A New EDF Feasibility Test
25
0
0
Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. A New EDF Feasibility Test Bruno Gaujal , Nicolas Navet. No 5125 February 27, 2004. ISSN 0249-6399. ISRN INRIA/RR--5125--FR+ENG. ` THEME 1. apport de recherche.
(3)
(4)
(5) "!$#&%&')(+*#-,.*0/ 1325476 '8/9*0: 2 *<;0=?> @BA CEDGF+HJILKJMONFQPSR8TUFEVWNX0NVCEDGFON Y Z[S\FEVJ@]KW^_ K"PS`a`b[cZVdaFeZFOfA FEZfA Fg5hi8HQj<ikImlnFEoaZRnPSZXpjrq0s j<t<tSuvIwj8Hx`nPSyrFON z|{~}OQ?ar< FG`aZFONFEgVPkg FEPSyr[cZVAaD[cZ VFONVgayVA FvFQPcNoaVX[SWPNFEV [S]g [cg8 ZFOfERaZZFEgVVPcN8Ne[cZ\[co N.]VAkZFQPSVDGFefO[cg NVZPSg-VNJNfA FOd8RaFOdRag daFEZVA Fxl`b[cfEX PSZFONVWxFQPcd8g F l~ZNV$.@BA Fe`aZ[c`b[<NFOdkFQPcNoaV X VFONVAnPcNWPv[¡FEZ"fO[cD¢`aF.T0V XkVAnPSg VA Fe`aZFE£[cR NX 0g [Q]gVFONVQ¤@BA Fx¥ ZNVWNVFE`p[SVA F5PSyr[cZVAaD¦N]V[GfO[cg NVZR fEVJVA Fe§"PcNNF d8¨PSy<ZPSD[SVA F]NFEV[SbVPcN0N]A FEZFVA FB`nPSZV¨PSn[cZdaFEZ NdaF.¥ g FOd¢oX5VA FBg fER N[cg+ZFE¨PSV[cg [cg©VA FVPcN8NQª@BA FEg3sBVA FPSyr[cZVAaD«fO[cg NVZR fEVN+VA FpNA [cZVFONV+`nPSVA¬g¬P yrFO[cDGFEVZfQPS ZFE`aZFONFEg-VPSV[cg)PSVFQPcfAFE£rFE3[SVA F§"PcNNF"d8¨PSy<ZPSD¤"FE`&FEg d8gay+[cgkVA FxD+P?T8D+PSN[c`&F [S FQPcfAp`nPSVA3saVA FeNFEV[SVPcN8N]N]FEVA FEZFQPcNoaF[Sg [cVQ @BA Fv[cZNVfQPcNF+fO[cD¢`aF.T0V X®[S¯VAaNxFQPcNoaVX)VFONVedaFE`&FEg daN[cgNVZR fEVRaZPS¤fAnPSZPcf. VFEZNVfONG[S"VA FkNFEVG[S\[co N¢Ng fOFkV¢NvVA FNRaD°[S"VA F FE£rFEN¢g©VA Fk§"PcNNFUd8¨PSy<ZPSD AnPSVFE£rFEZWVA FNFEVW[S&\[co NQs VAaNJNo&FEVVFEZWVAnPSg)VA Fe¯[cZNVfQPcNF5fO[cD¢`aF.T0VFONx[SF.T0NVgay PS`a`aZ[rPcfA FONQ laRaZVA FEZDG[cZF<sFB`aZ[¡£daFWPJ`aZ[conPSoaNVfJPSgnPSX0NN [SbVA F]fO[cD¢`aF.T0V Xv]A FEg VA FVPcN8NPSZF ZPSg da[cD5l [cZY±[cNN[cgPSZZ£cPSNePSg d®F.T8`b[cg FEgV¨PSX)d8NVZoaRaVFOd®¨PSVFEg fEFON¡s ¯FeNA [¡²VAnPSVVA FePcNX0D¢`aV[cVfefO[cD¢`aF.T8VX [cZWPcNNFONNgayvFQPcNoaV XUN O(N log(N )) ³ N¢PSgg-VFEZFONVgayoX-`aZ[0d8R fEVQsVA FkPSyr[cZVAaD´`aZ[¡£daFONGPg FEµ¯PQX¶V[®daFEZ£rFUVA F o&FONVN`&FOFOdaN[SVA Fv`aZ[0fOFONN[cZeN[kPcNV[UD¢gaD¢·QF¢VA FvV[cVPSFEg FEZy<X)fO[cg NRaD¢`aV[cg¶]AaF DGFOFEVgay VA F]daFQPcd8g F[SbFQPcfAGVPcN&±@BA F]F.T PcfEVfO[cD¢`aF.T0V X¢[SVAaNPSyr[cZVAaD¸NRao8fERaoaf g+VA F]g-RaD5obFEZB[SbVPcN0N$N [Q¯FEZVAnPSg|VA FfO[cD¢`aF.T0V XG[SPS VA FWPSyr[cZVAaDGN¯N[c£gay VA F N$PSDGFx`aZ[coaFEDs 0g [Q]goXUVA FePSRaVA [cZN¡ ¹ºr»&¼½5¾ ?¿~}< KJFQPS@BDGF¯ÀX8NVFEDGNQsrÀ0fA FOd8Ragay s<l FQPcNoaVXrsr"X0gnPSD¢fÁ[cVPSyrF¯À0fQPSgay s PSZFONVWxFQPcd8g Fel~ZNVQ ÂnÅ8Ãî~ÄJïÊÅrÏÆOÌ0Ç.ÈW ÈSÉbêÅ-ÔàâxÊË8Ì Ì¡ð"ÍEÎ]ØñEÏÔÐÑÈQÐÓÒå?ÔÔ]ÕrÕ?Ö Ê7Ô]×Ø$ÐÑØÐÓÙÓÔ.ÝSÌSÇÎ.ÞÚ$òàÛ.×ÛEÕ?Ü]Ô]ÅrÐÑÆOØ"Ç.È-óÌSØÝ¡ÆOÞÔ$Ø$Ì ÈSôßàÍEÔ.ôQáõÂnÎeâöØØ$ÙÑÈSÐãÕ?ÇOäÔÞå?å?ñOÈSÞÇ?Ô.á ÌaæÝ?ØÞå$ØçÈSØßÐéè~Ô$á"ÔÈS êɨâÐëÆOØ$Ç.ÙÓÈ-Ðãá ìíÞÃ~ÙÓßÇ$É ÐéØêá ÃØñEÔà× è~ÐÓÇ$ÞÙÑØQá ìéÞ ∗. †. ∗. †. Unit´e de recherche INRIA Lorraine Technopˆole de Nancy-Brabois, Campus scientifique, ` NANCY (France) 615 rue de Jardin Botanique, BP 101, 54600 VILLERS LES T´el´ephone : 03 83 59 30 30 - International : +33 3 3 83 59 30 30 T´el´ecopie : 03 83 27 83 19 - International : +33 3 83 27 83 19 Antenne de Metz, technopˆole de Metz 2000, 4 rue Marconi, 55070 METZ T´el´ephone : 03 87 20 35 00 - International: +33 3 87 20 35 00 T´el´ecopie : 03 87 76 39 77 - International : +33 3 87 76 39 77.
(6) j.
(7) "!$# &% '( ")+*,. Ê9Ã~ïÊÏ.
(8)
(9) 5
(10) O
(11)
(12) ]v . } [cR N`aZMONFEgV[cg N¯Rag+g [cRa£rFEbPSyr[cZVAaDGF"`&[cRaZVFONVFEZ¨PxPSNPSoaVMWd RagUFEg8 NFED oaFpdaFpV cfA FONUg [cg8ZMOfERaZZFEg-VFONk[cZda[cgagnPSg fOMOFON N[cR N|xlB 3FpVFONV+`aZ[c`&[<NM Rag F fO[cD¢`aF.T8VM`aR NkPSoaF"!R F F VFONVfO[cgag-R \R N#!R$% `aZMONFEgVQ P `aZFED¢CEZF¶MEVPS`&F daF &ÓPSyr[cZVAaDGFpFONVdaF fO[cg NVZRaZF F d8¨PSy<ZPSD¢DGFkdaFk§xPcNNFUdaFU&ëFEg NFED oaFkdaFONvV cfA FON¢[(' &ë[cZd8ZF `nPSZVFE±FONVJdaM.¥ ga`nPSZ¨PGZFE¨PSV[cgd g fER N[cg)NRaZFONV cfA FONQ) *ÓPSyr[cZVAaDGFfO[cg8 NVZRaV+PS[cZN¢F `aR NGfO[cRaZV+fA FED¢g¬d Rag F P(+O[cg yrMO[cDGMEVZ,!-R F`b[cRaZGfAnP(!-R Fkga£rFQPSR¬d8R d8¨PSy<ZPSD¢DGF¢daFv§xPcNNF<x g®[cg fEV[cg daF5¨P+`&FEg-VFvD+P?T8D+PSFdaFvfAnPcfERagdaFONx`aR N"fO[cRaZVN fA FED¢g NQsa&ëFEg NFED5oaF daFV cfA FONWFONVBPSNPSoaFe[cRkg [cg3 P]fO[cD¢`aF.T8VMd PSg N3F`aZFfQPcNdaME`bFEg dedaF fQPSZPcfEVMEZNV,!-R FON~NVZR fEVRaZFEFONdaF&ëFEg NFED oaF daFONeV cfA FON fQPSZ5FEFGFONVx[cg fEV[cg¶d8R g [cD oaZF|daF¢V cfA FON kfAnPcfERag¶daFON ga£rFQPSR8Td8Rd8¨P? y<ZPSD¢DGFkdaF §xPcNNF<"-JR FE!R F|N[cVv&ëFEg NFED5oaFkdaFUV cfA FONQs¤¨PfO[cD¢`aF.T8VMkd PSg NFU`aZF fQPcN"FONV`aR NJPSoaF !R F fOFEF5daFON"PS`a`aZ[0fA FONxF.T8NVPSg-VFONQW"F `aR NQsg [cR NW`aZMONFEgV[cg N"Rag F PSgnPSX8NF`aZ[conPSoaNVF.!RnPSg d)FON"fQPSZPcfEVMEZNV,!-R FONedaFONxV cfA FONeN[cgVPSMQPSV[cZFON¡ Y~[cRaZedaFON PSZZ£rMOFON~Y~[cNN[cgaFEgag FONFEV±daFON~MOfA MQPSg fOFONF.T8`b[cg FEgVFEFEDGFEg-Vd8NVZoaR MOFONQsSg [cR N±DG[cg-VZ[cg N !R FJ¨P fO[cD¢`aF.T8VMPcNXD¢`aV[cV,!R F"`b[cRaZBdaMOfEdaFEZdaFJ¨PPSNPSoaVMxFONV O(N log(N )) [(' N FONV]Fxg [cD oaZF daFeV cfA FONQ /Wg FD¢`afQPSV[cg¢g-VMEZFONN$PSg-VF]daF¯fOFEVPSyr[cZVAaDGF]FONV0!R$ -[cRaZgaVRag F¤g [cRa£rFEFP(+O[cg daFJfQPSfERaFEZFONDGFEFERaZFONB£0VFONNFON¤d8R|`aZ[fOFONNFERaZ`&[cRaZ¤D¢gaD¢NFEZ¨P fO[cg N[cD¢D+PSV[cg V[S VPSFBdëMEg FEZy<F]V[cRaVFEgvZFON`&FOfEVPSg-V¤FON MOfA MQPSg fOFONQ0 PxfO[cD¢`aF.T8VMF.T PcfEVF¯daFB&ÓPSyr[cZVAaDGF<s g8MEZFERaZF1 O(N ) sFONV]`aR N]PSoaF !R Fx¨P¢fO[cD¢`aF.T8VM5daFONWPSyr[cZVAaDGFONxfO[cgagR NW`nPSZFON PSRaVFERaZNW`b[cRaZ]ZMON[cR d8ZFFD32EDGF`aZ[coaCEDGF< 4 ¾ Q} ¼ 65 ÀX0NVCEDGFON @FED¢` NKWMOFEs _JZda[cgagnPSg fOFEDGFEg-VQs ³ d PS`aVPSV[cg "X0gnPSD¢,!-R F+d8R Á[cVPSyrF<sn¤PSZFONVJxFQPcd8g Fel~ZNVQ 3.
