exponential mean square stability of the be method for ... · the markovian switching, the be...

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Exponential Mean Square Stability of the BE Method for Linear Neutral Hybrid Stochastic Delay Differential Equations Haiyan Yuan 1,2* , Jihong Shen 1 1 College of Automation, Harbin Engineering University, Harbin, China. 2 Department of Mathematics, Heilongjiang Institute of Technology, Harbin, China. * Corresponding author. Tel.: +8615045061030; email: [email protected] Manuscript submitted March 10, 2014; accepted June 7, 2014. doi: 10.17706/ijapm.2016.6.3.104-111 Abstract: In this paper, the backward Euler (BE) method is introduced and analyzed for linear neutral hybrid stochastic delay differential equations. The exponential mean square stability of the BE method is considered for linear neutral hybrid stochastic delay differential equations. It is proved that, under the one-sided Lipschitz condition and the linear growth condition, for some positive stepsizes which depend on the Markovian switching, the BE method is exponential mean square stable. A numerical example is provided to illustrate the theoretical results. Key words: Linear neutral hybrid stochastic delay differential equations, exponential mean square stability, BE method, Markovian switching. 1. Introduction The practical systems which experience abrupt changes in their structure and parameters caused by phenomena, such as component failures or repairs, changing subsystem interconnections, and abrupt environmental disturbances must be modeled by the hybrid systems driven by continuous-time Markov chains [1]. The hybrid systems combine a part of the state that takes values continuously and another part of the state that takes discrete values. One of the important classes of hybrid systems is the stochastic delay differential equations (SDDEs) with Markovian switching. The natural extensions of this system are stochastic delay integro-differential equations (SDIDE) with Markovian switching and neutral stochastic delay differential equations (NSDDE) with Markovian switching. Although the theoretical study of stochastic differential delay equations with Markovian switching is considered in [2]-[4], the explicit solutions can hardly be obtained for the SDIDEs and NSDDEs with Markovian switching. Rathinasamy and Balachandran [5] analyzed the numerical MS-stability of second-order Runge-Kutta schemes for multi-dimensional stochastic differential systems with one multiplicative noise, the convergence and stability of the Euler method for a class of linear stochastic differential delay equations with Markovian switching is studied in [6]. The numerical method for SDIDEs with Markovian switching had been studied in [7], mean-square stability of Milstein method was obtained. For the neutral stochastic differential equations, Mao studied the exponential mean square stability of its analytical solution in [8], the delay-dependent exponential stability conditions for linear neutral stochastic differential systems was studied in [9]-[11]. Huang Chengming and Wang Wansheng studied the stability of semi-implicit Euler method and - method International Journal of Applied Physics and Mathematics 104 Volume 6, Number 3, July 2016

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Page 1: Exponential Mean Square Stability of the BE Method for ... · the Markovian switching, the BE method is exponential mean square stable. A numerical example is provided to illustrate

Exponential Mean Square Stability of the BE Method for Linear Neutral Hybrid Stochastic Delay Differential

Equations

Haiyan Yuan1,2*, Jihong Shen1

1 College of Automation, Harbin Engineering University, Harbin, China. 2 Department of Mathematics, Heilongjiang Institute of Technology, Harbin, China. * Corresponding author. Tel.: +8615045061030; email: [email protected] Manuscript submitted March 10, 2014; accepted June 7, 2014. doi: 10.17706/ijapm.2016.6.3.104-111

Abstract: In this paper, the backward Euler (BE) method is introduced and analyzed for linear neutral

hybrid stochastic delay differential equations. The exponential mean square stability of the BE method is

considered for linear neutral hybrid stochastic delay differential equations. It is proved that, under the

one-sided Lipschitz condition and the linear growth condition, for some positive stepsizes which depend on

the Markovian switching, the BE method is exponential mean square stable. A numerical example is

provided to illustrate the theoretical results.

Key words: Linear neutral hybrid stochastic delay differential equations, exponential mean square stability, BE method, Markovian switching.

