基于移动服务的扩展不确定性投入产出模型的稳定性分析_英文_

第27卷 第4期2010年8月

黑龙江大学自然科学学报

J OURNAL OF NATURAL SC IENCE O F HE I LONG JI ANG UN IVERS I TY

V o l 27N o 4A ugust ,2010

Stability of extended uncertai n Leontief i nput out put model on mobile services

Z HAO X i ao hong 1,3

, XU Fu yuan 2

, TANG B i ng yong

3

(1.Busi ness and A d m i n istration Co ll ege ,Zhe jiang N o r m al U niversity ,Jinhua 321004,Chi na ;2.Schoo l o fM anag e m ent ,Shangha iU n i versity o fT echno l ogy ,Shangha i 200093,China ;3.G l o rious Sun Schoo l of Busi ness and M anag e m ent ,Don

ghua U n i versity ,Shangha i 200051,Ch i na)

Abst ract :A m odel to descri b e co m pan ies i n put-output process on m ob ile serv ices is developed and tested .Through the theo ries fro m Leontief i n put-output econo m ics ,an ex tended si n gu l a r dyna m ic i n put-outputm odel on m obile ser v ice is provided .The prob le m of stability o f th is k i n d o f input-outpu t m odel on mobile serv ices is researched .A ne w m athe m aticalm ethod is app li e d to study the si n gular sys te m s w ithou t converting t h e m into genera l syste m s .The para m eter uncerta i n ties are considered .A ne w stability cr iterion for the ex tended si n gular dyna m ic input-outputm ode l is g i v en to ensure the stability o f i n put-outputm ode.l

K ey w ords :stab ility ;extended Leontief i n put-outputm ode;l uncerta i n ,m ob ile serv ice CLC nu m ber :O189 1 D ocum ent code :A A rticle I D :1001-7011(2010)04-0430-05

R ecei ved date :2010-04-28

Foundati on ite m:Supported by t h e N ati on al Nat u re Sci en ce Foundati on of Ch i na(70472057);

t h e Third Peri od Key D i sci p li n e Foundation of

Shanghai (S30504)

Bi ography :M rs .ZHAO X i ao-hong(1973-),lect or ,Ph. D.,

i nterested i n:m ob ile serv i ce ,el ectron i c co mm erce ,E -ma i :l zxh57@z j nu .cn

0 I ntroducti o n

Peop le s use o fm obile co mm un icati o n dev ices is co mm on ,and dev ices based on m ob ile techno logy are i n

creasi n g rapidly in everyday life .So ,it is decisive l y i m portant that co m pan ies should interact w ith consum es and prov i d e services by m ob ile dev ices for custo m s .No w m obile services are m ore and m o re i m portant for fir m s .H o w ever ,fir m s can no t invest a ll o f resources in develop i n g m ob ile services .In fac,t custo m s use m ob il e devicesm ainly for the si m plest serv i c es ,such as tex tm essag i n g or vo ice serv ices .Then,understand i n g the a m ount pu tti n g i n m o b ile services and the quantity o f output is very necessary .E spec ially ,the i n fluence of econo m ic crisis has not dis appeared.Co m panies hope that they can gather the i n for m ation of fir m s process and auto m atically control the quantity of co mpan i e s i n put to keep a w ay fro m co llapse .To prevent t h e spread o f econo m ic crisis ,econo m ists w an ted to study the current state of a company and research how various polic i e s can be used to m ove t h e syste m fro m its present status to a future m ore desirab le state w hen they tr i e d to dea lw ith those dyna m ic syste m s .A num ber of funda m enta lnotions and m ethods based on t h e theory of econo m ical cybernetics have been ex tended to the ar ea o f econo m ics .W hen the research focused on i n put-output ana lysis o f the econo m ic syste m s ,m any m ode lsw ere estab lished to descri b e the real econo m i c s .Professor Leon tief put for w ar d the Leonti e f input-output !m ode,l w hich m eans t h e foundati o n of input-outpu t econo m ics [1]

