Stochastic Passivity of Uncertain Neural Networks with Time-Varying Delays

Abstract

The passivity problem is investigated for a class of stochastic uncertain neural networks with time-varying delay as well as generalized activation functions. By constructing appropriate Lyapunov-Krasovskii functionals, and employing Newton-Leibniz formulation, the free-weighting matrix method, and stochastic analysis technique, a delay-dependent criterion for checking the passivity of the addressed neural networks is established in terms of linear matrix inequalities (LMIs), which can be checked numerically using the effective LMI toolbox in MATLAB. An example with simulation is given to show the effectiveness and less conservatism of the proposed criterion. It is noteworthy that the traditional assumptions on the differentiability of the time-varying delays and the boundedness of its derivative are removed.

1. Introduction

During the last two decades, many artificial neural networks have been extensively investigated and successfully applied to various areas such as signal processing, pattern recognition, associative memory, and optimization problems [1]. In such applications, it is of prime importance to ensure that the designed neural networks are stable [2].

In hardware implementation, time delays are likely to be present due to the finite switching speed of amplifiers and communication time. It has also been shown that the processing of moving images requires the introduction of delay in the signal transmitted through the networks [3]. The time delays are usually variable with time, which will affect the stability of designed neural networks and may lead to some complex dynamic behavior such as oscillation, bifurcation, or chaos [4]. Therefore, the study of stability with consideration of time delays becomes extremely important to manufacture high quality neural networks [5]. Many important results on stability of delayed neural networks have been reported, see [1–10] and the references therein for some recent publications.

It is also well known that parameter uncertainties, which are inherent features of many physical systems, are great sources of instability and poor performance [11]. These uncertainties may arise due to the variations in system parameters, modelling errors, or some ignored factors [12]. It is not possible to perfectly characterize the evolution of an uncertain dynamical system as a deterministic set of state equations [13]. Recently, the problem on robust stability analysis of uncertain neural networks with delays has been extensively investigated, see [11–14] and the references therein for some recent publications.

Just as pointed out in [15], in real nervous systems, synaptic transmission is a noisy process brought on by random fluctuations from the release of neurotransmitters and other probabilistic causes. In the implementation of artificial neural networks, noise is unavoidable and should be taken into consideration in modelling. Therefore, it is of significant importance to consider stochastic effects to the dynamical behavior of neural networks [16]. Some recent interest results on stability of stochastic neural networks can be found, see [15–26] and the references therein for some recent publications.

On the other hand, the passivity theory is another effective tool to the stability analysis of nonlinear system [27]. The main idea of passivity theory is that the passive properties of system can keep the system internal stability [27]. Thus, the passivity theory has received a lot of attention from the control community since 1970s [28–31]. Recently, the passivity theory for delayed neural networks was investigated, some criteria checking the passivity were provided for certain or uncertain neural networks, see [32–38] and references therein. In [32], the passivity-based approach is used to derive stability conditions for dynamic neural networks with different time scales. In [33–36], authors investigated the passivity of neural networks with time-varying delay. In [37, 38], stochastic neural networks with time-varying delays were considered, several sufficient conditions checking the passivity were obtained. It is worth pointing out that, the given criteria in [33–37] have been based on the following assumptions: (1) the time-varying delays are continuously differentiable; (2) the derivative of time-varying delay is bounded and is smaller than one; (3) the activation functions are bounded and monotonically nondecreasing. However, time delays can occur in an irregular fashion, and sometimes the time-varying delays are not differentiable. In such a case, the methods developed in [33–38] may be difficult to be applied, and it is therefore necessary to further investigate the passivity problem of neural networks with time-varying delays under milder assumptions. To the best of our knowledge, few authors have considered the passivity problem for stochastic uncertain neural networks with time-varying delays as well as generalized activation functions.

Motivated by the above discussions, the objective of this paper is to study the passivity of stochastic uncertain neural networks with time-varying delays as well as generalized activation functions by employing a combination of Lyapunov functional, the free-weighting matrix method and stochastic analysis technique. The obtained sufficient conditions require neither the differentiability of time-varying delays nor the monotony of the activation functions, and are expressed in terms of linear matrix inequalities (LMIs), which can be checked numerically using the effective LMI toolbox in MATLAB. An example is given to show the effectiveness and less conservatism of the proposed criterion.

2. Problem Formulation and Preliminaries

In this paper, we consider the following stochastic uncertain neural networks with time-varying delay:

𝑑𝑥(𝑡)=[−(𝐶+Δ𝐶(𝑡))𝑥(𝑡)+(𝐴+Δ𝐴(𝑡))𝑓(𝑥(𝑡))+(𝐵+Δ𝐵(𝑡))𝑓(𝑥(𝑡−𝜏(𝑡)))+𝑢(𝑡)]𝑑𝑡+𝜎(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)))𝑑𝜔(𝑡) (2.1) for 𝑡≥0, where 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇∈𝑅^𝑛 is the state vector of the network at time 𝑡, 𝑛 corresponds to the number of neurons; 𝐶=diag(𝑐₁,𝑐₂,…,𝑐_𝑛) is a positive diagonal matrix, 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛, and 𝐵=(𝑏_𝑖𝑗)_𝑛×𝑛 are known constant matrices; Δ𝐶(𝑡), Δ𝐴(𝑡) and Δ𝐵(𝑡) are time-varying parametric uncertainties; 𝜎(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)))∈𝑅^𝑛×𝑛 is the diffusion coefficient matrix and 𝜔(𝑡)=(𝜔₁(𝑡),𝜔₂(𝑡),…,𝜔_𝑛(𝑡))^𝑇 is an 𝑛-dimensional Brownian motion defined on a complete probability space (Ω,𝐹,{𝐹_𝑡}_𝑡≥0,𝒫) with a filtration {𝐹_𝑡}_𝑡≥0 satisfying the usual conditions (i.e., it is right continuous and 𝐹₀ contains all 𝑃-null sets); 𝑓(𝑥(𝑡))=(𝑓₁(𝑥₁(𝑡)),𝑓₂(𝑥₂(𝑡)),…,𝑓_𝑛(𝑥_𝑛(𝑡)))^𝑇 denotes the neuron activation at time 𝑡; 𝑢(𝑡)=(𝑢₁(𝑡),𝑢₂(𝑡),…,𝑢_𝑛(𝑡))^𝑇∈𝑅^𝑛 is a varying external input vector; 𝜏(𝑡)>0 is the time-varying delay, and is assumed to satisfy 0≤𝜏(𝑡)≤𝜏, where 𝜏 is constant.

