Almost sure stability with general decay rate of neutral stochastic delayed hybrid systems with Lévy noise

Nonlinear Analysis: Hybrid Systems 24 (2017) 171–185

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Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs

Almost sure stability with general decay rate of neutral stochastic delayed hybrid systems with Lévy noise Mengling Li, Feiqi Deng ∗ School of Automation Science and Engineering, South China University of Technology, Guangzhou 510640, China

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Article history: Received 12 September 2016 Accepted 5 January 2017

Keywords: ψ -function Neutral stochastic delayed hybrid systems Lévy noise Nonnegative semi-martingale convergence theorem M-matrix

abstract This paper focuses on neutral stochastic delayed hybrid systems with Lévy noise (NSDHSsLN). A kind of ψ -function is introduced and the almost sure stability with general decay rate is investigated, including the exponential stability and the polynomial stability. By virtue of Lyapunov function and nonnegative semi-martingale convergence theorem, we propose sufficient conditions for the almost sure stability of the NSDHSs-LN. Moreover, we give an upper bound of each coefficient at any mode according to the theory of M-matrix. Especially, the coefficients of considered systems can be allowed to be high order nonlinear. Finally, two examples are given to show the effectiveness of our results. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction Due to the influence of many complicated factors in real world, the stability issue of stochastic system has become a hot topic [1–11]. And with the development of the stochastic analysis, Markovian switching and Poisson jump are considered by many researchers, see [12–16]. On the other hand, Markov chain and jump–diffusion systems are more suitable to describe the random failures, abrupt change or sudden disturbances which arise in many physical systems. Generally speaking, a Markovian switching system is a hybrid system which is composed of two processes representing the system state and jumping parameter respectively. For a simple example, European option pricing has been considered in [17], one can see that the underlying price process is a continuous one, while the economic state of the world is described by finite modes Markov chain. In particular, there could be just only two modes representing ‘bull’ and ‘bear’. As we know, there exist time delays in many fields such as mechanical engineering, chemistry and chemical engineering, life sciences, finance, etc. Meanwhile, time delays are often the cause of instability which motivates several studies on the stability of switching diffusions with time delays, see [18–20]. So it is necessary to consider the action of time delays in systems. In this paper we study a class of time delay systems depending on past and present values but that involves derivatives with delays as well as the function itself. Such systems historically have been referred to as neutral systems. There are numerous literatures on the almost sure exponential stability of stochastic differential equations, for instance, [5,12,21,22]. Meanwhile there also exists an extensive literature concerning almost sure polynomial stability, here we only mention [23,24]. The difference between the almost sure exponential stability and the almost sure polynomial stability is the speed with which the solutions decay to zero. But these stability concepts could be generalized to the stability with general decay rate, see [25–28]. And to the best of our knowledge, the almost sure stability with general decay rate of neutral stochastic delayed hybrid systems with Lévy noise has not been completely studied, which motivates the present work.

∗

Corresponding author. E-mail address: [email protected] (F. Deng).

http://dx.doi.org/10.1016/j.nahs.2017.01.001 1751-570X/© 2017 Elsevier Ltd. All rights reserved.

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M. Li, F. Deng / Nonlinear Analysis: Hybrid Systems 24 (2017) 171–185

The existence and uniqueness of the global solution is the basic guarantee of the stability study, see [29–31], and the local Lipschitz condition and the linear growth condition can guarantee a unique global solution, but as is well-known, the linear growth condition is somewhat restrictive actually. Hence, in this paper, we retain the local Lipschitz condition but use a weaker condition to replace the linear growth condition. Based on the method of Lyapunov function, we can also know that there is a unique global solution. Then we introduce a kind of ψ -type function which will be defined in Section 2. By means of ψ -type function, we can obtain a kind of ψ -type stability which is a more encompassing stability, including exponential stability and polynomial stability. Notice that the Markov switching system has its own property, for example, it can describe the randomness of the discrete systems and if the switched continuous system with one mode is unstable, then the switching system also could be stable. With respect to the problem, we will briefly illustrate in our numerical simulations. Note that our systems are hybrid systems, the coefficients are different at different mode. Therefore, in this paper, our main aims are: 1. The system almost surely admits a global solution under some loose conditions. 2. The trivial solution of the system is almost surely ψ -type stable. 3. The upper bound of each coefficient at any mode is obtained. To summarize, the main contributions of our paper can be outlined to the following three points. Firstly, the considered neutral stochastic hybrid model driven by Lévy noise has not been studied before. Secondly, a more general almost sure stability problem has been explored. Finally, we try to allow the coefficients of the considered systems to be high-order nonlinear although both the model and the problem become more general. The structure of this paper is organized as follows: in Section 2, the model and several definitions are presented. In Section 3, we give two theorems which are related to the existence and uniqueness of the global solution and the sufficient conditions for ψ -type stability respectively. In Section 4, we apply the theory of M-matrix to obtain the result about the upper bound of each coefficient at any mode. Finally, we provide two examples to verify the effectiveness of our work. 2. Model and preliminaries Throughout this paper, unless otherwise specified, 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 right continuous and F0 contains all P-null sets) and | · | be either the matrix trace norm or the Euclidean norm. Denote by diag(ζ1 , . . . , ζN ) a diagonal matrix with diagonal entries ζ1 , . . . , ζN and C ([−τ , 0]; Rn ) the family of continuous functions φ from [−τ , 0] to Rn with the norm ∥φ∥ = sup−τ ≤s≤0 |φ(s)|. Denote by L2Ft ([−τ , 0]; Rn ) the family of Ft -measurable C ([−τ , 0]; Rn )-valued random variables ξ = {ξ (s) : −τ ≤ s ≤ 0} such

