pth Moment exponential stability of stochastic delayed hybrid systems with Lévy noise

Accepted Manuscript pth Moment Exponential Stability of Stochastic Delayed Hybrid Systems with Lévy Noise Wuneng Zhou, Jun Yang, Xueqing Yang, Anding Dai, Huashan Liu, Jian’an Fang PII: DOI: Reference:

S0307-904X(15)00028-1 http://dx.doi.org/10.1016/j.apm.2015.01.025 APM 10376

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Appl. Math. Modelling

Please cite this article as: W. Zhou, J. Yang, X. Yang, A. Dai, H. Liu, J. Fang, pth Moment Exponential Stability of Stochastic Delayed Hybrid Systems with Lévy Noise, Appl. Math. Modelling (2015), doi: http://dx.doi.org/ 10.1016/j.apm.2015.01.025

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pth Moment Exponential Stability of Stochastic Delayed Hybrid Systems with L´evy Noise Wuneng Zhou1,∗, Jun Yang1,2,∗, Xueqing Yang1 , Anding Dai1 , Huashan Liu1 , Jian’an Fang1

Abstract In this paper, the problem of pth moment exponential stability analysis is considered for stochastic delayed hybrid systems with L´evy noise. By the technique of stochastic analysis as well as M-matrix theories, we propose some sufficient conditions to guarantee the exponential stability of the system. The criterion of mean square exponential stability for delayed neural networks with L´evy noise is derived as well. A numerical example is provided to show the usefulness of the proposed exponential stability criterion. Keywords: exponential stability, stochastic hybrid systems, L´evy noise, generalized Itˆo’s formula, M-matrix, neural networks 1. Introduction The study of stability problem regarding jump diffusion systems [1, 2, 3, 4, 5, 6, 7] or so-called systems with L´evy noise [8, 9, 10, 11] has become an increasing interest in the past few years. Exponential or asymptotic stability conditions have been presented for these stochastic systems. These two kinds of systems are actually unified, as Applebaum shows in his book [12], by L´evy-Itˆo decomposition, L´evy noise can be decomposed into a continuous part and a jump part which respectively correspond to the diffusion and ∗

Corresponding author Email addresses: [email protected] (Wuneng Zhou), [email protected] (Jun Yang) 1 College of Information Sciences and Technology, Donghua University, Shanghai 201620, China 2 School of Mathematics and Statistics, Anyang Normal University, Anyang 455000, China Preprint submitted to Elsevier

January 21, 2015

jump term in systems. In the meantime, stability issues of stochastic systems with Markovian switching have received tremendous research attention, see [13, 14, 15, 16, 17], to name but a few. A Markovian switching system is a hybrid system with state vector that has two components x(t) and r(t). The first one is regarded as the state while the second one as the mode. Governed by a Markov chain with a finite state space, the system switches from one mode to another in a random way [18]. This switching manner is more suitable for the description of random failures, abrupt changes or sudden disturbances arising in many real systems. Nowadays, stability analysis for jump diffusion systems with Markovian switching [2, 3, 4, 6, 7, 19] or hybrid systems with jump [18] tends to be a new research focus. On the other hand, time delays, which commonly appear in practical systems, are often the cause of instability. Hence the stability of stochastic delay systems is always the hot area in many researches [10, 17, 18, 20, 21, 22]. To the best of our knowledge, up to now, the pth moment exponential stability of stochastic hybrid delayed systems with L´evy noise has not been completely studied. Yuan and Mao [18] studied the same problem only in the case of p = 2. In [6], a general exponential stability condition was derived for switching jump diffusion systems without delay. Ning and He etc. presented some sufficient conditions guaranteeing the exponential stability for neutral systems without Markovian switching. Therefore our aim in this paper is to obtain the pth moment exponential stability criteria for stochastic hybrid delayed systems with L´evy noise. By the technique of stochastic analysis and M-matrix approach, we extend Mao’s work [15] to the case with Poisson jump as well as Yang’s conclusion [6] to the case with delay. The rest of this paper is organized as follows: in Section 2, we give formulation and preliminaries for our main results. The main results in Section 3 are divided into two parts, where the first part investigates a general exponential stability theorem and an easy-to-test criterion which employs M-matrix theories and Theorem 1 in the case of p ≥ 2. In the second part, an exponential stability criterion is proposed for neural networks in the case of p = 2. Hereafter, in Section 4, We provide a numerical example to show the usefulness of our results. 2. Model and Preliminaries Let (Ω, F, {Ft }t≥0 , P) be a complete probability space with a filtration {Ft }t≥0 satisfying the usual conditions (i.e., it is increasing and right contin2