1. Introduction

The practical systems which experience abrupt changes in their structure and parameters caused by

phenomena, such as component failures or repairs, changing subsystem interconnections, and abrupt

environmental disturbances must be modeled by the hybrid systems driven by continuous-time Markov

chains [1]. The hybrid systems combine a part of the state that takes values continuously and another part

of the state that takes discrete values. One of the important classes of hybrid systems is the stochastic delay

differential equations (SDDEs) with Markovian switching. The natural extensions of this system are

stochastic delay integro-differential equations (SDIDE) with Markovian switching and neutral stochastic

delay differential equations (NSDDE) with Markovian switching. Although the theoretical study of

stochastic differential delay equations with Markovian switching is considered in [2]-[4], the explicit

solutions can hardly be obtained for the SDIDEs and NSDDEs with Markovian switching. Rathinasamy and

Balachandran [5] analyzed the numerical MS-stability of second-order Runge-Kutta schemes for

multi-dimensional stochastic differential systems with one multiplicative noise, the convergence and

stability of the Euler method for a class of linear stochastic differential delay equations with Markovian

switching is studied in [6]. The numerical method for SDIDEs with Markovian switching had been studied in

[7], mean-square stability of Milstein method was obtained. For the neutral stochastic differential equations,

Mao studied the exponential mean square stability of its analytical solution in [8], the delay-dependent

exponential stability conditions for linear neutral stochastic differential systems was studied in [9]-[11].

Huang Chengming and Wang Wansheng studied the stability of semi-implicit Euler method and - method

International Journal of Applied Physics and Mathematics

104 Volume 6, Number 3, July 2016

Page 2: Exponential Mean Square Stability of the BE Method for ... · the Markovian switching, the BE method is exponential mean square stable. A numerical example is provided to illustrate

for nonlinear neutral stochastic delay differential equations in [12], [13].

To best of our knowledge, no results on the numerical methods for linear NSDDEs with Markovian

switching have been presented in the literatures. Thus, it is necessary to develop appropriate numerical

methods and to study the properties of these approximate schemes for NSDDEs with Markovian switching.

In this paper we consider the linear d -dimensional neutral stochastic delay differential equation with

Markovian switching of the form

(1.1)

where ( )r t is a Markov chain taking values in {1,2, , }S N and ( )A , ( )B , ( )C , ( )D and

( ) RE , N is the d -dimensional square matrices.

In Section 2, we will introduce some necessary notations and assumptions. In Section 3, the definition of

exponential mean squares stability is defined, the BE method will be used to produce the numerical

solutions and the main result will be shown and proved in this section. A numerical example is given in

Section 4.

2. Notation and Preliminaries

Throughout this paper, unless otherwise specified, we use the following notations. If is a vector or

matrix, its transpose is denoted by . Let | · | denote both the Euclidean norm in and the trace (or

Frobenius) norm in (denoted by trace ). represents and

denotes . Let be a complete probability space with a filtration satisfying the

usual conditions, that is, it is right continuous and increasing while contains all P-null sets. Let be a

d-dimensional Brownian motion defined on this probability space, and is a positive fixed delay.

Let , be a right-continuous Markov chain on the probability space taking values in a finite state

space with generator given by

where . Here is the transition rate from to if while .We assume that the

Markov chain is independent of the Brownian motion . It is known that almost every sample path of

is a right-continuous step function with a finite number of simple jumps in any finite subinterval of .

As for , the following lemma is satisfied.

Lemma 2.1 [6]. Given , then is a discrete Markov chain with the one-step

transition probability matrix .

Since the are independent of , the paths of Markov chain can be generated independent of and,

in fact, before computing .

For the purpose of stability, assume that .This shows that (1.1) admits a trivial

solution. We assume that and satisfy the local Lipschitz condition:

]0,[),()(

,0),()())(()())((

)())(()())(())](()([

tttx

ttdwtxtrEtxtrD

dttxtrBtxtrAtxNtxd

A

A dR

ddR )(trace AAA ba },max{ ba ba

},min{ ba P)F,,( 0}F{ tt

0F )(tw

)(tr 0t

},,2,1{ NS NNijΓ

, if )(01

, if )(0Δ})()({P

ji

jiitrjtr

ii

ij

0 0ij i j ji

ij

ijii

)(r )(w

)(tr R

)(tr

0h ,2,1,0),( nnhrr hn

NNij ehh

)(P)(P

ij x r x

x

0),0,0,0(),0,0,0( igif

f g

International Journal of Applied Physics and Mathematics

105 Volume 6, Number 3, July 2016

Page 3: Exponential Mean Square Stability of the BE Method for ... · the Markovian switching, the BE method is exponential mean square stable. A numerical example is provided to illustrate

(H1): Both and satisfy the local Lipschitz condition. That is, for each there is an such

that

(2.1)

for all and and those with .

