ֻ43जֻ4௹上海交通大学学报 2009୍4ᄅ ໓ᅣщݼ:1006-2467(2009)04-0592-04ࣜ࠶༢֥ളթັٳѰᒮ杨忠直1, 张世英2, 陈炳富3(1.ഈݚࢌ๙ն࿐νࣜ࠶აܵ࿐ჽ,ഈݚ200052;2.฿ࣃն࿐ܵ࿐ჽ,฿ࣃ300072;3.ଲषն࿐ݓ࠽അ࿐ჽ,฿ࣃ300073)ᅋ ေ:定义了系统的生存域和多系统的共生存域,给出了具有凹增长生存函数的多个系统在资源稀缺的共生存域中的生存冲突定理.在生存函数分析的基础上,分别建立了以共生存函数和边际共生存函数为目标,以微分包含为动态系统约束的多系统微分生存博弈模型,导出了对应的最优生存博弈求解方程.所得结论可用于人类经济组织的各种系统.ܱՍ:共生存函数;共生存域;微分包含;生存微分博弈;费马原理ᇏٳোݼ: ໓ངѓ്:AViableDifferentialGamesofEconomicSystemsYANGZ1hong-zhi, ZH23ANGShi-ying, CHENBing-fu(,ShanghaiJiaotongUniversity,Shanghai200052,China;,TianjinUniversity,Tianjin300072,China;,NankaiUniversity,Tianjin300073,China)Abstract:Theviabledomainofsystemandthecoviabledomainofmultisystemsweredefined,,theviabledifferentialgamemodelsofmult-isystemswithcoviablefunctionandmarginalcoviablefunctionasobjectivefunctionsunderthedynamicsys-temconstraintofdifferentialinclusionwerebuiltrespectively,:coviablefunction;coviabledomain;differentialinclusion;viabledifferentialgame; مݓᇷള࿐ࡅ,୶Нغ၄࿐ࢂࠆ֤ᆀીႵളଁ֥)൞ุ֥ᄎࠣఃᄝᄎᇏ֥ཌྷቔMonod[1]ᆷԛ:დᇥᇏթᄝ֥ૄ၂۱൙൞ࠏ߶Ⴈ.ุ֥ᄎఃЌӻളթሑ,طุࡗ֥ބсေྟ֥ࢲݔ(Everythingexistingintheun-iཌྷቔႨఃᄝཌྷஆԇބཌྷ་ႄݖӱᇏЌӻࣩverseisthefruitofchanceandnecessity).ളթ൞ᆚ৯֥ཌྷؓޙ[2].οᅶؘ৯࿐ֻקੰ,ุაሱಖࢸაದোഠ߶֥၂۱ቋ௴ђ֥൙ൌ,დᇥຣߌၛุࠣაุᆭࡗ֥ཌྷቔႨ৯(ࣩᆚ৯)न၂֊թᄝ,ఃളթაؿᅚၹ(҂ં൞Ⴕളଁ֥ߎ൞֩,ࣩᆚ֥ކ৯ູਬ.൬۠ರ௹:2008-03-30ࠎࣁཛଢ:ݓࡅሱಖ॓࿐ࠎࣁሧᇹཛଢ(69874028)ቔᆀࡥࢺ:ဗᇑᆰ(1956-),ଳ,༆ڶದ,Ѱൖ,࢝൱,Ѱൖള֝ഽ,ᇶေ࣮ٚཟ:ඌࣜ࠶,ሧӁܵ,࿖ߌࣜ࠶.߅(Tel.):021-52301179;E-mail:zzyang@.
