Content-Type: multipart/mixed; boundary="-------------0512270728408" This is a multi-part message in MIME format. ---------------0512270728408 Content-Type: text/plain; name="05-440.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-440.comments" E-mail: Armen.Shirikyan@math.u-psud.fr ---------------0512270728408 Content-Type: text/plain; name="05-440.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-440.keywords" Exact controllability in projections, 3D Navier-Stokes system, Agrachev-Sarychev method ---------------0512270728408 Content-Type: application/postscript; name="article.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="article.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.95a Copyright 2005 Radical Eye Software %%Title: article.dvi %%Pages: 24 %%PageOrder: Ascend %%BoundingBox: 0 0 595 842 %%DocumentFonts: CMR17 CMR12 CMR9 CMBX9 MSBM10 CMR6 CMBX12 CMBX10 CMR10 %%+ CMMI10 CMSY10 CMR7 CMTI10 CMSS10 CMMI7 CMR8 CMSY7 CMMI5 CMEX10 CMSY5 %%+ CMBXTI10 CMR5 CMMI8 CMMI6 CMSY6 CMSY8 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o article.ps article.dvi %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2005.12.27:1322 %%BeginProcSet: tex.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: texps.pro 0 0 %! 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All Rights Reserved) readonly def /FullName (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 48 /zero put dup 49 /one put dup 50 /two put dup 52 /four put readonly def /FontBBox{-341 -250 1304 965}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBXTI10 %!PS-AdobeFont-1.1: CMBXTI10 1.0 %%CreationDate: 1991 Aug 18 17:46:30 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBXTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMBXTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 110 /n put readonly def /FontBBox{-29 -250 1274 754}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 58 /colon put dup 65 /A put dup 67 /C put dup 69 /E put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 120 /x put dup 121 /y put dup 123 /endash put readonly def /FontBBox{-53 -251 1139 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY5 %!PS-AdobeFont-1.1: CMSY5 1.0 %%CreationDate: 1991 Aug 15 07:21:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 49 /infinity put dup 63 /perpendicular put readonly def /FontBBox{21 -944 1448 791}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 8 /braceleftbig put dup 9 /bracerightbig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 80 /summationtext put dup 82 /integraltext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 91 /uniondisplay put dup 98 /hatwide put readonly def /FontBBox{-24 -2960 1454 772}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI5 %!PS-AdobeFont-1.1: CMMI5 1.100 %%CreationDate: 1996 Aug 02 08:21:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 70 /F put dup 84 /T put dup 112 /p put readonly def /FontBBox{37 -250 1349 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA06DA87FC7163A5A2A756A598FAB07633 89DE8BB201D5DB4627484A80A431B6AFDBBBF23D4157D4AFE17E6B1C853DD417 25F84CD55402AB88AB7EEFDEDBF2C2C731BD25567C53B474CCF739188A930039 098A197F9C4BE7594D79442B2C8A67447DE44698321145D7689B91EF235EA80E B600AA8E238064F154284096C4C2554EFE8DDF13AFF8D3CE30E0999375C0FEE6 F992DEA5FC3897E2CC8B7A90238E61E41622DE80F438DD994C73275CC52249D9 F6686F87F394FB7BB668138B210BEC9E46415A1B58C990B81E7D7DD301143517 4C2A259D2A0A1E200F8101469C10D7D537B0D4D39296A9AB3F132DA9A3B459B0 F850E2B3A03BDCB35AEF82285D19C38F474FB414F8EC971B994D1C7DD753B271 2B71549DF497C665DF0F266988209D9EB616E4D9BA229FF984E7A886DB01FD21 48ED2E4859FD6416C2CE52537464EA884C8C9C2D1083E2B83BE4B766474C23B6 6E8EC5003200AB10514BB44D14CA700416AB6B2683E80862E7D5B49A05526A32 554BB23AB8B0824BBA198E3825CE82380CC0FECF46651E3E5D77F09465E73164 20342822F29572BC7F73F2C3BF95ED3BB6FDEADC20C6AC866C4F2C679594D7E8 8D944704A3C5D771DC39503BECAB89F34D8CDB8FDB91AFE21F3F0260D05E90C5 73E2C13DFA022C4522E5918EE25038A0498FBB530DA33B0AE238B1C6ED03FC04 2BFED8236E07820C5BAB411EAE1B31D93A2FA7C374B1725FEC359ABCB88E2C89 214529A263D795AACB0B95A3AB2F4E08EF350C282CE521716DBB06E5B8291B3F 5D4ACA230FA192F64BC902A4C8842C0F916F92FBD002ADD408BF0401D0284FBB F05D4C6DB631420747CC902C5E1617E6573612FB26C8378DF41FFB5048D3CF06 4893DBA48EF4B043D760F60C75712169D16C83EE020C45369E443E853E1809DD F395B812067D6FDBD26111B34F42C21036AF952D0D767FD17F6959D9FDD46005 D64FFF54772B50BB9B173AE79702981F58F9F235C591F476A31852174DF0619C A470359153DC32610E782B204E7945515464DACE9099B81EEECC7EBD4B5126AF C3FD9DDFB329AF1C95C41FA4A5F6958869509A23BD7210386329771FA46FF926 0E54AC35106253EE140449425A8670E1F92B178A02A58EB57540F4BD8110E548 BB584EA6D625C5F5FE0124A98E49915F1A1B95D2125874360EED1C4379FEF3C6 90E5780C20309F11F2F23FAD635C44BA030B39EFF083A3ECCDD2641DCA8AF5D3 38B381954F72A516C2992AB4AE55D3E0C33C7D0BB5ABE6C75ADA357635B5D2DF 4B6DF92EFAEF26F28EAD20BBE7816A89C69D78151ACF6EADC34654A2846803DB 929F1313E7300B7C17D327A1D2B68049FFA5D9B115E5BA843284FA1606DA8A00 E1A68468423ABA6A54D9F7BBEDC25B29121EE7A69FD08B077D3F126524E7BCA8 4AC3FEFDA3C8EA77C38F5617B51C80974275ADB8DA58553B1C3492C4D98074A1 5F94D0E2B2FDE4FB906EBE8C98D9B120B43CDBE0BBF7D8D638B70A8EADB88AB7 39D6D842FD780E998D9FE63382029432F5282B0925C04BBB8CEEDA12DB033B33 26F40D016AC61055F19A06458E476134745BCAACE116F76E43115F47919BDB99 B2831DF9BA68072086AED4DE09F3A7063EA8C7A2513F65B81D0C341A14D66377 3698F89FF3ABB495171498C2349D0A1244EF752E1DC49EA96E30BDFCB2C20147 CAF16E7FFE60CA776B0161255667C06C12DF2CD32F2E368C7C49AD33DAE22405 7FE5611B00A2DBA68BFDF7CFCED99A0EDEC97F316D7345E463D695CD03199C64 D387A19FDF5EB733FE9946857154E3080B055BC42FDA1851121686492D14AD57 A5006AE9D3604BB65087B451F5C6255947BA697CC02B889141556AD77398F668 D328F09AD770556331422DCF8CF0EE9A882BE028BAE6AADC626F082708655852 5FAD761554C51FCB334F3C5D86F8027FCC5A765B95D822B46DBA0E399EF8E4CE 1F2980016CCB908A6D772284BA88365155DD41C02487F371C1D4B56E0F5F068D 7EFC6E0D578F05F8B6E0059FC3ADA5E8E122EAA492FFD9B64AFF3B8BE30631B4 F36399E13E9489CD87BFC5DA1A5AD7D839C338CD2ACDDDB8A1DF9D3CCEB86B2F 0A879F278E58D725346349ECCDBC861413D4E406467331C192348E858377BC76 D4EB6B2A1D85000AF7D97C6BAA4AE0B4FA3B6E0F0565CFD080CB4F802E3831B2 B33EFBAEFD4A6873F03B8FD183DE58ADDC9C871E0EB36F6AB8577223B0CAF8F7 E00702C8707D9C0FACFAD80EBDD660DB62D914899AD5B18C9C2266839ED140EB 0A6F453A2667343DD44867E72E78C2979D796079A9C4E8503BB9B757832BF195 BD99DEB263BF43ED593EF86E7B8ECD00A7E1A12C51786BA218A1CB665B24FAAA 3E66F9F333F281E0050A1A6C6DDDE341F7E10CE1C2D4D619609B31808B6348A1 B64602DC399D8EE18D66E7FFAFA96610E591D2455C022F88A5539D89F421661F 5D93488928E44C9D09922CA5A48529D3F478AB8B54B982DBFAF6DCDF6015161B D23DB73D23B0E44A2B30F31257 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 21 /greaterequal put dup 49 /infinity put dup 50 /element put dup 63 /perpendicular put dup 67 /C put dup 75 /K put dup 88 /X put readonly def /FontBBox{-15 -951 1252 782}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D251491EBF65A98C9FE2B1CF8D725A70281949 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 2 /Theta put dup 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 65 /A put dup 68 /D put dup 69 /E put dup 73 /I put dup 78 /N put dup 82 /R put dup 83 /S put dup 87 /W put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 123 /endash put readonly def /FontBBox{-36 -250 1070 750}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 14 /delta put dup 16 /zeta put dup 17 /eta put dup 23 /nu put dup 34 /epsilon put dup 59 /comma put dup 61 /slash put dup 64 /partialdiff put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 72 /H put dup 74 /J put dup 76 /L put dup 78 /N put dup 80 /P put dup 82 /R put dup 84 /T put dup 85 /U put dup 86 /V put dup 88 /X put dup 89 /Y put dup 100 /d put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put readonly def /FontBBox{0 -250 1171 750}readonly def currentdict end 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSS10 %!PS-AdobeFont-1.1: CMSS10 1.0 %%CreationDate: 1991 Aug 20 17:33:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSS10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMSS10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 80 /P put dup 81 /Q put readonly def /FontBBox{-61 -250 999 759}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 83 /S put dup 84 /T put dup 85 /U put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put readonly def /FontBBox{-163 -250 1146 969}readonly def currentdict end currentfile eexec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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 82 /R put dup 84 /T put readonly def /FontBBox{-55 -420 2343 920}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 61 /equal put dup 94 /circumflex put dup 99 /c put dup 108 /l put dup 111 /o put readonly def /FontBBox{-27 -250 1122 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 9 /circleminus put dup 14 /openbullet put dup 15 /bullet put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 26 /propersubset put dup 27 /propersuperset put dup 29 /greatermuch put dup 33 /arrowright put dup 49 /infinity put dup 50 /element put dup 55 /mapsto put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 75 /K put dup 82 /R put dup 83 /S put dup 88 /X put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 107 /bardbl put dup 114 /nabla put readonly def /FontBBox{-29 -960 1116 775}readonly def currentdict end currentfile