(13) u.
(14) . kº
(15) 9¾ ®¾ º ¾ { 5 º @3[eD¢`b[cZVPSgV`aZ[coaFEDGNZFE¨PSVFOd¢V[xVA F]NfA FOd8Ragay [S PvNFEVW[S~g daFE`&FEg daFEg-V\[co N]Rag daFEZWxl PSZF Pcdad8ZFONNFOdkgVAaNWNVR d8XH¡BVA FxFQPcNoaVX PSgnPSX8NNWPSg djrVA FD¢gaD¢·¡PSV[cg[SVA FeFEg FEZy<XkfO[cg NRaD¢`aV[cg3 T0NVgay N[cRaV[cg NUAnPO£rF)NA [¡]g¬VAnPSV|VA FONFp`aZ[coaFEDGN PSZF)fE[<NFEX fO[cZZFE¨PSVFOd"VA F [c`aVD+PS"FEg FEZy<X D¢gaD¢·¡PSV[cg PSyr[cZVAaD Z[c D Pc[¶FEV|PS JFED o&FOdaN|VA F FQPcNoaVX PSgnPSX8NNZ[cD À`aRaZ! Ó"i JPSVA [cRay<A o&[cVA ZFONRaVN¢AnPQ£rFkobFOFEg daFE£rFE[c`&FOd g daFE`&FEg daFEg-VXr @BAaN]fAnPSZPcfEVFEZNVfVAnPSV]VA FxN[cRaV[cg N][S~VA FJV [¢`aZ[coaFEDGNJPSZFJVy<AVX|garFOd ZFED+PSg N VZR Fe]VAVA Fxg FE PSyr[cZVAaDGN"daFE£rFE[c`bFOdkgkVAaNJNVR d8Xr #%$& }O ' &( ½5¾ ) ^* g Ó"i sJPSg²PSyr[cZVAaD«[cZ VFONVgay VA FpFQPcNoaV X¬[SPNFEV [S\[co N NfA FOd8RaFOd¬Rag daFEZUxl N+`aZFONFEgVFOd "V|]JobFfQPSFOd©VA F À`aRaZ&ëNUPSyr[cZVAaD«g¬VA F [c[Q]gay)Ng fOF¢V5ZFEFON [cg Pk@BA FO[cZFED NVPSVFOd oX À`aRaZJ@BA FO[cZFE D +8Ói `a,` +-+ .g Ó"i 9. @BA FWfO[cD¢`aF.T0V XG[S&VAaN¯PSyr[cZVAaDN O(N ) ]A FEZF N NVA FfQPSZd8gnPS&[SbVA FJNFEV[S8\[co N¡ @BA F|NVR d8X `aRaoaNA FOd¶/g " ¯`aZ[¡£daFON¡s~[cZ5g daFE`bFEg daFEgVVPcN0N5]VAdaFQPcd8g FON¡sVA F ¤"l NfA FOd8RaFkVAnPSVGD¢gaD¢·QFON|FEg FEZy<XR N$PSyrF ]AaFFEg NRaZgay VAnPSV+daFQPcd8g FONUPSZF g [cV D¢NNFOd @BAaN5PSyr[cZVAaDsVFEZDGFOdVA F Pc[ ëNPSyr[cZVAaDs~ZRag N5g O(N ) g¶VA F+[cZNV fQPcNFJ7[cZ±DG[cZF¯daFEVPSN [cg5VA F¯fO[cD¢`aF.T8VX5PSgnPSX8NNQsS`aFQPcNF¯ZF.FEZV[JVA FFEg d[S&À0FOfEV[cg¢jr. @BA F Pc[ ëNePSyr[cZVAaDPSN[U`aZ[¡£daFON PGFQPcNoaVX)VFONVxRag daFEZxl¬Ng fOF<s&gVRaV£rFEXrs3 [cZ"DGFOFEVgaydaFQPcd8g FONxVA 1F 0¯Y / N`&FOFOdpg FOFOdaNxV[Uo&F y<ZFQPSVFEZVAnPSg®VN"D+P?T8D RaD N`&FOFOd VA FEgpVA FNFEVW[Sn\[co N]N]g [cVFQPcNoaF< KWFOfOFEg-VXrs P g FE PS`a`aZ[rPcfA [cZ¯N[c£0gayob[cVAU`aZ[coaFEDGNBRag daFEZBVA FWy<AV[S±P NA [cZVFONV `nPSVA)`aZ[coaFED AnPcNWo&FOFEg)`aZ[c`b[<NFOd2g +" ~oaRaV"VJNJZFONVZfEVFOdV[+VPcN0NJ]VA)l^àl±_ ZFQPS VDGFfO[cg NVZPSgVNv a ≤ a → d ≤ d .l [cZ"NR fA)VPcN8NQsaFQPcNoaV XVFONVgaykPSg dpFEg FEZy<X D¢gaD¢·¡PSV[cg)fQPSgo&FN[c£rFOdkg O(N log(N )) 3 4 ¾ Q {03 94¾ ¾ 5º º 6 @BAaN¯[cZeN±VA FfO[cgVg-RnPSV[cg+[S& +" 8]A FEZF¤VA F¯N`&FOfE¨PS fQPcNFG[Sl~^àl±_µfO[cg NVZPSg-VNAnPcNeo&FOFEg¶VZFQPSVFOd §"FEZF<s~PkN[cRaV[cg¶V[VA FGyrFEg FEZPS¯fQPcNFGN `aZ[¡£daFOd 8s 7+ $VA FUl~^àl±_¦PcNNRaD¢`aV[cgN ZFEDG[¡£rFOd [cVF|VAnPSVVA F|fO[cD¢`aF.T8VX[SVA F PSyr[cZVAaDGNdaFE`&FEg daN±[cgNVZR fEVRaZPS8`aZ[c`bFEZVFON~[S VA F¤NFEV±[S<\[co NQ@BA FF.TaPcfEVfO[cD¢`aF.T0VFON [SJVA FPSyr[cZVAaDGNG[cg Pp`nPSZVfERa¨PSZGNFEVv[S¤\[co NvN`aZ[Q£0daFOd PSg d s]AnPSVFE£rFEZ¢VA F NFEV¢[S \[co NQsnVWNJobFEVVFEZJVAnPSgVA FfO[cD¢`aF.T0VFONx[SF.T8NVgay PS`a`aZ[rPcfA FONGNRao8 !RnPcd8ZPSVf gVA F gRaD o&FEZ[S&VPcN8N[cZFQPcNoaV XGPSg dGNRao8fERaoaf][cZFEg FEZy<XvD¢gaD¢·¡PSV[cg.laRaZVA FEZDG[cZF<s ¯Fv`aZ[Q£0daFGP|`aZ[conPSoaNVf+PSgnPSX8NN[SVA F¢fO[cD¢`aF.T8VFON]A FEg VA FVPcN√8NePSZF5ZPSg da[cD FkNA [Q VAnPSV¢VA FPO£rFEZPSyrFpfO[cD¢`aF.T0V XN|PS¯POX8NGND+PSFEZ¢VAnPSV O(N N ) §J[¡FE£rFEZ VAaNBo&[cRag d NBg [cV]Vy<A-VWobFOfQPSR NF"Vda[0FONBg [cV]VPSrFxg-V[GPcfOfO[cRagVVA F"PcfEV]VAnPSVdaFQPcd8g FON . 2. 3. i. j. i. j. Ê9Ã~ïÊÏ.
