Combined Convex Technique on Delay-Distribution-Dependent Stability for Delayed Neural Networks

Abstract

Together with the Lyapunov-Krasovskii functional approach and an improved delay-partitioning idea, one novel sufficient condition is derived to guarantee a class of delayed neural networks to be asymptotically stable in the mean-square sense, in which the probabilistic variable delay and both of delay variation limits can be measured. Through combining the reciprocal convex technique and convex technique one, the criterion is presented via LMIs and its solvability heavily depends on the sizes of both time-delay range and its variations, which can become much less conservative than those present ones by thinning the delay intervals. Finally, it can be demonstrated by four numerical examples that our idea reduces the conservatism more effectively than some earlier reported ones.

1. Introduction

In past decades, neural networks have been applied to various signal processing problems, such as optimization, image processing, associative memory design, and other engineering fields. In those applications, the key feature of the designed neural network is to be globally stable. Meanwhile, since there inevitably exists communication delay which can induce the oscillation and instability in various dynamical systems, great efforts have been made to analyze the dynamics of delayed systems including delayed neural networks (DNNs) and many elegant results have been reported; see [1–35]. In practical applications, though it is difficult to describe the form of time-delay precisely, the bounds of time-delay and its variation rates still can be measured. Since the Lyapunov functional approach imposes no restriction on delay variation and presents some simple stability criteria, the Lyapunov-Krasovskii functional (LKF) has been widely utilized due to that its analysis can fully make use of the information on time-delay of DNNs as much as possible. Thus recently, the delay-dependent stability has become an important topic of primary significance, in which the main purpose is to derive an allowable delay upper bound guaranteeing the global stability for addressed DNNs in [4–9, 16, 17, 21–35]. Furthermore, in recent years, since the delay-partitioning ideas have been proven to be more effective in reducing the conservatism than some previously reported techniques and received much research attention in [21–27], yet these convex ideas above still need some further improvements since they cannot effectively tackle interval variable delay or cannot fully utilize every delay subintervals, which have been fully addressed in [28].

Meanwhile, it can be seen from many existing references that only the deterministic time-delay case was concerned, and the stability criteria were derived based on the information of delay and its variation range. Actually, time-delay in some DNNs is often existent in a stochastic fashion. In practice, to propagate and control the stochastic signals through universal learning networks, a probabilistic universal leaning network (PULN) was proposed. In a PULN, the output signal of the node is transferred to another node by multibranches with arbitrary time-delays which are random and its probability often can be measured by the statistical methods. For this case, if some values of time-delay are very large but its probability of the delay taking such large values is very small, it may lead to a more conservative result if only the information of delay variation range is considered. Thus, recently, some researchers have considered the stability for various systems including DNNs with probability-distributed delays [29–39]. In [36–39], the authors have analyzed the stability and its applications for networked control systems, uncertain linear systems, and T-S fuzzy systems, in which probability delay has been fully considered. As for discrete-time DNNs with probabilistic delay, the global stability has been considered and some pretty results have been proposed in [29–33]. Yet, it has come to our attention that, though some works have studied the dynamics of continuous-time DNNs with probabilistic delay [34, 35], the lower limits of delay variation have not been considered and, in fact, such available information could play an important role in extending the results' application area, which has been illustrated in [40] and yet not taking delay distribution probability into consideration. Presently, as for time-variable delay, the reciprocal convex approach in [41] has been proven to be more effective in reducing the conservatism than some earlier convex techniques [28, 42]. Yet to the authors' best knowledge, few authors have used the combination of the reciprocal convex technique and general convex ones to tackle the global stability for DNNs with probabilistic time-varying delay, which constitutes the main focus of this present work.

Together with taking both bounds on probabilistic time-delay and its time variations into consideration, we make some great efforts to investigate the mean-squared stability for DNNs, in which an improved delay-partitioning idea is utilized and a novel Lyapunov functional is chosen. Through combining the reciprocal convex technique and the general convex one, one less conservative condition is given in terms of LMIs, which can present the pretty delay dependence and computational efficiency. Finally, we give four numerical examples to illustrate that our derived results can be less conservative than some existent ones.

The notations in the paper are really standard. For symmetric matrices 𝑋,𝑌,𝑋>𝑌 (resp., 𝑋≥𝑌) means that 𝑋−𝑌>0(𝑋−𝑌≥0) is a positive-definite (resp., positive-semidefinite) matrix; and ∗ denotes the symmetric term in a symmetric matrix, that is,   _𝑌^𝑋𝑌_𝑇𝑍=^𝑋𝑌_∗𝑍.

2. Problem Formulations and Preliminaries

Consider the delayed neural networks as follows: 𝑧̇(𝑡)=−𝐶𝑧(𝑡)+𝐴𝑔(𝑧(𝑡))+𝐵𝑔(𝑧(𝑡−𝜏(𝑡)))+I, (2.1) where 𝑧=[𝑧₁,…,𝑧_𝑛]^𝑇∈𝐑^𝑛 is a real 𝑛-vector denoting the state variables associated with the neurons, 𝑔(𝑧)=[𝑔₁(𝑧₁),…,𝑔_𝑛(𝑧_𝑛)]^𝑇 represents the neuron activation function, I=[I₁,…,I_𝑛]^𝑇∈𝐑^𝑛 is a constant input vector, 𝐶=diag{𝑐₁,…,𝑐_𝑛}>0, and 𝐴,𝐵 are the appropriately dimensional constant matrices.

The following assumptions on the system (2.1) are made throughout this paper.

Assumption 2.1. The time-varying delay 𝜏(𝑡) satisfies 0≤𝜏₁≤𝜏(𝑡)≤𝜏₃. Moreover, consider the information of probability distribution of 𝜏(𝑡), two sets and functions are defined as Ω₁={𝑡,𝜏(𝑡)∈[𝜏₁,𝜏₂)}, Ω₂={𝑡,𝜏(𝑡)∈[𝜏₂,𝜏₃]}, and 𝜏₁(𝑡)=𝜏(𝑡),for𝑡∈Ω₁,𝜏₁,for𝑡∈Ω₂,𝜏₂(𝑡)=𝜏(𝑡),for𝑡∈Ω₂,𝜏₂,for𝑡∈Ω₁,𝜇₁≤𝜏̇₁(𝑡)≤𝜇₂,𝜇₃≤𝜏̇₂(𝑡)≤𝜇₄, (2.2)(2.3) where 𝜏₂∈[𝜏₁,𝜏₃], 𝜏₁∈[𝜏₁,𝜏₂), and 𝜏₂∈[𝜏₂,𝜏₃]. It is easy to check that 𝑡∈Ω₁ means that the event 𝜏(𝑡)∈[𝜏₁,𝜏₂) occurs and 𝑡∈Ω₂ means that the event 𝜏(𝑡)∈[𝜏₂,𝜏₃] occurs. Therefore, a stochastic variable 𝛿(𝑡) can be defined as 𝛿(𝑡)=1,for𝑡∈Ω₁,0,for𝑡∈Ω₂. (2.4)

Assumption 2.2. 𝛿(𝑡) is a Bernoulli distributed sequence with Prob{𝛿(𝑡)=1}=𝐄{𝛿(𝐭)}=𝛿₀,Prob{𝛿(𝑡)=0}=1−𝐄{𝛿(𝐭)}=1−𝛿₀, (2.5) where 0≤𝛿₀≤1 is a constant and 𝐄{𝛿(𝐭)} is the mathematical expectation of 𝛿(𝑡). It is easy to check that 𝐄{𝛿(𝐭)−𝛿₀}=0.