eexec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% Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 8 /Phi put dup 9 /Psi put dup 11 /alpha put dup 12 /beta put dup 14 /delta put dup 16 /zeta put dup 17 /eta put dup 21 /lambda put dup 23 /nu put dup 32 /psi put dup 34 /epsilon put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 88 /X put dup 89 /Y put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put readonly def /FontBBox{-32 -250 1048 750}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 1 /Delta put dup 2 /Theta put dup 4 /Xi put dup 5 /Pi put dup 11 /ff put dup 12 /fi put dup 14 /ffi put dup 19 /acute put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 91 /bracketleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 126 /tilde put readonly def /FontBBox{-251 -250 1009 969}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F 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cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-301 -250 1164 946}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX9 %!PS-AdobeFont-1.1: CMBX9 1.0 %%CreationDate: 1991 Aug 20 16:36:25 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 58 /colon put dup 65 /A put dup 75 /K put dup 77 /M put dup 83 /S put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 105 /i put dup 106 /j put dup 108 /l put dup 110 /n put dup 111 /o put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 119 /w put dup 121 /y put readonly def /FontBBox{-58 -250 1195 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR9 %!PS-AdobeFont-1.1: CMR9 1.0 %%CreationDate: 1991 Aug 20 16:39:59 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 14 /ffi put dup 19 /acute put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 57 /nine put dup 58 /colon put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 88 /X put dup 94 /circumflex put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont TeXDict begin 39139632 55387786 1000 600 600 (article.dvi) @start /Fa 163[47 62[71 29[{}2 66.4176 /CMSY8 rf /Fb 206[61 49[{}1 49.8132 /CMSY6 rf /Fc 140[29 30[36 5[48 78[{}3 49.8132 /CMMI6 rf /Fd 138[41 9[37 2[41 15[58 7[45 1[56 1[48 1[39 1[58 2[52 1[50 1[53 5[20 49[43 9[{}14 66.4176 /CMMI8 rf /Fe 203[28 1[28 28 28 48[{}4 41.511 /CMR5 rf /Ff 145[54 110[{}1 83.022 /CMBXTI10 rf /Fg 132[56 1[59 59 1[59 62 44 44 46 59 62 56 62 93 31 59 34 31 62 56 34 51 62 50 62 54 11[86 1[62 2[77 1[88 4[42 3[74 1[81 8[31 5[56 56 56 56 2[31 33[62 12[{}39 99.6264 /CMBX12 rf /Fh 192[45 13[57 49[{}2 41.511 /CMSY5 rf /Fi 157[46 6[92 46 1[120 5[39 1[88 60[61 61 8[48 48 6[38 38{}12 83.022 /CMEX10 rf /Fj 143[31 27[35 13[36 70[{}3 41.511 /CMMI5 rf /Fk 167[48 12[51 7[36 3[52 12[45 66 27[52 19[20 52{}9 58.1154 /CMSY7 rf /Fl 132[35 1[37 37 51 37 39 27 28 28 37 39 35 39 59 20 37 1[20 39 35 22 31 39 31 39 35 3[20 1[20 3[72 3[39 52 3[53 4[25 3[48 54 2[53 10[35 35 35 35 35 35 35 1[20 1[20 2[27 27 27[39 9[55 2[{}48 66.4176 /CMR8 rf /Fm 137[33 39 25 31 31 30 34 2[59 21 35 27 23 4[35 10[39 54 1[39 45 39 1[50 1[42 1[53 1[46 1[37 1[54 1[42 49 54 48 50 1[35 2[34 1[20 24[31 10[33 5[34 30 1[30 14[{}37 58.1154 /CMMI7 rf /Fn 174[61 53 80[{}2 83.022 /CMSS10 rf /Fo 132[42 34 40 39 55 38 45 28 34 35 38 42 42 47 68 21 38 25 25 42 38 25 38 42 38 38 42 11[62 59 47 2[56 64 62 74 52 2[32 1[64 54 56 63 59 58 62 6[25 4[42 42 42 42 42 2[25 30 25 30[49 47 51 11[{}55 83.022 /CMTI10 rf /Fp 171[55 1[60 82[{}2 83.022 /MSBM10 rf /Fq 144[33 2[19 8[30 4[33 32[51 5[33 33 33 33 33 33 33 33 4[51 1[26 26 40[{}16 58.1154 /CMR7 rf /Fr 141[69 6[42 23 2[42 42 9[55 3[59 4[50 70 6[63 4[60 44 64 44 11[0 4[55 83 15[83 3[83 1[65 65 4[65 65 2[65 1[42 42 4[65 6[65 23 65{}30 83.022 /CMSY10 rf /Fs 135[47 59 40 48 30 39 37 37 42 2[73 25 43 34 29 48 40 41 39 43 36 9[48 69 1[48 57 49 51 63 66 53 63 67 81 57 71 46 36 69 65 53 61 69 59 63 62 44 1[65 42 65 23 23 18[54 4[39 1[54 8[41 1[48 3[41 36 1[37 1[47 53 1[51 55 8[{}62 83.022 /CMMI10 rf /Ft 129[42 2[42 37 44 44 60 44 46 32 33 33 44 46 42 46 69 23 44 25 23 46 42 25 37 46 37 46 42 1[23 42 23 1[23 1[62 1[85 62 62 60 46 61 1[57 65 62 76 52 1[43 30 62 65 54 57 63 60 59 62 3[65 1[23 23 42 42 42 42 42 42 42 42 42 42 42 23 28 23 65 1[32 32 23 19[42 4[69 1[46 48 5[62 55 1[65 69 1[{}83 83.022 /CMR10 rf /Fu 134[50 50 69 50 53 37 38 39 1[53 48 53 80 27 50 1[27 53 48 29 44 53 42 53 46 12[66 3[65 1[75 91 57 2[36 4[73 69 68 72 6[27 48 48 48 48 48 48 48 48 48 48 1[27 4[37 37 27[53 12[{}48 83.022 /CMBX10 rf /Fv 135[71 2[75 52 53 55 1[75 67 75 112 37 2[37 75 1[41 61 75 60 1[65 12[94 1[100 1[92 2[128 81 2[50 5[97 1[102 6[37 4[67 67 67 67 67 67 1[37 46[{}33 119.552 /CMBX12 rf /Fw 200[30 30 30 30 30 30 30 30 48[{}8 49.8132 /CMR6 rf /Fx 173[54 82[{}1 74.7198 /MSBM10 rf /Fy 134[47 1[64 1[49 34 35 36 2[44 49 1[25 1[27 25 3[41 49 39 49 43 13[49 5[84 1[69 9[67 6[25 45[49 12[{}22 74.7198 /CMBX9 rf /Fz 132[38 1[41 41 55 41 43 30 30 30 41 43 38 43 64 21 41 23 21 43 38 23 34 43 34 43 38 2[38 5[58 79 1[58 55 43 1[60 52 60 58 70 48 2[28 2[50 52 59 55 54 58 60 5[21 38 3[38 38 38 38 38 38 1[21 26 21 24[38 4[64 1[43 12[{}60 74.7198 /CMR9 rf /FA 134[51 6[38 3[54 81 1[51 1[27 54 2[43 3[49 13[54 17[73 65[{}11 99.6264 /CMR12 rf /FB 132[66 1[70 70 1[70 73 51 52 51 70 73 66 73 111 36 70 40 36 73 1[40 58 73 58 73 66 13[73 4[99 8[90 23[43 45[{}28 143.462 /CMR17 rf end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: A4 end %%EndSetup %%Page: 1 1 TeXDict begin 1 0 bop 847 872 a FB(Exact)43 b(con)l(trollabilit)l(y)f (in)h(pro)7 b(jections)44 b(for)721 1054 y(three-dimensional)f(Na)l (vier{Stok)l(es)g(equations)1584 1295 y FA(Armen)33 b(Shiriky)m(an)1429 1469 y Fz(Lab)r(oratoire)27 b(de)e(Math)n(\023)-36 b(ematiques)1245 1561 y(Univ)n(ersit)n(\023)g(e)26 b(de)f(P)n(aris-Sud)h(XI,)f(B^)-38 b(atimen)n(t)26 b(425)1488 1652 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y(cussion)g(on)g(the)h(sub)5 b(ject.)639 1220 y Fu(Notation)p Ft(.)64 b(W)-7 b(e)37 b(denote)f(b)n(y)h Fp(R)1710 1232 y Fq(+)1801 1220 y Ft(the)g(half-line)g([0)p Fs(;)14 b Ft(+)p Fr(1)p Ft(\))36 b(and)g(b)n(y)h Fs(J)2942 1232 y Fm(T)3031 1220 y Ft(the)g(in)n(ter-) 515 1320 y(v)-5 b(al)24 b([0)p Fs(;)14 b(T)e Ft(].)35 b(If)25 b Fs(s)e Fr(\025)g Ft(1)i(and)f Fs(r)i Fr(\025)d Ft(0)h(are)g(some)g(in)n(tegers,)h(then)g(w)n(e)g(set)g Fs(J)2708 1332 y Fm(T)2760 1320 y Ft(\()p Fs(r)n(;)14 b(s)p Ft(\))24 b(=)e([)p Fs(t)3099 1332 y Fm(r)3136 1320 y Fs(;)14 b(t)3203 1332 y Fm(r)r Fq(+1)3324 1320 y Ft(\),)515 1420 y(where)30 b Fs(t)788 1432 y Fm(r)854 1420 y Ft(=)e Fs(r)r(T)9 b(=s)p Ft(.)47 b(Let)31 b Fs(J)37 b Fr(\032)28 b Fp(R)1583 1432 y Fq(+)1669 1420 y Ft(b)r(e)k(a)e(closed)h(in)n(terv) -5 b(al,)31 b(let)g Fs(D)g Fr(\032)d Fp(R)2810 1390 y Fq(3)2879 1420 y Ft(b)r(e)j(a)g(b)r(ounded)515 1519 y(domain,)24 b(let)g Fs(X)30 b Ft(b)r(e)25 b(a)e(Banac)n(h)g(space)g(with)h(a)f (norm)h Fr(k)11 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b(follo)n(wing)f(result)h(is)g(a)g(simple)g (consequence)g(of)g(the)515 4821 y(Brou)n(w)n(er)25 b(theorem.)1926 5255 y(4)p eop end %%Page: 5 5 TeXDict begin 5 4 bop 515 523 a Fu(Prop)s(osition)30 b(1.1.)40 b Fo(Supp)l(ose)30 b(ther)l(e)g(is)g(a)g(c)l(onstant)f Fs(")23 b Fr(2)g Ft(\(0)p Fs(;)14 b(R)q Ft(\))30 b Fo(such)g(that)1247 675 y Fr(k)p Fs(\010)p Ft(\()p Fs(u)p Ft(\))19 b Fr(\000)f Fs(u)p Fr(k)1648 687 y Fm(F)1725 675 y Fr(\024)23 b Fs(")84 b Fo(for)31 b(any)f Fs(u)23 b Fr(2)g Fs(B)2440 687 y Fm(F)2495 675 y Ft(\()p Fs(R)q Ft(\))p Fs(:)562 b Ft(\(1.1\))515 828 y Fo(Then)30 b Fs(\010)p Ft(\()p Fs(B)881 840 y Fm(F)937 828 y Ft(\()p Fs(R)q Ft(\)\))24 b Fr(\033)e Fs(B)1271 840 y Fm(F)1326 828 y Ft(\()p Fs(R)e Fr(\000)e Fs(")p Ft(\))p Fo(.)515 970 y(Pr)l(o)l(of.)43 b Ft(Let)23 b(us)g(\014x)g(an)n (y)g(p)r(oin)n(t)28 b(^)-47 b Fs(u)23 b Fr(2)g Fs(B)1711 982 y Fm(F)1766 970 y Ft(\()p Fs(R)10 b Fr(\000)f Fs(")p Ft(\))24 b(and)f(consider)f(the)h(con)n(tin)n(uous)f(mapping)1221 1122 y 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Fs(B)t Ft(\()p Fs(w)r Ft(\))i(+)e Fs(B)t Ft(\()p Fs(v)s(;)c(w)r Ft(\))20 b(+)e Fs(B)t Ft(\()p Fs(w)r(;)c(v)s Ft(\))2551 4249 y Fi(\001)2614 4316 y Ft(=)22 b Fs(f)9 b Ft(\()p Fs(t)p Ft(\))p Fs(;)340 b Ft(\(1.3\))2423 4440 y Fs(w)r Ft(\(0\))24 b(=)e Fs(w)2760 4452 y Fq(0)2798 4440 y Fs(;)387 b Ft(\(1.4\))515 4602 y(where)19 b Fs(L)804 4614 y Fm(E)882 4602 y Ft(=)k Fn(Q)p Fs(L)p Ft(,)e Fs(B)t Ft(\()p Fs(v)s(;)14 b(w)r Ft(\))24 b(=)f(\005)p Fr(f)p Ft(\()p Fs(v)s(;)14 b Fr(r)p Ft(\))p Fs(w)r Fr(g)p Ft(,)22 b(and)d Fs(v)27 b Fr(2)c Fs(L)2336 4572 y Fq(4)2373 4602 y Ft(\()p Fs(J)2451 4614 y Fm(T)2503 4602 y Fs(;)14 b(V)19 b Ft(\))h(and)f Fs(f)32 b Fr(2)23 b Fs(L)3020 4572 y Fq(2)3057 4602 y Ft(\()p Fs(J)3135 4614 y Fm(T)3188 4602 y Fs(;)14 b(E)3291 4572 y Fk(?)3347 4602 y Ft(\))515 4701 y(are)26 b(giv)n(en)h(functions.)37 b(W)-7 b(e)28 b(set)590 4854 y Fr(X)649 4866 y Fm(T)725 4854 y Ft(=)22 b Fs(C)6 b Ft(\()p Fs(J)955 4866 y Fm(T)1008 4854 y Fs(;)14 b(V)19 b Ft(\))g Fr(\\)f Fs(L)1293 4820 y 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1563 y Fr(\000)33 b Fi(b)-61 b Fs(w)2841 1575 y Fq(0)2879 1563 y Fr(k)2921 1575 y Fm(V)3001 1563 y Fr(\024)22 b Fs(";)58 b Ft(\(1.5\))722 1746 y Fo(pr)l(oblem)37 b Ft(\(1.3\))p Fo(,)30 b Ft(\(1.4\))g Fo(has)g(a)g(unique)g(solution)g Fs(w)25 b Fr(2)f(X)2490 1758 y Fm(T)2542 1746 y Ft(\()p Fs(E)5 b Ft(\))p Fo(.)553 1912 y Fu(\(ii\))41 b Fo(L)l(et)1110 2011 y Fr(R)23 b Ft(:)g Fs(L)1306 1977 y Fq(4)1343 2011 y Ft(\()p Fs(J)1421 2023 y Fm(T)1474 2011 y Fs(;)14 