df ( x )= df ( x )= f ( x ) ;f ( x ) y ( x ;u ) y ( x ;u ) x (0)= x y t> 0 x (0)= x y t> 0 x Ix u 2 U U x 1 ;x 2 2 M h =[ h :::h ] x 2 M y = h ( x )1 • i • p x _ = f ( x )+ g ( x ) u X y =[ y :::y ] m u =[ u :::u ] p

Testo completo

(1) 

(2)      "!$#%'&)(*'&)(,+-".0/21%354767+-89&;:<&;=;:<&)>?+-"@A!5+B.%!53C676<+-D%A+B474<35:FE;&;G%"@"H6JIK8L3M:NA47"476<35(,A.%+B. @"8".%+B"3O!J&)#%:7\]QPS".*RT6JU&)@"@'"3(,>?H+;6<':7&;(*(,&^+V4<+;&).%@"@"+_+W35474<678%19:7D%35"47+*4<LD9LU4747""47476<67353M(,(,LUD%.%L+B.%.X+B@"4F"6<.%:73M+_&;":7.`U67"3M.:735>?47+B4<:F3;(WPa&,RbD%+B.%Y47!5"+BD9.Z356<:7:7'+;&)@"(,@'+A+ [ &;@"@"+B:<&c1%. 4<"4F6<35.%(*3M@"&d@'&e(_>?+;1%:7@H(*6<HE;&A&;f :7'&)G%"@"3!5+B. m ".%=;:7354<47 u = [u . . . u ] 3 p 1%47!MH673 y = [y . . . y ] x˙ = f (x) + g (x)u g R]P"hji R]PlkBi y = h (x) 1 ≤ i ≤ p g D%oK+O'EZ&)3(*h+0'=.%"p5[h'&;@"(,. .35..`h673^] &;@"\W!51%@'.%&3q(*D%&;3Cr98%.%89"&pM":7+;3M.9=;Q+BP @'&;:F3WD%3M@"@Qm 1%47!MH6<&XD%35@n47"476<35(*&3 x ∈ M P st`u<vnw*vQxzyq{9|*}C~€;~‚ x1, x2 ∈ M }$-ƒ;‚„< )†‡ ˆ".%D%"4F6<".%=B1%"G%"@",‰€{Z}CQ;Š* d‹"d†‡ )~€;Œ „J‚ ) )†%†~‚’<|Ž )‹”„J‹– jTŠ];%Še)Š0~‚~€" }J}CxQ—5”I‹–x ‘T ‹7˜ }O|C|J™)‘K )‹–9{ |C’ˆ‚ j†‡ 7“;| †9e„J9 )|$†9’ ~Q’< )‹”‹' u^∈%;U’$~Q•’<‰W|,„JƒB );† ‹•‹' *U }$~š•)†A~›}C W‚|$•ŠW†9|dQƒ`;|C‹'|   ’ y ^%;t’$~Q>•’<|00ƒZ)‹•‹' ,}C~€;~š W•†% Q;‹"|   _ | F ‹ ˜ C | ; ™ j – ‹ { ‚  )  † | %  C |   x(0) = x x(0) = x }O )† *QƒZ|C†9~Q‚„Jœ%|^†‡|$‹ ‹' )’< Wƒ`y )Še•†9Q Vt ƒ;>aƒ`0|QN†9  ‚ j†‡|Mž Ÿ@Q m 1%35:,=B19EZ&;35=B:7@"H'r9&)!M.%&;pO:7&X3D%@Qm35"@".%@"3VD%"D%4F6<1%".%3V=B1%1%47"!5G%H"6<@"3H6JIŽD% D%1%34F6J&)6< \D%1%.%/`1%3.%3M!5354<4<&;:7"+ˆEZ35:7Hr9!M&;:73 :7"!5+B:747+*&;=B@"¢476<:F19(,35.Z6< D%35@"@Qm &;@"=B35G%y(x:<&0D%, u)£ "3B3 P y(x , u)PdŸ 35:_>?&;:73V!5"¡cG%"47+B=B.9&>Q&):73 1. m. T. p. 1. m X. i. i. i=1. i. p. 1. 1. i. T. 2. 1. 1. 2. 2. 1. 2. st`u<vnw*v'¤yq{9|¥J{2†  ‚ )†9 f (x), f (x) }Cƒ;Q„J )†‡ ¦".%D%"835.%D%3M.Z6<W}O| „J )™;|C~Q~š )’$ | df (x) = O} )† _•†ƒ) 9|C†ƒZ|C†9~Qn‰€•†{2†eL{A†%~› , _”†{A†”†A}C‚|CŠ*|  ž df (x) = hBhBh 1. ∂ f1 ∂x. 2. ∂ f2 ∂x. 1. 2.