(16) i. 7
(17) 7
(18) 7 . AnPQ£rF]V[xo&F¨PSZyrFEZ¤VAnPSg+PSZZ£cPSaVDGFONQ^ g¢VA F]fQPcNFB[S&Y~[cNN[cg+PSZZ£<PSN¤PSg dGF.T8`&[cg FEg-V¨PSX d8NVZoaRaVFOd ¨PSVFEg fEFONQsc¯FBNA [QVAnPSV±VA F¯PcNX0D¢`aV[cVf¯fO[cD¢`aF.T8VXe[cZPcNNFONNgayWFQPcNoaVX N O(N log(N )) 4 ¾ ¿ º 5 ¾ º } » }O º ´ FfO[cg NdaFEZpP Ngay<F 0¯Y*/ daFOd8fQPSVFOd V[VA F®F.TaFOfERaV[cg [S]PkP|¥ gaVF¢NFEVe[S¯N[cDGFvg [cg8ZFOfERaZZFEg-Vg daFE`bFEg daFEgVeVPcN8NG[cZ \[co N àJ]VA ZFQPSVDGF fO[cg NVZPSg-VNQ@BA 1F 0¯Y*/ `aZ[0fOFONNgay N`&FOFOd fQPSg)£cPSZX[Q£rFEZeP+fO[cgVg-R [cR NZPSgayrF5Z[cD V[ 1 9VA F+D+P?T8D+PS]N`bFOFOd[SWVA F|`aZ[fOFONN[cuZ5V[¥aTaFOd¶V[HU]VA g [)[<NN[SWyrFEg FEZPSV 4X 0 VAaN|fQPSg©o&FPcfAaFE£rFOd¬o-X©P®ZF.NfQPSgay [SVA FpN·QFp[SxVA F¢\[co N$.ª@BA FpNFEV+[SVPcN0NGN PSg dªFQPcfA VPcN NUfAnPSZPcfEVFEZ·QFOd oXªP VZ`aFEV ]A FEZF T = {τ , . . . , τ } VA F !RnPSg-VVFON a , s , d ZFON`bFOfEV£rFEXkτdaFEg [cVFVA FePSZZ£<PS3VDGF<s8VA FeN·QFG(a9V,DGsFx, dg FO)FOdaFOdpV[ F.TaFOfERaVFVA FxVPcNkPSVVA FxD+P?T8D RaD¸N`bFOFOdnPSg d VA FedaFQPcd8g F [SVPcN τ ¢ (
(19) 8 74¾ ¸¾ *º º 6 @BA F Pc[ ëNpPSyr[cZVAaD [cZFEg FEZy<X¬D¢gaD¢·¡PSV[cg N ZFED¢g daFOdpg)À0FOfEV[cg)j8BÀ0FOfEV[cg +vNWdaFE£r[cVFOdV[G[cRaZ`aZ[c`&[<NPS[SFQPcNoaVX VFONVWRag daFEZ VA FUxl²`&[cfEXr)@BA F|fO[cZZFOfEVg FONN¢[SVA F PSyr[cZVAaD´N5`aZ[Q£rFOd PSg dVN5fO[cD¢`aF.T8VX N PcNNFONNFOd5ob[cVAvdaFEVFEZD¢gaNVfQPSXGPSg d`aZ[conPSoaNVfQPSXr l~gnPSXrs<g¢NFOfEV[cgvu srPSg¢[cZy<gnPS PSyr[cZVAaD«V[ VA Fp`aZ[coaFED«[S D¢gaD¢·Ogay VA F)FEg FEZy<X©fO[cg NRaD¢`aV[cg Rag daFEZ|ZFQPSVDGF fO[cg NVZPSg-VNN`aZ[c`b[<NFOd eab b b ^ gvVAaNÀ0FOfEV[cg3sFF.T8`a¨PSg¢VA F Pc[ ëN¤PSyr[cZVAaD [cZFEg FEZy<XvD¢gaD¢·¡PSV[cg3 FfA [<NF]V[ `aZFONFEg-V"VAaNJfO[cg NVZR fEV[cg)]VA9Ra~daFEVPSNJN[+VAnPSVWVA F5d8 &FEZFEg fOFON"]VA[cRaZPS`a`aZ[rPcfA fQPSgo&FFQPcNX Aay<Aay<A-VFOdGPSg d¢PSN["Ng fOFNFE£rFEZPS8fO[cg fOFE`aVNPSg d5g [cVPSV[cg N `aZFONFEgVFOdA FEZF ]o&FR NFOd)g®NRao NF !-R FEgV À0FOfEV[cg NQ [cVFVAnPSV"VA F+À`aRaZ&ëNePSyr[cZVAaD [cZ"FQPcNoaVX NVA F"¥ ZNVVFEZPSV[cgp[SVA F[0[c`[S~VA F PSyr[cZVAaD¦gk`nPSZPSy<ZPS`aA)j8Ój8 @BA F¢N`bFOFOd [S¯VA F 0¯Y*/µPSVVDGF NdaFEg [cVFOd FvZFOfQPSVA F¢fO[cg NVZR fEV[cg [S VA F [c`aVD+PSRag fEV[cg v (t) y<£rFEgpo-tX Pc[|FEVJPS¯v(t) g " ¯@BAaNJfO[cg NVZR fEV[cg)D¢gaD¢·QFON FEg FEZy<X®fO[cg NRaD¢`aV[cg ]AaF¢FEg NRaZgayVAnPSVeg [kdaFQPcd8g F¢] o&FvD¢NNFOd ]PcNfQPSXrsVA F PSyr[cZVAaD [cZ8N5o-X daFEg-VX0gaypVA FGVDGF|g-VFEZ£<PSs~VFEZDGFOd¶VA F|fEZVfQPS¤gVFEZ£cPSs [¡£rFEZ ]AafA VA FvAay<A FONV`aZ[0fOFONNgaykN`&FOFOdNxZF !RaZFOd @BA F5[¡FONVPcd8D¢NNoaFZF !-R FEg fEXN fO[cD¢`aRaVFOdk[cZ]VAaN]gVFEZ£cPSs VA FxVPcN0NobFE[cgay<gay¢V[¢VAaNBgVFEZ£cPSB9ëF<PSZZ£cPS±d PSVFePSg d daFQPcd8g F¯g NdaFVA Fg-VFEZ£<PS7±PSZFVA FEgZFEDG[Q£rFOd9VAaNNVFE`NVFEZDGFOd VDGF¯fO[cD¢`aZFONN[cg gkVA F[c[¡]gayPSg dpPvNRao8`aZ[coaFED¦NfO[cg NVZR fEVFOdk]VApVA FZFED+PSgagay+VPcN8NQ . 1. N i. . i. i. i. i. i. i. i. . . . . . . !#" $. &%. '. ï±ï È OôQõð$ô *). (.
(20)
(21) .
(22) . "§ FEZFG¯F|F.T8`a¨PSgA [QV[pNRa`a`aZFONN5VA F|g-VFEZ£<PS [a, d] Z[cD VA F+VDGF.g FkPSg dA [QµV[ DG[0d8X|VA FeNFEV[S~VPcN8NWPcfOfO[cZd8gay<Xr¯^V]AnPcNWobFOFEgfA [<NFEgV[+daFEg [cVFVA Fo&[cRag daNW[SVA F gVFEZ£cPSoX a PSg d d Ng fOF<sg)VA F5[c[¡]gay s a PSg d d ]~g FOfOFONNPSZXpobFvPSZZ£cPSd PSVFON PSg dpdaFQPcd8g F d PSVFONZFON`bFOfEV£rFEXr@BA Fg [cVPSV[cg τ ⊆ [a, d] DGFQPSg NWVAnPSVVPcN τ o&FE[cgayrN V[ . A FEgNRa`a`aZFONNgay [a, d] sVA F+VPcNNFEV5N DG[0d8¥nFOd¶g VA F [c[a,[Q]d]gay+aD+PS≥gag a,FEZ d ≤ d VA FEg τ N]ZFEDG[¡£rFOd s • τ ⊆ [a, d] [cZWPS3ZFED+PSgagay|VPcN8NQs s N]Rag fAnPSgayrFOd)PSg d • a ∈ [a, d] VA FEg a := a s FENFx a ≥ d VA FEg a := a − (d − a) s d ∈ [a, d] VA FEg d := a s FENFx d ≥ d VA FEg d := d − (d − a) @BA FeVDGF.fO[cD¢`aZFONN[cgp`aZ[0fOFOd8RaZFN]R NVZPSVFOdkgpl~y<RaZFGH< i. i. i. i. i. i. i. i. i. i. i. i. i. i. i. i. i. a d ai di ai+1 di+1 ai+2 di+2 ai+3 di+3. ai+4 di+4 t. ai di ai+1 di+1 di+3 ai+4 di+4 ai+3. t. l~y<RaZFH- TaPSD¢`aF [S]VDGF|fO[cD¢`aZFONN[cg3 A FEgNRa`a`aZFONNgaypVA F+g-VFEZ£<PS VPcN τ N]ZFEDG[Q£rFOd9Z[cD VA F NFEV[S~VPcN0NQ . [a, b]. s~VA F. i+2. . !"# %$&'(*)+. @BA 8F Pc[ ëN]PSyr[cZVAaDfQPSgUobFJdaFONfEZo&FOdGgkPefO[cg-£rFEgaFEgVB¯POX|R Ngay VA F[c[¡]gay-,EfO[cDv `&[<NV[cg,[S~Rag fEV[cg NQ 3FEV f PSg d g o&FxZFQPS3Rag fEV[cg N]VAPS`a`aZ[c`aZ¨PSVFda[cD+PSg NJPSg d FEV [a, d] o&FePVDGFg-VFEZ£<PSs VA FEg9Rag fEV[cg g ∧ f NWdaF.¥ g FOdPcNB[c[QN [a,d]. Ê9Ã~ïÊÏ.
(23) q. 7
(24) 7
(25) 7 . t < a s8VA FEg g ∧ f (t) = g(t) s s VA FEg g ∧ f (t) = f (t) s • t ∈ [a, d] saVA FEg g ∧ f (t) = g(t − (d − a)) • t>d [cVF VAnPSVJVAaNx[c`&FEZPSV[cgNJg [cVxPcNN[fE¨PSV£rFPSg dpFQPQ£rFONxD¢`afEVJVA Fda[cD+PSg®[SPS Rag fEV[cg Ng£r[c£rFOd ¤^~¯FedaF.¥ g F h := g ∧ f VAaNDGFQPSg N]VAnPSV f NdaF.¥ g FOdp[cg [a, d] s N¯daF.¥ g FOdU[¡£rFEZPSg|gVFEZ£cPS&g fER d8gay a s0NPQX [b, c] saPSg d h N¯daF.¥ g FOd [Q£rFEZ [b, c + d − a] g Pc[kFEV PS5daF.¥ g F¢VA F ,.gVFEg NV%X ,¢[S]PSg g-VFEZ£<PSPcNxVA Fv[cZ0[rPcd oaZ[cRay<A-V o-X)VA F VPcN0NobFE[cgay<gayvV[5VA F"g-VFEZ£<PS&d8£0daFOdUo-XGVA F"FEgay<VAk[S3VA F"g-VFEZ£<PS^ g-VRaV£rFEXrs8V¯N VA FxND+PSFONVBPSDG[cRagV[S[cZ|VAnPSV¯AnPcN¯V[o&FJda[cg Fxd8RaZgayvVA F"g-VFEZ£<PS&[cZ¯DGFOFEVgayvVA F VD¢gay|fO[cg NVZPSg-VNQ@BA Fg-VFEg NVXk[Q£rFEZ [a, d] NdaFEg [cVFOd W PSg dkV]NF !RnPS3V[ •. [a,d]. [a,d]. [a,d]. [a,d]. [a,d]. W[a,d] =. P. {τk ⊆[a,d]} sk. @BA FBPSyr[cZVAaD `aZ[c`&[<NFOdg afO[cg NVZR fEVN VA F9Rag fEV[cg ]AafA¢N~VA F][c`aVD+PS N`bFOFOd|[cZ¯D¢gaD¢·Ogay+VA FJV[cVPS3FEg FEZy<X fO[cg NRaD¢`aV[cg]AaFxvZFON`b(t)FOfEVgayGdaFQPcd8g FONQ¤@BAaN PSyr[cZVAaD¸fQPSgkobFdaFOfO[cD¢`&[<NFOdpgkVA Fx[c[¡]gay+NVFE` N H< n := 1 j8% 0¯[cD¢`aRaVFeVA FfEZVfQPS±g-VFEZ£<PS I := [a , d ] ]A FEZF a PSg d d PSZFeNR fAVAnPSV H¡ W = max W . ^g H¡Esb[cgaX£cPSR FON"[S a PSg d d VAnPSVePSZF5ZFON`&FOfEV£rFEXPSZZ£cPSd PSVFONxPSg ddaFQPcd8g F d PSVFONAnPQ£rFV[o&F]fO[cg NdaFEZFOd+Ng fOF W `&[<NNFONNFON[fQPSnD+P?T0D+P[cg|NR fAG`&[cg-VN¡ +8J_J£rFEZVA FefEZVfQPS3gVFEZ£cPSPSVNVFE` s saVA F"9Rag fEV[cg f NfO[cg NVPSg-V n I = [a , d ] d−a. !#" $. n. i. [ai ,dj ]. j. i. 0≤a≤d. j. [a,d]. [a,d] n. i. j. n. fn (t) := W[ai ,dj ] .. u /JNgaykVDGF.fO[cD¢`aZFONN[cg3s I NZFEDG[Q£rFOd s±FEV n := n + 1 s ZFEVRaZgV[kNVFE`¶jU]PSV FQPcNVW[cg FxVPcNUZFED+PSg NW[cZWyr[vV[GNVFE`kuG[cVA FEZ]NF< i8laRag fEV[cg v NfO[cD¢`aFEVFEXkdaF.¥ g FOd v := f ∧ (· · · ∧ f ) n. !#" $. ï±ï È OôQõð$ô *). !#" $. n. In−1. I1. 1.