(H2): For every , there are constants nonnegative constants , and such that

, (2.2)

, (2.3)

for all and .

We also assume that: exists a positive constant , such that .

Considering equation (1.1), we can match in the following form:

and .

If we take , , , ,then the hypotheses (H1) and

(H2) are satisfied naturally for Eq.(1.1). Then for any given initial data . Eq. (1.1) has a

unique continuous solution .

3. Math Stability Analysis of the BE Method

Applying the Backward Euler(BE)method to the linear neutral delay differential system with Markovian

switching (1.1) leading to a numerical process of the following type:

, (3.1)

where is a stepsize which satisfies for some positive integer and , ,

is an approximation to , if , we have , the increments ,

are independent - distributed

Gaussian random variables. Further, we assume that is -measurable at the mesh points .

and positive constant , such that , for with , any application of the

method to problem (1.1) generates a numerical approximation , which satisfies

(3.2)

f g ,,2,1 k 0kh

)(),,,(),,,(),,,(),,,( yyxxhityxgityxgityxfityxf k

0t Sidyxyx R,,, kyxyx

Si Ria i i i

22),,,()]([2 yxaityxfyNx ii

222),,,( yxityxg ii

dyx R, 0t

2

10 N

)())(()())(())(,),(),(( txtrBtxtrAtrttxtxf

)())(()())(())(,),(),(( txtrEtxtrDtrttxtxg

)()()2( iBiAai )()( iBiAi )(iDi )(iEi

)R],0,([0

dbFC

)(tx

nmnhnn

hnmn

hnn

hnmnnmnn wxrExrDhxrBxrANxxNxx )()()()( 111111

0h mh m nhtn Strr nhn )(

nx nn xtx )( 0nt )( nn tx )()( 1 kkk twtww

,,2,1,0 k ),0( hN

nxnt

Fnt

0),,,(0 iiii EDBAh ),0( 0hhm

h

}{ kx

2Elog

1suplim

2

k

tx

t

International Journal of Applied Physics and Mathematics

106 Volume 6, Number 3, July 2016

Definition 3.1 A numerical method is said to be exponential mean square stable, if there exists a

Page 4: Exponential Mean Square Stability of the BE Method for ... · the Markovian switching, the BE method is exponential mean square stable. A numerical example is provided to illustrate

Theorem 3.1 Assume that there are four nonnegative constants , , , , such that conditions (2.2)

–( 2.4) hold, and for any , ,where and .Then the BE

method is exponential mean square stable.

Proof. Let , then from conditions (2.2)-(2.4), we have

where .

Then we have

(3.3)

Note that for all .Taking expectation on the both sides of inequality (3.3), we have

For any , we have

(3.4)

which implies

where we used the inequality .

Note that

ia i i i

Si 03 iii EDA )(),( iDDiAA ii )(iEEi

mkkk NxxZ

hkmkikikkmkiki

kkmkkkkkmkkk

kkmkkkmkkkkk

mhxxZZhxxa

witxxgZZhitxxfZ

witxxghitxxfZZZ

2

1])([

2

1)(

2

1

),,,(,),,,(

),,,(),,,(,

2222

1

2

1

2

1

11111

1111

2

1

kmkkkkmkkhk wikhxxgZhwikhxxgm

),,,(2)(),,,(

22

hkmkikimkikikk mhxxhxxaZZ )()(

222

1

2

1

22

1

0 hkm 0k

hxxhxxaZZ mkikimkikikk )()(222

1

2

1

22

1

1C

hkmkiki

hkmkikik

khhkk

khk

hk

hCxx

hCxxaZCCZCZC

)1(22

)1(2

1

2

1

2)1(22

1)1(

)(

)()(][

21

0

10

2

21

0

0

2

1

1

0

)1(2

1

1

0

)1(2

0

2

)]1)(1([

)]1)(1([

mj

k

j

jhhhi

j

k

j

jhhhi

mj

k

j

hjij

k

j

hjik

kh

xCChC

xCChC

xChxChaZZC

2

0

22

0

22)1( mjjmjjj xxNxxZ

22

0

21

0

2

1

2

1

1

0

)1(

kkh

j

k

j

jh

j

k

j

jhj

k

j

hj

xCxxC

xCxC

International Journal of Applied Physics and Mathematics

107 Volume 6, Number 3, July 2016

Page 5: Exponential Mean Square Stability of the BE Method for ... · the Markovian switching, the BE method is exponential mean square stable. A numerical example is provided to illustrate

and

We therefore have

(3.5)

where

, (3.6)

.

