Finite-time non-fragile control for synchronization of fractional-order stochastic neural networks

This study concerned with the synchronization problem over a finite-time domain for a class of fractional-order stochastic neural networks via non-fragile controller with discontinuous activation functions. Specifically, the suggested state feedback controller sequentially cope with non-fragile scheme for gain scheduling process. Notably, the main objective of this work is to obtained the finite-time synchronization criterion for the resulting synchronized error system over finite bound with the developed non-fragile controller. For that, some consignment of sufficient conditions is derived systematically by implementing Lyapunov’s indirect method and finite-time stability theory which ensures the finite-time stochastic synchronization of the stipulated neural networks. Later on, the controller gain fluctuations that obeying certain white noise sequels derived from Bernoulli distribution are formulated in terms of linear matrix inequalities. Eventually, the illustrated study are substantiated through two numerical examples and the simulation results manifest the advantage and accuracy of the proposed synchronization criteria.

scale by analogy reasoning, based on this, many scientific researches show that fractional-order approach is the finest description for plentiful natural phenomenon. Moreover, the fractional-order systems has raised promising research attention due to its elegant reflection of dynamical behavior and practical values which confers accurate results (Huang and Bae 2019;. The significant aspect of fractional-order systems are the prominent hereditary properties of diverse materials and superiority of memory ). In the present situation, the better understanding of fractional calculus led to the increasing amount of works on stability and stabilization of fractional-order control systems (see Ding et al. 2018;Ji et al. 2018) and the references cited therein).
It is worth noticed that neural networks (NNs) has been extensively inspected owing to its successful applications in parameter estimation, parallel computation, prediction and data analysis, automatic control, artificial intelligence, and so on ; Li et al. (2019; Nazemi and Mortezaee (2020); Xiao et al. (2013). Owing to their viability, NNs has become a powerful tool to delineate the neurodynamics in human brains with higher quality. Generally, the accuracy can be increased while analyzing the dynamical behavior of NNs by incorporating the fractional calculus. As mentioned from the above facts and infinite memory property, a huge amount of significant results on the dynamics of fractional-order NNs (FNNs) systems embracing stability, stabilization and synchronization have been reported, for example see Li et al. 2019Yang et al. 2018). Namely, ) examined the asymptotic stability analysis of Riemann-Liouville derivative fractional-based NNs in the presence of distributed and discrete delays. Based on the theory of impulsive and fractional differential equation,  presented global synchronization of Mittag-Leffler derivative-based FNNs.
Synchronization is an universal phenomenon which was proposed first by Pecora and Carroll; subsequently, it is enhanced to analyze diverse fields, namely cryptography, secure communication, image encryption, information sciences and so on Bae (2014) Lee et al. (2016Lee et al. ( , 2019. Consequently, synchronization of two or more systems adjust each other and may lead to common dynamical behavior. However, the study on synchronization of FNNs cannot be ignored and a lot of interesting results have been reported through various control approaches Hu et al. 2019). For instance, the authors proposed globally asymptotically synchronization analysis for FNNs subject to multiple time-delay in . As it is well-known that the stochastic noise is usually unavoidable in practice, and it is one of the crucial factor which affects the system performance. So far, the synchronization of fractional-order stochastic neural networks (FSNNs) have not been examined fully. This stimulates our research attention toward synchronization of FSNNs.
It should be pointed out that in reality, the imprecision are frequently inevitable in the controller deployment while designing a practical systems. Obviously, these unmodeled drifts may probably cause performance degradation, round off errors in numerical computation, random perturbation in controller and inherent inaccuracies in simulations (Zhao et al. 2019;. To overwhelm this consequences, it is fundamental to implement a controller that will be able to incorporate some tuning parameters to tolerate the possible gain variations during the synchronization of the underlying system. Desperately, such form of controller are pointed as non-fragile (or resilient), and some important results has been found (see Sakthivel et al. 2018;Shu et al. 2019;Peng et al. 2018; and references therein). To be more precious, in Peng et al. (2018), the synchronization issue with non-fragile control problem for discontinuous NNs subject to randomly occurring controller gain fluctuation and time-varying delay has been investigated, whereas the global asymptotic synchronization has been achieved.
An another important front line in research is finite-time horizon, which is an effective framework for faster synchronization in control systems, and it provokes optimality in convergence time. Further, the introduction of finite-time synchronization criterion has revealed the faster convergence rate, better robustness and disturbance rejection performance (Liu et al. 2016;Wang et al. 2016;Aouiti and Miaadi 2018;Aouiti et al. 2017). In recent years, a huge enumerates of interesting results based on finite-time synchronization have been reported; for instance, see (Zheng et al. 2017;Duan et al. 2019). Recently, in Duan et al. (2019), the issue of synchronization in finite-time has been concerned for fuzzy cellular NNs under differential inclusions framework. Despite that seldom results concerning the synchronization of FSNNs under non-fragile controller design has not yet been inspected thoroughly over finite-time domain which provokes us to commence the present study.
Accordant from the abovesaid observations, we develop herein the non-fragile controller design for finite-time synchronization of FSNNs with nonlinear activation function over finite interval. Moreover, the crucial benefactions of this study can be encapsulated in the following form: • Under non-fragile control strategy, the finite-time synchronization is explored first time for the FSNNs subject to uncertain parameters and discontinuous activation function, respectively.

