Stability Analysis of Stochastic Markovian Jump Neural Networks with Different Time Scales and Randomly Occurred Nonlinearities Based on Delay-Partitioning Projection Approach

Abstract

In this paper, the mean square asymptotic stability of stochastic Markovian jump neural networks with different time scales and randomly occurred nonlinearities is investigated. In terms of linear matrix inequality (LMI) approach and delay-partitioning projection technique, delay-dependent stability criteria are derived for the considered neural networks for cases with or without the information of the delay rates via new Lyapunov-Krasovskii functionals. We also obtain that the thinner the delay is partitioned, the more obviously the conservatism can be reduced. An example with simulation results is given to show the effectiveness of the proposed approach.

1. Introduction

The human brain is made up of a large amount of neurons and their interconnections. An artificial neural network is an information processing system that has certain characteristics in common with biological neural networks. During the past decades, neural networks have been used for a wide variety of applications, for example, associative memories, pattern recognition, signal processing, and the other fields [1–4]. As is well known, the stability of neural networks plays an important role in modern control theories for these applications. However, time delays are often attributed as the major sources of instability in various engineering systems. Therefore, how to find sufficient conditions to guarantee the stability of neural networks with time delays is an important research topic [5–15].

Markovian jump system is an important class of stochastic models, which can be described by a set of nonlinear systems with the transitions between models determined by a Markovian chain in a finite mode set. This kind of system has been extensively applied to study the stability of neural networks. In real life, neural networks have a phenomenon of information latching, and it is recognized that the best way for modeling this class of neural networks is Markovian jump system [16–18]. Obviously, the Markovian jump system is more complex and challenging than the system without Markovian jump parameters, in which many authors are interested [12, 19–24].

On the other hand, it should be pointed out that lots of practical systems are influenced by additive randomly occurred nonlinear disturbances which are caused by environmental circumstances. In today’s networked environment, such nonlinear disturbances may be subject to random abrupt changes, which may result from abrupt phenomena, such as random failures and repairs of the components, environmental disturbance, and so forth. In other words, the nonlinear disturbances may occur in a probabilistic way, but they are randomly changeable in terms of their types and/or intensity. The stochastic nonlinearities, which are then named as randomly occurred nonlinearities (RONs), have recently attracted much attention [25–28].

Recently, by introducing free-weighting matrices [29, 30], model transformation method [31], linear matrix inequality (LMI) approach [32, 33], and adopting the concept of delay partitioning [9, 11, 34], stability criteria have been obtained for some neural networks. Sufficient conditions for the robust stability of uncertain stochastic system with interval time-varying delay were derived in [29] by employing the delay partitioning approach. Delay-dependent conditions on mean square asymptotic stability of stochastic neural networks with Markovian jumping parameters are presented by using the delay partitioning method which is different from the existing ones in the literature and convex combination method in [12]. The RONs model and the sensor failure model were introduced in [26]. In [9], better delay-dependent stability criteria for continuous systems with multiple delay components were established by utilizing a delay-partitioning projection approach. On the other hand, it is worth mentioning that neural networks on time scales have been presented and studied [35–37], which can unify the continuous and discrete situations. To the best of the authors’ knowledge, the delay-partitioning projection approach to stability analysis of stochastic Markovian jump neural networks with different time scales and randomly occurred nonlinearities has never been tackled in the previous literature. This motivates our research.

In this paper, the problem of stability analysis of stochastic Markovian jump neural networks with different time scales and RONs is considered. The paper is organised as follows. Section 2 introduces model description and preliminaries. RONs are introduced to model a class of sector-like nonlinearities whose occurrence is governed by a Bernoulli distributed white sequence with a known conditional probability. In Section 3, we derive the stability results based on delay-partitioning projection approach for stochastic Markovian jump neural networks with RONs. The results include two cases, one with a specified delay rates and the other with arbitrary delay rates. In addition to delay dependence, the obtained conditions are also dependent on the partitioning size; we verify that the conservatism of the conditions is a nonincreasing function of the number of partitions. A numerical example is presented to illustrate the effectiveness of the obtained criteria in Section 4. And finally, conclusions are drawn in Section 5.

Notations. Throughout this paper, the notation is fairly standard. 𝑅^𝑛 denotes the 𝑛-dimensional Euclidean space. 𝑅^𝑛×𝑚 stands for real matrix 𝑅 of size 𝑛×𝑚 (simply abbreviated 𝑅_𝑛 when 𝑚=𝑛). 𝑃>(<)0 is used to define a real symmetric positive definite (negative definite) matrix. For real symmetric matrices 𝑋 and 𝑌, the notation 𝑋≥𝑌 (resp., 𝑋>𝑌) means that the matrix 𝑋−𝑌 is positive semidefinite (resp., positive definite). The symmetric terms in a symmetric matrix are denoted by ∗ and diag{⋯} denotes a block-diagonal matrix. The superscripts 𝐴^𝑇 and 𝐴⁻¹ stand for the transpose and inverse of matrix 𝐴. (Ω,ℱ,{ℱ_𝑡}_𝑡≥0,𝒫) denotes a complete probability space with a filtration {ℱ_𝑡}_𝑡≥0, where Ω is a sample space, ℱ is the 𝜎-algebra of subset of the sample space, and 𝒫 is the probability measure on ℱ. 𝐸{𝑥} stands for the expectation of the stochastic variable 𝑥.

