Asymptotic stability of nonlinear hybrid stochastic systems driven by linear discrete time noises

Nonlinear Analysis: Hybrid Systems 33 (2019) 336–352

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

Asymptotic stability of nonlinear hybrid stochastic systems driven by linear discrete time noises✩ ∗

Lichao Feng a,b , Jinde Cao a , , Lei Liu c,a , Ahmed Alsaedi d a

The Jiangsu Provincial Key Laboratory of Networked Collective Intelligence, and School of Mathematics, Southeast University, Nanjing 210096, China b College of Science, North China University of Science and Technology, Tangshan 063210, China c College of Science, Hohai University, Nanjing 210098, China d Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia

article

info

Article history: Received 10 February 2018 Accepted 24 March 2019 Available online xxxx Keywords: Hybrid stochastic differential systems Discrete time noises H∞ stability Asymptotic stability Exponential stability

a b s t r a c t Author in the reference Mao (2016) designed a linear discrete time noise to exponentially stabilize an unstable differential system with the global Lipschitz condition, which essentially opened a new field of stochastic systems driven by discrete time noises. However, the global Lipschitz condition is over strict and the result of exponential convergence is excessively rapid for many systems. Hence we see the necessity to loosen it to the local Lipschitz condition and develop some new stability results. In the present paper, by the method of Lyapunov functionals, we not only investigate the exponential stability for hybrid stochastic differential systems driven by linear discrete time noises with the local Lipschitz condition, but also study the H∞ stability and asymptotic stability using some new techniques. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Many dynamic systems in various branches of science and industry are usually modeled as differential systems. In practice, due to the existence of phenomena such as component failures or repairs and abrupt disturbances, these systems may experience abrupt changes in their structures and parameters. The continuous time Markovian chain is a useful method to describe these abrupt changes [1]. Hence, hybrid differential systems (also known as differential systems with Markovian switching) are introduced to model these complex dynamic systems with abrupt changes, which have been applied widely in power systems, financial systems and other areas (e.g. [2,3]), with serious focus being placed on the analysis of the stability of the solutions over past years (e.g. [4–8]). Stochastic noises are inevitable in real environments. The dynamic systems have to bear the effects of stochastic noises. Therefore, hybrid stochastic differential systems, abbreviated as hybrid SDSs, are introduced to describe these systems with stochastic effects. In general, an n-dimensional hybrid SDS has the form dx(t) = f (x(t), t , r(t))dt + g (x(t), t , r(t))dB(t)

(1.1)

✩ No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work. For full disclosure statements refer to https://doi.org/10.1016/j.nahs.2019.03.008. ∗ Corresponding author. E-mail addresses: [email protected] (L. Feng), [email protected] (J. Cao). https://doi.org/10.1016/j.nahs.2019.03.008 1751-570X/© 2019 Elsevier Ltd. All rights reserved.

L. Feng, J. Cao, L. Liu et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 336–352

337

on t ≥ 0, where x(t) stands for the state while r(t) for the mode, B(t) for an m-dimensional Brownian motion independent of r(t), and please see the meanings of the others notations in the next section. In the meanwhile, since time delays are also unavoidable in practice, hybrid stochastic delay (functional) differential systems, abbreviated as hybrid SDDSs (SFDSs), are developed to describe the case of time delays. As a hot topic, the stability of hybrid stochastic systems under different settings has been discussed by more and more researchers. For example, Basak [9] discussed the asymptotic stability of semi-linear hybrid SDSs. Mao and his coauthors [10–12] investigated the stability of nonlinear hybrid stochastic systems with the linear growth condition. Recently, Hu [13] established some general results on asymptotic stability for hybrid SDDSs, which was extended to the case of non-autonomous hybrid SFDEs by Feng [14]. As to the external literature on this topic, please refer to [15–27] and the references therein. Obviously, a common feature of all literature above about stochastic systems is that all of the diffusion coefficients are based on the continuous time observations of state x(t) for t ≥ 0, i.e., these stochastic systems are driven by continuous time noises. In particular, the literature [24–27] are about the topic of stochastic stabilization, which mainly design continuous time noises to stabilize unstable systems. However, from the point of practical engineering, one always observes a system not continuously but only at some discrete times, such as 0, τ , 2τ , . . ., where τ > 0 is the time lag between two consecutive observations (e.g. [28]). Hence, introducing continuous time noises to stabilize an unstable system is expensive and sometimes impossible to achieve. Motivated by the idea above, Mao [29] designed a linear noise C x([t /τ ]τ )dB1 (t) based on the discrete observations at times 0, τ , 2τ , . . . to almost surely exponentially stabilize an unstable differential system with the global Lipschitz condition, where C ∈ Rn×n , B1 (t) is a scalar Brownian motion and [t /τ ] = j when jτ ≤ t < (j + 1)τ , j = 0, 1, . . .. As far as we know, the work [29] is the first paper to study the stability problem of such stochastic systems. In essence, the work [29] opens a new research field of stochastic systems driven by discrete time noises. However, the global Lipschitz condition required in [29] is over restrict, and the result of exponential convergence is excessively rapid for many systems. Hence, it is necessary to loosen the global Lipschitz condition and develop some new results. Based on these analysis, our paper is to loosen these constraints of [29] to study the stability of nonlinear hybrid SDSs driven by linear discrete time noises. To our best knowledge, the present paper is the first paper to investigate the H∞ stability, moment asymptotic stability and almost sure asymptotic stability of nonlinear hybrid SDSs driven by linear discrete time noises. The significant characters of the present paper which is different from those in [29] are as follows. ■ In [29], Mao mainly dealt with the stability of SDSs driven by linear discrete time noises, while our paper will discuss the case of hybrid SDSs. ■ In [29], Mao only discussed the problem of almost sure exponential stability. However, the form of exponential convergence is excessively rapid, i.e., the almost sure exponential stability is not suitable for many systems. Hence, it is necessary to consider other forms of stability. In the present paper, we will investigate the H∞ stability, moment asymptotic stability and almost sure asymptotic stability. ■ The main assumption in [29] is that the coefficients are globally Lipschitz continuous. But the global Lipschitz condition is over strict for many systems. We will loosen this condition to the local Lipschitz condition. Therefore, our results can cover more systems than [29]. ■ The key technique in [29] is the method of comparison principle. Mao mainly compared the stochastic system dx(t) = f 1 (x(t), t)dt + C x([t /τ ]τ )dB1 (t)

(1.2)

driven by a discrete time noise with the stable stochastic system dy(t) = f 1 (y(t), t)dt + C y(t)dB1 (t) driven by a continuous time noise. The results on stability of system (1.2) are entirely dependent on the stability of the underlying continuous time system. While, in the present paper, we will use the method of Lyapunov functionals to study hybrid SDSs driven by discrete time noises directly. Of course, we have to develop some new techniques to deal with the term x([t /τ ]τ ) based on the discrete time observations. The organization of this paper is as follows. Section 2 describes some necessary notations and preliminary results. In Sections 3 and 4, we mainly propose main assertions on H∞ stability, asymptotic stability and exponential stability of nonlinear hybrid SDSs driven by linear discrete time noises. In Section 5, we specially study the case of linear hybrid SDSs. In the final section, we give two illustrative examples. 2. Preliminaries and notations Let (Ω , F , {Ft }t ≥0 , P) be a complete probability space with a filtration {Ft }t ≥0 satisfying the usual conditions, w(t) = (w1 (t), w2 (t), . . . , wm (t))T an m-dimensional Brownian motion defined on this probability space and mutually independent of √each other, and τ > 0. For two constants a and b, a ∨ b = max{a, b}. For a vector or matrix B, denote by BT and |B| = trace(BT B) its transpose and trace norm, respectively. B ≫ 0 means all elements of B are positive. For a symmetric matrix D, denote by λmax (D) its maximum eigenvalue. Furthermore, let r(t) : R+ → S be a right-continuous Markovian chain on (Ω , F , {Ft }t ≥0 , P) with generator Γ = (γij )N ×N defined by { γij ∆ + o(∆) if i ̸ = j, P {r(t + ∆) = j|r(t) = i} = 1 + γii ∆ + o(∆) if i = j,

338

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where S = {1, 2, . . . , N } is its finite state space, ∆ > 0, γij is the transition rate from i to j, if i ̸ = j, while γii = − Assume r(t) is independent of w (t). Consider an n-dimensional nonlinear hybrid SDS driven by linear discrete time noises dx(t) = f (x(t), t , r(t))dt +

m ∑

Al (r(t))x([t /τ ]τ )dwl (t),

t≥0

∑

j ̸ =i

γij .