ֻ4௹杨忠直,等:经济系统的生存微分博弈 593Aubin[3]֥༢ളթં൞:ഡק၂۱თ,ӫ(nົ)ࠢӫቔ෮Ⴕሰ༢֥܋ളթшࢸ,࠺ቔ5P,ູളթთ,֒༢ሑࣉೆളթთ(҂ܵ༢ሑԩ5P=H5Dq,xႿളթთ֥હֹٚ),ಪູ༢൞ളթ;֥֒༢קၬ4 ഡP(t)<X@TؓթᄝtI[ti,tf]<ሑषളթთ,ࠇᄝളթთᆭຓ,ಪູ༢ԩႿඵT,֤ؓႿx(t)IP(t),Uq(x(t))>0,qIQӮ৫,ວሑ.Aubin[4]ᄎႨളթંٳ༅ࣜ࠶࿐໙ᄵӫP(t)<X@Tູ෮Ⴕ༢֥ളթܵ;ؓႿି,ีԩ༎ಌሧჷ֥ٳބ൧ӆनޙ໙ี.Uq(x(t))>0,qIQӮ৫֥x(t)IP(t)ܒӮ֥(n+1ߎ۳ԛਔ૭ඍ༢ᄎྛሑ(ԩႿളթࠇ൞҂ିົ)ࠢӫቔ෮Ⴕ༢ളթ֥ܵшࢸ૫(ࠇܵу),࠺ቔളթ)֥ॖളթྟݦඔ[3,4],ໃקၬ؇ਈ༢ᄝള5P(t),ࠧx(t)I5P(t).թთᇏ҂ׄളթඣ֥֮ۚളթݦඔ.ᄝ໓ངק1 ളթთDqູоࠢ,ᄵ܋ളթთP[4,5]ᇏקၬঔᅚݦඔቔູቔႨਈӧඍਔٮઔჰ,=HDqູоࠢ;5Dqູоࠢ,ՖطႵ5P=H5Dq൞ીႵ۳ԛঔᅚݦඔ֥ྟᇉ,ၛؓऎุࣜ࠶༢qIQqIQܒᄯఃളթݦඔ,္ၛႨٮઔჰ࣮ࣜ࠶༢ູоࠢ.֥ቋႪളթ໙ק2 ֒༢֥ളթݦඔູχᄹӉݦඔൈ,.ีᄵളթთD൞ܱႿჰׄx=0֥ࠢ.Ч໓ᄝ࣮ࡓ۽ӱഽაӵЇഅ֥Ѱᒮ໙ี֥ࠎԤഈ[6],קၬਔြᇶაӵЇഅ֥܋ളթݦඔק3 ؓPtI[ti,tf],U(x(t))൞ܱႿx֥χބ܋ളթთᄹӉݦඔ,ᄵD(t)൞ܱႿ၍ჰׄx(t)=0(t)֥,ࡹ৫ਔၛັٳЇݣູჿඏ֥ࣜ࠶ളթሼ.༢֥ቋႪളթѰᒮଆࠣఃࢳ่ࡱ,࣮ࢲݔؓႿ؟༢֥Ѱᒮࡹଆބ൫ဒิ܂ਔંק4 ۲༢֥ളթݦඔUq(t)ູχᄹӉݦඔ,Dqູоࠢ,P=HDqູо܋ളթთ,ᄵ၇ऌqIQ.۲༢֥ളթ(ݦඔ)сಖؿളԊ.1 ༢ളթંࠎԤളթԊ൞ᆷ:ᄝ၂۱ऎႵ༎ಌྟሧჷ֥܋ളթთᇏഡႵ2۱༢AބB,ೂݔ༢A֥ളթඣ ࣜ࠶༢൞Ⴎದࠣఃࣜ࠶ࠃ෮ቆӮ,ೂݓࡅaთaఒြaࡅ๖֩,ԩႿߌဆэᇏ֥ࣜ࠶༢ิۚ,ࠧUA(x){,ᄵ༢B֥ളթඣсಖࢆఃଢ֥ᄝളթؿᅚ.༢აߌ֥ཌྷቔႨ֮,ࠧUB(x)|,ࠇཌྷّ.ၹՎ,؟۱༢ᄝऎႵ༎ಌ֝ᇁ༢֥ൡႋྟളթ,༢ᆭࡗ֥ཌྷቔႨაࣩྟሧჷ֥܋ളթთᇏсಖؿളളթࣩᆚ,ູࠧਔളᆚᄵ֝ᇁ༢֥ࣩᆚྟളթ.༢෮ԩߌ൞༢թطࣩᆚ.֥ളթთaളթॢࡗࠇളթ่ࡱ,༢ᄝఃഈ֥ളթק5 ؓႿtI[ti,tf],۲༢֥ളթݦඔඣႨളթݦඔট؇ਈU(x(t))ູx֥χᄹӉݦඔ,Dq(t)ູ༢q֥ള.թሼ,P(t)=HDq(t)ູ෮Ⴕ༢֥оളթ,ܵ ༢֥ളթთᄵ۲༢ᄝᄎᇏ֥ളթсಖؿളԊ,ູࠧളթ קၬ1 ഡ༢֥ളթݦඔູU(x(t),u(t),طࣩᆚ.t):֒x(t)ID@T<X@Tൈ,U(x(t),u(t),t)>0Ӯ৫;֒ ؟