2. Model Description and Preliminaries

In this paper, the stochastic Markovian jump neural networks with different time scales RONs are considered: 𝜀𝑥̇(𝑡)(𝑟(𝑡𝑥(𝑡)(𝑟(𝑡𝑥((𝑡(𝑡)(𝑥(𝑡+[𝐶(𝑟(𝑡𝑥(𝑡)(𝑟(𝑡𝑥((𝑡𝑊(𝑡),𝑥(𝑡)(𝑡)[],=−𝐴))+𝐵))𝑡−𝜏))+𝜉𝐸𝑓))))+𝐷))𝑡−𝜏))]=𝜙,∀𝑡∈−𝑑,0 (1) where 𝑥(𝑡)∈𝑅^𝑛 is the state vector. 𝑊(𝑡) is a scalar zero mean Gaussian white noise process. 𝜙(𝑡) is a real-valued initial condition. 𝜀>0 is the time scale. {𝑟(𝑡)} is a right-continuous Markov chain on the probability space (Ω,ℱ,𝒫) taking values in a given finite set 𝑆={1,2,…,𝑁}. The transition probability matrix of system (1) is given by 𝑃𝑟()(𝑡)=𝑞𝑡+Δ𝑡=𝑗∣𝑟=𝑖^𝑖𝑗Δ𝑡+𝑜(Δ𝑡)if𝑖≠𝑗,1+𝑞_𝑖𝑖Δ𝑡+𝑜(Δ𝑡),if𝑖=𝑗, (2) where Δ𝑡>0 and lim_Δ𝑡→0𝑜(Δ𝑡)/Δ𝑡=0, 𝑞_𝑖𝑗≥0 is the transition rate from mode 𝑖 at time 𝑡 to mode 𝑗 at time 𝑡+Δ𝑡, and 𝑞_𝑖𝑖=−∑_𝑖≠𝑗𝑞_𝑖𝑗. 𝐴(𝑟(𝑡))=diag{𝑎_𝑖1,𝑎_𝑖2,…,𝑎_𝑖𝑛} is a positive diagonal matrix. 𝐵(𝑟(𝑡)), 𝐶(𝑟(𝑡)), 𝐷(𝑟(𝑡)), and 𝐸 are known matrices. For the sake of simplicity, in the sequel, each possible value of 𝑟(𝑡) is denoted by 𝑖,𝑖∈𝑆 and 𝐴(𝑟(𝑡)), 𝐵(𝑟(𝑡)), 𝐶(𝑟(𝑡)), 𝐷(𝑟(𝑡)) are abbreviated as 𝐴_𝑖, 𝐵_𝑖, 𝐶_𝑖, 𝐷_𝑖, respectively. 𝑑(𝑡) denotes the time-varying delay satisfying 0≤𝑑(𝑡)≤𝑑,𝑑̇(𝑡)≤𝜇, (3) where 𝑑 and 𝜇 are constant real numbers.

Finally, 𝑓(𝑥(𝑡)) stands for the mismatched external nonlinearity. The stochastic variable 𝜉(𝑡)∈𝑅, which accounts for the phenomena of RONs, is a Bernoulli distributed white noise sequence specified by the following distribution laws: Prob{𝜉(𝑡)=1}=𝐸{𝜉(𝑡)}={𝜉(𝑡)}{𝜉(𝑡𝜉,Prob=0=1−𝐸)}=1−𝜉. (4) Furthermore, we can show that 𝐸𝜉(𝑡)−𝜉=0,𝐸𝜉(𝑡)−𝜉²=𝜉1−𝜉, (5) where 𝜉∈[0,1] is a constant.

Before proceeding, we make the following assumption.

Assumption 1. For all 𝑥∈𝑅^𝑛, the nonlinear function 𝑓(𝑥) is assumed to satisfy the following sector-bounded condition: 𝑓(𝑥)−𝐾₁𝑥^𝑇𝑓(𝑥)−𝐾₂𝑥≤0, (6) where 𝐾₁ and 𝐾₂ are known real matrices of appropriate dimensions and 𝐾₁−𝐾₂≥0.

Remark 2. The nonlinear function 𝑓(𝑥) satisfying (6) is customarily said to belong to sector [𝐾₁,𝐾₂]. Because such a nonlinear condition is quite general which includes the usual Lipschitz condition as a special case, the systems with sector-bounded nonlinearities have been intensively studied in [25, 26].

Recall that the time derivative of a Wiener process is a white noise [27, 28]. We establish 𝑑𝑤(𝑡)=𝑊(𝑡)𝑑𝑡, where 𝑤(𝑡) is a scalar Wiener process on a probability space (Ω,ℱ,{ℱ_𝑡}_𝑡≥0,𝒫), which is independent from the Markov chain {𝑟(𝑡),𝑡≥0}. It is further assumed that 𝑤(𝑡) and the stochastic variable 𝜉(𝑡) are mutually independent. Besides, 𝑤(𝑡) satisfies 𝐸{𝑑𝑤(𝑡)}=0,𝐸𝑑𝑤²(𝑡)=𝑑𝑡. (7) Hence, the network (1) is rewritten as the following stochastic differential equations: 𝑑𝑥(𝑡)=𝜀⁻¹(𝑟(𝑡𝑥(𝑡)(𝑟(𝑡𝑥((𝑡(𝑡)(𝑥(𝑡−𝐴))+𝐵))𝑡−𝜏))+𝜉𝐸𝑓))𝑑𝑡+𝜀⁻¹[𝐶(𝑟(𝑡𝑥(𝑡)(𝑟(𝑡𝑥((𝑡(𝑡),𝑥(𝑡)(𝑡)[].))+𝐷))𝑡−𝜏))]𝑑𝑤=𝜙,∀𝑡∈−𝑑,0 (8)

Remark 3. The stochastic Markovian neural network with different time scales and RONs (8) is general enough to include many models as its special cases. For example, if we do not consider the RONs, time scales and remove the noise perturbations, then (8) is reduced to those studied in [9, 23, 38–41], whereas the authors in [9, 38–41] ignore the Markovian jump parameters. If we do not consider the RONs and time scales, then (8) is discussed in [29] and its references. If we do not take the noise perturbations and time scales into account and let Prob{𝜉(𝑡)=1}=1, 𝐵(𝑟(𝑡))=0, (8) is just the one in [42]. If we do not consider the RONs, the Markovian jump parameters, and the noise perturbations, then (8) is studied in [37]. Furthermore, if we remove the time scales, (8) is the one in [43]. Thus, our model generalizes and improves greatly many previous works and is therefore very significant.

Remark 4. In [44], the delay-partitioning approach was introduced to study the discrete-time recurrent neural networks. As an extension of the approach, we use two different partitions to deal with the continuous Markovian jump neural networks with different time scales in this paper.