b(H)1587 1977 y Fq(1)1624 2011 y Ft(\))k Fr(\002)g Fs(L)1814 1977 y Fq(2)1851 2011 y Ft(\()p Fs(J)1929 2023 y Fm(T)1982 2011 y Fs(;)c(E)2085 1977 y Fk(?)2141 2011 y Ft(\))k Fr(\002)g Ft(\()p Fs(V)38 b Fr(\\)19 b Fs(E)2532 1977 y Fk(?)2588 2011 y Ft(\))k Fr(!)g(X)2808 2023 y Fm(T)2861 2011 y Ft(\()p Fs(E)5 b Ft(\))722 2161 y Fo(b)l(e)45 b(an)g(op)l(er)l(ator)h(that)e(is)i(de\014ne)l(d)f(on)f(the)h(set)g(of) g(functions)g Ft(\()p Fs(v)s(;)14 b(f)t(;)g(w)3071 2173 y Fq(0)3109 2161 y Ft(\))45 b Fo(satis-)722 2260 y(fying)50 b Ft(\(1.5\))41 b Fo(and)g(takes)h(e)l(ach)g(triple)g Ft(\()p Fs(v)s(;)14 b(f)t(;)g(w)2235 2272 y Fq(0)2273 2260 y Ft(\))41 b Fo(to)g(the)h(solution)f Fs(w)47 b Fr(2)d(X)3196 2272 y Fm(T)3248 2260 y Ft(\()p Fs(E)5 b Ft(\))722 2360 y Fo(of)52 b Ft(\(1.3\))o Fo(,)34 b Ft(\(1.4\))p Fo(.)48 b(Then)33 b Fr(R)g Fo(is)g(uniformly)h(Lipschitz)g (c)l(ontinuous,)f(and)h(its)e(Lips-)722 2460 y(chitz)f(c)l(onstant)e (do)l(es)h(not)f(exc)l(e)l(e)l(d)h Fs(C)6 b Fo(.)639 2626 y Ft(W)-7 b(e)33 b(no)n(w)f(consider)f(Eq.)h(\(1.3\))g(in)h(whic)n (h)f Fs(E)37 b Ft(is)c(a)f(\014nite-dimensional)g(v)n(ector)f(space)515 2725 y(spanned)d(b)n(y)h(some)f(eigenfunctions)h(of)f(the)i(Stok)n(es)e (op)r(erator)f Fs(L)p Ft(.)40 b(Namely)-7 b(,)29 b(let)g Fr(f)p Fs(e)3189 2737 y Fm(j)3223 2725 y Fr(g)g Ft(b)r(e)515 2825 y(a)j(complete)h(set)g(of)g(normalised)f(eigenfunctions)h(for)f Fs(L)p Ft(,)j(let)e Fs(H)2595 2837 y Fm(N)2691 2825 y Ft(b)r(e)g(the)h(v)n(ector)d(span)515 2924 y(of)i Fr(f)p Fs(e)696 2936 y Fm(j)730 2924 y Fs(;)14 b Ft(1)32 b Fr(\024)g Fs(j)37 b Fr(\024)32 b Fs(N)9 b Fr(g)p Ft(,)34 b(and)f(let)g Fs(H)1649 2894 y Fk(?)1642 2947 y Fm(N)1739 2924 y Ft(b)r(e)g(the)h (orthogonal)d(complemen)n(t)i(of)g Fs(H)3065 2936 y Fm(N)3161 2924 y Ft(in)h(the)515 3024 y(space)29 b Fs(H)7 b Ft(.)43 b(W)-7 b(e)31 b(denote)e(b)n(y)h Fn(P)1467 3036 y Fm(N)1560 3024 y Ft(and)f Fn(Q)1784 3036 y Fm(N)1877 3024 y Ft(the)i(orthogonal)c (pro)5 b(jections)29 b(in)h Fs(H)37 b Ft(on)n(to)29 b(the)515 3124 y(subspaces)d Fs(H)963 3136 y Fm(N)1054 3124 y Ft(and)h Fs(H)1291 3094 y Fk(?)1284 3147 y Fm(N)1347 3124 y Ft(,)h(resp)r(ectiv) n(ely)-7 b(.)639 3223 y(Let)28 b(us)g(consider)e(the)i(equation)1252 3406 y(_)-49 b Fs(w)21 b Ft(+)d Fs(\027)5 b(L)1492 3418 y Fm(N)1555 3406 y Ft(\()p Fs(w)21 b Ft(+)e Fs(v)s Ft(\))f(+)g Fn(Q)1988 3418 y Fm(N)2052 3406 y Fs(B)t Ft(\()p Fs(w)j Ft(+)d Fs(v)s Ft(\))24 b(=)e Fs(f)9 b Ft(\()p Fs(t)p Ft(\))p Fs(;)541 b Ft(\(1.6\))515 3589 y(where)27 b Fs(L)812 3601 y Fm(N)897 3589 y Ft(=)c 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515 623 y(\(2.2\))o(.)49 b(Prop)r(osition)30 b(1.2)h(with)h Fs(E)j Ft(=)30 b Fr(f)p Ft(0)p Fr(g)g Ft(and)h Fs(v)i Fr(\021)d Ft(0)h(implies)h(that)g Fs(D)r Ft(\()p Fr(R)p Ft(\))g(is)g(an)f(op)r(en)515 722 y(subset)g(of)g Fs(V)39 b Fr(\002)20 b Fs(L)1100 692 y Fq(2)1137 722 y Ft(\()p Fs(J)1215 734 y Fm(T)1268 722 y Fs(;)14 b(H)7 b Ft(\),)31 b(and)g Fr(R)h Ft(is)e(lo)r(cally)h(Lipsc)n(hitz)f(con)n(tin)n(uous)g (on)h Fs(D)r Ft(\()p Fr(R)p Ft(\).)48 b(F)-7 b(or)515 822 y(an)n(y)27 b Fs(t)c Fr(2)g Fs(J)849 834 y Fm(T)901 822 y Ft(,)28 b(w)n(e)f(denote)h(b)n(y)f Fr(R)1527 834 y Fm(t)1557 822 y Ft(\()p Fs(u)1637 834 y Fq(0)1674 822 y Fs(;)14 b(\021)s Ft(\))28 b(the)g(restriction)e(of)i Fr(R)p Ft(\()p Fs(u)2600 834 y Fq(0)2637 822 y Fs(;)14 b(\021)s Ft(\))28 b(to)g(the)g(time)g Fs(t)p Ft(.)639 922 y(Let)e Fs(X)j Fr(\032)23 b Fs(L)1029 891 y Fq(2)1065 922 y Ft(\()p Fs(J)1143 934 y Fm(T)1196 922 y Fs(;)14 b(H)7 b Ft(\))25 b(b)r(e)h(an)f(arbitrary)e(v)n(ector)h(space,)h(not)g 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Fm(T)2125 2482 y Fs(;)14 b(E)5 b Ft(\))30 b Fo(is)h(UA)n(C)e(in)h(time)g Fs(T)12 b Fo(.)639 2648 y Ft(Theorem)29 b(2.5)g(will)h(b)r(e)g(established)g (in)g(Section)g(3.)43 b(Here)29 b(w)n(e)g(sho)n(w)g(that)h(the)h(exact) 515 2748 y(con)n(trollabilit)n(y)39 b(in)j(a)f(pro)5 b(jection)40 b(is)h(a)g(simple)h(consequence)e(of)h(UA)n(C;)h(in)f(the) h(next)515 2848 y(subsection,)27 b(w)n(e)g(deduce)h(some)f(corollaries) e(from)i(Theorems)g(2.2)f(and)i(2.5.)515 3014 y Fo(Pr)l(o)l(of)j(of)f (The)l(or)l(em)h(2.2.)43 b Ft(Let)35 b(us)f(\014x)h(a)f(time)h Fs(T)45 b(>)34 b Ft(0,)i(an)f(initial)f(p)r(oin)n(t)h Fs(u)2958 3026 y Fq(0)3029 3014 y Fr(2)g Fs(V)19 b Ft(,)36 b(and)515 3113 y(a)c(pro)5 b(jection)32 b Fn(P)1042 3125 y Fm(F)1130 3113 y Ft(:)g Fs(H)39 b Fr(!)32 b Fs(H)40 b Ft(on)n(to)32 b(a)h(\014nite-dimensional)f(subspace)h Fs(F)44 b Fr(\032)31 b Fs(H)7 b Ft(.)53 b(Recall)515 3213 y(that)33 b Fs(B)763 3225 y Fm(F)819 3213 y Ft(\()p Fs(R)q Ft(\))g(stands)g(for)g(the)h(closed)e(ball)h(in)h Fs(F)45 b Ft(of)33 b(radius)g Fs(R)h Ft(cen)n(tred)f(at)g(origin)f(and) 515 3313 y(denote)27 b(b)n(y)h Fs(C)34 b Ft(the)28 b(norm)f(of)g Fn(P)1498 3325 y Fm(F)1577 3313 y Ft(:)c Fs(H)30 b Fr(!)23 b Fs(H)7 b Ft(.)37 b(Let)28 b(us)f(\014x)h(an)n(y)f Fs(R)d(>)f(C)34 b Ft(and)27 b(c)n(ho)r(ose)g Fs(\016)f(>)c Ft(0)515 3412 y(so)27 b(small)g(that)1558 3512 y(sup)1484 3585 y Fq(^)-38 b Fm(u)p Fk(2)p Fm(B)1613 3593 y Fj(F)1660 3585 y Fq(\()p Fm(R)p Fq(\))1776 3512 y Fr(k)p Fs(e)1857 3478 y Fk(\000)p Fm(\016)r(L)1996 3512 y Ft(^)-48 b Fs(u)18 b Fr(\000)24 b Ft(^)-48 b Fs(u)p Fr(k)22 b(\024)2349 3479 y Fq(1)p 2349 3493 34 4 v 2349 3540 a(2)2392 3512 y Fs(:)793 b Ft(\(2.6\))515 3737 y(Denote)30 b(b)n(y)f Fr(K)j Ft(the)e(image)f(of)h Fs(B)1561 3749 y Fm(F)1616 3737 y Ft(\()p Fs(R)q Ft(\))h(under)e Fs(e)2051 3707 y Fk(\000)p Fm(\016)r(L)2185 3737 y Ft(.)44 b(This)30 b(is)f(a)h(compact)f(subset)h(of)g Fs(V)19 b Ft(,)515 3836 y(and)27 b(b)n(y)g(Theorem)g(2.5,)g(there)g(is)h(a)f 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b(is)h(said)f(to)h(b)r(e)515 2117 y Fo(solid)t(ly)44 b Fn(P)841 2129 y Fm(F)896 2117 y Fo(-c)l(ontr)l(ol)t(lable)g(in)e(time)h Fs(T)52 b Ft(if)42 b(for)e(an)n(y)h Fs(R)46 b(>)g Ft(0)40 b(and)i Fs(u)2723 2129 y Fq(0)2805 2117 y Fr(2)k Fs(V)60 b Ft(there)41 b(is)g(a)515 2217 y(constan)n(t)25 b Fs(")e(>)g Ft(0)j(and)g(a)g (compact)f(set)i Fr(C)g(\032)c Fs(L)1969 2187 y Fq(2)2006 2217 y Ft(\()p Fs(J)2084 2229 y Fm(T)2136 2217 y Fs(;)14 b(E)5 b Ft(\))17 b Fr(\\)f Ft(\002)2424 2229 y Fm(T)2476 2217 y Ft(\()p Fs(h;)e(u)2641 2229 y Fq(0)2677 2217 y Ft(\))27 b(suc)n(h)f(that,)h(for)f(an)n(y)515 2317 y(con)n(tin)n(uous)g (mapping)i Fs(S)f Ft(:)c Fr(C)28 b(!)23 b Fs(F)40 b Ft(satisfying)27 b(the)h(inequalit)n(y)1358 2487 y(sup)1361 2557 y Fm(\021)r Fk(2C)1497 2487 y Fr(k)p Fs(S)5 b Ft(\()p Fs(\021)s Ft(\))19 b Fr(\000)f Fn(P)1858 2499 y Fm(F)1913 2487 y Fr(R)1983 2499 y Fm(T)2036 2487 y Ft(\()p Fs(u)2116 2499 y Fq(0)2153 2487 y Fs(;)c(\021)s Ft(\))p Fr(k)2308 2499 y Fm(F)2386 2487 y Fr(\024)22 b 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b(and)g(subspace)f Fs(F)40 b Fr(\032)27 b Fs(H)7 b Ft(.)515 3329 y(As)20 b(w)n(as)g(sho)n(wn)g(in)g(the)h(pro)r(of)f(of)h(Theorem)e (2.2)h(\(see)g(Section)h(2.2\),)g(there)g(is)f(a)g(con)n(tin)n(uous)515 3428 y(mapping)27 b Fs(\011)32 b Ft(:)23 b Fs(B)1048 3440 y Fm(F)1103 3428 y Ft(\()p Fs(R)d Ft(+)e(2\))23 b Fr(!)g Fs(L)1561 3398 y Fq(2)1597 3428 y Ft(\()p Fs(J)1675 3440 y Fm(T)1728 3428 y Fs(;)14 b(E)5 b Ft(\))19 b Fr(\\)g Ft(\002)2021 3440 y Fm(T)2072 3428 y Ft(\()p Fs(h;)14 b(u)2237 3440 y Fq(0)2274 3428 y Ft(\))28 b(suc)n(h)f(that)1379 3599 y(sup)1263 3672 y Fq(^)-38 b Fm(u)p Fk(2)p Fm(B)1392 3680 y Fj(F)1439 3672 y Fq(\()p Fm(R)p Fq(+2\))1639 3599 y Fr(k)p Fn(P)1734 3611 y Fm(F)1789 3599 y Fr(R)1859 3611 y Fm(T)1912 3599 y Ft(\()p Fs(u)1992 3611 y Fq(0)2029 3599 y Fs(;)14 b(\011)9 b Ft(\()c(^)-47 b Fs(u)p Ft(\)\))19 b Fr(\000)k Ft(^)-47 b Fs(u)o Fr(k)23 b(\024)f Ft(1)p Fs(:)572 b Ft(\(2.9\))515 3836 y(Let)39 b(us)f(set)h Fr(C)46 b Ft(=)41 b Fs(\011)9 b Ft(\()p Fs(B)1284 3848 y Fm(F)1340 3836 y Ft(\()p Fs(R)27 b Ft(+)e(2\)\).)70 b(Since)39 b(dim)15 b Fs(F)53 b(<)41 b Fr(1)e Ft(and)g Fs(\011)47 b Ft(is)39 b(con)n(tin)n(uous,)h(w)n(e)515 3936 y(conclude)30 b(that)g Fr(C)35 b Ft(is)30 b(a)g(compact)g(subset)g (of)h Fs(L)2025 3906 y Fq(2)2061 3936 y Ft(\()p Fs(J)2139 3948 y Fm(T)2192 3936 y Fs(;)14 b(E)5 b Ft(\))21 b Fr(\\)f Ft(\002)2488 3948 y Fm(T)2540 3936 y Ft(\()p Fs(h;)14 b(u)2705 3948 y Fq(0)2742 3936 y Ft(\).)45 b(Let)31 b Fs(S)h Ft(:)27 b Fr(C)33 b(!)27 b Fs(F)515 4036 y Ft(b)r(e)35 b(an)f(arbitrary)e(con)n(tin)n(uous)i(mapping)g(suc)n(h)h(that)f (\(2.8\))h(holds)f(with)h Fs(")f Ft(=)h(1.)57 b(Then)515 4135 y(it)36 b(follo)n(ws)f(from)h(\(2.9\))g(that)g(the)g(mapping)g Fs(S)29 b Fr(\016)23 b Fs(\011)46 b Ft(:)38 b Fs(B)2354 4147 y Fm(F)2409 4135 y Ft(\()p Fs(R)25 b Ft(+)e(1\))37 b Fr(!)g Fs(F)48 b Ft(satis\014es)36 b(the)515 4235 y(inequalit)n(y) 1492 4334 y(sup)1375 4408 y Fq(^)-37 b Fm(u)p Fk(2)p Fm(B)1505 4416 y Fj(F)1552 4408 y Fq(\()p Fm(R)p 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b(w)n(e)f(ha)n(v)n(e)g(the)g(follo)n(wing)g(result.) 