Assumption 2.3. For the constants 𝜌⁺_𝑗,𝜌⁻_𝑗, the nonlinear function 𝑔_𝑗(⋅) in (2.1) satisfies the following condition: 𝜌⁻_𝑗≤𝑔_𝑗(𝛼)−𝑔_𝑗𝛽𝛼−𝛽≤𝜌⁺_𝑗,∀𝛼,𝛽∈𝐑,𝛼≠𝛽,𝑗=1,2,…,𝑛. (2.6) Here, we denote Σ=diag{𝜌⁺₁,…,𝜌⁺_𝑛}, Σ=diag{𝜌⁻₁,…,𝜌⁻_𝑛}, and Σ₁=diag𝜌⁺₁𝜌⁻₁,…,𝜌⁺_𝑛𝜌⁻_𝑛,Σ₂𝜌=diag⁺¹+𝜌⁻₁2𝜌,…,^+𝑛+𝜌⁻_𝑛2. (2.7) It is clear that under Assumptions 2.1–2.3, system (2.1) has one equilibrium point 𝑧^∗=[𝑧^∗₁,…,𝑧^∗_𝑛]^𝑇. For convenience, we shift the equilibrium point 𝑧^∗ to the origin by letting 𝑥=𝑧−𝑧^∗, 𝑓(𝑥)=𝑔(𝑥+𝑧^∗)−𝑔(𝑧^∗), and the system (2.1) can be converted to 𝑥̇(𝑡)=−𝐶𝑥(𝑡)+𝐴𝑓(𝑥(𝑡))+𝐵𝑓(𝑥(𝑡−𝜏(𝑡))), (2.8) where 𝑓(𝑥)=[𝑓₁(𝑥₁),…,𝑓_𝑛(𝑥_𝑛)]^𝑇. Based on the methods in [37–39], the system above can be equivalently converted to 𝑥̇(𝑡)=−𝐶𝑥(𝑡)+𝐴𝑓(𝑥(𝑡))+𝛿(𝑡)𝐵𝑓𝑥𝑡−𝜏₁(𝑡)+(1−𝛿(𝑡))𝐵𝑓𝑥𝑡−𝜏₂(𝑡). (2.9) It is easy to check that the function 𝑓_𝑗(⋅) satisfies 𝑓_𝑗(0)=0, and 𝜌⁻_𝑗≤𝑓_𝑗(𝛼)𝛼≤𝜌⁺_𝑗,∀𝛼∈𝐑,𝛼≠0,𝑗=1,2,…,𝑛. (2.10) Then, the problem to be addressed in the paper can be formulated as developing a condition ensuring that the system (2.9) is asymptotically stable.
In order to obtain the stability criterion for system (2.9), the following lemmas are introduced.

Lemma 2.4 (see [27]). For any constant matrix 𝑋∈𝐑^𝑛×𝑛, 𝑋=𝑋^𝑇≥0, a scalar functional ℎ∶=ℎ(𝑡)≥0, and a vector function 𝑥̇∶[−ℎ,0]→𝐑^𝑛 such that the following integration is well defined, then −ℎ∫^ℎ_𝑡−ℎ𝑥̇^𝑇(𝑠)𝑋𝑥̇(𝑠)𝑑𝑠≤[𝑥(𝑡)−𝑥(𝑡−ℎ)]^𝑇𝑋[𝑥(𝑡)−𝑥(𝑡−ℎ)].

Lemma 2.5 (see [41]). Let the functions 𝑓₁(𝑡),𝑓₂(𝑡),…,𝑓_𝑁(𝑡)∶𝐑^𝑚→𝐑 have the positive values in an open subset 𝐃 of 𝐑^𝑚 and satisfy (1/𝛼₁)𝑓₁(𝑡)+(1/𝛼₂)𝑓₂(𝑡)+⋯+(1/𝛼_𝑁)𝑓_𝑁(𝑡)∶𝐃→𝐑 with 𝛼_𝑖>0 and ∑^𝑁_𝑖=1𝛼_𝑖=1, then the reciprocal convex technique of 𝑓_𝑖(𝑡) over the set 𝐃 satisfies _𝑖1𝛼_𝑖𝑓_𝑖(𝑡)≥_𝑖𝑓_𝑖(𝑡)+_𝑖≠𝑗𝑔_𝑖,𝑗(𝑡),∀𝑔_𝑖,𝑗(𝑡)∶𝐑^𝑚𝑓→𝐑,^𝑖(𝑡)𝑔_𝑖,𝑗(𝑡)𝑔^𝑇_𝑖,𝑗(𝑡)𝑓_𝑗(𝑡)≥0. (2.11)

3. Delay-Distribution-Dependent Stability

Firstly, we can rewrite the system (2.9) as 𝑥̇(𝑡)=−𝐶𝑥(𝑡)+𝐴𝑓(𝑥(𝑡))+𝛿₀𝐵𝑓𝑥𝑡−𝜏₁(𝑡)+1−𝛿₀𝐵𝑓𝑥𝑡−𝜏₂(𝑡)+𝛿(𝑡)−𝛿₀×𝐵𝑓𝑥𝑡−𝜏₁(𝑡)−𝑓𝑥𝑡−𝜏₂(𝑡). (3.1) Now letting 𝑙,𝑚 be positive integers, we, respectively, divide the delay intervals [𝜏₁,𝜏₂] and [𝜏₂,𝜏₃] into 𝑙,𝑚 segments averagely. Moreover, we introduce the following denotations: 𝛿₁=𝜏₂−𝜏₁𝑙,𝛿₂=𝜏₃−𝜏₂𝑚,𝜌₁(𝑡)=𝜏₁(𝑡)−𝜏₁𝑙,𝜌₂(𝑡)=𝜏₂(𝑡)−𝜏₂𝑚. (3.2) Then, based on (2.10) and (3.2), we can construct the following Lyapunov-Krasovskii functional candidate: 𝑉𝑥_𝑡=𝑉₁𝑥_𝑡+𝑉₂𝑥_𝑡+𝑉₃𝑥_𝑡, (3.3) where 𝑉₁𝑥_𝑡=𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+⁰_−𝜏1^𝑡_𝑡+𝜃𝜏₁𝑥̇^𝑇(𝑠)𝑄𝑥̇(𝑠)𝑑𝑠𝑑𝜃+2^𝑛_𝑖=1𝑘_𝑖^𝑥𝑖₀𝑓_𝑖(𝑠)−𝜌⁻_𝑖𝑠𝑑𝑠+2^𝑛_𝑖=1𝑙_𝑖^𝑥𝑖₀𝜌⁺_𝑖𝑠−𝑓_𝑖(𝑠𝑉)𝑑𝑠,₂𝑥_𝑡=^𝑡_𝑡−𝜏1𝑥(𝑠)𝑓(𝑥(𝑠))^𝑇𝑃₁𝐻₁∗𝑄₁𝑥(𝑠)𝑓(𝑥(𝑠+))𝑑𝑠_{2𝑖=1𝑡−𝜏𝑖𝑡−𝜏𝑖−𝛿𝑖}𝜎_𝑖(𝑠)ℎ_𝑖𝜎_𝑖(𝑠)^𝑇𝑃_𝑖𝐻_𝑖∗𝑄_𝑖𝜎_𝑖(𝑠)ℎ_𝑖𝜎_𝑖(𝑠)+𝑑𝑠_{𝑙𝑖=1𝑡−𝜏1−(𝑖−1)𝛿1𝑡−𝜏1−()𝛿𝑖−11−𝜌1(𝑡)}𝑥(𝑠)𝑓(𝑥(𝑠))^𝑇𝑋_1𝑖𝑌_1𝑖∗𝑍_1𝑖𝑥(𝑠)𝑓(𝑥(𝑠+))𝑑𝑠_{𝑙𝑖=1𝑡−𝜏1−(𝑖−1)𝛿1−𝜌1(𝑡)𝑡−𝜏1−𝑖𝛿1}𝑥(𝑠)𝑓(𝑥(𝑠))^𝑇𝑋_2𝑖𝑌_2𝑖∗𝑍_2𝑖𝑥(𝑠)𝑓(𝑥(𝑠+))𝑑𝑠_{𝑚𝑖=1𝑡−𝜏2−(𝑖−1)𝛿2𝑡−𝜏2−()𝛿𝑖−12−𝜌2(𝑡)}𝑥(𝑠)𝑓(𝑥(𝑠))^𝑇𝑋_3𝑖𝑌_3𝑖∗𝑍_3𝑖𝑥(𝑠)𝑓(𝑥(𝑠+))𝑑𝑠_{𝑚𝑖=1𝑡−𝜏2−(𝑖−1)𝛿2−𝜌2(𝑡)𝑡−𝜏2−𝑖𝛿2}𝑥(𝑠)𝑓(𝑥(𝑠))^𝑇𝑋_4𝑖𝑌_4𝑖∗𝑍_4𝑖𝑥(𝑠)𝑓(𝑥(𝑠𝑉))𝑑𝑠,₃𝑥_𝑡=^𝑙_𝑖=1^{−𝜏1−(𝑖−1)𝛿1}_{−𝜏1−𝑖𝛿1}^𝑡_𝑡+𝜃𝛿₁𝑥̇^𝑇(𝑠)𝑊_1𝑖𝑥̇(𝑠)+𝑑𝑠𝑑𝜃_{𝑚𝑖=1−𝜏2−(𝑖−1)𝛿2−𝜏2−𝑖𝛿2𝑡𝑡+𝜃}𝛿₂𝑥̇^𝑇(𝑠)𝑊_2𝑖𝑥̇(𝑠)𝑑𝑠𝑑𝜃, (3.4) with 𝐾=diag{𝑘₁,…,𝑘_𝑛},𝐿=diag{𝑙₁,…,𝑙_𝑛}, 𝑛×𝑛 constant matrices 𝑃,𝑄,𝑃₁,𝐻₁,𝑄₁,𝑋_𝑗𝑖,𝑌_𝑗𝑖,𝑍_𝑗𝑖(𝑗=1,2,3,4), 𝑊_𝑗𝑖(𝑗=1,2), 𝑙𝑛×𝑙𝑛 constant matrices 𝑃₁,𝑄₁,𝐻₁, 𝑚𝑛×𝑚𝑛 constant matrices 𝑃₂,𝑄₂,𝐻₂, and 𝜎^𝑇₁(𝑠)=𝑥^𝑇(𝑠)𝑥^𝑇𝑠−𝛿₁⋯𝑥^𝑇𝑠−(𝑙−1)𝛿₁,ℎ^𝑇₁𝜎₁(𝑠)=𝑓^𝑇(𝑥(𝑠))𝑓^𝑇𝑥𝑠−𝛿₁⋯𝑓^𝑇𝑥𝑠−(𝑙−1)𝛿₁,𝜎^𝑇₂(𝑠)=𝑥^𝑇(𝑠)𝑥^𝑇𝑠−𝛿₂⋯𝑥^𝑇𝑠−(𝑚−1)𝛿₂,ℎ^𝑇₂𝜎₂(𝑠)=𝑓^𝑇(𝑥(𝑠))𝑓^𝑇𝑥𝑠−𝛿₂⋯𝑓^𝑇𝑥𝑠−(𝑚−1)𝛿₂. (3.5) Denoting a parameter set Φ={𝑃,𝑄,𝐾,𝐿,𝑊_1𝑖,𝑊_2ℎ,^𝑃1𝐻1_∗𝑄1,𝑃_𝑗𝐻_𝑗∗𝑄_𝑗,^{𝑋𝑗𝑖𝑌𝑗𝑖}_{∗𝑍𝑗𝑖},^{𝑋𝑘ℎ𝑌𝑘ℎ}_{∗𝑍𝑘ℎ}, 𝑗=1,2,𝑖=1,…,𝑙;𝑘=3,4,ℎ=1,…,𝑚}, then we give one proposition which is essential in the following deduction.