uous while F0 contains all P-null sets). Let | · | denote the Euclidean norm as well as the matrix trace norm. Denote by λmax (A) the largest eigenvalue of matrix A. The shorthand diag(ζ1 , · · · , ζN ) stands for a diagonal matrix with diagonal entries ζ1 , · · · , ζN . Let τ > 0 and p > 0. Denote by C([−τ, 0]; Rn ]) the family of continuous functions ϕ from [−τ, 0] to Rn with the norm ∥ϕ∥ = sup−τ ≤θ≤0 |ϕ(θ)|. Denote by LpFt ([−τ, 0]; Rn ]) the family of Ft -measurable C([−τ, 0]; Rn ])-valued random variables ξ = {ξ(θ) : −τ ≤ θ ≤ 0} such that E∥ξ∥p < ∞. Denote by C 2,1 (Rn × R+ × S; R+ ) the family of positive real-valued functions defined on Rn × R+ × S which are continuously twice differentiable in x ∈ Rn and once differentiable in t ∈ R+ . For a, b ∈ R, a ∨ b (respectively, a ∧ b) means the maximum (respectively, minimum) of a and b. Let B(t) = (B1 (t), B2 (t), ..., Bm (t))T be an m-dimensional Ft -adapted Brownian motion and N (t, z) be an l -dimensional Ft -adapted Poisson ran˜ (t, z) which satisfies N ˜ (dt, dz) = dom measure on [0, +∞)×Rl with compensator N N (dt, dz) − ν(dz)dt = (N1 (dt, dz1 )− ν1 (dz1 )dt, · · · , Nl (dt, dzl ) − νl (dzl )dt)T , where {Ni , i = 1, · · · , l} are independent 1-dimensional Poisson random measures with characteristic measures {νi , i = 1, · · · , l} coming from l independent 1-dimensional Poisson point processes. Let {r(t), t ≥ 0} be a right-continuous Markov chain on the probability space taking values in a finite state space S = {1, 2, · · · , N } with generator Γ = (γij )N ×N given by { γij ∆ + o(∆) if i ̸= j P{r(t + ∆) = j|r(t) = i} = 1 + γii ∆ + o(∆) if i = j where ∆∑> 0. Here γij ≥ 0 is the transition rate from i to j if i ̸= j while γii = − j̸=i γij . Consider the n-dimensional stochastic delay hybrid system with jumps dx(t) =f (x(t), x(t − δ(t)), t, r(t))dt + g(x(t), x(t − δ(t)), t, r(t))dB(t) ∫ (1) + h(x(t− ), x((t − δ(t))− ), t, r(t), z)N (dt, dz) Rl

on t ∈ R+ , where x(t− ) = lims↑t x(s). Here δ : R+ → [0, τ ] is a Borel measurable function which stands for the time lag, while f : Rn × Rn × R+ × S → Rn , g : Rn × Rn × R+ × S → Rn×m and h : Rn × Rn × R+ × S → Rn×l . We assume that the initial data are given by {x(θ) : −τ ≤ θ ≤ 0} = ξ(θ) ∈ 3

LpF0 ([−τ, 0]; Rn ]) , r(0) = r0 . We note that each column h(k) of the n × l matrix h = [hij ] depends on z only through the kth coordinate zk , i.e. h(k) (x, y, t, i, z) = h(k) (x, y, t, i, zk ),