(3) W§ ¨a©*ªMª5«`¬O­%®%¯ °?±Q°”²M³ ´šµˆ¶9·;¸›¹<º"»M¼;½'·;¸7¾;¿ ¶¾5¸KÀZ¾5¸7ºHÁ9»M·;¸7¾0½Q Ã%Ä;Ã9·;ÄB½"º'·;µ%ÅM·VÆ%ºNÆ%Ã%¾eÇ?Ã%µ%ÅMº"¼;µ%ºNº"µˆ¼BÄBµ%º º"È7¹J·)µ`¹<¾ t¿ Ç?É Ã%È7µ%ÃAÅ5Ê*º"¼B»Mµ%º"º¢¾5µZÈ<¼;¹<¾0µ9¼0»5¼BǕÃ%µAµ9Ç?¸7Å5¼;º"µ`¼B¹Jµ%·)º ¸7·;¾0µ9ºY·;½"ÀBºH·)¹<º"½"»J¼BÌ%¸7º ¾jÍJÆ%Înº Ï]¹<ÃA¼O¹7ÀZ¹<¾-¾eÈJ½"·)¾0¸<Ð ½"¼B¸7¼Æ%¾5¸7ºHÀ;·)¹<¾0º"µ t = 0 Ë È7¾0½"¾0Æ%Ã%¾ yi (0) = hi (x0 ) P P i i i y˙i (0) = ∂h x˙ |0 = ∂h f (x) + j ∂h g (x)uj |0 = Lf hi (x0 ) + j Lgj hi uj (0) ∂x ∂x ∂x j y¨i (0) = . = P P P Lf Lf hi + j Lf Lgj hi uj (0) + j Lgj Lf hi uj (0) + j Lgj Lgj hi uj (0) + . . .. Ñ º ¶%Ã%Ò0Á9µÆL·0¼;¸<·0¾Mµ`Ã%µ%»5º'·;¸7¾qÃ%µ9·0¶%¸7º"Ó*·0Æ%¾5Á9µ%º"Å5º"¼Bµ%¾-Æ9º¢¼BÈ7È<¾5¸FÀ;·)Ô%º"½'ºH¹<ÐAÕ Ö «`×<¨a§W¨'ØÚٕÛÜCÙ'ÜCݚÞCß,àáeâOÜJÜMÞCã$ä)àZåCٕÛ'Þ-ÜMÞ x Ix ٕßSæۖقç<àçJè9Þ x = x é ´šÆ%µ ¾5½"½"¸7¾ ¾M·;»5½H¼B¹Jµ%ÐXÆ%È<º"ÅMº'·)º"¼;Ó*µ%¼cº9Æ%º"µ`º9¹7¸<¾M·)¸7µ%¾5ÄBÈ7¼qÈJ·jÈ<¹<Ã*ºTÓ*·c·jÆ9¹<·)¸7¸7º"¾V»5ºQÎëÃ%µ9ê ·c¾5¸NÆ%ÇQ¾C·;Á9¸7¾ìµ9º"»MÅ5º"º"Òq¼Bµ%Æ%¾V¼BÔ%Æ%Ô%ºbº'·;¼BÓ,È7È7¼q¾M¸›¶À;·;¾5¸7Ô%Ò^º"½"ºH·;¹J»5ÐX»5¼B¹<µ`¸F¼O¹7À;¾MµZ·;¹Jµ%·;Æ%¸š¼ í »5·;ºYµ%»JÆ9Ì%ºY¾0»5¼B½'·µ%Æ%Æ%º"¾CÅ5Á9º"¼Bµ%µ%º"ºUÅ5º"À;¼B·;µ%½"¾0º"Æ%Æ%¾eºN½"¼`º"µ%»MÆ%·;º"½"È7Ó,¹7º"¾5µ%µ`ÄB¹<Ã%¾;º"¿Ô%¹<º"¸<½"ºH·j¹J¹7Ð ¹J·;µ%ÆLÆ%·V¼BÈ7º"ºYµZÆ%¹<¾MºYµ%È<Æ%º"ÈF¾5¹<¸7¾5È7Ó*ºNº"ºYµˆµ%¼BÈ7¾5µcµ%È<½"º"¼µ%¾M½"¼`·;¸7»OºQ·)ÎK½"¾Bî]¿¢½"»M½"¼;¼;¸<µ2· í È7º"Æ%¾M¸<·)µ%Æ%¼ »Mº"¼ É È7¼B½"¼ »5¼BµZ¹<¸7¼;½'½"ºS¹J·)½"ºK»JÌ%¾B¿ ¶É ¾M¸V¶%º"»5»5¼B½"º]º"µ`¹<¾5¸FÀ;·;½"½"ºS¹<¾5Ó,¶¼B¸7·;½"ºQ¿Sµ%¼Bµ ÇQÈ7·)¾M»MÄ;»5Ã%º'¾M·;µZµ%¹<¼ï¾q·;Æ%½"¾C½"Á9¼BµZµ%¹Jº"Å5·;º"µ9¼B·;µ%¸F¾q¾cÆ%½"º¢¼ðº"µ9È7¹JÆ%·jº"ÈF¹<¹<¼ º"µ%x(t) ÄBÃ%º"Ô%º"Æ9½"ºH·ð¹JÐAÃ%Î µñº"µ`¹<¼;¸7µ%¼ïÆ%º x ÎòÏ]º'·;Ó,¼ïÆ%Ã%µ%ó`Ã%¾c½'· Öæ9ÞC«`ã,×<â<¨aü;§Wø9Ù¨HôÚç<â)õ-ø9ÝQö9ã<Þâ)ەÜ$Û"âXݚà)çJ݂ÙâOÜCx݀à;,ø%xÝ›Þ ∈ V ⊂ù çJMâ)ø ÜCÙq÷;ٕø2قçJÜCâjقÞCø‡ß*â Þ*º'µ%÷`Æ%Þ$º"Ù ÈF¹<çJº"â)µ%ø9ÄBÝQÃ%ã<º"â)Ô%۔ۖº"Ùì½"ºWà;ùßexß0IxÙ'Ü<ÜCúÙQå5ٕÜOۖÞJÙ û u∈U U ú çJè%Þ*æ9Þ$ß,ã à;ø9ݚÞCø`üZà àXæAÛ"à;Þ0ã$ÜMÝQٕâ)ã<ۖÞKöBý5÷Zقà)â)ەø%Û'â-Ù x(t, û < ç ) â 9 ø š Ý C Þ 9 ø 2 ö š Ý , Þ • Ù ø ] û 7 Û þ C Þ ; ä ) â l Û Z ö C ý ‚ Ù ) ⠇ ø Þ x , u) x(t, x , u) V Ü$ݚà)ݛâqٕø9Ù"ýCÙQà;Û'Þ x(0) = x ÞSÛFþ ÞCä;âjۖöZýCقâ)ø‡Þ y æ9ÞCã t > 0 y t>0 ÷`àVÞ?