The initial condition associated with model (2.1) is given by 𝑥(𝑠)=𝜙(𝑠),𝑠∈[−𝜏,0]. (2.2)

Let 𝑥(𝑡,𝜙) denote the state trajectory of model (2.1) from the above initial condition and 𝑥(𝑡,0) the corresponding trajectory with zero initial condition.

Throughout this paper, we make the following assumptions.

(H1) [33] The time-varying uncertainties Δ𝐶(𝑡), Δ𝐴(𝑡) and Δ𝐵(𝑡) are of the form

Δ𝐶(𝑡)=𝐻₁𝐺₁(𝑡)𝐸₁,Δ𝐴(𝑡)=𝐻₂𝐺₂(𝑡)𝐸₂,Δ𝐵(𝑡)=𝐻₃𝐺₃(𝑡)𝐸₃, (2.3) where 𝐻₁, 𝐻₂, 𝐻₃, 𝐸₁, 𝐸₂, and 𝐸₃ are known constant matrices of appropriate dimensions, 𝐺₁(𝑡), 𝐺₂(𝑡), and 𝐺₃(𝑡) are known time-varying matrices with Lebesgue measurable elements bounded by

𝐺^𝑇₁(𝑡)𝐺₁(𝑡)≤𝐼,𝐺^𝑇₂(𝑡)𝐺₂(𝑡)≤𝐼,𝐺^𝑇₃(𝑡)𝐺₃(𝑡)≤𝐼. (2.4)

(H2) [10] For any 𝑗∈{1,2,…,𝑛}, 𝑓_𝑗(0)=0 and there exist constants 𝐹⁻_𝑗 and 𝐹⁺_𝑗 such that

𝐹⁻_𝑗≤𝑓_𝑗(𝛼₁)−𝑓_𝑗(𝛼₂)𝛼₁−𝛼₂≤𝐹⁺_𝑗 (2.5) for all 𝛼₁≠𝛼₂.

(H3) [15] There exist two scalars 𝜌₁>0, 𝜌₂>0 such that the following inequality:

trace[𝜎^𝑇(𝑡,𝑢,𝑣)𝜎(𝑡,𝑢,𝑣)]≤𝜌₁𝑢^𝑇𝑢+𝜌₂𝑣^𝑇𝑣 (2.6) holds for all (𝑡,𝑢,𝑣)∈𝑅×𝑅^𝑛×𝑅^𝑛.

Definition 2.1 (see [33]). System (2.1) is called globally passive in the sense of expectation if there exists a scalar 𝛾>0 such that 2𝐸{^𝑡𝑝₀𝑓^𝑇(𝑥(𝑠))𝑢(𝑠)𝑑𝑠}≥−𝐸{𝛾^𝑡𝑝₀𝑢^𝑇(𝑠)𝑢(𝑠)𝑑𝑠} (2.7) for all 𝑡_𝑝≥0 and for all 𝑥(𝑡,0), where 𝐸{⋅} stands for the mathematical expectation operator with respect to the given probability measure 𝒫.

To prove our results, the following lemmas that can be found in [39] are necessary.

Lemma 2.2 (see [39]). For given matrices 𝐻, 𝐸, and 𝐹 with 𝐹^𝑇𝐹≤𝐼 and a scalar 𝜀>0, the following holds: 𝐻𝐹𝐸+(𝐻𝐹𝐸)^𝑇≤𝜀𝐻𝐻^𝑇+𝜀⁻¹𝐸^𝑇𝐸. (2.8)

Lemma 2.3 (see [39]). For any constant matrix 𝑊∈𝑅^𝑚×𝑚, 𝑊>0, scalar 0<ℎ(𝑡)<ℎ, vector function 𝜔∶[0,ℎ]→𝑅^𝑚 such that the integrations concerned are well defined, then (₀^ℎ(𝑡)𝜔(𝑠)𝑑𝑠)^𝑇𝑊(₀^ℎ(𝑡)𝜔(𝑠)𝑑𝑠)≤ℎ(𝑡)₀^ℎ(𝑡)𝜔^𝑇(𝑠)𝑊𝜔(𝑠)𝑑𝑠. (2.9)

Lemma 2.4 (see [39]). Given constant matrices 𝑃, 𝑄, and 𝑅, where 𝑃^𝑇=𝑃, 𝑄^𝑇=𝑄, then 𝑅𝑃𝑅_𝑇−𝑄<0 (2.10) is equivalent to the following conditions: 𝑄>0,𝑃+𝑅𝑄⁻¹𝑅^𝑇<0. (2.11)

3. Main Results

For presentation convenience, in the following, we denote

𝐹₁=diag(𝐹⁻₁𝐹⁺₁,𝐹⁻₂𝐹⁺₂,…,𝐹⁻_𝑛𝐹⁺_𝑛),𝐹₂𝐹=diag(⁻¹+𝐹⁺₁2,𝐹⁻₂+𝐹⁺₂2𝐹,…,^−𝑛+𝐹⁺_𝑛2). (3.1)