z = (z1 , · · · , zl )T ∈ Rl , i ∈ S

We further assume that B(t), N (t, z), r(t) in system (1) are independent. For the purpose of stability study in this paper we impose the following assumptions. Assumption 1. Assume that the system (1) has a unique solution on t ≥ −τ which is denoted by x(t, ξ). The functions f, g and h satisfy f (0, 0, t, i) ≡ 0, g(0, 0, t, i) ≡ 0, h(0, 0, t, i, z) ≡ 0 for each (t, i) ∈ R+ × S and z ∈ Rl . Assumption 2. Assume that δ is differentiable and its derivative is bounded by a constant δ¯ satisfying 0 ≤ δ¯ < 1, namely δ˙ ≤ δ¯ < 1 , ∀t ≥ 0. One can immediately derive from Assumption 1 that (1) admits a trivial solution x(t; 0) ≡ 0 which is necessary for the following definition of exponential stability. Definition 1. The trivial solution of (1) is said to be exponentially stable in pth moment if 1 lim sup log(E|x(t; ξ)|p ) < 0 t→∞ t for any ξ ∈ LpF0 ([−τ, 0]; Rn ]). When p=2, it is said to be exponentially stable in mean square. Given V ∈ C 2,1 (Rn × R+ × S; R+ ), we define the operator LV [18] by LV (x, y, t, i) =Vt (x, t, i) + Vx (x, t, i)f (x, y, t, i) 1 + trace[g T (x, y, t, i)Vxx (x, t, i)g(x, y, t, i)] 2 ∫ ∑ l + [V (x + h(k) (x, y, t, i, zk ), t, i)

(2)

R k=1

− V (x, t, i)]νk (dzk ) +

N ∑

γij V (x, t, j)

j=1

where Vx (x, t, i) =

( ∂V (x, t, i) ∂x1

,··· ,

( ∂ 2 V (x, t, i) ) ∂V (x, t, i) ) , Vxx (x, t, i) = ∂xn ∂xu ∂xv n×n 4

Then the generalized Itˆo’s formula can be given as follows. V (x, t, r(t))

∫

t

=V (x(0), 0, r0 ) + LV (x(s), x(s − δ(s)), s, r(s))ds 0 ∫ t + Vx (x(s), s, r(s))g(x(s), x(s − δ(s)), s, r(s))dB(s) +

0 l ∫ t ∑ k=1

0

∫

(3) −

−

−

[V (x(s ) + h (x(s ), x((s − δ(s)) ), s, (k)

R

˜ (ds, dzk ) r(s), zk ), s, r(s)) − V (x(s− ), s, r(s))]N ∫ t∫ + [V (x(s− ), s, r0 + c(r(s), u)) − V (x(s− ), s, r(s))]µ(ds, du). 0

R

The details of the function c and the martingale measure µ(ds, du) can be seen in [15] (p. 46-48). Obviously (3) holds if we replace 0 and t with bounded stopping time τ1 and τ2 , respectively. Thus the following lemma is derived. Lemma 1. [15] Let τ1 , τ2 be bounded stopping times such that 0 ≤ τ1 ≤ τ2 a.s. If V (x(t), t, r(t)) and LV (x(t), x(t − δ(t)), t, r(t)) are bounded on t ∈ [τ1 , τ2 ] with probability 1, then EV (x(τ2 ), τ2 , r(τ2 )) = EV (x(τ1 ), τ1 , r(τ1 )) ∫ τ2 +E LV (x(s), x(s − δ(s), s, r(s))ds

(4)

τ1

Proof. Replace 0 and t in (3) with τ1 and τ2 , then (4) follows from taking expectation on both side of (3). We also need the definition and some properties of M-matrix as well as the Yong inequality. Definition 2. [15] A square matrix A = (aij )n×n is called a nonsingular M-matrix if A can be expressed in the form A = sI − G with some G ≥ 0 and s > ρ(G), where I is the identity n × n matrix and ρ(G) the spectral radius of G.