ÿëæ%à)ø9ã$Ù"݂ýCٕقã<â)Þø‡Þ÷Zé à;ەÛ"âXÜCݚà)ݛâcٕø9Ù"ý5Ù?à;Û'Þ x(0) = x ÜOâjø‡âdÙQ÷`Þ$ø9ÝQقçJè9ÞVø‡ÞCÛbÛ'âjãJâˆ÷`â)ßeٕø9قâŽ÷)Ù î]½"¼`½"»O½"·)¼B½"¸<Ó,·_¾MÈ7µZº'·;¹<Ó,¾;Î ¼e¶%¸7¼BµZ¹<º¢·_Æ9·;¸7¾KÃ%µ9·_Æ%¾5Á9µ%º"Å5º"¼Bµ%¾KÆ%º¼BÈ<È7¾5¸FÀ;·;Ô%º"½"ºH¹JÐ_µ%¼BµV½'º"µ%¾M·;¸F¾]À;·;½"º"Æ9· Ö «`×<÷)¨aÙ §W¨ ٕøJÛAçCÜCö2Ù'Ü$Ù'ûaݛÞ$æ9ß*ÞCãVà,â<á]ü;÷`ø%ÞCÙìÝQݚٕâø9ݚ½"â)¼2ã$»Mø·;â½"Ó,¾5µ`¹7¾ ¼;È<È7¾M÷;¸›Ù À;·;Ô%º"û]½"¾ Û"àXº"µ ã<Þ$xÛ"àOý5ÜOقÞKâjø‡ÞJÜCÞ Ù'Ü$Ý›Þ öAøVٕٕø9ßSݚæâ)ã$ۖقøç<àâ W V ⊂M x x Ix çJè%Þ x x= x é Þ_ٕۢÜCÙ'ÜCݚÞCß,àXá_Û'âOç<à;ÛlßWÞCø%ݛÞ,âOÜ<ÜOÞCã$ä)àBå5ٕÛ'Þ æ9ÞCã,â7ü;ø9Ù x û à;ەÛ'âjã<àá0÷`ÞCÝQݚ⠽"¼`»O·)½"Ó,¾MµZ¹<¾ ¼BÈ7È7¾M¸FÀ;·)Ô%º"½"¾ é ÖçJâ)«`ø%×<݂¨aÙQÞC§Wø‡Þ ¨ õ_ÞQÿëø%ÙQà;ß*Þìâ ÝQö2Ñ ÝQ¶9ݛ·;ÞTÅ5Û'º"Þì¼0Û'Æ%â)ã<ºâq¼BÈ7÷`È<ÞC¾5ã$¸FٕÀ;äj·;à;Å5ݚº"ÞS¼B÷;µ%ٕ¾ ã<Þ<Û'ýCâ_قâ)܀æ%øàOà)ý5ۖ٠قâe÷;٠ٕۖ ø‡Ù‚Þ7Þ à;ã<Þ^ۖö2÷;ø`Ù ü`Jâ-ö2Û'ø2Þìý5ÝQقã7âjà;ø9ÙQÙaÞC݂çJݚè%â Þ ã$قÞe÷ZÞCۇÜCÙ'hÜCݚÞC,ß,. .àj.ûì, hçCقâOá L L . . . L h ûìçJâ)ø x = f, g , g , . . . , g , k = 1, 2, . . . Jé øWæã7à)݂قç7à 1. 2. 1. 2. 1. 1. 2. 1. 1. 2. 2. 1. 1. 2. 2. . 0. 0. 1. . 0. 1. 2. 2. 0. . . . 1.