that ∥ξ ∥2 < ∞. 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 W (t ) = (W1 (t ), W2 (t ), . . . , Wm (t ))T be an m-dimensional Ft -adapted Brownian motion defined on the complete probability space (Ω , F , {Ft }t ≥0 , P ). Denote by N (t , z ) a Ft -adapted Poisson random measure on [0, +∞) × R with a σ -finite intensity measure π (dz ), and then the compensator martingale measure N˜ (t , z ) satisfies N˜ (dt , dz ) = N (dt , dz ) − π(dz )dt. Let {r (t ), t ≥ 0} be a right-continuous Markov chain defined on the probability space taking values in a finite state space S = {1, 2, . . . , N } with generator Γ = (γij )N ×N given by P {r (t + ∆) = j|r (t ) = i} =



γij ∆ + o(∆) 1 + γii ∆ + o(∆)

if i ̸= j if i = j,

where ∆ > 0, and γij ≥ 0 is the transition rate from i to j if i ̸= j while γii = −Σi̸=j γij . Consider the following neutral stochastic delayed hybrid systems with Lévy noise (NSDHSs-LN) d[x(t ) − D(x(t − τ ), r (t ))] = f (x(t ), x(t − τ ), t , r (t ))dt + g (x(t ), x(t − τ ), t , r (t ))dW (t )



h(x(t ), x(t − τ ), t , r (t ), z )N (dt , dz )

+

(1)

R

on t ≥ 0, where τ > 0 is a constant and f : Rn × Rn × R+ × S → Rn , g : Rn × Rn × R+ × S → Rn×m , h : Rn × Rn × R+ × S × R → Rn are continuous functions. We also need to know the initial data, assume that they are given by {x(θ ) : −τ ≤ θ ≤ 0} = ξ ∈ C ([−τ , 0]; Rn ), for simplify, we denote x˜ (t ) = x(t ) − D(x(t − τ ), r (t )). We further assume that W (t ), N (t , z ) and r (t ) are independent of each other. And for stability analysis, we assume that D(0, i) = f (0, 0, t , i) = g (0, 0, t , i) = 0 for all (t , i) ∈ R+ × S and h(0, 0, t , i, z ) = 0 for all (t , i, z ) ∈ R+ × S × R which admit x(t ) = 0 is the trivial solution. In this paper, we consider the asymptotic stability with general decay rate, so we give the following definition of ψ -type function which will be used as the decay function. Definition 1. The function ψ : R → (0, ∞) is said to be ψ -type function if the function satisfies the following three conditions:

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173

(i) It is continuous and nondecreasing in R and differentiable in R+ . ψ ′ (t )

(ii) ψ(0) = 1, ψ(∞) = ∞ and α = supt ≥0 [ ψ(t ) ] < ∞. (iii) For any s, t ≥ 0, ψ(t ) ≤ ψ(s)ψ(t − s). Remark 1. Obviously, if a, b > 0, t + = t ∨ 0, the functions ψ(t ) = eat and ψ(t ) = (1 + t + )b are all ψ -type functions since they satisfy the above three conditions. Next, we give the definition of the almost sure stability with general decay rate based on Definition 1. Definition 2. Let the function ψ(t ) ∈ C (R+ ; R+ ) be a ψ -type function. Then for any initial data ξ , the system is said to be almost surely stable with decay ψ(t ) of order γ if log |x(t , ξ )|

lim sup

log ψ(t )

t →∞

≤ −γ

a.s.