5

It is easy to see that a nonsingular M-matrix has nonpositive off-diagonal and positive diagonal entries, that is aii > 0 while aij ≤ 0, i ̸= j. We cite the notation by letting Z n×n = {A = (aij )n×n : aij ≤ 0, i ̸= j}. Lemma 2. [15] If A ∈ Z n×n , then the following statements are equivalent: 1) A is a nonsingular M-matrix. 2) All of the principle minors of A are positive. 3) Every real eigenvalue of A is positive. 4) A is inverse-positive, that is, A−1 exists and A−1 ≥ 0, which means each element of A−1 is non-negative. Lemma 3. (Yong inequality)[15] |a|β |b|1−β ≤ β|a| + (1 − β)|b| ∀a, b ∈ R and ∀β ∈ [0, 1]. 3. Main results In what follows, we will present the exponential p-stability condition for system (1), then propose the M-matrix approach to achieve this condition. An application in neural networks will be put forward subsequently. 3.1. Stability of hybrid systems Theorem 1. Let Assumptions 1-2 holds. Assume that there exist a function V ∈ C 2,1 (Rn × R+ × S; R+ ) and positive constants p, a1 , a2 , b1 , b2 such that ¯ b1 >b2 /(1 − δ), a1 |x|p ≤ V (x, t, i) ≤ a2 |x|p LV (x, y, t, i) ≤ −b1 |x|p + b2 |y|p

(5) (6)

for all x, y ∈ Rn , t ≥ 0 and i ∈ S. Then the system (1) is exponentially stable in pth moment. More precisely, 1 lim sup log(E|x(t; ξ)|p ) ≤ −λ, ∀ξ ∈ LpF0 ([−τ, 0]; Rn ]) t→∞ t

(7)

¯ is the unique root to the equation where λ ∈ (0, b1 − b2 /(1 − δ)) b2 eλτ = b1 − λa2 . 1 − δ¯

6

(8)

Proof. Fix any ξ and write x(t; ξ) = x(t). Set U (x(t), t, r(t)) = eλt V (x(t), t, r(t)), then we get LU = eλt (λV + LV ). Applying Lemma 1 to U and then using conditions (5) and (6) we can show that a1 ebt E|x(t)|p ≤ EU (x(t), t, r(t)) ∫ t = EU (ξ(0), 0, r0 ) + E LU ds 0 ∫ t = EV (ξ(0), 0, r0 ) + E eλs (λV + LV )ds 0 ∫ t p ≤ a2 E|ξ(0)| + E eλs [(λa2 − b1 )|x(s)|p + b2 |x(s − δ(s))|p ]ds

(9)

0

By Assumption 2 and ∥ξ∥ = sup−τ ≤θ≤0 |ξ(θ)|, we compute ∫

∫

t

e |x(s − δ(s))| ds ≤ e λs

0

p

t

eλ(s−δ(s)) |x(s − δ(s))|p ds 0 ∫ t eλτ ≤ eλu |x(u)|p du 1 − δ¯ −τ ∫ 0 ∫ t eλτ λu p = ( e |x(u))| du + eλu |x(u)|p du) ¯ 1 − δ −τ 0 λτ λτ ∫ t τe e ≤ E∥ξ∥p + eλu |x(u)|p du . ¯ 1−δ 1 − δ¯ 0 λτ

Substituting this into (9) and then making use of (8) we obtain that a1 eλt E|x(t)|p ≤ a2 E|ξ(0)|p +

τ b2 eλτ E∥ξ∥p ¯ 1−δ

Noting that E∥ξ∥p <∞. Dividing both side by a1 eλt and then letting t → ∞, we obtain the required assertion (7). Remark 1. The simplified proof of Theorem 1 can be supplemented by the technique of establishing stopping time and then taking limit. Here we follow 7

the proof pattern in [15]. Remark 2. Theorem 1 extends Mao’s conclusion (see [15]), which is for hybrid systems only, to hybrid systems with Poisson jumps. In particular, when h(x(t), x(t − δ(t), t, r(t), z)) ≡ 0 holds in Theorem 1, our results is consistent with Mao’s. Remark 3. In [6], the conclusion about exponential p-stability of regimeswitching jump diffusion without delay, i.e. Theorem 4.2, is a special case arising in our result when b2 = 0 is derived from that the time delay δ(t) ≡ 0. We now apply M-matrix approach to achieving the exponential p-stability (p ≥ 2) condition in Theorem 1. The assumption with regard to system (1) below is essential. Assumption 3. For each i ∈ S, there exist constants αi , βi , ρi , ηi , σi , πi such that xT f (x, y, t, i) ≤ αi |x|2 + βi |y|2 |g|2 ≤ ρi |x|2 + ηi |y|2 ∫ ∑ l (|x + h(k) |p − |x|p )νk (dzk ) ≤ σi |x|p + πi |y|p