(26)
(27) . @BA F+PSob[¡£rF+PSyr[cZVAaDN fERaoaf¢gVA F¢gRaD o&FEZ5[SBVPcN0N VA FEZF+PSZF|PSVDG[<NV .F TaFOfERaV[cg|[SNVFE`|j]AafA|ZRag Ng O(N ) Ng fOF[cZFQPcfAkPSZZN£cPSbd PSVFVA FEZFxPSZFJPSV¤DG[<NNV daFQPcd8g FON]V[vfO[cg NdaFEZQ^ g " s0VA FePSRaVA [cZNfE¨PSDs8]VA [cRaV]DG[cZFdaFEVPSNQs8VAnPSVBR Ngay àNPxNRaVPSoaF]d PSVPxNVZR fEVRaZFNR fA+PcNVA F]NFEy<DGFEg-VVZFOF s-VA FBZRagagagayeVDGF]fQPSg¢o&FBZFOd8R fOFOd V[ O(N log (N )) §J[¡FE£rFEZQs&VAaNx¯PcN"g FE£rFEZxZFQPSXPcfAaFE£rFOdPcNJDGFEgV[cg FOd®,g Ñq F PcfEVRnPSXda[+g [cV0g [Q A [¡ V[+[coaVPSg®PSgD¢`aFEDGFEgVPSV[cg ]VApVA FEZxPSyr[cZVAaD¦gFONN VAnPSg O(N ) PSg d ¯F5PSZFxg [cVJPQ¯PSZF5[S PSgXU`nPS`&FEZ]gVA FxVFEZPSVRaZF VAnPSVAnPcNWPcdad8ZFONNFOd VA Fx`aZ[coaFED¦]VAPvfO[cD¢`aF.T8VXk[Q¯FEZWVAnPSg O(N ) @BA F ZFONVG[SJVA F `nPS`&FEZ¢NGPSDG[<NVGFEg-VZFEX daFE£r[cVFOd V[®VA F `aZFONFEg-VPSV[cg©[SePg FE PSyr[cZVAaD ]VA¬P NRao8 !RnPcd8ZPSVffO[cD¢`aF.T8VX V[fO[cD¢`aRaVF)P9Rag fEV[cg¬]AafA©fO[cg fEdaF ]VA v [cg©VA FfEZVfQPSJgVFEZ£cPS @BAaN+]J`aZ[Q£0daF)P FQPcNoaV X VFONV+Rag daFEZUxl g)NRao8 !RnPcd8ZPSVf5VDGF ]AafA®NxPSgpD¢`aZ[Q£rFEDGFEgVe[Q£rFEZxF.T0NVgay PSyr[cZVAaDGN"ZRagagagay g !RnPcd8ZPSVfVDGF+ 8s"i 9.
(28) b
(29)
(30) O
(31) @BA F|PSyr[cZVAaD `aZFONFEg-VFOd¶A FEZF¢ZFEFON [cg VA F+fO[cg NVZR fEV[cg [SBVA FG§xPcNNFvd8¨PSy<ZPSD [SP `nPSZV¨PSX|[cZdaFEZFOd|NFEVx9`b[<NFEV$¤PSg d|[cg|VA FJN[cRaV[cgU[S±P NA [cZVFONV¯`nPSVA|`aZ[coaFEDVAnPSV¯AnPcN o&FOFEg`aZ[c`&[<NFOdg + @BA FONFeV¯[+`&[cg-VNW]o&Fx¥ ZNVZFED¢g daFOd F VA FEgp`aZ[¡£daF VA F fO[cZF[S0VA F¯PSyr[cZVAaDsc`aZ[Q£rF¤VNfO[cZZFOfEVg FONNPSg de¯F¤FE£<PSRnPSVFVNPSyr[cZVAaD¢f¯fO[cD¢`aF.T0V X o&[cVAkgPvdaFEVFEZD¢gaNVfPSg dkgpPv`aZ[conPSoaNVfe¯POXr 2. . . 2. 3. 3. !#" $. . . + % &' %. FEVWR NfO[cg NdaFEZJVA Fe`&[<NFEV P [cgVA FeNFEVW[S VPcN8N]]VApVA Fe`nPSZV¨PS±[cZdaFEZWo&FEgay|daF.¥ g FOd oXVA F NVZfEVg fER N[cg [SWVA FUg-VFEZ£<PSN P = {[a , d ], (} s ]A FEZF [a, d] ( [a , d ] PSg d d > d ³ ¥ gaVFG`&[<NFEV fQPSg obFGZFE`aZFONFEg-VFOd o-X¶Pk§xPcNNF+d8¨PSy<ZPSD NFOF a <a [cZWDG[cZFdaFEVPSN$JPSg dpVA F FE£rFEN"gVA F5d8¨PSy<ZPSD PSZFdaFEg [cVFOd L , · · · , L ]A FEZF K N VA FvAay<A FONVFE£rFE¢_Jg F¢daFEg [cVFONeo-X L(i) VA FvFE£rFE[SVA FvgVFEZ£cPS [a , d ] g P s&V] o&F"fQPSFOd VA FJFE£rFE [SVPcN τ @BA FxNFEVB[S±VPcN0N¯VAnPSVB]&obF"fO[cg NdaFEZFOdU[cZR NVZPSV[cg `aRaZ`&[<NFxgVA F"[c[Q]gay|N 0. i. 0. 0. i. 1. 0. K. i. i. i. Ê9Ã~ïÊÏ.
(32) 7
(33) 7
(34) 7 . t d<j j s+ j HQt H u H i H+ u H H H<H HQi j HQi jSu H @BA FU§xPcNNF+d8¨PSy<ZPSD PcNN[fE¨PSVFOd ]VAVAaNNFEV[SWVPcN0N5N y<£rFEg g l~y<RaZF jpoe]AaF l~y<RaZF j¢Pr]NA [QN]A [¡ VA FxgVFEZ£cPSN [a , d ] [Q£rFEZ¨PS`3 a. τ1 τ2 τ3 τ4 τ5 τ6 τ7. . i. i. τ1 τ2 τ3 τ4. a). τ5 τ5 τ7. τ3. τ6. Level 3. τ5. Level 2. τ4 τ2. b). τ1. τ7. Level 1. l~y<RaZF j BÀRao8¥ y<RaZFPrBZFE`aZFONFEg-VNJVA Fe[Q£rFEZ¨PS`a`agay [S VA Fg-VFEZ£<PSN [a , d ] ]AaF NRao8 ¥ y<RaZFeo¯NA [QNWVA FefO[cZZFON`&[cg d8gay+§"PcNNFd8¨PSy<ZPSD Fxg [Q F.T8`a¨PSgA [Q V[¢fO[cD¢`aRaVFVA FxFE£rFENW[S PS VA FVPcN8NBgkVA F§"PcNNF"d8¨PSy<ZPSD g O(N log(N )) FEFEDGFEg-VPSZX[c`&FEZPSV[cg NQ ³ gPSZZPOX ] o&FR NFOd®V[ NV[cZFvVA FvFE£rFENe[SVA FvVPcN01N NVA FFE£rFE¤[S τ ³ gUPSZZPQX T NVL[cZFONVA FW£<PSR Fx[S3¨PSZyrFONV]daFQPcd8g F"[cgkP y<£rFEg|FE£rFEL[i] @BA FxPSZZPOX8N PSg d PSZFxVFEZPSV£rFEXfO[cg NVZR fEVFOdpd8RaZgayGVA FeF.TaFOfERaV[cgp[SVA FePSyr[cZVAaD y<£rFEgo&FE[QL T H<KWFO[cZdaFEZVA FJNFEV[S3VPcN0NN[eVAnPSVBPSnVA FWg daF.T8FON¯[S3VA FVPcN8NPSZFJy<£rFEgUPcfOfO[cZd8gay V[¢VA FePSZZ£<PS3VDGFON a ]A FEZFeVFONWPSZFoaZ[crFEgR Ngay+VA FedaFQPcd8g FONQ i. i. i. i. ï±ï È OôQõð$ô *).
(35) HQt.