Remark 2. If we replace ψ(t ) by et , 1 + t, then it leads to the usual almost sure stability with exponential, polynomial decays, respectively. So our results will be more general than [3,18,23,32] since we have a wide choice for ψ -type functions. Let C 2,1 (Rn × R+ × S ; R+ ) denote the family of all functions V (x, t , i) on Rn × R+ × S which are continuously twice differentiable in x and once in t, moreover, define

 ∂ V (x, t , i) ∂ V (x, t , i) ,..., , ∂ x1 ∂ xn  2  ∂ V (x, t , i) Vxx (x, t , i) = , ∂ xk ∂ xl n×n ∂ V (x, t , i) Vt (x, t , i) = . ∂t Vx (x, t , i) =



Then, we can define an operator LV : Rn × Rn × R+ × S → R for the function V (x, t , i) ∈ C 2,1 (Rn × R+ × S ; R+ ) by

LV (x, y, t , i) = Vx (˜x, t , i)f (x, y, t , i) +

+ Vt (˜x, t , i) +



1 2

trace[g T (x, y, t , i)Vxx (˜x, t , i)g (x, y, t , i)]

[V (˜x + h(x, y, t , i, z ), t , i) − V (˜x, t , i)]π (dz ) + R

N 

γij V (˜x, t , j)

j=1

where t ∈ R+ . Based on this, we can cite the generalized Itô formula V (˜x(t ), t , r (t )) = V (˜x(0), 0, r0 ) +



t

LV (x(s), x(s − τ ), s, r (s))ds + Gt

(2)

0

where t



Vx (˜x(s), s, r (s))g (x(s), x(s − τ ), s, r (s))dW (s)

Gt = 0

 t

[V (˜x(s) + h(x(s), x(s − τ ), s, r (s), z )) − V (˜x(s), s, r (s))]N˜ (ds, dz )

+ 0

R

 t

[V (˜x(s), s, r0 + c (r (s), u) − V (˜x(s), s, r (s))]µ(ds, du).

+ 0

R

The detailed representation of the functions µ and c can be found in [33]. And µ(ds, du) is a martingale measure. In fact, if a stochastic process is a martingale, then it is a local martingale. Hence, we can easily know that {Gt }t ≥0 is a local martingale. 3. Global solution and general decay stability Before we state our main results in this section, the following two standing hypotheses are imposed. Assumption 1 (Contractility Condition). For each i ∈ S , D(0, i) = 0 and there is a constant ki ∈ [0, 1) such that

|D(y, i)| ≤ ki |y|, for all y ∈ Rn .

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M. Li, F. Deng / Nonlinear Analysis: Hybrid Systems 24 (2017) 171–185

Assumption 2 (Local Lipschitz Condition). For arbitrary x1 , x2 , y1 , y2 ∈ Rn and |x1 | ∨ |x2 | ∨ |y1 | ∨ |y2 | ≤ n, there is a positive constant kn such that

|f (x1 , y1 , t , i) − f (x2 , y2 , t , i)|2 ∨ |g (x1 , y1 , t , i) − g (x2 , y2 , t , i)|2  ∨ |h(x1 , y1 , t , i, z ) − h(x2 , y2 , t , i, z )|2 π (dz ) ≤ kn (|x1 − x2 |2 + |y1 − y2 |2 ), R

where n ∈ N , t ∈ R , i ∈ S. +

+

The above hypotheses are associated with the coefficients of the systems (1), Assumption 1 is a normal hypothesis. Generally, the local Lipschitz condition and the linear growth condition can guarantee a unique global solution to the system. Here, we only retain the local Lipschitz condition which only guarantees a unique maximal local solution. Meanwhile, we note that the local Lipschitz condition can allow the coefficients f , g , h to be high order nonlinear. In fact, we use the following Assumption 3 to replace the classical linear growth condition. Assumption 3. There are several nonnegative numbers C , Kj , pj (1 ≤ j ≤ J ) for some positive integer J and a Lyapunov function V ∈ C 2,1 (Rn × R+ × S ; R+ ) such that lim

inf V (˜x, t , r (t )) = ∞

|˜x|→∞ 0≤t <∞

LV (x, y, t , i) ≤ C +

J 

Kj (|y|pj − |x|pj ),

j =1

for any x, y ∈ R , t ≥ 0, i ∈ S. n

According to the above assumptions, let us state the following existence and uniqueness theorem: Theorem 1. Assume that Assumptions 1–3 hold, then for any initial data ξ ∈ C ([−τ , o]; Rn ), there is a unique global solution x(t , ξ ) on t > 0 to the system (1). Proof. Applying the standing truncation technique [33], Assumptions 1 and 2 admit a unique maximal local solution to the system (1). Let x(t )(t ∈ [−τ , τe )) be the maximal local solution to the system (1) and τe be the explosion time. And let k0 ∈ R+ be sufficiently large for ∥ξ ∥ ≤ k0 . For any integer k ≥ k0 , define

τk = inf{t ∈ [0, τe ) : |x(t )| ≥ k} where inf ∅ = ∞. Obviously, the sequence {τk } is increasing. So we have a limit τ∞ = limk→∞ τk , whence τ∞ ≤ τe . If we can show that τ∞ = ∞ a.s., then we have τe = ∞ a.s.. Therefore, we only need to devote to prove τ∞ = ∞ a.s. which is equivalent to proving that P (τk ≤ t ) → 0 as k → ∞ for any t > 0. By Assumption 3, applying indicator function and the generalized Itô formula, we have E (Iτk ≤t V (˜x(τk ), τk , r (τk ))) ≤ EV (˜x(t ∧ τk ), t ∧ τk , r (t ∧ τk ))

= EV (˜x(0), 0, r0 ) + E

t ∧τk



LV (x(s), x(s − τ ), s, r (s))ds

0

≤ EV (˜x(0), 0, r0 ) + Ct +

J 

J

 j =1

[|x(s − τ )|pj − |x(s)|pj ]ds

0

j =1

≤ EV (˜x(0), 0, r0 ) + Ct +

t ∧τk

 Kj E



0

|x(s)|pj ds =: f (t ).

