Further Stability Criterion on Delayed Recurrent Neural Networks Based on Reciprocal Convex Technique

Abstract

Together with Lyapunov-Krasovskii functional theory and reciprocal convex technique, a new sufficient condition is derived to guarantee the global stability for recurrent neural networks with both time-varying and continuously distributed delays, in which one improved delay-partitioning technique is employed. The LMI-based criterion heavily depends on both the upper and lower bounds on state delay and its derivative, which is different from the existent ones and has more application areas as the lower bound of delay derivative is available. Finally, some numerical examples can illustrate the reduced conservatism of the derived results by thinning the delay interval.

1. Introduction

Recently, various classes of neural networks have been increasingly studied in the past few decades, due to their practical importance and successful applications in many areas such as optimization, image processing, and associative memory design. In those applications, the key feature of the designed neural network is to be convergent. Meanwhile, since there inevitably exist communication delay which is the main source of oscillation and instability in various dynamical systems, great efforts have been made to analyze the dynamical behaviors of time-delay systems including delayed neural networks (DNNs), and many elegant results have been reported; see [1–31] and the references therein. In practical application, though it is difficult to describe the form of delay precisely, the ranges of time delay and its variation rate can be measured. Since Lyapunov functional approach imposed no restriction on delay and its derivative and presented the simple stability results, Lyapunov-Krasovskii functional (LKF) one has been widely utilized due to that it can fully utilize the information on time-delay system. Thus recently, the delay-dependent stability or delay-derivative-dependent one for DNNs has become an important topic of primary significance, in which its main purpose was to derive the maximum allowable upper bound on time delay such that the system can be convergent [7–9, 11–16]. Meanwhile, since a neural network usually has a spatial nature due to the presence of an amount of parallel pathways of a variety of axon sizes and lengths, it is desired to model them by introducing a distributed delay over a certain duration of time such that the distant past has less influence compared to the recent behavior of the state. In other words, when studying stability for DNNs, the distributed delay should be taken into consideration simultaneously [18–26].

Presently, during tackling the effect of time delay, the delay-partitioning idea has been verified to be more effective in reducing the conservatism and widely employed [10–17]. In [11], one delay-partitioning idea has been used to tackle time-varying delay of DNNs based on the improved idea in [10] and in [12, 13], and some researchers also put forward the other novel delay-partitioning idea to tackle constant delay, which can be more evident and concise than the idea based on the one in [10]. Later, though the idea was extended to time-varying delay case [14, 15, 17], the improved idea cannot deal with the interval varying delay efficiently, especially as the lower bound of delay is greater than 0. Meanwhile, as for time-varying delay, because convex combination technique can play an important role in reducing the conservatism, it has received much attention and achieved some great improvements in studying the stability for time-delay systems including DNNs [15–18, 28–30]. Its basic idea is to approximate the integral terms of quadratic quantities into a convex combination of quadratic terms of the LMIs. However, owing to the inversely weighted nature of coefficients in Jensen inequality approach or the limitation of free-weighting matrix method, when estimating the derivative of Lyapunov-Krasovskii functional, some important terms still have been ignored based on the convex combination technique [15–18]. In [31], together with integral inequality lemma, the authors put forward the reciprocal convex technique, which can consider these previously ignorant terms. Yet, we have noticed that the reciprocal convex technique could not tackle the case that the lower and upper bound of time delay can be measured simultaneously, which still needs the convex technique. Now, together with the improved delay-partitioning idea in [17] and combining reciprocal convex technique with convex combination one, no researcher has investigated the stability for DNNs as the lower bound of delay derivative is available, which motivates the focus of this presented work.

In the paper, we make some great efforts to investigate asymptotical stability for recurrent neural networks with both time-varying and continuously distributed delay, in which both the upper and lower bounds of time delay and its derivative are treated. Through applying an improved delay-partitioning idea, one LMI-based condition is derived based on combination of reciprocal convex technique and convex one, which can present the pretty delay dependence and computational efficiency. Finally, we give three numerical examples to illustrate the reduced conservatism.

Notations
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; Tr(𝐴) denotes the trace of the matrix 𝐴; _𝑌^𝑋𝑌_𝑇𝑍=^𝑋𝑌_∗𝑍 with ∗ denotes the symmetric term in a symmetric matrix.

2. Problem Formulations

Consider the DNNs with continuously distributed delay of the following form: 𝑧̇(𝑡)=−𝐶𝑧(𝑡)+𝐴𝑔(𝑧(𝑡))+𝐵𝑔(𝑧(𝑡−𝜏(𝑡)))+𝐷^𝑡_−∞𝐾(𝑡−𝑠)𝑔(𝑧(𝑠))𝑑𝑠+𝐽, (2.1) where 𝑧(𝑡)=[𝑧₁(𝑡)⋯𝑧_𝑛(𝑡)]^𝑇∈𝐑^𝑛 is the neuron state vector; 𝐶=diag{𝑐₁,…,𝑐_𝑛}>0 and 𝐴,𝐵,𝐷 are 𝑛×𝑛 known constant matrices; 𝑔(𝑧(⋅))=[𝑔₁(𝑧₁(⋅))⋯𝑔_𝑛(𝑧_𝑛(⋅))]^𝑇∈𝐑^𝑛 stands for the neuron activation function, 𝐽∈𝐑^𝑛 is a constant input vector, and 𝐾(𝑡−𝑠)=[𝑘_𝑖𝑗(𝑡−𝑠)]_𝑛×𝑛; here the delay kernel 𝑘_𝑖𝑗(⋅) is a real-valued nonnegative continuous function defined on [0,+∞) and satisfies ∫^∞₀𝑘_𝑖𝑗(𝜃)𝑑𝜃=1 for 𝑖,𝑗=1,2,…,𝑛.