(36) . j8 T [0] := ∞ s T [1] := d s L[1] := 1 PSg d [cZWPS k = 1..N NFEV T [k] := 0 8+ l [cZ i Z[cD¦jV[ N da[ Pr "fA [cV[cD XàÀ0FQPSZfA [S d gªVA F)fERaZZFEg-V T y<£rFON k s]VA F)g daF.T¬NR fAªVAnPSV T [k] > d ≥ T [k + 1] 9o T [k + 1] := d L[i] = k + 1 [cVFxVAnPSVVA F d8fA [cV[cD¢fNFQPSZfAN£cPSdko&FOfQPSR NF NN[cZVFOdkgkg fEZFQPcNgay|[cZdaFEZQ @BA F fO[cZZFOfEVg FONNJ[S VAaNJfO[cg NVZR fEV[cg)fO[cDGFONWZ[cD VA FTPcfEVVAnPSVJPS±g-VFEZ£<PS±[cZdaFEZNvPcN A FEZF¡PSZFxV[|d8DGFEg N[cgnPS¯NFOF"[cZF.T PSD¢`aF Ój 9. @BA FpV[cVPSxfO[cD¢`aF.T0V X¬[SeVA F®PSyr[cZVAaD«N O(N log(N )) )VA FpgaV¨PSN[cZV|NUg PSg d¶FQPcfA d8fA [cV[cD5XR NFON O(log(N )) [c`&FEZPSV[cg N5PSg d¶VeNZFE`&FQPSVFOd N O(N log(N )) VDGFON¡ 1. i. i. i. . + + ) ) )+
(37) )
(38) ) . ^ g +" s]A FEZFVA FJfQPcNF K = 1 N¤VZFQPSVFOd)l^àl±_ VPcN8N$.s-VA FJPSyr[cZVAaDg-£r[c£rFON¯VA FWfO[cg8 NVZR fEV[cg+[SbVA FWNA [cZVFONV`nPSVA|PSDG[cgayPS VA F]`nPSVA N o&FEV¯FOFEg 0 PSg d T ]VA+y<£rFEgGRa`a`bFEZ PSg d¶[Q¯FEZ5o&[cRag daNQk@BA F+fO[cg NVZR fEV[cg g VA FGyrFEg FEZPSBfQPcNF|PSN[pR NFONeVAaNe9ZPSDGFE¯[cZbs oaRaVQsVAaN¯VDGF<s0FJg FOFOd+V[fO[cg NVZR fEV]NR fAkP`nPSVAU[cZFQPcfAUFE£rFEbgUVA FW§xPcNNFWd8¨PSy<ZPSD l [cZ fO[cg-£rFEgaFEg fOF<s[cg FGPcNNRaDGFONVAnPSV a = 0 PS¤d PSVFONefQPSgo&F¢NAaVFOd®V[ VA F¢F.V$ ln[cZ|PSBVPcN0Nv]VA FE£rFEWg [cVGND+PSFEZ¢VAnPSg g VA Fk§"PcNNF PSg d FEV d8¨PSy<ZPSDTs=F¢fO[cmax g NVZR {dfEV }V[k9Rag fEV[cg N A (t) PSg d D (t) [Q£rFEZ VA Fvg-VFEZk£<PS [0, T ] R Ngay VA F"[c[Q]gayUdaF.¥ gaV[cg X 1. def. i. i. k. Ak (t) =. L(i)≥k. Dk (t) =. X. k. si · 1[ai <t] ,. si · 1[di ≤t] .. B@ A F9Rag fEV[cg N A (t) PSg d D (t) PSZFWNVPSZfQPcNFRag fEV[cg N" 7" $~`aFOfOF.]NF"fO[cg NVPSg-VQs0]VA P ¥ gaVFgRaD o&FEZU[Se`aFOfOFON$. ³ N[ g [cVFVAnPSV A N+F.9VfO[cgVg-R [cR N ]AaF D N+Zy<A-V fO[cgVg-R [cR NQ 3FEV g obF¯PSgvPSZoaVZPSZX g [cg8daFOfEZFQPcNgayePSg d5NVZfEVXefO[cg£rF.TeRag fEV[cg3~^ ge`&[Q¯FEZ PQ¯PSZF NfA FOd8Ragay scVAaN9Rag fEV[cgNyrFEg FEZPSX5fO[cg NdaFEZFOde[cZDG[0daFEgay"VA FFEg FEZy<XefO[cg NRaD¢`aV[cg [S]VA F¢`aZ[0fOFONN[cZQ @BA FGfO[cg NRaD¢`aV[cg PSV5VDGF t daFE`&FEg daN[cgVA FG`aZ[0fOFONN[cZ N`bFOFOd u(t) PSg d [cZeg NVPSg fOFG]VA¶VA F 0®_À®VFOfAag [c[cy<XrsVX0`afQPSXrs s ]A FEZF 1 ≤ γ ≤ 3, α ≥ 0, C ≥ 0 s NFO1F u [cZDG[cZFedaFEVPSN¡ g(u(t)) ≈ αCu(t) L(i)≥k. k. k. 1+2/(γ−1). Ê9Ã~ïÊÏ.
(39) H<H. 7
(40) 7
(41) 7 . 4 8 º 8 r 5e ¾ ( S
(42)
(43) R. . Z. -7 . Z. . Z. º¾ º 7 . t 0 t 0. T 0. Pk . g(u∗k (s))ds. u∗k (t) ≥ 0. 7
(44) 7 .
(45)
(46)
(47) ( '7 . u∗k : [0, T ] →. jr. 7 57 7 57 . +r 9u ir. ∀t ∈ [0, T ],. u∗k (s)ds ≤ Ak (t). ∀t ∈ [0, T ],. u∗k (s)ds ≥ Dk (t). ∀t ∈ [0, T ].. !
(48)
(49) 7 " .
(50)
(51)
(52) # 7%$'& (.
(53)
(54)
(55) #) '7
(56)
(57) '7 )
(58)
(59) k /#0 . Pk. 7 7+* ," ). . . 2 7. 73$
(60) 4 5
(61) ∗ 7 ) '7
(62)
(63) '7 )
(64)
(65) 7u68k 7 ∗ 7
(66) 9 ) '7
(67) P k
(68) '7 g
(69) 9 76 Pk 7 5: u'k7 . 7 71. 777 ! . #) '7
(70)
(71) '7 . 9u 9¶ir$. u∗k. '7 ; <
(72)
(73) 7 7 *
(74) 7 >=
(75)
(76)
(77) # '7 . '7 7 7 5 +r u u (t),. sup0≤t≤T u∗k (t) ≤ sup0≤t≤T. k. k. 7 ! . Rt. 7 ?
(78) . ) @ 7
(79)
(80) U ∗ def = 0 u∗k (s)ds k
(81) & A 9 D , 7 B&:
(82)
(83) 7
(84)
(85) '7 0 A. 0. . T.
(86) ( " 7. 27 . º¾ º 7
(87) D
(88) !7
(89) 7 '7 @$F&8($ u (t) @BA F+PSyr[cZVAaDV[fO[cg NVZR fEV `aZFONFEgVFOd¶g +" Ng N`aZFOd®Z[cD VA FGg FQPSZ VDGF PSyr[cZVAaD¸fO[cD¢`aRaVgay+VA FfO[cg£rF.TuA-Ra[S n [cZdaFEZFOdk`b[cgVNgkVA F`a¨PSg F< GIH Ô ÞÔ.Ìa× H Ôê>å J¢#Ç J<ÔàÞØ×ÇÞ]ê×Ø$ÈSÕ?êìÇÞ¯× H ÔßÔàÈQ×ÞØÐ&êå>J¡ÞÔâ¯å?âxÌ8êÙÓÈSßàÔ"ØÐÑÐìåSÈ?ßà×ÙÓÇ.È?êBØÞÔWÇ.È?ÐëÆ¢Õ?Ô'KcÈSÔÕ Ø$ÐÓâ]Ç.ê×ÔñEÔ ÞMÆ L H ÔàÞÔ.á k. k. C , $'&8( 0 1 D '7 .
(90) 9
(91) 7. E) 7 '7 A k ∗. k. ∗ k. ï±ï È OôQõð$ô *). . k.
(92).
(93) HQj.
(94) . .
(95) () )+ % $*) %$&''() + . J_ g FxZFOfQPSNBVAnPSV K NBVA FxAay<A FONV]FE£rFE g VA F§"PcNNFxd8¨PSy<ZPSD FfQPS T , 1 ≤ k ≤ K VA F¤NFEV3[SaPS-VA F VPcN8N3PSV3FE£rFEN k PSg dAay<A FEZQ~@BA F¤PSyr[cZVAaD²[cZdaFOfEd8gayVA F FQPcNoaV X NVA F"[c[¡]gay H<l [cZJPS k ∈ {1 · · · , K} Pr 0¯[cg NVZR fEV u (t) VA F[c`aVD+PS±N[cRaV[cgp[S P 9o xF.¥ g F r := sup u (t) j8@BA FeNFEV[SVPcN8NBNBFQPcNoaF PSg dp[cgaX| sup r ≤ 1 ³ gUF.TaPSD¢`aFJ[S3VAaNfO[cg NVZR fEV[cgUNy<£rFEg+g l±y<RaZF + PSg d|l±y<RaZFJu R Ngay VA F"NPSDGF NFEVJ[SVPcN8NJPcNJgl~y<RaZFvj8@BA FeRag fEV[cg N U = R u (x)dx PSZFd8N`a¨POXrFOdg NVFQPcd)[S g[cZdaFEZV[GNA [Q²]AX NR fAkRag fEV[cg NWPSZFefQPSFOdNA [cZVFONV`nPSVA NQ u (t) l±y<RaZF +)ZFE`aZFONFEg-VN U (t) PSg d U (t) ]AaFkl~y<RaZF uNA [QN¢VA FkfO[cg NVZR fEV[cg©[S FyrFEV r = 1/2 s r = 7/11 PSg d r = 12/22 [cVFpVAnPSV|VA FpD+P?T0D5RaD U (t) s NBZFQPcfA FOdko-X [Q£rFEZ]VA Fxg-VFEZ£<PS VAnPSV]g fER daFON]VPcN8N PSg d r = 7/11 ³ fEVRnPSXrs0VAaNN¤VA FJfEZVfQuPSgVFEZ£cPS¤^gU[cRaZF.T [4,PSD¢15]`aF<sVA FJNFEV[S3VPcN0NN τFQPc, NτoaF"Ng τfOF r = 7/11 < 1 k. ∗ k. k. k. 0≤t≤T. ∗ k. K k=1 k. ∗ i. ∗ i. ∗ 3. ∗ 1. 3. ∗ i. ∗ 2. ∗ 2. 2. t 0. 2. 1. 3. 4. 6. 2. . '
(96) ) # )+ %$&'(*)+ . @BAaN NFOfEV[cgªN daFE£r[cVFOdªV[¶VA F)`aZ[0[S5[SeVA F®fO[cZZFOfEVg FONN [S VA F PSyr[cZVAaD ^g¬VA F [c[Q]gay s I daFEg [cVFON VA F)fEZVfQPSxg-VFEZ£<PSPSg d W sBVA FgVFEg NVXª[Q£rFEZUVA F®fEZVfQPS gVFEZ£cPS NxdaFEg [cVFOd W FvPSN[UdaF.¥ g FvVA F5Rag fEV[cg v s&PcNxVA FPc[ ëNfO[cg NVZR fEV[cg [¡£rFEZVA FeNFEV[SVPcN8N T , 1 ≤ k ≤ K @BA FEZF.[cZF<s v = v
(97) ¤º !