Proposition 3.1. If the parameter set Φ satisfies the following condition: 𝑃>0,𝑄>0,𝐾>0,𝐿>0,𝑊_1𝑖>0,𝑊_2ℎ𝑃>0,₁𝐻₁∗𝑄₁𝑃>0,^𝑗𝐻_𝑗∗𝑄_𝑗𝑋>0,^𝑗𝑖𝑌_𝑗𝑖∗𝑍_𝑗𝑖𝑋>0,_𝑘ℎ𝑌_𝑘ℎ∗𝑍_𝑘ℎ>0,𝑗=1,2,𝑘=3,4;𝑖=1,…,𝑙;ℎ=1,…,𝑚, (3.6) then the Lyapunov-Krasovskii functional (3.3) is definitely positive.

Moreover, in order to simplify the subsequent proof, we also give some notations in the following: 𝜇₁=𝜇₂−𝜇₁,𝜇₃=𝜇₄−𝜇₃,Σ_𝑖=diagΣ_𝑖,…,Σ_{𝑖𝑙𝑛×𝑙𝑛},Σ_𝑖=diagΣ_𝑖,…,Σ_{𝑖𝑚𝑛×𝑚𝑛},𝑋_𝑖=diag𝑋_𝑖1,…,𝑋_𝑖𝑙,𝑌_𝑖=diag𝑌_𝑖1,…,𝑌_𝑖𝑙,𝑍_𝑖=diag𝑍_𝑖1,…,𝑍_𝑖𝑙𝑆,𝑖=1,2,₁=diag𝑆₁₁,…,𝑆_1𝑙,𝑊₁=diag𝑊₁₁,…,𝑊_1𝑙,𝑉₁=diag𝑉₁₁,…,𝑉_1𝑙,𝑅₁=diag𝑅₁₁,…,𝑅_1𝑙,𝑋_𝑗=diag𝑋_𝑗1,…,𝑋_𝑗𝑚,𝑌_𝑗=diag𝑌_𝑗1,…,𝑌_𝑗𝑚,𝑍_𝑗=diag𝑍_𝑗1,…,𝑍_𝑗𝑚𝑆,𝑗=3,4,₂=diag𝑆₂₁,…,𝑆_2𝑚,𝑊₂=diag𝑊₂₁,…,𝑊_2𝑚,𝑉₂=diag𝑉₂₁,…,𝑉_2𝑚,𝑅₂=diag𝑅₂₁,…,𝑅_2𝑚. (3.7)(3.8)