The following assumptions on system (2.1) are utilized throughout this paper. (H1) The delay 𝜏(𝑡) denotes one continuous function satisfying 0≤𝜏₀≤𝜏(𝑡)≤𝜏_𝑚,𝜇₀≤𝜏̇(𝑡)≤𝜇_𝑚, (2.2) in which 𝜏₀,𝜏_𝑚,𝜇₀,𝜇_𝑚 are constants. Here we denote 𝜏_𝑚=𝜏_𝑚−𝜏₀ and 𝜇_𝑚=𝜇_𝑚−𝜇₀. (H2) For the constants 𝛿⁺_𝑗,𝛿⁻_𝑗, the functions 𝑔_𝑗(⋅) in (2.1) are bounded and satisfy the following condition: 𝛿⁻_𝑗≤𝑔_𝑗(𝑥)−𝑔_𝑗𝑦𝑥−𝑦≤𝛿⁺_𝑗,∀𝑥,𝑦∈𝑅,𝑥≠𝑦,𝑗=1,2,…,𝑛. (2.3) In what follows, we denote Σ=diag{𝛿⁻₁,…,𝛿⁻_𝑛}, Σ=diag{𝛿⁺₁,…,𝛿⁺_𝑛} and set Σ₁=ΣΣ, Σ₂=(Σ+Σ)/2, respectively.

Remark 2.1. As pointed out in [22], the constants 𝛿⁺_𝑖,𝛿⁻_𝑖 in (H2) are allowed to be positive, negative, or zero. Thus, some previously used Lipschitz conditions are just special cases of (H2), which means that the activation functions are of more general descriptions than these previous ones.
Suppose 𝑧^∗=[𝑧^∗₁⋯𝑧^∗_𝑛]^𝑇 is the equilibrium point of the system (2.1). In order to prove the result, we will shift the equilibrium to the origin by changing the variable 𝑥(⋅)=𝑧(⋅)−𝑧^∗. Then the system (2.1) can be transformed into 𝑥̇(𝑡)=−𝐶𝑥(𝑡)+𝐴𝑓(𝑥(𝑡))+𝐵𝑓(𝑥(𝑡−𝜏(𝑡)))+𝐷^𝑡_−∞𝐾(𝑡−𝑠)𝑓(𝑥(𝑠))𝑑𝑠, (2.4) where 𝑥(𝑡)=[𝑥₁(𝑡),…,𝑥_𝑛(𝑡)]^𝑇, and 𝑓(𝑥)=[𝑓₁(𝑥₁),…,𝑓_𝑛(𝑥_𝑛)]^𝑇 with 𝑓_𝑗(𝑥_𝑗)=𝑔_𝑗(𝑥_𝑗+𝑧^∗_𝑗)−𝑔_𝑗(𝑧^∗_𝑗),𝑗=1,2,…,𝑛. Note that the function 𝑓_𝑗(⋅) satisfies 𝑓_𝑗(0)=0, and 𝛿⁻_𝑗≤𝑓_𝑗(𝑥)𝑥≤𝛿⁺_𝑗,∀𝑥∈𝑅,𝑥≠0. (2.5)
In order to establish the stability criterion, firstly, the following lemmas are introduced.

Lemma 2.2 (see [27]). For any constant matrix W∈𝐑^𝑛×𝑛,W=W^𝑇>0, scalar functional 0≤𝑟(𝑡)≤𝑟_𝑀, and a vector function 𝑒̇∶[−𝑟_𝑀,0]→𝐑^𝑛 such that the following integration is well defined, then −𝑟_𝑀∫⁰_{−𝑟(𝑡)}𝑒^𝑇̇(𝑡+𝑠)W𝑒(̇𝑡+𝑠)𝑑𝑠≤[𝑒(𝑡)−𝑒(𝑡−𝑟(𝑡))]^𝑇W[𝑒(𝑡)−𝑒(𝑡−𝑟(𝑡))].

Lemma 2.3 (see [31]). 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 1𝛼₁𝑓₁(𝑡)+1𝛼₂𝑓₂(𝑡)1+⋯+𝛼_𝑁𝑓_𝑁(𝑡)∶𝐃⟶𝐑^𝑛 (2.6) with 𝛼_𝑖>0 and ∑_𝑖𝛼_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^𝑚→𝐑^𝐧,𝑔_𝑖,𝑗(𝑡)=̇𝑔_𝑗,𝑖(𝑡),𝑓_𝑖(𝑡)𝑔_𝑖,𝑗(𝑡)𝑔_𝑗,𝑖(𝑡)𝑓_𝑗(𝑡)≥0. (2.7)

Then the problem to be addressed in next section can be formulated as developing a condition ensuring that the DNNs (2.4) are asymptotically stable.

