Exponentially Mean Stability Analysis of Positive Markov Jump Neural Networks with Time Delay

A new result for the analysis of positive Markov jump neural networks (PMJNNs) with time delay is described in this paper. By rewriting the PMJNNs with time delay in both continues-time and discrete-time domains into equivalent positive neural networks(PNNs) and analyzing their stability issues, two delay-dependent sufﬁcient conditions are presented to ensure that the continuous-time and the discrete-time PMJNNs with time delay are exponentially mean stable(EMS) through using the inequality technique. All conditions obtained in the paper are in terms of standard linear programming, which reduces the conservatism. Finally, two numerical examples are provided to verify the validity of our results.

tical fields, such as pattern recognition [1], signal processing [2], finance [3] and so forth. It is public knowledge that many practical systems only experience nonnegative variables(see, for example, [4][5][6]), which gives rise to exploration of positive neural networks(PNNs). By using some novel comparison techniques, the work in [7] has analysed the global exponential stability of PNNs with time-varying delay and a testable condition has been derived to guarantee the uniqueness of positive equilibrium point. The authors in [8] have finished off the work where the exponential stability issue of PNNs in bidirectional associative memory(BAM) model with multiple time-varying delays was solved. The filter design with l 1 -gain disturbance attenuation performance has been implemented for discrete-time PNNs in [9].
On another research front, Markov jump systems(MJSs) [10][11][12][13] that can be used to explore the practical system with random changes in structure and parameters were a special class of stochastic hybrid systems. In fact, there exist random mutations in structure for NNs due to component failures, sudden environmental disturbances or changing subsystem interconnections, which can be modeled as a Markov model and brings about the discussion of Markov jump neural networks(MJNNs). Up to date, some substantial results of MJNNs have been explored in a large body of literature on stability analysis [14,15], synchronization control [16][17][18], state estimation [19][20][21], filter design [22][23][24] and so forth. Very recently, some initial efforts have been committed to the research of positive Markov jump neural networks(PMJNNs) [25]. The work in [25] has investigated the finite-time stabilization of uncertain PMJNNs and a finite-time stabilizable controller has also been designed, where all obtained conditions exist in form of linear matrix inequality(LMI). As far as we know, only a few limited works were devoted to PMJNNs and the existing results are highly conservatism. How to reduce the conservatism of the results may make the stability analysis of the PMJNNs with time delay more complex and the corresponding work remains to be studied, which is more challenging and meaningful.
In accordance with the analysis of the aforementioned results, the main purpose of this paper is to study the exponentially mean stability of the PMJNNs with time delay and to obtain less conservative conditions. The major contributions of this paper are as below: (1)For the first time, the linear programming(LP) method is applied to explore the stability issues of PMJNNs with time delay in both continues-time and discrete-time domains. Compared with the LMI method in [25], it is on reducing computational complexity that the LP method is obviously better than the LMI method and the relation between the LP method and the LMI method has been studied in detail in [26].