۱༢֥܋ളթݦඔx(t)I/D@T<X@Tൈ,U(x(t),u(t),t)<0Ӯ৫ ᄝ၂ߌᇏ,؟۱༢ંࣩᆚೂޅࠗਛ,;֒x(t)I5D@T<X@Tൈ,U(x(t),҂ॖିᆺႵ၂۱༢ֆളթ,ཌྷّᇭ؟༢൞ཌྷu(t),t)=0Ӯ৫,ᄵӫD<Xູ༢֥ളթთ,ఃᇏ5DູളթთD֥шࢸ၇թ,ఃࣩᆚ൞ູਔղ֞ބ܋ԩa܋ളթބࣉ.קၬ2 ഡD߄.ᄝሧჷ༎ಌ֥၂ߌᇏ,ᇭ؟༢ᆜุളթඣ(t)<X@T,ؓႿtI[ti,tf]<T֤թᄝေႨ܋ളթݦඔট؇ਈ.x(t)ID(t),U(x(t),u(t),t)>0Ӯ৫,ᄵӫDקၬ5 ഡ۲༢֥ളթݦඔູ(t)<X@Tູ༢֥ളթሼ;ؓႿିUUq(x(t)),x(t)IDq(t),qIQ,ᄵקၬބൔ܋ളթݦ(x(t),u(t),t)=0,Ӯ৫֥෮Ⴕx(t)ID(t)ܒӮ֥(n+1ົ)ࠢӫູ༢ളթሼ֥шࢸ૫ඔູ(ࠇуU(x(t))=EAqUq(x(t))),࠺ቔ5D(t),ࠧx(t)I5D(t).qIQקၬ3 ഡDqIX,qIQֻູq۱༢֥ളթx(t)IP(t)=HDq(t)(1)თqIQ,ᄵӫP=HDqູQᇏ෮Ⴕ༢֥܋ളթთ,ѩqIQିЌᆣթᄝxIP<X֤Uq(x)>0,qIQӮ৫;EAq=1, AqI(0,1)qIQؓႿିUq(x)=0,qIQӮ৫֥෮ႵxIPܒӮ֥ؓႿ2۱༢,ః܋ളթݦඔູ
594上 海 交 通 大 学 学 报ֻ43ज U(x(t))=A1U1(x(t))+A1U(x(t))ٳѰᒮଆູx(t)IP(t)=D1(t)HD2(t)(2)maxU(x(t))=FUAqq(x(t))AqIQ1+A2=1, A1,A2I(0,1) קၬ6 ഡ۲༢֥ളթݦඔູÛ(t)IF(x(t))U F(x(t))B={f(x(t)),uq(t),qIQ}q(x(t)),x(t)IDq(t),qIQ,ᄵקၬࠒൔ܋ളթݦඔູ x(t)IP(t)B=HDq(t)(7)qIQUA(x( uq(t)IUq(x(t)), qIQt))=FUqq(x(t))qIQ tI[ti,tf]x(t)IP(t)=HDq(t)(3)qIQ EAq=1, AqI(0,1)qIQEAq=1, AqI(0,1)qIQࠧ෮ႵQ۱༢֥ളթݦඔܒӮ֥ૢࡆಃ֥QົؓႿ2۱༢,ః܋ളթݦඔุູࠒቋն,֒ಖࣼ൞ૢࡆಃ֥ሹളթඣቋնࠇڞU(x(t))=UA11(x(t))@UA22(x(t))০ඣቋն.x(t)IP(t)=D1(t)HD2(t)(4)ೂݔႨࣜ࠶࿐ژݼקၬш࠽ളթݦඔ,ބൔшA1+A2=1, A1,A2I(0,1)࠽܋ളթݦඔູMU(x(t))=EaqMUq(x(t))(8)2 ؟༢֥ളթࣩᆚqIQᄝሧჷႵཋ֥၂ളթߌᇏ,ᇭ؟༢ູਔࠒൔш࠽܋ളթݦඔູak۲ሱ֥ളթބؿᅚطᅚषࣩᆚ,ૄ۱༢ᄝࣩᆚᇏ MU(x(t))=EMUk(x(t))FUAqq(x(t))(9)kIQqIQ\kീᅚҦሱ֥࠭ളթ(ݦඔ)ඣิ.