Mean almost periodicity and moment exponential stability of semi-discrete random cellular neural networks with fuzzy operations

By using the semi-discretization technique of differential equations, the discrete analogue of a kind of cellular neural networks with stochastic perturbations and fuzzy operations is formulated, which gives a more accurate characterization for continuous-time models than that by Euler scheme. Firstly, the existence of at least one p-th mean almost periodic sequence solution of the semi-discrete stochastic models with almost periodic coefficients is investigated by using Minkowski inequality, Hölder inequality and Krasnoselskii’s fixed point theorem. Secondly, the p-th moment global exponential stability of the semi-discrete stochastic models is also studied by using some analytical skills and the proof of contradiction. Finally, a problem of stochastic stabilization for discrete cellular neural networks is studied.


Introduction
Cellular neural networks (CNNs) [1] have been widely applied in psychophysics, parallel computing, perception, robotics associative memory, image processing pattern recognition and combinatorial optimization. Most of these applications heavily depend on the (almost) periodicity and global exponential stability. Specifically, there are many scholars focusing on the study of the equilibrium points, (almost) periodic solutions and global exponential stability of CNNs with time delays in literatures [2][3][4][5][6][7]. For instance, Xu [7] considered the following CNNs with time delays: where n denotes the number of units in a neural network, x i (t) corresponds to the state of the ith unit at time t, a i > 0 represents the passive decay rates at time t, f j and g j are the neuronal output signal functions, b ij (t) and c ij (t) denote the strength of the jth unit on the ith unit at time t, I i (t) denotes the external inputs at time t, the continuous function τ ij (t) corresponds to the transmission delay at time t, i, j = 1, 2, . . ., n. In [7], the author studied the existence and exponential stability of anti-periodic solutions of system (1). In real world applications, most of the problems are uncertain. They should be described by uncertain models and studied by using the research techniques for uncertain models. Stochastic and fuzzy theories are the most general and practical techniques for the research of uncertain models. On one hand, in the actual situations, uncertainties have a consequence on the performance of neural networks. The connection weights of the neurons depend on certain resistance and capacitance values that include modeling errors or uncertainties during the parameter identification process. Therefore, many neural network models described by stochastic differential equations [8,9] have been widely studied over the last two decades, see [10][11][12][13][14][15][16][17]. On the other hand, fuzzy theory was conceived in the 1960s by L.A. Zadeh, it took about 20 years until the broader use of this theory in practice. Fuzzy technology joined forces with artificial neural networks and genetic algorithms under the title of computational intelligence or soft computing. In recent years, the research on the dynamical behaviours of fuzzy neural networks has attracted much attention, see [18][19][20][21][22]. To summarize, we consider the following CNNs with stochastic perturbations and fuzzy operations: where α ij , β ij , T ij and S ij are elements of fuzzy feedback MIN, MAX template, fuzzy feed forward MIN and MAX template, respectively; V and W denote the fuzzy AND and fuzzy OR operation, respectively; d ij , η ij and σ j are similarly specified as that in system (1), w j is the standard Brownian motion defined on a complete probability space, i, j = 1, 2, . . ., n.