Kj E −τ

Therefore, we have P (τk ≤ t )EV (˜x(τk ), τk , r (τk )) ≤ f (t ). Note the |˜x(τk )| = k by the definition of stopping time τk and f (t ) is independent of k. Thus we can derive lim P (τk ≤ t ) = 0,

k→∞

this implies that there exists a unique global solution x(t , ξ ) for the system (1).



Remark 3. The existence and uniqueness of the global solution of the systems (1) is the basic guarantee of the stability issue. These is no sense in discussing stability problem if we cannot insure that there is a unique global solution for the considered systems.

M. Li, F. Deng / Nonlinear Analysis: Hybrid Systems 24 (2017) 171–185

175

Next, we want to obtain sufficient conditions of the almost sure stability with general decay rate. In order to elaborate clearly, we need the following lemmas Lemma 1 ([34]). Let {Mt }t ≥0 be a local martingale and {θt }t ≥0 be a locally bounded predictable process, then the stochastic t integral 0 θs dMs is also a local martingale. Lemma 2 ([35]). Assume {At } and {Ut } are two continuous predictable increasing processes vanishing at t = 0 a.s and {Mt } is a real-valued continuous local martingale with M0 = 0 a.s. Let {Xt } be a nonnegative adapted process and ξ be a nonnegative F0 -measurable random variable satisfying Xt ≤ ξ + At − Ut + Mt for t ≥ 0. If A∞ := lim At < ∞

a.s.

t →∞

then, we have lim sup Xt < ∞

a.s.

t →∞

Actually, Lemma 1 is to make preparation for Lemma 2 and Lemma 2 is the well-known nonnegative semi-martingale convergence theorem. With these above assumptions and lemmas, we can now state our result about the almost sure stability with general decay rate. Theorem 2. Let Assumptions 1 and 2 hold. If there are positive numbers c1 , c2 , αi (1 ≤ i ≤ 4), q > 2 such that the functions V (˜x, t , i) and LV (x, y, t , i) satisfy c1 |˜x|2 ≤ V (˜x, t , i) ≤ c2 |˜x|2

(3)

LV (x, y, t , i) ≤ −α1 |x|2 + α2 |y|2 − α3 |x|q + α4 |y|q . If α2 < α1 and α4 < α3 , then there must exist a sufficiently small ε > 0 such that

α1 − 2c2 αε − (α2 + 2c2 k2 αε)ψ ε (τ ) > 0 α3 − α4 ψ ε (τ ) > 0

(4)

k2 ψ ε (τ ) < 1, where k = maxi∈S {ki } ∈ (0, 1), τ is the time delay, ψ(·) denotes the ψ -type function and ψ ε (τ ) represents that ψ(τ ) raised to the power of ε . Therefore, for any initial data ξ , the inequality lim sup

log |x(t , ξ )|

t →∞

log ψ(t )

< −ε,

(5)

holds, that is, the trivial solution is almost surely stable with decay ψ(t ) of order ε . Proof. Note that the inequality (3) is stronger than Assumption 3, so there is a unique global solution for the system (1). Let

ψ(t ) be a ψ -type function, and applying the generalized Itô formula to ψ ε (t )V (˜x(t ), t , r (t )) yields  t ψ ′ (s) ε ψ (s)V (˜x(s), s, r (s))ds ψ ε (t )V (˜x(t ), t , r (t )) = V (˜x(0), 0, r0 ) + ε ψ(s) 0  t + ψ ε (s)LV (x(s), x(s − τ ), s, r (s))ds + Mt , 0

t

ε

where Mt = 0 ψ (s)dGs is a real-valued continuous local martingale with M0 = 0 by Lemma 1. By inequality (3) and the definition of ψ -type function, we have

 t ψ ε (t )V (˜x(t ), t , r (t )) ≤ V (˜x(0), 0, r0 ) + εα ψ ε (s)V (˜x(s), s, r (s))ds 0  t  t − α1 ψ ε (s)|x(s)|2 ds + α2 ψ ε (s)|x(s − τ )|2 ds 0

− α3

0 t

 0

ψ ε (s)|x(s)|q ds + α4

 0

t

ψ ε (s)|x(s − τ )|q ds + Mt .