ۚႮႿളթൔᇏ,༢q֥ш࠽ളթݦඔקၬູMUq(x(t))=ሧჷ༎ಌ,ᆃсಖ֝ᇁః༢֥ളթ(ݦඔ)ඣdUq(x(t)),طQ۱༢֥ш࠽܋ളթݦඔקၬູࢆ֮dx(t),ᆃဢᇭ؟༢ԩႿളթ҂ޙሑ,ളթࣩᆚ(ᆚ)ࡼ҂؎ࣉྛ,/ބ܋ԩ0ჰᄵࡼିᇭ؟༢MUdU(x(t))(x(t))=.Ⴟ൞,༢֥ളթັٳѰᒮᄝ၂༎ಌሧჷߌᇏԩႿളթޙ,۲ٚdx(t)൳ၭࠇാԩႿन֩ሑଆሇ߄ູ.2op tMU(x(t))=EAqMUq(x(t)).1 ༢ളթັٳѰᒮଆqIQ༢֥ളթᄎັٳЇݣູ.Ûx(t)IF(x(t))xÛ(t)IF(x(t))F(x(t))B={f(x(t)),uq(t),qIQ}F(x(t))B={f(x(t)),uq(t),qIQ}x(t)IP(t)B=HDq(t)q(10)IQx(t)IP(t)B=HDq(t)qIQ(5)uq(t)IUq(x(t)), qIQuq(t)IUq(x(t)), qIQtI[ti,tf]tI[ti,tf]ൔᇏ,uq(t)ູ༢q֥ളթҦ.༢ऎႵބൔ܋EAq=1, AqI(0,1)qIQളթݦඔ֥ളթັٳѰᒮଆູop tMU(x(t))=maxU(x(t))=EAqUq(x(t))k qIQEMUAk(x(t))FUAqq(x(t))kIQqIQ\.Ûx(t)IF(x(t)).Ûx(t)IF(x(t)) F(x(t))B={f(x(t)),uq(t),qIQ}F(x(t))B={f(x(t)),uq(t),qIQ} x(t)IP(t)B=HDq(t)(6)(11)qIQx(t)IP(t)B=HDq(t)qIQ uq(t)IUq(x(t)), qIQuq(t)IUq(x(t)), qIQ tI[ti,tf]tI[ti,tf] EAq=1, AqI(0,1)qIQࠧ෮ႵQ۱༢֥ཌྟࡆಃ֥EAq=1, AqI(0,1)ሹളթඣቋնࠇqIQڞ০ඣቋն.༢ऎႵࠒൔ܋ളթݦඔ֥ളթັ ഈඍଆᇏ,
ֻ4௹杨忠直,等:经济系统的生存微分博弈 595 opt=maxmin, Q1GQ2=Q, Q1GQ2=5 ଆ֥ࢳٚӱ(13)ބ(16)ژކؘ৯࿐ֻ ༢ളթັٳѰᒮଆ֥ࢳק,ੰູࠧਔ၂ሧჷطࣩᆚ֥۲༢֥ࣩᆚ৯ᆭק6ބູ0,ఃࣩᆚԩႿޙሑ,ሧჷ֥ٳղ֞ ഡHamiltonianݦඔູH܄ࢸ,෮Ⴕ༢֥ሹڞ০ඣቋն.(#)=U(x(t))+KT(t)F(x(t))(12)ᄵקࢳ่ࡱ:ູ3 ࢲ ე05HI=dU5x+5U+5uqdx5uq5uqЧ໓࣮֥؟۱ࣜ࠶༢ᄝঠၛളթ֥၂༎ಌሧჷߌᇏ֥ളթࣩᆚ໙,ีॖႋႨႿေ൧ӆa KTdF5x+5Fdx5uq5uqӁ൧ӆބᆟҦمܿ֩ჿඏ༯ᇭ؟ఒြ֥ളթࣩ ÛKI-5H=-dU-KTdF(13)ᆚaކቔࣩᆚaބ܋ԩა܋ؿᅚ֩໙ี,္ॖၛ5xdxdxႋႨႿ۽ӱࡹഡ༢aֹࣜ࠶ބݓࣜ࠶֩۲ᇕxÛIF(x(t)), 0dUI-Kࣜ࠶༢ᆭࡗ֥ࣩᆚაؿᅚ֩໙.ีႋႨ֥ܱᄝdxuႿᆌؓૄ၂োࣜ࠶ቆᆮܒᄯఃളթݦඔބᄎັٳqI@Uq(x(t)), tI[ti,tf]qIQ ק7 ሰࠢ{
