multiobjective optimization in hydrodynamic stability control
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Multiobjective optimization in hydrodynamic stability control Frank Strauß, Jean-Antoine Désidéri, Régis Duvigneau and Vincent Heuveline. N° 6608 Juillet 2008. ISSN 0249-6399. apport de recherche. ISRN INRIA/RR--6608--FR+ENG. Thème NUM.
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(120) , 0 @ ,%4 :. CF G. ut. 4 + '<0 7. ®\^xwy] }TbU1[2bsn«vsw«]¬ prxxb8b 1nÃw-b,¨© }}Tu`¨ w^wypr¸Tpy]`]`bsbUb
(121) n }®:Jn«py}mbs]``b bslLwV @Yx{Xn
(122) !± wyoprbs¦nsnr±¹!bUwr x®:¨©}o[2bs °v u¬ n
(123) ®mxb l}wtªny `u`prwr]`x¸tb*b b,ox¨©nb8u`ny`t b!wyµprxoxt`}Tprb8tx}Tnt°®O xl}t tny}xtn«pyb*xt`zOÁu}¨owÁtO!p
(124) }Ta'uny:x}Ttnyxpyn«x}lmt°a,a,}¨b!!pywruol ¼ F. ³xpy]§ }tOpywr}I:}xtTpn. NG. b(t) =. (3). bj Bj (t). t [2bswytn«pybsxtÁ:}lmt`}a,x{{n j=0. b0 , . . . , b3 ∈ R 2 (2). }w^pr]`b!}Ta':}Tnyxpyx}t³ b ]?¸Tbe¨©}Tw. Bj (t) =. ^. 3 X. . . 2 j. tj (1 − t)2−j ,. t ∈ [0, 1].. i = 0, 1. T} t pg nUTb!}TxttOpypywr]`}x{¹n: }Txn«tTb pn ¨©}wpy]`b* }a,:}Tnyx¶prx}Tt}¨2pq³2}§!u`®x [2vs¬ xb w!uwy¸Tbsn xtªpy]`b,µwn«p-zTu1owtOp
(125) ³ b t*¨©u`w«pr]`b w b(t) =. 3 X. b3i+j Bj3. j=0. b0 = (0, h),. . t − ti ti+1 − ti. b3 = (x1 , y1 ),. b1 = (l/6, h),. Ö. efe g/h,h,i,j. . t ∈ [ti , ti+1 ].. b6 = (l, 0). b5 = (l, h/6).
(126) (0 ! (0 & ! !/- !K ,+, ' (, +
(127) ! !K' f!!/ ! rpny}a,}mT}upy]w!tO}prtb t`bbs!`pyx Ix}b tUwrb ³tOxpypyx{]®xx¸x¶pqb l´tn«p }:b t ± \^]`bp :}x»OtT³pn]b wrb t wybÁ±=]`Tãt}On«pybs]`t x{n^py}ª!}Tb tTt1prb!n«·mu`p wrb³2b nyb!}mp b U!xb tOprn t »t ±\^]`b
(128) Twyx{1}TxtOp x{nt`}pµ`·obsÁtobs1bst`n}Ttpr]`b]`}Tx b}¨ \^]b
(129) ¸}u`a,b xne``wr}?·oxapybs*®mlUpy]bwrbs3u`tobswpy]`b }tOpywr}0:}lm}Tt}Tt*pr]`bxtTprb wr¸ ± TÀplmxb {`n^xtT}m}m`wy}?·oxapyx}tUpy]`b
(130) ¸T}u`a,b}¨pr]`bob8ny wyx®1b8*}T®o~qbs!ps± . .