(2.1)

l=1

with the initial data x(0) = x0 ∈ Rn and r(0) = r0 ∈ S, where f : Rn × R+ × S → Rn , Al : S → Rn×n and [t /τ ] = j when jτ ≤ t < (j + 1)τ , j = 0, 1, . . .. For the aim of our paper, we always assume that f (0, t , i) = 0 for all t ∈ R+ , i ∈ S such that system (2.1) has the trivial equilibrium solution x(t) ≡ 0.

∑m

Remark 1. It is necessary to point out that hybrid SDS (2.1) driven discrete time noises l=1 Al (r(t))x([t /τ ]τ )dwl (t) ∑by m is distinguished from the one driven by continuous time noises l=1 Al (r(t))x(t)dwl (t), since x(t) ̸ = x([t /τ ]τ ) when t ∈ (jτ , (j + 1)τ ). For the term of x([t /τ ]τ ), the existing methods on hybrid stochastic systems driven by continuous time noises (e.g. [9–27,30]) cannot be employed to study the stability of hybrid SDS (2.1). From the point of original motivation of [29], for an unstable nonlinear differential system with the global Lipschitz condition (implying the linear growth condition), designing linear discrete time noises is simpler and more achievable than designing nonlinear ones. Borrowing the idea of [29], the present paper is to discuss the nonlinear stochastic system driven by linear discrete time noises, i.e., system (2.1). Of course, stochastic systems driven by nonlinear discrete time noises are complex and need further discussions. Remark 2. The assumption of f (0, t , i) = 0 for all t ∈ R+ , i ∈ S is conventional for stochastic systems (e.g. [10–26,29–31]). This assumption here is just to guarantee that there exists the zero equilibrium solution for hybrid SDS (2.1). If this assumption does not hold, the equilibrium solution of system (2.1) always can be transformed to the zero equilibrium solution of a new system. For example, for a recurrent neural network, it is easy to transform its equilibrium solution to the zero equilibrium solution of a new neural network, please see [32]. Furthermore, we propose the following assumptions for system (2.1). Assumption 2.1 (Local Lipschitz Condition). For each L = 1, 2, . . ., there exist some positive constants dL > 0 such that

|f (x, t , i) − f (y , t , i)| ≤ dL |x − y |, for all t ∈ R+ , i ∈ S and x, y ∈ Rn with |x| ∨ |y | ≤ L. Remark 3. This assumption is to loosen the global Lipschitz condition in [29] to the local Lipschitz condition. The local Lipschitz condition is more general than the global one essentially. For example, function sin(x2 ) not satisfies the global Lipschitz condition but satisfies the local Lipschitz condition. Of course, the global Lipschitz condition implies the local one. Assumption 2.2 (Linear Growth Condition). There is a positive constant k > 0 such that

|f (x, t , i)| ≤ k|x|, for all t ∈ R+ , i ∈ S and x ∈ Rn . Let C 2,1 (Rn × R+ × S ; R+ ) be the family of all continuous nonnegative functions V (x, t , i) which are continuously twice differentiable in x and once in t. For the hybrid SDS (1.1), we give the generalized Itô’s formula [30]: if V ∈ C 2,1 (Rn × R+ × S ; R+ ), then for any t ≥ 0, V (x(t), t , r(t)) = V (x(0), 0, r(0)) +

t

∫

L V (x(s), s, r(s))ds + 0

∫ t∫

Vx (x(s), s, r(s))g(x(s), s, 0

V (x(s), s, i0 + h(r(s−), l)) − V (x(s), s, r(s))µ(ds, dl),

r(s))dB(s) + 0

t

∫

R

where µ(ds, dl) is a martingale measure which is relate to r(t) but not B(t) by the independence of r(t) and B(t), and the operator L : Rn × R+ × S → R is defined by L V (x, t , i) =

N ∑

γij V (x, t , j) + Vt (x, t , i) + Vx (x, t , i)f (x, t , i)

j=1

1

+ trace[gT (x, t , i)Vxx (x, t , i)g(x, t , i)], 2

Vt (x, t , i) =

∂ V (x, t , i) ∂ V (x, t , i) ∂ V (x, t , i) ∂ 2 V (x, t , i) , Vx (x, t , i) = ( ,···, ), Vxx (x, t , i) = ( )n×n . ∂t ∂ x1 ∂ xn ∂ xi xj

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339

In essence, system (2.1) is a hybrid SDDS whose delay is a variable function with a boundedness. For this view, we can see [t /τ ]τ ≜ δ (t) = t − δ1 (t), where δ1 (t) = t − jτ when jτ ≤ t < (j + 1)τ , j = 0, 1, . . .. Then system (2.1) can be rewritten as dx(t) = f (x(t), t , r(t))dt +

m ∑

Al (r(t))x(t − δ1 (t))dwl (t).

t≥0

(2.2)

l=1

From [30], we can get that system (2.1) (i.e., system (2.2)) with Assumptions 2.1 and 2.2 has a unique global solution x(t , x0 , r0 ) and that E |x(t , x0 , r0 )|2 < ∞ for all t > 0. Here we omit its proof. To our best knowledge, the traditional results on the stability of hybrid SDDSs always require that the variable delay be differential and its derivative be less than 1 (e.g. [13], Page 285 in [30]). But the delay δ1 (t) in our paper is non-differential at points jτ , and the derivative δ1′ (t) = 1 for jτ ≤ t < (j + 1)τ , j = 0, 1, . . .. Hence, the already existing results on the stability of hybrid SDDSs cannot be applied to system (2.1) directly, and it is necessary to use some new techniques to investigate some new theory for system (2.1). 3. H∞ Stability and asymptotic stability For the stability purpose of this paper, let x(t) = x(0) = x0 , r(t) = r(0) = r0 for −2τ ≤ t ≤ 0, and f (x, t , i) = f (x, 0, i) for all t ∈ [−2τ , 0], i ∈ S and x ∈ Rn . Furthermore, we impose another assumption for system (2.1). There are some constants αi such that

Assumption 3.1.

x f (x, t , i) ≤ αi |x|2 , T

(3.1) n

for all t ∈ R+ , i ∈ S and x ∈ R . Remark 4. This assumption for functions f (x, t , i), i ∈ S includes many cases, such as bi x, bi sin(x), bi x − b′i |x|2 x with constants b′i ≥ 0 and bi . Next, under Assumption 3.1, we give the first result on the H∞ stability (as for its concrete definition, please refer to [31]) for system (2.1) in the following theorem. Theorem 3.1. Let Assumptions 2.1, 2.2 and 3.1 hold. If