(98)
(99) 7 $ '7 c. Ic. c. 1. !#" $. k. . . k. !#" $. !#" $. 76 supt v (t) = Wc 7
(100)
(101) 7 " # 7 2$ 7
(102)
(103) ($ 7 7 6 7 7 7 6 v
(104) (t)
(105)
(106) ≤k 1 vk 7
(107) '7
(108)
(109) P ,I k 7 6 9
(110)
(111) k k−1. . 7
(112) ) '7
(113) 0 . 1 < k ≤ K supt v (t) ≤ supt v (t). J@BA F5`aZ[[S¤[S `&[cg-VN 7 N"P+d8ZFOfEV"fO[cg NF !-R FEg fOF[SVA F5fO[cg NVZR fEV[cg®[S NFOF 9.xY~[cg-V7 7 AnPcN"obFOFEg®NA [¡]g)gÓi" xY~[cg-V17 7 7&[c[QN"Z[cD VA FePcfEVJVAnPSVePS~vFQPcNoaF NfA FOd8RagayU`b[cfEFONW]VA)P¢`aZ[0fOFONN[cZW]VAN`&FOFOd u NA [cRadNPSVNXkfO[cg NVZPSg-VN +8s uUPSg d i8 ³ N [cZ 7 ¡srVAaNN¤ND¢`aX¢obFOfQPSR NF T ⊆ T PSg d|VA FD+P?T8D+PSngVFEg NVXGy<Z[¡N¯]A FEg g FE VPcN0NPSZF5PcdadaFOd 8 . !#" $. " . k. k−1. Ê9Ã~ïÊÏ.
(114) H+. 7
(115) 7
(116) 7 . a). A3 (t). 3. U3∗ (t). D3 (t). 1. 9 7 6. A2 (t). b) D2 (t). U2∗ (t) 2 1. l~y<RaZF1+ %0¯[cg NVZR fEV[cg[S u (t) PSg d u (t) eÀRao8¥ y<RaZFGPrJNA [¡N PSg dpNRao8¥ y<RaZFo U (t) = R v (x)dx ∗ 2. ∗ 3 t 0 2. ∗ 2. U3∗ (t) =. A1 (t). D1 (t). U1∗ (t). u∗2 (t) > u∗1 (t). l~y<RaZFu ¯0 [cg NVZR fEV[cg[S U (t) = R ∗ 1. ï±ï È OôQõð$ô *). t 0. u∗1 (x)dx. Rt 0. u∗3 (x)dx.
(117) HEu.
(118) . ¤º C
(119)
(120)
(121) t ∈ [0, T ], sup u (t) ≤ sup v (t) 8 J@BAaN]NWPvd8ZFOfEV"fO[cg NF !R FEg fOFe[S0 FED¢D+PkHr 7 7 73BPSg d@BA FO[cZFED Hr 7 7 7. sup r ≥ W
(122) ¤º 8 F fO[cg NdaFEZxPS±VA F VPcN8Ng fER daFOdgpVA FfEZVfQPS~g-VFEZ£<PS PcNW[cg F NRa`bFEZVPcN τ ]AafA N·QF s NVA F¢NRaD[SBVA FGN·QFONe[SPSVPcN8Ng IfER =daFO[ad ,g d I] ]AaF VN"PSZZ£cPSN a PSg dVNJdaFQPcd8g FeN d [cVFVAnPSVJVA FeFE£rFE L [S VAaNg FE VPcN|NWVA F [¡FONVWFE£rFE[S PSVPcN0Ng I XGdaF.¥ gaV[cgk[S s8PSg [cg fEZVfQPS&VPcN¢g PSZFWg [cV¯g fER daFOd|g ~g [cg fEZVfQPS VPcN0NNVPSZVgayeobF.[cZF ττ FEg d¢o&F.[cZF τ PSg dGg [cg+TfEZVfQPSnVPcN8N NVPSZVgay5obF.τ[cZFBVA FFEg dG[S AnPQ£rFePSZZ£rFOdpP?9VFEZ τ @BAaNBDGFQPSg N]VAnPSV D (d ) − A (a ) ≥ s @BA FfO[cg NVZR fEV[cg τ [S u D¢`aFONWVAnPSV sup u (t) ≥ D (d ) − A (a )/d − a ≥ W l±gnPSXrs sup r ≥ r ≥ sup u (t) ≥ W ! " 7 $ 7
(123) 7 9
(124) 7 sup r ≤ 1 º¾ º 8 J@BAaN]NWPvd8ZFOfEV"fO[cg NF !R FEg fOFe[S0 FED¢D+PcN Hr 7.s3Hr 7 7.s j¢PSg d +8 [cVFUVAnPSV u 6= v g yrFEg FEZPS¶§"[Q¯FE£rFEZQs¯PcNvNA [¡]g©PSob[¡£rF<sVA F F !RnPSVX¶A [cdaN g NdaFxVA FefEZVfQPSgVFEZ£cPS [cZN[cDGF k
(125). . ∗ k. t. t. k. c. k k. c. c. c. c. c. c. c. c. c. %. c. Lc. c. c. c. c. ∗ Lc. t∈Ic. . Lc. ∗ Lc. Lc. k k. c. c. Lc. t∈Ic. #. ∗ Lc. c. c. Lc. Lc. c. c. c. c. c. c. c.
(126). . 0≤k≤K k. ∗ k. . . k. !#" $. ) ( )
(127) %$ - )+
(128) $ *) . ³ N NRaDGF"VAnPSV]VPcN8NBAnPO£rF5PSZFQPcd8XUo&FOFEgkN[cZVFOdkoX|VA FEZWPSZZ£<PS3d PSVFePcN]FE~PcNBo-XUVA FEZ daFQPcd8g FON¡ ³ NNRaDGFGVAnPSVVA F|FE£rFEN gVA F|§xPcNNF+d8¨PSy<ZPSD´PSZF PSN[py<£rFEg3p@BA F|VDGF fO[cD¢`aF.T8VX[S fO[cg NVZR fEVgay u NWg FQPSZWgVA Feg-RaD5obFEZx[S VPcN8NJPSVWFE£rFE k PSg dAay<A FEZ PcN"NA [¡]g)2g +" 0¯[cD¢`aRaVgay r NPSN[+g FQPSZ"g)VA F5gRaD o&FEZ[S¤VPcN8N"PSV"FE£rFEN"¨PSZyrFEZ VAnPSg @BA FEZF.[cZF<s"VA F fO[cD¢`aF.T8VX [S5VA FPSyr[cZVAaD V[ fO[cg NVZR fEV v N9Ra` V[ P D5RaV`akfQPSV£rF5fO[cg NVPSg-V$ ∗ k. k. 1. N X. L(τi ) =. K X. Ni · k. ] A FEZF N N"VA FvgRaD o&FEZ [S¤VPcN8NePSVxFE£rFE i @BA FAay<A FONVefO[cD¢`aF.T8VX®fO[cZZFON`b[cg daNeV[ VA FfQPcNFx]A FEZFVA FEZFN][cg FVPcN PSVWFQPcfApFE£rFE [QFEZJVAnPSg K PSg d N = N − (K − 1) VPcN0NPSVFE£rFE K @BA FefO[cD¢`aF.T8VX N]VAR N O(KN − K /2) [cVFxVAnPSV]VAaN]VDGF fO[cD¢`aF.T0V XUNBg FQPSZWPS VPcN8NPSZFe[cgkFE£rFEHG ]AafA N"VA Fo&FONVJ`&[<NNoaFfQPcNF<x@BA F[cZNFvfQPcNF¢PSZNFON"]A FEg K = N ]VA 1 KVPcN=0Nx1[cg FQPcfA i=1. k=1. i. 2. K. Ê9Ã~ïÊÏ.
(129) HQi. 7
(130) 7
(131) 7 . FE£rFEsb]A FEZF5VA FvfO[cD¢`aF.T0V Xpo&FOfO[cDGFON N /2 J§"[Q¯FE£rFEZQs VJN"FQPcNXkV[|`nPSVfA®VAaN"[cZNF fQPcNFUPSg dN[c£rF|VA F+`aZ[coaFED gg FQPSZVDGF|gVAnPSVfQPcNF|o-X¶DGFEZFEX¶VFONVgay r ≤ 1 ®POX0o&FNR fAP`nPSVfAF.T8NVNBgPS !RnPcd8ZPSVfefQPcNFON¡ Àg fOF K ≤ N snVAaNWDGFQPSg NJVAnPSVJVA F5fO[cD¢`aF.T0V Xp[S VA F5PSyr[cZVAaD `aZFONFEgVFOd)PSo&[Q£rF NJPSVDG[<NV !RnPcd8ZPSVf< ³ gkg-VFEZFONVgayU`aZ[coaFEDs g£rFONVy-PSVFOdo&FE[QesnN]V[GFONVD+PSVFeVA F £<PSR F[S K [cZ]VX0`afQPS±NFEV[SVPcN8NQ 2. 1. $ )
(132) % %$ # )+
(133) $ *) . . . ^ gkVAaN"NFOfEV[cg3saF `aZ[Q£0daF5PSg)FONVD+PSV[cg[S K ]A FEgpVA FefAnPSZPcfEVFEZNVfONx[SVA FeVPcN0N PSZF+fA [<NFEgZPSg da[cD¢XrkKWFOfQPSVAnPSV NeVA F¢gRaD o&FEZ[S]FE£rFEN5gVA F+§xPcNNF¢d8¨PSy<ZPSD [S5VA F N VPcN0NQ ³ g [cVA FEZ ¯PQX¬V[ KdaF.¥ g F)VQsNUV[ NFOF)V PcNUVA F)[cgayrFONVkg fEZFQPcNgay NRao NF !-R FEg fOFegkVA F`nPSZV¨PS[cZdaFEZW[SVA FeNFEV P ^VN¤P"fE¨PcNNfQPSn`aZ[coaFED V[FONVD+PSVFBVA FPO£rFEZPSyrFWFEgay<VAG[SbVA FB[cgayrFONVNRao NF !R FEg fOF g PSg¶[cZdaFEZFOdNFEVQ ^]VA F|PSZZ£<PSN a PSZF|fA [<NFEg ZPSg da[cD¢X PSg dVA F|daFQPcd8g FON d PSZF PSN[efA [<NFEgUZPSg da[cD¢X+PSg d|PSZFWg daFE`bFEg daFEgV¯[S VA F"PSZZ£cPSNQs-VA FEg|[cg FWAnPcNVA F][c[Q]gay ZFONRaVQ
(134) ¤º √ √ E(K) = 2 N + O( N ). §J[¡FE£rFEZQs3¯F¢fQPSg yrFEVePUD R fA [¡FEZ FONVD+PSV[cg¶[cg®VA FGPO£rFEZPSyrF o-XpVPS0gaykg-V[ PcfOfO[cRagVWVA FxPcfEV]VAnPSVVA FedaFQPcd8g FON"PSZFg [cVg daFE√`&FEg daFEg-V"[S~VA F PSZZK£<PSN5 d ≥ a .^ g VAnPSVJfQPcNF<s8VA F PO£rFEZPSyrF K ND R fA)ND+PSFEZ]VAnPSg N l [cZJF.T PSD¢`aF<snA FEZF NWVA F `aZ[[S gVA F fQPcNF VA F5PSZZ£<PS`aZ[0fOFONNN"Y±[cNN[cg)PSg dpVA F ZFE¨PSV£rF daFQPcd8g FON5 d − a ]PSZFF.T0`&[cg FEg-V¨PSXd8NVZoaRaVFOd
(135) ¤º 1 7
(136) ) 8 7 7
(137)
(138) '7 -
(139) 7 5 )E 76 '7+
(140)
(141) -7 7 7 9 )
(142) E(K) = O(log(N )) 8 Jl~ZNVQs±g [cVF+VAnPSV5VA F+`aZ[coaFED fQPSg¶o&F+NFOFEg PcNPV¯[d8DGFEg N[cgnPS¯`aZ[coaFED o-X `aRay<y<gay a [cg VA F x P?T0NePSg d −d [cg VA F y P?T8NQ ^ g VAaNxg FE 9ZPSDGFE¯[cZbs K NxVA F [cgayrFONVNF !R FEg fOF[S fO[cD¢`b[cg FEgV]NF¤g fEZFQPcNgay`b[cgVN (a , −d ) < · · · < (a , −d ) [¡esr&FfO[cg NdaFEZ¯PSg-Xvy<£rFEgG`&[cg-V (a , −d ) s-VA FB[cgayrFONVg fEZFQPcNgayNF !-R FEg fOF S NVPSZVgay¢gUVAaNB`&[cg-V]N]fEFQPSZXUND+PSFEZ]VAnPSV]VA F"V[cVPS g-RaD5obFEZ [S`&[cg-VN]]AafApPSZF ¨PSZyrFEZ VAnPSg (a , −d ) ¢@BA F¢ZFONV5[S¯VA F¢`aZ[0[SBN P d8ZFOfEV daFEZ£<TPSV[cg¶y<£0gayPSg Ra`a`bFEZ i. i.
