H ∞ Synchronization of Semi-Markovian Jump Neural Networks with Randomly Occurring Time-Varying Delays

Based on the Lyapunov stability theory, this paper mainly investigates theH∞ synchronization problem for semi-Markovian jump neural networks (semi-MJNNs) with randomly occurring time-varying delays (TVDs). The continuous-time semi-MJNNs, where the transition rates are dependent on sojourn time, are introduced to make the issue under our consideration more general. One of the main characteristics of our work is the handling of TVDs. In addition to using the improved Jensen inequality and the reciprocal convexity lemma to deal with the integral inequality, we also employ Schur complement and the projection lemma to achieve the decoupling between the square term of TVDs. Finally, we verify the validity and feasibility of our method by a couple of simulation examples.


Introduction
In recent years, with the unceasingly thorough research on large data and artificial intelligence, the theory and application of neural networks have been greatly developed.It has tremendous application prospect, especially in robotics [1], pattern recognition [2,3], associative memory [4][5][6], identification [7,8], and combinatorial optimization [9][10][11][12].Neural networks can be simply divided into the deterministic neural networks and stochastic neural networks based on whether they are disturbed by outside noise [13].When the system is undisturbed, the deterministic neural network can describe the actual system accurately [13].Nevertheless, as far as we know, the actual system is generally uncertain and most of the physical system will be affected by random parameter variation and structure change [14][15][16].These changes may be caused by some sudden phenomena, such as components or connection failure and the deviation of parameter.In this circumstance, the stochastic neural networks can be described by a hybrid model, where a discrete stochastic variable called mode or pattern is attached to continuous state variables to describe the random jump of system parameters as well as the appearance of discontinuous points.It allows policymakers to respond to discrete events, which significantly perturb or alter the normal working condition of the system, by combining the empirical knowledge of events and the statistical information of their rates, adequately [17,18].
Markovian jump neural networks (MJNNs), as we all know, as a kind of typical hybrid dynamic systems are widely used in the field of aerospace, industrial production, and biological, medical, and social construction in the past few decades due to its strong modeling ability and therefore draw great attention from researchers.For instance, the stability analysis, state estimation, filter design, passivity analysis, and stochastic synchronization for MJNNs were discussed in [19,20], respectively.But one obvious drawback of MJNNs is that its jump time obeys the exponential distribution, which is a memoryless distribution and makes the transition probability of jump system an invariant function matrix; that is, the transition probability of the system obeys a stochastic process which is not relevant with the mode of the past [21].Because of this limitation, it brings great restriction to the application of MJNNs.Therefore, semi-MJNNs were put forward later, where a transition probability matrix and a fixed dwell time probability density function matrix are used to represent the stochastic neural networks [22].It has a wider range of application background due to the relaxation to the constraint condition where the probability density distribution function obeys exponential distribution.Compared with the abundant research achievements on MJNNs, the research efforts devoted to semi-MJNNs are relatively scarce.The robust stochastic stability condition for semi-MJNNs was derived in [23] and the relevant controller was also designed there.The synchronization controller for the semi-MJNNs was designed in [24] where the semi-MJNNs were transformed into associated MJNNs.An exponential passive filter was designed, and a cone complementarity linearization method was applied to manage the nonconvex feasibility issue in [25].As mentioned above, the semi-MJNNs have more extensive application, such as in complex medical procedures [26].
