The stability of memristive multidirectional associative memory neural networks with time-varying delays in the leakage terms via sampled-data control

In this paper, we propose a new model of memristive multidirectional associative memory neural networks, which concludes the time-varying delays in leakage terms via sampled-data control. We use the input delay method to turn the sampling system into a continuous time-delaying system. Then we analyze the exponential stability and asymptotic stability of the equilibrium points for this model. By constructing a suitable Lyapunov function, using the Lyapunov stability theorem and some inequality techniques, some sufficient criteria for ensuring the stability of equilibrium points are obtained. Finally, numerical examples are given to demonstrate the effectiveness of our results.


Introduction
Associative memory is one of the most important activities of human brains. It includes oneto-many association, many-to-one association and many-to-many association. Due to the complexity of human brains, many-to-many associative memory is more suitable for simulating the associative memory process of human brains than one-to-many association or manyto-one association.
Multidirectional associative memory neural networks(MAMNNs) were proposed by Japanese scholars in 1990 [1]. They are used to realize many-to-many association. Moreover, MAMNNs are the extension of bidirectional associative memory neural networks(BAMNNs), and they are similar in structure, i.e. there is no connection between the neurons in the same field, but there exist interconnections between the neurons from different fields. In recent PLOS  years, some studies have analyzed and dealt with MAMNNs in [2][3][4]. In [2], the authors proposed a multi-valued exponential associative memory model, and they analyzed the stability of this system. The global exponential stability of MAMNNs with time-varying delays were analyzed in [3]. In addition, MAMNNs with almost periodic coefficients and continuously distributed delays were studied in [4]. So far, there have been few results on the stability of MAMNNs, therefore, it is significant to analyze the stability of MAMNNs. Due to the characteristics of a memristor, it has been found to be the best device for simulating variable synaptic weights of human brains. Therefore, according to using the memristors in neural networks(NNs) instead of resistors, memristive neural networks(MNNs) was designed in [5,6]. Since then, the dynamic behaviors of MNNs have attracted the attention of many researchers in [7][8][9][10], and they have been widely applied to associative memory [11], medical image processing [12], etc. Meanwhile, BAMNNs as a special case of MAMNNs, memristive bidirectional associative memory neural networks(MBAMNNs) have been extensively studied in [13][14][15][16][17]. As an extension of MBAMNNs, the study of memristive multidirectional associative memory neural networks(MMAMNNs) have attracted the attention of researchers [18]. However, it is worth noting that, because of the complexity of MMAMNNs, their research results are few. Thus, it is meaningful to analyze the dynamic behaviors of MMAMNNs.
It is well known that stability of systems plays an important role due to their potential applications to image encryption [19,20], associative memory [11], medical image processing [12], information storage [18], etc. In the past few years, the stability of MNNs and MBAMNNs have attracted the attention of many researchers [21][22][23][24]. Global exponential stability of MNNs with impulse time window and time-varying delays was discussed in [21]. The problem of exponential stability for switched MNNs with time-varying delays was studied in [22]. The theoretical results on the global asymptotic stability and synchronization of a class of fractional-order MNNs with multiple delays were analyzed in [23]. Based on above discussions, the existence, uniqueness and exponential stability for complex-valued MBAMNNs with time delays were studied in [24]. As we all know, a stable equilibrium or a periodic solution is stored as an associative memory pattern. The storage capacity of a system is the collection of associative memory patterns. In other words, the more equilibrium points, the larger the storage capacity. Recently, some results about the multistability of MNNs have been found in [25,26]. At present, there are few literatures about the stability of MMAMNNs, accordingly, stability and multistability of MMAMNNs are still a problem that deserves investigation.
Delays play an important role in the system. The time-varying delays are inevitable in the hardware implementation due to the switching of amplifiers [27][28][29][30][31][32]. The leakage delays (or forgetting delays) exist in the negative feedback of NNs [33,34]. These two delays have great impact on the dynamical behaviors of the systems. Simultaneously, time delays can cause oscillation and instability of a system. So it is necessary to adopt some control strategies to stabilize a system. Various types of control methods, such as output-feedback control [35], switching control [36], adaptive control [37] and sampled-data control [38][39][40][41][42][43] are often considered. In practical applications, the system cannot be in a stable state for a long time, and it is difficult to ensure that the state variables are continuous. Thus, we choose periodic sampling control, which has good flexibility and easy maintenance.
Motivated by the above discussions, the main contributions of this paper can be summarized in the following:

Preliminaries
In this section, we consider the following MMAMNNs with time-varying leakage delays: where x ki (t) denotes the voltage of the ith neuron in the field k at time t. m is the number of fields in system (1) and n p corresponds to the number of neurons in the field p. d ki (x ki (t)), a pjki (x ki (t)), b pjki (x ki (t)) are connection weights. f ki (x) and g ki (x) are activation functions. The time delays γ ki (t) and τ pjki (t) are leakage delays and time-varying delays, respectively. I ki (t) represents the sampled-data state feedback inputs of the ith neuron in the field k.

Remark 1.
According to the definitions of connection weights, d ki (x ki (t)), a pjki (x ki (t)) and b pjki (x ki (t)) are varying with the state of memristance of system (1). Therefore, we consider the MMAMNNs with time-varying leakage delays as state-dependent switching system. When d ki (x ki (t)), a pjki (x ki (t)) and b pjki (x ki (t)) are constants, system (1) becomes a general MAMNNs.
Because d ki (x ki (t)), a pjki (x ki (t)) and b pjki (x ki (t)) are discontinuities, the solutions considered in this paper are defined in the sense of Filippov. co½x; x represent the convex closure on ½x; x . A column vector is defined as colðx ki Þ ¼ ðx 11 is the upper right Dini derivative of k(t), and defined as . Some notations are defined as follows: In the Banach space, all sets of continuous functions are expressed as C([−τ, 0], R n ). The initial condition of system (1) are given as follows: 0ðsÞ ¼ ð0 11 ðsÞ; 0 12 ðsÞ; Á Á Á ; 0 1n 1 ðsÞ; 0 21 ðsÞ; Á Á Á ; 0 mn m ðsÞÞ T 2 Cð½À t; 0; R n Þ, in which By applying the set-valued mapping theorem and the differential inclusion theorem, we define the following equations cof a pjki ; a pjki g; jx ki ðtÞj ¼ G ki ; a pjki ; jx ki ðtÞj > G ki ; Obviously, cof d ki ; Remark 2. In [44], the effect of leakage delay on stability was discussed. It was shown that larger leakage delay can lead to instability of a system. In order to reduce the effect of leakage delay, we will use the sampled-data control method to ensure the stability of the system.
In this paper, we consider the following sampled-data controller: where L ki denotes the sampled-data feedback control gain matrix,

Remark 3.
Due to the existence of the discrete term I ki (t) = L ki x ki (t l ), it is difficult to analyze the stability of system (4). The input delay method was proposed in [45]. By applying it, system (4) will be changed into a continuous system.
The input delay method is applied, we define where 0 Δ(t) < Δ and the sampled-data controller can be written as Then we get Some preliminaries are introduced as follows.
then, the constant vector j0ðsÞ À x Ã j; then, the equilibrium point x Ã of system (1) is globally exponential stable. Lemma 1 Let Assumption 1 be valid. Then there is at least one local solution x(t) of system (1) with the initial condition ϕ(s), s 2 [−τ, 0], which is bounded in [46]. Furthermore, the local solution x(t) of system (1) can be extended to the interval [0, + 1) in the sense of Filippov.