Periodicity often appears in implicit ways in various natural phenomena. Though one can deliberately periodically fluctuate environmental parameters in laboratory experiments, fluctuations in nature are hardly periodic. Almost periodicity is more likely to accurately describe natural fluctuations [23][24][25][26][27][28][29][30]. The concept of mean almost periodicity is important in probability especially for investigations on stochastic processes. In particular, mean almost periodicity enables us to understand the impact of the noise or stochastic perturbation on the corresponding recurrent motions, is of great concern in the study of stochastic differential equations and random dynamical systems. The notion of almost periodic stochastic process was proposed in the 1980s and since then almost periodic solutions to stochastic differential equations driven have been studied by many authors. On the other hand, the problem of stability analysis of dynamic systems has a rich, long history of literature [31][32][33][34][35]. All the applications of such stochastic dynamical systems depend on qualitative behavior such as stability, existence and uniqueness, convergence and so on. In particular, exponential stability is a significant one in the design and applications of neural networks. Therefore, the mean almost periodicity and moment exponential stability of various kinds of stochastic neural networks has been reported in [36][37][38][39][40][41].
The discrete-time neural networks become more important than the continuous-time counterparts when implementing the neural networks in a digital way. In order to investigate the dynamical characteristics with respect to digital signal transmission, it is essential to formulate the discrete analog of neural networks. A large number of literatures have been obtained for the dynamics of discrete-time neural networks formulated by Euler scheme [42][43][44][45][46]. Mohamad and Gopalsamy [47, 48] proposed a novel method (i.e., semi-discretization technique) in formulating a discrete-time analogue of the continuous-time neural networks, which faithfully preserved the characteristics of their continuous-time counterparts. In [47], the authors employed computer simulations to show that semi-discrete models give a more accurate characterization for the corresponding continuous-time models than that by Euler scheme. With the help of the semi-discretization technique [47], many scholars obtained the semi-discrete analogue of the continuous-time neural networks and some meaningful results were gained for the dynamic behaviours of the semi-discrete neural networks, such as periodic solutions, almost periodic solutions and global exponential stability, see [49][50][51][52][53][54][55]. For instance, Huang et al.
where k 2 Z, i = 1, 2, . . ., n. The authors [53] derived the existence of locally exponentially convergent 2 N almost periodic sequence solutions of system (4). Kong and Fang [50] in 2018 investigated a class of semi-discrete neutral-type neural networks with delays and some results are acquired for the existence of a unique pseudo almost periodic sequence solution which is globally attractive and globally exponentially stable. However, the disquisitive models in literatures [49-55] are deterministic. Stimulated by this point, we should consider random factors in the studied models, such as system (2). By using the semi-discretization technique [47], Krasnoselskii's fixed point theorem and stochastic theory, the main aim of this paper is to establish some decision theorems for the existence of p-th mean almost periodic sequence solutions and p-th moment global exponential stability for the semi-discrete analogue of uncertain system (2). The work of this paper is a continuation of that in [52-55] and the results in this paper complement the corresponding results in [52-55]. The main contributions of this paper are summed up as: (1) The semi-discrete analogue is established for stochastic fuzzy CNNs (2); (2) A Volterra additive equation is derived for the solution of the semi-discrete stochastic fuzzy CNNs; (3) The existence of p-th mean almost periodic sequence solutions is obtained; (4) A decision theorem is acquired for the p-th moment global exponential stability; (5) A problem of stochastic stabilization for discrete CNNs is proposed and researched.
Throughout this paper, we use the following notations. Let R denote the set of real numbers. R n denotes the n-dimensional real vector space. Let ðO; F ; PÞ be a complete probability space. Denote by BCðZ; L p ðO; R n ÞÞ the vector space of all bounded continuous functions from Z to L p ðO; R n Þ, where L p ðO; R n Þ denotes the collection of all p-th integrable R n -valued random variables. Then BCðZ; L p ðO; R n ÞÞ is a Banach space with the norm kXk p ¼ sup k2Z jXj p ,