.Leonti e f i n put-output m odel is a kind o f linear and deter m i n ate m ode.l H o w ever ,the input-outpu tm odels o f co m pan ies wh ich pr ov i d e m ob ile ser v ices i n rea l society are si n gu lar li n ear syste m s ,wh ich are much harder to so lve than genera l linear syste m s .S i n gu lar syste m s are also

re ferred to as i m plicit syste m s ,descriptor syste m s or se m i-state syste m s [2]

.Singu lar syste m s i n econo m ics are

genera lly converted i n to genera l li n ear syste m s by m eans of the selection of state vector ,con tro l vector and output vector .L ittle research is about the d irect disposa lm et h od to si n gular li n ear i n put-outputm odels of fir m s on m ob ile services .Further m ore ,real econo m ic syste m s have uncerta i n and unpredictab le factors .These factorsm ust be con si d ered when i n put-output proble m s are researched .I n th is paper w e w ill directly deal w ith a k i n d o f ex tended si n gular Leontief i n pu t-output syste m on mobile serv ices instead of converti n g the m into general syste m s and con si d er the uncerta i n ties ex isting i n rea l co m pan ies .

1 Econo m ics mode l

In th i s section ,the classic dyna m ic Leonti e f i n put-ou t p u tm ode lw ill be ex tended to a k i n d of singu lar and stochastic input-outpu tm odel to descr i b e the i n put-output a m ount o f fir m s on m ob ile serv ices .1 1 Proble m for m ulation

Consi d er the classi c Leontief input-outpu tm odel described by

x (k )=Ax (k )+B [x (k +1)-x (k )]+Y(k ).

(1-1)

The vector x (k )=x 1(k )

?

x i (k )

?

x n (k

)

T

i s the total output vector and x i (k )is the to tal output fro m

sector i .Y(k )is the fi n al net pr oduct vector .The m atri x A is the d irect consumpti o n coeffic ientm atrix ,B is the capital coeffic ientm atri x .This m odel i s d iscrete-ti m e input-outpu t syste m.The d iscrete-ti m e input-output m ode ls are used w ide l y because nati o ns m ake the econo m ic plans one ti m e every year or every m onth .H owever ,so m e econo m ic actions ,such as the pr oducti o n o f a facto r y ,are o ften conti n uous .So the continuous-ti m e dyna m ic i n put-output mode l should be i n vesti g ated .Ifw e replace t h e difference [x (k +1)-x (k )]w it h the d ifferenti a l x (t),t h e classic Leontief input-output syste m (1-1)w ill be t u rned into conti n uous-ti m e dyna m ic Leontief i n put -outputm ode:l

x (t)=Ax (t)+B x (t)+Y(t).

(1-2)

The vecto r Y(t)can be considered as the Leonti e f input-output mode l s con tro l vecto r because w e can affect the quantity of fi n al net product by m any econo m ic m anage m ent m ethods such as contr o lli n g the scale of i n vest m en.t Then x (t)can be treated as the state vector .Thus the dyna m ic Leontief i n put-outpu t m odel (1-2)can be t u rned into a state space m ode.l I n real co m pan ies on m ob ile serv ices ,cap ita l coeffic ientm atr i x B is not a l w ays i n vertible ,because the pr oducts o f so m e secto rs o f co m panies ,such as after-sale serv ice depart m ents ,can no t be treat as cap ita l product and applied to i n ves.t So the dyna m ic Leon tief i n put-ou tputm odel (1-2)can be re w rit ten as

B x (t)=(I -A )x (t)-Y(t),

(1-3)

w here rank B =r

In rea l econo m ic syste m s ,there are m any unpred icted facto rs .W hen the disturbing factors are considered ,the trad iti o na l input-output m odel should be extended to describe the rea l econo m i c syste m s .The para m eter uncer tainty is i n troduced and the para m eters are considered as the M arkov ian jump i n g para m eters .I n other w or ds ,the coeffic ientm atrices A and B w ill be consi d ered to M arkov ian j u m pi n g para m eters .Furt h er m ore ,the invest m ent to a co m pany can not have the effect at once .So the ti m e lag need be consi d ered .Then the syste m (1-3)tur ns into

B (r(t)) x (t)=(I -A (r(t)))x (t)+Ad (r(t))x (t -1)-Y(t).