(131) . . . b0. t = 0, t1 = 1/2 0 cl. . b6 (1 · l, m · h). t2 = 1. . b2. . . b4 m ∈ [0, −1]. b3. b3. ch. b3 = (x1 , y1 ) = (cl · l, ch · h).. V. [0, l]. V. =. 6 X. (bi,x − bi−1,x )(bi−1,y + bi,y )/2. i=1. . 5 m − 12 6. . . 5 7m + 72 72. . 7 ch lh. 12. \^^x]b'±` }± tn«pywruprx}Ttxt}Tt`b'zOuTowtOptLnyb ¸Tb w`bsnyxTtng¨©}w
(132) !}TtnqptOpg¸T}u`a,b'wyb'ob x!pybsxt =. cl +. 0.6. 1 b. 0. 0.5. 0.8. b. 1. 0.6. b2. 0.4. 0.4 b3. 0.2 Height. Height. +. 0.3 b4. 0 −0.2. 0.2. −0.4 0.1. b. −0.6. 5. −0.8. b6. 0. −1 −1. ³^xxpy]§u`wr b
(133) }tn«prtT}tpn«¸Tpy}wruu`!a,pyx}b t*áwyxxt]Opyp]`b± snybg}¨!u®`x ny`xt`b
(134) ``wr}?·mxaprx}Tt ©b!¨×p ^tox Ib wrb tOpny]:bsn ¦ xpy]xt-py]`x{n+!}tOprb!·mppr]`b^n«]1b }oprxa,x¬spyx}t`wr}®bsa ³2b2³2tTppy}n«}T¸TbÃx{n¨©}Twya3u`pyb8n4¨©}}³n n ± ps± M? 0. I. #. 0.1. F. 0.2. 0.3 Length. 0.4. 0.5. 0.6. −0.5. 0 Length. F. 0.5. G. G. 1. . minq J(q) = FD (q). F G. Re(λmin (A(S(q)))) ≥ 0, V (q) = V ∗ , q ≤ q ≤ q¯.. ³®ml*]b nywr`b xtb x{n¨©u`toprb8xn«}Txttns±¸dewrnyxu®`blT»o¸T³ bsb
(135) pr} }wt ny}xt`nyb xw^n«pypyx]t`b µ}¸¨b ¼Àoxwra,a,b tb!nypyxbs}wrtn ¨©:}T¸Twbspy!]`pyb}w `bsnr!wrxoprx}TtÁ}¨py]`b®:}ool q. q = (l, h, m, cl , ch ). Õ eÕ HL. NM.
(136) prx`t]`u` bbu!pr`}T}UxtTt`wyprb8wynq}Tpypr]wy:x{b'}prxxb tO}Ttptnpy]}T±3tÁ\^py]`]tb3b'¸T]`b bs}Tbs!xa,pyT}]Ob!wppyn wrl*}}¨¹¨Ãtpypy]`]`b3b,®1®:wr}o}mb'o`lTl}±e³2\^nb ]`w³ bbst¸T¹}Tu`u`nga,`ny:b}Tb 1w b ®:}}¨+u`pyt]`tb3nL®1¨©1}o}ow}On«lÁpyx]`pyx{xb,ne}t nrnywru`a,a,bsb!pybspy}wr} n¨ ®:b!}Ttn«prtOpp¸u`b ± 2 - (L ,&)+ ( Á .4!. "! ". Tãt$V *X1 tmu`a,b wrx{ `nr]`bsa'b^®Tn«b8-}Tt'py]`b^µtx¶prbb b a,b tOpa,b!pr]`}o,xn+`wr}:}Tnybs-py}ny}¸b py]b]Oloowr}¼ `tlOUt1xa,¨prx]`b»:py}]`ob'prxb a,xxbs¬stOpr¸x}TtUu`b'`wr}t®`§b pya ]`b'x{n^}on«}Tprxa,¸Tbsx¬su1pyn«x}xt`tL-bswy¸}T®`ubsaÁprx±}Tt\^nÃ]`}x{¨npyx]tmx¸Tn^}|øºgbsne ]`a'bs?}o¸mol§bs:!}Tpy]`a,b `xuopyb prwpyprx}x}Ttt n pra,xwy}Tb }ot pr!b8¨©a,ny}nw3}oxopynebs]¸Ibb pywr}U}Tlwyxprxxa,`twyb3}?÷o |Ãx}aºetny u`pybea,a,pyx]`t`}obo±^b ¨©u \^tpy]`}ªb prxwrn«}Tb!:t¨©b }Tnb8wy}bT¨+»u`³2o wbpy]`wrbb