3. Delay-Derivative-Dependent Stability

In the section, through utilizing the reciprocal convex technique idea in [31], we present one novel delay-derivative-dependent stability criterion for the system (2.4) in terms of LMIs.

Theorem 3.1. Given a positive integer 𝑙 and setting 𝛿=𝜏_𝑚/𝑙,Σ₁=diag{Σ₁,…,Σ₁}_𝑙×𝑙,Σ₂=diag{Σ₂,…,Σ₂}_𝑙×𝑙, then the system (2.4) satisfying (2.2) and (2.5) is globally asymptotically stable if there exist 𝑛×𝑛 matrices 𝑃>0,𝑃₁>0,𝐻₁,𝑄₁>0,𝑉>0,𝑊_𝑗>0(𝑗=1,…,𝑙),𝑆_𝑗(𝑗=1,…,𝑙),𝐸_𝑖(𝑖=1,2), 𝑛×𝑛 diagonal matrices 𝐾>0,𝐿>0,𝑇>0,𝑈_𝑖>0(𝑖=1,2,3),𝑉_𝑗>0,𝑅_𝑗>0(𝑗=1,…,𝑙), 𝑛×𝑛 matrices 𝑋_𝑖𝑗>0,𝑌_𝑖𝑗,𝑍_𝑖𝑗>0(𝑖=1,2;𝑗=1,…,𝑙), and ln×ln constant matrices 𝑃>0,𝑄>0,𝐻 such that the LMIs in (3.1)-(3.2) hold: 𝑃₁𝐻₁∗𝑄₁𝑃𝐻∗𝑄𝑋>0,>0,^𝑖𝑗𝑌_𝑖𝑗∗𝑍_𝑖𝑗𝑊>0,^𝑗𝑆_𝑗∗𝑊_𝑗≥0,𝑖=1,2;𝑗=1,…,𝑙, (3.1)Υ₁ΘΥ^𝑇₁+Υ₂ΞΥ^𝑇₂+𝜇_𝑚𝑙^𝑙_𝑖=1I^𝑇_1𝑖𝑋_𝑘𝑖𝑌_𝑘𝑖∗𝑍_𝑘𝑖I_1𝑖<0,𝑘=1,2, (3.2) where I_1𝑖=^{0𝑛⋅(𝑙+1+𝑖)𝑛−𝐼𝑛0𝑛⋅(2𝑙+1)𝑛−𝐼𝑛0𝑛⋅(𝑙−𝑖+4)𝑛}_{0𝑛⋅(𝑙+1+𝑖)𝑛𝐼𝑛0𝑛⋅(2𝑙+1)𝑛𝐼𝑛0𝑛⋅(𝑙−𝑖+4)𝑛}, and ΘΘ=¹¹𝑉0Θ₁₄00Θ₁₇0𝐸^𝑇₁𝐵𝐸^𝑇₁𝐷∗Θ₂₂00−𝐻₁00000∗∗−𝑈₂Σ₁00𝑈₂Σ₂0000∗∗∗Θ₄₄00Θ₄₇000∗∗∗∗−𝑄₁00000∗∗∗∗∗−𝑈₂0000∗∗∗∗∗∗Θ₇₇0𝐸^𝑇₂𝐵𝐸^𝑇₂𝐷∗∗∗∗∗∗∗−𝑈₃Σ₁𝑈₃Σ₂0∗∗∗∗∗∗∗∗−𝑈₃01∗∗∗∗∗∗∗∗∗−𝑛𝑇,Υ₁=𝐼_𝑛∗0_𝑛⋅𝑛𝐼_𝑛∗0_{𝑛⋅(𝑙+1)𝑛}𝐼_𝑛∗0_{𝑛⋅(2𝑙+2)𝑛}𝐼_𝑛∗0_{𝑛⋅(2𝑙+3)𝑛}𝐼_𝑛∗0_{𝑛⋅(3𝑙+3)𝑛}𝐼_𝑛∗0_{𝑛⋅(4𝑙+3)𝑛}𝐼_𝑛∗0_{𝑛⋅(4𝑙+4)𝑛}𝐼_𝑛∗0_{𝑛⋅(4𝑙+5)𝑛}𝐼_𝑛∗0_{𝑛⋅(4𝑙+6)𝑛}𝐼_𝑛∗Ξ,Ξ=¹¹𝑆𝑊₁−𝑆Ξ₁₄00∗Ξ₂₂𝑊−𝑆00−𝐻∗∗Ξ₃₃00Ξ₃₆∗∗∗Ξ₄₄00∗∗∗∗−𝑄0∗∗∗∗∗Ξ₆₆,Υ₂=0_ln⋅𝑛𝐼_ln∗0_ln⋅2𝑛𝐼_ln∗0_{ln⋅(𝑙+2)𝑛}𝐼_ln∗0_{ln⋅(2𝑙+3)𝑛}𝐼_ln∗0_{ln⋅(2𝑙+4)𝑛}𝐼_ln∗0_{ln⋅(3𝑙+4)𝑛}𝐼_ln∗ (3.3) with ∗ representing the appropriately dimensional 0 matrix making Υ_𝑖(𝑖=1,2) of (4𝑙+8)𝑛 columns, 𝑋_𝑖=diag{𝑋_𝑖1,…,𝑋_𝑖𝑙}, 𝑌_𝑖=diag{𝑌_𝑖1…𝑌_𝑖𝑙}, 𝑍_𝑖=diag{𝑍_𝑖1,…,𝑍_𝑖𝑙},𝑖=1,2, 𝑊=diag{𝑊₁,…,𝑊_𝑙}, 𝑆=diag{𝑆₁,…,𝑆_𝑙}, 𝑉=diag{𝑉₁,…,𝑉_𝑙}, 𝑅=diag{𝑅₁,…,𝑅_𝑙}, and Θ₁₁=−𝐸^𝑇₁𝐶−𝐶^𝑇𝐸₁+𝑃₁−𝑉−𝑈₁Σ₁,Θ₁₄=𝐸^𝑇₁𝐴+𝐻₁+𝑈₁Σ₂,Θ₁₇=𝑃−Σ𝐾+Σ𝐿−𝐸^𝑇₁−𝐶^𝑇𝐸₂,Θ₂₂=−𝑃₁Θ−𝑉,₄₄=−𝑈₁+𝑄₁+Tr(𝑇)𝐼_𝑛,Θ₄₇=𝐾−𝐿+𝐴^𝑇𝐸₂,Θ₇₇=−𝐸^𝑇₂−𝐸₂+𝜏²₀𝑉+^𝑙_𝑖=1𝛿²𝑊_𝑖,Ξ₁₁=𝑃+𝑋₁−𝑊−𝑉Σ₁,Ξ₁₄=𝐻+𝑉Σ₂+𝑌₁,Ξ₂₂=−𝑃−𝑊,Ξ₃₃=−2𝑊+𝑆+𝑆^𝑇−𝑅Σ₁,Ξ₃₆=𝜇1−^𝑚𝑙𝑌₂−𝜇1−⁰𝑙𝑌₁−𝑅Σ₂,Ξ₄₄=𝑄+𝑍₁−Ξ𝑉,₆₆=𝜇1−^𝑚𝑙𝑍₂−𝜇1−⁰𝑙𝑍₁−𝑅. (3.4)