{

k2 τ 2 + min 2αi − 4 i∈S

m ∑

} |Al (i)|2 τ + 1 ≥ 0

(3.2)

l=1

holds and A (τ ) = diag(∆1 , ∆2 , . . . , ∆N ) − Γ

is a nonsingular M-matrix, then for any initial data x0 and r0 , system (2.1) is stable in the sense of H∞ stability, i.e., the global solution x(t , x0 , r0 ) of system (2.1) satisfies

∫

∞

E |x(t , x0 , r0 )|2 dt < ∞,

(3.3)

0

[ ] ∑m where ∆i = − 2αi + 4 l=1 |Al (i)|2 + τ k2 , i ∈ S. Proof. The existence and uniqueness of the global solution x(t , x0 , r0 ) follows from [30]. For simplicity, write x(t) = x(t , x0 , r0 ). Since A (τ ) is a nonsingular M-matrix, then, by Theorem 2.10 on Page 68 in [30], there exists c⃗ = (c1 , c2 , . . . , cN ) ≫ 0 satisfying A (τ )c⃗ ≫ 0. Set

[

V (x(t), t , r(t)) = cr(t) |x(t)|2 +

1

∫

τ

t t −τ

∫ t(

τ |f (x(v ), v, r(v ))|2 + |

s

l=1

Applying the generalized Itô’s formula to V (x, t , i), we yield L V (x(t), t , i) 2 m [ ∑ ≤ ci 2xT (t)f (x(t), t , i) + 2| Al (i)x(δ (t))| + τ |f (x(t), t , i)|2 l=1

−

1

τ

∫

t t −τ

(

τ |f (x(s), s, r(s))|2 + |

m ∑ l=1

m ∑

2

Al (r(s))x(δ (s))|

) ] ds

2

Al (r(v ))x(δ (v ))|

)

]

dv ds .

(3.4)

340

L. Feng, J. Cao, L. Liu et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 336–352 N ∑

+

j=1

2 ∫ ∫ t( m [ ) ] ∑ 1 t γij cj |x(t)|2 + τ |f (x(v ), v, r(v ))|2 + | Al (r(v ))x(δ (v ))| dv ds τ t −τ s l=1

m

[

≤ ci 2αi |x(t)|2 + 2

∑

|Al (i)|2 |x(δ (t))|2 + τ k2 |x(t)|2

l=1

−

1

t

∫

τ

τ |f (x(s), s, r(s))| + | 2

t −τ

N ∑

+

(

[(

2

t

∫ [ 2 γij cj |x(t)| +

(

τ |f (x(s), s, r(s))| + | 2

t −τ

2αi + 4

−

1

τ

+

t

m ∑

(

m ∑

2

Al (r(s))x(δ (s))|

) ] ds

m ) ] ∑ |Al (i)|2 + τ k2 |x(t)|2 + 4 |Al (i)|2 |x(t) − x(δ (t))|2 l=1 2

m

τ |f (x(s), s, r(s))|2 + |

t −τ

N ∑

ds

l=1

l=1

∫

) ]

Al (r(s))x(δ (s))|

l=1

j=1

≤ ci

m ∑

∑

Al (r(s))x(δ (s))|

)

ds

l=1 t

∫ [ γij cj |x(t)|2 +

(

τ |f (x(s), s, r(s))|2 + |

t −τ

j=1

m ∑

2

Al (r(s))x(δ (s))|

) ] ds

l=1

m N m [ ( ) ∑ ] ∑ ∑ ≤ ci 2αi + 4 |Al (i)|2 + τ k2 + γij cj |x(t)|2 + 4ci |Al (i)|2 |x(t) − x(δ (t))|2 l=1

j=1

N ( 1 ∑ γij cj − ci − τ

)∫

t

l=1

2 m ( ) ∑ 2 τ |f (x(s), s, r(s))| + | Al (r(s))x(δ (s))| ds

t −τ

j=1

l=1

Note that A (τ )c⃗ ≫ 0 can be rewritten, for i = 1, 2, . . . , N,

[

c¯i ≜ −ci 2αi + 4

m ∑

N ] ∑ |Al (i)|2 + τ k2 − γij cj > 0.

l=1

(3.5)

j=1

Substitute this into the above inequality, then L V (x(t), t , i)

≤ −¯ci |x(t)|2 + 4ci

m ∑

|Al (i)|2 |x(t) − x(δ (t))|2

l=1 2 m ( 1 ∑ )∫ t ( ) ∑ − ci − γij cj τ |f (x(s), s, r(s))|2 + | Al (r(s))x(δ (s))| ds τ t −τ N

j=1

≤ 4ci

m ∑

l=1

[

|Al (i)| |x(t) − x(δ (t))| − 2 2

2

l=1

− c¯i |x(t)|2 + 8ci

m ∑ l=1

2

|Al (i)|

∫

t

δ (t)

(

∫

t

δ (t)

(

τ |f (x(s), s, r(s))| + | 2

m ∑

2

Al (r(s))x(δ (s))|

l=1

τ |f (x(s), s, r(s))| + | 2

m ∑

2

Al (r(s))x(δ (s))|

l=1

2 m N ( 1 ∑ )∫ t ( ) ∑ − ci − γij cj τ |f (x(s), s, r(s))|2 + | Al (r(s))x(δ (s))| ds τ t −τ j=1

l=1

m

≤ −¯ci |x(t)|2 + 4ci

∑ l=1

∫ [ |Al (i)|2 |x(t) − x(δ (t))|2 − 2

t

δ (t)

(

τ |f (x(s), s, r(s))|2

)

ds

) ] ds

L. Feng, J. Cao, L. Liu et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 336–352

341

2 m m N ) ] [ ∑ ( 1 ∑ )] ∑ +| Al (r(s))x(δ (s))| ds + 8ci |Al (i)|2 − ci − γij cj τ l=1

l=1

j=1

2

m

t

∫

( ) ∑ τ |f (x(s), s, r(s))|2 + | Al (r(s))x(δ (s))| ds.

· t −τ

(3.6)

l=1

] ∑ ∑m 2 2 1 2 |Al (i)|2 τ + 1 ≥ 0 holds for i ∈ S. Then, 8 m l=1 |Al (i)| − τ ≤ 2αi + 4 l=1 |Al (i)| + τ k ∑m ∑ N holds for i ∈ S. Furthermore, from (3.5), 8ci l=1 |Al (i)|2 < ci τ1 − j=1 γij cj holds for i ∈ S. Noting that [

From (3.2), k2 τ 2 + 2αi − 4

∑m

l=1

E |x(s) − x(δ (s))|2 s

∫

(

= E|

f (x(v ), v, r(v ))dv +

δ (s)

Al (r(v ))x(δ (v )) dw (v )|

2

δ (s) s

∫

)

l=1

s

∫ ≤ 2E |

2

m ∑

≤ 2E δ (s)

s

∫

f (x(v ), v, r(v ))dv| + 2E |

m ∑

δ (s) l=1

2

Al (r(v ))x(δ (v ))dw (v )|

2 m ( ) ∑ 2 τ |f (x(v ), v, r(v ))| + | Al (r(v ))x(δ (v ))| dv,

(3.7)

l=1

we have t

∫

L V (x(s), s, r(s))ds

E 0

i∈S

m { ∑

t

∫

E |x(s)|2 ds + 4 max ci

≤ − min ci

i∈S

0 s

∫ t∫ − 2E

δ (s)

0

(

− cr(s)

1

τ

E |x(s) − x(δ (s))|2 ds 0

l=1

l=1

∑

γr(s)j cj

)] ∫ t ∫

j=1

0

s

(

l=1 2

m

τ |f (x(v ), v, r(v ))|2 + |

s−τ

∑

Al (r(v ))x(δ (v ))|

)

dv ds

l=1

t

∫

E |x(s)|2 ds.