(137) t x*tOpr{pyb !]wrubsbn«n«pypyabsx}ÁtxnstÁbs± n«µp^¦ tbs`bxTxt`b xtOtmpy¸bstu`ª}mb}opy±¹}ªÁ\^!]`}T`bst`tn«wrpy³2}?wr·oub
(138) xa p3t ¼ bstmx¶u`pra3]`b ®1w-bswywu`}t°¨¨©pru`]`tb!³pyx}]`tÁ}bb ¸}`upyxa'pyxx}¬stnspy±x}tª`wr}o!bsnrng¨©}w
(139) pr]`bUa,b!pa,}oob Ã}wny³x¶p]pr}§x¶p
(140) ¨©}Tw-§!bsw«pxt ¨©:\^u}]tb lmwrprt`bx}}Ta,twrn x{b*»IIt`nyb xbstT¸Tu`prb wb wwr1Ãt`}T³^b!pq?py³2lox}}n
(141) tU)w ¨©Uo}Ta,nw'b nypytu]` }o] `UOnswr»`xtxwrt`bg`±u1n«wyTãb8}?t*·o}xnya,u`xtw py bxT}pyn«t´]`bTbs»1Tlwn-]bT`?±x1¸±b4:®®1}Tbsn«b lmx{tnt`}¨©¨©}ua,u`tx{t1prUx}Tµprt}3p«n py®:x»It`b³1¸]`»b x{wrwrT]Llox{swy2b!u`®tw}nytoprxbn¼ ¨©x{n}Tw
(142) ®xTtOn«prb8b Lwr1}T}TtÁpyx}`t°pxtª®T]`n«xb']ª}¨ oxa'bst1ny}Tx}TxtOtpn n ányb b,bT± 1± V³X ]!±*b wr\^b3]`prb]`b,!}T}twrxnqTprwyxt1u1+pyxa,}t°}m`}b ¨2¹pyx{]n
(143) bUb ¸``uwr}?py·obsxa±'\^pyx}]`tb ``wr}?·oxaprx}Tt }¨t}Twyxxt1¨©utprx}Tt xn`bsnr!wrx®:bsn2¨©}T}³n
(144) !"!$#&%
(145) '() *' %! +,-
(146) %. !/(0 l. h q. b3. m. (cl , ch ). q¯. V. V∗. . F. xi ∈ R n f. ND. f˜. NG. ˜ = f(x). N X. ci Φ(x − xi ),. w³]`b xwrb :®1nyxn2¨©u`t1pyx}t x{n-Áwox{¨©u`tprx}Tt± ¦ b]`}m}Tnyb3pr]`b¨©u`t1pyx}t pr}®1bUpy]`b unrn«x{t ny}tuoprpyx]`}Tb-t}¨4wpy]a,b
(147) b!xprt`b bsw wxn«nlon«spybsab8Áp«prb tmupyx}t¨á!py}w8± \^]b-!}mb U xbstTpn wrbob!prb wra'xt`b8®mlpy]`b pr³³2]`b-]b^b ]1awr?b ¸pyb-wrx· tLtxn+:lO}Tprnyxxspyx¸b^b!·oo`b!wrµbstnrx¶n«prxb}tÁt¨©3}Twe]`tprb ]`t1b'!b^py`]`b wrwr}?b2·ob xa·oxn«prprn+x}T t n«}Tu`³ pyb'x} t't°}¨± pr^{]`n«}x{}n¹wooxb!t`x{prnqb8prb xwrt1w¹a'n«pxlot`n«`b3pybspaw-±¹:ojm}xb xxttOtO!prprbnn ³³]xpyx]Á]*wrbs ny}1u`b8{p^®:pybe} unybsUxxnt*-x¸b wto®mxl b tOpy¼Q®Tn«b8'}oprxa,x¬sprx}TtUa,b!pr]`}o0± Tãtob b8,py]bgw«prx1obswyx¸prx¸Tb i=1. Φ(x) = φ(kxk). Φ. Φ(x) = e−x. 2. /a2. ,. . a. ci. Ac = F,. (Aji )i,j=1,...,ND = Φ(xj − xi ) A. Fi = f (xi ), i = 1, . . . , ND f˜. x(k). N. X 2 2 ∂f (k) = ci e−x /a (−2(x(k) − xi )/a2 ). (k) ∂x i=1. Ö. efe g/h,h,i,j.
(148) . (0 ! (0 & !