Theorem 3.1. Under assumptions (H1)–(H3), model (2.1) is passive in the sense of expectation if there exist two scalars 𝛾>0, 𝜆>0, three symmetric positive definite matrices 𝑃_𝑖 (𝑖=1,2,3), two positive diagonal matrices 𝐿 and 𝑆, and matrices 𝑄_𝑖 (𝑖=1,2,3,4) such that the following two LMIs hold: 𝑃₁Ω<𝜆𝐼,Ω=¹Ω₂∗Ω₃<0, (3.2)(3.3) where Ω₁=Ω₁₁Ω₁₂Ω₁₃𝑄₂𝐵𝑄₃0𝑄₂∗Ω₂₂𝑄₁𝐴𝑄₁𝐵00𝑄₁∗∗Ω₃₃000−𝐼∗∗∗Ω₄₄𝐹₂𝑆00∗∗∗∗Ω₅₅𝑄₄0∗∗∗∗∗−𝑃₂0,Ω∗∗∗∗∗∗−𝛾𝐼₂=𝑄₂𝐻₁𝑄₂𝐻₂𝑄₂𝐻₃𝑄₃𝑄₃𝑄₄0000000000𝑄₁𝐻₁𝑄₁𝐻₂𝑄₁𝐻₃000000000000000000000000000000𝑄₄,Ω00000000000000000000₃=diag{−𝜀₄𝐼,−𝜀₅𝐼,−𝜀₆𝐼,−𝜏𝑃₃,−𝑃₁,−𝑃₁,−𝜀₁𝐼,−𝜀₂𝐼,−𝜀₃𝐼,−𝜏𝑃₃}, (3.4) in which Ω₁₁=𝑃₂−𝑄₂𝐶−𝐶𝑄^𝑇₂+(𝜀₁+𝜀₄)𝐸^𝑇₁𝐸₁−𝑄₃−𝑄^𝑇₃−𝐹₁𝐿+(1+𝜏)𝜆𝜌₁𝐼, Ω₁₂=𝑃₁−𝐶𝑄^𝑇₁−𝑄₂, Ω₁₃=𝑄₂𝐴+𝐹₂𝐿, Ω₂₂=−𝑄₁−𝑄^𝑇₁+𝜏𝑃₃, Ω₃₃=(𝜀₂+𝜀₅)𝐸^𝑇₂𝐸₂−𝐿, Ω₄₄=(𝜀₃+𝜀₆)𝐸^𝑇₃𝐸₃−𝑆, Ω₅₅=−𝑄₄−𝑄^𝑇₄−𝐹₁𝑆+(1+𝜏)𝜆𝜌₂𝐼.