≤ − min ci i∈S

t

}[∫

2 m m ( ) ] [ ∑ ∑ τ |f (x(v ), v, r(v ))|2 + | Al (r(v ))x(δ (v ))| dv ds + E 8cr(s) |Al (r(s))|2 N

−

|Al (i)|2

(3.8)

0

By the generalized Itô’s formula, it follows that, for t ≥ 0, V (x(t), t , r(t)) = V (x0 , 0, r0 ) +

t

∫

L V (x(s), s, r(s))ds + M(t),

(3.9)

0

where V (x0 , 0, r0 ) is a constant which will be given later, M(t) is a local martingale whose explicit expression is of no use here. Taking the expectation on the both sides of (3.9), then EV (x(t), t , r(t)) = V (x0 , 0, r0 ) + E

t

∫

L V (x(s), s, r(s))ds 0

≤ V (x0 , 0, r0 ) − min ci i∈S

where V (x0 , 0, r0 ) = cr0 |x0 |2 + t → ∞. □

∫t ∫t

1 c 2 r0

∫

t

E |x(s)|2 ds,

(3.10)

0

∑ 2 τ (τ |f (x0 , 0, r0 )|2 + | m l=1 Al (r0 )x0 | ). This implies our desired assertion (3.3), as ∑m

2

Remark 5. The term t −τ s (τ |f (x(v ), v, r(v ))|2 + | l=1 Al (r(v ))x(δ (v ))| )dv ds of Lyapunov functional (3.4) introduced here is to deal with the term of x([t /τ ]τ ), which plays the key role in this proof. Remark 6. As far as we know, on the stability of stochastic systems driven by discrete time noises, the key technique of the existing literature [29] is the method of comparison principle. The method of Lyapunov functional (3.4) in our paper provides a new approach to study the stability of such stochastic systems driven by discrete time noises. Generally speaking, the H∞ stability does not imply the asymptotic stability in the sense of moment. However, in our case, we can obtain the following result on the pth moment asymptotic stability, p ∈ (0, 2].

342

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Theorem 3.2. If all the conditions of Theorem 3.1 hold, then for any initial data x0 and r0 , system (2.1) is pth moment asymptotically stable, i.e., the global solution x(t , x0 , r0 ) of system (2.1) satisfies lim E |x(t , x0 , r0 )|p = 0,

p ∈ (0, 2].

t →∞

(3.11)

Proof. For simplicity, write x(t) = x(t , x0 , r0 ). From (3.10) in the proof of Theorem 3.1, we have that EV (x(t), t , r(t)) ≤ V (x0 , 0, r0 ), where V (x0 , 0, r0 ) is a constant given in the proof of Theorem 3.1. Further, from the definition of V (x(t), t , r(t)), it is easy to derive that, for any t ≥ 0, E |x(t)|2 ≤ C ≜

V (x0 , 0, r0 ) maxi∈S ci

.

(3.12)

It follows from the generalized Itô’s formula that 2

2

t

∫

2x (s)f (x(s), s, r(s)) + | T

E |x(t)| = |x(0)| +

m ∑

0

2

Al (r(s))x(δ (s))| ds.

(3.13)

l=1

Then, for any 0 ≤ t1 < t2 < ∞, we have E |x(t2 )|2 − E |x(t1 )|2 = E

t2

∫

2xT (s)f (x(s), s, r(s)) + |

t1

∫

t2

≤

m ∑

2

Al (r(s))x(δ (s))| ds

l=1

2αr(s) E |x(s)|2 +

t1

≤ max(2αi + i∈S

m ∑

|Al (r(s))|2 E |x(δ (s))|2 ds

l=1 m ∑

|Al (i)|2 )C (t2 − t1 ),

l=1

2

i.e., E |x(t)| is uniformly continuous on R+ . By Barbalat Lemma, it follows from (3.3) that lim E |x(t)|2 = 0.

t →∞

Furthermore, by the Lyapunov inequality, yield that lim E |x(t)|p = 0,

t →∞

p ∈ (0, 2).

Hence, the required assertion (3.11) is proved. □ Theorem 3.2 shows the asymptotic stability in the sense of moment for hybrid SDS (2.1). However, asymptotic stability in the sense of moment does not guarantee the one in the sense of sample path. Hence, using some other analysis techniques, we further present the following result on asymptotic stability in the sense of sample path for system (2.1). Theorem 3.3. If all the conditions of Theorem 3.1 hold, then for any initial data x0 and r0 , system (2.1) is almost surely asymptotically stable, i.e., the global solution x(t , x0 , r0 ) of system (2.1) satisfies lim x(t , x0 , r0 ) = 0.

a. s .

t →∞

(3.14)

Proof. For simplicity, write x(t) = x(t , x0 , r0 ). It is obvious to see that the assertion (3.14) is equivalent to lim |x(t)|2 = 0.

t →∞

a. s .

(3.15)

Furthermore, since lim inft →∞ |x(t)|2 = 0 holds naturally, the assertion (3.15) is equivalent to lim sup |x(t)|2 = 0.

a.s.

t →∞

(3.16)

Next, we mainly claim that the assertion (3.16) holds. If (3.16) does not hold, then

{

}

P lim sup |x(t)|2 > 0 t →∞

> 0.

Hence, we can choose a sufficient small constant ϵ > 0 for P(B) > 3ϵ , where B =

{

} ω : lim supt →∞ |x(t)|2 > 2ϵ .

L. Feng, J. Cao, L. Liu et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 336–352

343

Let h > |x0 |. Define a stopping time by µh = inf{t ≥ 0 : |x(t)| ≥ h}. From (3.12), we easily obtain that E |x(t ∧ µh )|2 ≤ C ,

(3.17)

where C is a constant. Since E |x(t ∧ µh )|2 ≥ E(|x(µh )|2 I{µh ≤t } ) ≥ h2 P(µh ≤ t), we have that, as t → ∞ and taking h sufficient large, P(µh ≤ t) ≤ hC2 ≤ ϵ , which implies that P(B1 ) > 1 − ϵ, where B1 = {supt ∈R+ |x(t)| < h}. Then we can easily calculate that P(B ∩ B1 ) > 2ϵ . Define a sequence of stopping times by

ν1 = inf{t ≥ 0 : |x(t)|2 ≥ 2ϵ}, ν2l = inf{t ≥ ν2l−1 : |x(t)|2 ≤ ϵ},

l = 1, 2, . . .

ν2l+1 = inf{t ≥ ν2l : |x(t)|2 ≥ 2ϵ}. l = 1, 2, . . . We observe that when ω ∈ B ∩ B1 , then µh = ∞, νl < ∞, l = 1, 2, . . ., i.e. B ∩ B1 ⊂ {µh = ∞, νl < ∞, l = 1, 2, . . .}.