1905 5255 y(10)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 515 523 a Fu(Prop)s(osition)26 b(2.8.)37 b Fo(Supp)l(ose)27 b(that)g(the)g(c)l(onditions)g(of)h(The)l (or)l(em)g(2.2)g(ar)l(e)f(ful\014l)t(le)l(d,)h(and)515 623 y(let)j Fs(X)37 b Fo(b)l(e)31 b(a)h(ve)l(ctor)f(sp)l(ac)l(e)h (dense)f(in)g Fs(L)1770 593 y Fq(2)1807 623 y Ft(\()p Fs(J)1885 635 y Fm(T)1938 623 y Fs(;)14 b(E)5 b Ft(\))p Fo(.)42 b(Then)32 b(for)g(any)f(p)l(ositive)i(c)l(onstants)d Fs(T)515 722 y Fo(and)f Fs(R)q Fo(,)h(any)g(initial)g(function)f Fs(u)1570 734 y Fq(0)1630 722 y Fr(2)24 b Fs(V)18 b Fo(,)30 b(any)g(subsp)l(ac)l(e)f Fs(F)35 b Fr(\032)23 b Fs(H)36 b Fo(with)30 b Ft(dim)14 b Fs(F)35 b(<)23 b Fr(1)p Fo(,)29 b(and)515 822 y(any)34 b(pr)l(oje)l(ction)i Fn(P)1121 834 y Fm(F)1207 822 y Ft(:)31 b Fs(H)37 b Fr(!)31 b Fs(H)41 b Fo(onto)34 b Fs(F)12 b Fo(,)36 b(ther)l(e)e(is)g(a)h(b)l(al)t(l)g Fs(B)j Fo(in)c(a)h(\014nite-dimensional)515 922 y(subsp)l(ac)l(e)30 b Fs(Y)42 b Fr(\032)22 b Fs(X)36 b Fo(such)30 b(that)1515 1104 y Fn(P)1568 1116 y Fm(F)1623 1104 y Fr(R)1693 1116 y Fm(T)1746 1104 y Ft(\()p Fs(u)1826 1116 y Fq(0)1863 1104 y Fs(;)14 b(B)t Ft(\))23 b Fr(\033)g Fs(B)2173 1116 y Fm(F)2228 1104 y Ft(\()p Fs(R)q Ft(\))p Fs(:)788 b Ft(\(2.10\))515 1287 y Fo(In)29 b(p)l(articular,)i(we)f(c)l(an)g(take)g Fs(X)g Ft(=)22 b Fs(C)1731 1257 y Fk(1)1802 1287 y Ft(\()p Fs(J)1880 1299 y Fm(T)1933 1287 y Fs(;)14 b(E)5 b Ft(\))p Fo(.)515 1453 y(Pr)l(o)l(of.)43 b Ft(By)18 b(Prop)r(osition)f(2.7,)i (Eq.)f(\(2.1\))g(with)h Fs(\021)26 b Fr(2)e Fs(L)2197 1423 y Fq(2)2233 1453 y Ft(\()p Fs(J)2311 1465 y Fm(T)2364 1453 y Fs(;)14 b(E)5 b Ft(\))19 b(is)f(solidly)g Fn(P)2897 1465 y Fm(F)2952 1453 y Ft(-con)n(trollable)515 1553 y(in)27 b(time)h Fs(T)12 b Ft(.)36 b(Let)28 b Fs(")23 b(>)f Ft(0)27 b(and)h Fr(C)f(\032)c Fs(L)1665 1522 y Fq(2)1702 1553 y Ft(\()p Fs(J)1780 1565 y Fm(T)1832 1553 y Fs(;)14 b(E)5 b Ft(\))19 b Fr(\\)f Ft(\002)2124 1565 y Fm(T)2176 1553 y Ft(\()p Fs(h;)c(u)2341 1565 y Fq(0)2378 1553 y Ft(\))28 b(b)r(e)f(the)h(corresp)r(onding)e(con-)515 1652 y(stan)n(t)31 b(and)g(compact)f(set)h(en)n(tering)g(De\014nition)g (2.6.)47 b(It)32 b(follo)n(ws)e(from)g(Prop)r(osition)g(1.2)515 1752 y(with)d Fs(E)i Ft(=)22 b Fr(f)p Ft(0)p Fr(g)k Ft(and)h Fs(v)f Fr(\021)d Ft(0)k(that)g(\002)1660 1764 y Fm(T)1712 1752 y Ft(\()p Fs(h;)14 b(u)1877 1764 y Fq(0)1914 1752 y Ft(\))27 b(is)g(an)g(op)r(en)g(subset)g(of)g Fs(L)2776 1722 y Fq(2)2813 1752 y Ft(\()p Fs(J)2891 1764 y Fm(T)2944 1752 y Fs(;)14 b(E)5 b Ft(\).)37 b(There-)515 1851 y(fore)27 b(there)g(is)h Fs(\016)e(>)c Ft(0)28 b(suc)n(h)f(that)967 2034 y Fs(O)1030 2046 y Fm(\016)1067 2034 y Ft(\()p Fr(C)5 b Ft(\))23 b(=)1291 1967 y Fi(\010)1339 2034 y Fs(\021)k Fr(2)c Fs(L)1542 2000 y Fq(2)1579 2034 y Ft(\()p Fs(J)1657 2046 y Fm(T)1710 2034 y Fs(;)14 b(E)5 b Ft(\))23 b(:)g(dist\()p Fs(\021)s(;)14 b Fr(C)5 b Ft(\))23 b Fr(\024)g Fs(\016)2393 1967 y Fi(\011)2465 2034 y Fr(\032)f Ft(\002)2617 2046 y Fm(T)2669 2034 y Ft(\()p Fs(h;)14 b(u)2834 2046 y Fq(0)2871 2034 y Ft(\))p Fs(;)515 2217 y Ft(where)27 b(w)n(e)g(set)1367 2316 y(dist\()p Fs(\021)s(;)14 b Fr(C)5 b Ft(\))23 b(=)31 b(inf)1806 2371 y Fm(\020)t Fk(2C)1938 2316 y Fr(k)p Fs(\021)21 b Fr(\000)d Fs(\020)6 b Fr(k)2209 2331 y Fm(L)2255 2315 y Fe(2)2287 2331 y Fq(\()p Fm(J)2350 2339 y Fj(T)2395 2331 y Fm(;H)t Fq(\))2504 2316 y Fs(:)515 2516 y Ft(F)-7 b(urthermore,)36 b(since)e Fs(X)42 b Ft(is)35 b(dense)f(in)i Fs(L)1836 2486 y Fq(2)1872 2516 y Ft(\()p Fs(J)1950 2528 y Fm(T)2003 2516 y Fs(;)14 b(E)5 b Ft(\),)37 b(w)n(e)e(can)f(\014nd)i (a)e(\014nite-dimensional)515 2615 y(subspace)27 b Fs(Y)41 b Fr(\032)23 b Fs(X)34 b Ft(suc)n(h)27 b(that)1457 2798 y(sup)1459 2868 y Fm(\021)r Fk(2C)1596 2798 y Fr(k)p Fs(P)1691 2810 y Fm(Y)1748 2798 y Fs(\021)22 b Fr(\000)c Fs(\021)s Fr(k)1980 2813 y Fm(L)2026 2796 y Fe(2)2057 2813 y Fq(\()p Fm(J)2120 2821 y Fj(T)2166 2813 y Fm(;E)s Fq(\))2291 2798 y Fr(\024)k Fs(\016)o(;)730 b Ft(\(2.11\))515 3046 y(where)35 b Fs(P)816 3058 y Fm(Y)909 3046 y Ft(denotes)g(the)g (orthogonal)e(pro)5 b(jection)35 b(in)g Fs(L)2354 3016 y Fq(2)2391 3046 y Ft(\()p Fs(J)2469 3058 y Fm(T)2522 3046 y Fs(;)14 b(E)5 b Ft(\))35 b(on)n(to)g Fs(Y)18 b Ft(.)60 b(It)36 b(follo)n(ws)515 3146 y(that)1463 3246 y Fs(P)1516 3258 y Fm(Y)1574 3246 y Fr(C)27 b(\032)c Fs(O)1796 3258 y Fm(\016)1833 3246 y Ft(\()p Fr(C)5 b Ft(\))23 b Fr(\032)g Ft(\002)2122 3258 y Fm(T)2174 3246 y Ft(\()p Fs(h;)14 b(u)2339 3258 y Fq(0)2375 3246 y Ft(\))p Fs(:)737 b Ft(\(2.12\))515 3395 y(By)28 b(Prop)r(osition)e(1.2,)i(the)h (op)r(erator)e Fr(R)p Ft(\()p Fs(u)1881 3407 y Fq(0)1918 3395 y Fs(;)14 b Fr(\001)p Ft(\))24 b(:)h(\002)2147 3407 y Fm(T)2199 3395 y Ft(\()p Fs(h;)14 b(u)2364 3407 y Fq(0)2401 3395 y Ft(\))24 b Fr(!)g(X)2623 3407 y Fm(T)2704 3395 y Ft(is)29 b(lo)r(cally)e(Lipsc)n(hitz)515 3495 y(con)n(tin)n(uous.)51 b(Therefore,)33 b(taking)g Fs(\016)h(>)e Ft(0)g(su\016cien)n(tly)h (small,)h(w)n(e)e(deduce)h(from)g(\(2.11\))515 3594 y(and)27 b(\(2.12\))g(that)1238 3715 y(sup)1241 3785 y Fm(\021)r Fk(2C)1377 3715 y Fr(kR)1489 3727 y Fm(T)1541 3715 y Ft(\()p Fs(u)1621 3727 y Fq(0)1658 3715 y Fs(;)14 b(P)1748 3727 y Fm(Y)1806 3715 y Fs(\021)s Ft(\))19 b Fr(\000)f(R)2054 3727 y Fm(T)2107 3715 y Ft(\()p Fs(u)2187 3727 y Fq(0)2224 3715 y Fs(;)c(\021)s Ft(\))p Fr(k)2379 3727 y Fm(V)2459 3715 y Fr(\024)2570 3659 y Fs(")p 2557 3696 66 4 v 2557 3772 a(C)2632 3715 y(;)515 3920 y Ft(where)32 b Fs(C)38 b Ft(is)32 b(the)h(norm)f(of)g Fn(P)1467 3932 y Fm(F)1555 3920 y Ft(regarded)e(as)i(an)g(op)r(erator)f(from)h Fs(V)51 b Ft(to)32 b Fs(H)7 b Ft(.)51 b(Th)n(us,)34 b(the)515 4020 y(mapping)29 b Fs(S)5 b Ft(\()p Fs(\021)s Ft(\))27 b(=)f Fn(P)1193 4032 y Fm(F)1248 4020 y Fr(R)1318 4032 y Fm(T)1371 4020 y Ft(\()p Fs(u)1451 4032 y Fq(0)1488 4020 y Fs(;)14 b(P)1578 4032 y Fm(Y)1636 4020 y Fs(\021)s Ft(\))30 b(satis\014es)f(\(2.8\))o(.)43 b(Hence,)31 b(b)n(y)e(Prop)r (osition)f(2.7,)h(w)n(e)515 4120 y(ha)n(v)n(e)f Fn(P)761 4132 y 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Fs(H)7 b Fo(.)41 b(Then)31 b(ther)l(e)g(is)g(an)f(inte)l(ger)h Fs(k)c Fr(\025)d Ft(1)30 b Fo(dep)l(ending)i(on)f Fs(")p Fo(,)g Fs(u)2870 2572 y Fq(0)2907 2560 y Fo(,)g(and)g Fr(K)h Fo(such)515 2660 y(that)d(Eq.)i Ft(\(2.1\))e Fo(with)i Fs(\021)26 b Fr(2)e Fs(C)1426 2629 y Fk(1)1496 2660 y Ft(\()p Fs(J)1574 2672 y Fm(T)1627 2660 y Fs(;)14 b(E)1725 2672 y Fm(k)1766 2660 y Ft(\))30 b Fo(is)g(uniformly)h Fs(")p Fo(-c)l(ontr)l(ol)t(lable.)639 2811 y Ft(If)22 b(Prop)r(ositions)d(3.1)i({)g(3.3)f(are)g(established,)i(then)g(for)f (an)n(y)f Fs(")j(>)g Ft(0,)f Fs(u)2821 2823 y Fq(0)2881 2811 y Fr(2)h Fs(V)c Ft(,)k(and)e Fr(K)j(\032)515 2910 y Fs(V)46 b Ft(w)n(e)26 b(\014rst)h(use)g(Prop)r(osition)e(3.3)h(to)h (\014nd)h(an)e(in)n(teger)g Fs(k)g Fr(\025)d Ft(1)j(suc)n(h)h(that)g (Eq.)g(\(2.1\))g(with)515 3010 y Fs(\021)35 b Fr(2)d Fs(C)743 2980 y Fk(1)814 3010 y Ft(\()p Fs(J)892 3022 y Fm(T)944 3010 y Fs(;)14 b(E)1042 3022 y Fm(k)1083 3010 y Ft(\))33 b(is)g(uniformly)g(\()p Fs(";)14 b(u)1777 3022 y Fq(0)1813 3010 y Fs(;)g Fr(K)q Ft(\)-con)n(trollable.)52 b(Com)n(bining)32 b(this)h(prop)r(ert)n(y)515 3110 y(with)e(Prop)r (ositions)e(3.2)h(and)h(3.1)f(in)h(whic)n(h)g Fs(E)j Ft(=)28 b Fs(E)2219 3122 y Fm(k)q Fk(\000)p Fq(1)2345 3110 y Ft(,)k(w)n(e)f(conclude)g(that)g(Eq.)f(\(2.1\))515 3209 y(with)41 b Fs(\021)49 b Fr(2)c Fs(C)972 3179 y Fk(1)1043 3209 y Ft(\()p Fs(J)1121 3221 y Fm(T)1174 3209 y Fs(;)14 b(E)1272 3221 y Fm(k)q Fk(\000)p Fq(1)1398 3209 y Ft(\))41 b(is)g(uniformly)g(\()p Fs(";)14 b(u)2116 3221 y Fq(0)2152 3209 y Fs(;)g Fr(K)q Ft(\)-con)n(trollable.)76 b(Rep)r(eating)41 b(this)515 3309 y(argumen)n(t)27 b Fs(k)21 b Fr(\000)e Ft(1)27 b(times,)i(w)n(e)f(see)f(that)i(the)f(same) g(prop)r(ert)n(y)f(is)h(true)g(for)f(Eq.)h(\(2.1\))g(with)515 3409 y Fs(\021)34 b Fr(2)c Fs(C)740 3378 y Fk(1)811 3409 y Ft(\()p Fs(J)889 3421 y Fm(T)942 3409 y Fs(;)14 b(E)5 b Ft(\).)50 b(Since)33 b Fs(")p Ft(,)g Fs(u)1515 3421 y Fq(0)1552 3409 y Ft(,)g(and)f Fr(K)h Ft(are)f(arbitrary)-7 b(,)31 b(this)h(completes)g(the)h(pro)r(of)e(of)515 3508 y(Theorem)c(2.5.)639 3608 y(T)-7 b(o)29 b(pro)n(v)n(e)f(the)h(ab)r(o)n (v)n(e)f(prop)r(ositions,)h(w)n(e)f(rep)r(eat)h(the)h(sc)n(heme)f(used) g(in)g([Shi06])g(\(see)515 3707 y(Sections)34 b(2.2,)i(3.2,)g(and)f (3.3\).)57 b(The)35 b(imp)r(ortan)n(t)g(p)r(oin)n(t)g(no)n(w)f(is)h (that)g(w)n(e)f(ha)n(v)n(e)g(to)g(fol-)515 3807 y(lo)n(w)i(carefully)g (the)i(dep)r(endence)f(of)g(con)n(trols)f(on)h(the)g(\014nal)g(state)42 b(^)-47 b Fs(u)o Ft(.)66 b(The)37 b(pro)r(ofs)f(of)515 3907 y(Prop)r(ositions)22 b(3.1)i({)g(3.3)g(are)g(carried)f(out)h(in)h (next)g(three)f(subsections.)36 b(Here)24 b(w)n(e)g(form)n(u-)515 4006 y(late)29 b(a)f(lemma)h(on)g(uniform)g Fs(")p Ft(-con)n (trollabilit)n(y;)f(it)i(will)f(b)r(e)h(used)f(in)g(Sections)g(3.2)f({) h(3.4.)515 4106 y(As)36 b(b)r(efore,)i(w)n(e)e(\014x)g(a)f(constan)n(t) h Fs(")g(>)h Ft(0,)h(an)e(initial)g(p)r(oin)n(t)g Fs(u)2532 4118 y Fq(0)2606 4106 y Fr(2)i Fs(V)19 b Ft(,)38 b(and)e(a)g(compact) 515 4206 y(set)27 b Fr(K)e(\032)d Fs(V)d Ft(.)515 4357 y Fu(Lemma)33 b(3.4.)41 b Fo(L)l(et)30 b Fs(X)r(;)14 b(Y)43 b Fr(\032)25 b Fs(L)1536 4327 y Fq(2)1573 4357 y Ft(\()p Fs(J)1651 4369 y Fm(T)1704 4357 y Fs(;)14 b(H)7 b Ft(\))30 b Fo(b)l(e)h(ve)l(ctor)h(sp)l(ac)l(es)f(such)g(that)g Fs(X)37 b Fo(is)31 b(c)l(ontaine)l(d)515 4456 y(in)f(the)h(closur)l(e)f (of)i Fs(Y)49 b Fo(and)31 b(Eq.)g Ft(\(2.1\))f Fo(with)h Fs(\021)c Fr(2)e Fs(X)37 b Fo(is)30 b(uniformly)i Fs(")p Fo(-c)l(ontr)l(ol)t(lable.)41 b(Then)515 4556 y(ther)l(e)28 b(is)h(a)g(\014nite-dimensional)h(subsp)l(ac)l(e)e Fs(Y)1934 4568 y Fq(0)1995 4556 y Fr(\032)22 b Fs(Y)48 b Fo(such)28 b(that)h(Eq.)g Ft(\(2.1\))f Fo(with)h Fs(\021)d Fr(2)e Fs(Y)3254 4568 y Fq(0)3320 4556 y Fo(is)515 4656 y(uniformly)30 b Fs(")p Fo(-c)l(ontr)l(ol)t(lable.)639 4807 y Ft(T)-7 b(o)34 b(pro)n(v)n(e)e(this)j(lemma,)g(it)g(su\016ces)e(to)h(rep)r(eat) g(the)g(argumen)n(t)f(used)h(in)g(the)h(pro)r(of)515 4907 y(of)j(Prop)r(osition)f(2.8;)43 