(10) (11) (12)

R k=1

for all (x, y, t) ∈ Rn × Rn × R+ and zk ∈ R. Moreover, We set p−1 ηi ) 2 p(p − 1) ζi = pαi + ρi + σi + (p − 2)ωi 2 A = −diag(ζ1 , · · · , ζN ) − Γ (q1 , · · · , qN )T = A−1⃗1, ωi = 0 ∨ (βi +

(13) (14) (15) (16)

where ⃗1 = (1, · · · , 1)T . Theorem 2. Let Assumptions 1-3 hold and p ≥ 2. If A is a nonsingular M-matrix and ¯ ∀i ∈ S (πi + 2ωi )qi < 1 − δ, (17) then system (1) is exponentially stable in pth moment. 8

Proof. It follows from Lemma 2 that A−1 exists and A−1 ≥ 0, which means that the sum of each row of A−1 is positive. Hence by (16), it can be deduced that qi > 0, ∀i ∈ S. Define the function V : Rn × R+ × S → R+ by V (x, t, i) = qi |x|p . Clearly V obeys (5) with a1 = mini∈S qi and a2 = maxi∈S qi . (11) yields that |xT g|2 ≤ ρi |x|4 + ηi |x|2 |y|2 .

(18)

We compute the operator LV from Rn × Rn × R+ × S to R by conditions (10)-(13) and (18) as follows: p p(p − 2) LV =pqi |x|p−2 xT f + qi |x|p−2 |g|2 + qi |x|p−4 |xT g|2 2 2 ∫ ∑ N l ∑ p + γij qj |x| + qi (|x + h(k) |p − |x|p )νk (dzk ) j=1

R k=1

pqi ρi p pqi ηi p−2 2 |x| + |x| |y| 2 2 p(p − 2)qi ρi p p(p − 2)qi ηi p−2 2 + |x| + |x| |y| 2 2 N ∑ + σi qi |x|p + πi qi |y|p + γij qj |x|p

≤pqi αi |x|p + pqi βi |x|p−2 |y|2 +

j=1

∑ p(p − 1)ρi =[(pαi + + σi )qi + γij qj ]|x|p 2 j=1 N

p(p − 1)ηi ]qi |x|p−2 |y|2 2 N ∑ p(p − 1)ρi ≤[(pαi + + σi )qi + γij qj ]|x|p 2 j=1 + πi qi |y|p + [pβi +

+ πi qi |y|p + pωi qi |x|p−2 |y|2 By virtue of Lemma 3, |x|p−2 |y|2 = (|x|p )

p−2 p

2

(|y|p ) p ≤ 9

p−2 p 2 p |x| + |y| . p p

(19)

Substituting this and (14) into (19), noting that pωi qi ≥ 0, we have ∑ p(p − 1)ρi LV ≤[(pαi + + σi + (p − 2)ωi )qi + γij qj ]|x|p + (πi + 2ωi )qi |y|p 2 j=1 N

=(ζi qi +

N ∑

γij qj )|x|p + (πi + 2ωi )qi |y|p

j=1 p

≤ − b1 |x| + b2 |y|p (20) where b1 = 1, b2 = maxi∈S {(πi + 2ωi )qi }. ¯ holds. Hence all the By condition (17), the inequality b1 > b2 /(1 − δ) conditions of Theorem 1 have been verified, so system (1) is exponentially stable in pth moment. 3.2. Stability of neural networks As an application of Theorem 2, we discuss the mean square exponential stability of delayed neural networks with L´evy noise and Markovian switching. Consider the neural network of this form: dx(t) =[−F (r(t)x(t) + D(r(t))s1 (x(t)) + E(r(t))s2 (x(t − δ(t)))]dt + g(x(t), x(t − δ(t)), t, r(t))dB(t) ∫ + h(x(t− ), x((t − δ(t))− ), t, r(t), z)N (dt, dz)

(21)

Rl

where F is a diagonal positive definite matrix, D and E are respectively the connection weight matrix and the delayed connection weight matrix, sj , j = 1, 2 stand for the neuron activation function with sj (0) = 0, j = 1, 2, and what the other symbols denote are the same as those in system (1). We need more assumption to study the stability of neural network (21). Assumption 4. 1) The neuron activation functions sj , (j = 1, 2) satisfy the Lipschitz condition |sj (u) − sj (v)| ≤ |Gj (u − v)|