(6)

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M. Li, F. Deng / Nonlinear Analysis: Hybrid Systems 24 (2017) 171–185

Recall the elementary inequality : (a + b)2 ≤ 2a2 + 2b2 for any a, b ∈ R and the inequality (3), we can compute t



ε

ψ (s)V (˜x(s), s, r (s))ds ≤ c2

t



ψ ε (s)|˜x(s)|2 ds

0

0

t



ψ ε (s)(|x(s)|2 + |D(x(s − τ ), r (s))|2 )ds  t  t ψ ε (s)|x(s − τ )|2 ds. ψ ε (s)|x(s)|2 ds + 2c2 k2 ≤ 2c2

≤ 2c2

0

0

0

Substituting this inequality into the inequality (6) yields



t

ψ ε (s)|x(s)|2 ds ψ (t )V (˜x(t ), t , r (t )) ≤ V (˜x(0), 0, r0 ) − (α1 − 2c2 αε) 0  t ψ ε (s)|x(s − τ )|2 ds + (α2 + 2c2 k2 αε) 0  t  t ψ ε (s)|x(s − τ )|q ds + Mt ψ ε (s)|x(s)|q ds + α4 − α3 0 0  t ≤ const − [α1 − 2c2 αε − (α2 + 2c2 k2 αε)ψ ε (τ )] ψ ε (s)|x(s)|2 ds ε

0

− (α3 − α4 ψ ε (τ ))

t



ψ ε (s)|x(s)|q ds + Mt ,

(7)

0

where const is a constant which is not important in this paper. Consequently, by inequality (4), the inequality (7) implies

ψ ε (t )V (˜x(t ), t , r (t )) ≤ const + Mt . Applying Lemma 2, we have lim sup ψ ε (t )V (˜x(t ), t , r (t )) < ∞

a.s.

t →∞

therefore, there exists a positive number h such that for any t > 0,

ψ ε (t )V (˜x(t ), t , r (t )) ≤ h a.s..

(8)

Moreover, for any δ ∈ (0, 1), we have

ψ ε (t )|x(t )|2 ≤ (1 − δ)−1 ψ ε (t )|˜x(t )|2 + δ −1 ψ ε (t )|D(x(t − τ ), r (t ))|2 ≤ (1 − δ)−1 ψ ε (t )|˜x(t )|2 +

k2

δ

ψ ε (t )|x(s − τ )|2 ,

then for any T > τ , we obtain sup ψ ε (t )|x(t )|2 ≤ (1 − δ)−1 sup ψ ε (t )|˜x(t )|2 + 0≤t ≤T

0 ≤t ≤T

≤ (1 − δ)−1 sup ψ ε (t )|˜x(t )|2 + 0 ≤t ≤T

k2

δ k2

δ

sup ψ ε (t )|x(s − τ )|2 0≤t ≤T

ψ ε (τ ) × [ sup |ξ (t )|2 + sup ψ ε (t )|x(t )|2 ], −τ ≤t ≤0

0≤t ≤T

which implies sup ψ ε (t )|x(t )|2 ≤ 0≤t ≤T

δ k2 ψ ε (τ ) sup ψ ε (t )|˜x(t )|2 + sup |ξ (t )|2 2 ε (1 − δ)(δ − k ψ (τ )) 0≤t ≤T δ − k2 ψ ε (τ ) −τ ≤t ≤0

where we take δ ∈ (k2 ψ ε (τ ), 1). By inequality (8), we have lim sup ψ ε (t )|x(t )|2 ≤ t →∞

h k2 ψ ε (τ ) δ + sup |ξ (t )|2 < ∞, 2 ε (1 − δ)(δ − k ψ (τ )) c1 δ − k2 ψ ε (τ ) −τ ≤t ≤0

which implies the assertion inequality (5).



Remark 4. If we take ψ(t ) = e , our result will be degenerated to the almost sure exponential stability. And if we take ψ(t ) = 1 + t (t > 0), our result will be degenerated to the almost sure polynomial stability. Different almost sure stability can be obtained when ψ(t ) has different form satisfying the three conditions of Definition 1. t

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177

4. ψ -type stability Obviously, the condition (3) in Theorem 2 is somewhat inconvenient in applications since it is not related to the coefficients f , g , h explicitly and our systems are also complex systems. So, we will give the visualized conditions for the coefficients f , g , h. Before we give the main theorem in this section, it is necessary to make some preparations. The systems are hybrid systems, the technique we adopt here is the theory of M-matrices. Before we give the definition and properties of M-matrices, we introduce the class of Z -matrices firstly. A square matrix A = [aij ]N ×N is called a Z -matrix if its off-diagonal entries are less than or equal to zero, namely aij ≤ 0

for all i ̸= j.

