#### Abstract

This paper is concerned with the passivity problem of memristive bidirectional associative memory neural networks (MBAMNNs) with probabilistic and mixed time-varying delays. By applying random variables with Bernoulli distribution, the information of probability time-varying delays is taken into account. Furthermore, we consider the probability distribution of the variation and the extent of the delays; therefore, the results derived are less conservative than in the existing papers. In particular, the leakage delays as well as distributed delays are all taken into consideration. Based on appropriate Lyapunov-Krasovskii functionals (LKFs) and some useful inequalities, several conditions for passive performance are established in linear matrix inequalities (LMIs). Finally, numerical examples are given to demonstrate the feasibility of the presented theories, and the results reveal that the probabilistic and mixed time-varying delays have an unstable influence on the system and should not be ignored.

#### 1. Introduction

Bidirectional associative memory neural networks (BAMNNs) are a class of two-layer neural systems, which were first introduced by Kosko in 1987. The neurons in the first layer are connected to another layer, and in the same layer, the neurons are not interconnected [1–3]. Owing to their special structure, BAMNNs have displayed many good features in various areas such as signal processing, image processing, and optimization problems [4–6]. In 2015, the stability of inertial BAMNNs with time-varying delay via impulsive control was discussed in [7]. Zhang et al. considered the exponential stability of BAMNNs with time-varying delays in [8]. Wang et al. addressed the global asymptotic stability of impulsive fractional-order BAMNNs with time delay in [9].

Memristor, a combination of a resistor and memory, has received increasing attention in many fields [10–13]. By applying the nonvolatile feature of the memristor, researchers were able to develop MBAMNN models. Because of the pinched hysteresis effects, MBAMNNs have a memory function, which can be used to imitate the human brain [14, 15]. In 2015, nonfragile synchronization of MBAMNNs with random feedback gain fluctuations was investigated in [16]. Based on functional differential inclusions, Jiang et al. obtained the dynamic behaviors for MBAMNNs with time-varying delays in [17].

Moreover, passivity is a special case of a broader theory of dissipativity, which plays a significant role in the stability analysis of dynamical systems, nonlinear control, and other areas. The main innate character of passivity theory is that the passive characteristics can make the system internally stable [18–20]. In recent years, many researchers have proposed passivity analysis for memristive neural networks (MNNs). Liu and Xu investigated the passivity analysis of MNNs with different state-dependent memductance functions and mixed time-varying delays in [21]. In 2016, the passivity of MBAMNNs with uncertain delays and different memductance was investigated in [22]. Nevertheless, there are few people to study the passivity of MBAMNNs, which encourages our idea.

In the human brain, the transmission of information in neurons is often accompanied by a time delay, so time delay is inevitable in the neural networks, which is the origin of oscillation, divergence, and so forth [23–33]. Sometimes, the value of delay may be very large, but the probability of such delay is very small. Therefore, we use the probability distribution of time delay in the interval to reflect an actual situation better. Furthermore, it is clearer to describe the probabilistic time-varying delays through introducing random variables with Bernoulli distribution. In recent years, some researchers have discussed the probabilistic time-varying delays in the neural networks [34, 35]. In 2016, Pradeep et al. investigated the robust stability analysis of stochastic neural networks with probabilistic time-varying delays in [36]. Li et al. considered passivity analysis of memristive neural networks with probabilistic time-varying delays in [37]. Hence, it is of great importance to research the passivity of MBAMNNs with probabilistic time-varying delays.

In addition, there also exist two types of time-varying delays named leakage delays (or forgetting delays) and distributed time delays. The research of leakage time delay can be traced back to the early 90s of the last century; researchers found out that, due to the delay in switching time or signal transmission, there is a time delay in the negative feedback term of the network system; this delay is named leakage time delay. As is well known, leakage delays exist in many real systems such as population dynamics and neural networks [38, 39]. Moreover, leakage delay also has a significant influence on the dynamics of neural networks because it has been shown that such kind of time delay in the leakage term has a tendency to destabilize a system. Under the influence of leakage and additive time-varying delays, robust passivity analysis for neural networks was addressed in [40]. In 2016, the robust stability analysis for discrete-time neural networks with leakage delays was studied in [41].