Problem statement and preliminaries
Consider the FSNNs as shown in Fig. 1 can be characterized by the following fractional differential equation: where C D ν denotes the Caputo fractional derivative which takes the commensurate order ν (0 < ν ≤ 1); x(t) = [x 1 (t), x 2 (t), . . . , x n (t)] T ∈ R n represents the n-neuronal state vector; φ(x(t) = [φ 1 (x 1 (t), φ 2 (x 2 (t)), . . . , φ n (x n (t))] T denotes the nonlinear neuron activation function with φ(0) = 0. Moreover, A = a i > 0 is the self-connection diagonal matrix and B = (b lk ) n×n is the interconnection weight is time-varying matrix that satisfies G T (t)G(t) ≤ I for all t ≥ 0. The noise free system (1) is the drive system and its corresponding response system can be governed by where C depicts the input weight matrix, u(t) ∈ R p represents the control input, and g(t, y(t)) is the noise intensity function. It is a well-known fact that design of non-fragile controller is established to reduce some amount of parameter variations and fluctuations of the controller without loss of its robustness. Due to clear engineering insights, the non-fragile state feedback control incorporated in this study is represented as where K is the state feedback controller gain which is to be computed, K (t) represents the controller gain fluctuations and possess the norm-bounded additive configuration as K (t) = U (t)W , where U and W are known real constant matrices and unknown norm-bounded function (t) which is limited as T (t) (t) ≤ I . Further, it is to be noted that the synchronization error signal can be denoted asx(t) = x(t)−y(t). Thus, the dynamical error system from (1) and (2) with the proposed control strategy (3) can be expressed as follows: where (x(t)) = φ(y(t)) − φ(x(t)). Further, the stochastic noise is defined over the Wiener process probability space ( , F, P), where , F and P reveal the sample space, algebra of the events and the probability measure which is defined on [0, T ] with 1/2 < H < 1. The function g(t,x(t)) : R × R n × R n → R n is the vector-valued noise intensity function and B H (t) denotes the fractional Brownian motion of Hurst parameter H . A centered zero mean one-parameter family of Gaussian process B H (t) on the probability space ( , F, P) with filtration t has covari- Hereinafter, the following definitions, lemmas and assumption are mandatory for smooth development of finite-time synchronization criterion.
Definition 1 Doye et al. (2015) The Riemann-Liouville fractional integral with ν order for the univariate function x(t) is defined as follows: where k is the first-order integer which is bigger than ν. i.e., k − 1 < ν < k, k ∈ Z + and x (k) (.) is the classical k th derivative.

Definition 3 Ma et al. (2016) The Mittag-Leffler function for one parameter
Definition 4 For a given positive definite matrix Q and positive constants c 1 , c 2 , T f with c 1 < c 2 , the fractional-order error system (4) under the controller (3) Lemma 1 Sakthivel et al. (2015) For any real matrix D, S and F of appropriate dimensions and F T F ≤ I . Then, for any scalar > 0 and vectors x, y ∈ R n , one has the following inequality: Assumption 1 For any distinct x, y ∈ R, x = y, the neuron activation function φ i (.)(i = 1, . . . , n) in (1) is assumed to be continuously bounded, nondecreasing and globally Lipschitz and accomplishes . . , n, where χ − i and χ + i are known real constants which takes either positive, negative or zero. For the purpose of proclamation, we express Assumption 2 Given 1 > 0, 2 > 0, the matrix-valued functions A(t) and K (t) are assumed to be normbounded such that where ζ a (t) ∈ R n and ζ k (t) ∈ R n with ζ(t) < 1.