Definition 3 ([36] (M-matrix)). Let A be a N × N real Z -matrix. And the matrix A is also a nonsingular M-matrix if it can be expressed in the form A = sI − B while all the elements of B = [bij ] are nonnegative and s ≥ ρ(B), where I is an identity matrix and ρ(B) the spectral radius of B. Based on the definition, it is easy for us to know that if A = [aij ]N ×N is an M-matrix, then it has positive diagonal entries and nonpositive off-diagonal entries, that is aii ≥ 0

while aij ≤ 0, i ̸= j.

Lemma 3 ([36]). If A = [aij ]N ×N is a Z -matrix, then the following statements are equivalent: (1) A is a nonsingular M-matrix. (2) Every real eigenvalue of A is positive. (3) All of the principle minors of A are positive. (4) A−1 exists and its elements are all non-negative. Assumption 4. Let q > 2 and for each i ∈ S and any x, y ∈ Rn , we impose the following conditions on the coefficients f , g , h. (1) there exist constants αi1 , αi2 , αi3 such that xT



f ( x, y , t , i ) +





h(x, y, t , i, z )π (dz )

≤ αi1 |x|2 − αi2 |x|q + αi3 |y|q .

R

(2) there exist constants βi1 , βi2 , βi3 such that

     f (x, y, t , i) + h(x, y, t , i, z )π (dz ) ≤ βi1 |x| + βi2 |x|q−1 + βi3 |y|q−1 .   R

(3) there exist constants ηi1 , ηi2 , ηi3 , ηi4 such that

|g (x, y, t , i)|2 ≤ ηi1 |x|2 + ηi2 |y|2 + ηi3 |x|q + ηi4 |y|q . (4) there exist constants ρi1 , ρi2 , ρi3 , ρi4 such that



|h(x, y, t , i, z )|2 π (dz ) ≤ ρi1 |x|2 + ρi2 |y|2 + ρi3 |x|q + ρi4 |y|q . R

Remark 5. Note that q > 2 in Assumption 4 which means that we can allow the coefficients f , g , h of the systems (1) to be high-order nonlinear. However, the existing literatures [3,5,14,21] only allow the coefficients f , g to be quasi linear at most. Assumption 5. According to the definition of M-matrix, let

A := −diag (2(α11 + kβ11 ) + η11 + ρ11 , . . . , 2(αN1 + kβN1 ) + ηN1 + ρN1 ) − k∗ Γ be a nonsingular M-matrix. Moreover, by the fourth equivalent condition in Lemma 3, we have

− → (θ1 , . . . , θN )T := A−1 1 > 0, − → where 1 = (1, . . . , 1)T . That is, θi > 0 for all i ∈ S.

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M. Li, F. Deng / Nonlinear Analysis: Hybrid Systems 24 (2017) 171–185

Theorem 3. Let Assumptions 1, 2, 4 and 5 hold. Those constants satisfy the following inequalities 0 < a ≤ a¯ < 1, 0 < b, 0 < c ≤ c¯ < b, where a = min{(ηi2 + ρi2 − αi1 − ηi1 − ρi1 )θi − 1}, i∈S

a¯ = max{(ηi2 + ρi2 − αi1 − ηi1 − ρi1 )θi − 1}, i∈S

 

b = min θi i∈S

 

c = min θi i∈S

2αi2 −

2k

2αi3 +

2k

 

c¯ = max θi i∈S

q q

2αi3 +

βi2 − 2kβi3 − ηi3 − ρi3 βi2 + 2kβi3 + ηi4 + ρi4

2k q

 

βi2 + 2kβi3 + ηi4 + ρi4

, ,



,

k = max{ki }. i∈S

Then for any given initial data ξ , there is a unique global solution x(t ) and the trivial solution is almost surely ψ -type stable. Proof. Let us define a function V : Rn × R+ × S → R+ by V (˜x, t , i) = θi |˜x|2 , where x˜ = x − D(y, i). Observe that there are two positive numbers c1 = mini∈S {θi }, c2 = maxi∈S {θi } such that c1 |˜x|2 ≤ V (˜x, t , i) ≤ c2 |˜x|2 .

(9)

By the definition of operator LV before, we have

LV (x, y, t , i) = 2θi x˜ T f (x, y, t , i) + θi |g (x, y, t , i)|2 +

N 

γij θj |x − D(y, i)|p

j =1

+ θi



[|˜x + h(x, y, t , i, z )|2 − |˜x|2 ]π (dz )    ≤ 2θi x˜ T f + h(x, y, t , i, z )π (dz ) + θi |g (x, y, t , i)|2 R

R

+ θi



|h(x, y, t , i, z )|2 π (dz ) + R

N 

γij θj |x − D(y, i)|p

j =1

   ≤ 2θi xT f + h(x, y, t , i, z )π (dz ) R      + 2θi |D(y, i)| f + h(x, y, t , i, z )π (dz ) + θi |g (x, y, t , i)|2 R

+ θi



|h(x, y, t , i, z )|2 π (dz ) + R

N 

γij θj |x − D(y, i)|p .