Proof. Let 𝑦(𝑡)=−(𝐶+Δ𝐶(𝑡))𝑥(𝑡)+(𝐴+Δ𝐴(𝑡))𝑓(𝑥(𝑡))+(𝐵+Δ𝐵(𝑡))𝑓(𝑥(𝑡−𝜏(𝑡)))+𝑢(𝑡), 𝛼(𝑡)=𝜎(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡))), then model (2.1) is rewritten as 𝑑𝑥(𝑡)=𝑦(𝑡)𝑑𝑡+𝛼(𝑡)𝑑𝜔(𝑡). (3.5) Consider the following Lyapunov-Krasovskii functional as 𝑉(𝑡,𝑥(𝑡))=𝑥^𝑇(𝑡)𝑃₁𝑥(𝑡)+^𝑡_𝑡−𝜏𝑥^𝑇(𝑠)𝑃₂𝑥(𝑠)𝑑𝑠+⁰_−𝜏^𝑡_𝑡+𝜃𝑦^𝑇(𝑠)𝑃₃+𝑦(𝑠)𝑑𝑠_{0−𝜏𝑡𝑡+𝜃}trace[𝛼^𝑇(𝑠)𝑃₁𝛼(𝑠)]𝑑𝑠𝑑𝜃. (3.6) By It𝑜̂ differential rule, the stochastic derivative of 𝑉(𝑡) along the trajectory of model (3.5) can be obtained as 𝑑𝑉(𝑡,𝑥(𝑡))={2𝑥^𝑇(𝑡)𝑃₁𝑦(𝑡)+trace[𝛼^𝑇(𝑡)𝑃₁𝛼(𝑡)]+𝑥^𝑇(𝑡)𝑃₂𝑥(𝑡)−𝑥^𝑇(𝑡−𝜏)𝑃₂𝑥(𝑡−𝜏)+𝜏𝑦^𝑇(𝑡)𝑃₃𝑦(𝑡)−^𝑡_𝑡−𝜏𝑦^𝑇(𝑠)𝑃₃𝑦(𝑠)𝑑𝑠+𝜏trace[𝛼^𝑇(𝑡)𝑃₁𝛼(𝑡)]−^𝑡_𝑡−𝜏trace[𝛼^𝑇(𝑠)𝑃₁𝛼(𝑠)]𝑑𝑠}𝑑𝑡+[𝑥^𝑇(𝑡)𝑃₁𝛼(𝑡)+𝛼^𝑇(𝑡)𝑃₁𝑥(𝑡)]𝑑𝜔(𝑡). (3.7) From the definition of 𝑦(𝑡), we have 0=2(𝑦^𝑇(𝑡)𝑄₁+𝑥^𝑇(𝑡)𝑄₂)[−𝑦(𝑡)−(𝐶+Δ𝐶(𝑡))𝑥(𝑡)+(𝐴+Δ𝐴(𝑡))𝑓(𝑥(𝑡))+(𝐵+Δ𝐵(𝑡))𝑓(𝑥(𝑡−𝜏(𝑡)))+𝑢(𝑡)]. (3.8) By assumption (H1) and Lemma 2.2, we get −2𝑦^𝑇(𝑡)𝑄₁Δ𝐶(𝑡)𝑥(𝑡)≤𝜀₁⁻¹𝑦^𝑇(𝑡)𝑄₁𝐻₁𝐻^𝑇₁𝑄^𝑇₁𝑦(𝑡)+𝜀₁𝑥^𝑇(𝑡)𝐸^𝑇₁𝐸₁𝑥(𝑡),2𝑦^𝑇(𝑡)𝑄₁Δ𝐴(𝑡)𝑓(𝑥(𝑡))≤𝜀₂⁻¹𝑦^𝑇(𝑡)𝑄₁𝐻₂𝐻^𝑇₂𝑄^𝑇₁𝑦(𝑡)+𝜀₂𝑓^𝑇(𝑥(𝑡))𝐸^𝑇₂𝐸₂𝑓(𝑥(𝑡)),2𝑦^𝑇(𝑡)𝑄₁Δ𝐵(𝑡)𝑓(𝑥(𝑡−𝜏(𝑡)))≤𝜀₃⁻¹𝑦^𝑇(𝑡)𝑄₁𝐻₃𝐻^𝑇₃𝑄^𝑇₁𝑦(𝑡)+𝜀₃𝑓^𝑇(𝑥(𝑡−𝜏(𝑡)))𝐸^𝑇₃𝐸₃𝑓(𝑥(𝑡−𝜏(𝑡))),−2𝑥^𝑇(𝑡)𝑄₂Δ𝐶(𝑡)𝑥(𝑡)≤𝜀₄⁻¹𝑥^𝑇(𝑡)𝑄₂𝐻₁𝐻^𝑇₁𝑄^𝑇₂𝑥(𝑡)+𝜀₄𝑥^𝑇(𝑡)𝐸^𝑇₁𝐸₁𝑥(𝑡),2𝑥^𝑇(𝑡)𝑄₂Δ𝐴(𝑡)𝑓(𝑥(𝑡))≤𝜀₅⁻¹𝑥^𝑇(𝑡)𝑄₂𝐻₂𝐻^𝑇₂𝑄^𝑇₂𝑥(𝑡)+𝜀₅𝑓^𝑇(𝑥(𝑡))𝐸^𝑇₂𝐸₂𝑓(𝑥(𝑡)),2𝑥^𝑇(𝑡)𝑄₂Δ𝐵(𝑡)𝑓(𝑥(𝑡−𝜏(𝑡)))≤𝜀₆⁻¹𝑥^𝑇(𝑡)𝑄₂𝐻₃𝐻^𝑇₃𝑄^𝑇₂𝑥(𝑡)+𝜀₆𝑓^𝑇(𝑥(𝑡−𝜏(𝑡)))𝐸^𝑇₃𝐸₃𝑓(𝑥(𝑡−𝜏(𝑡))). (3.9) It follows from (3.8) and (3.9) that 0≤𝑥^𝑇(𝑡)[−2𝑄₂𝐶+(𝜀₁+𝜀₄)𝐸^𝑇₁𝐸₁+𝜀₄⁻¹𝑄₂𝐻₁𝐻^𝑇₁𝑄^𝑇₂+𝜀₅⁻¹𝑄₂𝐻₂𝐻^𝑇₂𝑄^𝑇₂+𝜀₆⁻¹𝑄₂𝐻₃𝐻^𝑇₃𝑄^𝑇₂]𝑥(𝑡)+2𝑥^𝑇(𝑡)[−𝐶𝑄^𝑇₁−𝑄₂]𝑦(𝑡)+2𝑥^𝑇(𝑡)𝑄₂𝐴𝑓(𝑥(𝑡))+2𝑥^𝑇(𝑡)𝑄₂𝐵𝑓(𝑥(𝑡−𝜏(𝑡)))+2𝑥^𝑇(𝑡)𝑄₂𝑢(𝑡)+𝑦^𝑇(𝑡)[−2𝑄₁+𝜀₁⁻¹𝑄₁𝐻₁𝐻^𝑇₁𝑄^𝑇₁+𝜀₂⁻¹𝑄₁𝐻₂𝐻^𝑇₂𝑄^𝑇₁+𝜀₃⁻¹𝑄₁𝐻₃𝐻^𝑇₃𝑄^𝑇₁]𝑦(𝑡)+2𝑦^𝑇(𝑡)𝑄₁𝐴𝑓(𝑥(𝑡))+2𝑦^𝑇(𝑡)𝑄₁𝐵𝑓(𝑥(𝑡−𝜏(𝑡)))+2𝑦^𝑇(𝑡)𝑄₁𝑢(𝑡)+(𝜀₂+𝜀₅)𝑓^𝑇(𝑥(𝑡))𝐸^𝑇₂𝐸₂𝑓(𝑥(𝑡))+(𝜀₃+𝜀₆)𝑓^𝑇(𝑥(𝑡−𝜏(𝑡)))𝐸^𝑇₃𝐸₃𝑓(𝑥(𝑡−𝜏(𝑡))). (3.10) Integrating both sides of (3.5) from 𝑡−𝜏(𝑡) to 𝑡, we have 𝑥(𝑡)−𝑥(𝑡−𝜏(𝑡))−^𝑡_{𝑡−𝜏(𝑡)}𝑦(𝑠)𝑑𝑠−^𝑡_{𝑡−𝜏(𝑡)}𝛼(𝑠)𝑑𝜔(𝑠)=0. (3.11) Hence, −2𝑥^𝑇(𝑡)𝑄₃[𝑥(𝑡)−𝑥(𝑡−𝜏(𝑡))−^𝑡_{𝑡−𝜏(𝑡)}𝑦(𝑠)𝑑𝑠−^𝑡_{𝑡−𝜏(𝑡)}𝛼(𝑠)𝑑𝜔(𝑠)]=0. (3.12) By