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: () the time-varying delays are continuously differentiable; () the derivative of time-varying delay is bounded and is smaller than one; () 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:
for , where is the state vector of the network at time , corresponds to the number of neurons; is a positive diagonal matrix, , and are known constant matrices; , and are time-varying parametric uncertainties; is the diffusion coefficient matrix and is an -dimensional Brownian motion defined on a complete probability space () with a filtration satisfying the usual conditions (i.e., it is right continuous and contains all -null sets); denotes the neuron activation at time ; is a varying external input vector; is the time-varying delay, and is assumed to satisfy , where is constant.
The initial condition associated with model (2.1) is given by
Let denote the state trajectory of model (2.1) from the above initial condition and the corresponding trajectory with zero initial condition.
Throughout this paper, we make the following assumptions.
(H1) [33] The time-varying uncertainties , and are of the form
where , , , , and are known constant matrices of appropriate dimensions, , and are known time-varying matrices with Lebesgue measurable elements bounded by
(H2) [10] For any , and there exist constants and such that
for all .
(H3) [15] There exist two scalars , such that the following inequality:
holds for all .
Definition 2.1 (see [33]). System (2.1) is called globally passive in the sense of expectation if there exists a scalar such that
for all and for all , where stands for the mathematical expectation operator with respect to the given probability measure .
To prove our results, the following lemmas that can be found in [39] are necessary.
Lemma 2.2 (see [39]). For given matrices , and with and a scalar , the following holds:
Lemma 2.3 (see [39]). For any constant matrix , , scalar , vector function such that the integrations concerned are well defined, then
Lemma 2.4 (see [39]). Given constant matrices , and , where , , then
is equivalent to the following conditions:
3. Main Results
For presentation convenience, in the following, we denote
Theorem 3.1. Under assumptions (H1)–(H3), model (2.1) is passive in the sense of expectation if there exist two scalars , , three symmetric positive definite matrices (), two positive diagonal matrices and , and matrices () such that the following two LMIs hold:
where
in which , , , , , , .
Proof. Let , , then model (2.1) is rewritten as
Consider the following Lyapunov-Krasovskii functional as
By It differential rule, the stochastic derivative of along the trajectory of model (3.5) can be obtained as
From the definition of , we have
By assumption (H1) and Lemma 2.2, we get
It follows from (3.8) and (3.9) that
Integrating both sides of (3.5) from to , we have
Hence,
By Lemmas 2.2 and 2.3, and noting , we get
Integrating both sides of (3.5) from to , we have
Similarly, by using of the same way, and noting , we get
From assumption (H2), we have
which are equivalent to
where denotes the unit column vector having 1 element on its th row and zeros elsewhere. Let
then
that is
Similarly, one has
It follows from (3.7), (3.10), (3.13), (3.15), (3.20) and (3.21) that
By assumption (H3) and inequality (3.2), we get
From the proof of [19], we have
Taking the mathematical expectation on both sides of (3.22), and noting (3.24), we get
where