(3.18)

∑m From Assumption 2.1 and f (0, t , i) = 0 , we have that |f (x(t), t , i)| ∨ | l=1 Al (i)x(δ (t))| ≤ Ch for all t ∈ R+ , i ∈ S, when |x(t)| ∨ |x(δ (t))| ≤ h, where Ch > 0. By the Hölder’s inequality and Burkholder–Davis–Gundy inequality, we have that, for any T1 > 0, ( ) 2 E I{µh ∧ν2l−1 <∞} sup |x(ν2l−1 + t) − x(ν2l−1 )| 0≤t ≤T1

( ≤ 2E I{µh ∧ν2l−1 <∞}

∫ sup |

ν2l−1 +t ν2l−1

0≤t ≤T1

( ∫ + 2E I{µh ∧ν2l−1 <∞} sup | 0≤t ≤T1

≤

2T1 Ch2 (T1

2

f (x(s), s, r(s))ds|

ν2l−1 +t ν2l−1

m ∑

)

2

Al (r(s))x(δ (s))dwl (s)|

)

l=1

+ 4).

Since the function |x|2 is uniformly continuous in the closed ball {x ∈ Rn : |x| ≤ h}, there is a sufficiently small ϵ > 0 such that ||x|2 −|y |2 |< ϵ , when |x| ∨ |y | ≤ h and |x − y | ≤ ϵ ′ . 2h

ϵ′ =

By the Chebyshev’s inequality and (3.18), taking sufficiently small T1 > 0 with

({ P

}

µh = ∞, ν2l−1 < ∞ ∩

{

sup |x(ν2l−1 + t) − x(ν2l−1 )| < ϵ ′

2T1 Ch2 (T1 +4) ϵ ′2

< ϵ , we have that

})

0≤t ≤T1

({ =P

µh = ∞, ν2l−1 < ∞

})

({ } { − P µh = ∞, ν2l−1 < ∞ ∩ sup |x(ν2l−1 + t) 0≤t ≤T1

− x(ν2l−1 )|≥ ϵ ′

({ =P

})

µh = ∞, ν2l−1 < ∞

})

(

− E I{µh =∞,µh ∧ν2l−1 <∞} sup |x(ν2l−1 + t) − x(ν2l−1 )| ≥ ϵ

′

)

0≤t ≤T1

≥ P(B ∩ B1 ) − ≥ 2ϵ − ≥ ϵ.

E(I{µh ∧ν2l−1 <∞} sup0≤t ≤T1 |x(ν2l−1 + t) − x(ν2l−1 )|2 )

2T1 Ch2 (T1 ′2

ϵ ′2 + 4)

ϵ

{ From the relation of

sup0≤t ≤T1 |x(ν2l−1 + t) − x(ν2l−1 )| ≤ ϵ ′

}

{ ⊂

sup0≤t ≤T1 ||x(ν2l−1 + t)|2 −|x(ν2l−1 )|2 |< ϵ

}

⊂ { } { } { } ν2l − ν2l−1 ≥ T1 , we can claim that, for any ω ∈ µh = ∞, ν2l−1 < ∞ ∩ sup0≤t ≤T1 |x(ν2l−1 + t) − x(ν2l−1 )| < ϵ ′ ,

344

L. Feng, J. Cao, L. Liu et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 336–352

we have ∞

∫

|x(s)|2 ds ≥

E

( ∞ ∑

0

∫

E I{µh =∞,ν2l−1 <∞}

l=1

≥ϵ

ν2l

|x(s)|2 ds

)

ν2l−1

( ∞ ∑

E I{µh =∞,ν2l−1 <∞} (ν2l − ν2l−1 )

)

l=1

≥ ϵ T1

( ∞ ∑

)

E I{µh =∞,ν2l−1 <∞}∩{sup0≤t ≤T

|x(ν2l−1 +t)−x(ν2l−1 )|<ϵ ′ } 1

l=1

≥ ϵ T1

∞ ∑

ϵ

l=1

= ∞, which implies a contradiction with (3.3). Therefore, we can prove the assertions (3.16) and (3.15) as required. □ Remark 7. For the term of x([t /τ ]τ ), we fail to study the H∞ stability and asymptotic stability of hybrid SDS (2.1) using the existing methods on hybrid stochastic systems driven by continuous time noises (e.g. [9,13,14,16,19,21,30]). Therefore, we creatively design a new class of Lyapunov functionals to explore the problems of H∞ stability and asymptotic stability. 4. Exponential stability In the above section, we have discussed the H∞ stability and asymptotic stability of hybrid SDS (2.1) driven by linear discrete time noises. But all of those above results do not consider the speed at which the global solution converges to the zero equilibrium solution. Next, we give the result of stability at the exponential speed. Theorem 4.1. If all the conditions of Theorem 3.1 hold, then for any initial data x0 and r0 , system (2.1) is pth moment exponentially stable, i.e., the global solution x(t , x0 , r0 ) of system (2.1) satisfies log(E |x(t , x0 , r0 )|p )

p ≤ − ε¯ , p ∈ (0, 2] t 2 where ε¯ is a positive constant defined by (A.2). lim

t →∞

(4.1)

To guarantee the readability of the whole paper, we defer the detailed proof of Theorem 4.1 to the section of Appendix. Theorem 4.1 shows the assertion about pth moment exponential stability of the global solution of system (2.1). Furthermore, Theorem 8.8 on Page 309 in [30] shows that, for a hybrid SDDS with the linear growth condition, the exponential stability in the sense of moment implies the one in the sense of sample path. Hence, we can give the following theorem about almost sure exponential stability directly but omit its proof. Theorem 4.2. If all the conditions of Theorem 3.1 hold, then for any initial data x0 and r0 , system (2.1) is almost surely exponentially stable, i.e., the global solution x(t , x0 , r0 ) of system (2.1) satisfies log |x(t , x0 , r0 )|

1 ≤ − ε¯ , a.s. t 2 where ε¯ is the same defined in Theorem 4.1. lim

t →∞

(4.2)

Remark 8. Compared with [29], the method of Lyapunov functional (A.3) in our paper is direct to study the problem of exponential stability, without considering the underlying stable stochastic systems driven by continuous time noises. Remark 9. Similar to Remark 7, the methods in the existing references about hybrid stochastic systems driven by continuous time noises (e.g. [10–14,19,20,24–26,30]) cannot be applied directly to deal with the term of x([t /τ ]τ ) and study the exponential stability of hybrid SDS (2.1). Hence, the new class of Lyapunov functionals is designed creatively to investigate the exponential stability of system (2.1). Remark 10. It is necessary to point out that the main assertions of the present paper are about H∞ stability and asymptotic stability in Section 3. This section is just to prove that, compared with [29], our method of Lyapunov functional (A.3) can also be applied to investigate the issue of exponential stability of hybrid SDS (2.1). Hence, this section designed here does not contradict with the claim of "exponential convergence is excessively rapid for many systems" mentioned in Section 1. Remark 11. Similar to Theorems 3.1–3.3, 4.1 and 4.2, we can use the method of Lyapunov functionals here to discuss the stability of stochastic systems driven by discrete time noises without Markovian switching.