Lemmas 2.2 and 2.3, and noting 𝜏(𝑡)≤𝜏, we get 0=−2𝑥^𝑇(𝑡)𝑄₃𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑄₃𝑥(𝑡−𝜏(𝑡))+2𝑥^𝑇(𝑡)𝑄₃^𝑡_{𝑡−𝜏(𝑡)}𝑦(𝑠)𝑑𝑠+2𝑥^𝑇(𝑡)𝑄₃^𝑡_{𝑡−𝜏(𝑡)}𝛼(𝑠)𝑑𝜔(𝑠)≤−2𝑥^𝑇(𝑡)𝑄₃𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑄₃𝑥(𝑡−𝜏(𝑡))+𝜏𝑥^𝑇(𝑡)𝑄₃𝑃₃⁻¹𝑄^𝑇₃𝑥(𝑡)+^𝑡_{𝑡−𝜏(𝑡)}𝑦^𝑇(𝑠)𝑃₃𝑦(𝑠)𝑑𝑠+𝑥^𝑇(𝑡)𝑄₃𝑃₁⁻¹𝑄^𝑇₃𝑥(𝑡)+(^𝑡_{𝑡−𝜏(𝑡)}𝛼(𝑠)𝑑𝜔(𝑠))^𝑇𝑃₁(^𝑡_{𝑡−𝜏(𝑡)}𝛼(𝑠)𝑑𝜔(𝑠)). (3.13) Integrating both sides of (3.5) from 𝑡−𝜏 to 𝑡−𝜏(𝑡), we have 𝑥(𝑡−𝜏(𝑡))−𝑥(𝑡−𝜏)−^{𝑡−𝜏(𝑡)}_𝑡−𝜏𝑦(𝑠)𝑑𝑠−^{𝑡−𝜏(𝑡)}_𝑡−𝜏𝛼(𝑠)𝑑𝜔(𝑠)=0. (3.14) Similarly, by using of the same way, and noting 𝜏−𝜏(𝑡)≤𝜏, we get 0=−2𝑥^𝑇(𝑡−𝜏(𝑡))𝑄₄(𝑥(𝑡−𝜏(𝑡))−𝑥(𝑡−𝜏)−^{𝑡−𝜏(𝑡)}_𝑡−𝜏𝑦(𝑠)𝑑𝑠−^{𝑡−𝜏(𝑡)}_𝑡−𝜏𝛼(𝑠)𝑑𝜔(𝑠))≤−2𝑥^𝑇(𝑡−𝜏(𝑡))𝑄₄𝑥(𝑡−𝜏(𝑡))+2𝑥^𝑇(𝑡−𝜏(𝑡))𝑄₄𝑥(𝑡−𝜏)+𝜏𝑥^𝑇(𝑡−𝜏(𝑡))𝑄₄𝑃₃⁻¹𝑄^𝑇₄𝑥(𝑡−𝜏(𝑡))+^{𝑡−𝜏(𝑡)}_𝑡−𝜏𝑦^𝑇(𝑠)𝑃₃𝑦(𝑠)𝑑𝑠+𝑥^𝑇(𝑡)𝑄₄𝑃₁⁻¹𝑄^𝑇₄𝑥(𝑡)+(^{𝑡−𝜏(𝑡)}_𝑡−𝜏𝛼(𝑠)𝑑𝜔(𝑠))^𝑇𝑃₁(^{𝑡−𝜏(𝑡)}_𝑡−𝜏𝛼(𝑠)𝑑𝜔(𝑠)). (3.15) From assumption (H2), we have (𝑓_𝑖(𝑥_𝑖(𝑡))−𝐹⁻_𝑖𝑥_𝑖(𝑡))(𝑓_𝑖(𝑥_𝑖(𝑡))−𝐹⁺_𝑖𝑥_𝑖(𝑡))≤0,𝑖=1,2,…,𝑛, (3.16) which are equivalent to 𝑥_𝑖𝑓(𝑡)_𝑖(𝑥_𝑖(𝑡))^𝑇𝐹⁻_𝑖𝐹⁺_𝑖𝑒_𝑖𝑒^𝑇_𝑖−𝐹⁻_𝑖+𝐹⁺_𝑖2𝑒_𝑖𝑒^𝑇_𝑖−𝐹⁻_𝑖+𝐹⁺_𝑖2𝑒_𝑖𝑒^𝑇_𝑖𝑒_𝑖𝑒^𝑇_𝑖𝑥_𝑖𝑓(𝑡)_𝑖(𝑥_𝑖(𝑡))≤0,𝑖=1,2,…,𝑛, (3.17) where 𝑒_𝑟 denotes the unit column vector having 1 element on its 𝑟th row and zeros elsewhere. Let 𝐿=diag{𝑙₁,𝑙₂,…,𝑙_𝑛},𝑆=diag{𝑠₁,𝑠₂,…,𝑠_𝑛}, (3.18) then ^𝑛_𝑖=1𝑙_𝑖𝑥_𝑖𝑓(𝑡)_𝑖(𝑥_𝑖(𝑡))^𝑇𝐹⁻_𝑖𝐹⁺_𝑖𝑒_𝑖𝑒^𝑇_𝑖−𝐹⁻_𝑖+𝐹⁺_𝑖2𝑒_𝑖𝑒^𝑇_𝑖−𝐹⁻_𝑖+𝐹⁺_𝑖2𝑒_𝑖𝑒^𝑇_𝑖𝑒_𝑖𝑒^𝑇_𝑖𝑥_𝑖𝑓(𝑡)_𝑖(𝑥_𝑖(𝑡))≤0, (3.19) that is 𝑥(𝑡)𝑓(𝑥(𝑡))^𝑇𝐹₁𝐿−𝐹₂𝐿−𝐹₂𝐿𝐿𝑥(𝑡)𝑓(𝑥(𝑡))≤0. (3.20) Similarly, one has 𝑥(𝑡−𝜏(𝑡))𝑓(𝑥(𝑡−𝜏(𝑡)))^𝑇𝐹₁𝑆−𝐹₂𝑆−𝐹₂𝑆𝑆𝑥(𝑡−𝜏(𝑡))𝑓(𝑥(𝑡−𝜏(𝑡)))≤0. (3.21) It follows from (3.7), (3.10), (3.13), (3.15), (3.20) and (3.21) that 𝑑𝑉(𝑡,𝑥(𝑡))≤{𝑥^𝑇(𝑡)[𝑃₂−2𝑄₂𝐶+(𝜀₁+𝜀₄)𝐸^𝑇₁𝐸₁+𝜀₄⁻¹𝑄₂𝐻₁𝐻^𝑇₁𝑄^𝑇₂+𝜀₅⁻¹𝑄₂𝐻₂𝐻^𝑇₂𝑄^𝑇₂+𝜀₆⁻¹𝑄₂𝐻₃𝐻^𝑇₃𝑄^𝑇₂−2𝑄₃+𝜏𝑄₃𝑃₃⁻¹𝑄^𝑇₃+𝑄₃𝑃₁⁻¹𝑄^𝑇₃+𝑄₄𝑃₁⁻¹𝑄^𝑇₄−𝐹₁𝐿]𝑥(𝑡)+2𝑥^𝑇(𝑡)[𝑃₁−𝐶𝑄^𝑇₁−𝑄₂]𝑦(𝑡)+2𝑥^𝑇(𝑡)[𝑄₂𝐴+𝐹₂𝐿]𝑓(𝑥(𝑡))+2𝑥^𝑇(𝑡)𝑄₂𝐵𝑓(𝑥(𝑡−𝜏(𝑡)))+2𝑥^𝑇(𝑡)𝑄₂𝑢(𝑡)+2𝑥^𝑇(𝑡)𝑄₃𝑥(𝑡−𝜏(𝑡))+𝑦^𝑇(𝑡)[−2𝑄₁+𝜀₁⁻¹𝑄₁𝐻₁𝐻^𝑇₁𝑄^𝑇₁+𝜀₂⁻¹𝑄₁𝐻₂𝐻^𝑇₂𝑄^