Note that it is easy to prove that there exists at least one equilibrium point for (8) by employing the well-known Brouwer’s fixed point theorem. To end this section, the lemmas which are necessary for the proof of our main results are introduced as follows.

Lemma 5. For any constant matrix 𝑅>0, any scalars 𝑎 and 𝑏 with 𝑎<𝑏, and a vector function 𝑥(𝑡)∶[𝑎,𝑏]→𝑅^𝑛 such that the integrations concerned are well defined, the following inequality holds: ^𝑏_𝑎𝑥(𝑠)𝑑𝑠^𝑇𝑅^𝑏_𝑎𝑥(𝑠)𝑑𝑠≤(𝑏−𝑎)^𝑏_𝑎𝑥^𝑇(𝑠)𝑅𝑥(𝑠)𝑑𝑠. (9)

Lemma 6 (see [9]). Let 𝑍∈𝑅^𝑛×𝑛 and the bidiagonal upper triangular block matrix 𝐽_𝐾(𝑍)=𝐼_𝑛−𝑍0⋱⋱⋱−𝑍0𝐼_𝑛∈𝑅^{𝐾𝑛×𝐾𝑛}. (10) If 𝑌=(𝐽_𝐾(𝑍)𝐹)∈𝑅^{𝐾𝑛×(𝐾𝑛+𝑚)} with 𝐹=^𝐹1⋮_𝐹𝑘∈𝑅^{𝐾𝑛×𝑚}, 𝐹_𝑗∈𝑅^𝑛×𝑚(𝑗=1,…,𝐾), then 𝑌^⟂=col−^𝐾_𝑗=1𝑍^𝑗−1𝐹_𝑗,−^𝐾_𝑗=2𝑍^𝑗−2𝐹_𝑗,…,−𝐹_𝐾,𝐼_𝑚. (11)

Lemma 7 (Finsler’s lemma). Suppose that 𝑥∈𝑅^𝑛, 𝑀=𝑀^𝑇∈𝑅^𝑛×𝑛 and 𝐵∈𝑅^𝑚×𝑛 such that 𝐵 has full row rank. Then, the following statements are equivalent:(1) there exists a vector 𝑥∈𝑅^𝑛 such that 𝑥^𝑇𝑀𝑥<0 and 𝐵𝑥=0,(2) there exists a scalar 𝜇∈𝑅 such that 𝜇𝐵^𝑇𝐵−𝑀>0,(3)∃𝑋∈𝑅^𝑛×𝑚 such that 𝑀+𝑋𝐵+𝐵^𝑇𝑋^𝑇<0,(4)𝐵^⟂𝑇𝑀𝐵^⟂<0, where 𝐵^⟂ is the orthogonal complement of 𝐵.

Remark 8. It should be pointed out that various problems in control theory have been solved by combing Lyapunov control approach with Finsler’s lemma. In lots of applications, Finsler’s lemma is referred to Elimination lemma, which devotes to eliminate the redundant variables in matrix inequalities [45].

3. Main Results

In this section, we will give the mean square asymptotic stability for the system (8) in terms of LMI approach. The main results are stated as follows.

Theorem 9. For given constants 𝑑 and 𝜇, and two positive integers 𝑚 and 𝑀, the stochastic system (8) is mean square asymptotically stable if there exist matrices 𝑃_𝑖>0, 𝑄_𝑘>0(𝑘=1,…,𝑚), 𝑅>0, 𝑋₁>0, 𝑋₂>0 and 𝑊=𝑊^𝑇=𝑊₁₁⋯𝑊_1𝑀𝑊⋮⋱⋮_𝑇1𝑀⋯𝑊_𝑀𝑀>0, (12) and positive scalar 𝑙>0 such that the following LMI holds: 𝐵^⟂𝑇Ξ₁+Ξ₂+Ξ₃0∗Ξ₄𝐵^⟂<0, (13) where 𝐵^⟂∈𝑅^{(2𝑚+2𝑀+3)𝑛×(𝑚+𝑀+3)𝑛} is all arbitrary but fixed matrix whose columns form a basis of the kernel space of 𝐵_{(𝑚+𝑀)𝑛×(2𝑚+2𝑀+3)𝑛}=(𝐽_𝑚+𝑀(𝐼_𝑛)𝐹),

𝐹=000−𝐼_𝑛0⋯0⋮⋮⋮⋱⋱⋮000⋱⋱0−𝐼_𝑛000⋯0−𝐼_𝑛∈𝑅^{(𝑚+𝑀)𝑛×(𝑚+𝑀+3)𝑛},Ξ₁=Ω₁0⋯𝜀⁻¹𝑃_𝑖𝐵_𝑖0𝑊_1𝑀⋯𝑊₁₂Δ₁∗Ω₂⋯000⋯00⋮⋮⋱⋮⋮⋮⋱⋮⋮∗∗…Ω_𝑚+100⋯00∗∗…∗Γ_𝑀+1−𝑊^𝑇_{(𝑀−1)𝑀}⋯−𝑊^𝑇_1𝑀0∗∗…∗∗Γ_𝑀⋯𝑊^𝑇_2𝑀−𝑊^𝑇_1(𝑀−1)0⋮⋮⋱⋮⋮⋮⋱⋮⋮∗∗…∗∗∗⋯Γ₂0∗∗…∗∗∗⋯∗Δ₂,Ω₁=−2𝜀⁻¹𝑃_𝑖𝐴_𝑖+^𝑁_𝑗=1𝑞_𝑖𝑗𝑃_𝑗+𝑊₁₁+^𝑚_𝑖=1𝑄_𝑖−𝑙𝐾₁,Δ₁=−𝑙𝐾₂+𝜉𝜀⁻¹𝑃_𝑖𝐸,Δ₂=−𝑙𝐼+𝑅+𝐸^𝑇𝜀⁻²𝜉−𝜉²ℎ(𝑑−ℎ)𝑋₁+𝜀⁻²𝜉−𝜉²𝑑𝜎𝑋₂Ω𝐸,_𝑘=(𝑘−1)𝜇𝑄𝑚−1^𝑘−1,(),Γ𝑘=2,…,𝑚+1_𝑘=𝑊_𝑘𝑘−𝑊_{(𝑘−1)(𝑘−1)},𝑊_{(𝑀+1)(𝑀+1)}(),Ξ=0,𝑘=2,…,𝑀+1₂=−𝐴_𝑖0⋯0𝐵_𝑖0⋯0𝜉𝐸^𝑇𝜀⁻²ℎ(𝑑−ℎ)𝑋₁+𝜀⁻²𝑑𝜎𝑋₂−𝐴_𝑖0⋯0𝐵_𝑖0⋯0,Ξ𝜉𝐸₃=𝐶_𝑖0⋯0𝐷_𝑖0⋯0^𝑇𝜀⁻²(𝑑−ℎ)𝑋₁+𝜀⁻²𝑑𝑋₂+𝜀⁻²𝑃_𝑖𝐶_𝑖0⋯0𝐷_𝑖,Ξ0⋯0₄=diag𝜇−1𝑅,−𝑋₂,…,−𝑋₂,−𝑚⁻¹𝑋₂,−𝑋₁,…,−𝑋₁,𝐾₁=𝐾^𝑇₁𝐾₂+𝐾^𝑇₂𝐾₁2,𝐾₂𝐾=−^𝑇1+𝐾^𝑇₂2. (14)