It is easy to see that , and for any ,

hence, there exists unique constant such that .

Thus (3.5) yields

. (3.7)

In view of (3.6), for any bounded initial condition , is nonnegative -measurable random variable,

thus the right-hand side of (3.7) converges to a finite random variable as , that is,

Define , then , and we also define .

21

1

21

0

21

1

2

1

)(2

1

1

0

)1(

j

k

mkj

jhmhj

k

j

jhmhj

mj

jhmh

j

mk

mj

hmjmj

k

j

hj

xCCxCCxCC

xCxC

21

21

0

21

21

)(21

0

i

k

mki

ihmhi

k

i

ihmhi

mi

ihmh

i

mk

mi

hmimi

k

i

ih

xCCxCCxCC

xCxC

21

0

2

0

2

0

2)( i

k

i

ihik

kh xChCxhaZXZC

21

1

10

2 ]/)1)(1([ i

mi

ihhhii xChChCCX

Ch

CC

h

CCCaC

hh

i

hh

iii

1)1(

1)1()(

10

20

0

1

i

ia 031 iiiiiii EDAa 0)( C 1C

),1(

1

*

i

ih

aC 0)( * hC

2

0

2

0

2* xhaZXZC ik

kh

h

X 0F

k

2

0

2

0

2* suplimsuplim xhaZXZC ik

k

kh

hk

Cu log )2,0(

*lim hohu

International Journal of Applied Physics and Mathematics

108 Volume 6, Number 3, July 2016

Page 6: Exponential Mean Square Stability of the BE Method for ... · the Markovian switching, the BE method is exponential mean square stable. A numerical example is provided to illustrate

So, for any , there exists which is the solution of , such that

for any , .

Since we earlier obtained that , so we get , for any , that

.

Consequently, for any , there exists an integer , such that .

Applying the elementary inequality with and

, as well as the condition (2.4), we obtain

, (3.8)

then, for any integer ,

,

implying that

. (3.9)

Letting in (3.9), we obtain

,

which gives .

Thus the estimate (3.8) yields

,

so, foe any , we obtain .

Consequently, , which gives (3.2).

4. Numerical Example

)2,0( ),,,(00 iiii EDBAhh 0)( * hC

),0( 0hh 2* hu

2*suplim k

kh

hk

ZC ),0( 0hh

2)2(suplim k

kh

kZe

),0()2(log

1

e 1k 1

2, kkZe k

kh

,0,1,0, ),()1()( 11 cpbabcacba ppppp2p

)1( c

2212)1( mk

kh

k

kh

k

khxeZexe

12 kk

2)(1212

212121

sup)()1(sup)()1(sup mk

hmk

kkkmk

kh

kkkk

kh

kkk

xeexexe

221

2111

supsup)()1( k

kh

kkkk

kh

kkmk

xeexee

2112

1121

sup)()1()1(sup k

kh

kkmkk

kh

kkk

xeeexe

2k

2112

111

sup)()1()1(sup k

kh

kkmkk

kh

kk

xeeexe

2lim k

kh

kxe

2)(12suplim)()1(suplim mk

hmk

kk

kh

kxeexe

),0()2(log

1

e )()1()1(suplim 112

exe k

kh

k

kh

x

kh

xe k

k

k

kh

k

logsuplim2

logsuplim0

2

International Journal of Applied Physics and Mathematics

109 Volume 6, Number 3, July 2016

Page 7: Exponential Mean Square Stability of the BE Method for ... · the Markovian switching, the BE method is exponential mean square stable. A numerical example is provided to illustrate

We shall discuss an example to illustrate our theory. Let )(tw be a scalar Brownian motion. Let )(tr be a

right continuous Markov chain taking values in }2,1{S with the generator

11

22ij .

We assume )(tw and )(tr are independent.

Consider the following one-dimensional linear neutral stochastic delay differential equation with

Markovian switching

]0,1[,1)(

,0),()1())(()())((

)1())(()())(())]1(()([41

tttx

ttdwtxtrEtxtrD

dttxtrBtxtrAtxtxd

(4.1)

Let 25

(1) 2, (1) 1, (1) 0, (1)A B D E and 45

(2) 9, (2) 4, (2) 0, (2)A B D E .