j =1

By Assumption 4, we have

LV (x, y, t , i) ≤ 2θi (αi1 |x|2 − αi2 |x|q + αi3 |y|q ) + 2kθi |y|(βi1 |x| + βi2 |x|q−1 + βi3 |y|q−1 )

+ θi (ηi1 |x|2 + ηi2 |y|2 + ηi3 |x|q + ηi4 |y|q ) + θi (ρi1 |x|2 + ρi2 |y|2 + ρi3 |x|q + ρi4 |y|q ) +

N 

γij θj |x − D(y, i)|p .

j =1

Recall the Young inequality: ac b1−c ≤ ca + (1 − c )b for any a, b > 0, c ∈ (0, 1) and the inequality:

|x − D(y, i)|2 ≤

1 1 − ki

|x|2 +

1 ki

|D(y, i)|2 ≤

1 1 − ki

|x|2 + ki |y|2 ≤ k∗ |x|2 + k∗ |y|2 ,

M. Li, F. Deng / Nonlinear Analysis: Hybrid Systems 24 (2017) 171–185

179

where k∗ = maxi∈S { 1−1k }, then we obtain i



LV (x, y, t , i) ≤ 2θi (αi1 |x|2 − αi2 |x|q + αi3 |y|q ) + 2θi βi1



1 2

    1 q 1 q−1 q |x|2 + |y|2 + βi2 |y| + |x| + βi3 |y|q 2

q

q

+ θi (ηi1 |x|2 + ηi2 |y|2 + ηi3 |x|q + ηi4 |y|q ) + θi (ρi1 |x|2 + ρi2 |y|2 + ρi3 |x|q + ρi4 |y|q ) + k∗

N 

γij θj (|x|p + |y|p )

j =1

 = θi (αi1 + 2kβi1 + ηi1 + ρi1 ) + k

∗

N 





γij θj |x| + θi (2kβi1 + ηi2 + ρi2 ) + k

∗

2

j=1



− θi 2αi2 −

2k q

N 

 γij θj |y|2

j =1





βi2 − 2kβi3 − ηi3 − ρi3 |x|q + θi 2αi3 +

2k q

 βi2 + 2kβi3 + ηi4 + ρi4 |y|q .

By Assumption 5, we have

LV (x, y, t , i) ≤ −|x|2 + [θi (ηi2 + ρi2 − αi1 − ηi1 − ρi1 ) − 1]|y|2



− θi 2αi2 −

2k q





βi2 − 2kβi3 − ηi3 − ρi3 |x| + θi 2αi3 + q

2k q



βi2 + 2kβi3 + ηi4 + ρi4 |y|q

≤ −|x|2 + a¯ |y|2 − b|x|q + c¯ |y|q ,

(10)

where a¯ < 1, c¯ < b. By Assumptions 1, 2, inequality (10) and Theorem 1, there is a unique global solution for any initial data. By inequality (10) and Theorem 2, the trivial solution is almost surely ψ -type stable.  5. Numerical simulations For the sake of simplicity of computation, we consider two scalar neutral stochastic delayed systems with Lévy noise and 2-state Markovian switching, d[x(t ) − D(x(t − τ ), r (t ))] = f (x(t ), x(t − τ ), t , r (t ))dt + g (x(t ), x(t − τ ), t , r (t ))dW (t )



h(x(t ), x(t − τ ), t , r (t ), z )N (dt , dz ),

+

(11)

R

where we uniformly take τ = 1 and W (t ), N (t , z ) are all one-dimensional. The character measure π of Poisson jump satisfies π (dz ) = aω(dz ), where a denotes the intensity of Poisson distribution and ω is the probability intensity of the standard normal distributed variable z. Assume the initial data is ξ (t ) = 1 + t (−1 ≤ t ≤ 0). Example 1. In this example, we take a = 0.238,

D(y, 1) = 0,

D(y, 2) =

y; 2 f (x, y, t , 2) = −1.2x3 + 0.3y3 ;

f (x, y, t , 1) = −8.12x − 2.77x3 , g (x, y, t , 1) = 1.4495x + 0.506y2 , h(x, y, t , 1, z ) = 1.6398y2 z , and Γ =



−1 2

1

1

g (x, y, t , 2) = 0.7558y + 0.1897x2 ;

h(x, y, t , 2, z ) = 3.557yz ,



−2 .

Therefore, we can compute 7.937 A= −8



 −2 8

and θ1 = 0.2105, θ2 = 0.3355. Then, we can choose the following two Lyapunov functions V (x, y, t , 1) = θ1 |x|2 ,



2

 1  V (x, y, t , 2) = θ2 x − y . 2

(12)

180

M. Li, F. Deng / Nonlinear Analysis: Hybrid Systems 24 (2017) 171–185

Fig. 1. Poisson jump process.