b(w)n(e)38 b(shall)f(not)i(dw)n (ell)f(on)g(it.)70 b(Also)38 b(note)g(that)g(an)g(analogue)f(of)515 5006 y(Lemma)27 b(3.4)g(is)g(true)h(for)f(Eq.)g(\(3.1\).)1905 5255 y(13)p eop end %%Page: 14 14 TeXDict begin 14 13 bop 515 523 a Fg(3.2)112 b(Extension)38 b(principle:)51 b(pro)s(of)37 b(of)h(Prop)s(osition)g(3.1)515 676 y Ft(W)-7 b(e)34 b(need)h(to)f(sho)n(w)g(that)g(if)h(Eq.)f(\(3.1\)) g(with)h(\()p Fs(\021)s(;)14 b(\020)6 b Ft(\))35 b Fr(2)g Fs(C)2391 646 y Fk(1)2461 676 y Ft(\()p Fs(J)2539 688 y Fm(T)2592 676 y Fs(;)14 b(E)28 b Fr(\002)22 b Fs(E)5 b Ft(\))35 b(is)f(uniformly)515 776 y Fs(")p Ft(-con)n(trollable,)f (then)i(so)f(is)g(Eq.)f(\(2.1\))h(with)h Fs(\021)i Fr(2)e Fs(C)2248 746 y Fk(1)2318 776 y Ft(\()p Fs(J)2396 788 y Fm(T)2449 776 y Fs(;)14 b(E)5 b Ft(\).)57 b(Since)34 b Fs(C)2952 746 y Fk(1)3023 776 y Ft(\()p Fs(J)3101 788 y Fm(T)3154 776 y Fs(;)14 b(E)5 b Ft(\))34 b(is)515 876 y(dense)27 b(in)g Fs(L)894 845 y Fq(2)931 876 y Ft(\()p Fs(J)1009 888 y Fm(T)1061 876 y Fs(;)14 b(E)5 b 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2053 y Fs(:)711 b Ft(\(3.6\))515 2227 y(W)-7 b(e)30 b(no)n(w)f(de\014ne)h(a)g(sequence)f(of)h(con)n(tin) n(uous)f(mappings)h Fs(\011)2438 2239 y Fm(k)2505 2227 y Ft(:)d Fr(K)h(!)f Fs(L)2813 2196 y Fq(2)2850 2227 y Ft(\()p Fs(J)2928 2239 y Fm(T)2981 2227 y Fs(;)14 b(E)5 b Ft(\))30 b(b)n(y)g(the)515 2326 y(follo)n(wing)c(rule:)639 2485 y Fr(\017)41 b Ft(for)27 b(an)n(y)32 b(^)-47 b Fs(u)23 b Fr(2)g(K)29 b Ft(and)f Fs(k)e Fr(\025)c Ft(1,)28 b(set)1167 2659 y Fs(v)1207 2671 y Fm(k)1248 2659 y Ft(\()p Fs(t;)20 b Ft(^)-48 b Fs(u)p Ft(\))23 b(=)g Fs(')1592 2671 y Fm(k)1633 2659 y Ft(\()p Fs(t)p Ft(\))p Fs(\020)6 b Ft(\()p Fs(t;)21 b Ft(^)-49 b Fs(u)q Ft(\))18 b(+)g Fn(P)2123 2638 y Fi(b)2103 2659 y Fr(R)2173 2671 y Fm(t)2203 2659 y Ft(\()p Fs(u)2283 2671 y Fq(0)2320 2659 y Fs(;)2369 2638 y Fi(b)2357 2659 y Fs(\011)9 b Ft(\()c(^)-47 b Fs(u)p Ft(\)\))p Fs(;)97 b(t)23 b Fr(2)h Fs(J)2859 2671 y Fm(T)2911 2659 y Ft(;)274 b(\(3.7\))639 2864 y Fr(\017)41 b Ft(denote)28 b(b)n(y)f Fs(w)1164 2876 y 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3901 y Fm(k)2661 3889 y Ft(\))h Fr(\000)f Fs(h)p Ft(\))p Fs(:)269 b Ft(\(3.10\))515 4094 y(W)-7 b(e)33 b(claim)g(that)g(for)g(su\016cien)n(tly)g(large)f Fs(k)j Fr(\025)c Ft(1)i(the)h(function)f Fs(\011)2617 4106 y Fm(k)2691 4094 y Ft(is)g(w)n(ell)g(de\014ned)g(and)515 4194 y(con)n(tin)n(uous)26 b(and)i(satis\014es)f(\(2.5\))o(.)37 b(Indeed,)28 b(let)g(us)g(write)1018 4378 y Fs(v)s Ft(\()p Fs(t;)20 b Ft(^)-48 b Fs(u)p Ft(\))24 b(=)e Fn(P)1423 4357 y Fi(b)1404 4378 y Fr(R)1474 4390 y Fm(t)1504 4378 y Ft(\()p Fs(u)1584 4390 y Fq(0)1621 4378 y Fs(;)1669 4357 y Fi(b)1658 4378 y Fs(\011)9 b Ft(\()c(^)-47 b Fs(u)p Ft(\)\))p Fs(;)97 b(w)r Ft(\()p Fs(t;)20 b Ft(^)-48 b Fs(u)p Ft(\))24 b(=)e Fn(Q)2414 4357 y Fi(b)2394 4378 y Fr(R)2464 4390 y Fm(t)2494 4378 y Ft(\()p Fs(u)2574 4390 y Fq(0)2611 4378 y Fs(;)2660 4357 y Fi(b)2648 4378 y Fs(\011)9 b Ft(\()c(^)-47 b Fs(u)p Ft(\)\))p Fs(:)515 4551 y Ft(Then)28 b Fs(w)r Ft(\()p Fr(\001)p Fs(;)20 b Ft(^)-48 b Fs(u)p Ft(\))23 b Fr(2)h(X)1126 4563 y 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Ft(\()p Fr(\001)p Fs(;)d Ft(^)-48 b Fs(u)p Ft(\))p Fr(\000)p Fs(f)9 b Ft(\()p Fr(\001)p Fs(;)19 b Ft(^)-47 b Fs(u)o Ft(\))p Fr(k)2103 2918 y Fm(L)2149 2901 y Fe(2)2181 2918 y Fq(\()p Fm(J)2244 2926 y Fj(T)2289 2918 y Fm(;H)t Fq(\))2398 2903 y Ft(+)p Fr(k)p Fs(v)2545 2915 y Fm(k)2585 2903 y Ft(\()p Fr(\001)p Fs(;)20 b Ft(^)-48 b Fs(u)p Ft(\))p Fr(\000)p Fs(v)s Ft(\()p Fr(\001)p Fs(;)20 b Ft(^)-48 b Fs(u)p Ft(\))p Fr(k)3079 2918 y Fm(L)3125 2901 y Fe(4)3157 2918 y Fq(\()p Fm(J)3220 2926 y Fj(T)3265 2918 y Fm(;V)14 b Fq(\))3368 2836 y Fi(\001)3407 2903 y Fs(;)515 3077 y Ft(where)21 b Fs(C)29 b(>)23 b Ft(0)e(do)r(es)h(not)f (dep)r(end)i(on)e Fs(k)k Ft(and)i(^)-47 b Fs(u)o Ft(.)35 b(Com)n(bining)21 b(this)h(inequalit)n(y)g(with)g(\(3.13\))515 3176 y(and)27 b(\(3.14\))o(,)h(w)n(e)f(deriv)n(e)1165 3350 y(sup)1165 3420 y Fq(^)-38 b Fm(u)p Fk(2K)1309 3350 y Fr(k)p Fs(w)1410 3362 y Fm(k)1451 3350 y Ft(\()p Fr(\001)p Fs(;)19 b Ft(^)-47 b Fs(u)o Ft(\))19 b Fr(\000)f Fs(w)r Ft(\()p Fr(\001)p Fs(;)i 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1392 y Ft(\()p Fs(T)7 b(;)19 b Ft(^)-47 b Fs(u)o Ft(\))p Fr(k)2215 1404 y Fm(V)2296 1392 y Fr(\024)22 b Fs(")c Fr(\000)24 b Ft(^)-48 b Fs(";)582 b Ft(\(3.17\))515 1618 y(and)32 b(the)h(mapping)k(^)-47 b Fs(u)30 b Fr(7!)h Ft(\()p Fs(\021)s Ft(\()p Fr(\001)p Fs(;)20 b Ft(^)-48 b Fs(u)q Ft(\))p Fs(;)14 b(\020)6 b Ft(\()p Fr(\001)p Fs(;)20 b Ft(^)-48 b Fs(u)p Ft(\)\))33 b(is)f(con)n(tin)n(uous)g(as)f(an)i(op)r(erator)d(from)i Fr(K)i Ft(to)515 1718 y(the)28 b(space)f Fs(L)937 1687 y Fq(2)973 1718 y Ft(\()p Fs(J)1051 1730 y Fm(T)1104 1718 y Fs(;)14 b(E)5 b Ft(\))19 b Fr(\002)f Fs(L)1398 1687 y Fq(4)1434 1718 y Ft(\()p Fs(J)1512 1730 y Fm(T)1565 1718 y Fs(;)c(E)5 b Ft(\).)639 1817 y(T)-7 b(o)23 b(construct)f Fs(u)p Ft(\()p Fs(t;)d Ft(^)-47 b Fs(u)o Ft(\),)24 b(one)f(could)f(try) h(to)f(apply)h(the)g(argumen)n(t)e(used)i(in)g(Section)g(3.3)515 1917 y(of)37 b([Shi06)o(])g(for)f(appro)n(ximating)f(individual)i (solutions.)64 b(Unfortunately)-7 b(,)39 b(it)f(do)r(es)e(not)515 2016 y(w)n(ork)41 b(b)r(ecause)g(it)i(is)f(di\016cult)h(to)g(ensure)e (that)i(the)g(resulting)e(con)n(trol)g(functions)i Fs(\021)515 2116 y Ft(and)30 b Fs(\020)36 b Ft(con)n(tin)n(uously)30 b(dep)r(end)g(on)36 b(^)-48 b Fs(u)p Ft(.)45 b(T)-7 b(o)29 b(o)n(v)n(ercome)f(this)j(di\016cult)n(y)-7 b(,)31 b(w)n(e)f(\014rst)g (appro)n(xi-)515 2216 y(mate)21 b Fs(\021)757 2228 y Fq(1)795 2216 y Ft(\()p Fs(t;)e Ft(^)-47 b Fs(u)o Ft(\))22 b(b)n(y)f(a)g(family)g(of)h(piecewise)f(constan)n(t)f(con)n(trols)26 b(~)-48 b Fs(\021)2532 2228 y Fq(1)2570 2216 y Ft(\()p Fs(t;)19 b Ft(^)-47 b Fs(u)o Ft(\))22 b(with)g(range)e(in)i(the)515 2315 y(con)n(v)n(ex)27 b(en)n(v)n(elop)r(e)g(of)i(a)f(\014nite)h(set)f (not)h(dep)r(ending)f(on)34 b(^)-47 b Fs(u)28 b Ft(\(cf.)h(Section)f (12.3)f(in)i([AS05]\).)515 2415 y(W)-7 b(e)32 b(next)g(rep)r(eat)f(the) i(sc)n(heme)e(of)h([Shi06)o(])g(to)g(construct)f(an)h(appro)n(ximation) e(for)h(solu-)515 2515 y(tions)e(~)-48 b Fs(u)762 2527 y Fq(1)799 2515 y Ft(\()p Fs(t;)20 b Ft(^)-48 b Fs(u)p Ft(\))24 b(corresp)r(onding)e(to)30 b(~)-48 b Fs(\021)1671 2527 y Fq(1)1708 2515 y Ft(\()p Fs(t;)20 b Ft(^)-48 b Fs(u)p Ft(\).)36 b(A)24 b(di\016cult)h(p)r(oin)n(t)f(of)g(the)g(pro)r (of)f(is)h(to)g(follo)n(w)515 2614 y(the)34 b(dep)r(endence)g(of)f(the) h(con)n(trol)e(functions)i(on)k(^)-47 b Fs(u)o Ft(.)55 b(In)33 b(what)h(follo)n(ws,)g(w)n(e)f(shall)g(omit)515 2714 y(the)28 b(tilde)g(from)f(the)h(notation.)639 2859 y(The)j(realisaion)e(of)i(the)g(ab)r(o)n(v)n(e)f(sc)n(heme)g(is)h (divided)g(in)n(to)f(sev)n(eral)f(steps.)47 b(W)-7 b(e)31 b(b)r(egin)515 2959 y(with)d(a)f(generalisation)f(of)h(the)h(concept)f (of)h(uniform)g(appro)n(ximate)e(con)n(trollabilit)n(y)-7 b(.)639 3081 y Fo(Step)39 b(1)p Ft(.)67 b(Let)37 b Fs(A)j Ft(=)f Fr(f)p Fs(\021)1415 3051 y Fm(l)1412 3102 y Fq(1)1449 3081 y Fs(;)14 b(l)40 b Ft(=)f(1)p Fs(;)14 b(:)g(:)g(:)f(;)h(m)p Fr(g)39 b(\032)g Fs(E)2200 3093 y Fq(1)2275 3081 y Ft(b)r(e)f(a)f (\014nite)h(set.)66 b(F)-7 b(or)37 b(an)n(y)f(in)n(te-)515 3181 y(ger)30 b 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Ft(\()c(^)-47 b Fs(u)p Ft(\))34 b Fr(2)f Ft(\002)1253 1033 y Fm(T)1305 1021 y Ft(\()p Fs(h;)14 b(u)1470 1033 y Fq(0)1507 1021 y Ft(\))34 b(for)f(an)n(y)39 b(^)-48 b Fs(u)33 b Fr(2)h(K)q Ft(,)i(and)e(\(3.16\))f(holds.)55 b(A)34 b(pro)r(of)f(of)h(the)515 1121 y(follo)n(wing)29 b(lemma)h(is)g(based)g(on)g(a)g(standard)f(argumen)n(t)g(of)h(the)h (con)n(trol)e(theory)g(and)h(is)515 1220 y(giv)n(en)d(in)g(the)h(App)r (endix.)515 1384 y Fu(Lemma)35 b(3.5.)42 b Fo(L)l(et)33 b(us)f(assume)h(that)g(Eq.)h Ft(\(3.2\))e Fo(with)i Fs(\021)e Fr(2)d Fs(C)2536 1354 y Fk(1)2607 1384 y Ft(\()p Fs(J)2685 1396 y Fm(T)2737 1384 y Fs(;)14 b(E)2835 1396 y Fq(1)2873 1384 y Ft(\))33 b Fo(is)g(uniformly)515 1484 y Fs(")p Fo(-c)l(ontr)l(ol)t(lable.)53 b(Then)34 b(ther)l(e)h(is)f(a)h(\014nite) f(set)f Fs(A)f Ft(=)e Fr(f)p Fs(\021)2300 1454 y Fm(l)2297 1504 y Fq(1)2334 1484 y Fs(;)14 b(l)33 b Ft(=)d(1)p Fs(;)14 b(:)g(:)g(:)g(;)g(m)p Fr(g)30 b(\032)h Fs(E)3053 1496 y Fq(1)3124 1484 y Fo(and)k(an)515 1583 y(inte)l(ger)28 b Fs(s)23 b Fr(\025)f Ft(1)28 b Fo(such)g(that)f(Eq.)i Ft(\(3.2\))f Fo(with)g Fs(\021)e Fr(2)e Fs(P)2085 1595 y Fm(s)2121 1583 y Ft(\()p Fs(J)2199 1595 y Fm(T)2251 1583 y Fs(;)14 b(A)p Ft(\))28 b Fo(is)h(uniformly)g Fs(")p Fo(-c)l(ontr)l(ol)t(lable.)639 1747 y Ft(Let)22 b Fs(\011)833 1759 y Fq(1)893 1747 y Ft(:)h Fr(K)i(!)e Fs(P)1186 1759 y Fm(s)1222 1747 y Ft(\()p Fs(J)1300 1759 y Fm(T)1352 1747 y Fs(;)14 b(A)p Ft(\))23 b(b)r(e)f(the)g(function)g(constructed)g (in)g(Lemma)f(3.5.)34 b(W)-7 b(e)22 b(write)1360 1994 y Fs(\011)1411 2006 y Fq(1)1448 1994 y Ft(\()5 b(^)-47 b Fs(u)p Ft(\))24 b(=)e Fs(\021)1712 2006 y Fq(1)1750 1994 y Ft(\()p Fs(t;)d Ft(^)-47 b Fs(u)o Ft(\))24 b(=)2070 1890 y Fm(m)2039 1915 y Fi(X)2047 2094 y Fm(l)p Fq(=1)2173 1994 y Fs(')2227 2006 y Fm(l)2253 1994 y Ft(\()p Fs(t;)19 b Ft(^)-47 b Fs(u)p Ft(\))p Fs(\021)2476 1959 y Fm(l)2473 2014 y Fq(1)2510 1994 y Fs(:)515 2253 y Ft(The)37 b(de\014nition)g(of)f (the)h(space)f Fs(P)1612 2265 y Fm(s)1648 2253 y Ft(\()p Fs(J)1726 2265 y Fm(T)1779 2253 y Fs(;)14 b(A)p Ft(\))37 b(and)g(of)f(its)h(metric)g(imply)g(that)g(the)g(func-)515 2352 