(2)By virtue of the augmentation system approach, the PMJNNs with time delay in this paper are rewritten as the augmented systems and the equivalence between the PMJNNs with time delay and the augmented systems are further discussed. And by means of the inequality technique, two delay-dependent sufficient conditions that make PMJNNs with time delay exponentially mean stable(EMS) are provided. Besides, numerical examples further verify that the stability of the PMJNNs is related to the size of time-delay.
The rest of the paper is as follows. In Section II, some essential Lemmas, Definitions, Assumption and the system formulae are given. The main results are discussed in Section III and verified by numerical examples in Section IV. Finally, the conclusion of the paper is given in Section V.
Notations: The set of real numbers and n-dimensional (positive) Euclidean space are expressed as R, R n (R n + ), respectively. R n⇥m denotes the space of n ⇥ m matrix. A ⌫ 0 and A 0 indicate that all elements of matrix A are nonnegative and nonpositive. A 0(A 0) means that every element of matrix A is positive (negative). a ij represents the element of the i-th row and j-th column of the matrix A = {a ij }. E{.} stands for the mathematical expectation and ⌦ denotes Kronecker product. The 1-norm and the transposition of the matrix A are described as ||A|| 1 and A T , respectively. I n denote the identity matrix of n ⇥ n.
Concerning the discrete-case, we examine a kind of discrete-time PMJNNs with time delay as below where the system state x(k) 2 R n + is controlled by n-neurons. Markov sequence {r(k), k 0} take values in set N={1, 2,...,N} with transition probability matrix Π = {π pq } defined by Clearly, for all p, q 2 N, π pq 2 [0, 1], and for all p 2 N, ∑ M q=1 π pq = 1. For r(k)=p 2 N, D p , A p , B p and J p indicate the corresponding matrixes of the pth mode of the system Σ 2 . ϕ(θ ) is the initial value with ϕ(θ )=φ (0) for θ 2 [ β , 0] and β is a given constant time delay. Assumption 1. we assume that the activation function f u (.) and g u (.) are continuous and bounded on R + for u 2 [1, 2,...,n] such that the following conditions holds for x, y 2 R + , x 6 = y. We describe Assume that x e is equilibrium point of system Σ 1 , let y(t)=x(t) x e , Σ 1 can be rewritten as: Obviously, h u (.) and κ u (.) also satisfy the Assumption 1.
Assume that x 1e is equilibrium point of system Σ 2 , let y(k)=x(k) x 1e , Σ 2 can be rewritten as: where ϕ y (θ )=ϕ(θ ) x 1e . Remark 1. For systems Σ 3 and Σ 4 , according to assumption 1 and the analysis in [25], we know that the external input vector J i and J p are removed by y(t)=x(t) x e and y(k)=x(k) x 1e .
and Assumption 1 hold, the system Σ 1 and Σ 3 are positive for nonnegative initial state. Lemma 2. If D p ⌫ 0, A p ⌫ 0, B p ⌫ 0, J p ⌫ 0 and Assumption 1 hold, the system Σ 2 and Σ 4 are positive for nonnegative initial state. Definition 1. The system Σ 1 and Σ 3 are exponentially mean stable(EMS) if there exist two positive constants ε and γ such that for any nonnegative initial condition. Definition 2. The system Σ 2 and Σ 4 are EMS if there exist two positive constants ε 1 and 0 < ξ < 1 such that for any nonnegative initial condition. Lemma 3.
[11] Consider a stochastic process { f (t), r(t),t  0} such that the jumping process r(t) is a homogeneous Markov chain with right-continuous trajectories When M={1} and N={1}, the Σ 3 and Σ 4 are transformed into the continuous-time PNNs Γ 1 and the discrete-time PNNs Γ 2 respectively, as shown below Definition 3. The system Γ 1 is exponentially stable(ES) if there exist two positive constants ε and γ such that for any nonnegative initial condition. Definition 4. The system Γ 2 are ES if there exist two positive constants ε 1 and 0 < ξ < 1 such that for any nonnegative initial condition.