Theorem 3.2. For two given positive integers 𝑙,𝑚, and time-delay satisfying (2.2), the delayed neural network (3.1) is globally asymptotically stable in the mean square sense, if there exists one parameter set Φ satisfying Proposition 3.1, 𝑛×𝑛 matrices 𝐸_𝑖(𝑖=1,2), and 𝑛×𝑛 diagonal matrices 𝑈_𝑖>0(𝑖=1,2,3,4,5),𝑉_1𝑖>0,𝑅_1𝑖>0,𝑆_1𝑖(𝑖=1,…,𝑙) making ^{𝑊1𝑖𝑆1𝑖}_∗𝑊1𝑖≥0, 𝑉_2𝑗>0,𝑅_2𝑗>0,𝑆_2𝑖(𝑗=1,…,𝑚) making ^{𝑊2𝑖𝑆2𝑖}_∗𝑊2𝑖≥0 such that the following LMIs in (3.9) hold: Υ₁ΘΥ^𝑇₁+Υ₂ΞΥ^𝑇₂+𝜇₁𝑙^𝑙_𝑖=1I^𝑇_1𝑖𝑋_𝑒𝑖𝑌_𝑒𝑖∗𝑍_𝑒𝑖I_1𝑖+𝜇₃𝑚^𝑚_𝑖=1I^𝑇_2𝑖𝑋_𝑓𝑖𝑌_𝑓𝑖∗𝑍_𝑓𝑖I_2𝑖<0,𝑒=1,2;𝑓=3,4, (3.9) where I_1𝑖=0_{𝑛⋅(𝑚+𝑙+1+𝑖)𝑛}−𝐼_𝑛0_{𝑛⋅(2𝑙+2𝑚+1)𝑛}−𝐼_𝑛∗0_{𝑛⋅(𝑚+𝑙+1+𝑖)𝑛}𝐼_𝑛0_{𝑛⋅(2𝑙+2𝑚+1)𝑛}𝐼_𝑛∗,I_2𝑖=0_{𝑛⋅(𝑚+2𝑙+1+𝑖)𝑛}−𝐼_𝑛0_{𝑛⋅(2𝑙+2𝑚+1)𝑛}−𝐼_𝑛∗0_{𝑛⋅(𝑚+2𝑙+1+𝑖)𝑛}𝐼_𝑛0_{𝑛⋅(2𝑙+2𝑚+1)𝑛}𝐼_𝑛∗,ΘΘ=¹¹𝑄00Θ₁₅000Θ₁₉00Θ_1,12Θ_1,13∗Θ₂₂000−𝐻₁0000000∗∗Θ₃₃000𝑈₂Σ₂000000∗∗∗Θ₄₄000𝑈₃Σ₂00000∗∗∗∗Θ₅₅000Θ₅₉0000∗∗∗∗∗−𝑄₁0000000∗∗∗∗∗∗−𝑈₂000000∗∗∗∗∗∗∗−𝑈₃0000∗∗∗∗∗∗∗∗Θ₉₉00Θ_9,12Θ_9,13∗∗∗∗∗∗∗∗∗−𝑈₄Σ₁0𝑈₄Σ₂0∗∗∗∗∗∗∗∗∗∗−𝑈₅Σ₁0𝑈₅Σ₂∗∗∗∗∗∗∗∗∗∗∗−𝑈₄0∗∗∗∗∗∗∗∗∗∗∗∗−𝑈₅,Ξ=Ξ₁Ξ₂∗Ξ₃,Ξ₁=Ξ₁₁000𝑆₁0∗Ξ₂₂00𝑆₁0∗∗Ξ₃₃00𝑆₂∗∗∗Ξ₄₄0𝑆₂∗∗∗∗Ξ₅₅0∗∗∗∗∗Ξ₆₆,Ξ₂=diagΞ₁₇,Ξ₂₈,Ξ₃₉,Ξ_4,10,Ξ_5,11,Ξ_6,12,Ξ₃=diagΞ₇₇,Ξ₈₈,Ξ₉₉,Ξ_10,10,Ξ_11,11,Ξ_12,12,Υ₁=𝐼_𝑛∗0_𝑛⋅𝑛𝐼_𝑛∗0_{𝑛⋅(𝑙+1)𝑛}𝐼_𝑛∗0_{𝑛⋅(𝑚+𝑙+1)𝑛}𝐼_𝑛∗0_{𝑛⋅2(𝑚+𝑙+1)𝑛}𝐼_𝑛∗0_{𝑛⋅(2𝑚+2𝑙+3)𝑛}𝐼_𝑛∗0_{𝑛⋅(2𝑚+3𝑙+3)𝑛}𝐼_𝑛∗0_{𝑛⋅3(𝑚+𝑙+1)𝑛}𝐼_𝑛∗0_{𝑛⋅4(𝑚+𝑙+1)𝑛}𝐼_𝑛∗0_{𝑛⋅(4𝑚+4𝑙+5)𝑛}𝐼_𝑛∗0_{𝑛⋅(4𝑚+4𝑙+6)𝑛}𝐼_𝑛∗0_{𝑛⋅(4𝑚+4𝑙+7)𝑛}𝐼_𝑛∗0_{𝑛⋅(4𝑚+4𝑙+8)𝑛}𝐼_𝑛,Υ₂=0_{𝑙𝑛⋅𝑛}𝐼_𝑙𝑛∗0_{𝑙𝑛⋅2𝑛}𝐼_𝑙𝑛∗0_{𝑚𝑛⋅(𝑙+1)𝑛}𝐼_𝑚𝑛∗0_{𝑚𝑛⋅(𝑙+2)𝑛}𝐼_𝑚𝑛∗0_{𝑙𝑛⋅(𝑚+𝑙+2)𝑛}𝐼_𝑙𝑛∗0_{𝑚𝑛⋅(𝑚+2𝑙+2)𝑛}𝐼_𝑚𝑛∗0_{𝑙𝑛⋅(2𝑚+2𝑙+3)𝑛}𝐼_𝑙𝑛∗0_{𝑙𝑛⋅(2𝑚+2𝑙+4)𝑛}𝐼_𝑙𝑛∗0_{𝑚𝑛⋅(2𝑚+3𝑙+3)𝑛}𝐼_𝑚𝑛∗0_{𝑚𝑛⋅(2𝑚+3𝑙+4)𝑛}𝐼_𝑚𝑛∗0_{𝑙𝑛⋅(3𝑚+3𝑙+4)𝑛}𝐼_𝑙𝑛∗0_{𝑚𝑛⋅(3𝑚+4𝑙+4)𝑛}𝐼_𝑚𝑛∗, (3.10) with the notations ∗ in 𝐼_𝑗𝑖(𝑗=1,…,6) and Υ_𝑔(𝑔=1,2) denoting the appropriately dimensional zero matrices making them (4𝑙+4𝑚+9)𝑛 columns: Θ₁₁=−𝐸^𝑇₁𝐶−𝐶^𝑇𝐸₁+𝑃₁−𝑄−𝑈₁Σ₁,Θ₁₅=𝐸^𝑇₁𝐴+𝐻₁+𝑈₁Σ₂,Θ₁₉=𝑃−Σ𝐾+Σ𝐿−𝐸^𝑇₁−𝐶^𝑇𝐸₂,Θ_1,12=𝛿₀𝐸^𝑇₁Θ𝐵,_1,13=1−𝛿₀𝐸^𝑇₁𝐵,Θ₂₂=−𝑃₁−𝑄,Θ₃₃=−𝑈₂Σ₁,Θ₄₄=−𝑈₃Σ₁,Θ₅₅=−𝑈₁+𝑄₁,Θ₅₉=𝐾−𝐿+𝐴^𝑇𝐸₂,Θ₉₉=−𝐸^𝑇₂−𝐸₂+𝜏²₁𝑄+^𝑙_𝑖=1𝛿²₁𝑊_1𝑖+^𝑚_𝑖=1𝛿²₂𝑊_2𝑖,Θ_9,12=𝛿₀𝐸^𝑇₂𝐵,Θ_9,13=1−𝛿₀𝐸^𝑇₂Ξ𝐵,₁₁=𝑃₁+𝑋₁−𝑊₁−𝑉₁Σ₁,Ξ₂₂=−𝑃₁−𝑋₂−𝑊₁,Ξ₃₃=𝑃₂+𝑋₃−𝑊₂−𝑉₂Σ₁,Ξ₄₄=−𝑃₂−𝑋₄−𝑊₂,Ξ₅₅=𝜇1−²𝑙𝑋₂−𝜇1−¹𝑙𝑋₁−2𝑊₁−𝑅₁Σ₁,Ξ₆₆=𝜇1−⁴𝑚𝑋₄−𝜇1−³𝑚𝑋₃−2𝑊₂−𝑅₂Σ₁,Ξ₇₇=𝑄₁+𝑍₁−𝑉₁,Ξ₈₈=−𝑄₁−𝑍₂,Ξ₉₉=−𝑉₂+𝑄₂+𝑍₃,Ξ_10,10=−𝑄₂−𝑍₄,Ξ_11,11=𝜇1−²𝑙𝑍₂−𝜇1−¹𝑙𝑍₁−𝑅₁,Ξ_12,12=𝜇1−⁴𝑚𝑍₄−𝜇1−³𝑚𝑍₃−𝑅₂,Ξ₁₇=𝐻₁+𝑌₁+𝑉₁Σ₂,Ξ₂₈=−𝐻₁−𝑌₂,Ξ₃₉=𝐻₂+𝑌₃+𝑉₂Σ₂,Ξ_4,10=−𝐻₂−𝑌₄,Ξ_5,11=𝜇1−²𝑙𝑌₂−𝜇1−¹𝑙𝑌₁+𝑅₁Σ₁,Ξ_6,12=𝜇1−⁴𝑙𝑌₄−𝜇1−³𝑙𝑌₃+𝑅₂Σ₂. (3.11)