y(tions)27 b Fs(')772 2364 y Fm(l)826 2352 y Ft(ha)n(v)n(e)f(the)i (form)1473 2606 y Fs(')1527 2618 y Fm(l)1553 2606 y Ft(\()p Fs(t;)19 b Ft(^)-47 b Fs(u)p Ft(\))23 b(=)1844 2503 y Fm(s)p Fk(\000)p Fq(1)1842 2528 y Fi(X)1844 2703 y Fm(r)r Fq(=0)1976 2606 y Fs(c)2012 2618 y Fm(lr)2070 2606 y Ft(\()5 b(^)-47 b Fs(u)p Ft(\))p Fs(I)2218 2618 y Fm(r)n(;s)2303 2606 y Ft(\()p Fs(t)p Ft(\))p Fs(;)747 b Ft(\(3.18\))515 2858 y(where)27 b Fs(c)791 2870 y Fm(lr)872 2858 y Ft(:)c Fr(K)i(!)e Fp(R)k Ft(are)g(non-negativ)n(e)f(con)n(tin)n(uous)g (functions)i(suc)n(h)g(that)1430 2996 y Fm(m)1399 3021 y Fi(X)1406 3200 y Fm(l)p Fq(=1)1533 3100 y Fs(c)1569 3112 y Fm(lr)1627 3100 y Ft(\()5 b(^)-47 b Fs(u)p Ft(\))23 b(=)g(1)82 b(for)27 b(an)n(y)32 b(^)-47 b Fs(u)23 b Fr(2)g(K)r Fs(:)515 3367 y Ft(Since)j Fs(\021)774 3337 y Fm(l)771 3387 y Fq(1)831 3367 y Fr(2)e(F)8 b Ft(\()p Fs(E)d Ft(\),)27 b(b)n(y)e(Lemma)h(3.3)f(in)h([Shi06)o(],)h(there)f(are)e(v)n(ectors)h Fs(\021)2759 3337 y Fm(l)2784 3367 y Fs(;)14 b(\020)2863 3337 y Fq(1)p Fm(l)2922 3367 y Fs(;)g(:)g(:)g(:)g(;)g(\020)3149 3337 y Fm(k)q(l)3235 3367 y Fr(2)23 b Fs(E)515 3466 y Ft(and)k(non-negativ)n(e)f(constan)n(ts)h Fs(\025)1581 3478 y Fq(1)p Fm(l)1640 3466 y Fs(;)14 b(:)g(:)g(:)f(;)h(\025)1872 3478 y Fm(k)q(l)1962 3466 y Ft(suc)n(h)28 b(that)2330 3404 y Fi(P)2417 3491 y Fm(j)2466 3466 y Fs(\025)2514 3478 y Fm(j)s(l)2594 3466 y Ft(=)22 b(1)28 b(and)885 3744 y Fs(B)t Ft(\()p Fs(u)p Ft(\))18 b Fr(\000)g Fs(\021)1209 3709 y Fm(l)1206 3764 y Fq(1)1267 3744 y Ft(=)1396 3640 y Fm(k)1355 3665 y Fi(X)1357 3842 y Fm(j)s Fq(=1)1488 3744 y Fs(\025)1536 3756 y Fm(j)s(l)1593 3676 y Fi(\000)1631 3744 y Fs(B)t Ft(\()p Fs(u)1778 3756 y Fq(1)1834 3744 y Ft(+)g Fs(\020)1959 3709 y Fm(j)s(l)2016 3744 y Ft(\))g(+)g Fs(\027)5 b(L\020)2294 3709 y Fm(j)s(l)2351 3676 y Fi(\001)2407 3744 y Fr(\000)18 b Fs(\021)2534 3709 y Fm(l)2643 3744 y Ft(for)27 b Fs(u)c Fr(2)g Fs(V)c(:)515 4012 y Ft(It)28 b(follo)n(ws)e(that)i(the)g(function)g Fs(u)1573 4024 y Fq(1)1610 4012 y Ft(\()p Fr(\001)p Ft(\))c(=)e Fr(R)p Ft(\()p Fs(u)1958 4024 y Fq(0)1996 4012 y Fs(;)14 b(\011)2084 4024 y Fq(1)2121 4012 y Ft(\()5 b(^)-47 b Fs(u)p Ft(\)\))28 b(is)f(a)h(solution)f(of)g(the)h(equation)598 4299 y Fs(@)642 4311 y Fm(t)671 4299 y Fs(u)719 4311 y Fq(1)774 4299 y Ft(+)18 b Fs(\027)5 b(Lu)1008 4311 y Fq(1)1063 4299 y Ft(+)1188 4195 y Fm(k)1146 4220 y Fi(X)1149 4397 y Fm(j)s Fq(=1)1311 4195 y Fm(m)1280 4220 y Fi(X)1287 4398 y Fm(l)p Fq(=1)1414 4299 y Fs(\025)1462 4311 y Fm(j)s(l)1519 4299 y Fs(')1573 4311 y Fm(l)1598 4299 y Ft(\()p Fs(t;)20 b Ft(^)-48 b Fs(u)p Ft(\))1777 4231 y Fi(\000)1815 4299 y Fs(B)t Ft(\()p Fs(u)1962 4311 y Fq(1)2018 4299 y Ft(+)18 b Fs(\020)2143 4264 y Fm(j)s(l)2200 4299 y Ft(\))h(+)f Fs(\027)5 b(L\020)2479 4264 y Fm(j)s(l)2535 4231 y Fi(\001)2029 4602 y Ft(=)22 b Fs(h)p Ft(\()p Fs(t)p Ft(\))d(+)2402 4498 y Fm(k)2360 4523 y Fi(X)2363 4700 y Fm(j)s Fq(=1)2524 4498 y Fm(m)2494 4523 y Fi(X)2501 4702 y Fm(l)p Fq(=1)2628 4602 y Fs(\025)2676 4614 y Fm(j)s(l)2732 4602 y Fs(')2786 4614 y Fm(l)2812 4602 y Ft(\()p Fs(t;)h Ft(^)-48 b Fs(u)p Ft(\))p Fs(\021)3035 4568 y Fm(l)3061 4602 y Fs(:)83 b Ft(\(3.19\))p 515 4771 1146 4 v 607 4825 a Fw(6)642 4848 y Fl(W)-6 b(e)24 b(emphasise)g(that,)h(in)e(con)n(trast)i(to)f (De\014nition)g(2.4)g(in)f(whic)n(h)h(the)g(v)n(ector)h(space)g Fd(X)j Fl(is)23 b(endo)n(w)n(ed)515 4927 y(with)f(the)g(norm)g(of)g Fd(L)1103 4904 y Fw(2)1137 4927 y Fl(\()p Fd(J)1203 4938 y Fc(T)1252 4927 y Fd(;)11 b(H)5 b Fl(\),)23 b(the)f(mapping)h Fd(\011)1867 4936 y Fw(1)1923 4927 y Fl(is)f(required)f(to)i(b)r(e)f (con)n(tin)n(uous)i(with)e(resp)r(ect)g(to)h(the)515 5006 y(top)r(ology)i(of)e Fd(P)930 5014 y Fc(s)963 5006 y Fl(\()p Fd(J)1029 5017 y Fc(T)1078 5006 y Fd(;)11 b(A)p Fl(\),)24 b(whic)n(h)g(is)f(stronger)g(than)i(that)g(of)e Fd(L)2226 4983 y Fw(2)2261 5006 y Fl(\()p Fd(J)2327 5017 y Fc(T)2376 5006 y Fd(;)11 b(H)5 b Fl(\).)1905 5255 y Ft(17)p eop end %%Page: 18 18 TeXDict begin 18 17 bop 515 523 a Ft(Indexing)25 b(the)g(pairs)g(\()p Fs(j;)14 b(l)r Ft(\))25 b(b)n(y)g(a)g(single)g(sequence)f Fs(i)f Ft(=)g(1)p Fs(;)14 b(:)g(:)g(:)f(;)h(q)s Ft(,)26 b(w)n(e)e(can)h(write)g(\(3.19\))g(as)556 776 y Fs(@)600 788 y Fm(t)630 776 y Fs(u)678 788 y Fq(1)726 776 y Ft(+)12 b Fs(\027)5 b(L)920 659 y Fi(\022)981 776 y Fs(u)1029 788 y Fq(1)1077 776 y Ft(+)1198 668 y Fm(q)1154 697 y Fi(X)1160 874 y Fm(i)p Fq(=1)1288 776 y Fs( )1342 788 y Fm(i)1369 776 y Ft(\()p Fs(t;)20 b Ft(^)-48 b Fs(u)p Ft(\))p Fs(\020)1590 742 y Fm(i)1618 659 y Fi(\023)1691 776 y Ft(+)1812 668 y Fm(q)1768 697 y Fi(X)1774 874 y Fm(i)p Fq(=1)1902 776 y Fs( )1956 788 y Fm(i)1983 776 y Ft(\()p Fs(t;)20 b Ft(^)-48 b Fs(u)p Ft(\))p Fs(B)t Ft(\()p Fs(u)2309 788 y Fq(1)2359 776 y Ft(+)12 b Fs(\020)2478 742 y Fm(i)2505 776 y Ft(\))24 b(=)e Fs(h)p Ft(\()p Fs(t)p Ft(\))12 b(+)g Fs(\021)s Ft(\()p Fs(t;)20 b Ft(^)-48 b Fs(u)p Ft(\))p Fs(:)42 b Ft(\(3.20\))515 1042 y(Here)37 b Fs(\020)763 1012 y Fm(i)830 1042 y Fr(2)j Fs(E)5 b Ft(,)39 b Fs(i)g Ft(=)g(1)p Fs(;)14 b(:)g(:)g(:)f(;)h(q)s Ft(,)40 b(are)c(some)h(v)n(ectors,)h Fs(\021)s Ft(\()p Fr(\001)p Fs(;)20 b Ft(^)-48 b Fs(u)p Ft(\))38 b(denotes)f(the)g(sum)h (on)f(the)515 1142 y(righ)n(t-hand)26 b(side)i(of)34 b(\(3.19\))o(,)28 b(and)1467 1404 y Fs( )1521 1416 y Fm(i)1549 1404 y Ft(\()p Fs(t;)20 b Ft(^)-48 b Fs(u)p Ft(\))23 b(=)1841 1300 y Fm(s)p Fk(\000)p Fq(1)1839 1325 y Fi(X)1840 1500 y Fm(r)r Fq(=0)1972 1404 y Fs(d)2015 1416 y Fm(ir)2076 1404 y Ft(\()5 b(^)-47 b Fs(u)p Ft(\))p Fs(I)2224 1416 y Fm(r)n(;s)2309 1404 y Ft(\()p Fs(t)p Ft(\))p Fs(;)741 b Ft(\(3.21\))515 1662 y(where)27 b Fs(d)798 1674 y Fm(ir)881 1662 y Fr(2)d Fs(C)6 b Ft(\()p Fr(K)q Ft(\))29 b(are)d(non-negativ)n(e)g(functions)i(suc)n(h)f(that) 2534 1600 y Fi(P)2622 1687 y Fm(i)2663 1662 y Fs(d)2706 1674 y Fm(ir)2790 1662 y Fr(\021)c Ft(1.)639 1787 y Fo(Step)31 b(2)p Ft(.)40 b(W)-7 b(e)29 b(no)n(w)f(appro)n(ximate)e Fs(u)1776 1799 y Fq(1)1813 1787 y Ft(\()p Fs(t;)14 b Fr(\001)p Ft(\))29 b(b)n(y)f(solutions)g(of)h(Eq.)f(\(3.1\))o(.)40 b(T)-7 b(o)28 b(this)h(end,)515 1887 y(w)n(e)h(\014rst)h(assume)f(that) h(there)g(is)f(only)h(one)f(in)n(terv)-5 b(al)30 b(of)h(constancy)-7 b(,)31 b(that)g(is,)g Fs(s)e Ft(=)f(1.)46 b(In)515 1986 y(this)30 b(case,)h(the)f(sums)h(in)f(\(3.18\))g(and)g(\(3.21\))g(con)n (tain)g(only)g(one)g(term,)h(and)f(Eq.)g(\(3.20\))515 2086 y(tak)n(es)c(the)i(form)602 2318 y Fs(@)646 2330 y Fm(t)675 2318 y Fs(u)723 2330 y Fq(1)778 2318 y Ft(+)18 b Fs(\027)5 b(L)978 2201 y Fi(\022)1039 2318 y Fs(u)1087 2330 y Fq(1)1142 2318 y Ft(+)1269 2210 y Fm(q)1225 2239 y Fi(X)1231 2416 y Fm(i)p Fq(=1)1359 2318 y Fs(d)1402 2330 y Fm(i)1430 2318 y Ft(\()g(^)-47 b Fs(u)p Ft(\))p Fs(\020)1584 2284 y Fm(i)1612 2201 y Fi(\023)1691 2318 y Ft(+)1818 2210 y Fm(q)1774 2239 y Fi(X)1781 2416 y Fm(i)p Fq(=1)1908 2318 y Fs(d)1951 2330 y Fm(i)1979 2318 y Ft(\()5 b(^)-47 b Fs(u)p Ft(\))p Fs(B)t Ft(\()p Fs(u)2238 2330 y Fq(1)2294 2318 y Ft(+)18 b Fs(\020)2419 2284 y Fm(i)2447 2318 y Ft(\))23 b(=)g Fs(h)p Ft(\()p Fs(t)p Ft(\))c(+)f Fs(\021)s Ft(\()p Fs(t;)h Ft(^)-47 b Fs(u)p Ft(\))p Fs(;)87 b Ft(\(3.22\))515 2583 y(where)33 b Fs(d)804 2595 y Fm(i)864 2583 y Fr(2)g Fs(C)6 b Ft(\()p Fr(K)q Ft(\))35 b(and)e Fs(\021)j Fr(2)d Fs(C)6 b Ft(\()p Fr(K)q Fs(;)14 b(L)1767 2553 y Fq(2)1804 2583 y Ft(\()p Fs(J)1882 2595 y Fm(T)1935 2583 y Fs(;)g(E)5 b Ft(\)\).)55 b(Let)33 b(us)h(\014x)f(an)g(in)n(teger)g Fs(k)i Fr(\025)d Ft(1)h(and,)515 2683 y(follo)n(wing)28 b(a)g(classical)g(idea)h(in)g(the)h(con)n(trol)e (theory)-7 b(,)28 b(de\014ne)i(a)e(sequence)h(of)g(con)n(tin)n(uous)515 2783 y(mappings)e Fs(\020)925 2795 y Fm(k)989 2783 y Ft(:)c Fr(K)i(!)e Fs(L)1286 2752 y Fq(4)1322 2783 y Ft(\()p Fs(J)1400 2795 y Fm(T)1453 2783 y Fs(;)14 b(H)1566 2752 y Fq(2)1603 2783 y Ft(\))28 b(as)1570 2965 y Fs(\020)1606 2977 y Fm(k)1647 2965 y Ft(\()p Fs(t;)19 b Ft(^)-47 b Fs(u)p Ft(\))23 b(=)g Fs(\020)6 b Ft(\()p Fs(k)s(t=T)h(;)18 b Ft(^)-46 b Fs(u)o Ft(\))p Fs(;)843 b Ft(\(3.23\))515 3148 y(where)27 b Fs(\020)6 b Ft(\()p Fr(\001)p Fs(;)20 b Ft(^)-48 b Fs(u)p Ft(\))28 b(is)f(a)h(1-p)r(erio)r(dic)e(function)i (on)g Fp(R)f Ft(suc)n(h)h(that)585 3331 y Fs(\020)6 b Ft(\()p Fs(s;)20 b Ft(^)-48 b Fs(u)p Ft(\))23 b(=)g Fs(\020)968 3296 y Fm(i)1079 3331 y Ft(for)k(0)c Fr(\024)f Fs(s)d Fr(\000)f Ft(\()p Fs(d)1574 3343 y Fq(1)1612 3331 y Ft(\()5 b(^)-47 b Fs(u)o Ft(\))19 b(+)f Fr(\001)c(\001)g(\001)k Ft(+)h Fs(d)2067 3343 y Fm(i)p Fk(\000)p Fq(1)2179 3331 y Ft(\()5 b(^)-47 b Fs(u)p Ft(\)\))24 b Fs(<)e(d)2477 3343 y Fm(i)2505 3331 y Ft(\()5 b(^)-47 b Fs(u)p Ft(\),)28 b Fs(i)23 b Ft(=)f(1)p Fs(;)14 b(:)g(:)g(:)g(;)g(q)r(:)71 b Ft(\(3.24\))515 3513 y(It)27 b(is)f(easy)g(to)g(see)h(that)g Fr(f)p Fs(\020)1359 3525 y Fm(k)1399 3513 y Ft(\()p Fr(\001)p Fs(;)20 b Ft(^)-48 b Fs(u)p Ft(\))p Fs(;)20 b Ft(^)-48 b Fs(u)23 b Fr(2)g(K)q Fs(;)14 b(k)27 b Fr(\025)22 b Ft(1)p Fr(g)k Ft(is)g(a)h(b)r(ounded)g(subset)f(in)h Fs(L)3020 3483 y Fk(1)3090 3513 y Ft(\()p Fs(J)3168 3525 y Fm(T)3221 3513 y Fs(;)14 b(E)5 b Ft(\).)515 3613 y(Let)27 b(us)h(rewrite)f(\(3.22\))g(in)h(the)g(form)587 3795 y Fs(@)631 3807 y Fm(t)660 3795 y Fs(u)708 3807 y Fq(1)763 3795 y Ft(+)19 b Fs(\027)5 b(L)p Ft(\()p Fs(u)1030 3807 y Fq(1)1085 3795 y Ft(+)18 b Fs(\020)1204 3807 y Fm(k)1245 3795 y Ft(\()p Fs(t;)h Ft(^)-47 b Fs(u)p Ft(\)\))19 b(+)f Fs(B)t Ft(\()p Fs(u)1705 3807 y Fq(1)1760 3795 y Ft(+)g Fs(\020)1879 3807 y Fm(k)1921 3795 y Ft(\()p