On the other hand, due to the presence of multiple parallel paths with a variety of neuronal synapses’ lengths and sizes, there is a spatial width of the network, and then there may exist either a distribution of the transmission voltage in these parallel paths or a distribution of transmission delays over a period of time. Hence, the distribution delay is used to describe this phenomenon [42, 43]. In 2015, Du et al. investigated the passivity of neural networks with discrete and distributed time-varying delays in [44]. In 2016, Yang et al. considered finite-time stabilization of uncertain neural networks with distributed time-varying delays in [45]. However, to the best of our knowledge, there are few results on the passivity of MBAMNNs with probabilistic, leakage, and distributed time-varying delays. Thus, it is significant to study the passivity of MBAMNNs with these time-varying delays.

Motivated by the main points discussed above, the contribution of this paper lies in three aspects.

(1) This is the first attempt to discuss the passivity analysis of MBAMNNs with probabilistic and mixed time-varying delays. In particular, the leakage delays as well as distributed delays are all taken into consideration.

(2) The LKFs that we designed include double and triple integral terms, and by applying some helpful inequalities, the passivity analysis of MBAMNNs becomes less conservative than the existing results [19, 21].

(3) After using MATLAB LMI control toolbox, all the derived results are expressed in LMIs, and a feasible solution can be easily obtained.

The rest of the paper is structured as follows. In Section 2, we introduce the model of the MBAMNNs with probabilistic time-varying delays. In Section 3, the main results on passivity analysis of MBAMNNs with probabilistic and mixed time-varying delays are derived. In Section 4, some numerical simulations are provided to demonstrate the feasibility of our results. In Section 5, the conclusion is shown.

#### 2. Model Description and Preliminaries

In this paper, we propose the MBAMNN with probabilistic time-varying delays as follows: or it can be rewritten as follows: where and denote the state variables related to the th and th neurons. and are the connection weight matrices, and are the activation functions, and and are the delayed connection weight matrices. The self-feedback connection weights and are positive diagonal matrices. and represent the continuous external inputs; the nonnegative continuous variables and correspond to the time-varying delays.

*Assumption 1. *The functions and are bounded and continuous and satisfy the conditions as follows: with , , , and , .

Based on the current-voltage characteristic and the feature of memristor, the memristive connection weights , , , and will change with time. Then, we let in which , , , and are constants and the switching jumps , .

Based on Figures 1 and 2, it is clear that , , , and are piecewise continuous functions; the solutions of the systems are indicated in Filippov’s sense and the interval is represented by . Set for , . indicates the convex closure of . Obviously, the set-valued maps are defined as

As a matter of convenience, we make the following assumptions.

*Assumption 2. *We define the probability distribution of time delays and as follows: where and are constants.

Thus, the random variables and can be defined as

Then, it can be derived that and are Bernoulli distributed sequences with

According to Assumption 2, it is easy to see that

Now, time-varying delays , , , and are introduced as

*Assumption 3. *Here, constants , , , , , , , and exist, such that

*Assumption 4. *The leakage delays and and distributed delays and satisfy

By employing the theories of set-valued maps, differential inclusions, stochastic variables , , and new functions , , , and , system (1) becomes Or it can be rewritten as follows: equivalently,

*Definition 5. *System (15) is called passive if there exists a scalar such that for all solutions of (15) with and and for all .

*Remark 6. *Passivity analysis originates from circuit theory and it uses the input-output description method based on energy to design and analyze a system. The physical meaning of passivity is reflected in Definition 5 where the energy growth of the system is always less than or equal to the total energy of the external inputs; this means that the passive system is always accompanied by the loss of energy. In fact, the storage function of the passive system can be used as a Lyapunov function under certain conditions. Thus, both Lyapunov stability theory and passivity theory can be used to research the stability of the system.

Lemma 7. *For any scalars , matrix , , and vector function , the inequalities hold as follows:*

#### 3. Main Results

For derivation convenience, we denote

Theorem 8. *Under Assumptions 1–3, system (16) is passive, if there exist any appropriately dimensional matrices , , a scalar , and symmetric positive definite matrices , , , , , and , such that the following LMIs hold: where*

*Proof. *Consider the following LKF candidate: whereThen, we define the infinitesimal generator of as Taking the mathematical expectation of , we getAccording to Assumption 3, we obtainIt is obvious thatUsing Lemma 7, we haveSimilarly,Moreover, we obtainUsing (18), we have Similarly,To derive a less conservative criterion, we add the following inequations with any matrices , , , and of appropriate dimensions:Based on Assumption 1 [20], we have and there exist positive diagonal matrices