Proof. For presentation convenience, in the following, we denote 𝑦(𝑡)(𝑟(𝑡𝑥(𝑡)(𝑟(𝑡𝑥((𝑡(𝑡)(𝑥(𝑡,𝑚(𝑡)(𝑟(𝑡𝑥(𝑡)(𝑟(𝑡𝑥((𝑡,=−𝐴))+𝐵))𝑡−𝑑))+𝜉𝐸𝑓))=𝐶))+𝐷))𝑡−𝑑)) (15) then, the system (8) becomes 𝑑𝑥(𝑡)=𝑦(𝑡)𝑑𝑡+𝑚(𝑡)𝑑𝑤(𝑡). (16) By employing the idea of delay partitioning approach to the time-delay 𝑑(𝑡), we choose the following Lyapunov-Krasovskii functional candidate: 𝑉(𝑥(𝑡),𝑡,𝑖)=𝑥^𝑇(𝑡)𝑃_𝑖𝑥(𝑡)+^𝑚_𝑘=1^𝑡_{𝑡−𝛼𝑘(𝑡)}𝑥^𝑇(𝑠)𝑄_𝑘𝑥(𝑠)+𝑑𝑠_{𝑡𝑡−ℎ}𝛾^𝑇(𝑠)(𝑠)+𝑊𝛾𝑑𝑠_{𝑡(𝑡)𝑡−𝑑}𝑓^𝑇(𝑥(𝑠))𝑅𝑓(𝑥(𝑠))𝑑𝑠+ℎ^−ℎ_−𝑑^𝑡_𝑡+𝜃𝑦^𝑇(𝑠)𝑋₁𝑦(𝑠)+𝑑𝑠𝑑𝜃_{−ℎ−𝑑𝑡𝑡+𝜃}𝑚^𝑇(𝑠)𝑋₁𝑚(𝑠)𝑑𝑠𝑑𝜃+𝜎⁰_−𝑑^𝑡_𝑡+𝜃𝑦^𝑇(𝑠)𝑋₂𝑦(𝑠)+𝑑𝑠𝑑𝜃_{0−𝑑𝑡𝑡+𝜃}𝑚^𝑇(𝑠)𝑋₂𝑚(𝑠)𝑑𝑠𝑑𝜃, (17) where 𝛾^𝑇(𝑡)=𝑥^𝑇(𝑡)𝑥^𝑇(𝑡−ℎ)⋯𝑥^𝑇(()ℎ),𝜎(𝑡)=𝑑(𝑡)𝑡−𝑀−1𝑚𝑑,ℎ=𝑀𝑑,𝜎=𝑚,𝛼_𝑘(𝑡)=𝑘𝜎(𝑡),(𝑘=1,…,𝑚). (18) Let ℒ be the weak infinitesimal generator of the random process {𝑥(𝑡),𝑟(𝑡),𝑡≥0} along the trajectory of the system (8). Then, by Itô differential formula, we have ℒ𝑉(𝑥(𝑡),𝑡,𝑖)=2𝑥^𝑇(𝑡)𝑃_𝑖𝑦(𝑡)+𝑥^𝑇(𝑡)^𝑁_𝑗=1𝑞_𝑖𝑗𝑃_𝑗𝑥(𝑡)+𝑚^𝑇(𝑡)𝑃_𝑖𝑚(𝑡)+^𝑚_𝑖=1𝑥^𝑇(𝑡)𝑄_𝑖𝑥(𝑡)−^𝑚_𝑖=11−𝛼̇_𝑖(𝑡)𝑥^𝑇𝑡−𝛼_𝑖(𝑡)𝑄_𝑖𝑥𝑡−𝛼_𝑖(𝑡)+𝛾^𝑇(𝑡)𝑊𝛾(𝑡)−𝛾^𝑇(𝑡−ℎ)𝑊𝛾(𝑡−ℎ)+𝑓^𝑇(𝑥(𝑡(𝑥(𝑡−𝑑̇(𝑡)𝑓))𝑅𝑓))1−^𝑇(𝑥((𝑡(𝑥((𝑡()𝑦𝑡−𝑑)))𝑅𝑓𝑡−𝑑)))+ℎ𝑑−ℎ^𝑇(𝑡)𝑋₁𝑦(𝑡)−ℎ^𝑀_𝑘=2^{𝑡−(𝑘−1)ℎ}_{𝑡−𝑘ℎ}𝑦^𝑇(𝑠)𝑋₁𝑦(𝑠)+()𝑚𝑑𝑠𝑑−ℎ^𝑇(𝑡)𝑋₁𝑚(𝑡)−^𝑀_𝑘=2^{𝑡−(𝑘−1)ℎ}_{𝑡−𝑘ℎ}𝑚^𝑇(𝑠)𝑋₁𝑚(𝑠)𝑑𝑠+𝜎𝑑𝑦^𝑇(𝑡)𝑋₂𝑦(𝑡)−𝜎^𝑚_𝑖=1^{𝑡−𝛼𝑖−1(𝑡)}_{𝑡−𝛼𝑖(𝑡)}𝑦^𝑇(𝑠)𝑋₂𝑦(𝑠)𝑑𝑠−𝜎^{𝑡−𝑑(𝑡)}_𝑡−𝑑𝑦^𝑇(𝑠)𝑋₂𝑦(𝑠)𝑑𝑠+𝑑𝑚^𝑇(𝑡)𝑋₂𝑚(𝑡)−^𝑚_𝑖=1^{𝑡−𝛼𝑖−1(𝑡)}_{𝑡−𝛼𝑖(𝑡)}𝑚^𝑇(𝑠)𝑋₂𝑚(𝑠)−𝑑𝑠_{𝑡−𝑑(𝑡)𝑡−𝑑}𝑚^𝑇(𝑠)𝑋₂𝑚(𝑠)𝑑𝑠. (19) By virtue of (3), (7), and (9), it can be verified that 𝐸{ℒ𝑉(𝑥(𝑡),𝑡,𝑖)}≤𝐸2𝑥^𝑇(𝑡)𝑃_𝑖𝑦(𝑡)+𝑥^𝑇(𝑡)^𝑁_𝑗=1𝑞_𝑖𝑗𝑃_𝑗𝑥(𝑡)+𝑚^𝑇(𝑡)𝑃_𝑖𝑚(𝑡)−^𝑚_𝑖=1𝑖1−𝑚𝜇𝑥^𝑇𝑡−𝛼_𝑖(𝑡)𝑄_𝑖𝑥𝑡−𝛼_𝑖(𝑡)+^𝑚_𝑖=1𝑥^𝑇(𝑡)𝑄_𝑖𝑥(𝑡)+𝛾^𝑇(𝑡)𝑊𝛾(𝑡)−𝛾^𝑇(𝑡−ℎ)𝑊𝛾(𝑡−ℎ)+𝑓^𝑇(𝑥(𝑡(𝑥(𝑡−𝑓))𝑅𝑓))1−𝜇^𝑇(𝑥((𝑡(𝑥((𝑡()𝑦𝑡−𝑑)))×𝑅𝑓𝑡−𝑑)))+ℎ𝑑−ℎ^𝑇(𝑡)𝑋₁𝑦(𝑡)+(𝑑−ℎ)𝑚^𝑇(𝑡)𝑋₁𝑚(𝑡)+𝜎𝑑𝑦^𝑇(𝑡)𝑋₂𝑦(𝑡)+𝑑𝑚^𝑇(𝑡)𝑋₂𝑚(𝑡)−^𝑀_𝑘=2^{𝑡−(𝑘−1)ℎ}_{𝑡−𝑘ℎ}𝑦(𝑠)𝑑𝑠^𝑇𝑋₁×^{𝑡−(𝑘−1)ℎ}_{𝑡−𝑘ℎ}𝑦(𝑠)−𝑑𝑠_{𝑀𝑘=2𝑡−(𝑘−1)ℎ𝑡−𝑘ℎ}𝑚(𝑠)𝑑𝜔(𝑠)^𝑇𝑋₁×^{𝑡−(𝑘−1)ℎ}_{𝑡−𝑘ℎ}𝑚(𝑠)(𝑠)−𝑑𝜔_{𝑚𝑖=1𝑡−𝛼𝑖−1(𝑡)𝑡−𝛼𝑖(𝑡)}𝑦(𝑠)𝑑𝑠^𝑇𝑋₂×^{𝑡−𝛼𝑖−1(𝑡)}_{𝑡−𝛼𝑖(𝑡)}𝑦(𝑠)−1𝑑𝑠𝑚^{𝑡−𝑑(𝑡)}_𝑡−𝑑𝑦(𝑠)𝑑𝑠^𝑇𝑋₂×^{𝑡−𝑑(𝑡)}_𝑡−𝑑𝑦(𝑠)−𝑑𝑠_{𝑚𝑖=1𝑡−𝛼𝑖−1(𝑡)𝑡−𝛼𝑖(𝑡)}𝑚(𝑠)𝑑𝜔(𝑠)^𝑇𝑋₂×^{𝑡−𝛼𝑖−1(𝑡)}_{𝑡−𝛼𝑖(𝑡)}𝑚(𝑠)(𝑠)−1𝑑𝜔𝑚^{𝑡−𝑑(𝑡)}_𝑡−𝑑𝑚(𝑠)𝑑𝜔(𝑠)^𝑇𝑋₂×^{𝑡−𝑑(𝑡)}_𝑡−𝑑𝑚(𝑠)𝑑𝜔(𝑠). (20) On the other hand, note that (6) is equivalent to 𝑥𝑓(𝑥)^𝑇𝐾₁𝐾₂𝑥∗𝐼𝑓(𝑥)≤0, (21) which implies −𝑙𝑓^𝑇(𝑥(𝑡))𝑓(𝑥(𝑡))+𝑓^𝑇(𝑥(𝑡))𝐾^𝑇₂𝑥(𝑡)+𝑥^𝑇(𝑡)𝐾₂𝑓(𝑥(𝑡))+𝑥^𝑇(𝑡)𝐾₁𝑥(𝑡)≥0, (22) where 𝑙 is a positive constant.