Due to the finite signal transition speed as well as the limited switching speed of hardware facilities, the time-delay phenomenon exists in various practical industrial control systems widely, such as chemical system, process control system, and network control system [27][28][29][30][31][32][33][34][35][36].It is known that delay argument existing in the system is often unknown or time-varying and the occurrence of delay tends to be random, which makes the analysis and control of the system more difficult.Also, the existence of time-delay tends to result in the degradation of the performance index and can even make the system unstable [37].Therefore, it has important theoretical significance and practical applying value to study the system with time-varying delays (TVDs).As the system that considers that time-delay is more in line with the actual situation, an increasing number of researches have been made on the time-delay systems in recent years, and considerable results have been presented.To mention a few, Gun and Niculescu discussed the problem of stability analysis for the systems with time-delay and gave a summary on literature about the stability analysis and controller design of systems with time-delay in [38].Park and Wan Ko studied the stability and robust stability criteria for TVD systems in [39], and then the reciprocally convex approach and the second-order reciprocally convex approach were proposed for stability analysis of TVD systems in [40].
Synchronization refers to two or more dynamic systems whose properties are identical or close to each other.Through the interaction between the systems, the state of the dynamic system that evolves under different initial conditions is gradually close to each other and finally reaches the same.Synchronization analysis is particularly important in many dynamic behaviors of neural networks and therefore has been widely studied.Exponential synchronization, adaptive synchronization, finite-time H ∞ synchronization, mixed H ∞ /passive synchronization, and new delay-dependent exponential H ∞ synchronization were considered in [41,42], respectively.This paper mainly studies the H ∞ synchronization of semi-MJNNs with randomly occurring TVDs.First of all, by Lyapunov stability theory, we can get that the key point to establish a Lyapunov functional is to contain more useful information about the delays, which is useful to obtain the results with less conservatism [43].As a result of the existence of TVDs, some novel inequality techniques derived from the Park inequality and the improved Jensen inequality [44] are employed to handle the time-varying items.At the same time, considering the existence of the square term of TVDs in the formulas, we use the projection lemma to achieve the decoupling between time-varying items.By using convex optimization techniques, the synchronization control of semi-MJNNs is investigated in this paper.The corresponding main results are presented by three theorems: Theorem 1 provides sufficient conditions for the stochastic stability and H ∞ synchronization of the closed-loop dynamic error system; Theorem 2 conducts the decoupling arithmetic; and Theorem 3 is then presented to get strict LMI-based conditions, and a numerical method to calculate controller gains is presented, which is simple and easily conducted.
Compared with the existing literature, this article has the following characteristics: (1) Different from the previous literature, a more general system model is introduced in this paper, in which both the semi-MJNNs and the random TVDs are taken into account simultaneously; (2) with the introduction of some advanced inequalities, combining with Schur complement lemma and projection lemma, H ∞ synchronization conditions with less conservatism are derived; (3) we use the LMI control toolbox to carry out the relevant simulation, and the corresponding controller can be obtained which can verify the correctness and feasibility of the proposed method.Throughout this work, the notations used are standard.