∀u, v ∈ Rn

(22)

where Gj , j = 1, 2 are known constant matrices. 2) g(0, 0, t, i) ≡ 0 and h(0, 0, t, i, z) ≡ 0 hold for all (t, i) ∈ R+ ×S and z ∈ Rl . 10

3) The function g satisfies (11), and h satisfies (12) in the case of p = 2, i.e. ∫ ∑ l R k=1

(|x + h(k) |2 − |x|2 )νk (dzk ) ≤ σi |x|2 + πi |y|2

(23)

For each i ∈ S, we now set |Ei ||G2 | |Ei ||G2 | αi = λmax (−Fi ) + |Di ||G1 | + , βi = 2 2 ηi ωi = 0 ∨ (βi + ), ζi = 2αi + ρi + σi 2 A = −diag(ζ1 , · · · , ζN ) − Γ (q1 , · · · , qN )T = A−1⃗1

(24)

where ⃗1 = (1, · · · , 1)T . Theorem 3. Let Assumption 2 and Assumption 4 hold, if A is a non-singular ¯ then the neural network is exponentially M-matrix and (πi + 2ωi )qi ≤ 1 − δ, stable in mean square. Proof. Let f (x(t), x(t − δ(t)), t, r(t)) = − F (r(t)x(t) + D(r(t))s1 (x(t)) + E(r(t))s2 (x(t − δ(t)))

(25)

Comparing with Theorem 2 in the case of p = 2, we only need to show that (10) holds. According to the conditions sj (0) = 0 and (22), we get |sj (u)| ≤ |Gj u| j = 1, 2 ∀u ∈ Rn

(26)

By (25) and (26), we compute xT f (x, y, t, i) =xT (−Fi )x + xT Di s1 (x) + xT Ei s2 (y) ≤λmax (−Fi )|x|2 + |Di ||G1 ||x|2 + |Ei ||G2 ||x||y| |Ei ||G2 | |Ei ||G2 | 2 ≤(λmax (−Fi ) + |Di ||G1 | + )|x|2 + |y| 2 2

11

(27)

By (24), we obtain xT f (x, y, t, i) ≤ αi |x|2 + βi |y|2 as required. It then follows from Theorem 2 that the neural network (21) is exponentially stable in mean square. 4. Numerical Simulation Consider a two-neuron delayed neural network (21) with Poisson jump and 2-state Markovian switching, where [ ] [ ] [ ] 12 0 −2 1 −1 1 F1 = , D1 = , E1 = , 0 10 1 −1 1 2 x+y yz s1 (x) = tanh(x), g(x, y, t, 1) = , h(x, y, t, 1, z) = , 2] [ ] [ [ ] 2 10 0 1 −1.2 1 0 F2 = , D2 = , E2 = , 0 10 −1 1.5 1.5 1 x−y s2 (y) = tanh(y), g(x, y, t, 2) = , h(x, y, t, 2, z) = −x + yz. 4 The time delay δ(t) = 0.6 sin(t) + 0.4, which means that τ = 1 and ˙δ ≤ δ¯ = 0.6. B(t) and N (t, z) are all one dimensional. The character measure µ of Poisson jump satisfies µ(dz) = ςϕ(dz), where ς = 1.5 is the intensity of Poisson distribution and ϕ is the probability intensity of the standard normal distributed variable z. We set [ ] −2 2 S = {1, 2}, Γ= 1 −1 as the state space and transition rate matrix with respect to the Markovian switching. By (11), (22), (23) and (24), We can get α1 = −6.0314, β1 = 1.3229, ρ1 = 0.5, η1 = 0.5, σ1 = 1.5, π1 = 0.75, ω1 = 1.5729, ζ1 = −10.0628, α2 = −6.5839, β2 = 1.0308, ρ2 = 0.125, η2 = 0.125, σ2 = −1.5, π2 = 1.5, ω2 = 1.0933, ζ2 = −14.5428,