Letting q = 4 in Assumption 4, we can compute

α11 = −8.12 α12 = 2.77 α13 = 0; α21 = 0 α22 = 1.125 α23 = 0.225; β11 = 8.12 β12 = 2.77 β13 = 0; β21 = 0 β22 = 1.2 β23 = 0.3; η11 = 2.183 η12 = 0 η13 = 0 η14 = 1.2395; η21 = 0 η22 = 0.8991 η23 = 0 η24 = 0.333 ρ11 = 0 ρ12 = 0 ρ13 = 0 ρ14 = 0.67; ρ21 = 0 ρ22 = 2.68 ρ23 = 0

ρ24 = 0.

By Theorem 3, we derive that a = 0.2009,

a¯ = 0.25,

b = 0.5536,

c = 0.4641,

c¯ = 0.5478.

It can be concluded that the hybrid system with the above coefficients is stable in theory. The emulate experiment also illustrates the validity of our results. Figs. 1 and 2 show Poisson point process and 2-state Markovian switching respectively. Fig. 3 shows the state trajectory of the two subsystems and it can be seen that the first subsystem is almost surely stable but the second subsystem is unstable. However, the state of the whole system tends to zero which verifies that the neutral stochastic Markov switching system with the above coefficients is almost surely stable according to Fig. 4. Example 2. In this example, we take a = 0.5,

D(y, 1) =

1 2

y,

D(y, 2) =

f (x, y, t , 1) = −8.22x − 12.12x3 ,

and Γ =



−1 3

1



−3 .

Therefore, wan can compute 13.8392 A= −6



 −3 5.9154

3

y;

f (x, y, t , 2) = −0.05x − 26.64x3 ;

g (x, y, t , 1) = −3.08x + 4.82y2 sin(y), h(x, y, t , 1, z ) = −2.85y2 z ,

2

g (x, y, t , 2) = 0.44y2 sin(y);

h(x, y, t , 2, z ) = 0.05x − 3.59yz ,

(13)

M. Li, F. Deng / Nonlinear Analysis: Hybrid Systems 24 (2017) 171–185

181

Fig. 2. Markov chain.

Fig. 3. State trajectory of two subsystems.

and θ1 = 0.1396, θ2 = 0.3106. Then, we can choose the following two Lyapunov functions

 2  1   V (x, y, t , 1) = θ1 x − y , 2

 2  2   V (x, y, t , 2) = θ2 x − y . 3

Letting q = 4 in Assumption 4, we can compute

α11 = −8.22 α12 = 12.12 α13 = 0; α21 = −0.025 α22 = 26.64 α23 = 0; β11 = 8.22 β12 = 12.12 β13 = 0; β21 = 0.025 β22 = 26.64 β23 = 0; η11 = −5.3592 η12 = 0 η13 = 0 η14 = 8.3868; η21 = 0 η22 = 0 η23 = 0 η24 = 0.1936 ρ11 = 0 ρ12 = 0 ρ13 = 0 ρ14 = 4.0612; ρ21 = 0.0013 ρ22 = 6.444 ρ23 = 0

ρ24 = 0.

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M. Li, F. Deng / Nonlinear Analysis: Hybrid Systems 24 (2017) 171–185

Fig. 4. State trajectory of whole system.

Fig. 5. Poisson jump process.

By Theorem 3, we derive that a = 0.8956,

a¯ = 0.9937,

b = 0.2.8199,

c = 2.3017,

c¯ = 2.8187.

It can be concluded that the hybrid system with the above coefficients is stable in theory. The emulate experiment also illustrates the validity of our results. Figs. 5 and 6 show the Poisson jump process and the 2-state Markovian switching respectively. Fig. 7 shows the state trajectory of the two subsystems and it can be seen that the two subsystems are both almost surely stable. Moreover, the state of the whole system tends to zero which verifies that the neutral stochastic Markov switching system with the above coefficients is also almost surely stable according to Fig. 8. 6. Conclusion In this paper, the existence and uniqueness of solution of the neutral stochastic delayed hybrid systems with Lévy noise under the local Lipschitz condition and some loose condition has been established. There was no need for the linear growth condition, so we could deal with the problems that the coefficients of the equation are of highly order. Moreover,

M. Li, F. Deng / Nonlinear Analysis: Hybrid Systems 24 (2017) 171–185

183

Fig. 6. Markov chain.

Fig. 7. State trajectory of two subsystems.

we introduce the concept of ψ -type function and investigate the almost sure ψ -type stability which is a wider stability, including the exponential stability and polynomial stability. Applying the theory of M-matrix, we can obtain the upper bound of the coefficients at each mode. For further work, we can obtain the upper bounds of LV and the coefficients are time-varying. And moreover, we can let the time delay be time-varying.

Acknowledgments This work was supported by the National Natural Science Foundation of China under Grants 61503142, 61573156, and the Fundamental Research Funds for the Central Universities under Grant x2zdD2153620. Appendix A. Supplementary data Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.nahs.2017.01.001.

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Fig. 8. State trajectory of whole system.

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