Results
In this section, the stability of one equilibrium point will be studied. By constructing a suitable Lyapunov function, some sufficient criteria for exponential stability and asymptotic stability are obtained. Theorem 1. Under Assumption 1, letd ki " g ki < 1, and there exist positive constants Z 11 ; Z 12 ; Á Á Á ; Z mn m , for t > 0, such that the system (9) is globally exponentially stable if Proof. Due to the characteristics of the memristor, the theorem will be proved in three cases.
① |x ki (t)| < Γ ki . According to the set-valued mapping theorem and the differential inclusion theorem, system (9) can be rewritten as Then for ω > 0, we define a continuous function as follows According to the condition of Theorem 1, we have Because F ki (ω) is continuous, there exists a small positive constant λ, fulfilling the following in equality We construct a suitable Lyapunov function as follows Calculating the upper right Dini derivative of (18), we obtain Let O ¼ sup . Then for t > 0, we claim that the above formula also holds. The proof will be given as follows.
In conclusion, the system (9) is exponentially stable under the condition of Theorem 1. Remark 4. According to Assumption 1, it is obvious that (0, 0, Á Á Á, 0) T is a equilibrium point of the system (9). Remark 5. According to the definition of a convex closure, d ki (x ki (t)), a pjki (x ki (t)) and b pjki (x ki (t)) in system (9) are in a interval. Based on the analysis of the first two cases, we know that system (9) is exponentially stable in this interval.
Proof. Due to the characteristics of the memristor, the theorem will be proved in three cases. The proof process is similar to Theorem 1, and we will not described here.
Theorem 2. Under Assumption 1, if there exists a constant λ satisfies Then, the solution of system (4) is globally asymptotically stable under the sampled-data controller I ki (t) = L ki x ki (t − Δ(t)) − λ ki x ki (t).
Proof. We construct a suitable Lyapunov function as follows Due to the characteristics of the memristor, the theorem will be proved in three cases. ① |x ki (t)| < Γ ki .
According to the set-valued mapping theorem and the differential inclusion theorem, system (9) can be rewritten as follows dx ki ðtÞ dt ¼ L ki x ki ðt À DðtÞÞ À l ki x ki ðtÞ À d ki x ki ðt À g ki ðtÞÞ þ Calculating the upper right Dini derivative of (32), we have According to Assumption 1, we yield j b pjki jr pj jx pj ðt À tðtÞÞj þ jx ki ðtÞjL ki jx ki ðt À DðtÞÞj À l ki x 2 ki ðtÞ: By the mean-value inequality, we get jx ki ðtÞj j a pjki js pj jx pj ðtÞj jx ki ðtÞj jx ki ðtÞjL ki jx ki ðt À DðtÞÞj According to (34) and the mean-value inequality, we get an inequation as follows Then we have According to the condition of Theorem 2, we get D + V ki (t)<0. From the Lyapunov stability throrem, the solution of system (9) is globally asymptotically stable.
Under the same parameters, on the one hand, according to Figs 1 and 2, we know that whatever the initial value of each field is, it eventually approximates a straight line. The corresponding value of the line is the equilibrium point value of each field, i.e. no matter what the initial value of each field is, the equilibrium point ultimately converges to zero. In other words, whatever the initial value is, an arbitrary local solution x(t) is gradually approaching the equilibrium point x Ã = (0, 0, Á Á Á, 0) T .   On the other hand, compared with the MMAMNNs without sample-data control, MMAMNNs with sample-data control converge to the equilibrium point faster. Hence, it is valuable to study the MMAMNNs with time-varying delays in leakage terms via sampled-data feedback control. Remark 6. Since the system cannot be in a stable state for a long time, and it is also a huge consumption to continuously acquire the state of the system, thus, we use sampled-data control method in this paper, which has good flexibility and easy maintenance.
On the other hand, we know that the leakage delays have an effect on the stability of the system. Compared with Figs 6 and 7, it is clear that the curve of MMAMNNs with leakage terms has a significant change. However, the leakage delays are inevitable, so it is significant to study MMAMNNs with leakage terms.
In the simulation experiment, we set the sampling period to 0.02s, and the specific sampling controller action diagram is shown in Fig 4 (after partial enlargement). As can be seen from Figs 3 and 8, the value of the controller remains unchanged during the sampling period until the next sampling period. As time goes on, the system gradually stabilizes and the controller values tend to zero. Compared to continuous control methods, the sampled-data control method reduces energy consumption to a certain extent. At the same time, because the system cannot be in a stable state for a long time, the state of the interval control system is more realistic.

Conclusion
In this paper, we propose a new model of MMAMNNs with time-varying delays in leakage terms via sampled-data control. Compared with some continuous control methods, the sample-data control method is more effective and realistic. So we turn the sampling system into a continuous time-delay system by using sampled-data control. Then the exponential stability and asymptotic stability of the equilibrium points for this model are analyzed. By constructing a suitable Lyapunov function, using Lyapunov stability theorem and some inequality techniques, some sufficient criteria are obtained to guarantee the stability of the system. Some numerical examples are given to demonstrate the effectiveness of the proposed theories. These results will be further applied in the areas such as associative memory of brain-like systems, intelligent thinking for intelligent robots, mass storage, medical image processing etc.