12

[

and A = −diag(ζ1 , ζ2 ) − Γ =

12.0628 −1

−2 15.5428

]

is a non-singular M-matrix. q = [q1 , q2 ]T = A−1⃗1 = [0.0946, 0.0704]T . Hence we can verify (π1 + 2ω1 )q1 = 0.3685 < 0.4 = 1 − δ¯ ¯ (π2 + 2ω2 )q2 = 0.2596 < 0.4 = 1 − δ. It then follows from Theorem 3 that the two-neuron neural network (21) is exponentially stable in mean square. Figs. 1-4 show the 2-state Markov chain, Poisson point process with normally distributed variable z, the state trajectory and response of state’s norm square, respectively. We can see from Fig. 3 that the state of system tends to zero at about 2 second, which verifies the stability of two neuron network (21). For equation (8), we can find its root λ = 0.0749, which means that the Lyapunov exponent in mean square for neural network (21) is not greater than −0.0749. In Fig. 4, the two curves show the evolution of the norm square concerning system state (solid line) and the negative exponent (dotted line) respectively with time t. The solid line is lower than the dotted one from t = 2, which interprets the exponential convergence rate of the neural network (21).

13

2−state Markov chain 3

2.5

r(t)

2

1.5

1

0.5

0 0

2

4 6 Time(second)

8

10

Figure 1: 2-state Markov chain.

Poisson point process with normally distributed jump 7

Random jump amplitude

6 5 4 3 2 1 0 −1 0

5

10 Time(second)

15

Figure 2: Poisson point process.

14

20

Responses of neuron dynamics to initial value: 6, −6 8 x x

6

1 2

4

x(t)

2 0 −2 −4 −6 −8 0

1

2

3 4 Time(second)

5

6

7

Figure 3: State trajectory.

Norm square of neuron dynamics 50 2

|x(t)| 45

−0.0749t

e

40 35

|x(t)|

2

30 25 20 15 10 5 0 0

2

4

6

8

10

t

Figure 4: State’s norm square trajectory.

5. Conclusion We have dealt with the problem of exponential p-stability analysis for stochastic hybrid systems with jumps and delay. The exponential stability criteria have been obtained through stochastic analysis and M-matrix approach. As an application of our results, the condition of mean square exponential stability for hybrid neural networks has been derived as well. An example has been used to demonstrate the effectiveness of the main results in this paper. 15

6. Acknowledgement This work is partially supported by the Innovation Program of Shanghai Municipal Education Commission (12zz064, and 13zz050), the Key Foundation Project of Shanghai (Grant No. 12ZR1440200), and the Specialized Research Fund for the Doctoral Program of Higher Eduction (20120075120009). References [1] In-Suk Wee. Stability for multidimensional jump-diffusion processes. Stochastic Processes and their Applications, 80:193–209, 1999. [2] F. B. Xi. On the stability of jump-diffusions with Markovian switching. Journal of Mathematical Analysis and Applications, 341(1):588–600, 2008. [3] Fubao Xi. Asymptotic properties of jump-diffusion processes with state-dependent switching. Stochastic Processes and their Applications, 119(7):2198–2221, 2009. [4] G. Yin and F. B. Xi. Stability of regime-switching jump diffusions. Siam Journal on Control and Optimization, 48(7):4525–4549, 2010. [5] D. P. Siu and G. S. Ladde. Stochastic hybrid system with nonhomogeneous jumps. Nonlinear Analysis: Hybrid Systems, 5(3):591–602, 2011. [6] Z. X. Yang and G. Yin. Stability of nonlinear regime-switching jump diffusion. Nonlinear Analysis-Theory Methods & Applications, 75(9):3854– 3873, 2012. [7] Fubao Xi and G. Yin. Almost sure stability and instability for switchingjump-diffusion systems with state-dependent switching. Journal of Mathematical Analysis and Applications, 400(2):460–474, 2013. [8] D. Applebaum and M. Siakalli. Asymptotic stability of stochastic differential equations driven by Levy noise. Journal of Applied Probability, 46(4):1116–1129, 2009. [9] D. Applebaum and M. Siakalli. Stochastic stabilization of dynamical systems using Levy noise. Stochastics and Dynamics, 10(4):509–527, 2010. 16

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