(4) . p. x1. x2. xk j. i. 1. 2. m. . !#"%$&

(5) (')#*

(6) ,+-)#*#./ %*#". 8&89. λ(x) : <n →. <1. "0#'!)#*1/234%"'5'!!6 ". f (x) 7.

(7) O = [h1 , . . . , hp , Lf h 1 , . . . , L f h p , ... Lfn−1 h1 , . . . , Lfn−1 hp , Lg1 h1 , . . . Lg1 hp , . . . , Lgm Lgm−1 hp , . . .] :<;>=?=@A ;%B CED?FGD?HICE;KJLHICEHNMGD?HPOKMGHIQSRETL;GJL;,U(VWULU(;,=?;,XYVECEFGD?HICEDPAE;G=?=?HNJLU@#U(HNMZE;K@3RER @3D?H3CEH CE;>=?=?;NVEJLM>D-UL;I[QSHI=-U(D?RE=?D?M>@

(8) U(;NR\;GTK=?;<XYVECEFGD?HICED\AE;G]I=?D\D?CE]IT/;>JLJLD!^`_CaRET(@

(9) U(D?M>@b=?HcJLR @3FGD?HcAED HIJLJ(;GT/d3@3FGD?HICE;eMGHID?CEMGD?AE;MGHICS=?HfJLR @3FGD?HfAE;G=?=?;ed3@

(10) TLD@3FGD?HICEDEAE;G=?=;eVEJLM>D-UL;MGHITLTLD?JLR\H3C AE;GCgU(D @3AhD?CE]ITL;GJLJ(D0MGHIJ/U@3CgU(Di@jU(T(@

(11) ULULDiAED0RED?MGM>H3=@kAEVET(@

(12) U@^alHnmoEp>qGr!sut3rNsmm>v%wx

(13) pgy%r{z|r{}~€ M>H3CPU(D?;GCE;U(VWULU(;u=?;XYVECEF>D?H3CED. h(x). ;u=?;u=H3TLH€AE;GTLD-d3@

(14) U(;uAED<HI]3CED<HITLAED?CE;3[M>@3=?MGHI=@#U(;. =?VEC ]3HS=?;1U(T(@

(15) D?;GULULHITLD?;fAE;G=‚J(D?J/U(;GQƒ@^. „j P†(‡,ˆ‰‡‹ŠŒcvY0Ž rYp3ƒsSz?pjMGHAED?J/U(TLD?EVEFGD?HICE;fAED‚HIJ(JL;GT/d3@3ED?=?D-U‘. dO = span ∂Lgm hp ,... ∂x. h. i. dO ’. s#ƒv>“. ∂Lg1 hp ∂Lf hp ∂Lg1 h1 p ∂Lf h1 gm h 1 ∂h1 , . . . , ∂h , ∂x , . . . , ∂x , ∂x , . . . , ∂x , . . . , ∂L∂x ,..., ∂x ∂x. = span[dλ(x), λ ∈ O]. ”Ž•‰p

(16) Ž r!v%wLp1o\r{– ’. s#NoEp

(17) }!}p#—. dO(q) = span[dH(q)|H ∈ O], q ∈ M ˜iCPVECEMGD@3QSH™H3T(@NVECšU(;GHIT/;>Q‰@NMZE;KM>DPA&‘<VEC @NMGHICEAED?FGD?HICE;KJ(VE=?=5› HIJLJL;>T/d3@

(18) ED?=?D-U‘œY=?HPM>@3=?;ž JLX{TLVWULU@3CEAEH•=@uAE;%B C D?FGD?HICE;ƒAEDKMGHPAED?JLU(TLD?EVEFGD?HICE;ƒAEDKHIJLJL;>T/d3@

(19) ED?=?D-U‘•@3RER\;GC @XYHIT/C D-U(@W^ Ÿ. H3QS;bOb=?;GM>D-U(Hƒ@

(20) U/U(;>CEAE;GTLJLD5[ U(T(@

(21) ULU(@3CEAEHƒJLD?J/U(;>QSDCEHICu=?D?CE;>@3TLD5[&=5› H3J(JL;>TdI@

(22) ED?=?D-U‘S=?HM>@3=?;. CEHICkDQSRE=?D?M>@=5› HIJLJ(;GT/d3@3ED?=?D-U‘]3=?HI @3=?;3^.  . P¡&¢ £¥¤§¦€¨/¡\©>©G P¢ªE¤E«­¬Y®Y¬‹¯°¥®{¡±3¤&®{ E²¥³&r5pµ´Ž¶m%rm%}v%Spµ´z|}5r{x

(23) p

(24) wr5pgyGr{z?v r{Ž·3w(v(mm%r. u = [u1 . . . um ]T. v. p. ´gm ’. r{}v. y = [y1 . . . yp ]. x˙ = f (x) +. m X. Ž¸vz{z?p<¹Gs