(149) !/- !K,+, '(, +
(150) ! !K'f!!/ !. \ }-ob pybswya,xt`bpyt]be®:bs¸?bsn«p u1pypypyb b tmpru]`bpybsx}wyt,wr}¨áw Tpy}Tw ®1³ b bpq³ }bsb ttÁnyx`py]b w b¸prbsu`n«b
(151) pyxt`}¨+3pyny]b!bp }}¨wrxTxt1I`¨©ou`xtpyx!}pyt1x}t :}xtOtpr n pr]`b`wy}?·oxapyx}t ¨©}Twox Ib wrb tOppypybstOu1pyx}t¨á!py}Twrns»oxJ± b±. (ˆ xj )j=1,...,NT. a ef (a). . f˜. NT. f. ef (a) = k(f (ˆ xj ) − f˜(ˆ xj ; a))j=1,...,NT k.. ¦nyb tOb,prbs!}a,®m`l§u`prpy]`b-bwn«a`x4bs®n«Tpn«x{bsnxT¨©b utmt¸prxu`}Ttb `wy±}?·o¦ xabupyxny}b,t1n^¨©}weprpy]`®b,nyob-w³ xpy] t1§py]b3:n«pr}x®tOxprn x¶pqlÁwrb `wrb!¼ ¨áTtpr}
(152) wn2prbsn«wypyb
(153) xt`ny-]`}nyb!³p2t}x¨ t$^x1±/1`± :}xtTpn ±¹\^]`bgb wrwy}Tw+¨©u`t!pyx}tn t ¨©T} w2`x Ib wrb tOp2p«prb tmuprx}Tt FD ND = 99 eλ. λmin. NT = 8. e FD. x i ∈ R5. −3. x 10 11 0.8. 10. 0.7. 9 8. e (a). 7 6. λ. D. 0.5. F. e (a). 0.6. 0.4. 5 4. 0.3. 3. 0.2. 2 0.1 1. xTu`wrbd1 +wywr}w2¨©}Twowrtb xbstO¸u`b
(154) ``wr}?·mxaprx}TtUunyxt`'pr]`b unrn2¨©u`t!pyx}t± ^u`wypy]`bsw!}a,`u`prprx}Ttnwrb 1bsw«¨©}Twya,b8*unyxt` t ± pr\bs} n«pyxxtt`!wrbsn«b Tpn«bÁxn^prx]`tObLpybstso!bsu`Áw tl´}³¨
(155) x 0®1 bu`T{`pyox}wrt1bsnrn n«»b8*tæxtbst`¨©uopyuwyTwyb b
(156) a,!}Tb tOa,pU`}uo¨gprprpy]`x}bLt`ns±pr®1nybn³2b enUpy]`b
(157) ! "
(158) )!2! .
(159) )+ \^]b*}u®1nyb2}}¨oprxwa3u`oa xb tOx{pyn3¼Q®}TTt`n«b8l ª1}Onyny}xwr®xpyb]`ax¶¨n³ xbÁUbn«prjwy]gp| n«u ¨áU bsxn'bstT!prbsw«l´p xt´}On«b*xa,py}ªx¶pxprpsx± }Ttdgn ±n«u}ltm» ¸Tb ³ wrbÁb ot1}°!b,t`py}} p Umt`}³ pr]`xn}Topyxa-u`a xt To¸t b±`Tãt `oxpyx}t4»py]bpywrbspya,b tOp}¨
(160) ¨©u`t!pyx}tn³]`x{]=a?læ®1b t³2}bt³2oxItTb p2wrb py},tOpy!x{}T®`tnybxoxbstLw!3bsw«nypb a,xt§x¼Àn«1py}T}oxtO]pTngnqsprxet§}obsprx§a,pyx¬s}Uxprtx}Tt*s!a,u`wb!prpr]`b}o*wrbs³nyu`]¶pxn ]±g\^uny]`bsb n wr¨©b!u`¨©}Ttwy!bTpy»x}xtt¸?}u`wubsnqprn2u}ot`xbsl n txa,npµtoxt`'py]b
(161) }®0}oprxa3u`aÁ± Tãt,wrbs b tOplb8wn+³ }T0w U]1n+®:b b t,¨©}o un«b8,}t}Twyxpy]a,npr]pÃwybxtny`xwyb8-¨©wr}a tpru`wo`]`bst`}a'¼ bs³t]`x=±] Tãt,a,py]xa,xnx{ ¨©wrn,pra']`bsb§³ n«}T}o0w !U-x{³ b®:³^b ]1?tO¸Op¹xpr}}-uw'nqpr}u¨gol3®`xpyw]` be²1|+}mw«Umprxxt` bjmány³2b bwyaV X !g±=opy\^xa,]`b§¬spyx}}Tt wyx©py| ]`j1a +x{n,pyb8®T]`n«t`b8x{ zOu`}Tbt » pr]`bÁnrn«u`a,oprx}Tt´pr]p,xt1ox¸mx{ou®`xw`n3b!·`]t`bxto¨©}Twyapyx}t ®1}Tuop'py]`bsxw,:}Tnyx¶prx}Tt»¸bs}o x¶pql 0.2. ^. 0.3. 0.4. 0.5. 0.6. 0.7. 0.8. 0.9. 1. 0.2. 0.3. 0.4. 0.5. a. 0.6. 0.7. 0.8. 0.9. 1. a.