Proof. Based on (2.5) and denoting 𝜌(𝑡)=(𝜏(𝑡)−𝜏₀)/𝑙, we construct the Lyapunov-Krasovskii functional candidate as 𝑉(𝑥(𝑡))=𝑉₁(𝑥(𝑡))+𝑉₂(𝑥(𝑡))+𝑉₃(𝑥(𝑡))+𝑉₄(𝑥(𝑡)), (3.5) where 𝑉₁(𝑥(𝑡))=𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+2^𝑛_𝑖=1𝑘_𝑖^𝑥𝑖₀𝑓_𝑖(𝑠)−𝛿⁻_𝑖𝑠𝑑𝑠+2^𝑛_𝑖=1𝑙_𝑖^𝑥𝑖₀𝛿⁺_𝑖𝑠−𝑓_𝑖(𝑠𝑉)𝑑𝑠,₂(𝑥(𝑡))=^𝑡_𝑡−𝜏0𝑥(𝑠)𝑓(𝑥(𝑠))^𝑇𝑃₁𝐻₁∗𝑄₁𝑥(𝑠)𝑓(𝑥(𝑠))𝑑𝑠+^𝑡−𝜏0_{𝑡−𝜏0−𝛿}𝜎(𝑠)ℎ(𝜎(𝑠))^𝑇𝑃𝐻∗𝑄𝜎(𝑠)ℎ(𝜎(𝑠+))𝑑𝑠_{𝑙𝑖=1𝑡−𝜏0−(𝑖−1)𝛿𝑡−𝜏0−()(𝑡)𝑖−1𝛿−𝜌}𝑥(𝑠)𝑓(𝑥(𝑠))^𝑇𝑋_1𝑖𝑌_1𝑖∗𝑍_1𝑖𝑥(𝑠)𝑓(𝑥(𝑠+))𝑑𝑠_{𝑙𝑖=1𝑡−𝜏0−(𝑖−1)𝛿−𝜌(𝑡)𝑡−𝜏0−𝑖𝛿}𝑥(𝑠)𝑓(𝑥(𝑠))^𝑇𝑋_2𝑖𝑌_2𝑖∗𝑍_2𝑖𝑥(𝑠)𝑓(𝑥(𝑠𝑉))𝑑𝑠,₃(𝑥(𝑡))=^𝑛𝑛_𝑖=1𝑗=1𝑡_𝑖^∞₀𝑘_𝑖𝑗(𝜃)^𝑡_𝑡−𝜃𝑓²_𝑗𝑥_𝑗(𝑠)𝑉𝑑𝑠𝑑𝜃,₄(𝑥(𝑡))=⁰_−𝜏0^𝑡_𝑡+𝜃𝜏₀𝑥̇^𝑇(𝑠)𝑉𝑥̇(𝑠)𝑑𝑠𝑑𝜃+^𝑙_𝑖=1^{−𝜏0−(𝑖−1)𝛿}_{−𝜏0−𝑖𝛿}^𝑡_𝑡+𝜃𝛿𝑥̇^𝑇(𝑠)𝑊_𝑖𝑥̇(𝑠)𝑑𝑠𝑑𝜃 (3.6) with 𝐾=diag{𝑘₁,…,𝑘_𝑛}>0, 𝐿=diag{𝑙₁,…,𝑙_𝑛}>0, 𝑇=diag{𝑡₁,…,𝑡_𝑛}>0 waiting to be determined, and 𝜎^𝑇(𝑠)=𝑥^𝑇(𝑠)⋯𝑥^𝑇(𝑠−(𝑙−1)𝛿),ℎ^𝑇(𝜎(𝑠))=𝑓^𝑇(𝑥(𝑠))⋯𝑓^𝑇(𝑥(𝑠−(𝑙−1)𝛿)). (3.7) Through directly calculating and using any 𝑛×𝑛 constant matrices 𝐸_𝑖(𝑖=1,2), the time derivative of functional (3.5) along the trajectories of system (2.4) yields 𝑉̇₁(𝑥(𝑡))=2𝑥^𝑇(𝑡)𝑃𝑥̇(𝑡)+2𝑓^𝑇(𝑥(𝑡))(𝐾−𝐿)+𝑥^𝑇(𝑡)(𝑡)𝑥Σ𝐿−Σ𝐾𝑥̇+2^𝑇(𝑡)𝐸^𝑇₁+𝑥̇^𝑇(𝑡)𝐸^𝑇₂−𝑥̇(𝑡)−𝐶𝑥(𝑡)+𝐴𝑓(𝑥(𝑡))+𝐵𝑓(𝑥(𝑡−𝜏(𝑡)))+𝐷^𝑡_−∞𝐾()𝑓(𝑥(𝑠,𝑉̇𝑡−𝑠))𝑑𝑠₂(𝑥(𝑡))=𝑥^𝑇(𝑡)𝑃₁𝑥(𝑡)+2𝑥^𝑇(𝑡)𝐻₁𝑓(𝑥(𝑡))+𝑓^𝑇(𝑥(𝑡))𝑄₁𝑓(𝑥(𝑡−𝑥))_𝑇𝑡−𝜏₀𝑃₁𝑥𝑡−𝜏₀+2𝑥^𝑇𝑡−𝜏₀𝐻₁𝑓𝑥𝑡−𝜏₀+𝑓^𝑇𝑥𝑡−𝜏₀𝑄₁𝑓𝑥𝑡−𝜏₀+𝜎^𝑇𝑡−𝜏₀𝑃𝜎𝑡−𝜏