Main results
In this section, we derive the finite-time synchronization criterion for FSNNs (1) under the proposed non-fragile controller. For carrying out the above consideration, the upcoming theorem substantiate a quantity of sufficient condition in respect of LMIs to ease the closed-loop error system (4) of FSNNs (1) to be stochastically synchronized over finite-time interval.
Furthermore, the anticipated non-fragile controller gain fluctuation is procured by employing the relation K = Y X −1 .
Proof To obtain the finite-time synchronization criterion for FSNNs (1), let us construct the Lyapunov function for synchronized error system (4) in the following configuration: Further, based on Property 1 in Doye et al. (2015), the fractional-order Caputo derivative function for V (t) is given as where R D ν signifies the Riemann-Liouville fractionalorder derivative. For representation convenience, we denote (1−ν) x(0) 2 > 0 and further replacing the Caputo fractional derivative in place of the Riemann-Liouville fractional derivative, it can be easy to attain that Further, based on Itô's formula with respect to fractional Brownian motion (Dai and Heyde 1996) and from the trajectory of synchronized error system (4), the above equation can be rephrased as

t)(−P(A + A(t))
− (A + A(t)) T P + PC(K + K (t)) Then, by taking mathematical expectation on both sides of (8), we can obtain that On the other hand, with the aid of Assumption 1, for the diagonal matrices β i > 0 i = 1, . . . , n, one has the following: Now, from Eqs. (9) and (10), we can attain

SG(t)J ) T P + PCU (t)W + (U (t)W ) T C T P.
Further, in light of Lemma (1) and Schur complement, can be remodeled as It is noted that the is not strictly in LMIs format, so we need to transform the inequality constraints into LMIs to attain the control gain fluctuation. Considering this fact, first pre-and post-multiplying the right-hand side of (12) with diag{P −1 , P −1 , I , I , I , I } and assume that P −1 = X . Secondly, denote a new variable as P −1 1 P −1 =ˆ 1 and Y = K X. Thus, it is easy to spot from right-hand side of (12) that left-hand side of (5) is arrived. Thus, if the condition (5) holds, then C D ν V (t) − δV (t) < 0. Suppose that there exists a nonnegative function L(t), the inequality can be rephrased as C D ν V (t) + L(t) − δV (t) = 0. Then, applying Laplace transform, we get Further, taking inverse Laplace transform to (13), we procure LetP = −1/2 P −1/2 , λ 1 = λ min (P) and λ 2 = λ max (P), we can obtain then the above inequality can be easily written as Further, from the inequalities (14), (15) and (16), we can attain Therefore, if the condition (6) holds, then it is clear that x T (t) x(t) < c 2 , for all t ∈ [0, T f ]. Thus, from Definition 4, it can be easily deduced FSNNs (1) is stochastically synchronized within finite-time interval through the controller (3).
In the forthcoming theorem, we derive the finite-time synchronization of FSNNs (1) by assuming the system parameter and control gain variation with norm bounded uncertainty which is defined in Assumption (2). (c 1 , c 2 , d 2 , T f , ), if there exist a matrix X , diagonal matrix 1 , 2 , , ν, c 1 , c 2 , δ and T f with c 2 > c 1 , such that the following conditions hold:

Theorem 2 With the assistance of Assumption 1 and 2, the FSNNs (1) under the controller (3) achieves the stochastic finite-time synchronization criterion with respect to
Moreover, the controller gain fluctuation is acquired by the relation K = Y X −1 .
Proof To derive the anticipated result, we assume that Assumption 2 holds for A(t) and K (t). Then, by following the similar lines of Theorem 1, (11) can be rewritten as where T (t)x(t) and θ 11 = −P A − A T P − δ P + P + K T C T P + PC K − χ 1 1 + 1 ζ a 2 + 2 ζ k 2 . Next, using Lemma (1) and Schur complement, 1 can be rewritten as Further, pre-and post-multiply the right-hand side of (20) with diag{P −1 , P −1 , I , I } and letting X = P −1 , P −1 1 P −1 =ˆ 1 and Y = K X, the right-hand side of (20) is arrived which is similar to the left-hand side of (17). Thus, if the conditions (17), (18) holds, then from Definition 4, it can be concluded that FSNNs (1) is stochastically synchronized over finite-time interval.
For the case neither parameter uncertainty nor control variations ( A(t) = 0 and K (t) = 0), the error dynamics of the FSNNs has the following framework: Corollary 1 Given some positive constants , ν, c 1 , c 2 , δ and T f with c 2 > c 1 , the FSNNs (21) achieves the finite-time synchronization criterion, if there exists X ,ˆ 1 and Y such that the following conditions hold: The non-fragile state feedback controller gain fluctuation can be computed as K = Y X −1 .
Proof In order to procure the required consequence, we consider the uncertain terms as A(t) = 0 and K (t) = 0, the conditions (22) and (23) can be secured easily from Theorem 2. The proof is similar to that of Theorem 2, and hence it is neglected.