L. Feng, J. Cao, L. Liu et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 336–352

345

5. Stability of linear hybrid stochastic system driven by linear discrete time noises In the above sections, we discussed the H∞ stability, asymptotic stability and exponential stability of nonlinear hybrid stochastic systems driven by linear discrete time noises with Assumptions 2.1, 2.2 and 3.1. Then, there is a natural question: How about the case of linear hybrid stochastic systems? That is to say, whether our method of Lyapunov functionals can be employed to explore the stability of linear hybrid stochastic systems driven by linear discrete time noises? This section is to answer this question. Consider an n-dimensional linear hybrid SDS driven by linear discrete time noises dx(t) = B(r(t))x(t)dt +

m ∑

Al (r(t))x([t /τ ]τ )dwl (t),

t≥0

(5.1)

l=1

with the initial data x0 and r0 , where B : S → Rn×n and the other notations refer to Section 2. By [30], system (5.1) has a unique global solution which admits the trivial equilibrium solution x(t) ≡ 0. Obviously, since

|B(i)x| ≤ |B(i)||x|, 1 1 T x [B(i) + B(i)T ]x ≤ λmax [B(i) + B(i)T ]|x|2 2 2 hold for all i ∈ S and x ∈ Rn , system (5.1) satisfies Assumptions 2.1, 2.2 and 3.1. Similar to Theorems 3.1–3.3, we propose the results on the H∞ stability, moment asymptotic stability and almost sure asymptotic stability for linear hybrid SDS (5.1) but omit its proof. xT B(i)x =

Corollary 5.1. If

{

(

min |B(i)|2 τ 2 + λmax [B(i) + B(i)T ] − 4 i∈S

m ∑

) } |Al (i)|2 τ + 1 ≥ 0

(5.2)

l=1

holds and B (τ ) = diag(β1 , β2 , . . . , βN ) − Γ

(5.3)

is a nonsingular M-matrix, then for any initial data x0 and r0 , we have the following assertions: (1) system (5.1) is stable in the sense of H∞ stability, i.e., ∞

∫

E |x(t , x0 , r0 )|2 dt < ∞,

(5.4)

0

(2) system (5.1) is pth moment asymptotically stable, i.e., lim E |x(t , x0 , r0 )|p = 0,

t →∞

p ∈ (0, 2]

(5.5)

(3) system (5.1) is almost surely asymptotically stable, i.e., lim x(t , x0 , r0 ) = 0,

a.s.

t →∞

(5.6)

[

where x(t , x0 , r0 ) is the global solution of system (5.1), βi = − λmax [B(i) + B(i)T ] + 4

∑m

l=1

] |Al (i)|2 + τ |B(i)|2 , i ∈ S.

Similar to Theorems 4.1 and 4.2, we also propose the results on moment exponential stability and almost sure exponential stability for linear hybrid SDS (5.1) but omit its proof. Corollary 5.2. Assume there is at least a constant ε > 0 such that min eετ |B(i)|2 τ 2 + eετ ε + λmax [B(i) + B(i)T ] − 4

{

(

i∈S

m ∑

) } |Al (i)|2 τ + 1 ≥ 0

(5.7)

l=1

holds and B ′ (τ , ε ) = diag(β1′ , β2′ , . . . , βN′ ) − Γ

(5.8)

[ ] ∑m is a nonsingular M-matrix, where βi′ = − ε + λmax [B(i) + B(i)T ] + 4 l=1 |Al (i)|2 + τ |B(i)|2 , i ∈ S. If B ′′ = {ε > 0 : { ( ) } ∑m mini∈S eετ |B(i)|2 τ 2 + eετ ε + λmax [B(i) + B(i)T ] − 4 l=1 |Al (i)|2 τ + 1 ≥ 0 and B ′ (τ , ε ) is a nonsingular M-matrix } ̸ = φ , then for any initial data x0 and r0 , we have the following assertions:

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L. Feng, J. Cao, L. Liu et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 336–352

(1) system (5.1) is pth moment exponentially stable, i.e., lim

log(E |x(t , x0 , r0 )|p ) t

t →∞

p

≤ − εˆ ,

p ∈ (0, 2]

2

(5.9)

(2) system (5.1) is almost surely exponentially stable, i.e., lim

log |x(t , x0 , r0 )|

1

≤ − εˆ ,

t

t →∞

2

a.s.

(5.10)

where x(t , x0 , r0 ) is the global solution of system (5.1), εˆ = sup{ε > 0 : ε ∈ B ′′ }. Remark 12. If parameter k in Assumption 2.1 is replaced by maxi∈S |B(i)|, for linear hybrid SDS (5.1), we can easily yield a series of results on the stability which are direct applications of Theorems 3.1–3.3, 4.1 and 4.2 for nonlinear hybrid SDS (2.1). But, researches show that those direct applications results are not accurate. Therefore, similar to the results (i.e., Theorems 3.1–3.3, 4.1 and 4.2) for nonlinear hybrid SDS (2.1), we propose the above results (i.e., Corollaries 5.1 and 5.2) on the stability for linear hybrid SDS (5.1) by some modifications. We leave the proofs of Corollaries 5.1 and 5.2 to the readers. Remark 13. In fact, Corollaries 5.1 and 5.2 have answered the question mentioned above in this section. That is to say, compared with nonlinear hybrid SDS (2.1), our method of Lyapunov functionals can also be employed to explore the stability of linear hybrid SDS (5.1). 6. Examples In this section, we shall discuss two illustrative examples to illustrate our results. Example 6.1. Let us consider a scalar linear hybrid stochastic system driven by a linear discrete time noise dx(t) = B(r(t))x(t)dt + A(r(t))x([t /τ ]τ )dw (t),

(6.1)

with the initial data x0 = 5 and r0 = 1, where w (t) is(a scalar Brownian ) motion, r(t) is a right-continuous Markovian −10 10 chain on the state space S = {1, 2} with generator Γ = , and B(1) = −3, B(2) = 0.6, A(1) = 0.6, A(2) = 7 −7 0.4, τ = 0.025. condition (5.2) holds and the matrix B (τ ) defined in Corollary 5.1 becomes B (τ ) = ( By simple computations, ) 14.335 −10 which is a nonsingular M-matrix. −7 5.151 Furthermore, the matrix B ′ (τ , ε ) defined in Corollary 5.2 becomes

(

14.335 − ε

−7

−10 5.151 − ε

) . By Theorem 2.10 on

Page 68 in [30], B ′ (τ , ε ) is a nonsingular M-matrix if and only if all the leading principal minors of B ′ (τ , ε ) are positive, i.e., 14.335 − ε > 0,

(14.335 − ε )(5.151 − ε ) − 70 > 0.

Simple computations show that condition (5.7) holds and B ′ (τ , ε ) is a nonsingular M-matrix for any ε ∈ (0, 0.199077], i.e., εˆ = sup{ε > 0 : ε ∈ B ′′ } = 0.199077. Hence, by [30], Corollaries 5.1 and 5.2 in our paper, we have that there is a unique global solution x(t) of system (6.1), and that x(t) has the following sample properties lim x(t) = 0,

a.s.

t →∞

lim

t →∞

log |x(t)| t

≤ −0.0995385,

(6.2) a.s.

(6.3)

and the following moment properties ∞

∫

E |x(t)|2 dt ≤ ∞,

(6.4)

0

lim E |x(t)|2 = 0,

(6.5)

t →∞

lim

t →∞

log(E |x(t)|2 ) t

≤ −0.199077.

(6.6)

By the Euler scheme with time step ∆ = 10−5 , we make some simulations for system (6.1) with the initial data x0 = 5 and r0 = 1 as follows. Fig. 1 illustrates the existence and uniqueness of the global solution x(t) and the assertions (6.2),

L. Feng, J. Cao, L. Liu et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 336–352

347

Fig. 1. Simulations of the trajectories of x(t), log(|x(t)|)/t on [0, 20] and r(t) on [0, 5] for system (6.1).

Fig. 2. Simulations of

∫t 0

E |x(t)|2 dt , E |x(t)|2 , log(E |x(t)|2 )/t on [0, 20] for system (6.1) and sample size 200.

(6.3) by the simulations of the trajectories of x(t), by the simulations of

∫t 0

E |x(t)| dt , E |x(t)| , 2

2

log(|x(t)|) t log(E |x(t)|2 ) for t

for system (6.1). Fig. 2 illustrates the assertions (6.4), (6.5), (6.6) system (6.1).