Їݣ,ᄝႵུ౦ঃ༯ླေࡹ৫֥ҵٳЇݣ.ࢳuq(t),qIQ}<@Uq(x(t))ିqIQ໙ีᄵᄝႿ༢ളթݦඔބᄎٚӱ֥ඔ࿐ྙൔࠣ༢֥ളթѰᒮခቋႪ݅ཌࠢ{x(t)}<P(t)ᄎఃॖັྟބ,ྟ္၇ঠႿ༢ളթݦඔބᄎ,ᄵсေ่ࡱູັٳЇݣ֥ಒקྟࠇෛࠏྟ.ᄝ۲ᇕ౦ঃ༯,ᆺေࡹMU(u
q(t),q1IQ1;uq(t),q2IQ2;12৫ਔЧ໓෮ิԛ֥ଆაࢳٚӱ,ࠧ۴ऌ҂౦Q1GQ2=Q)\MU(u
q(t),qIQ)\ঃҕॉህ֥ࢳ࠹ෘٚم,ࣼॖ֤֞໙ี֥ࢳ.MU(
uq(t),q1IQ1;
uq(t),q2IQ2;12Q1GQ2=Q)(14)ҕॉ໓ང:ൔᇏ:
uq(t)Ir(x(t))B={
uq(t),qIQ}[1] :Anessayonthenatu-uq(t)IUq(x(t)), qIQralphilosophyofmodernbiology[M].NewYork:ק8 ഡHamiltonianݦඔູVintage,(#)=MU(x(t))+KT(t)F(x(t))(15)[2] ဗᇑᆰ.ఒြളթࣩᆚაࣉ߄࣮[D].฿ࣃ:ଲषնᄵקࢳ่ࡱ࿐ݓ࠽അ࿐ჽѰൖު࣮Бۡ,1999.:ູ5HdMU5x5MU[3] [M].Boston:Birkhauser,0I=++5uqdx5uq5uq1991:113,319,377.[4] :Aviabilityap- KTdF5x+5Fdx5uq5uqproach[M].Berlin:Springer-Verlag,1997:409.[5] [M].NewÛKI-5H=-dMU-KTdF(16)5xdxdxYork:JohnWiley&Sons,Inc,2000:228-229,233.[6] ဗᇑᆰ.ࡹഡཛଢࡓଢѓܿ߃ა॥ᇅٚمં࣮ÛdMUxIF(x(t)), 0I-Kdx[D].฿ࣃ:฿ࣃն࿐ܵ࿐ჽ,@Uq(x(t)), tI[ti,tf]qIQ (ഈࢤֻ591်)[6] CostaLdaF,RodriguesFA,TraviesoG,[J].PhysRevCharacterizationofcomplexnetworks:AsurveyofE,2004,70:[J].AdvancesinPhysics,2007,56(1):[9] MisloveA,MarconM,GummadiKP,[C]//[7] XuXP,HuJH,[J].CHAOS,2007,17::ACMPress,2007:[8] ClausetA,NewmanMEJ,-29-42.