(2)The vector functi o n r (t)is a rando m function representing a conti n uous-ti m e d iscre te-state M ar kov ian process w hich takes value in a fi n ite set M ={1,2,3?,s }.I n econo m ic society ,one i m portant pr oble m is about the quan tity of fi n al net produc.t So m eti m es ,the fi n a lnet product is hoped to increase to a higher leve l severa l years later .H o w ever ,the quantity of fi n al net product should not i n crease forever because resources o f the earth are scarce .Scarcity m eans tha t soc i e ty has li m ited resources and therefore can no t put all of resources to one secto r .So it is hoped that the quantity of the fina lnet product re m a i n s i n a quite stab le leve.l So ,w e need research t h e stability o f t h e ex tended uncerta i n Leontief input-output m ode l (2)on m obile serv ices-syste m.

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431#第4期赵小红等:基于移动服务的扩展不确定性投入产出模型的稳定性分析

1 2 Notation

In this paper,R n is the n-d i m ensi o nal Eucli d ean space;R m?n denotes the m?n rea lm atrices space;I is the n?n i d entity m atrix;A T denotes the m atri x transposition;*!is used as the ter m that are introduced by sy mm e try;when m atrices X and Y are sy mm etric,X>Y m eans thatm atri x(Y-X)is positive-defi n ite; is a compact se.t

2 S t ab ility of input-output model

Th is secti o n w ill i n vestigate t h e stability o f t h e syste m(2).To d iscuss this pr oble m,the fo ll o w ing defi n ition is necessary.

D efinition1 The ex tended Leontief i n put-outputm ode l is described by equation(2).r(t)can be re w rit ten as i=r(t).

1)Syste m(2)is called to be regu lar if det(s B i-I+A i)and det(s B i-I+A i-Ad i)is not i d entically zero for any i.

2)Syste m(2)is i m pulse-free if deg(det(s B i-I+A i))=rankB i and deg(det(s B i-I+A i-Ad i))=rank

B i.

3)Syste m(2)is asy m ptotically stab le if equati o n B i

x(t)=(I-A i)x(t)+Ad i x(t-1)is asy m ptotica lly sta

b le for any i.

4)Syste m(2)is called to be asy mpto tica ll y adm issi b le if it i s regu lar,causal and asy mpto ticall y stable.

Then w e have the fo llo w i n g resu l.t

Theore m1 Syste m(2)is asy m ptoti c ally ad m issi b le if there ex istm atrices P i>0,Q i>0,Z>0,X11i%0, X12i%0,X22i%0,N1i,N2i(i=1,2,?,s),such that t h e fo llo w ing i n equa lity holds:

B T i P i=P T i B i%0(3)

=

1112Z-A

T

i

Z

*22Ad T i Z

**-Z

<0(4)

!=X11i X12i N1i

*X22i N2i

**Z

%0(5)

w here

11=2P i-P i A i+A T

i P i+Q i+N1i B i+B

T

i N

T

1i+X11i+

&s

j=1

?ij B T i P j

12=P i Ad i-N1i B i+B T

i N

T

1i+X12i

22=-N1i B i-B T

i N

T

1i+X22i-Q i.

Proof By the Schur co m ple m ent theory and Le mm a2i n reference[3],we have the fo ll o w i n g conclusion fro m i n equality(4):

0>11-12-122T12%11+12+T12+22%2P i-P i A i-A T i P+P i Ad i+Ad T i P+&s j=1?ij B T i P j.(6) By the Schur co m ple m ent theory,w e can get0>11fro m i n equa lity(4).Because Q i>0,X11i%0and B T i P i%0are defi n ed,Q i+X11i+&s j=1?ij B T i P j%0is righ.t Then,the fo llo w i n g inequality is true:

0>11=2P i-P i A i+A T i P i+Q i+N1i B i+B T i N T1i+X11i+&s j=1?ij B T i P j%2P i-P i A i+A T i P i+N1i B+B T i N T1i.