Remark 14. Obviously, linear system (6.1) satisfies the global Lipschitz condition. However, the condition (2.4) required by Theorem 2.1 in [29] holds for subsystem 1 but not for subsystem 2 of linear system (6.1). Hence, Theorem 2.1 in [29] cannot be applied to linear system (6.1) directly. Similar to [29], using the method of comparison principle to investigate the exponential stability of linear system (6.1) is another new problem, which needs further discussions. But, that is not our main aim. Note that there is the only literature [29] on the stability of stochastic systems driven by discrete time noises and that the global Lipschitz condition implies the local one. Therefore, for linear system (6.1), it is nature to appeal to the assertions of the present paper. Fortunately, our theory can work for this example. Example 6.2. Consider a two-dimensional nonlinear hybrid stochastic system driven by a linear discrete time noise dx(t) = f (x(t), t , r(t))dt + A′ (r(t))x([t /τ ]τ )dw (t),

(6.7)

with the initial data x0 = (x1 (0), x2 (0)) = (7, −5) and r0 = 1, where x(t) = (x1 (t), x2 (t)) is the state vector, w (t) is) a ( −2 2 scalar Brownian motion, r(t) is a Markovian chain on the state space S = {1, 2} with generator Γ = , 5 −5 T

348

L. Feng, J. Cao, L. Liu et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 336–352

Fig. 3. Simulations of the trajectories of x(t), log(|x(t)|)/t on [0, 15] and r(t) on [0, 5] for system (6.7).

( f (x, t , 1) =

τ = 0.002.

√

−√2 − 22

2 2

)

−2

cos (t)x, f (x, t , 2) = 2

(

1

0

0

1

)

′

sin(x), A (1) =

(

0.4

0

0

0.4

)

′

, A (2) =

(

0.5

0

0

0.5

) and

condition (3.2) holds and the matrix A (τ ) defined in Theorem 3.1 becomes A (τ ) = ( By simple computations, ) 5.342 −2 which is a nonsingular M-matrix. Furthermore, the matrix A ′ (τ , ε ) defined in Theorem 4.1 becomes −5 1.982 ( ) 5.342 − ε −2 . Similar to the analysis of Example 6.1, condition (A.1) holds and A ′ (τ , ε ) is a nonsingular −5 1.982 − ε M-matrix for any ε ∈ (0, 0.0811621], i.e., ε¯ = sup{ε > 0 : ε ∈ A ′′ } = 0.0811621. Hence, by [30], Theorems 3.1–3.3, 4.1 and 4.2 in our paper, we have that there is a unique global solution x(t) of system (6.7), and that x(t) has the following sample properties lim x(t) = (x1 (t), x2 (t))T = (0, 0)T ,

a.s.

t →∞

lim

t →∞

log |x(t)| t

≤ −0.04058105,

a.s.

(6.8)

(6.9)

and the following moment properties ∞

∫

E |x(t)|2 dt ≤ ∞,

(6.10)

0

lim E |x(t)|2 = 0,

(6.11)

t →∞

lim

t →∞

log(E |x(t)|2 ) t

≤ −0.0811621.

(6.12)

Similar to Example 6.1, by the Euler scheme with time step ∆ = 10−5 , we make some simulations (Figs. 3 and 4) for system (6.7) to illustrate the assertions (6.8)–(6.12). Remark 15. Similar to Remark 14, the global Lipschitz condition holds for nonlinear system (6.7). However, Assumption 3.2 required by Theorem 3.3 in [29] does not hold for all subsystems of nonlinear system (6.7). That is to say, Theorem 3.3 in [29] cannot work for nonlinear system (6.7). Except for [29], there is no other literature about stochastic systems driven by discrete time noises. Therefore, we resort to the present paper for nonlinear system (6.7). Fortunately, our assertions can work for this system. 7. Conclusions and further discussions Mao [29] has discussed the exponential stability for a new class of stochastic systems driven by linear discrete time noises, which has opened a new chapter of stochastic systems. However, the global Lipschitz condition required in [29]

L. Feng, J. Cao, L. Liu et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 336–352

Fig. 4. Simulations of

∫t 0

349

E |x(t)|2 dt , E |x(t)|2 , log(E |x(t)|2 )/t on [0, 15] for system (6.7) and sample size 200.

is over strict for many systems. Hence, it is necessary to loosen this condition. In the present paper, under the local Lipschitz condition and the linear growth condition, we use the method of Lyapunov functionals to study the H∞ stability, asymptotic stability and exponential stability of hybrid stochastic systems driven by linear discrete time noises. Of course, it is worth mentioning that our method provides a new approach to study such systems driven by discrete time noises. Since our new approach can be employed to deal with the term of x([t /τ ]τ ) effectively, we have a feeling that our new approach may play important roles on the stability of such nonlinear stochastic systems driven by discrete time noises. Acknowledgments This project is jointly supported by the National Natural Science Foundation of China (Nos. 11571024, 61833005, 61573096 and 61272530), China Postdoctoral Science Foundation (No. 2017M621588), Science and Technology Research Foundation of Higher Education Institutions of Hebei Province of China (No. QN2017116). Appendix

A.1. Proof of Theorem 4.1 Proof. Since condition (3.2) holds and A (τ ) is a nonsingular M-matrix, there is at least a sufficient small constant ε > 0 such that

[

eετ k2 τ 2 + eετ ε + min 2αi − 4

{

i∈S

m ∑

|Al (i)|2

}]

τ +1≥0

(A.1)

l=1

holds and A ′ (τ , ε ) = diag(∆′1 , ∆′2 , . . . , ∆′N ) − Γ

[

is also a nonsingular M-matrix, where ∆′i = − ε + 2αi + 4

ε¯ = sup{ε > 0 : ε ∈ A ′′ },

∑m

l=1

] |Al (i)|2 + τ k2 , i ∈ S. Define ε¯ by (A.2)

[ { }] ∑m where A = {ε > 0 : e k τ + e ε + mini∈S 2αi − 4 l=1 |Al (i)|2 τ + 1 ≥ 0 and A ′ (τ , ε ) is a nonsingular M-matrix }. ⃗ = (q1 , q2 , . . . , qN ) ≫ 0 satisfying A ′ (τ , ε)q⃗ ≫ 0, for ε ∈ A ′′ . Set By Theorem 2.10 on Page 68 in [30], there exists q ∫ ∫ [ 1 t εs t ( e τ |f (x(v ), v, r(v ))|2 W (x(t), t , r(t)) = qr(t) eεt |x(t)|2 + τ t −τ s 2 m ) ] ∑ +| Al (r(v ))x(δ (v ))| dv ds , (A.3) ′′

ετ 2 2

l=1

ετ

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L. Feng, J. Cao, L. Liu et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 336–352

for ε ∈ A ′′ . Applying the generalized Itô’s formula to W (x, t , i), we get that L W (x(t), t , i) 2 ∫ m [ ) t ∑ 1( = qi ε eεt |x(t)|2 + eεt L |x(t)|2 + τ |f (x(t), t , i)|2 + | Al (i)x(δ (t))| eεs ds τ t −τ l=1

−

1

eε t

τ eετ

t −τ

N

+

∑

2 m ( ) ] ∑ 2 τ |f (x(s), s, r(s))| + | Al (r(s))x(δ (s))| ds

t

∫

l=1

1

[

γij qj eεt |x(t)|2 +

τ

j=1

≤ qi eεt

[(

∫

2 ∫ t( m ) ] ∑ τ |f (x(v ), v, r(v ))|2 + | eεs Al (r(v ))x(δ (v ))| dv ds

t

s

t −τ

l=1 2

m

ε|x(t)|2 + 2xT (t)f (x(t), t , i) + |

∑

)