Problem Formulation
Firstly, given the following semi-MJNNs with randomly occurring TVDs Σ , T ∈ ℝ n is the system state vector which is associated with the n neurons; the neuron activation functions of the system, which is assumed to be bounded and satisfies where f q 0 = 0, a, b ∈ ℝ, a ≠ b l − q , and l + q are real known scalars, and they could be zero, positive, or negative.For the purpose of simplifying the symbols, we set I t stands for external input; θ t denotes the TVDs satisfying 0 ≤ θ 1 ≤ θ t ≤ θ 2 < ∞ and θ t ≤ μ < ∞, where the nonnegative scalars θ 1 and θ 2 refer to the minimum 2 Complexity and maximum time-delay, respectively.ξ t , σ t≥0 = ξ m , σ m m∈M ≥1 (M is a positive integer) represents a continuous-time and discrete-state homogeneous semi-Markovian process whose trajectories are right continuous.Assuming ξ t takes value in a finite state space ϒ = 1, 2, … , N , the transition rate matrix Λ σ ≜ ρ ij σ N ×N can be given by where ε > 0, lim ε→0 o ε /ε = 0 and ρ ij σ ≥ 0 ξ t = i, ξ t + ε = j ; i ≠ j stands for the transition rates from i to j, and ρ ii σ = −∑ N j=1,j≠i ρ ij σ β k is a Bernoulli-distributed white sequence that takes values of 0 and 1 and obeys the following probability distribution laws where β ∈ 0, 1 is a known constant.
Remark 1. Different from the previous literature, a more general system model is introduced in this paper, in which both the semi-MJNNs and the random TVDs are taken into account.The time-delay phenomenon, which occurs randomly and tends to be time-varying, exists in various practical neural networks.Therefore, the stochastic variable β t is introduced to express the randomly occurring TVDs in this paper to make the issue under consideration more practical and more reasonable.
In this paper, the slave system Σ could be represented as the following forms where x t and ẑ k are the response state vector and the response output, respectively; u t is the advisable control input.
For presenting a better explanation to the addressed problem, we introduce ϱ t = x t − x t as the synchronization error vector, ϱ t = z t − ẑ t = C ξ t ϱ t as the output error, and f ϱ t = f x t − f x t as the nonlinear error.Then, the following error system is obtained: The controller input u t for the error system is established as where the controller gain matrix K ξ t ∈ ℝ n×n will be designed in the sequel.In order to simplify the notation, A ξ t , B ξ t , B θ ξ t , C ξ t , D ξ t and K ξ t are denoted by A i , B i , B θi , C i , D i , and K i , for each i ∈ ϒ, respectively.Then, the closed-loop dynamic error system Σ can be obtained: Before making further derivation, we need the definition and lemmas as shown in the following.Lemma 1. [24] The following inequalities hold for any diagonal matrices Lemma 2. [45] (projection lemma) For a symmetric matrix F, two matrices U, V with the same column dimension of F, the problem can be solved with respect to matrix X if and only if where N U and N V are any basis of the nullspace of U, V.
Lemma 3. [30] (Schur complement lemma).Given constant matrices M 11 , M 12 , and M 22 , where M 11 = M T 11 < 0 and it is easy to obtain the following inequalities Lemma 5. [40] (reciprocal convexity lemma) For any vector ; and nonnegative real scalars α and β meeting α + β = 1, the following inequality holds The system Σ is said to be H ∞ synchronization with the disturbance attenuation γ under the condition that the following requirements are met: (1) The system Σ is stochastically stable when the disturbance input ω t is always equal to 0.
(2) For a positive scalar γ, the following inequality is satisfied under zero initial conditions