(143) . i. i. i. . . .
(144). . . . i. i. i1. h. iK. i1. h. *). h. iK. h. h. h. ï±ï È OôQõð$ô. i.
(145) H.
(146) . o&[cRag dk[cg T h. N. NJPZPSg da[cD `b[cgV. P(max Sh ≥ C) ≤ N P(SH ≥ C). ]A FEZF. h=1. H. SP ZZ£cPSNgPSg g-VFEZ£<PS±[S~FEgay<VA x))e dx Z DG[cZFVAnPSg C − 1 PSZZ£cPSN N e P( gPSgkg-VFEZ£<PS±[SFEgay<VA x)e dx. ≤ N. Z. ∞. e. −x. x=0. ∞ X. P(k. k=C−1. −kx. ∞. ≤. −x. x=0. −(C−1)x. ≤ N. Z. ∞. x=0. e−Cx. xC N dx = C! (C)C+1. [¡es& s&VAaNx`aZ[conPSoaV X)yr[0FON"V[U·QFEZ[PcN yr[FON"V[Ug8¥ gaVXr¢À0[UVA F V yr[)o&FEXr[cg d log(N ) [cZ ¨PSZyrF N ^ g¶`nPSNZVfERa¨PSZQsVAaN5D¢`aFONVAnPSV VPS¯[S K da[CFON5≥g [clog(N E(K) = O(log(N )) @BAaN¢DGFQPSg N¢VAnPSV¢VA FPO£rFEZPSyrFfO[cD¢`aF.T0V X N log(N )) ³ fEVRnPSXrs¤VAaN¢`aZ[0[S NA [QNPeNVZ[cgayrFEZ¤ZFONRaV ~]A FEg|VA F]g-RaD5obFEZB[SbVPcN0O(N N N yr[0FON V[g8¥ gaV XrsVA FWfO[cD¢`aF.T0V X [SVA FePSyr[cZVAaD NobFE[¡ N log(N ) ]VA`aZ[conPSoaV X[cg F< [cVFBVAnPSVVA FJPO£rFEZPSyrFWfO[cD¢`aF.T8VX¢[S&VA F]fE¨PcNNfQPSPSyr[cZVAaD[S3À`aRaZnNg FOfOFONNPSZX y<ZFQPSVFEZVAnPSg C((N + N )/2) [cZN[cDGF¢fO[cg NVPSg-V C ]AnPSVFE£rFEZeVA F¢d8NVZoaRaV[cg N5Ng fOF VAaN±N~VA F¤o&FONV±`b[<NNoaF¤fO[cD¢`aF.T8VXUfO[cZZFON`b[cg d8gayxV[JVA FfQPcNF¤]A FEZF¯VPcN0N±AnPO£rF¯l^ l~_ fO[cg NVZPSg-VN$. ~
(147) B ^ gpVAaN"NFOfEV[cg)FfO[cg NdaFEZ"VA Fe[c[Q]gay `aZ[coaFED ]fA [0[<NF5PSVxFQPcfA)VDGF t VA FN`&FOFOd g NR fA©P)¯POX PcNvV[ F.TaFOfERaVFpPS]VPcN8Nv]VAag VA FEZ+daFQPcd8g FfO[cg NVZPSgVNG]AaF u(t) D¢gaD¢·Ogay|VA FV[cVPS3FEg FEZy<XkfO[cg NRaD¢`aV[cg3 @BA FRag fEV[cg PcfEVRnPSXk`aZ[¡£daFONxPGN[cRaV[cgV[GVA F `aZ[coaFED¸[S D¢gaD¢·OgayUVA F V[cVPSFEg FEZy<X R NFOd v oX VA F`aZ[fOFONN[cZ]A FEg©VA F D¢DGFOd8¨PSVFpFEg FEZy<X fO[cg NRaD¢`aV[cg©[SeP `aZ[0fOFONN[cZ yr[cgay®PSVN`bFOFOd u sN5[SVA F¢[cZD g(u) s]A FEZF g N5fO[cg-£rF.T PSg d¶g fEZFQPcNgay @BAaNNVA FBZFONRaV¤fE¨PSDGFOd+g " ^g¢VA F][c[¡]gay s-¯F]`aZFONFEg-VV¯[ d8 FEZFEg-V¯PSyr[cZVAaDGN 2. . . !#" $. . Ê9Ã~ïÊÏ.
(148) H¡q. 7
(149) 7
(150) 7 . VAnPSV]FEgnPSoaF"V[vfO[cD¢`aRaVFxVA FxNF !-R FEg fOFx[S±[c`aVD+PS&ZF !R FEg fEFON]VApP [Q¯FEZfO[cD¢`aF.T0V X VAnPSg Pc[¢FEVJPS ëNWPSyr[cZVAaD . $&''()+. . ³ N VZPSy<AV[cZBPSZd¯POX¶V[pdaFEVFEZD¢g F|VA F+[c`aVD+PS¯`aZ[fOFONN[cZ5N`&FOFOdaNeN5V[VFEZPSVF+VA F PSyr[cZVAaD [cZPcNNFONNgayWFQPcNoaVX [cg fOFBVA FBfEZVfQPSag-VFEZ£<PSaN~daFEg-V¥nFOd¢VN~ZFEDG[Q£rFOd PSg dpVA FeFQPcNoaV XPSyr[cZVAaD NJPS`a`aFOd®[cgpVA F5ZFONRaVgay NRao8`aZ[coaFEDJ@BAaNJFQPcdaN"V[ VA F"[c[Q]gay PSyr[cZVAaD H<l [cZ k = 1..K fO[cg NVZR fEV u (t) " FEV r = sup u (t) ¤VA F fEZVfQPS]gVFEZ£cPS NNR fAVAnPSV W = sup r [a , b ] j8JÀ0FEV|VA FN`&FOFOd¬[cg [a , b ] F !RnPS"V[ W /JNgay VDGFfO[cD¢`aZFONN[cg3s]ZFEDG[Q£rF g-VFEZ£<PS [a , b ] KJFEVRaZgV[GNVFE`HxPSVFQPcNV[cg FeVPcN|ZFED+PSg N¡ ^ V±NA [cRad o&Fg [cVFOd5VAnPSV K s?VA F¤gRaD o&FEZ[S FE£rFEN~g5VA F§xPcNNFd8¨PSy<ZPSDs-fQPSgag [cV±g fEZFQPcNF P?VFEZ VA FfO[cD¢`aZFONN[cg|[S&VA FfEZVfQPS g-VFEZ£<PS ^ g daFOFOd¢bVPcN τ N g [cVNVZfEVX5g fER daFOd g®VPcN τ o&F.[cZF5VDGF¢fO[cD¢`aZFONN[cg3s VA FEg τ NJg [cVeNVZfEVXpg fER daFOd®g VPcN τ P?VFEZ fO[cD¢`aZFONN[cg3|_JgVA FG[cVA FEZ AnPSg d s3g [cVF¢VAnPSV K D+POX daFOfEZFQPcNFGo-X®VDGFGfO[cD¢`aZFONN[cg3 ^ g daFOFOd s τ NNVZfEVXg fER daFOd g τ obF.[cZFVDGF¢fO[cD¢`aZFONN[cg¶PSg d®VA F¢daFQPcd8g FON [cZ¯VA FxPSZZ£cPS&VDGFON$¤[So&[cVA|VPcN8N¤PSbg|VA F"fO[cD¢`aZFONNFOdUgVFEZ£cPSs0VA FEg VA FEX|PSZFJg [cV NVZfEVX|g fER daFOd)PSg-X0DG[cZF P?9VFEZWVDGFefO[cD¢`aZFONN[cg3 Àg fOF+VA FEZFUNvPSV5DG[<NV N fEZVfQPSBg-VFEZ£<PSNQsVA F+FQPcNoaV XPSyr[cZVAaD NVFE` H¡N g£r[crFOd+PSV DG[<NV N VDGFON¤PSg d¢VA FB[cZNVfQPcNFfO[cD¢`aF.T8VX¢[SbVA FPSyr[cZVAaDN O(KN ) ^ g VA F PO£rFEZPSyrF fQPcNF<s8]A FEZFVPcN8NPSZFfA [<NFEgpZPSg da[cD¢XU]VAY~[cNN[cgd8NVZoaRaV[cg N][cZ VA FvPSZZ£<PS±VDGFONxPSg dF.T8`&[cg FEg-V¨PS¨PSVFEg fEFONQsbVA FPQ£rFEZPSyrFvfO[cD¢`aF.T0V XpN O(N log N ) @BA F fO[cZZFOfEVg FONN[SVAaNPSyr[cZVAaD NkD¢`aFOd o-XªVA F fO[cZZFOfEVg FONNp[SVA F)FQPcNoaVX PSyr[cZVAaD NFOF@BA FO[cZFED +r. ∗ k. i. j. i. i. ∗ k. k k. [ai ,bj ]. i. 0≤t≤T. j. [ai ,bj ]. j. 1. 2. 1. 1. 2. 2. 2. 2. . $&''()+. . ³ g [cVA FEZ~¯POX5V[JdaF.¥ g F v NoXeR NgayP`aZ[0fOFOd8RaZFBfE[<NF¤V[JVA FB[cg F¤R NFOde[cZFQPcNoaV Xr 4 8 º 8 r 5 e ¾ ( S C
(151)
(152)
(153) 1 ≤ k ≤ K . 73 7
(154)
(155) . '7 !#" $. vk : [0, T ] → R. ï±ï È OôQõð$ô *). . EA .