Fs(t;)h Ft(^)-47 b Fs(u)o Ft(\)\))24 b(=)f Fs(h)p Ft(\()p Fs(t)p Ft(\))18 b(+)g Fs(\021)s Ft(\()p Fs(t;)i Ft(^)-48 b Fs(u)p Ft(\))19 b(+)f Fs(f)2852 3807 y Fm(k)2893 3795 y Ft(\()p Fs(t;)h Ft(^)-47 b Fs(u)o Ft(\))p Fs(;)73 b Ft(\(3.25\))515 3978 y(where)27 b Fs(f)796 3990 y Fm(k)836 3978 y Ft(\()p Fs(t;)20 b Ft(^)-48 b Fs(u)p Ft(\))23 b(=)g Fs(f)1167 3990 y Fm(k)q Fq(1)1241 3978 y Ft(\()p Fs(t;)c Ft(^)-47 b Fs(u)o Ft(\))19 b(+)f Fs(f)1562 3990 y Fm(k)q Fq(2)1636 3978 y Ft(\()p Fs(t;)h Ft(^)-47 b Fs(u)o Ft(\),)758 4231 y Fs(f)799 4243 y Fm(k)q Fq(1)873 4231 y Ft(\()p Fs(t;)19 b Ft(^)-47 b Fs(u)o Ft(\))24 b(=)e Fs(\027)5 b(L)1265 4114 y Fi(\022)1326 4231 y Fs(\020)1362 4243 y Fm(k)1403 4231 y Ft(\()p Fs(t;)20 b Ft(^)-48 b Fs(u)p Ft(\))19 b Fr(\000)1727 4123 y Fm(q)1684 4152 y Fi(X)1690 4329 y Fm(i)p Fq(=1)1818 4231 y Fs(d)1861 4243 y Fm(i)1888 4231 y Ft(\()5 b(^)-47 b Fs(u)p Ft(\))p Fs(\020)2042 4197 y Fm(i)2071 4114 y Fi(\023)2132 4231 y Fs(;)1012 b Ft(\(3.26\))758 4521 y Fs(f)799 4533 y Fm(k)q Fq(2)873 4521 y Ft(\()p Fs(t;)19 b Ft(^)-47 b Fs(u)o Ft(\))24 b(=)e Fs(B)t Ft(\()p Fs(u)1309 4533 y Fq(1)1346 4521 y Ft(\()p Fs(t;)e Ft(^)-48 b Fs(u)p Ft(\))19 b(+)f Fs(\020)1663 4533 y Fm(k)1704 4521 y Ft(\()p Fs(t;)i Ft(^)-48 b Fs(u)p Ft(\)\))19 b Fr(\000)2065 4417 y Fm(i)2017 4442 y Fi(X)2023 4619 y Fm(i)p Fq(=1)2151 4521 y Fs(d)2194 4533 y Fm(i)2221 4521 y Ft(\()5 b(^)-47 b Fs(u)p Ft(\))p Fs(B)t Ft(\()p Fs(u)2480 4533 y Fq(1)2518 4521 y Ft(\()p Fs(t;)19 b Ft(^)-47 b Fs(u)o Ft(\))19 b(+)f Fs(\020)2840 4487 y Fm(i)2868 4521 y Ft(\))p Fs(:)244 b Ft(\(3.27\))515 4776 y(W)-7 b(e)19 b(shall)f(need)h(the)g(follo)n(wing)f(result,)i(whic)n(h) f(will)g(b)r(e)g(pro)n(v)n(ed)e(in)i(the)g(next)g(steps.)34 b(Denote)515 4876 y(b)n(y)27 b Fs(B)693 4888 y Fm(V)751 4876 y Ft(\()p Fs(u;)14 b(r)r Ft(\))28 b(the)g(closed)f(ball)g(in)h Fs(V)47 b Ft(of)27 b(radius)g Fs(r)k Ft(cen)n(tred)c(at)g Fs(u)p Ft(.)1905 5255 y(18)p eop end %%Page: 19 19 TeXDict begin 19 18 bop 515 523 a Fu(Lemma)32 b(3.6.)41 b Fo(F)-6 b(or)31 b(any)g Fs(")1400 535 y Fq(0)1461 523 y Fs(>)25 b Ft(0)30 b Fo(ther)l(e)h(is)g(an)f(inte)l(ger)h Fs(k)2359 535 y Fq(0)2421 523 y Fr(\025)24 b Ft(1)30 b Fo(and)i(a)f(c)l(onstant)e Fs(\016)3186 535 y Fq(0)3248 523 y Fs(>)24 b Ft(0)515 623 y Fo(such)30 b(that)f(for)i(any)f Fs(k)c Fr(\025)d Fs(k)1365 635 y Fq(0)1402 623 y Fo(,)35 b Ft(^)-47 b Fs(u)23 b Fr(2)g(K)q Fo(,)31 b(and)f Fs(v)1927 635 y Fq(0)1988 623 y Fr(2)23 b Fs(B)2129 635 y Fm(V)2187 623 y Ft(\()p Fs(u)2267 635 y Fq(0)2304 623 y Fs(;)14 b(\016)2378 635 y Fq(0)2415 623 y Ft(\))p Fo(,)30 b(the)g(pr)l(oblem) 612 805 y Fs(@)656 817 y Fm(t)685 805 y Fs(v)22 b Ft(+)c Fs(\027)5 b(L)p Ft(\()p Fs(v)22 b Ft(+)c Fs(\020)1146 817 y Fm(k)1187 805 y Ft(\()p Fs(t;)i Ft(^)-48 b Fs(u)p Ft(\)\))19 b(+)f Fs(B)t Ft(\()p Fs(v)k Ft(+)c Fs(\020)1780 817 y Fm(k)1821 805 y Ft(\()p Fs(t;)i Ft(^)-48 b Fs(u)p Ft(\)\))23 b(=)g Fs(h)p Ft(\()p Fs(t)p Ft(\))c(+)f Fs(\021)s Ft(\()p Fs(t;)i Ft(^)-48 b Fs(u)p Ft(\))p Fs(;)99 b(v)s Ft(\(0\))23 b(=)g Fs(v)3032 817 y Fq(0)3167 805 y Ft(\(3.28\))515 987 y Fo(has)30 b(a)g(unique)g(solution)g Fs(v)1357 999 y Fm(k)1398 987 y Ft(\()p Fr(\001)p Fs(;)19 b Ft(^)-47 b Fs(u)p Ft(\))23 b Fr(2)g(X)1730 999 y Fm(T)1783 987 y Fo(,)30 b(which)h(satis\014es)f(the)g(ine)l(quality)1363 1168 y Fr(k)p Fs(v)1445 1180 y Fm(k)1486 1168 y Ft(\()p Fr(\001)p Fs(;)19 b Ft(^)-47 b Fs(u)o Ft(\))19 b Fr(\000)f Fs(u)1807 1180 y Fq(1)1844 1168 y Ft(\()p Fr(\001)p Fs(;)h Ft(^)-47 b Fs(u)p Ft(\))p Fr(k)2058 1183 y Fm(C)t Fq(\()p Fm(J)2173 1191 y Fj(T)2218 1183 y Fm(;V)14 b Fq(\))2344 1168 y Fr(\024)23 b Fs(")2471 1180 y Fq(0)2508 1168 y Fs(:)636 b Ft(\(3.29\))639 1350 y(In)29 b(particular,)g(taking)f Fs(")1449 1362 y Fq(0)1511 1350 y Ft(=)j(^)-48 b Fs(")p Ft(,)29 b(where)34 b(^)-47 b Fs(")29 b Ft(is)g(the)g(constan)n(t)f(in)i (\(3.16\))o(,)f(and)g(de\014ning)515 1450 y(the)f(op)r(erator)909 1611 y Fi(b)897 1632 y Fs(\011)948 1644 y Fm(k)1012 1632 y Ft(:)23 b Fr(K)h(!)f Fs(L)1308 1598 y Fq(2)1345 1632 y Ft(\()p Fs(J)1423 1644 y Fm(T)1476 1632 y Fs(;)14 b(E)5 b Ft(\))19 b Fr(\002)f Fs(L)1770 1598 y Fq(4)1806 1632 y Ft(\()p Fs(J)1884 1644 y Fm(T)1937 1632 y Fs(;)c(E)5 b Ft(\))p Fs(;)102 b Ft(^)-47 b Fs(u)23 b Fr(7!)g Ft(\()p Fs(\021)2442 1644 y Fm(k)2483 1632 y Ft(\()p Fr(\001)p Fs(;)d Ft(^)-48 b Fs(u)p Ft(\))p Fs(;)14 b(\020)2728 1644 y Fm(k)2769 1632 y Ft(\()p Fr(\001)p Fs(;)20 b Ft(^)-48 b Fs(u)p Ft(\)\))p Fs(;)515 1824 y Ft(w)n(e)27 b(conclude)g(that)1170 1803 y Fi(b)1158 1824 y Fs(\011)1209 1836 y Fm(k)1250 1824 y Ft(\()5 b(^)-47 b Fs(u)p Ft(\))23 b Fr(2)1473 1803 y Fi(b)1463 1824 y Ft(\002)1528 1836 y Fm(T)1580 1824 y Ft(\()p Fs(h;)14 b(v)1737 1836 y Fq(0)1775 1824 y Ft(\))28 b(for)f Fs(v)2002 1836 y Fq(0)2062 1824 y Fr(2)d Fs(B)2204 1836 y Fm(V)2261 1824 y Ft(\()p Fs(u)2341 1836 y Fq(0)2378 1824 y Fs(;)14 b(")p Ft(\))28 b(and)f Fs(k)f Fr(\025)d Fs(k)2875 1836 y Fq(0)2912 1824 y Ft(,)28 b(and)1046 2017 y(sup)1046 2087 y Fq(^)-38 b Fm(u)p Fk(2K)1189 2017 y Fr(k)1250 1996 y Fi(b)1231 2017 y Fr(R)1301 2029 y Fm(T)1354 2017 y Ft(\()p Fs(u)1434 2029 y Fq(0)1471 2017 y Fs(;)1519 1996 y Fi(b)1508 2017 y Fs(\011)1559 2029 y Fm(k)1599 2017 y Ft(\()5 b(^)-47 b Fs(u)p Ft(\)\))19 b Fr(\000)f Fs(u)1893 2029 y Fq(1)1930 2017 y Ft(\()p Fs(T)7 b(;)19 b Ft(^)-47 b Fs(u)o Ft(\))p Fr(k)2176 2029 y Fm(V)2256 2017 y Fr(\024)29 b Ft(^)-48 b Fs(")83 b Ft(for)27 b Fs(k)f Fr(\025)c Fs(k)2792 2029 y Fq(0)2830 2017 y Fs(:)515 2253 y Ft(Com)n(bining)27 b(this)h(with)g(\(3.16\))o(,) g(w)n(e)f(obtain)1166 2445 y(sup)1166 2515 y Fq(^)-37 b Fm(u)p Fk(2K)1310 2445 y Fr(k)1371 2424 y Fi(b)1352 2445 y Fr(R)1422 2457 y Fm(T)1474 2445 y Ft(\()p Fs(u)1554 2457 y Fq(0)1591 2445 y Fs(;)1640 2424 y Fi(b)1628 2445 y Fs(\011)1679 2457 y Fm(k)1720 2445 y Ft(\()5 b(^)-47 b Fs(u)p Ft(\)\))19 b Fr(\000)k Ft(^)-47 b Fs(u)o Fr(k)2055 2457 y Fm(V)2136 2445 y Fs(<)22 b(")83 b Ft(for)27 b Fs(k)f Fr(\025)d Fs(k)2672 2457 y Fq(0)2709 2445 y Fs(:)515 2686 y Ft(Hence,)g(Eq.)f(\(3.1\))h(with)f(\()p Fs(\021)s(;)14 b(\020)6 b Ft(\))25 b Fr(2)e Fs(L)1650 2656 y Fq(2)1687 2686 y Ft(\()p Fs(J)1765 2698 y Fm(T)1817 2686 y Fs(;)14 b(E)5 b Ft(\))j Fr(\002)g Fs(L)2090 2656 y Fq(4)2127 2686 y Ft(\()p Fs(J)2205 2698 y Fm(T)2258 2686 y Fs(;)14 b(E)5 b Ft(\))23 b(is)f(uniformly)g Fs(")p Ft(-con)n(trollable.)639 2810 y Fo(Step)41 b(3)p Ft(.)75 b(W)-7 b(e)41 b(no)n(w)e(pro)n(v)n(e)g (Lemma)h(3.5.)73 b(Literal)40 b(rep)r(etition)g(of)g(the)g(argumen)n (ts)515 2910 y(in)c([Shi06,)i(Section)e(3.3])g(\(see)g(Step)h(2\))f (sho)n(ws)f(that)i(the)f(required)f(assertion)g(will)i(b)r(e)515 3010 y(established)27 b(if)h(w)n(e)f(pro)n(v)n(e)f(the)i(con)n(v)n (ergence)578 3192 y(sup)578 3262 y Fq(^)-38 b Fm(u)p Fk(2K)707 3124 y Fi(\000)745 3192 y Fr(k)p Fs(K)858 3204 y Fm(\027)899 3192 y Fs(f)940 3204 y Fm(k)980 3192 y Ft(\()p Fr(\001)p Fs(;)20 b Ft(^)-48 b Fs(u)p Ft(\))p Fr(k)1194 3207 y Fm(C)t Fq(\()p Fm(J)1309 3215 y Fj(T)1354 3207 y Fm(;V)14 b Fq(\))1476 3192 y Ft(+)k Fr(k)p Fs(B)t Ft(\()p Fs(K)1771 3204 y Fm(\027)1812 3192 y Fs(f)1853 3204 y Fm(k)1893 3192 y Ft(\()p Fr(\001)p Fs(;)i Ft(^)-48 b Fs(u)p Ft(\)\))p Fr(k)2139 3207 y Fm(L)2185 3190 y Fe(2)2217 3207 y Fq(\()p Fm(J)2280 3215 y Fj(T)2326 3207 y Fm(;H)t Fq(\))2434 3124 y Fi(\001)2495 3192 y Fr(!)23 b Ft(0)83 b(as)27 b Fs(k)f Fr(!)d(1)p Fs(;)58 b Ft(\(3.30\))515 3423 y(where)1396 3559 y Fs(K)1467 3571 y Fm(\027)1507 3559 y Fs(f)9 b Ft(\()p Fs(t)p Ft(\))24 b(=)1762 3446 y Fi(Z)1845 3467 y Fm(t)1808 3635 y Fq(0)1888 3559 y Fs(e)1927 3525 y Fk(\000)p Fm(\027)t Fq(\()p Fm(t)p Fk(\000)p Fm(s)p Fq(\))p Fm(L)2226 3559 y Fs(f)9 b Ft(\()p Fs(s)p Ft(\))14 b Fs(ds:)515 3755 y Ft(F)-7 b(urthermore,)28 b(in)h(view)g(of)g(the)g(calculations)f(of)h(Steps)g(3{4)e(in)i ([Shi06,)g(Section)g(3.3],)g(it)515 3855 y(su\016ces)e(to)g(sho)n(w)g (that)1250 4037 y(sup)1250 4107 y Fq(^)-38 b Fm(u)p Fk(2K)1394 4037 y Fr(k)p Fs(F)1489 4049 y Fm(k)1529 4037 y Ft(\()p Fr(\001)p Fs(;)20 b Ft(^)-48 b Fs(u)p Ft(\))p Fr(k)1743 4052 y Fm(C)t Fq(\()p Fm(J)1858 4060 y Fj(T)1903 4052 y Fm(;H)t Fq(\))2035 4037 y Fr(!)23 b Ft(0)83 b(as)27 b Fs(k)e Fr(!)f(1)p Fs(;)518 b Ft(\(3.31\))515 4292 y(where)27 b Fs(F)808 4304 y Fm(k)849 4292 y Ft(\()p Fs(t;)20 b Ft(^)-48 b Fs(u)p Ft(\))23 b(=)1139 4225 y Fi(R)1194 4246 y Fm(t)1178 4322 y Fq(0)1237 4292 y Fs(f)1278 4304 y Fm(k)1319 4292 y Ft(\()p Fs(s;)c Ft(^)-47 b Fs(u)o Ft(\))14 b Fs(ds)p Ft(.)37 b(T)-7 b(o)28 b(this)g(end,)f(w)n(e)h (\014rst)f(note)h(that)2757 4262 y Fq(7)1057 4474 y Fr(k)p Fs(F)1152 4486 y Fm(k)1193 4474 y Ft(\()p Fr(\001)p Fs(;)19 b Ft(^)-47 b Fs(u)o Ft(\))p Fr(k)1406 4489 y Fm(C)t Fq(\()p Fm(J)1521 4497 y Fj(T)1566 4489 y Fm(;H)t Fq(\))1698 4474 y Fr(!)23 b Ft(0)83 b(as)27 b Fs(k)f Fr(!)d(1)k Ft(for)g(an)n(y)33 b(^)-48 b Fs(u)23 b Fr(2)g(K)r Fs(:)330 b Ft(\(3.32\))515 4656 y(Supp)r(ose)31 b(no)n(w)g(that)h(w)n(e)f(ha)n (v)n(e)g(pro)n(v)n(ed)f(the)i(uniform)f(equicon)n(tin)n(uit)n(y)g(of)h (the)f(family)h(of)515 4756 y(mappings)1309 4855 y Fu(f)1338 4867 y Fm(k)1402 4855 y Ft(:)23 b Fr(K)i(!)e Fs(L)1699 4821 y Fq(1)1736 4855 y Ft(\()p Fs(J)1814 4867 y Fm(T)1866 4855 y Fs(;)14 b(H)7 b Ft(\))p Fs(;)103 b Ft(^)-48 b Fs(u)23 b Fr(7!)g Fs(f)2349 4867 y Fm(k)2389 4855 y Ft(\()p Fr(\001)p Fs(;)d Ft(^)-48 b Fs(u)p Ft(\))p Fs(:)583 b Ft(\(3.33\))p 515 4929 1146 4 v 607 4983 a Fw(7)642 5006 y Fl(See)24 b(Step)h(5)f(in)f([Shi06,)g(Section)i(3.3].)1905 5255 y Ft(19)p eop end %%Page: 20 20 TeXDict