EMS of continuous-time PMJNNs
In this subsection, the continuous-time PMJNNs is rewritten as an equivalent continuoustime PNNs. By means of analyzing the exponential stability issue of the continuoustime PNNs, a sufficient condition is derived to ensure that the system Σ 3 is EMS. For simplicity of presentation, according to Lemma 3, introduce the following notations: where Ω (τ)={ρ ij (τ)} and ρ ij (τ)=P r {r(t + τ)= j|r(t)=i} for i, j 2 M, τ 0,t 0. And Ω (τ) satisfies the forward Kolmogorov differential equation: then, the system Σ 3 is positive and EMS for a given τ.
λ ji y y y j (t)dt (7) that isẏ y y i (t)= D i y y y i (t)+A i F(y y y i (t)) + B i M ∑ j=1 ρ ji (β )G(y y y j (t β )) + M ∑ j=1 λ ji y y y j (t) (8) On the basis of (5), (8) is written in vector form as follows where F(Y(t)) = [F(y y y 1 (t)), F(y y y 2 (t)), ··· , (y y y M (t))], G(Y(t τ)) = [G(y y y 1 (t τ)), G(y y y 2 (t τ)), ··· , G(y y y M (t τ))]. Clearly, the activation functions F(.) and G(.) are continuous on R Mn and satisfy Assumption 1. We define Then From (10) and Definition 3, we know that there exist positive constants ε and γ such that So, from Definition 1, we know that the system Σ 3 is EMS if the system Σ 5 is ES. Next, we will discuss the exponential stability of the system Σ 5 . Under Lemma 1, let Z(t)=e γ(t t 0 ) Y(t), taking the upper right derivative of Z(t), we obtain where ϖ ϖ ϖ = γI Mn . After substituting Z(t) for e γ(t t 0 ) Y(t) in Equ. (12), the following formula is obtained For t > 0, we assume that Z(t) Pl 0 . If this is not true, then there are corresponding t 0 > 0, which make Z(t 0 )=Pl 0 , D + Z(t 0 ) ⌫ 0. According to (13), we get let W =( D + ϖ ϖ ϖ + AL f + e γτ BL g )P Substituting (5) and (9) into (15) yields From (6) and (17), it is easy to get W =( D + ϖ ϖ ϖ + AL f + e γτ BL g )P 0, which means D + Z(t 0 ) 0. Obviously, this is a contradiction. Thus Z(t) Pl 0 for t 0. That is Then, we have where ε = Mn· max 1iM,1cn, {o i c }/o min . The proof is completed.
where Ξ q = ∑ N p=1 π pq D p P p , q 2 N. From (23) and (30), it's obvious that W 1 = ξ 1 {D + AL f + ξ β BL g }P P 0 is established, which means ∆ Z(k 0 ) 0. It's easy to see that this is a contradiction to the previous assertion. So, Z(k)=ξ (k k 0 ) Y(k) Pl 0 for k 0 holds. Consequently Let ε 1 = Nn· max 1pN,1cn {o p c }/o min , we get The proof is complete. Remark 2. In this paper, by utilizing the special properties of the PMJNNs itself, the augmentation system approach is applied to solve the stability problem of PMJNNs with time-delay, which can reduce the complexity of system analysis. The EMS conditions in terms of the standard linear programming are derived in Theorem 1 and Theorem 2. Although stability conditions for PMJSSs have been discussed in [25], there are only sufficient conditions in the form of LMI. Thus, the conservatism of the results in [25] is greatly reduced by the proposed method in the paper. Remark 3. Delay-dependent EMS conditions for continuous time PMJNNs and discrete time PMJNNs are given in Theorem 1 and Theorem 2. In this paper, the stability of the PMJNNS are directly affected by time delays, parameters γ and ξ , which also can be verified by the example at numerical example section.
4 Numerical example Example 1. Consider a class of the system Σ 3 with three operation modes described as follows: The initial states for the system Σ 3 is chosen as: φ y (0)=[5; 2.5; 3.5]. Fig. 1 shows mode evolution of the system Σ 3 . Fig. 2 depicts the state trajectories of the system Σ 3 , which can draw the conclusion that the systems Σ 3 is positive and EMS.   Table 1 and Table 2 show the effects of different time delays τ and different system parameters γ on the stability of the system Σ 3 .    The initial state is initialized to ϕ y (0)=[5; 2.5; 3.5]. Fig. 3-4 demonstrates the simulation results of the system Σ 4 , which can illustrate that the systems Σ 4 is positive and EMS.  Table 3 and Table 4 show the effects of different time delays β and different system parameters ξ on the stability of the system Σ 4 .

Conclusion
In this paper, the linear programming method, the augmentation system method and inequality technique are used to investigate the EMS issues of PMJNNs with time delay. Sufficient criteria in the form of linear programming are put forward to guarantee the EMS on PMJNNs with time delay in both continuous-time domain and discretetime domain. Further more, this paper reveals that the EMS of PMJNNs with time delay is influenced by the size of time-delay. Two numerical examples is provided to prove the validity of our theoretical discovery. Finally, the effect of time-varying delay for positive Markov jump neural networks will be discussed in future work.

Data Availability
Data sharing is not applicable to this article as no new data were created or analyzed in this study.