(7)

For any singu lar m atri x is congruentw it h the m atri x I0

00

,there ex ist nonsi n gu lar m atrices L i and R i such that

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432

#黑 龙 江 大 学 自 然 科 学 学 报 第27卷

L i B i R i=I

00

.Then w e can w rite:

L i(I-A i)R i=

A1i A2i

A3i A4i

, L i Ad i R i=

Ad?1i Ad?2i

Ad?3i Ad?4i

, L-T i P i R T i=

P1i P2i

P3i P4i

R i N1i L-1i=

!N11i!N12i

!N13i!N14i

, R i N2i L-1i=

!N21i!N22i

!N23i!N24i

.(8)

Applying equation(8)to inequa lity(3),w e can get

B T i P i=R-T i I0

00

L-T i L-T i

P1i P2i

P3i P4i

R-T i=R-T i

P1i P2i

00

R-T i

and P T i B i=R-T i P1i0

P2i0

R-T i.So,it can be derived that P2i=0.

Pre-mu lti p ly i n g equations(6)and(7)by R T i and post-m ulti p l y ing equati o ns(6)and(7)by R i,w e can get that

T1T2 T3T4<0 and

M1M2

M3M4

<0(9)

w here T4=P4i(A4i+Ad?4i)+(A4i+Ad?4i)T P4i,M4=P4i A4i+A T4i P4i.W e w ill proof that A4i is nonsi n gular.By t h e Schur co m ple m ent theo r y,inequality(9)m eans M4<0.If A4i is si n gular,there ex ist a vector#such that A4i #=0.Then w e have#T M4#=#T(P4i A4i+A T4i P4i)#=0.U sing the m et h od of reduction to absurdity,it can be easi l y derived that A4i is nonsi n gular.So,det(s B i-I+A i)is not iden tica ll y zer o and deg(det(s B i-I+A i))=rank B i.S i m ilarl y,det(s B i-I+A i-Ad i)is not i d enticall y zero and deg(det(s B i-I+A i-Ad i))=rankB i.Therefore, t h e ex tended Leontief input-output m odel is si n gu lar and i m pulse-free by Definiti o n1.Then,w e w ill prove the asy m ptotica lly stab ility.Consi d er the fo ll o w i n g Lyapunov f u nction:

V(x(t))=x T(t)B T i P i x(t)+(t t-1x T(?)Q i x(?)d?+(0-1(t t+% x T(?)B T i ZB i x(?)d?d%.

By the Lei b niz-N e w ton for mu la,there ex ists the follo w i n g equati o n:

2#T(t)N B

i x(t)-B i x(t-1)-(t t-

1

B i

x(?)d?=0

w here#(t)=

x(t)

x(t-1)

,N=N1i N2i.Obv iously,w e can get another inequality

#T(t)X#(t)-(t t-1#T(t)X#(t)d?%0 w here X=X11i X12i

X T12i X22i

.

If A is the w eak infi n itesi m a l generator of the rando m process{x(t),r(t)},for each r(t)=i,w e can obta i n:

AV(x(t))=li m

&)01

&{?[V(x(t+&))]-V(x(t))}

?2x T(t)P T i B i

x(t)+x T(t)&s j=1?ij B T i P j x(t)+x T(t)Q i x(t)-x T(t-1)Q i x(t-1) +

x T(t)B T i x(t)ZB T i

x(t)-(t t-1x T(?)B T i ZB i x(?)d?

?2x T(t)P T i B i

x(t)+x T(t)&s

j=1

?ij B T i P j x(t)+x T(t)Q i x(t)-x T(t-1)Q i x(t-1) +

x T(t)B T i x(t)ZB T i x(t)-(t t-1x T(?)B T i ZB i x(?)d?