𝑇₁+𝜀₃⁻¹𝑄₁𝐻₃𝐻^𝑇₃𝑄^𝑇₁+𝜏𝑃₃]𝑦^𝑇(𝑡)+2𝑦^𝑇(𝑡)𝑄₁𝐴𝑓(𝑥(𝑡))+2𝑦^𝑇(𝑡)𝑄₁𝐵𝑓(𝑥(𝑡−𝜏(𝑡)))+2𝑦^𝑇(𝑡)𝑄₁𝑢(𝑡)+𝑓^𝑇(𝑥(𝑡))[(𝜀₂+𝜀₅)𝐸^𝑇₂𝐸₂−𝐿]𝑓(𝑥(𝑡))+𝑓^𝑇(𝑥(𝑡−𝜏(𝑡)))[(𝜀₃+𝜀₆)𝐸^𝑇₃𝐸₃−𝑆]𝑓(𝑥(𝑡−𝜏(𝑡)))+2𝑓^𝑇(𝑥(𝑡−𝜏(𝑡)))𝐹₂𝑆𝑥(𝑡−𝜏(𝑡))+𝑥^𝑇(𝑡−𝜏(𝑡))(−2𝑄₄+𝜏𝑄₄𝑃₃⁻¹𝑄^𝑇₄−𝐹₁𝑆)𝑥(𝑡−𝜏(𝑡))+2𝑥^𝑇(𝑡−𝜏(𝑡))𝑄₄𝑥(𝑡−𝜏)−𝑥^𝑇(𝑡−𝜏)𝑃₂𝑥(𝑡−𝜏)+(1+𝜏)trace[𝛼^𝑇(𝑡)𝑃₁−𝛼(𝑡)]_{𝑡𝑡−𝜏}trace[𝛼^𝑇(𝑠)𝑃₁𝛼(𝑠)]𝑑𝑠+(^{𝑡−𝜏(𝑡)}_𝑡−𝜏𝛼(𝑠)𝑑𝜔(𝑠))^𝑇𝑃₁(^{𝑡−𝜏(𝑡)}_𝑡−𝜏𝛼(𝑠)𝑑𝜔(𝑠))+(^𝑡_{𝑡−𝜏(𝑡)}𝛼(𝑠)𝑑𝜔(𝑠))^𝑇𝑃₁(^𝑡_{𝑡−𝜏(𝑡)}𝛼(𝑠)𝑑𝜔(𝑠))}𝑑𝑡+[𝑥^𝑇(𝑡)𝑃₁𝛼(𝑡)+𝛼^𝑇(𝑡)𝑃₁𝑥(𝑡)]𝑑𝜔(𝑡). (3.22) By assumption (H3) and inequality (3.2), we get trace[𝛼^𝑇(𝑡)𝑃₁𝛼(𝑡)]≤𝜆[𝜌₁𝑥^𝑇(𝑡)𝑥(𝑡)+𝜌₂𝑥^𝑇(𝑡−𝜏(𝑡))𝑥(𝑡−𝜏(𝑡))]. (3.23) From the proof of [19], we have 𝐸{(^{𝑡−𝜏(𝑡)}_𝑡−𝜏𝛼(𝑠)𝑑𝜔(𝑠))^𝑇𝑃₁(^{𝑡−𝜏(𝑡)}_𝑡−𝜏𝛼(𝑠)𝑑𝜔(𝑠))}=𝐸{^{𝑡−𝜏(𝑡)}_𝑡−𝜏trace[𝛼^𝑇(𝑠)𝑃₁𝛼(𝑠)]𝑑𝑠},𝐸{(^𝑡_{𝑡−𝜏(𝑡)}𝛼(𝑠)𝑑𝜔(𝑠))^𝑇𝑃₁(^𝑡_{𝑡−𝜏(𝑡)}𝛼(𝑠)𝑑𝜔(𝑠))}=𝐸{^𝑡_{𝑡−𝜏(𝑡)}trace[𝛼^𝑇(𝑠)𝑃₁𝛼(𝑠)]𝑑𝑠}. (3.24) Taking the mathematical expectation on both sides of (3.22), and noting (3.24), we get 𝐸{𝑑𝑉(𝑡,𝑥(𝑡))−2𝑓^𝑇(𝑥(𝑡))𝑢(𝑡)𝑑𝑡−𝛾𝑢^𝑇(𝑡)𝑢(𝑡)𝑑𝑡}≤𝐸{𝜉^𝑇(𝑡)Π𝜉(𝑡)𝑑𝑡}, (3.25) where 𝜉(𝑡)=(𝑥^𝑇(𝑡),𝑦^𝑇(𝑡),𝑓^𝑇(𝑥(𝑡)),𝑓^𝑇(𝑥(𝑡−𝜏(𝑡))),𝑥^𝑇(𝑡−𝜏(𝑡)),𝑥^𝑇(𝑡−𝜏),𝑢^𝑇(𝑡))^𝑇, and ΠΠ=¹𝑃₁−𝐶𝑄^𝑇₁−𝑄₂𝑄₂𝐴+𝐹₂𝐿𝑄₂𝐵𝑄₃0𝑄₂∗Π₂𝑄₁𝐴𝑄₁𝐵00𝑄₁∗∗Π₃000−𝐼∗∗∗Π₄𝐹₂𝑆00∗∗∗∗Π₅𝑄₄0∗∗∗∗∗−𝑃₂0∗∗∗∗∗∗−𝛾𝐼 (3.26) with Π₁=𝑃₂−𝑄₂𝐶−𝐶𝑄^𝑇₂+(𝜀₁+𝜀₄)𝐸^𝑇₁𝐸₁+𝜀₄⁻¹𝑄₂𝐻₁𝐻^𝑇₁𝑄^𝑇₂+𝜀₅⁻¹𝑄₂𝐻₂𝐻^𝑇₂𝑄^𝑇₂+𝜀₆⁻¹𝑄₂𝐻₃𝐻^𝑇₃𝑄^𝑇₂−𝑄₃−𝑄^𝑇₃+𝜏𝑄₃𝑃₃⁻¹𝑄^𝑇₃+𝑄₃𝑃₁⁻¹𝑄^𝑇₃+𝑄₄𝑃₁⁻¹𝑄^𝑇₄−𝐹₁𝐿+(1+𝜏)𝜆𝜌₁𝐼, Π₂=−𝑄₁−𝑄^𝑇₁+𝜀₁⁻¹𝑄₁𝐻₁𝐻^𝑇₁𝑄^𝑇₁+𝜀₂⁻¹𝑄₁𝐻₂𝐻^𝑇₂𝑄^𝑇₁+𝜀₃⁻¹𝑄₁𝐻₃𝐻^𝑇₃𝑄^𝑇₁+𝜏𝑃₃, Π₃=(𝜀₂+𝜀₅)𝐸^𝑇₂𝐸₂−𝐿, Π₄=(𝜀₃+𝜀₆)𝐸^𝑇₃𝐸₃−𝑆, Π₅=−𝑄₄−𝑄^𝑇₄+𝜏𝑄₄𝑃₃⁻¹𝑄^𝑇₄−𝐹₁𝑆+(1+𝜏)𝜆𝜌₂𝐼.
It is easy to verify the equivalence of Π<0 and Ω<0 by using Lemma 2.4. Thus, one can derive from (3.3) and (3.25) that
𝐸{𝑑𝑉(𝑡,𝑥(𝑡))}𝑑𝑡−𝐸{2𝑓^𝑇(𝑥(𝑡))𝑢(𝑡)+𝛾𝑢^𝑇(𝑡)𝑢(𝑡)}≤0. (3.27) From (3.27) and the definition of 𝑉(𝑡,𝑥(𝑡)), we can get 2𝐸{^𝑡𝑝₀𝑓^𝑇(𝑥(𝑠))𝑢(𝑠)𝑑𝑠}≥−𝛾𝐸{^𝑡𝑝₀𝑢^𝑇(𝑠)𝑢(𝑠)𝑑𝑠}. (3.28) From Definition 2.1, we know that the stochastic neural networks (2.1) are globally passive in the sense of expectation, and the proof of Theorem 3.1 is then completed.