Substituting (5), (7), (15), and (22) into (20), we can obtain 𝐸{ℒ𝑉(𝑥(𝑡),𝑡,𝑖)}≤𝐸2𝑥^𝑇(𝑡)𝑃_𝑖−𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑥(𝑡−𝑑(𝑡))−𝑙𝑥^𝑇(𝑡)𝐾₁𝑥(𝑡)+2𝑥^𝑇(𝑡)𝜀⁻¹𝜉𝑃_𝑖𝐸−𝑙𝐾₂×𝑓(𝑥(𝑡))+𝑥^𝑇(𝑡)^𝑁_𝑗=1𝑞_𝑖𝑗𝑃_𝑗𝑥(𝑡)+ℎ^𝑇(𝑡)𝑃_𝑖ℎ(𝑡)−^𝑚_𝑖=1𝑖1−𝑚𝜇𝑥^𝑇×𝑡−𝛼_𝑖(𝑡)𝑄_𝑖𝑥𝑡−𝛼_𝑖(𝑡)+^𝑚_𝑖=1𝑥^𝑇(𝑡)𝑄_𝑖𝑥(𝑡)+𝛾^𝑇(𝑡)𝑊𝛾(𝑡)−𝛾^𝑇(𝑡−ℎ)𝑊𝛾(𝑡−ℎ)+𝑓^𝑇(𝑥(𝑡)𝑓(𝑥(𝑡()𝑦))(𝑅−𝑙𝐼))+ℎ𝑑−ℎ^𝑇(𝑡)𝑋₁𝑦(𝑡)−1−𝜇𝑓^𝑇×(𝑥((𝑡(𝑥((𝑡+()𝑚𝑡−𝑑)))𝑅𝑓𝑡−𝑑)))𝑑−ℎ^𝑇(𝑡)𝑋₁𝑚(𝑡)+𝜎𝑑𝑦^𝑇(𝑡)𝑋₂𝑦(𝑡)+𝑑𝑚^𝑇(𝑡)𝑋₂𝑚(𝑡)−^𝑀_𝑘=2^{𝑡−(𝑘−1)ℎ}_{𝑡−𝑘ℎ}𝑦(𝑠)𝑑𝑠^𝑇𝑋₁×^{𝑡−(𝑘−1)ℎ}_{𝑡−𝑘ℎ}𝑦(𝑠)−𝑑𝑠_{𝑀𝑘=2𝑡−(𝑘−1)ℎ𝑡−𝑘ℎ}𝑚(𝑠)𝑑𝜔(𝑠)^𝑇𝑋₁×^{𝑡−(𝑘−1)ℎ}_{𝑡−𝑘ℎ}𝑚(𝑠)(𝑠−𝑑𝜔)_{𝑚𝑖=1𝑡−𝛼𝑖−1(𝑡)𝑡−𝛼𝑖(𝑡)}𝑦(𝑠)𝑑𝑠^𝑇𝑋₂×^{𝑡−𝛼𝑖−1(𝑡)}_{𝑡−𝛼𝑖(𝑡)}𝑦(𝑠)−1𝑑𝑠𝑚^{𝑡−𝑑(𝑡)}_𝑡−𝑑𝑦(𝑠)𝑑𝑠^𝑇𝑋₂×^{𝑡−𝑑(𝑡)}_𝑡−𝑑𝑦(𝑠)−𝑑𝑠_{𝑚𝑖=1𝑡−𝛼𝑖−1(𝑡)𝑡−𝛼𝑖(𝑡)}𝑚(𝑠)𝑑𝜔(𝑠)^𝑇𝑋₂×^{𝑡−𝛼𝑖−1(𝑡)}_{𝑡−𝛼𝑖(𝑡)}𝑚(𝑠)(𝑠)−1𝑑𝜔𝑚^{𝑡−𝑑(𝑡)}_𝑡−𝑑𝑚(𝑠)𝑑𝜔(𝑠)^𝑇𝑋₂×^{𝑡−𝑑(𝑡)}_𝑡−𝑑𝑚(𝑠)(𝑠𝜉𝑑𝜔)=𝐸^𝑇(𝑡)Ξ₁+Ξ₂+Ξ₃0∗Ξ₄𝜉(𝑡), (23) where ^𝜉(𝑡𝜁^{)=𝜁𝑇1(𝑡)𝜁𝑇2(𝑡)𝑓𝑇(𝑥(𝑡𝑓))𝑇(𝑥((𝑡𝜁𝑡−𝑑)))𝑇(𝑡)𝑇,}₁(𝑡)=_𝑥^𝑇_(𝑡)𝑥^𝑇_{𝑡−𝛼1(𝑡)⋯𝑥}^𝑇_{𝑡−𝛼𝑚(𝑡)}^𝑇,𝜁₂(𝑡)=_𝑥^𝑇_{()𝑥𝑡−𝑑}^𝑇_{(()ℎ)𝑡−𝑀−1⋯𝑥}^𝑇_()𝑡−ℎ^𝑇,_{𝜁(𝑡)=𝛿1⋯𝛿𝑚∫𝑡−𝑑(𝑡)𝑡−𝑑𝑦𝑇(𝑠)𝑑𝑠+∫𝑡−𝑑(𝑡)𝑡−𝑑𝑚𝑇(𝑠)(𝑠)𝜃𝑑𝜔𝑀−1𝜃1𝑇,𝛿𝑘=∫𝑡−𝛼𝑘−1(𝑡)𝑡−𝛼𝑘(𝑡)𝑦𝑇(𝑠)𝑑𝑠+∫𝑡−𝛼𝑘−1(𝑡)𝑡−𝛼𝑘(𝑡)𝑚𝑇(𝑠)(𝑠),(),𝜃𝑑𝜔𝑘=1,…,𝑚𝑘=∫𝑡−𝑘ℎ𝑡−(𝑘+1)ℎ𝑦𝑇(𝑠)∫𝑑𝑠+𝑡−𝑘ℎ𝑡−(𝑘+1)ℎ𝑚𝑇(𝑠)(𝑠),().𝑑𝜔𝑘=1,…,𝑀−1} (24) In addition, it follows from the Newton-Leibniz formula that 𝑥(𝑡)−𝑥𝑡−𝛼₁(𝑡)−^𝑡_{𝑡−𝛼1(𝑡)}𝑦(𝑠)−𝑑𝑠_{𝑡𝑡−𝛼1(𝑡)}𝑚(𝑠)(𝑠)⋮𝑥𝑑𝜔=0,𝑡−𝛼_𝑚−1(𝑡)−𝑥𝑡−𝛼_𝑚(𝑡)−^{𝑡−𝛼𝑚−1(𝑡)}_{𝑡−𝛼𝑚(𝑡)}𝑦(𝑠)−𝑑𝑠_{𝑡−𝛼𝑚−1(𝑡)𝑡−𝛼𝑚(𝑡)}𝑚(𝑠)(𝑠)𝑥𝑑𝜔=0,𝑡−𝛼_𝑚(𝑡)−𝑥(𝑡−𝑑)−^{𝑡−𝛼𝑚(𝑡)}_𝑡−𝑑𝑦(𝑠)−𝑑𝑠_{𝑡−𝛼𝑚(𝑡)𝑡−𝑑}𝑚(𝑠)(𝑠)𝑥()(()ℎ)+𝑑𝜔=0,𝑡−𝑑−𝑥𝑡−𝑀−1^{𝑡−(𝑀−1)ℎ}_𝑡−𝑑𝑦(𝑠)+𝑑𝑠_{𝑡−(𝑀−1)ℎ𝑡−𝑑}𝑚(𝑠)(𝑠)⋮𝑥()()+𝑑𝜔=0,𝑡−2ℎ−𝑥𝑡−ℎ^𝑡−ℎ_𝑡−2ℎ𝑦(𝑠)+𝑑𝑠_{𝑡−ℎ𝑡−2ℎ}𝑚(𝑠)𝑑𝜔(𝑠)=0 (25) which are equivalent to 𝐵𝜉(𝑡)=𝐽_𝑚+1𝐼_𝑛𝐹𝜉(𝑡)=𝐼_𝑛−𝐼_𝑛0⋯000−𝐼_𝑛0⋯00𝐼_𝑛−𝐼_𝑛⋯0000−𝐼_𝑛⋯0⋮⋮⋱⋱⋮⋮⋮⋮⋮⋱⋮00⋯𝐼_𝑛−𝐼_𝑛0000⋯−𝐼_𝑛𝜉(𝑡)=0. (26) The full column rank matrix representation of the right orthogonal complement of 𝐵 is denoted by 𝐵^⟂, and Lemma 6 offers a computation method with 𝑍=𝐼_𝑛, 𝐵^⟂=col{−∑^𝐾_𝑗=1𝐹_𝑗,−∑^𝐾_𝑗=2𝐹_𝑗,…,−𝐹_𝐾,𝐼_𝑚}. According to Lemma 7, 𝐸{ℒ𝑉(𝑥(𝑡),𝑡,𝑖)} is negative as long as 𝜉^𝑇(𝑡)𝐵^⟂Ξ𝐵−¹+Ξ₂+Ξ₃0∗Ξ₄𝜉(𝑡)>0 (27) holds, which, in other words, can be expressed as (13). Hence, the stochastic system (8) is asymptotical stable in the mean square, and it completes the proof.