+2#T(t)N B

i x(t)-B i x(t-1)-(t t-1B i x(?)d?+#T(t)X#(t)-(t t-1#T(t)X#(t)d?

?#T(t)(#(t)-(t t-1)T(t)!)(t)d?

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第4期赵小红等:基于移动服务的扩展不确定性投入产出模型的稳定性分析

w here)(t)=[x T(t)x T(t-1)

x T(?)B T i]and(=11+(I-A i)T Z(I-A i)12+(I-A i)T

ZAd i

(12+(I-A i)T ZAd i)T22+Ad T i ZAd i

.

Together w it h(4)and(5),there is the conc l u si o n that AV(x(t))<0for any)(t)+0.So syste m(2)is asy m ptotica lly stab le accor d i n g to Defi n ition1.Then the ex tended Leonti e f input-output m odel(2)is regu lar, i m pulse-free and asy m pto tica ll y stable.There f o re,it is asy m pto tica ll y adm issible.Th is co m p letes t h e proo.f

4 Conclus i o n

Th is artic le develops and tests a kind o f input-outpu tm ode l to descri b e co m panies i n put-output process on m ob ile services.B ased on the classic Leontief input-output t h eo ris,an ex tended uncertai n singu lar dyna m ic i n put -outputm odel on m ob ile serv ice is prov i d ed.Firstly,the classi c Leonti e f i n pu t-output m odel is i m proved.The para m eter uncertainties are consi d ered and the ne w extended i n put-outputm ode l o fm ob ile ser v ices fir m s is de rived.Secondly,the stab ility of ne w m ode l is analyzed.A ne w judge cond ition under w hich ex tended Leontief i n put-outputm ode l is asy m ptotically ad m issible is proved.

R eferences

[1] LEONT I EF W.I npu t-output econo m ics[M].Be iji ng:C o mm erci al Press,1980.

[2] DAI L.S i ngu lar control sys t e m s[M].B erli n:Spri nger-Verl ag,1989.

[3] XIE L,SOUZ A C E.Robu st H-i n fi n ite control for li near s yste m s w i th nor m-bounder ti m e-varyi ng un certai n ties[J].IEEE T ransacti on s on Au

to m atic Con tro,l1992,14:1188-1191.

[4] LU H Q,Z HOU W,XU Y,et a.l S tochasti c stab ility analys i s for uncertai n si ngu l ar s yste m w it h M arkovian j um p i ng para m et ers[C].CCDC2008,

Yanta,i2008:1077-1080.

[5] WU Z,Z HOU W.Delay-dependen t robu st H i n fi n ite contro l f or un certai n si ngu lar ti m e-del ay s yste m s[J].I ET C ontrol Th eory A pp,l2007

(1):1234-1241.

[6] X U S,LAM J.Robu st stab ility and stab ilizati on of d iscrete s i ngu l ar sys t e m s:an equ i val en t charact eriz ati on[J].A sian J Con tro,l2003(5):399

-405.

基于移动服务的扩展不确定性投入产出模型的稳定性分析

赵小红1,3, 徐福缘2, 汤兵勇3

(1.浙江师范大学经济与管理学院,金华321004; 2.上海理工大学管理学院,上海200093;3.东华大学旭日工商管理学院,上海200051)

摘 要:研究移动服务企业的投入产出过程。基于经典的Leontief投入产出经济学的基本理论,提出一种扩展的广义动态投入产出模型,并研究了其稳定性问题。传统的投入产出模型稳定性研究方法是把广义系统转换为一般系统,针对移动服务的短过程特点,应用新方法直接处理广义模型。考虑参数的不确定性,提出了新的稳定性判断方法以确保移动服务过程中投入产出模型的稳定性。

关键词:稳定性;扩展的投入产出模型;不确定性;移动服务

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#黑 龙 江 大 学 自 然 科 学 学 报 第27卷