30
with First of all, we define

32
where g i σ is the probability density function of sojourn time σ resting on mode i.Before further analysis, we consider the weak infinitesimal operator ℒ as the following forms and we have Complexity Considering the integral items in (35) and (36), we can get that (37), by using Lemma 4, we can obtain that Similarly, it is easy for us to get the following conditions: Synthesizing the above results, we can get that Considering the single integral items in (41) and (42), we can get that (43), by using Lemma 4, we can obtain that By using a similar method, it is not difficult to get that Considering the double integral items in ( 41) and ( 42), we can get that 47), by using Lemma 4, we can obtain that It is easy to find that the first integral term in (49) can be reduced to a single integral.So we can use Lemma 4 to handle it: For the second integral term in (49), we firstly divide it into three parts: Then, we consider using Lemma 4 to deal with (51) and the following inequalities can be obtained: Let us define α = θ t − θ 1 /θ and β = θ 2 − θ t /θ Using Lemma 5, we can obtain the following relation from inequalities (45), (46), and (53): Synthesizing inequalities ( 41), ( 42), ( 44), ( 48), ( 50), (52), and (54), we can get that After that, some free-weight matrices are utilized.For any scalars δ 1 and δ 2 and the matrix Y i of appropriate dimensions, we have 9 Complexity On the other hand, considering the nonlinear part and according to Lemma 1, one has that for any matrices V ki > 0, k = 1, 2: Furthermore, considering the H ∞ synchronization, in view of Definition 1, we can get Above all, we can get that According to (20), one can obtain that With ω t = 0, we can get from (59) that ℒV ϱ s t , ξ t < 0, which implies that the system Σ is stochastically stable.
Furthermore, considering the system Σ , from (60) one could obtain the following inequality under zero initial conditions for any γ > 0: Therefore, the condition in Definition 1 holds.
In view of the above, we can conclude that the considered system is stochastically stable and satisfies H ∞ synchronization with the disturbance attenuation γ.
From the stochastic stability and H ∞ synchronization analysis criterion for the system in Theorem 1, it is not difficult to find that the inequalities may not be easily verified as the existence of TVD term θ t and nonlinear delay term θ 2 t , especially the quadratic term ∏ θ t ∑ N k=1 ρ ik P k ∏ T θ t .So, we utilize projection lemma and Schur complement to achieve matrix decoupling.
Remark 2. For the quadratic term ∏ θ t ∑ N k=1 ρ ik P k ∏ T θ t , it is not difficult to find that ρ ii , i ∈ ϒ are negative terms while ρ ij , i, j ∈ ϒ, i ≠ j are positive terms.Therefore, we use different methods to deal with this problem.When i ≠ j, ρ ij > 0, we can use Schur complement to handle it.When i = j, ρ ii < 0, it is quite clear that Schur complement cannot be applied to it directly.So we use projection lemma to cope with it.By using this way, we can achieve the decoupling between nonlinear terms properly.Remark 3. From stochastic Lyapunov functional (29) as well as inequality (16), sufficient conditions which guarantee the stochastically stable and H ∞ synchronization of system Σ are obtained.The term used to include more information about state delays.The semi-Markovian parameters are introduced to V 2 ϱ s t , ξ t , V 4 ϱ s t , ξ t , and V 5 ϱ s t , ξ t for the consideration of mode-dependent Lyapunov functional.Inequality ( 16) is used to deal with the integral terms.By using this way, we can obtain sufficient conditions with less conservatism.
Proof 2. From Theorem 1, we can know that when conditions ( 20), ( 21), ( 22), ( 23), ( 24), (25), and ( 26) are satisfied, the considered system Σ is stochastically stable.In order to achieve decoupling between nonlinear timedelay terms in (20), further processing will be made for inequality (20).First of all, the inequality (20) can be rewritten as where From inequality (65) we can also get from quadratic form knowledge that Ξ i < 0, then we can obtain the following inequality where It is not difficult to know that PN P = 0, QN Q = 0 Therefore, N P and N Q are the elements of the nullspace of P and Q.
Then, by using projection lemma to inequalities (65) and (67), we can know that Ξ i + sym P T XQ < 0, 68 which means Then, we use Schur complement to deal with the coupling term Π θ t ∑ N k=1,k≠i ρ ik P k Π T θ t in (69) and we can obtain that
Since a stochastic stability and H ∞ synchronization analysis criterion for the system in Theorem 2 is developed, the procedure of H ∞ synchronization can be developed in Theorem 3 in the further analysis.Therefore, in this section we derive H ∞ synchronization conditions for the driveresponse dynamic systems.The main result about the design of a desired controller will be presented, and the controller gain will be presented.
0, and β ∈ 0, 1 , the considered system Σ is stochastically stable and H ∞ synchronized, if there exist symmetric matrices P i ∈ ℝ 3n > 0, positive definite matrices , matrices Y i and K i , and S ij ∈ ℝ n , i, j ∈ ϒ, such that for each i ∈ ϒ, and k = 1, 2,, the conditions ( 21), ( 22), ( 23), ( 24), (25), and ( 26) and the following inequality hold: where then the system Σ is stochastically stable.In this regard, the available gain of the controller can be calculated by Proof 3. We define By using this way, the freedom variable Y i and the controller gain K i can be merged, which makes sure inequality (72) can be solved by dealing with a convex optimization problem.This completes the proof.