Al (i)x(δ (t))|

( + τ |f (x(t), t , i)|2

l=1 2 2 ∫ t ( m m ) ] ) ∑ ∑ 1 2 τ | f (x(s) , s , r(s)) | + | A (r(s))x( δ (s)) | ds +| Al (i)x(δ (t))| − l ετ τe t −τ l=1

l=1

+

N ∑

(

τ |f (x(s), s, r(s))| + | 2

t −τ

j=1

≤ q i eε t

t

∫ [ γij qj eεt |x(t)|2 +

[(

m ∑

2

) ]

Al (r(s))x(δ (s))|

ds

l=1

ε|x(t)|2 + 2αi |x(t)|2 + 2

m ∑

|Al (i)|2 |x(δ (t))|2 + τ k2 |x(t)|2

)

l=1

−

+

∫

1

τ eετ N ∑

t

(

τ |f (x(s), s, r(s))|2 + |

t −τ

m ∑

2

) ]

Al (r(s))x(δ (s))|

ds

l=1 t

∫ [ γij qj e |x(t)|2 + εt

(

τ |f (x(s), s, r(s))| + | 2

t −τ

j=1

m ∑

2

) ]

Al (r(s))x(δ (s))|

ds

l=1

m N m [ ( ) ∑ ] ∑ ∑ ≤ qi ε + 2αi + 4 |Al (i)|2 + τ k2 + γij qj eεt |x(t)|2 + 4qi |Al (i)|2 eεt |x(t) − x(δ (t))|2 l=1

−(

j=1

N

1

∑

τe

j=1

q − ετ i

γij qj )eεt

∫

t

(

l=1 2

m

τ |f (x(s), s, r(s))|2 + |

∑

t −τ

Al (r(s))x(δ (s))|

)

ds.

l=1

⃗ ≫ 0 can be rewritten, for i = 1, 2, . . . , N, Note that A ′ (τ , ε )q

(

q′i ≜ −qi ε + 2αi + 4

m ∑

N ) ∑ |Al (i)|2 + τ k2 − γij qj > 0.

l=1

(A.4)

j=1

Substitute this into the above inequality, then L W (x(t), t , i)

≤ −q′i eεt |x(t)|2 + 4qi

m ∑

|Al (i)|2 eεt |x(t) − x(δ (t))|2 − (

l=1

·eεt

t

∫

1

N ∑

τe

j=1

q − ετ i

γij qj )

2 m ( ) ∑ τ |f (x(s), s, r(s))|2 + | Al (r(s))x(δ (s))| ds

t −τ

l=1

m

∫ [ ∑ ≤ 4qi |Al (i)|2 eεt |x(t) − x(δ (t))|2 − 2 l=1

− q′i eεt |x(t)|2 + 8qi

m ∑ l=1

|Al (i)|2 eεt

∫

t

δ (t)

t

δ (t)

(

τ |f (x(s), s, r(s))| + | 2

m ∑

2

Al (r(s))x(δ (s))|

l=1

2 m ( ) ∑ τ |f (x(s), s, r(s))|2 + | Al (r(s))x(δ (s))| ds l=1

) ] ds

L. Feng, J. Cao, L. Liu et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 336–352

−(

1

qi − τ eετ

N ∑

γij qj )eεt

∫

t

(

τ |f (x(s), s, r(s))|2 + |

t −τ

j=1

≤ −q′i eεt |x(t)|2 + 4qi

m ∑

351

2

m ∑

Al (r(s))x(δ (s))|

)

ds

l=1

∫ [ |Al (i)|2 eεt |x(t) − x(δ (t))|2 − 2

t

(

δ (t)

l=1

τ |f (x(s), s, r(s))|2

2 m m N ) ] [ ∑ ] ∑ ∑ 1 +| Al (r(s))x(δ (s))| ds + 8qi |Al (i)|2 − ( ετ qi − γij qj ) τe l=1

·eεt

l=1

t

∫

(

τ |f (x(s), s, r(s))|2 + |

m ∑

t −τ

j=1

2

Al (r(s))x(δ (s))|

)

ds.

l=1

] ∑ 2 1 |Al (i)|2 τ + 1 ≥ 0 holds for i ∈ S. Then, 8 m l=1 |Al (i)| − τ eετ ≤ ∑ ∑ ∑m m N ε + 2αi + 4 l=1 |Al (i)|2 + k2 τ holds for i ∈ S. Furthermore, by (A.4), 8 l=1 |Al (i)|2 qi < τ e1ετ qi − j=1 γij qj holds for i ∈ S. [

From (A.1), eετ k2 τ 2 + eετ ε + 2αi − 4

∑m

l=1

In view of the inequality (3.7), we have t

∫

L W (x(s), s, r(s))ds

E 0

′

t

∫

εs

2

e E |x(s)| ds + 4 max

≤ − min qi i∈S

i∈S

0 t

∫

s

∫

eεs

− 2E

(

δ (s)

0

+E 8

2

|Al (i)| qi

t

}[∫

eεs E |x(s) − x(δ (s))|2 ds

0

l=1

τ |f (x(v ), v, r(s))|2 + |

m ∑

2

Al (r(s))x(δ (v ))|

)

dv ds

]

l=1

m

[ ∑

m {∑

|Al (r(s))|2 qr(s) − (

l=1

1

qr(s) − τ eετ

N ∑

γr(s)j qj )

t

]∫ 0

j=1

eε s

∫

s

(

τ |f (x(v ), v, r(s))|2

s−τ

2

m

) ∑ +| Al (r(s))x(δ (v ))| dv ds l=1

′

t

∫

≤ − min qi i∈S

eεs E |x(s)|2 ds.

(A.5)

0

By the generalized Itô’s formula, it follows that, for t ≥ 0, W (x(t), t , r(t)) = W (x0 , 0, r0 ) +

t

∫

L W (x(s), s, r(s))ds + M1 (t),

(A.6)

0

where W (x0 , 0, r0 ) is a constant given later, M1 (t) is a local Martingale (its explicit expression is of no use, we do not state it here). Taking the expectation for (A.6), we have EW (x(t), t , r(t)) ≤ W (x0 , 0, r0 ) − min q′i i∈S

t

∫

eεs E |x(s)|2 ds

0

≤ W (x0 , 0, r0 ), where W (x0 , 0, r0 ) ≜ C1 = qr0 |x0 | − 2

εt

(A.7) 1

q [τ e τ ε r0

−ετ

1

− ε (1 − e

−ετ

)](τ |f (x0 , 0, r0 )| + | 2

∑m

l=1

max qi e E |x(t)| ≤ C1 . 2

i∈S

Hence, for any t ≥ 0, E |x(t)|2 ≤

C1 maxi∈S qi

e−εt ,

which implies the assertion limt →∞

log(E |x(t)|2 ) t

≤ −ε. Then we can claim that, as ε → ε¯ ,

2

lim

log(E |x(t)| )

≤ −¯ε.

t By the Lyapunov inequality, yield that, for any p ∈ (0, 2), t →∞

p

E |x(t)|p ≤ (E |x(t)|2 ) 2 ≤ (

C1 q¯

p

p

) 2 e− 2 ε t ,

2

Al (r0 )x0 | ). This implies that

352

L. Feng, J. Cao, L. Liu et al. / Nonlinear Analysis: Hybrid Systems 33 (2019) 336–352

which yields the desired assertion lim

log(E |x(t)|p )

t →∞

t

p

≤ − ε¯ , 2

p ∈ (0, 2).

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