(156). Z. T. g(vk (s))ds 0. 7 57 7 7 . .
(157) H .
(158) . -7 . q< Z r v (s)ds ≤ A (t) ∀t ∈ [0, T ], Z r v (s)ds ≥ D (t) ∀t ∈ [0, T ]. ³ N¤[cZ)PSVA FED+PSVfQPS3YZ[cy<ZPSD H<s0[cg F"fQPSgUfA FOf+VAnPSV¯VA FW9Rag fEV[cg v N¤Raga,!R FJRa` V[GPGNFEV[S~DGFQPcNRaZF 0 9 v 7 73 v (t) = v (t), ∀0 ≤ t ≤ T º¾ º & ! '7 v 8 NrFEVfA 0¯[cg NdaFEZ|VA FfEZVfQPSWgVFEZ£cPS g VA Fpda[cD+PSgª[SxdaF.¥ gaV[cg [SePS VPcN0NQs D = [0, T ] ³ d8ZFOfEVeg d8R fEV[cg¶[cg K NA I[QNVAnPSV v (t) = sup u (t) [Q£rFEZVA F fEZVfQPSg-VFEZ£<PS I ¯X+R Ngay|@BA FO[cZFED +8s8VAaNNA [¡NVAnPSV v (t) = v (t) t ∈ I [¡es¯FxfO[cD¢`aZFONNBVDGFWoX FxfO[cg NdaFEZBVA FJg FE NFEV¯[S±VPcN0N fO[cD¢`aZFONNFOd oX VA F¯g-VFEZ£<PS I ~@BA Fg FE da[cD+IPSgG[SbdaF.¥ gaV[cgGNdaFEg [cVFOd D\I ~^ Tf \INPSg-X 9Rag fEV[cg daF.¥ g FOd+[cg [cZ VA FNFEV[S&VPcN0N s<¯FdaFEg [cVF]o-X VA F]NPSDGF¯Rag fEV[cg+fO[cg NVZR fEVFOd [cg D\I [cDZVA FfO[cD¢`aZFONNFOdNFEV[ST~VPcN8N T \I f \I F¢¥ ZNVfA FOf o-X g d8R fEV[cg [cg K VAnPSV5VA FGRag fEV[cg v = (v \I ) ∧ v k^]g [ fEZVfQPSWVPcN0NGPSZFPSV¢FE£rFEeH<sVA FEg v = (v \I ) ∧ v o-Xg d8R fEV[cg3 laRaZVA FEZDG[cZF<s [Q£rFEZ I o&FOfQPSR NF|g [pVPcN8N[cg FE£rFE][cg F PSZF|fEZVfQPS®@BAaNDGFQPSg NVAnPSV v = v = v ~^3N[cDGFWfEZVfQPSbVPcN8N¤PSZF"[cg+FE£rFE[cg F<sFW`a¨POXGVA FJNPSDGFVZf+PcNg|VA F (v \I ) ∧ v `aZ[0[S][S¯FED¢D+P +kPSg dZFE`a¨PcfOF+PSfEZVfQPS¤VPcN8Nxo-X [cg F¢NRa`&FEZVPcN®[S¯FE£rFEJH<¢@BA FEg3s [¡£rFEZ D\I PSg d v = W sVA FZPSVF5[SVA F5fEZVfQPSg-VFEZ£<PS [Q£rFEZ I "@BAaN v = (v \I ) NWd8ZFOfEVo-XUR NgayGVA FefO[cg NVZPSg-VNR NFOdkV[¢daF.¥ g F v ³ N[voXUfO[cg NVZR fEV[cg3s VNfEFQPSZWVAnPSV v = (v \I ) ∧ v l±gnPSXrsSR NgayJVA FPcfEV3VAnPSV v \I PSg d v \I PSN[WfO[cg fEdaF¯[Q£rFEZVA F¯fEZVfQPS0g-VFEZ£<PS g D\I NA [QNVAnPSV v PSg d v fO[cg fEdaFJ[¡£rFEZ¤o&[cVA+fEZVfQPSngVFEZ£cPSN¡ XVFEZPSVgayVAaN fO[cg NVZR fEV[cg3sn[cg FNFOFON]VAnPSV v PSg d v PSZFF !RnPS[¡£rFEZJPS [0, T ] @BAaNxy<£rFONePSg¶PSgnPSXVfQPS BPOX)V[UdaF.¥ g FGVA FfO[cg NVZPSg-VNx[cZ"FQPcNoaV Xr [Qes3A FEZF NJP`b[<NNoaFfO[cg NVZR fEV[cg[S v H<l [cZ k Z[cD K da[Q]g-V[Hda[ Pr 0¯[cg NVZR fEV u (t) ¯À0FEV v := u vk (t) ≥ vk+1 (t). t. k. k. k. k. ∀t ∈ [0, T ],. 0. t. 0. k. . . 1. 1. c. c. 1. %. k. !#" $. 1. ∗ k. c. c. c. c. c. c. c. c. 1. 2. 1. 2. 1. 1. c. 2. c. Ic. 1. c. Ic. 1. 2. c. 1. Ic 1. 1. c. c. 1. ic. c. 1. !#" $. !#" $. c. !#" $. c. 1. !#" $. c. c. Ic. !#" $. %. 1. !#" $. 1. 1. ∗ k. k. ∗ k. Ê9Ã~ïÊÏ.
(159) H. 7
(160) 7
(161) 7 . . 9o ³ N~[cgay PcNVA FEZFBF.T8NVNPSgvg-VFEZ£<PS I [Q£rFEZ ]AafA v (t) < v (t) s<ZFEDG[Q£rFPS VPcN8N3g fER daFOd5g I [SaFE£rFE0g [cV±ND+PSFEZ±VAnPSg k+1 PSg deR NF¤VDGF.fO[cD¢`aZFONN[cg [cZ I +x[ VAaN[cZ5PS¤NR fAgVFEZ£cPSNQ PSg dfO[cg NVZR fEV u \I PSy-PSg [Q£rFEZ VA F fO[cD¢`aZFONNFOdNFEV[S~VPcN8NWPSg dNFEV v := (u \I) ∧ v j8KWFEVRaZg v ³ g F.TaPSD¢`aF[S~fO[cg NVZR fEV[cgkN¯y<£rFEgUgkl~y<RaZFi5R Ngay5VA FxNPSDGFxNFEVB[S±VPcN0NBPcNBg l~y<RaZFj8 @BA FxfO[cg NVZR fEV[cgkyr[0FONBPcN[c[QN PSg dUVA FEZF.[cZFxVN¯g [cV]d8N`a¨POXrFOd A FEZF®NFOF|l±y<RaZF +8ÓPr.)^VvPSN[AnPS`a`&FEg NVAnPSV vv ==u u @BA F|g-VFEy<ZPS V Nd8N`a¨POXrFOd g l~y<RaZFpi¶PcN¢FEJPcN U ©Àg fOF v ≥ u [¡£rFEZGVA FkgVFEZ£cPS I = [4, 15] sBFAnPQ£rF V[pfO[cD¢`aRaVF u \[4, 15] [Q£rFEZ VA F|NFEV5[S]VPcN8N fO[cD¢`aZFONNFOdoX [4, 15] @BA FG9Rag fEV[cg v NR fAªVAnPSV v = v [Q£rFEZ [4, 15] PSg d v = u \[4, 15] [cg VA FpZFONVQsNQs]o-X¬daF.¥ gaV[cg3s [Qes0F"fA FOfGVAnPSV PSg d+VA F"PSyr[cZVAaDµNV[c` NQ@BA F v = (u \[4, 15]) ∧ gVFEy<ZPS V [S v Nd8N`av¨PQXrFOdgl~y<RaZF i8 v ≥ v k. ∗ k. ∗ k. k. k+1. I. k+1. 1. ∗ 3. 3. ∗ 1. ∗ 1. 1. ∗ 1. 1. 2. 2. 1. [4,15] 2. 1. 2. ∗ 1. ∗ 2. 2. c. 1. ∗ 1. 1. 2. 1. A1 (t). D1 (t) V1 (t) U1∗ (t). V2 (t). v2 (t) > u∗1 (t). l~y<RaZF i 0¯[cg NVZR fEV[cg[S v (t) ¤@BA Fx9Rag fEV[cg V (t) = R v (x)dx N]ZFE`aZFONFEgVFOd @BA F PSyr[cZVAaD¢f fO[cD¢`aF.T8VXk[SVAaNWfO[cg NVZR fEV[cgNgVA F[cZNFefQPcNF 1. 1. C(. K−1 X k=1. ï±ï È OôQõð$ô *). nk nk+1 ). t 0 1.
Documents relatifs
Dans chaque ville et pour toutes les lignes de tramway considérées, il faudrait disposer des trafics avant et après ouverture d’une ligne et de plusieurs projections d’évolution
Le fait d’écrire à partir d’une oeuvre d’art visuelle ressemble à une forme de contrainte; nous croyons par ailleurs que cette pratique a eu un effet durable sur l’écriture
variables d'ordre psychologique: soit f.amiliale ou personnelle, soit so- ciale, soit professionnelle, soit une combinaison de ces facteurs. Ce patient a récupéré
For the two coolest objects in Figure 3.6, HD 4539 and PG 1716+426, both at Ta < 27, 000 K, there is a good match between the predicted synthetic spectra and the FUSE data: the
lui inculquer le registre li'lquistioue, les l1abitudes alimen- taires et vestimentaires ainsi que le comportement des bour- geois de Québec. Pierrette Paul doit
La conciliation emploi-famille : des mesures à développer dans les milieux de travail Tableau 4 : Les dispositions favorisant la conciliation travailfamille dans les
Figure 2.4 Predicted (continuous lines) and observed values (filled circles: stands aged from trees with root collar; open circles: stands aged from trees without root collar)
Cette posture permet de dégager deux idées centrales : le poids des milieux professionnels locaux dans l’intermédiation et le transit des idées, modèles et pratiques