Remark 3.2. Assumption (H2) was first proposed in [10]. The constants 𝐹⁻_𝑗 and 𝐹⁺_𝑗 (𝑖=1,2,…,𝑛) in assumption (H2) are allowed to be positive, negative or zero. Hence, Assumption (H2) is weaker than the assumption in [27–37]. In addition, the conditions in [32–37] that the time-varying delay is differentiable and the derivative is bounded or smaller than one have been removed in this paper.

Remark 3.3. In [36], authors considered the passivity of uncertain neural networks with both discrete and distributed time-varying delays. In [37], authors considered the passivity for stochastic neural networks with time-varying delays and random abrupt changes. It is worth pointing out that, the method in this paper can also analyze the passivity for models in [36, 37].

Remark 3.4. It is known that the obtained criteria for checking passivity of neural networks depend on the constructed Lyapunov functionals or Lyapunov-Krasovskii functionals in varying degrees. Constructing proper Lyapunov functionals or Lyapunov-Krasovskii functionals can reduce conservatism. Recently, the delay fractioning approach has been used to investigate global synchronization of delayed complex networks with stochastic disturbances, which has shown the potential of reducing conservatism [22]. Using the delay fractioning approach, we can also investigate the passivity of delayed neural networks. The corresponding results will appear in the near future.

Remark 3.5. When we do not consider the stochastic effect, model (2.1) turns into the following model: 𝑑𝑥(𝑡)𝑑𝑡=−(𝐶+Δ𝐶(𝑡))𝑥(𝑡)+(𝐴+Δ𝐴(𝑡))𝑓(𝑥(𝑡))+(𝐵+Δ𝐵(𝑡))𝑓(𝑥(𝑡−𝜏(𝑡)))+𝑢(𝑡). (3.29) Furthermore, model (3.29) also comprises the following neural network model with neither stochastic effect nor uncertainty 𝑑𝑥(𝑡)𝑑𝑡=−𝐶𝑥(𝑡)+𝐴𝑓(𝑥(𝑡))+𝐵𝑓(𝑥(𝑡−𝜏(𝑡)))+𝑢(𝑡) (3.30) which have been considered in [33, 34]. For models (3.29) and (3.30), one can get the following results.