(25) w‰pI“. s#Ž ’. m. gi (x)ui. i=1. º5»¼½¾

(26) ¿#»º!À½iÁKÂÃÄ-ÅÆÇ{Æ<ÈÄ|Ã!ÂÉÄ?ÊË3ÆÌ ÂKÈÄ. λ(x). Ì ÍËGÎGÊ. f (x). ÏÐ Ñ%Ò. ½/Ó!ÔÕ» Ñ. ¿#½. n. Lf λ(x) =. X ∂λ ∂λ f (x) = fi (x) ∂x ∂xi i=1. Ü Ô ½ ½ÓÔÕ/» ¿#½\Ôº!º À»ÔݼÞ#¿

(27) ßGÞ#½\Ô%¼`Þ#¿

(28) Ԃà?Þ#¿#Õ/» ¿#½\º!À/Ô ÔÓ5½\½\Ô¼0Þ#¿iÀ/Ôá ½ã5ã Ï  Ñ Ò Ñ Ñ Ñ Ï Ò3ÑÝâ à?Þ#¿#Õ/» # ¿ ½eº5À/Ô ÔÓ!½ ԙ¼½/Ó5» ÔãÔN¼» Ó!½/Õ/» ¿

(29) Ô ½KåÓ!» ½ã!» æ#» ½%ç¼Þ#¿

(30) ßGÞ#½ ½/Ó0½/º5½/á Ñ Ï Øeä â Ñ Ï Ò Ï Ò Ò ¼½/Ó!» Ô(ãÔ¼» â λ(x) Ï Þ#¿#è Ñ f (x) ½iº5Þ

(31) ÀÀ½Lº!ºY» â Ôá½/¿Gã!½ Ï Þ#¿#è Ñ g(x) º5» Ñ ã5ã!» ½/¿#½ Ï ÔK¿>Þ. Lg Lf λ(x) =. ∂(Lf λ(x)) g(x) ∂x. ֋×ÙØ Ú%Û. ». Ó!»Ô Ñ Ñ. Ñ(â. ç. Ï. ½&Þ#¿

(32) Ô¿>Þ. Ñ(â. Ó5½/¿

(33) ¼½/¿

(34) ¼. Ò Ôià?Þ#¿#Õ/». Ñ. ¿#½%é. ÑNÏ. Ô Ô. ֋×ÙØ êGÛ. ¦&¦ë.

(35) ìƒíiî‰ï>ïGðPñòEóEô­õYö5õ‹÷>ø. yi = hi (x) 1 ≤ i ≤ p. . ùPú#û3ü. h = [h1 . . . hp ]T. ü.     . ýbþ?ÿ. ‰ÿ. ü Pú. Eÿ. . þ-ÿ. 

(36)  . ücùgü. þ{þ. 

(37) 

(38) . ùPü. ÿ. þ. dO = n. ü. ‰ÿ. ü. x∈M. . !#"%$ ?;@ <;:;A

(39)  CBD .  & 

(40) 

(41)  (')+*-,' /.10+23. )434 .1576 ,8  ' .:9 ;: <;&= <;> E F== <;>E E ;> x0 7 <;> ; 

(42) E;  <EG x0

(43) 

(44) 3

(45)  H . 'I*,J43) 0 *1K L0 K 0NM K 0+2 )  4 )%', 2 ) 6 ,'O,  = k < n  M K 0 2 )   4 0 O)%',QP 5 R2S $T 0 ) 0UM )434  ,  ) . 4 *1'K .-57. ',V')<*-,' . )%434 .1576 ,dO 8  ' R2S ÿ3þ{þ. ú. Lÿ. Lÿ. ü. {þ. û. úú. ‰ÿSý. iÿ. þ. ú. ù. {þ ù. WXWY. ü. ý. ü. ú. ü. ü. ú. ý.

(46)

figura

Updating...

Riferimenti

Updating...

Argomenti correlati :