Remark 10. The delay-partitioning projection technique employed in this paper constitutes the major improvement from most existing results in the literature. Firstly, it should be pointed out that such technique is very rational. The reasons are twofold. (1) The properties of subinterval delays may be sharply different in many practical situations. Thus, it is not reasonable to combine them together. (2) When 𝑑(𝑡) reaches its upper bound, we do not necessarily have every subinterval delay reaches its maximum at the same time. That is to say, if we use an upper bound to bound the delay 𝑑(𝑡) we have to use the sum of the maxima of subinterval delays; however, 𝑑(𝑡) does not achieve this maximum value usually. Therefore, by adopting the delay-partitioning projection approach, less conservative conditions can be proposed. Secondly, in [46], the central point of variation of the delay was introduced to study the stability for time-delay systems, which is called the DCP method. As an extension of the method, we divide the delay into several subintervals which permits employing one slightly different function for different subintervals. This treatment makes us utilize more information on the time delay and thus may present better criteria with less conservatism. Thirdly, two different partitions are made to deal with the time-varying delay in this paper, which are different from the approach in [43]. The parameters 𝑚 and 𝑀 refer to the number of delay partitioning, and it indicates that the solution can be searched in a wider space, which leads to the reduction of conservatism. Finally, using the similar analysis method of [7], it is easy to verify that the conservatism of the conditions is a nonincreasing function of the number of delay partitions. However, as we all know, the computational complexity will be increased as the partitioning becomes thinner. Therefore, the delay-partitioning projection approach can provide the flexibility that allows us to trade off between complexity and performance of the stability analysis.