Discrete analogue and preliminaries
The semi-discretization model For the sake of gaining the discrete analogue of system (2) with the semi-discretization technique [47], the following uncertain CNNs with piecewise constant arguments corresponding to system (2) have been taken into account: where [t] denotes the integer part of t, i = 1, 2, . . ., n. Here the discrete analogue of the stochastic parts of system (2) is obtained by Euler scheme, i.e., dw For each t, there exists an integer k such that k � t < k + 1. Then the above equation becomes where i = 1, 2, . . ., n. Integrating the above equation from k to t and letting t ! k + 1, we achieve the discrete analogue of system (2) as follows: where k 2 Z, i = 1, 2, . . ., n.

Volterra additive equation for the solution of system (5) Lemma 1. X = {x i } is a solution of system (5) if and only if
. . . ; n: where i = 1, 2, . . ., n, k 2 Z. By the above equations, we can easily derive (6). If X = {x i } satisfies (6), then e a i ðsÞ ½1 À e À a i ðvÞ � a i ðvÞ F i ðv; xÞ; which implies that where i = 1, 2, . . ., n, k 2 Z. Therefore, X = {x i } is a solution of system (5). This completes the proof.

Some lemmas
Lemma 5. Assume that fxðkÞ : k 2 Zg is real-valued stochastic process and w(k) is the standard Brownian motion, then where C p is defined as that in Lemma 4, p > 0.

Proof. By Lemma 4, it follows that
where k 2 Z. This completes the proof.
A stochastic process X, which is 2-nd mean almost periodic sequence will be called squaremean almost periodic sequence. Like for classical almost periodic functions, the number ω will be called an �-translation of X. Lemma 7. ([58]) Assume that Λ is a closed convex nonempty subset of a Banach space X. Suppose further that B and C map Λ into X such that 1. B is continuous and BL is contained in a compact set, 2. x, y 2 Λ implies that Bx þ Cy 2 L, 3. C is a contraction mapping.
Then there exists a z 2 Λ such that z ¼ Bz þ Cz.
Throughout this paper, we always assume that the following conditions are satisfied: (H 2 ) There are several positive constants L f j , L g j and L s j such that Assume that all coefficients in system (5) excluding the Brownian motions are almost periodic sequences, (H 1 )-(H 2 ) hold and the following condition is satisfied: Then there exists a p-th mean almost periodic sequence solution X of system (5) with Proof. Let L � BCðZ; L p ðO; R n ÞÞ be the collection of all p-th mean almost periodic sequences X = {x i } satisfying kXk p � β p .
Firstly, X = {x i } is described by where i = 1, 2, . . ., n, k 2 Z. Obviously, (10) is well defined and satisfies (6). So we define FXðkÞ ¼ BXðkÞ þ CXðkÞ, where ðCXÞ i ðkÞ ¼ where i = 1, 2, . . ., n, k 2 Z. where i = 1, 2, . . ., n, k 2 Z. By Minkoswki inequality in Lemma 2, we have From Lemma 6 and Hölder inequality in Lemma 3, it gets from the above inequality that It follows from (11), (12) and (13) that e À a i ðsÞ ½1 À e À a i ðvÞ � a i ðvÞ jx j ðv À Z ij ðvÞÞDw j ðvÞj which yields from Lemma 3 that Hence, 8X = {x i } 2 Λ, it leads from (14) and (16) to Similar to the argument as that in (17), it is easy to verify that BL is uniformly bounded and continuous. Together with the continuity of B, for any bounded sequence {φ n } in Λ, we know that there exists a subsequence fφ n k g in Λ such that fBðφ n k Þg is convergent in BðLÞ. Therefore, B is compact on Λ. Then condition (1) of Lemma 7 is satisfied.
In view of (H 3 ), C is a contraction mapping. Hence condition (3) of Lemma 7 is satisfied. Therefore, all conditions in Lemma 7 hold. By Lemma 7, system (5) has a p-th mean almost periodic sequence solution. This completes the proof.