₀+2𝜎^𝑇𝑡−𝜏₀𝐻ℎ𝜎𝑡−𝜏₀+ℎ^𝑇𝜎𝑡−𝜏₀𝑄ℎ𝜎𝑡−𝜏₀−𝜎^𝑇𝑡−𝜏₀−𝛿𝑃𝜎𝑡−𝜏₀−𝛿+2𝜎^𝑇𝑡−𝜏₀−𝛿𝐻ℎ𝜎𝑡−𝜏₀−𝛿+ℎ^𝑇𝜎𝑡−𝜏₀×𝜎−𝛿𝑄ℎ𝑡−𝜏₀+𝜎−𝛿_𝑇𝑡−𝜏₀𝑋₁𝜎𝑡−𝜏₀+𝜎^𝑇𝑡−𝜏₀(𝑡)𝜏̇(𝑡)−𝜌1−𝑙𝑋₂−𝑋₁𝜎𝑡−𝜏₀−𝜌(𝑡)−𝜎^𝑇𝑡−𝜏₀−𝛿𝑋₂𝜎𝑡−𝜏₀+−𝛿2𝜎^𝑇𝑡−𝜏₀𝑌₁ℎ𝜎𝑡−𝜏₀+2𝜎^𝑇𝑡−𝜏₀(𝑡)𝜏̇(𝑡)−𝜌1−𝑙𝑌₂−𝑌₁×ℎ𝜎𝑡−𝜏₀−𝜌(𝑡)−2𝜎^𝑇𝑡−𝜏₀−𝛿𝑌₂ℎ𝜎𝑡−𝜏₀+ℎ−𝛿_𝑇𝜎𝑡−𝜏₀𝑍₁ℎ𝜎𝑡−𝜏₀+ℎ^𝑇𝜎𝑡−𝜏₀(𝑡)𝜏̇(𝑡)−𝜌1−𝑙𝑍₂−𝑍₁×ℎ𝜎𝑡−𝜏₀−𝜌(𝑡)−ℎ^𝑇𝜎𝑡−𝜏₀−𝛿𝑍₂ℎ^𝑇𝜎𝑡−𝜏₀,𝑉̇−𝛿₃(𝑥(𝑡))=^𝑛𝑛_𝑖=1𝑗=1𝑡_𝑖^∞₀𝑘_𝑖𝑗(𝜃)𝑓²_𝑗𝑥_𝑗(𝑡)𝑑𝜃−^𝑛𝑛_𝑖=1𝑗=1^∞₀𝑘_𝑖𝑗(𝜃)𝑓²_𝑗𝑥_𝑗(≤𝑡−𝜃)𝑑𝜃_𝑛𝑖=1𝑡_𝑖𝑓^𝑇(𝑥(𝑡))𝑓(𝑥(𝑡))−^𝑛𝑛_𝑖=1𝑗=1^∞₀𝑘_𝑖𝑗(𝜃)𝑓_𝑗𝑥_𝑗(𝑡−𝜃)𝑑𝜃²≤Tr(𝑇)𝑓^𝑇(𝑥(𝑡))𝑓(𝑥(𝑡))−^𝑡_−∞𝐾(𝑡−𝑠)𝑓(𝑥(𝑠))𝑑𝑠^𝑇1𝑛𝑇×^𝑡_−∞𝐾()𝑓(𝑥(𝑠,𝑉̇𝑡−𝑠))𝑑𝑠₄(𝑥(𝑡))=𝑥̇^𝑇(𝑡)𝜏²₀𝑉+^𝑙_𝑖=1𝛿²𝑊_𝑖𝑥̇(𝑡)−^𝑡_𝑡−𝜏0𝜏₀𝑥̇^𝑇(𝑠)(𝑠)−𝑉𝑥̇𝑑𝑠_{𝑙𝑖=1𝑡−𝜏0−(𝑖−1)𝛿𝑡−𝜏0−𝑖𝛿}𝛿𝑥̇^𝑇(𝑠)𝑊_𝑖𝑥̇(𝑠)𝑑𝑠. (3.8)(3.9)(3.10)(3.11) Moreover, together with Lemmas 2.2 and 2.3 and (3.1), we can estimate −∑^𝑙_𝑖=1∫^{𝑡−𝜏0−(𝑖−1)𝛿}_{𝑡−𝜏0−𝑖𝛿}×𝛿𝑥̇^𝑇(𝑠)𝑊_𝑖𝑥̇(𝑠)𝑑𝑠 as follows: −^𝑙_𝑖=1^{𝑡−𝜏0−(𝑖−1)𝛿}_{𝑡−𝜏0−𝑖𝛿}𝛿𝑥̇^𝑇(𝑠)𝑊_𝑖𝑥̇(𝑠)𝑑𝑠≤−^𝑙_𝑖=1𝜏_𝑚𝜏_𝑚(𝑡)𝑥−𝜏𝑡−𝜏₀−(𝑖−1)𝛿−𝜌(𝑡)−𝑥𝑡−𝜏₀−𝑖𝛿^𝑇×𝑊_𝑖𝑥𝑡−𝜏₀−(𝑖−1)𝛿−𝜌(𝑡)−𝑥𝑡−𝜏₀−𝑖𝛿−𝜏_𝑚𝜏(𝑡)−𝜏₀×𝑥𝑡−𝜏₀−(𝑖−1)𝛿−𝑥𝑡−𝜏₀−(𝑖−1)𝛿−𝜌(𝑡)^𝑇×𝑊_𝑖𝑥𝑡−𝜏₀−(𝑖−1)𝛿−𝑥𝑡−𝜏₀−(𝑖−1)𝛿−𝜌(𝑡)≤−^𝑙_𝑖=1𝑥𝑡−𝜏₀−()𝛿𝑥𝑖−1𝑡−𝜏₀−()(𝑡)𝑥𝑖−1𝛿−𝜌𝑡−𝜏₀−𝑖𝛿^𝑇𝑊_𝑖𝑆_𝑖−𝑊_𝑖−𝑆_𝑖∗2𝑊_𝑖−𝑆_𝑖−𝑆^𝑇_𝑖𝑆^𝑇_𝑖−𝑊_𝑖∗∗𝑊_𝑖×𝑥𝑡−𝜏₀−()𝛿𝑥𝑖−1𝑡−𝜏₀−()(𝑡)𝑥𝑖−1𝛿−𝜌𝑡−𝜏₀𝜎−𝑖𝛿=−𝑡−𝜏₀𝜎𝑡−𝜏₀(𝑡)𝜎−𝜌𝑡−𝜏₀−𝛿^𝑇𝑊𝑆𝑆𝑆−𝑊−∗2𝑊−𝑆−^𝑇𝑆^𝑇−𝑊𝑊𝜎∗∗𝑡−𝜏₀𝜎𝑡−𝜏₀(𝑡)𝜎−𝜌𝑡−𝜏₀.