Remark 1
The authors in  discussed the synchronization problem for FNNs with discrete and distributed delays. The problem of synchronization criterion for a class of uncertain FNNs subject to multiple time-delays has been investigated (Hu et al. 2019). In the above said works, only synchronization criterion for FNNs has been discussed, whereas the finite-time synchronization analysis has not been yet addressed for the FNNs. Moreover, in our present study, we have considered finite-time synchronization criterion for FNNs through non-fragile control design with discontinuous activation function. It is noticed that finite-time synchronization differs from the synchronization problem in Zhang et al.

Simulation results
This section confers two numerical examples with its simulations for FSNNs (1) to validate the applicability of the developed theoretical approach through finite-time synchronization criterion under the Theorem 1.
Example 1 Consider the FSNNs (1) with two neuron function and the dynamics of parameter coefficients are retrieved as follows: The activation function revealed for the FSNNs (1) is φ(x(t)) = [0.2 tanh(x(t)); 0] and the anticipated values from Assumption (1) are χ 1 = 0.10 and χ 2 = 0.50. Moreover, the fractional-order here is taken ν = 0.93, and the rest of the parameters involved in the inequality (5) Fig. 2 delineates the response of state trajectory curve, i.e., Fig. 2 portrays that the response system exactly traces the drive system. Further, the phase portrait of the drive and its response system is delineated in Fig. 3. Moreover, the corresponding synchronized error state is shown in Fig. 4. Figure 5 shows the time history x T (t) x(t) of the considered system (1), wherein it is noted that the state trajectories of all states indeed satisfy the condition x T (t) x(t) < 5.6414 for all  Fig. 6. Finally, Table 1 displays the optimal bound value of c 2 for various values of c 1 . From Table 1, it is noticed that the optimum bound value of c 2 increasing when the optimum bound value of c 1 is increasing.   At this point, the nonlinear neuron activation functions is spotted as φ(x(t)) = [0.5 tanh(x(t)); 0; 0], from which we can hold the values χ 1 = 0.10 and χ 2 = 0.50 satisfying Assumption 1. Further, the other parameters engaged in the simulation process are stipulated as ν = 0.93, = 0.9, c 1 = 1.4, δ = 0.1 and T f = 10. Moreover, by resorting the LMIs formulated in Theorem 2 with the prior mentioned parameter values, the feasibility can be endorsed and the corresponding state feedback gain fluctuation is quantified as K = [−10.2966 − 11.2866 − 13.4298]. Further, by substituting the obtained feedback gain fluctuation, the simulation results are drawn in Figs. 8,9,10,11 and 12, whereas the initial conditions are randomly chosen as x T (0) = [1 2 1] and y T (0) = [2 3 4]. To mention in short, 8 and 9 illustrate the trajectories of drive and its response state which trace exactly each other within a specified finite-time period. Furthermore, the synchronized error state is shown in Fig. 10. Figure 11 spots the response of the controller. Further, the finite-time synchronization criterion is shown in Fig. 12, whereas the time evolution of x T (t) x(t) does not exceeds the optimum bound value c 2 = 7.8881 and contained within the finite interval. Additionally, the optimal bound value of c 2 for various values of c 1 are established and listed in Table 2. Thus, Table 2 reveals that the value of c 1 increases, the bound value of c 2 will also increases subsequently.
Thus, it is conferred from the above simulation results, the finite-time synchronization criterion under the proposed non-fragile control strategy for the addressed FSNNs (1) is achieved with respect to (c 1 , c 2 , d 2 , T f , ) at specified finite-time interval.

Conclusion
In this paper, we have examined the synchronization problem in finite-time domain for a class of fractional-order stochastic neural networks with discontinuous activation functions via a non-fragile controller. Precisely, the phenomenon of randomness in the controller is characterized by stochastic variables that satisfies the Bernoulli distributed sequences with white noise. Further, by utilizing indirect Lyapunov method along with linear matrix inequalities, a novel set of adequate conditions is procured, and it assures the finite-time stochastic synchronization criterion over specified finite domain. Moreover, the desired form of non-fragile state feedback controller gain fluctuation has been exhibited by using aforementioned sufficient conditions. At last, two numerical examples with its simulations have been proffered to validate the superiority and potency of the synchronization criterion proposed in this study.
Funding The authors have not disclosed any funding.
Data Availability Enquiries about data availability should be directed to the authors.

Declarations
Informed Consent There is no individual participant included in the study.

Conflict of interest
The authors declare that they have no conflict of interest.
human participants The article does not contain any studies with human participants or animals performed by any of the authors.