It is interesting to regard Eq. (4.1) as the result of the following two equations:

)()1()1()(2))]1(()([52

41 tdwtxdttxtxtxtxd (4.2)

and

)()1()1(4)(9))]1(()([54

41 tdwtxdtttxtxtxd (4.3)

switching from one to the other according to the movement of the Markov chain )(tr . It is known that Eqs.

(4.2) and (4.3) are exponentially mean square stable, where 41 , ,3, 12

91 ua 5

211 ,0 , ,,

425

2479

2 ua

54

22 ,0 , 40 , we take 1 ,111

111100 ),,,( EDBAhh , 91

222200 ),,,( EDBAhh .

Thus, when a method of form (3.1) with stepsize ),0(111h is used to solve the above system, the

corresponding numerical solution is exponential mean-square stable by Theorem 3.1.

Acknowledgment

This work was supported by the Natural Science Foundation of Heilongjiang Province (A201418) and the

Creative Talent Project Foundation of Heilongjiang Province Education Department (UNPYSCT-2015102)

References

[1] Abolnikov, L., Dshalalow, J. H., & Treerattrakoon, A., (2008). On a dual hybrid queueing system.

Nonlinear Analysis: Hybrid Systems, 2, 96-109.

[2] Matasov, A., & Piunovkiy, A. B., et al. (2000). Stochastic differential delay equations with Markovian

switching. Bernoulli, 6, 73-90.

[3] Xuerong, M. (2000). Robustness of stability of stochastic differential delay equations with Markovian

switching. SACTA, 3, 48-61.

[4] Yuan, C., et al. (2006). Stochastic Differential Equations with Markovian Switching. London: Imperial

College Press.

[5] Rathinasamy, A., & Balachandran, K. (2008). Mean-square stability of second-order Runge-Kutta

methods for multi-dimensional linear stochastic differential systems. Journal of Computational and

Applied Mathematics, 219,170-197.

International Journal of Applied Physics and Mathematics

110 Volume 6, Number 3, July 2016

Page 8: Exponential Mean Square Stability of the BE Method for ... · the Markovian switching, the BE method is exponential mean square stable. A numerical example is provided to illustrate

[6] Ronghua, L., & Yingmin, H. (2006). Convergence and stability of numerical solutions to SDDEs with

Markovian switching. Applied Mathematics and Computation, 175, 1080-1091.

[7] Rathinasamy, A., & Balachandran, K. (2008). Mean-square stability of Milstein method for linear hybrid

stochastic delay integro-differential equations. Nonlinear Analysis: Hybrid Systems, 2, 1256-1263.

[8] Xuerong, M. (1995). Exponential stability in mean square of neutral stochastic differential functional

equations. Systems &Control Letters, 26, 245–251.

[9] Cong, S. (2013). On exponential stability conditions of linear neutral stochastic differential systems

with time-varying delay. International Journal of Robust and Nonlinear Control, 23(11), 1265–1276.

[10] Lirong, H., & Xuerong, M. (2009). Delay-dependent exponential stability of neutral stochastic delay

systems. IEEE Transactions on Astomatic Control, 54,147–152.

[11] Gouaisbaut, F., & Peaucelle, D. (2006). Delay-dependent stability analysis of linear time delay systems.

IFAC workshop on Time Delay System, 6(1), 54–59.

[12] Fuke, W., & Chengming, H. (2015). Exponential mean square stability of the theta approximations for

neutral stochastic differential delay equations. Journal of Computational and Applied Mathematics, 286,

172–185.

[13] Wansheng, W. (2011). Mean-square stability of semi-implicit Euler method for nonlinear neutral

stochastic delay differential equations. Applied Numerical Mathematics, 61, 696–701.

Yuan Haiyan was born in April 1982. She got the PhD of HIT and the postdoctor of

Harbin Engineering University. Her major interesting is the study of mathematics and

applied mathematics. She is also interested in the optimization algorithm and system

modeling in research work.

Shen Jihong was born in September 1966. He got the doctor of engineering. He is a

professor, Ph.D. supervisor. He is the dean of College of science, Harbin Engineering

University. His major interesting is the study of mathematics and applied mathematics.

He is also interested in the optimization algorithm and system modeling in research

work.

International Journal of Applied Physics and Mathematics

111 Volume 6, Number 3, July 2016