−𝛿 (3.12) From (2.5), the following inequality holds for any diagonal matrices 𝑈_𝑖>0(𝑖=1,2,3),𝑉_𝑗>0,𝑅_𝑗>0(𝑗=1,…,𝑙) with the compatible dimensions and setting 𝑉=diag{𝑉₁,…,𝑉_𝑙},𝑅=diag{𝑅₁,…,𝑅_𝑙}, 0≤−𝑥^𝑇(𝑡)𝑈₁Σ₁𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑈₁Σ₂𝑓(𝑥(𝑡))−𝑓^𝑇(𝑥(𝑡))𝑈₁𝑓(𝑥(𝑡+))−𝑥^𝑇𝑡−𝜏_𝑚𝑈₂Σ₁𝑥𝑡−𝜏_𝑚+2𝑥^𝑇𝑡−𝜏_𝑚𝑈₂Σ₂𝑓𝑥𝑡−𝜏_𝑚−𝑓^𝑇𝑥𝑡−𝜏_𝑚𝑈₂𝑓𝑥𝑡−𝜏_𝑚+−𝑥^𝑇(𝑡−𝜏(𝑡))𝑈₃Σ₁𝑥(𝑡−𝜏(𝑡))+2𝑥^𝑇(𝑡−𝜏(𝑡))𝑈₃Σ₂𝑓(𝑥(𝑡−𝜏(𝑡)))−𝑓^𝑇(𝑥(𝑡−𝜏(𝑡)))×𝑈₃𝑓(𝑥((𝑡+𝑡−𝜏)))−𝜎^𝑇𝑡−𝜏₀𝑉Σ₁𝜎𝑡−𝜏₀+2𝜎^𝑇𝑡−𝜏₀𝑉Σ₂ℎ𝜎𝑡−𝜏₀−ℎ^𝑇𝜎𝑡−𝜏₀𝑉ℎ𝜎𝑡−𝜏₀+−𝜎^𝑇𝑡−𝜏₀−𝜌(𝑡)𝑅Σ₁𝜎𝑡−𝜏₀−𝜌(𝑡)+2𝜎^𝑇𝑡−𝜏₀−𝜌(𝑡)𝑅Σ₂ℎ𝜎𝑡−𝜏₀−𝜌(𝑡)−ℎ^𝑇𝜎𝑡−𝜏₀−𝜌(𝑡)𝑅ℎ𝜎𝑡−𝜏₀−𝜌(𝑡). (3.13) Now, adding the terms on the right-hand side of (3.8)–(3.12) and employing inequality (3.13), we can deduce 𝑉̇(𝑥(𝑡))≤𝜁^𝑇(𝑡)Υ₁ΘΥ^𝑇₁+Υ₂ΞΥ^𝑇₂+𝜏̇(𝑡)−𝜇₀𝑙^𝑙_𝑖=1I^𝑇_3𝑖𝑋_1𝑖𝑌_1𝑖∗𝑍_1𝑖I_3𝑖+𝜇_𝑚−𝜏̇(𝑡)𝑙×^𝑙_𝑖=1I^𝑇_3𝑖𝑋_2𝑖𝑌_2𝑖∗𝑍_2𝑖I_3𝑖𝜁(𝑡)=̇𝜁^𝑇(𝑡)Λ(𝑡)𝜁(𝑡), (3.14) in which Θ,Ξ,Υ_𝑖(𝑖=1,2) are presented in (3.2), and 𝜁^𝑇(𝑡)=𝑥^𝑇(𝑡)𝜎^𝑇𝑡−𝜏₀𝑥^𝑇𝑡−𝜏₀−𝜏_𝑚𝜎^𝑇𝑡−𝜏₀−𝜌(𝑡)𝑓^𝑇(𝑥(𝑡ℎ))_𝑇𝜎𝑡−𝜏₀𝑓^𝑇𝑥𝑡−𝜏_𝑚ℎ^𝑇𝜎𝑡−𝜏₀−𝜌(𝑡)𝑥̇^𝑇(𝑡)𝑥^𝑇(𝑡−𝜏(𝑡))𝑓^𝑇(𝑥(𝑡−𝜏(𝑡)))^𝑡_−∞𝐾(𝑡−𝑠)𝑓(𝑥(𝑠))𝑑𝑠^𝑇. (3.15) Then by employing convex combination technique, the LMIs in (3.2) can guarantee Λ(𝑡)<0, which indicates that there must exist a positive scalar 𝜒>0 such that 𝑉̇(𝑥(𝑡))≤−𝜒‖𝑥(𝑡)‖²<0 for 𝑥(𝑡)≠0. Then it follows from Lyapunov-Krasovskii stability theory that the system (2.4) is asymptotically stable, and the proof is completed.