begin 20 19 bop 515 523 a Ft(In)28 b(this)h(case,)e(the)i (family)1379 456 y Fi(\010)1433 523 y Ft(^)-48 b Fs(u)24 b Fr(7!)1606 456 y Fi(R)1661 477 y Fk(\001)1645 553 y Fq(0)1699 523 y Fs(f)1740 535 y Fm(k)1781 523 y Ft(\()p Fs(s;)19 b Ft(^)-47 b Fs(u)o Ft(\))14 b Fs(ds;)g(k)27 b Fr(\025)d Ft(1)2302 456 y Fi(\011)2378 523 y Ft(is)k(uniformly)g (equicon)n(tin)n(uous)515 623 y(from)f Fr(K)i Ft(to)f Fs(C)6 b Ft(\()p Fs(J)1048 635 y Fm(T)1100 623 y Fs(;)14 b(V)19 b Ft(\).)38 b(Com)n(bining)27 b(this)g(prop)r(ert)n(y)g(with)h (\(3.32\))o(,)g(w)n(e)f(arriv)n(e)f(at)h(\(3.31\))o(.)639 747 y Fo(Step)k(4)p Ft(.)40 b(W)-7 b(e)29 b(no)n(w)f(sho)n(w)g(that)h (\(3.33\))f(is)h(uniformly)f(equicon)n(tin)n(uous.)39 b(The)29 b(explicit)515 846 y(form)n(ulas)24 b(\(3.26\))h(and)g (\(3.27\))f(and)i(standard)e(estimates)h(for)g(the)h(bilinear)e(form)h Fs(B)30 b Ft(sho)n(w)515 946 y(that)c(it)h(su\016ces)f(to)g(pro)n(v)n (e)f(that)h(the)h(function)32 b(^)-48 b Fs(u)23 b Fr(7!)g Fs(\020)2235 958 y Fm(k)2276 946 y Ft(\()p Fr(\001)p Fs(;)d 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b(are)d(non-negativ)n(e)g(functions)i(whose)f(sum)h(is)515 1572 y(equal)27 b(to)g(1.)639 1672 y(F)-7 b(or)27 b(an)n(y)g(in)n (teger)g Fs(s)c Fr(\025)f Ft(1,)28 b(let)g(us)f(set)1525 1906 y Fs(\011)1585 1872 y Fm(s)1576 1927 y Fq(1)1621 1906 y Ft(\()5 b(^)-47 b Fs(u)o Ft(\))24 b(=)1874 1802 y Fm(m)1843 1827 y Fi(X)1851 2006 y Fm(l)p Fq(=1)1977 1906 y Fs( )2031 1918 y Fm(ls)2088 1906 y Ft(\()p Fs(t;)c Ft(^)-48 b Fs(u)p Ft(\))p Fs(\021)2311 1872 y Fm(l)2308 1927 y Fq(1)2346 1906 y Fs(;)515 2173 y Ft(where)18 b Fs( )800 2185 y Fm(ls)857 2173 y Ft(\()p Fs(t;)i Ft(^)-48 b Fs(u)p Ft(\))23 b(=)1154 2151 y(~)1147 2173 y Fs(\020)1183 2185 y Fm(l)1209 2173 y Ft(\()p Fs(r)r(T)9 b(=s;)19 b Ft(^)-47 b Fs(u)o Ft(\))20 b(for)e Fs(t)23 b Fr(2)h Fs(J)1851 2185 y Fm(T)1903 2173 y Ft(\()p Fs(r)n(;)14 b(s)p Ft(\).)35 b(It)19 b(is)g(clear)f(that)h Fs(\011)2713 2143 y Fm(s)2704 2193 y Fq(1)2749 2173 y Ft(\()p Fr(\001)p Ft(\))h(is)e(a)h(con)n(tin)n (uous)515 2272 y(function)j(from)g Fr(K)h Ft(to)f Fs(P)1260 2284 y Fm(s)1295 2272 y Ft(\()p Fs(J)1373 2284 y Fm(T)1426 2272 y Fs(;)14 b(A)p Ft(\),)24 b(where)d Fs(A)i Ft(=)g Fr(f)p Fs(\021)2097 2242 y Fm(l)2094 2293 y Fq(1)2131 2272 y Fs(;)14 b(l)24 b Ft(=)f(1)p Fs(;)14 b(:)g(:)g(:)f(;)h(m)p Fr(g)p Ft(.)34 b(F)-7 b(urthermore,)22 b(since)515 2372 y Fr(K)i(\032)f Fs(V)46 b Ft(is)28 b(compact,)f(it)h(is)g(not)f (di\016cult)i(to)e(sho)n(w)g(that)1119 2544 y(sup)1119 2615 y Fq(^)-38 b Fm(u)p Fk(2K)1263 2544 y Fr(k)p Fs(\011)1365 2510 y Fm(s)1356 2565 y Fq(1)1400 2544 y Ft(\()5 b(^)-47 b Fs(u)o Ft(\))19 b Fr(\000)f Fs(\011)1664 2556 y Fq(1)1701 2544 y Ft(\()5 b(^)-47 b Fs(u)p Ft(\))p Fr(k)1855 2559 y Fm(L)1901 2543 y Fe(2)1933 2559 y Fq(\()p Fm(J)1996 2567 y Fj(T)2041 2559 y Fm(;H)t Fq(\))2173 2544 y Fr(!)23 b Ft(0)83 b(as)27 b Fs(s)c Fr(!)g(1)p Fs(:)515 2771 y Ft(Prop)r(osition)32 b(1.2)g(no)n(w)h(implies)h(that)g Fs(\011)1820 2740 y Fm(s)1811 2791 y Fq(1)1855 2771 y Ft(\()5 b(^)-47 b Fs(u)p Ft(\))33 b Fr(2)h Ft(\002)2154 2783 y Fm(T)2206 2771 y Ft(\()p Fs(h;)14 b(u)2371 2783 y Fq(0)2407 2771 y Ft(\))34 b(for)f(an)n(y)38 b(^)-47 b Fs(u)32 b Fr(2)i(K)h Ft(and)e(su\016-)515 2870 y(cien)n(tly)27 b(large)f Fs(s)p Ft(,)i(and)g(w)n(e)f(ha)n(v)n(e)929 3043 y(sup)929 3113 y Fq(^)-38 b Fm(u)p Fk(2K)1072 3043 y Fr(kR)1184 3055 y Fm(T)1237 3043 y Ft(\()p Fs(u)1317 3055 y Fq(0)1354 3043 y Fs(;)14 b(\011)1451 3008 y Fm(s)1442 3063 y Fq(1)1486 3043 y Ft(\()5 b(^)-47 b Fs(u)p Ft(\)\))19 b Fr(\000)f(R)1802 3055 y Fm(T)1855 3043 y Ft(\()p Fs(u)1935 3055 y Fq(0)1972 3043 y Fs(;)c(\011)2060 3055 y Fq(1)2097 3043 y Ft(\()5 b(^)-47 b Fs(u)o Ft(\)\))p Fr(k)2282 3055 y Fm(V)2363 3043 y Fr(!)23 b Ft(0)83 b(as)27 b Fs(s)c Fr(!)g(1)p Fs(:)515 3269 y Ft(Com)n(bining)29 b(this)h(with)g (\(3.16\),)g(w)n(e)f(conclude)h(that)g(Eq.)f(\(3.2\))h(with)g Fs(\021)2796 3281 y Fq(1)2860 3269 y Fr(2)d Fs(P)2995 3281 y Fm(s)3031 3269 y Ft(\()p Fs(J)3109 3281 y Fm(T)3162 3269 y Fs(;)14 b(A)p Ft(\))30 b(is)515 3368 y(uniformly)d Fs(")p Ft(-con)n(trollable.)515 3641 y Fv(References)515 3823 y Ft([AS05])92 b(A.)23 b(A.)h(Agrac)n(hev)d(and)i(A.)h(V.)g(Saryc) n(hev,)e Fo(Navier{Stokes)27 b(e)l(quations:)37 b(c)l(ontr)l(ol-)845 3923 y(lability)29 b(by)f(me)l(ans)f(of)h(low)h(mo)l(des)f(for)l(cing)p Ft(,)f(J.)d(Math.)i(Fluid)f(Mec)n(h.)g Fu(7)g Ft(\(2005\),)845 4022 y(108{152.)515 4184 y([AS06])p 845 4184 250 4 v 354 w(,)40 b Fo(Contr)l(ol)t(lability)h(of)e(2D)g(Euler)f(and)h (Navier{Stokes)h(e)l(quations)e(by)845 4284 y(de)l(gener)l(ate)30 b(for)l(cing)p Ft(,)f(Comm)n(un.)e(Math.)h(Ph)n(ys.)f(\(2006\),)f(to)h (app)r(ear.)515 4446 y([BS01])95 b(V.)42 b(Barbu)f(and)h(S.)g(S.)g (Sritharan,)i Fo(Flow)g(invarianc)l(e)h(pr)l(eserving)e(fe)l(e)l(db)l (ack)845 4545 y(c)l(ontr)l(ol)t(lers)33 b(for)g(the)g(Navier{Stokes)i (e)l(quation)p Ft(,)d(J.)e(Math.)h(Anal.)g(Appl.)h Fu(255)845 4645 y Ft(\(2001\),)26 b(no.)h(1,)g(281{307.)515 4807 y([BT04])81 b(V.)44 b(Barbu)f(and)g(R.)h(T)-7 b(riggiani,)46 b Fo(Internal)f(stabilization)h(of)g(Navier{Stokes)845 4907 y(e)l(quations)35 b(with)h(\014nite-dimensional)h(c)l(ontr)l(ol)t (lers)p Ft(,)f(Indiana)d(Univ.)h(Math.)g(J.)845 5006 y Fu(53)27 b Ft(\(2004\),)f(no.)h(5,)h(1443{1494.)1905 5255 y(23)p eop end %%Page: 24 24 TeXDict begin 24 23 bop 515 523 a Ft([CF96])86 b(J.-M.)26 b(Coron)f(and)i(A.)g(V.)g(F)-7 b(ursik)n(o)n(v,)25 b Fo(Glob)l(al)30 b(exact)f(c)l(ontr)l(ol)t(lability)i(of)f(the)f Ft(2)p Fo(D)845 623 y(Navier{Stokes)36 b(e)l(quations)f(on)g(a)h (manifold)h(without)e(b)l(oundary)p Ft(,)g(Russian)e(J.)845 722 y(Math.)27 b(Ph)n(ys.)g Fu(4)g Ft(\(1996\),)g(no.)g(4,)g(429{448.) 515 885 y([Cor96])65 b(J.-M.)21 b(Coron,)g Fo(On)i(the)h(c)l(ontr)l(ol) t(lability)j(of)e Ft(2)p Fo(-D)d(inc)l(ompr)l(essible)k(p)l(erfe)l(ct)f (\015uids)p Ft(,)845 985 y(J.)i(Math.)h(Pures)e(Appl.)i(\(9\))g Fu(75)f Ft(\(1996\),)g(no.)g(2,)g(155{188.)515 1148 y([Cor99])65 b(J.-M.)f(Coron,)73 b Fo(On)64 b(the)g(nul)t(l)g(asymptotic)i (stabilization)g(of)f(the)f(two-)845 1247 y(dimensional)32 b(inc)l(ompr)l(essible)h(Euler)e(e)l(quations)g(in)g(a)g(simply)h(c)l (onne)l(cte)l(d)e(do-)845 1347 y(main)p Ft(,)45 b(SIAM)d(J.)f(Con)n (trol)f(Optim.)i Fu(37)e Ft(\(1999\),)k(no.)d(6,)j(1874{1896)37 b(\(elec-)845 1446 y(tronic\).)515 1609 y([Cor96])65 b(J.-M.)28 b(Coron,)g Fo(On)j(the)g(c)l(ontr)l(ol)t(lability)i(of)f (the)f Ft(2)p Fo(-D)f(inc)l(ompr)l(essible)j(Navier{)845 1709 y(Stokes)44 b(e)l(quations)h(with)g(the)g(Navier)h(slip)g(b)l (oundary)f(c)l(onditions)p Ft(,)k(ESAIM)845 1809 y(Con)n(tr^)-42 b(ole)26 b(Optim.)i(Calc.)f(V)-7 b(ar.)27 b Fu(1)g Ft(\(1995/96\),)e (35{75)h(\(electronic\).)515 1971 y([F)n(C99])88 b(E.)27 b(F)-7 b(ern\023)-42 b(andez-Cara,)25 b Fo(On)k(the)h(appr)l(oximate)h (and)f(nul)t(l)g(c)l(ontr)l(ol)t(lability)h(of)g(the)845 2071 y(Navier{Stokes)i(e)l(quations)p Ft(,)f(SIAM)e(Rev.)h Fu(41)f Ft(\(1999\),)f(no.)i(2,)f(269{277)d(\(elec-)845 2171 y(tronic\).)515 2333 y([FE99])89 b(A.)42 b(V.)h(F)-7 b(ursik)n(o)n(v)41 b(and)h(O.)h(Y)-7 b(u.)43 b(Eman)n(uilo)n(v,)h Fo(Exact)g(c)l(ontr)l(ol)t(lability)h(of)g(the)845 2433 y(Navier{Stokes)55 b(and)f(Boussinesq)g(e)l(quations)p Ft(,)61 b(Usp)r(ekhi)54 b(Mat.)g(Nauk)g Fu(54)845 2533 y Ft(\(1999\),)26 b(no.)h(3)g(\(327\),)g(93{146.)515 2696 y([F)-7 b(ur95])74 b(A.)18 b(V.)h(F)-7 b(ursik)n(o)n(v,)19 b Fo(Exact)i(b)l(oundary)h(zer)l(o)g(c)l(ontr)l(ol)t(lability)h(of)g (thr)l(e)l(e-dimensional)845 2795 y(Navier{Stokes)32 b(e)l(quations)p Ft(,)e(J.)g(Dynam.)f(Con)n(trol)f(Systems)i Fu(1)f Ft(\(1995\),)f(no.)h(3,)845 2895 y(325{350.)515 3058 y([F)-7 b(ur01])74 b(A.)27 b(V.)g(F)-7 b(ursik)n(o)n(v,)26 b Fo(Stabilizability)32 b(of)e(two-dimensional)h(Navier{Stokes)f(e)l (qua-)845 3157 y(tions)e(with)g(help)i(of)f(a)f(b)l(oundary)h(fe)l(e)l (db)l(ack)h(c)l(ontr)l(ol)p Ft(,)c(J.)g(Math.)g(Fluid)h(Mec)n(h.)e Fu(3)845 3257 y Ft(\(2001\),)h(no.)h(3,)g(259{301.)515 3420 y([F)-7 b(ur04])74 b(A.)26 b(V.)g(F)-7 b(ursik)n(o)n(v,)25 b Fo(Stabilization)30 b(for)f(the)g(3D)f(Navier{Stokes)i(system)e(by)h (fe)l(e)l(d-)845 3519 y(b)l(ack)g(b)l(oundary)h(c)l(ontr)l(ol)p Ft(,)d(Discrete)g(Con)n(tin.)f(Dyn.)i(Syst.)f Fu(10)f Ft(\(2004\),)g(no.)g(1-2,)845 3619 y(289{314.)515 3782 y([Gla00])70 b(O.)21 b(Glass,)h Fo(Exact)i(b)l(oundary)i(c)l(ontr)l(ol) t(lability)g(of)f(3-D)f(Euler)h(e)l(quation)p Ft(,)e(ESAIM)845 3881 y(Con)n(trol)j(Optim.)i(Calc.)f(V)-7 b(ar.)27 b Fu(5)h Ft(\(2000\),)e(1{44)g(\(electronic\).)515 4044 y([Ima98])59 b(O.)19 b(Y)-7 b(u.)21 b(Iman)n(uvilo)n(v,)f Fo(On)j(exact)f(c)l(ontr)l(ol)t(lability)j(for)f(the)f(Navier{Stokes)h (e)l(qua-)845 4144 y(tions)p Ft(,)g(ESAIM)f(Con)n(trol)f(Optim.)h (Calc.)g(V)-7 b(ar.)22 b Fu(3)h Ft(\(1998\),)g(97{131)d (\(electronic\).)515 4307 y([Shi06])85 b(A.)22 b(Shiriky)n(an,)g Fo(Appr)l(oximate)k(c)l(ontr)l(ol)t(lability)h(of)e(thr)l(e)l (e-dimensional)i(Navier{)845 4406 y(Stokes)i(e)l(quations)p Ft(,)f(Comm)n(un.)g(Math.)f(Ph)n(ys.)g(\(2006\),)f(to)i(app)r(ear.)515 4569 y([T)-7 b(a)n(y97])63 b(M.)38 b(E.)g(T)-7 b(a)n(ylor,)40 b Fo(Partial)h(Di\013er)l(ential)g(Equations.)f(I{III)p Ft(,)g(Springer-V)-7 b(erlag,)845 4669 y(New)27 b(Y)-7 b(ork,)27 b(1996-1997.)515 4832 y([T)-7 b(em79])41 b(R.)50 b(T)-7 b(emam,)57 b Fo(Navier{Stokes)52 b(Equations)p Ft(,)58 b(North-Holland,)d(Amsterdam,)845 4931 y(1979.)1905 5255 y(24)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0512270728408--