p-th moment global exponential stability
Suppose that X = {x i } with initial value φ = {φ i } and X � ¼ fx � i g with initial value φ � ¼ fφ � i g are arbitrary two solutions of system (5). For convenience, let The 2-nd moment global exponential stability will be called square-mean global exponential stability. Theorem 2. Assume that (H 1 )-(H 3 ) hold, then system (5) is p-th moment globally exponentially stable, p > 1.
Together with Theorem 1, we have Theorem 3. Assume that all conditions in Theorem 1 hold, then system (5) admits a p-th mean almost periodic sequence solution, which is p-th moment globally exponentially stable. Further, if all coefficients in system (5) are periodic sequences, then system (5) admits at least one pth mean periodic sequence solution, which is globally exponentially stable.
Proof. The result can be easily obtained by Theorem 2, so we omit it. This completes the proof.
In system (5), if we remove the effects of uncertain factors, then the following deterministic model is obtained: where k 2 Z, i = 1, 2, . . ., n. Definer Corollary 1. Assume that (H 1 ) and (7) and (8) in (H 2 ) hold. Suppose further that all of coefficients of model (27) are almost periodic sequences, andr < 1, then model (27) admits at least one almost periodic sequence solution, which is globally exponentially stable. Moreover, if all of coefficients of model (27) are periodic sequences, then model (27) (27) with c ij � 0(i, j = 1, 2, . . ., n) and obtained some sufficient conditions for the existence of a unique almost periodic sequence solution which is globally attractive. In [53], they considered system (4) and studied the dynamics of 2 N almost periodic sequence solutions. But neither of them considered the uncertain factors. Therefore, the work in this paper complements the corresponding results in [52,53].
Remark 2. Assume that X(k) = (x 1 (k), x 2 (k), . . ., x n (k)) is a solution of (27), the length of X(k) is usually measured by kXk 1 ¼ sup k2R max 1�i�n jx i ðkÞj. However, if X(k) is a solution of stochastic system (5), its length should not be measured by kXk 1 because X(k) is a random variable. In this paper, we use norm kXk p ¼ max 1�i�n sup k2Z ðEjx i ðkÞj p Þ 1 p ðp > 1Þ for random variable X(k). Owing to the expectation E and order p in kXk p , the computing processes of this paper are more complicated than that in literatures [49][50][51][52][53][54][55]. It is worth mentioning that Minkoswki inequality in Lemma 2 and Hölder inequality in Lemma 3 are crucial to the computing processes. The facts above are obvious from the computations of (14), (15), (22) and (26) in Theorems 1 and 2. Further, the stochastic term d ij σ j Δw j (i, j = 1, 2, . . ., n) in system (5) also increases the complexity of computing. This point is also clear from the computations of (20) and (22) in Theorems 1 and 2.

Stochastic stabilization
In this section, we consider the following stochastic cellular neural networks: where w(t) is a standard Brownian motion, t 2 R, i = 1, 2, . . ., n.

Stability analysis of systems (SM) and (DM)
Assume that X = {x i } with initial value X 0 ¼ fx i0 g 2 R n and X � ¼ fx � i g with initial value X � 0 ¼ fx � i0 g 2 R n are arbitrary two solutions of system (SM) or (DM).

Definition 3. ([9]) System (SM) or (DM) is said to be exponential stability if
System (SM) or (DM) is said to be exponential instability if Lemma 8. ( [9]) Assume that w is a standard Brownian motion, then w(0) = 0 and Then system (SM) is exponential stability. This completes the proof. Let κ = 0 in Theorem 4, it has Theorem 5. Assume that (H 2 ) holds. Suppose further that Then system (DM) is exponentially stable. Similar to the argument as that in Theorem 4, the exponential instability of system (DM) is easily derived as follows: Theorem 6. Assume that (H 2 ) holds. Suppose further that Then system (DM) is exponentially instable. Definition 4. ([9]) Assume that system (DM) is exponential instability and there exists a suitable stochastic disturbance coefficient κ ensuring that system (SM) is exponential stable, then system (SM) is a stochastic stabilization system of system (DM).
Together with Theorems 4 and 6, it gains Theorem 7. Assume that (H 2 ), (H 4 ) and (H 6 ) are satisfied. Then system (SM) is a stochastic stabilization system of system (DM). Remark 3. If (H 6 ) is valid, (DM) is exponentially instable. Meanwhile, (H 5 ) is invalid. By viewing (H 4 ), one could select a suitable stochastic disturbance coefficient κ ensuring that (H 4 ) is satisfied, which yields system (SM) is exponentially stable. Therefore, stochastic disturbance could be a useful method, which brings unstable system to be stable. More details could be observed in Example 2.