Proof. Firstly, we show the uniqueness of the equilibrium point by the method of contradiction. Here we can denote the equilibrium point as 𝑧^∗ of DNNs (2.9), then we have 0=−𝐶𝑧^∗+𝐴𝑓𝑧^∗+𝛿(𝑡)𝐵𝑓𝑧^∗+(1−𝛿(𝑡))𝐵𝑓𝑧^∗=−𝐶𝑧^∗+(𝐴+𝐵)𝑓𝑧^∗. (3.12) Now we suppose that the other equilibrium point 𝑧^∗≠0 exists, then it follows that 2𝑧^∗𝑇𝐸^𝑇₁−𝐶𝑧^∗+(𝐴+𝐵)𝑓𝑧^∗−𝑧=0,_∗𝑇𝑈₁Σ₁𝑧^∗+2𝑧^∗𝑇𝑈₁Σ₂𝑓𝑧^∗−𝑓^𝑇𝑧^∗𝑈₁𝑓𝑧^∗≥0. (3.13)(3.14) Then combining (3.13) and (3.14) yields that 𝑧^∗𝑓𝑧^∗𝑇Π₁₁Π₁₂Π^𝑇₁₂Π₂₂𝑧^∗𝑓𝑧^∗≥0 (3.15) with Π₁₁=−𝐸^𝑇₁𝐶−𝐶^𝑇𝐸₁−𝑈₁Σ₁, Π₁₂=𝐸^𝑇₁(𝐴+𝐵)+𝑈₁Σ₂, and Π₂₂=−𝑈₁. Meanwhile, it is noted that 𝑧^∗𝑓𝑧^∗𝑇𝑃₁𝐻₁∗𝑄₁𝑧^∗𝑓𝑧^∗≥0. (3.16) Yet on the other hand, let 𝐼Ψ=^𝑛𝐼_𝑛0_{𝑛×(4𝑙+4𝑚+7)𝑛}0_{𝑛×(2𝑙+2𝑚+2)𝑛}𝐼_𝑛0_{𝑛×(2𝑙+2𝑚+4)𝑛}𝐼_𝑛𝐼_𝑛. (3.17) Then multiplying the term Υ₁ΘΥ^𝑇₁+Υ₂ΞΥ^𝑇₂ in (3.9) by Ψ and Ψ^𝑇 on its left hand and right one, respectively, and using (3.15) and (3.16), we can deduce that Π₁₁+𝑃₁Π₁₂+𝐻₁Π^𝑇₁₂+𝐻^𝑇₁Π₂₂+𝑄₁Π<0⇒Π=¹¹Π₁₂Π^𝑇₁₂Π₂₂𝑃<−¹𝐻₁𝐻^𝑇₁𝑄₁. (3.18) Thus we can derive Π<0, which contradicts with (3.15) and implies 𝑧^∗=0. This is to say, the origin of the DNNs (3.1) is the unique equilibrium point.
Next, through directly calculating and using the denotations in (3.7)-(3.8), the stochastic differential of 𝑉_𝑖(𝑥_𝑡)(𝑖=1,2) in (3.3) along the trajectories of system (3.1) yields ℒ𝑉₁𝑥_𝑡=2𝑥^𝑇(𝑡)𝑃𝑥̇(𝑡)+𝑥̇^𝑇(𝑡)𝜏²₁𝑄𝑥̇(𝑡)−^𝑡_𝑡−𝜏1𝜏₁𝑥̇^𝑇(𝑠)𝑄𝑥̇(𝑠)𝑓𝑑𝑠+2^𝑇(𝑥(𝑡))(𝐾−𝐿)+𝑥^𝑇(𝑡)Σ𝐿−Σ𝐾𝑥̇(𝑡)≤2𝑥^𝑇(𝑡)𝑃𝑥̇(𝑡)+𝑥̇^𝑇(𝑡)𝜏²₁𝑄𝑥̇(𝑡)−𝑥(𝑡)−𝑥𝑡−𝜏₁^𝑇𝑄𝑥(𝑡)−𝑥𝑡−𝜏₁+2𝑓^𝑇(𝑥(𝑡))(𝐾−𝐿)+𝑥^𝑇(𝑡)Σ𝐿−Σ𝐾𝑥̇(𝑡),ℒ𝑉₂𝑥_𝑡=𝑥^𝑇(𝑡)𝑃₁𝑥(𝑡)+2𝑥^𝑇(𝑡)𝐻₁𝑓(𝑥(𝑡))+𝑓^𝑇(𝑥(𝑡))𝑄₁𝑓(𝑥(𝑡−𝑥))_𝑇𝑡−𝜏₁𝑃₁𝑥𝑡−𝜏₁+2𝑥^𝑇𝑡−𝜏₁𝐻₁𝑓𝑥𝑡−𝜏₁+𝑓^𝑇𝑥𝑡−𝜏₁𝑄₁𝑓𝑥𝑡−𝜏₁+𝜎^𝑇₁𝑡−𝜏₁𝑃₁𝜎₁𝑡−𝜏₁+2𝜎^𝑇₁𝑡−𝜏₁𝐻₁ℎ₁𝜎₁𝑡−𝜏₁+ℎ^𝑇₁𝜎₁𝑡−𝜏₁𝑄₁ℎ₁𝜎₁𝑡−𝜏₁−𝜎^𝑇₁𝑡−𝜏₁−𝛿₁𝑃₁×𝜎𝑡−𝜏₁−𝛿₁+2𝜎^𝑇₁𝑡−𝜏₁−𝛿₁𝐻₁ℎ₁𝜎𝑡−𝜏₁−𝛿₁+ℎ^𝑇₁𝜎𝑡−𝜏₁−𝛿₁𝑄₁×ℎ₁𝜎𝑡−𝜏₁−𝛿₁+𝜎^𝑇₂𝑡−𝜏₂𝑃₂𝜎₂𝑡−𝜏₂+2𝜎^𝑇₂𝑡−𝜏₂𝐻₂ℎ₂𝜎₂𝑡−𝜏₂+ℎ^𝑇₂𝜎₂𝑡−𝜏₂𝑄₂ℎ₂𝜎₂𝑡−𝜏₂−𝜎^𝑇₂𝑡−𝜏₂−𝛿₂𝑃₂𝜎𝑡−𝜏₂−𝛿₂+2𝜎^𝑇₂𝑡−𝜏₂−𝛿₂𝐻₂ℎ₂𝜎𝑡−𝜏₂−𝛿₂+ℎ^𝑇₂𝜎𝑡−𝜏₂−𝛿₂𝑄₂ℎ₂𝜎𝑡−𝜏₂−𝛿₂+𝜎^𝑇₁𝑡−𝜏₁𝑋₁𝜎₁𝑡−𝜏₁+𝜎^𝑇₁𝑡−𝜏₁−𝜌₁(𝑡)𝜏̇1−¹(𝑡)𝑙𝑋₂−𝑋₁×𝜎₁𝑡−𝜏₁−𝜌₁(𝑡)−𝜎^𝑇₁𝑡−𝜏₁−𝛿₁𝑋₂𝜎₁𝑡−𝜏₁−𝛿₁+2𝜎^𝑇₁𝑡−𝜏₁𝑌₁ℎ₁𝜎₁𝑡−𝜏₁+2𝜎^𝑇₁𝑡−𝜏₁−𝜌₁(𝑡)×𝜏̇1−¹(𝑡)𝑙𝑌₂−𝑌₁×ℎ₁𝜎₁𝑡−𝜏₁−𝜌₁(𝑡)−2𝜎^𝑇₁𝑡−𝜏₁−𝛿₁𝑌₂ℎ₁𝜎₁𝑡−𝜏₁−𝛿₁+ℎ^𝑇₁𝜎₁𝑡−𝜏₁𝑍₁ℎ₁𝜎₁𝑡−𝜏₁+ℎ^𝑇₁𝜎₁𝑡−𝜏₁−𝜌₁(𝑡)𝜏̇1−¹(𝑡)𝑙×𝑍₂−𝑍₁ℎ₁𝜎₁𝑡−𝜏₁−𝜌₁(𝑡)−ℎ^𝑇₁𝜎₁𝑡−𝜏₁−𝛿₁𝑍₂ℎ^𝑇₁𝜎₁𝑡−𝜏₁−𝛿₁+𝜎^𝑇₂𝑡−𝜏₂𝑋₃𝜎₂𝑡−𝜏₂+𝜎^𝑇₂𝑡−𝜏₂−𝜌₂(𝑡)𝜏̇1−²(𝑡)𝑚𝑋₄−𝑋₃×𝜎₂𝑡−𝜏₂−𝜌₂(𝑡)−𝜎^𝑇₂𝑡−𝜏₂−𝛿₂𝑋₄𝜎₂𝑡−𝜏₂−𝛿₂+2𝜎^𝑇₂𝑡−𝜏₂𝑌₃ℎ₂𝜎₂𝑡−𝜏₂+2𝜎^𝑇₂𝑡−𝜏₂−𝜌₂(𝑡×𝜏̇)1−²(𝑡)𝑚×𝑌₄−𝑌₃ℎ₂𝜎₂𝑡−𝜏₂−𝜌₂(𝑡)−2𝜎^𝑇₂𝑡−𝜏₂−𝛿₂𝑌₄ℎ₂𝜎₂𝑡−𝜏₂−𝛿₂+ℎ^𝑇₂𝜎₂𝑡−𝜏₂𝑍₃ℎ₂𝜎₂𝑡−𝜏₂+ℎ^𝑇₂𝜎₂𝑡−𝜏₂−𝜌₂(𝑡)𝜏̇1−²(𝑡)𝑚×𝑍₄−𝑍₃ℎ₂𝜎₂𝑡−𝜏₂−𝜌₂(𝑡)−ℎ^𝑇₂𝜎₂𝑡−𝜏₂−𝛿₂𝑍₄ℎ^𝑇𝜎₂𝑡−𝜏₂−𝛿₂. (3.19) Moreover, we can compute out ℒ𝑉₃(𝑥_𝑡) as follows: ℒ𝑉₃𝑥_𝑡=^𝑙_𝑖=1𝑥̇^𝑇(𝑡)𝛿²₁𝑊_1𝑖𝑥̇(𝑡)−^𝑙_𝑖=1^{𝑡−𝜏1−(𝑖−1)𝛿1}_{𝑡−𝜏1−𝑖𝛿1}𝛿₁𝑥̇^𝑇(𝑠)𝑊_1𝑖𝑥̇(𝑠)+𝑑𝑠_𝑚𝑖=1𝑥̇^𝑇(𝑡)𝛿²₂𝑊_2𝑖𝑥̇(𝑡)−^𝑚_𝑖=1^{𝑡−𝜏2−(𝑖−1)𝛿2}_{𝑡−𝜏2−𝑖𝛿2}𝑥̇^𝑇(𝑠)𝛿₂𝑊_2𝑖𝑥̇(𝑠)𝑑𝑠. (3.20) Then, by resorting to Lemmas 2.4 and 2.5  ^{𝑊1𝑖𝑆1𝑖}_∗𝑊1𝑖≥0,^{𝑊2𝑖𝑆2𝑖}_∗𝑊2𝑖≥0, and using the donations 𝑊_𝑖,𝑆_𝑖(𝑖=1,2), the following inequalities can be derived: −^𝑙_𝑖=1^{𝑡−𝜏1−(𝑖−1)𝛿1}_{𝑡−𝜏1−𝑖𝛿1}𝛿₁𝑥̇^𝑇(𝑠)𝑊_1𝑖𝑥̇(𝑠)≤𝑑𝑠_𝑙𝑖=1−𝑥𝑡−𝜏₁−(𝑖−1)𝛿₁−𝜌₁(𝑡)−𝑥𝑡−𝜏₁−𝑖𝛿₁^𝑇𝑊_1𝑖×𝑥𝑡−𝜏₁−(𝑖−1)𝛿₁−𝜌₁(𝑡)−𝑥𝑡−𝜏₁−𝑖𝛿₁−2𝑥𝑡−𝜏₁−(𝑖−1)𝛿₁−𝜌₁(𝑡)−𝑥𝑡−𝜏₁−𝑖𝛿₁^𝑇𝑆_1𝑖×𝑥𝑡−𝜏₁−(𝑖−1)𝛿₁−𝑥𝑡−𝜏₁−(𝑖−1)𝛿₁−𝜌₁(𝑡)−𝑥𝑡−𝜏₁−(𝑖−1)𝛿₁−𝑥𝑡−𝜏₁−(𝑖−1)𝛿₁−𝜌₁(𝑡)^𝑇𝑊_1𝑖×𝑥𝑡−𝜏₁−(𝑖−1)𝛿₁−𝑥𝑡−𝜏₁−(𝑖−1)𝛿₁−𝜌₁(𝑡)=−𝜎^𝑇₁𝑡−𝜏₁−𝜌₁(𝑡)𝑊₁𝜎𝑡−𝜏₁−𝜌₁(𝑡)−2𝜎^𝑇₁𝑡−𝜏₁−𝜌₁(𝑡)×𝑆₁𝜎₁𝑡−𝜏₁−𝛿₁+𝜎^𝑇₁𝑡−𝜏₁−𝛿₁𝑊₁𝜎₁𝑡−𝜏₁−𝛿₁−𝜎^𝑇₁𝑡−𝜏₁−𝜌₁(𝑡)𝑊₁𝜎₁𝑡−𝜏₁−𝜌₁(𝑡)−2𝜎^𝑇₂𝑡−𝜏₂−𝜌₂(𝑡)𝑆₁𝜎₂𝑡−𝜏₂+𝜎^𝑇₂𝑡−𝜏₂𝑊₂𝜎₂𝑡−𝜏₂,−^𝑚_𝑖=1^{𝑡−𝜏2−(𝑖−1)𝛿2}_{𝑡−𝜏2−𝑖𝛿2}𝛿₂𝑥̇^𝑇(𝑠)𝑊_2𝑖𝑥̇(𝑠)𝜎𝑑𝑠≤−^𝑇₂𝑡−𝜏₂−𝜌₂(𝑡)𝑊₂𝜎𝑡−𝜏₂−𝜌₂(𝑡)−2𝜎^𝑇₂𝑡−𝜏₂−𝜌₂(𝑡)×𝑆₂𝜎₂𝑡−𝜏₂−𝛿₂+𝜎^𝑇₂𝑡−𝜏₂−𝛿₂𝑊₂𝜎₂𝑡−𝜏₂−𝛿₂−𝜎^𝑇₂𝑡−𝜏₂−𝜌₂(𝑡)𝑊₂𝜎₂𝑡−𝜏₂−𝜌₂(𝑡)−2𝜎^𝑇₂𝑡−𝜏₂−𝜌₂(𝑡)𝑆₂𝜎₂𝑡−𝜏₂+𝜎^𝑇₂𝑡−𝜏₂𝑊₂𝜎₂𝑡−𝜏₂. (3.21) Then, it follows from (3.21) that ℒ𝑉₃(𝑥_𝑡) satisfies ℒ𝑉₃𝑥_𝑡≤^𝑙_𝑖=1𝑥̇^𝑇(𝑡)𝛿²₁𝑊_1𝑖𝑥̇(𝑡)+^𝑙_𝑖=1𝑥̇^𝑇(𝑡)𝛿²₂𝑊_2𝑖𝑥̇(𝑡)−𝜎^𝑇₁𝑡−𝜏₁−𝜌₁(𝑡)𝑊₁𝜎𝑡−𝜏₁−𝜌₁(𝑡)−2𝜎^𝑇₁𝑡−𝜏₁−𝜌₁(𝑡)𝑆₁𝜎₁𝑡−𝜏₁−𝛿₁+𝜎^𝑇₁𝑡−𝜏₁−𝛿₁𝑊₁𝜎₁𝑡−𝜏₁−𝛿₁−𝜎^𝑇₁𝑡−𝜏₁−𝜌₁(𝑡)𝑊₁𝜎₁𝑡−𝜏₁−𝜌₁(𝑡)−2𝜎^𝑇₂𝑡−𝜏₂−𝜌₂(𝑡)𝑆₁𝜎₂𝑡−𝜏₂+𝜎^𝑇₂𝑡−𝜏₂𝑊₁𝜎₂𝑡−𝜏₂−𝜎^𝑇₂𝑡−𝜏₂−𝜌₂(𝑡)𝑊₂𝜎𝑡−𝜏₂−𝜌₂(𝑡)−2𝜎^𝑇₂𝑡−𝜏₂−𝜌₂(𝑡)×𝑆₂𝜎₂𝑡−𝜏₂−𝛿₂+𝜎^𝑇₂𝑡−𝜏₂−𝛿₂𝑊₂𝜎₂𝑡−𝜏₂−𝛿₂−𝜎^𝑇₂𝑡−𝜏₂−𝜌₂(𝑡)×𝑊₂𝜎₂𝑡−𝜏₂−𝜌₂(𝑡)−2𝜎^𝑇₂𝑡−𝜏₂−𝜌₂(𝑡)𝑆₂𝜎₂𝑡−𝜏₂+𝜎^𝑇₂𝑡−𝜏₂𝑊₂𝜎₂𝑡−𝜏₂. (3.22) From (2.10), for any 𝑛×𝑛 diagonal matrices 𝑈_𝑖>0(𝑖=1,2,3,4,5), 𝑉_1𝑖>0, 𝑅_1𝑖>0(𝑖=1,…,𝑙), 𝑉_2𝑗>0, 𝑅_2𝑗>0(𝑗=1,…,𝑚), and setting 𝑉₁,𝑅₁ in (3.7), 𝑉₂,𝑅₂ in (3.8), the following inequality holds: 0≤−𝑥^𝑇(𝑡)𝑈₁Σ₁𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑈₁Σ₂𝑓(𝑥(𝑡))−𝑓^𝑇(𝑥(𝑡))𝑈₁𝑓(𝑥(𝑡+))−𝑥^𝑇𝑡−𝜏₂𝑈₂Σ₁𝑥𝑡−𝜏₂+2𝑥^𝑇𝑡−𝜏₂𝑈₂Σ₂𝑓𝑥𝑡−𝜏₂−𝑓^𝑇𝑥𝑡−𝜏₂𝑈₂𝑓𝑥𝑡−𝜏₂+−𝑥^𝑇𝑡−𝜏₃𝑈₃Σ₁𝑥𝑡−𝜏₃+2𝑥^𝑇𝑡−𝜏₃×𝑈₃Σ₂𝑓𝑥𝑡−𝜏₃−𝑓^𝑇𝑥𝑡−𝜏₃𝑈₃𝑓𝑥𝑡−𝜏₃+−𝑥^𝑇𝑡−𝜏₁(𝑡)×𝑈₄Σ₁𝑥𝑡−𝜏₁(𝑡)+2𝑥^𝑇𝑡−𝜏₁(𝑡)𝑈₄Σ₂𝑓𝑥𝑡−𝜏₁(𝑡)−𝑓^𝑇𝑥𝑡−𝜏₁(𝑡)𝑈₄𝑓𝑥𝑡−𝜏₁(𝑡)+−𝑥^𝑇𝑡−𝜏₂(𝑡)𝑈₅Σ₁𝑥𝑡−𝜏₂(𝑡)+2𝑥^𝑇𝑡−𝜏₂(𝑡)𝑈₅Σ₂𝑓𝑥𝑡−𝜏₂(𝑡)−𝑓^𝑇𝑥𝑡−𝜏₂(𝑡)𝑈₅𝑓𝑥𝑡−𝜏₂(𝑡)+−𝜎^𝑇₁𝑡−𝜏₁𝑉₁Σ₁𝜎₁𝑡−𝜏₁+2𝜎^𝑇₁𝑡−𝜏₁𝑉₁Σ₂ℎ₁𝜎₁𝑡−𝜏₁−ℎ^𝑇₁𝜎₁𝑡−𝜏₁𝑉₁ℎ₁𝜎₁𝑡−𝜏₁+−𝜎^𝑇₁𝑡−𝜏₁−𝜌₁(𝑡)𝑅₁Σ₁𝜎₁𝑡−𝜏₁−𝜌₁(𝑡)+2𝜎^𝑇₁𝑡−𝜏₁−𝜌₁(𝑡)𝑅₁Σ₂ℎ₁𝜎₁𝑡−𝜏₁−𝜌₁(𝑡)−ℎ^𝑇₁𝜎₁𝑡−𝜏₁−𝜌₁(𝑡)𝑅₁ℎ₁𝜎₁𝑡−𝜏₁−𝜌₁(𝑡)+−𝜎^𝑇₂𝑡−𝜏₂𝑉₂Σ₁𝜎₂𝑡−𝜏₂+2𝜎^𝑇₂𝑡−𝜏₂𝑉₂Σ₂ℎ₂𝜎₂𝑡−𝜏₂−ℎ^𝑇₂𝜎₂𝑡−𝜏₂×𝑉₂ℎ₂𝜎₂𝑡−𝜏₂+−𝜎^𝑇₂𝑡−𝜏₂−𝜌₂(𝑡)𝑅₂Σ₁𝜎₂𝑡−𝜏₂−𝜌₂(𝑡)+2𝜎^𝑇₂𝑡−𝜏₂−𝜌₂(𝑡)×𝑅₂Σ₂ℎ₂𝜎₂𝑡−𝜏₂−𝜌₂(𝑡)−ℎ^𝑇₂𝜎₂𝑡−𝜏₂−𝜌₂(𝑡)𝑅₂ℎ₂𝜎₂𝑡−𝜏₂−𝜌₂(𝑡). (3.23) Moreover, together with (3.1) and any 𝑛×𝑛 constant matrices 𝐸₁,𝐸₂, one can deduce 0=𝑥^𝑇(𝑡)𝐸^𝑇₁+𝑥̇^𝑇(𝑡)𝐸^𝑇₂×−𝑥̇(𝑡)−𝐶𝑥(𝑡)+𝐴𝑓(𝑥(𝑡))+𝛿₀𝐵𝑓𝑥𝑡−𝜏₁(𝑡)+1−𝛿₀𝐵𝑓𝑥𝑡−𝜏₂(𝑡)+𝛿(𝑡)−𝛿₀𝐵𝑓𝑥𝑡−𝜏₁(𝑡)−𝑓𝑥𝑡−𝜏₂(𝑡). (3.24) Now adding the right terms (3.19) and (3.22)–(3.24) to ℒ𝑉(𝑥_𝑡) and taking the mathematical expectation on its both sides, we can deduce 𝐄ℒ𝑉𝑥_𝑡≤𝜁^𝑇(𝑡)Υ₁ΘΥ^𝑇₁+𝜏̇₁(𝑡)−𝜇₁𝑙^𝑙_𝑖=1I^𝑇_1𝑖𝑋_1𝑖𝑌_1𝑖∗𝑍_1𝑖I_1𝑖+𝜇₂−𝜏̇₁(𝑡)𝑙^𝑙_𝑖=1I^𝑇_1𝑖𝑋_2𝑖𝑌_2𝑖∗𝑍_2𝑖I_1𝑖+Υ₂ΞΥ^𝑇₂+𝜏̇₂(𝑡)−𝜇₃𝑚^𝑚_𝑖=1I^𝑇_2𝑖𝑋_3𝑖𝑌_3𝑖∗𝑍_3𝑖I_2𝑖+𝜇₄−𝜏̇₂(𝑡)𝑚^𝑚_𝑖=1I^𝑇_2𝑖𝑋_4𝑖𝑌_4𝑖∗𝑍_2𝑖I_2𝑖𝜁(𝑡)∶=𝜁^𝑇(𝑡)Λ(𝑡)𝜁(𝑡), (3.25) where Θ,Ξ,Υ_𝑙(𝑙=1,2),I_𝑘𝑖(𝑘=1,2) are presented in (3.9), and 𝜁^𝑇(𝑡)=𝑥^𝑇(𝑡)𝜎^𝑇₁𝑡−𝜏₁𝜎^𝑇₂𝑡−𝜏₂𝑥^𝑇𝑡−𝜏₃𝜎^𝑇𝑡−𝜏₁−𝜌₁(𝑡)×𝜎^𝑇𝑡−𝜏₂−𝜌₂(𝑡)𝑓^𝑇(𝑥(𝑡))ℎ^𝑇₁𝜎₁𝑡−𝜏₁ℎ^𝑇₂𝜎₂𝑡−𝜏₂𝑓^𝑇𝑥𝑡−𝜏₃×ℎ^𝑇₁𝜎₁𝑡−𝜏₁−𝜌₁(𝑡)ℎ^𝑇₂𝜎₂𝑡−𝜏₂−𝜌₂(𝑡×𝑥̇_𝑇)(𝑡)𝑥^𝑇𝑡−𝜏₁(𝑡)𝑥^𝑇𝑡−𝜏₂(𝑡)𝑓^𝑇𝑥𝑡−𝜏₁(𝑡)𝑓^𝑇𝑥𝑡−𝜏₂(𝑡). (3.26) Then utilizing the general convex technique in [28, 39], the LMIs described by (3.9) can guarantee Λ(𝑡)<0, which indicates that there must exist a positive scalar 𝜒>0 such that 𝐄{ℒ𝑉(𝑥_𝑡)}≤𝜁^𝑇(𝑡)Λ(𝑡)𝜁(𝑡)≤−𝜒‖𝑥(𝑡)‖²<0 for any 𝑥(𝑡)≠0. Then, it follows from the Lyapunov-Krasovskii stability theorem that the system (3.1) is asymptotically stable in the mean-square sense, and it completes the proof.