Numerical Examples
In this section, two numerical examples are given for demonstrating the feasibility and validity of the proposed method.In the first example, the existence of the desired H ∞ synchronization controller is verified.The second example gives a set of figures to reveal the correctness of our method.
Example 1.For the proposed system Σ and Σ with three modes i = 1, 2, 3 , the parameters are chosen as follows: Considering the activation functions f i s , i = 1, 2, 3 with the parameters l − q = 0 and l + q = 0 4, then it is easy to get that The expectation of β t is taken as β = 0 6 Assuming that I t = 0, and the TVDs θ t = 1 1 + sin 0 5t , then we can obtain that 0 1 ≤ θ t ≤ 2 1 and θ t ≤ 0 5, which means θ 1 = 0 1, θ 2 = 2 1, and μ = 0 5. Besides, the transition rates of ξ t are selected as the same as in [24], where the probability distribution function is also chosen as Weibull distribution.Therefore, the corresponding mathematical expectation of Λ σ is  22), ( 23), ( 24), ( 25), ( 26) and (72) by using the LMI control toolbox, the H ∞ synchronization controller gain matrixes can be obtained as follows: Then, we can find that when μ = 0 5, θ 1 = 0 1, the maximum allowable upper delay bound θ 2 in [24] is 0 984, while we set θ 2 = 2 1.The desired controller gains can also be obtained in this paper.This indicates that our method is superior to [24] and can lead to less conservatism for time-delay systems.
Example 2. The parameters of the systems Σ and Σ are given with N = 3 as follows:  The activation functions are assumed as f i s = tanh s , i ∈ ϒ with the parameters l − q = 0 and l + q = 1; then, it is easy to get that L 1 = 0 0 0 0 , L 2 = 0 5 0 0 0 5 82 The expectation of β t is chosen as β = 1 Assuming that I t = 0, ω t = 1/1 + t 2 1/1 + t 2 T and the TVDs θ t = 1 + 0 3 sin 2t , then we can obtain that θ 1 = 0 7, θ 2 = 1 3, and μ = 0 6 The mathematical expectation of the transition rate matrix is taken as the same as Example 1.Then, we set δ 1 = 1, δ 2 = 5, γ 2 = 0 4, and we can get the desired controller gain matrixes as follows: Under the obtained gain matrixes and x 0 = 0 2 0 25 T and x 0 = 0 05 −0 15 T , we simulate the system and obtain a set of figures.Figure 1 shows the mode transitions ξ t of the system.Figures 2 and 3 show the state vector's behaviors of systems Σ and Σ , respectively.Figure 4 shows the state responses of the system Σ .In view of Figure 4, we can observe that although under different initial conditions, the error between systems Σ and Σ gradually tends to zero which indicates that the gain matrixes derived from this paper can guarantee the synchronization between the systems Σ and Σ effectively.Therefore, it confirms the validity of the developed method.

Conclusion
In this paper, the H ∞ synchronization problem for semi-MJNNs, where the randomly occurring TVDs have been considered to make the neural networks under consideration more practical, has been investigated.By constructing an appropriate semi-Markovian Lyapunov-Krasovskii functional, combining with the derivation of an infinitesimal generator for the Lyapunov functional and the sufficient H ∞ synchronization condition for semi-MJNNs has been 14 Complexity established.To deal with the TVD items, some improved inequalities, together with Schur complement lemma and projection lemma, have been introduced.By using a linearization technique, the desired controller has been designed and the existence of the desired controller can be verified by the feasibility of a set of LMIs.Finally, two meaningful examples have been given to validate the feasibility and validity of the developed approach.Considering that the conditions established in this paper are sufficient, how to reduce the conservatism of current results will therefore be our future work.In addition, how to extend our results to cope with the controller design problem for more complex networks, such as genetic regulatory networks, will also be one of our future research directions.

Remark 4 .
When we set β = 1 and D 1 = D 2 = D 3 = 0 and do not consider the H ∞ performance index and the other parameters remain the same, then our system is the same as Example 13 in[24].In this condition, we can get the controller gain matrixes as follows:

Figure 4 :
Figure4: The synchronization error between the drive system and the response system.