppose thatθ1 and θ2 satisfy A5 and A7, respectively. Put the parameters of Remark 3 into A5 and A7, respectively, and we can get (47)minsupθ1,θ2=0.1778.
Next, we can getθ3=0.1774 in Theorem 1 by using MATLAB when other essential parameters are provided; then we obtain that(48)θ4=minθ3,minsupθ1,θ2=0.1774.
Moreover, since the increasing monotonicity of exponential function (26), the solution θ4=0.1774 also satisfies assumptions A5 and A7, respectively. Hence, the conditions in the assumptions and theorem are all satisfied. So if we take k=0.01, θ=0.1, the convergence stability of rt of NPRNN system (45) can be seen in Figure 6.
Furthermore, several unstable cases are given to illustrate.
In Figure7, where θ=0.1, k=1.02, which are not suitable for conditions in Theorem 1, the state trajectories of NPRNN (45) are unstable.
In Figure8, where k=0.01, θ=0.5, which are not fit for the conditions in Theorem 1, the state trajectories of NPRNN (45) are unstable.
In Figure9, where k=1.02, θ=0.5, which are unsuitable with the conditions in Theorem 1, the state trajectories of NPRNN (45) are unstable.Figure 5
Stable trajectory ofyt in (44).Figure 6
The EGS ofrt with k=0.01 and θ=0.1 in (45).Figure 7
Unstable trajectory ofrt of (45) with θ=0.1, k=1.02.Figure 8
Unstable trajectory ofrt of (45) with k=0.01, θ=0.5.Figure 9
Unstable trajectory ofrt of (45) with k=1.02, θ=0.5.


## 5. Conclusion

Throughout the paper, we mainly analysed the robustness of EGS of NPRNN. Firstly, we provided a leading Lemma1 to clarify the relationship between rδt and rt. Secondly, we respectively estimated the maximum interval length of the PCAs and the scope of NT compression coefficient that the disturbed NPRNN can tolerate to remain stable. Additionally, we theoretically testified that if the bounds of the interval length of PCAs and NT compression coefficient are all lower than the upper bounds derived herein, NPRNN will still remain stable. In the end, two examples were provided to show that if the real selected values are all lower than the derived values, the system will hold steady. Conversely, as long as one of the two selected values does not satisfy the theoretical result deduced, the system would be destabilized. In other words, we have successfully verified the robustness of the NPRNN model proposed herein. At the same time, we can see from the figures that the trajectory of NPRNN decays exponentially to stability; that is, we also have achieved the robustness of EGS of NPRNN. The criteria provided here are meaningful, which can provide more possibilities for future design and application of NPRNN or any other differential systems with neutral terms and piecewise constant arguments. For instance, future investigations may aim to consider the complex-valued type or fractional-order type of NPRNN with other disturbed factors.



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*Source: 9941881-2021-07-23.xml*