Remark 3.3. Transmitted delays are always existent in various dynamical networks due to the finite switching speed of amplifiers in electronic neural networks or the finite signal propagation time in biological networks. Furthermore, with the development of network delay tomography, the probability of time-delay distribution can be estimated. Thus, when the probability is available, it will be helpful to utilize such information to reduce the conservatism [29–35]. Yet, up till now, few authors have utilized the delay-partitioning idea to investigate the stability of DNNs with probabilistic time-varying delay, and, in this work, we applied one improved idea, which can fully consider the information of delay subinterval. Moreover, though the stability criterion in (3.9) is not presented in the forms of standard LMIs, it is still convenient and straightforward to check the feasibility without tuning any parameters by resorting to LMI in Matlab Toolbox.

Remark 3.4. Presently, in the literatures [21–28], various convex combination techniques have been widely employed and improved to tackle constant or time-varying delays owing to that they can help reduce the conservatism efficiently. In [41], the authors put forward the reciprocal convex approach, which can consider those important terms ignored and be more effective than the ones [21–28]. Yet it has come to our attention that the reciprocal convex one cannot efficiently tackle the case that both the bounds on delay derivatives are available. In this paper, we first combine the reciprocal convex technique and the convex ones to study the stability for DNNs with probabilistic time-varying delay.