We now consider that the time-varying delay is nondifferentiable or the bound of the delay is unknown, which means that the restriction on the delay is removed. For this goal, we modify the Lyapunov-Krasovskii functional as 𝑉(𝑥(𝑡),𝑡,𝑖)=𝑥^𝑇(𝑡)𝑃_𝑖𝑥(𝑡)+^𝑡_𝑡−ℎ𝛾^𝑇(𝑠)(𝑠)+𝑊𝛾𝑑𝑠_{𝑡𝑡−𝑑}𝑓^𝑇(𝑥(𝑠))𝑅𝑓(𝑥(𝑠))𝑑𝑠+ℎ^−ℎ_−𝑑^𝑡_𝑡+𝜃𝑦^𝑇(𝑠)𝑋₁𝑦(𝑠)+𝑑𝑠𝑑𝜃_{−ℎ−𝑑𝑡𝑡+𝜃}𝑚^𝑇(𝑠)𝑋₁𝑚(𝑠)𝑑𝑠𝑑𝜃+𝜎⁰_−𝑑^𝑡_𝑡+𝜃𝑦^𝑇(𝑠)𝑋₂𝑦(𝑠)+𝑑𝑠𝑑𝜃_{0−𝑑𝑡𝑡+𝜃}𝑚^𝑇(𝑠)𝑋₂𝑚(𝑠)𝑑𝑠𝑑𝜃. (28) The proof can then be derived by following a similar line of arguments as that in Theorem 9. Besides, 𝑓(𝑥(𝑡−𝑑(𝑡))) in 𝜉(𝑡) will turn out to be 𝑓(𝑥(𝑡−𝑑)).