Corollary 3.6. Under assumptions (H1)-(H2), model (3.29) is passive if there exist a scalar 𝛾>0, three symmetric positive definite matrices 𝑃_𝑖 (𝑖=1,2,3), two positive diagonal matrices 𝐿 and 𝑆, and matrices 𝑄_𝑖 (𝑖=1,2,3,4) such that the following LMI holds: ΩΩ=¹Ω₂∗Ω₃<0, (3.31) where Ω₁=Ω₁₁Ω₁₂Ω₁₃𝑄₂𝐵𝑄₃0𝑄₂∗Ω₂₂𝑄₁𝐴𝑄₁𝐵00𝑄₁∗∗Ω₃₃000−𝐼∗∗∗Ω₄₄𝐹₂𝑆00∗∗∗∗Ω₅₅𝑄₄0∗∗∗∗∗−𝑃₂0,Ω∗∗∗∗∗∗−𝛾𝐼₂=𝑄₂𝐻₁𝑄₂𝐻₂𝑄₂𝐻₃𝑄₃𝑄₃𝑄₄0000000000𝑄₁𝐻₁𝑄₁𝐻₂𝑄₁𝐻₃000000000000000000000000000000𝑄₄,Ω00000000000000000000₃=diag{−𝜀₄𝐼,−𝜀₅𝐼,−𝜀₆𝐼,−𝜏𝑃₃,−𝑃₁,−𝑃₁,−𝜀₁𝐼,−𝜀₂𝐼,−𝜀₃𝐼,−𝜏𝑃₃}, (3.32) in which Ω₁₁=𝑃₂−𝑄₂𝐶−𝐶𝑄^𝑇₂+(𝜀₁+𝜀₄)𝐸^𝑇₁𝐸₁−𝑄₃−𝑄^𝑇₃−𝐹₁𝐿, Ω₁₂=𝑃₁−𝐶𝑄^𝑇₁−𝑄₂, Ω₁₃=𝑄₂𝐴+𝐹₂𝐿, Ω₂₂=−𝑄₁−𝑄^𝑇₁+𝜏𝑃₃, Ω₃₃=(𝜀₂+𝜀₅)𝐸^𝑇₂𝐸₂−𝐿, Ω₄₄=(𝜀₃+𝜀₆)𝐸^𝑇₃𝐸₃−𝑆, Ω₅₅=−𝑄₄−𝑄^𝑇₄−𝐹₁𝑆.

Corollary 3.7. Under assumption (H2), model (3.30) is passive if there exist a scalar 𝛾>0, three symmetric positive definite matrices 𝑃_𝑖 (𝑖=1,2,3), two positive diagonal matrices 𝐿 and 𝑆, and matrices 𝑄_𝑖 (𝑖=1,2,3,4) such that the following LMI holds: ΠΠ=¹Π₂Π₃𝑄₂𝐵𝑄₃0𝑄₂𝑄₃𝑄₃𝑄₄0∗Π₄𝑄₁𝐴𝑄₁𝐵00𝑄₁0000∗∗−𝐿000−𝐼0000∗∗∗−𝑆𝐹₂𝑆000000∗∗∗∗Π₅𝑄₄0000𝑄₄∗∗∗∗∗−𝑃₂00000∗∗∗∗∗∗−𝛾𝐼0000∗∗∗∗∗∗∗−𝜏𝑃₃000∗∗∗∗∗∗∗∗−𝑃₁00∗∗∗∗∗∗∗∗∗−𝑃₁0∗∗∗∗∗∗∗∗∗∗−𝜏𝑃₃<0, (3.33) where Π₁=𝑃₂−𝑄₂𝐶−𝐶𝑄^𝑇₂−𝑄₃−𝑄^𝑇₃−𝐹₁𝐿, Π₂=𝑃₁−𝐶𝑄^𝑇₁−𝑄₂, Π₃=𝑄₂𝐴+𝐹₂𝐿, Π₄=−𝑄₁−𝑄^𝑇₁+𝜏𝑃₃, Π₅=−𝑄₄−𝑄^𝑇₄−𝐹₁𝑆.

4. An Example

Consider a two-neuron neural network (3.30), where

,𝑓𝐶=1000.9,𝐴=2.3−0.1−62.8,𝐵=−1.9−0.1−0.3−9.7₁(𝑧)=tanh(𝑧),𝑓₂(𝑧)=tanh(𝑧),𝜏(𝑡)=2|sin𝑡|,𝑢₁(𝑡)=−0.2𝑡cos𝑡,𝑢₂(𝑡)=−0.5sin𝑡. (4.1)

Figure 1 depicts the states of the considered network with initial conditions 𝑥₁(𝑡)=0.5, 𝑥₂(𝑡)=0.45, 𝑡∈[−2,0].

Figure 1 

State responses of 𝑥1(𝑡) and 𝑥2(𝑡).

It can be verified that assumption (H2) is satisfied, and 𝐹₁=0, 𝐹₂=diag{0.5,0.5}, 𝜏=2. By the Matlab LMI Control Toolbox, we find a solution to the LMI in (3.33) as follows:

𝑃₁=10⁻¹⁰0.67440.10080.10080.4522,𝑃₂=10⁻¹⁰,𝑃0.24900.07550.07550.1154₃=10⁻¹⁰,𝑄0.17350.01930.01930.1175₁=10⁻¹⁰0.17240.03420.02460.0149,𝑄₂=10⁻¹⁰,𝑄0.26150.06220.12160.0513₃=10⁻¹⁰0.47650.16390.06680.4203,𝑄₄=10⁻¹⁰,0.41260.02330.02160.3631𝐿=10⁻⁹0.1100000.0811,𝑆=10⁻⁹,0.2772000.2750𝛾=6.7369×10⁹. (4.2)

Therefore, by Corollary 3.7, we know that model (3.30) is passive. It should be pointed out that the conditions in [33–36] cannot be applied to this example since that require the differentiability of the time-varying delay.

5. Conclusions

In this paper, the passivity has been investigated for a class of stochastic uncertain neural networks with time-varying delay as well as generalized activation functions. By employing a combination of Lyapunov-Krasovskii functionals, the free-weighting matrix method, Newton-Leibniz formulation, and stochastic analysis technique, a delay-independent criterion for checking the passivity of the addressed neural networks has been established in terms of linear matrix inequalities (LMIs), which can be checked numerically using the effective LMI toolbox in MATLAB. The obtained results generalize and improve the earlier publications and remove the traditional assumptions on the differentiability of the time-varying delay and the boundedness of its derivative. An example has been provided to demonstrate the effectiveness and less conservatism of the proposed criterion.

We would like to point out that it is possible to generalize our main results to more complex neural networks, such as neural networks with discrete and distributed delays [10, 26], and neural networks of neutral-type [7, 20], neural networks with Markovian jumping [24, 25]. The corresponding results will appear in the near future.

Acknowledgments

The authors would like to thank the reviewers and the editor for their valuable suggestions and comments which have led to a much improved paper. This work was supported by the National Natural Science Foundation of China under Grants 60974132 and 50608072, and in part by Natural Science Foundation Project of CQ CSTC2008BA6038 and 2008BB2351, and Scientific & Technological Research Projects of CQ KJ090406, and the Ministry of Education for New Century Excellent Talent Support Program.