Remark 3.5. As for 𝑉₂(𝑥_𝑡) in (3.3), if we denote ^{𝑋2𝑖𝑌2𝑖}_∗𝑍2𝑖=^{𝑋4𝑖𝑌4𝑖}_∗𝑍4𝑖=0 (resp., ^{𝑋1𝑖𝑌1𝑖}_∗𝑍1𝑖=^{𝑋3𝑖𝑌3𝑖}_∗𝑍3𝑖=0), our results can be true as only 𝜇₂,𝜇₄ (resp., 𝜇₁,𝜇₃) are available. If we set ^{𝑋𝑗𝑖𝑌𝑗𝑖}_{∗𝑍𝑗𝑖}=0  (𝑗=1,2,3,4) in (3.3) simultaneously, Theorem 3.2 still holds as that 𝜇_𝑖(𝑖=1,2,3,4) are unknown, or 𝜏_𝑖(𝑡)(𝑖=1,2) are not differentiable. Moreover, the number of free-weighting matrices in Theorem 3.2 is much smaller than the ones of these present results [34, 37], and the reciprocal convex technique is used, it can induce much more computational simplicity in a mathematical point of view.

Remark 3.6. In view of the delay-partitioning idea employed in this work, with integers 𝑙,𝑚 increasing, the dimension of the derived LMIs will become higher and it will take more computing time to check them. Yet, if the lower bound of 𝜏₁(𝑡) is set and 𝑙≥5, the maximum allowable delay upper bound will become unapparently larger and approach to an approximate upper limitation [22–27]. Thus, if we want to employ the idea to real cases, we do not necessarily partition two delay intervals into too many segments.

Remark 3.7. In order to give the more general results, the delay intervals [𝜏₁,𝜏₂] and [𝜏₂,𝜏₃] are, respectively, divided into 𝑙,𝑚 subintervals in this work, which makes the condition of Theorem 3.2 seem to be very complicated. Yet if we set 𝑙=𝑚≤5, our results will avoid the complexity in some degree. Moreover, if we choose the simple Lyapunov-Krasovskii functional in (3.3) with 𝐻₁=𝑌_1𝑖=𝑌_2𝑖=𝑌_3𝑖=𝑌_4𝑖=0 and 𝐻_𝑖=0, the condition of Theorem 3.2 also will become much less complicated.

4. Numerical Examples

In this section, four numerical examples will be presented to illustrate the derived results. Firstly, we will utilize a numerical example to illustrate the significance of studying the lower bound of delay derivative.

Example 4.1. As a special case of 𝛿₀=1, we revisit the delayed neural networks considered in [21, 28] with 𝐶=,2002,𝐴=11−1−1,𝐵=0.88111,Σ=0000Σ=0.4000.8, (4.1) and 𝜏₁=0 is set. If we do not consider the existence of 𝜇₁, then, by utilizing Theorem 3.2 and Remark 3.5, the corresponding maximum allowable upper bounds (MAUBs) 𝜏_max for different 𝜇₂ derived by the results in [21] and in the paper can be summarized in Table 1, which demonstrates that Theorem 3.2 of 𝑙=1 is somewhat more conservative than the one in [21, 28]. Yet, if we set 𝜇₁=0.5, it is easy to verify that our results can yield much less conservative results than the ones in [21, 28], which can be shown in Table 2.
Based on Tables 1 and 2, it is indicated that the conservatism of stability criterion can be greatly deduced if we take the available 𝜇₁ into consideration. Moreover, though the delay-partitioning idea has been used in [28], the corresponding MAUBs 𝜏_max derived by [28] and Theorem 3.2 are summarized in Table 3, which shows that our idea can be more efficient than the one in [18] even for 𝑙=1,2.

Table 1 

Calculated MAUBs 𝜏_max for 𝑙=1, unavailable 𝜇₁ in Example 4.1.

Table 2 

Calculated MAUBs 𝜏_max for 𝑙=1 and 𝜇₁=0.5 in Example 4.1.

Table 3 

Calculated MAUBs 𝜏_max for various 𝑙 and 𝜇₁=0.5 in Example 4.1.

Example 4.2. Considering the special case of 𝛿₀=1, we consider the delayed neural networks (2.1) with 𝐶=diag{1.2769,0.6231,0.9230,0.4480},Σ=0_3×3,{},,,Σ=diag0.1137,0.1279,0.7994,0.2368𝐴=−0.03730.4852−0.33510.2336−1.60330.5988−0.32241.23520.3394−0.0860−0.3824−0.5785−0.13110.3253−0.9534−0.5015𝐵=0.8674−1.2405−0.53250.02200.0474−0.91640.03600.98161.84952.6117−0.37880.8428−2.04130.51791.1734−0.2775 (4.2) which has been addressed extensively; see [26, 27] and the references therein. Together with the delay-partitioning idea and for different 𝜇₂, the work [27] has calculated the MAUBs 𝜏_max such that the origin of the system is globally asymptotically stable for 𝜏₁(𝑡) satisfying 3=𝜏₁≤𝜏₁(𝑡)≤𝜏₂≤𝜏_max. By resorting to Theorem 3.2 and Remark 3.5, the corresponding results can be given Table 4, which indicates that our delay-partitioning idea can be more effective than the relevant ones in [27] even for 𝑙=1,2 and 𝜇₁=0.

Table 4 

Calculated MAUBs 𝜏_max for various 𝑙,𝜇₂ in Example 4.2.

Example 4.3. We still consider the DNNs with the following parameters [24, 28] by setting 𝛿₀=1: 𝐶=,4.18890000.71600001.9985,𝐵=−0.1052−0.5069−0.1121−0.0257−0.28080.02120.1205−0.21530.1315Σ=,0.41290003.89930001.016 (4.3)𝐴=Σ=0_3×3, and time-varying delay 𝜏(𝑡) satisfies 0≤𝜏(𝑡)≤𝜏₂,0≤𝜏̇(𝑡)≤𝜇₂. Then, the MAUBs for this example with time-varying delay 𝜏(𝑡) are given in Table 5 by using the delay-partitioning idea in [28] and the one in this paper, which can illustrate that our delay-decomposition idea is superior over the one by [28].

Table 5 

Calculated MAUBs 𝜏_max for 𝑙=2,𝜇₂ in Example 4.3.

Example 4.4. Consider DNN (2.3) of the following parameters, which has been considered in [33]: 𝐶=(𝑥)=7006,𝐴=0.2−40.10.3,𝐵=0.40.20.10.7,𝑓tanh−0.2𝑥₁tanh𝑥₂. (4.4) If we set 𝜏₁=0, 𝜏₂=0.4, 𝜇₁=𝜇₃=0.1, and 𝜇₂=𝜇₄, the relevant MAUBs 𝜏_max are computed and listed in Tables 6 and 7 by setting 𝑙=𝑚=1, for various 𝜇₂, and the delay probability distribution 𝛿₀=0.2, which can guarantee the addressed system to be asymptotically stable in the mean-squared sense.
Based on Tables 6 and 7, even for 𝑚=𝑙=1, our results still can be less conservative than the one in [33]. As for 𝑚=1, 𝑙=2, one verifies that our results can reduce the conservatism much more evidently.

Table 6 

Calculated MAUBs 𝜏_max for 𝑙=𝑚=1, various 𝜇₂, and 𝛿₀=0.2 in Example 4.4.

Table 7 

Calculated MAUBs 𝜏_max for 𝑙=𝑚=1, various 𝜇₂, and 𝛿₀=0.8 in Example 4.4.

5. Conclusions

This paper has investigated the asymptotical stability for DNNs with probability-distribution delay. Through employing an improved idea of delay partitioning and constructing one novel Lyapunov-Krasovskii functional, one stability criterion with significantly reduced conservatism has been established in terms of LMIs. The proposed stability condition benefits from both bounds of time-varying delay and variation and combined convex technique. Four numerical examples have been given to demonstrate the effectiveness of the derived criteria and the improvements over some existent ones.

Acknowledgments

This work is supported by the National Natural Science Foundation of China no. 60904020, no. 61004064, and no. 61004032 and the Special Foundation of China Postdoctoral Science Foundation Project no. 201003546.