Theorem 11. For given constant 𝑑 and two positive integers 𝑚 and 𝑀, the stochastic system (8) is mean square asymptotically stable if there exist matrices 𝑃_𝑖>0, 𝑅>0, 𝑋₁>0, 𝑋₂>0, and 𝑊=𝑊^𝑇=𝑊₁₁⋯𝑊_1𝑀𝑊⋮⋱⋮_𝑇1𝑀⋯𝑊_𝑀𝑀>0, (29) and positive scalar 𝑙>0 such that the following LMI holds: 𝐵^⟂𝑇Ξ₁+Ξ₂+Ξ₃0∗Ξ₄𝐵^⟂<0, (30) where

Ξ₁=Ω₁0⋯𝜀⁻¹𝑃_𝑖𝐵_𝑖0𝑊_1𝑀⋯𝑊₁₂Δ₁∗0⋯000⋯00⋮⋮⋱⋮⋮⋮⋱⋮⋮∗∗⋯000⋯00∗∗⋯∗Γ_𝑀+1−𝑊^𝑇_{(𝑀−1)𝑀}⋯−𝑊^𝑇_1𝑀0∗∗⋯∗∗Γ_𝑀⋯𝑊^𝑇_2𝑀−𝑊^𝑇_1(𝑀−1)0⋮⋮⋱⋮⋮⋮⋱⋮⋮∗∗⋯∗∗∗⋯Γ₂0∗∗⋯∗∗∗⋯∗Δ₂,Ω₁=−2𝜀⁻¹𝑃_𝑖𝐴_𝑖+^𝑁_𝑗=1𝑞_𝑖𝑗𝑃_𝑗+𝑊₁₁−𝑙𝐾₁,Ξ₄=diag−𝑅,−𝑋₂,…,−𝑋₂,−𝑚⁻¹𝑋₂,−𝑋₁,…,−𝑋₁, (31)

and 𝐵^⟂, Δ₁, Δ₂, Γ_𝑘, Ξ₂, Ξ₃, 𝐾₁, 𝐾₂ are the same as in Theorem 9.

Remark 12. Obviously, the delay rate is not specified in Theorem 11, which includes the constant time delay as a special case. Therefore, Theorem 11 is applicable to the special case to derive some results.

Remark 13. In previous work as [47], the authors have investigated the stability of neutral type neural networks with discrete and distributed delays. The reason why we do not take these neural networks into account is to make our idea more lucid and to avoid complicated notations. However, it is not difficult to extend our results to the neutral type neural networks with mixed time-varying delays. The results will appear in our following study.

4. A Numerical Example

One illustrative example is presented to show the effectiveness of the theoretical results. For simplicity, the system (8) with Markovian switching between two modes is taken into consideration. In addition, the state vector of each node is of dimension two. In other words, 𝑁=2, 𝑛=2. The mode switching is governed by the rate matrix −0.40.40.3−0.3, and the other parameters are assumed as 𝐴₁=3.3003.4,𝐴₂=,𝐵4.4004.5₁=−0.3−0.6−0.50.5,𝐵₂=,−2.2−0.5−1.21.2𝐸=−100−0.1,𝐶₁=,𝐶0.10.20.10.5₂=0.510.20.1,𝐷₁=,𝐷0.1−0.2−0.50.4₂=,0.2−0.5−0.10.2𝜉=0.56,𝜀=1,𝑑(𝑡)=0.8+0.5sin𝑡𝜋2(𝑡∈𝑍). (32) The sector-bounded nonlinear function 𝑓(𝑥(𝑡)) is given as follows: 𝑓(𝑥(𝑡=1))20.3𝑥₁(𝑡)+𝑥₂(𝑡)1+𝑥²₁(𝑡)+𝑥²₂(𝑡)+0.1𝑥₁(𝑡)+0.1𝑥₂(𝑡)0.3𝑥₁(𝑡)+0.3𝑥₂(𝑡)^𝑇, (33) which can be bounded by 𝐾₁=0.20.100.2,𝐾₂=−0.10−0.10.1. (34) We choose the parameters 𝑚=2 and 𝑀=2. It is easy to know 𝑑=1.3 and 𝜇=0.5.

By using Matlab LMI control toolbox, we can find the feasible solution of the LMI (13) as follows: 𝑃₁=,𝑃1.8392−0.8471−0.84714.5444₂=,𝑄2.0534−0.5930−0.59303.9335₁=,𝑄0.4422−0.3769−0.37691.8266₂=,𝑋7.6178−2.8997−2.899711.9753₁=,𝑋0.0594−0.1562−0.15620.4677₂=,,𝑊0.0663−0.1841−0.18410.5600𝑅=0.2487−0.0084−0.00840.5947₁₁=,𝑊1.1837−0.4007−0.40073.6472₁₂=,𝑊0.0111−0.1045−0.10450.4755₂₂=.0.7856−0.4878−0.48782.4023 (35) The simulation result is shown in Figure 1, which implies that all the expected system performance requirements are well achieved.

Figure 1 

State development of system (8) with parameters (32).

Remark 14. It is worth pointing out that if Prob{𝜉(𝑡)=1}=1, then the function 𝑓(𝑥(𝑡)) in (8) will be the neuron activation function. According to Assumption 1, 𝑓(𝑥(𝑡)) satisfies the sector-bounded condition which is more general than the Lipschitz condition. Therefore, the stability criteria in [42] can not apply to our example.

Remark 15. In our example, many factors such as noise perturbations, Markovian jump parameters, RONs, and different time scales are considered and the delay-partitioning projection approach is employed. Therefore, the results reported in [9, 23, 38–41] do not hold in our example. Moreover, our results are expressed by LMIs, which can be easily checked by using the powerful Matlab LMI Toolbox. Thus, our stability criteria are more computationally efficient than those given in [13, 40, 41].

5. Conclusions

In this paper, we have dealt with the mean square asymptotic stability problem for stochastic Markovian jump neural networks with different time scales and RONs. By using new Lyapunov-Krasovskii functionals and delay-partitioning projection approach, the LMI-based criteria have been developed to guarantee stability of such systems. It is also shown that the delay-partitioning projection approach can achieve the aim of reducing conservativeness. An numerical example is exploited to illustrate the effectiveness of the theoretical results.

Acknowledgments

This work was supported by the Fundamental Research Funds for the Central Universities (JUSRP51317B) and the National Natural Science Foundation of China under Grant 10901073 and 11002061.