Remark 3.2. Presently, the convex combination technique has been widely employed to tackle time-varying delay owing to the truth that it could reduce the conservatism more effectively than the previous ones, see [15–18, 28–30]. In [31], the authors put forward the reciprocal convex approach, which can reduce the conservatism more effectively than convex combination ones. Yet, it has come to our attention that no researchers have utilized both of them simultaneously to tackle the stability for DNNs.

Remark 3.3. One can easily check that the theorem in this work achieves some great improvements over the one in [17], which can be illustrated in the following. Firstly, some ignored terms in [17] have been fully considered in this paper when estimating the derivative of Lyapunov-Krasovskii functional in (3.11), such as the ignored ones.
Secondly, owing to the introduction of reciprocal convex approach, Theorem 3.1 can be much less complicated than the ones in [17], which will result in some computation simplicity in some degree. Thirdly, though reciprocal convex approach plays an important role in tackling the range of time delay, it cannot efficiently deal with the effect of lower and upper bound on delay derivative, which can be checked in (3.14). Thus we employ the convex combination technique to overcome this shortcoming.

Remark 3.4. When 𝜏(𝑡) is not differentiable or 𝜇₀ (resp., 𝜇_𝑚) is unknown, through setting ^{𝑋𝑖𝑗𝑌𝑖𝑗}_{∗𝑍𝑖𝑗}=0(𝑖=1,2) or ^{𝑋2𝑗𝑌2𝑗}_∗𝑍2𝑗=0 (resp., ^{𝑋1𝑗𝑌1𝑗}_∗𝑍1𝑗=0) in (3.5), our theorem still can be true.

Remark 3.5. Owing to the introduction of delay-partitioning idea in this work, the difficulty and complexity in checking the theorem will become more and more evident when the integer 𝑙 increases and the dimension of the LMIs in (3.2) will become much higher. Yet based on the results in [12–16], the maximum allowable upper bound of 𝜏_𝑚 would become unapparent enlarging as 𝑙≥5. Thus if we want to employ the idea to real cases, we do not necessarily partition the interval [𝜏₀,𝜏_𝑚] into more than 𝑙≥5 subintervals.

4. Numerical Examples

In the section, three numerical examples will be presented to illustrate that our results are superior over the ones by convex combination technique.

Example 4.1. We revisit the system considered in [9, 11, 17] with the following parameters: ,,𝐶=2002,𝐴=11−1−1,𝐵=0.88111𝐷=Σ=0000Σ=,0.4000.8 (4.1) in which 𝜏₀=0 is set. If we do not consider the existence of 𝜇₀, then by utilizing Theorem 3.1 and Remark 3.2, the corresponding maximum allowable upper bounds (MAUBs) 𝜏_max for different 𝜇_𝑚 derived by Theorem 3.1 in [17] and in the paper can be summarized in Table 1, which demonstrates that Theorem 3.1 in this paper of 𝑙=1 is less conservative than the theorem in [17]. Yet, if we set 𝜇₀=0.5, it is still easy to verify that our results can yield much less conservative results than the one in [17], which can be shown in Table 2.

Table 1 

Calculated MAUBs 𝜏_max for various 𝑙, unavailable 𝜇₀ in Example 4.1.

Table 2 

Calculated MAUBs 𝜏_max for various 𝑙 and 𝜇₀=0.5 in Example 4.1.

Based on Tables 1 and 2, it indicates that the conservatism of stability criterion can be greatly deduced if we take 𝜇₀ into consideration. Moreover, though the delay-partitioning idea has been used in [17], the corresponding MAUBs 𝜏_max derived by [17] and Theorem 3.1 are summarized in Table 3, which shows that the idea in this work can be more efficient than the one in [17] even for 𝑙=1,2.

Table 3 

Calculated MAUBs 𝜏_𝑚𝑎𝑥 for various 𝑙, 𝜇₀=0.5 in Example 4.1

Example 4.2. 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=−0.03730.4852−0.33510.2336−1.60330.5988−0.32241.23520.3394−0.0860−0.3824−0.5785−0.13110.3253−0.9534−0.5015,𝐵=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 [2, 15, 17] and the references therein. Together with the delay-partitioning idea and for different 𝜇_𝑚, the works [15, 17] have calculated the MAUBs 𝜏_max such that the origin of the system is globally asymptotically stable for 𝜏(𝑡) satisfying 3=𝜏₀≤𝜏(𝑡)≤𝜏_𝑚≤𝜏_max. By resorting to Theorem 3.1 and Remark 3.2, the corresponding results can be given in Table 4, which indicates that our delay-partitioning idea can be more effective than the relevant ones in [15, 17] for 𝑙=1,2 and 𝜇₀=0.

Table 4 

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

Example 4.3. Consider the delayed neural networks (2.4) with ,𝐾()=𝐶=4.2003.8,𝐴=1−1.660−1,𝐵=10−2.4751,𝐷=0.50.20.30.4𝑡−𝑠2𝑒^{−2(𝑡−𝑠)}3𝑒^{−3(𝑡−𝑠)}4𝑒^{−4(𝑡−𝑠)}2𝑒^{−2(𝑡−𝑠)}, (4.3) and 𝑓(𝑥_𝑖)=tanh(𝑥_𝑖),𝑖=1,2. For 𝜏₀=0.5, choosing various 𝜇₀,𝜇_𝑚 in Table 5 and applying Theorem 3.1 in our work and the one in [17], we can find the MAUBs on 𝜏_𝑚 for which the system remains asymptotically stable.

Table 5 

Calculated MAUBs 𝜏_𝑚𝑎𝑥 for various [𝜇₀,𝜇_𝑚], and 𝑙 in Example 4.3.

Based on Table 5, it can indicate that the delay-partitioning idea in our work can be less conservative than the ones in [17].

5. Conclusion

This paper has investigated the asymptotical stability for DNNs with continuously distributed delay. Through employing one improved delay-partitioning idea and combining reciprocal convex technique with convex combination one, one stability criterion with significantly reduced conservatism has been established in terms of LMIs. The proposed stability condition benefits from the partition of delay intervals and reciprocal convex technique. Three numerical examples have been given to demonstrate the effectiveness of the presented criteria and the improvements over some existent ones. Finally, it should be worth noting that the delay-partitioning idea presented in this work is widely applicable in many cases.

Acknowledgment

This work is supported by the National Natural Science Foundation of China no. 60875035, no. 60904020, no. 61004064, no. 61004032, and the Special Foundation of China Postdoctoral Science Foundation Projects no. 201003546.