(162) I. af = 0.7. aλ = 0.9. #. . . F. NG. F. G. Õ eÕ HL. NM.
(163)
(164) !"!$#&%
(165) '() *' %! +,-
(166) %. !/(0 . ,atx§wµprpyxt`}Tb8tUnyn py}t1wr§b Tpyx}T]tb3n2®:}b ¨]]`?x¸mTx]}u`µ`wpyt`}¨Ãb8nypyn ]`b'Jn«bs²b }oV UÁX !x±ngpr]`b t°xt`²u`b t1!bsLpy}xt!wrbsTn«b
(167) pr]`b,`wr}®®`xxpql}¨ prpr¦ ]`]`bwrbÁ}!u`prwr}Tt1]`~qn«}bsx{uo!opypÃb }w,pywr]`l °bg}n«}b ¨op'prbsx}Ta,¨]x¬s1prx}Twywypytpyx{x{!`!bwrbs}m»n Tbs?nr~qn u±=n«TãpytUxt`b8 pyx]p]xopyp,b lmw t}prta,x}Tn«pyxtUsxpynqu`prpyb l b upy]`nyxbÁprt`]`´nybe³2py:]`}TwybnyaÁxpyx± tox}¨©t¦}Twy}ab¨0³^prpy]`xbe}tOt=p31pr!wy}L}Tpyx{¨©!bs}T!bpy}b8³ x{n]t`b8*®ml*'¸bs}o x¶pqlU¸Tbs!py}w » F. NG. xi ∈ Rn. p. xki. k. vi. rp\^}-]prb]`¸bbs®1}ob8!nqxp^pql-1}O¸Tn«bsx!pypyx}}twÃxn ob!}pr¨4b pywr]`a'bxt`b8wypy®mx{!l-b pr]`bxt*¸bsxpybs}owr x¶pqpyl'x}xt t,pr]`betUwypybs]`¸Obx}®1u1b8n¹nqx¶p^prb b w¸Tb prw2x}TwytUb8nqpr]`b b8»Tpy:]`}Tbgnyx¶oprxxn«}Tprt t!b » aeb wrb ³ x]bstOxpr]'nexnÃtn«§b p¹nyb!pype} prpy}]bpy]¸?b-wrbu¸b2u`t}u`¨x¶b
(168) ¨©}TTpqwy±³21 a,}lÁtpy}*'`³]xn«?py]`¸wrxxb-®]'uoUx{pyn¹a'b8§wyb8b8tmot§u`ua- }bs¨®:'b n«w±unef§s!tb8}Tnywynybsx}¸T¸b b lw8t»`®m l-t§xt`¨ábswr!w«b-prpyx}Ts,w prb b8wrÁa ¨×prpyb wrwu
(169) n«x{nepg!T!bs}mw«`pbo,b8xt ¼ tm³u`xpya3]®1xtÁbsw+}}³ ¨bsxpyw2bswrtpyx}u`t`ns:±b \^w]`®:b^}u`}ot prn x}Tt}¨1pyt]` b^w«prwyx b8n«b8:nbs!tpyx
(170) ¸b pr]`lb^±+¸bs!}o}T!a,xpyx`bsn+b pyb
(171) wyb xt`x}Tpywyx{xpy]x ¬ a bs3wyb8wt`on }Tpya,]b tl Tn2¨©}}³ns± ± ^`}Tw »`xtx¶prxx¬ bpy]xb
(172) t`xxpyt`x{bsw«xpr¬sxb wrt»ony}b!a,p l§py]bU± }ospyx}t1n t1°py]`b¸Tb }m x¶prxb8n o±^¸uprb py]`b ¨©utprx}Tt¨©}TwbsT]w«prx bT± 1`±dI`prb
(173) xtb wypyx{,x¶¨+t`bs bsnrnywylT»TxQ± b± ± ?±dI`prb py]`b
(174) ®:bsn«p:}Tnyx¶prx}Tt ¨©}w1wypyx{!b tpy]b
(175) }®:®:bsn«p:}Tnyx¶prx}Tt ± @o± }a,`uoprb py]`b
(176) ¸Tb }o!xpyxbsn^}¨py]`b
(177) 1wypyx{!bsn xk+1 = xki + vik . i. x∗i. i. x∗. k. vik = ω k vik−1 + c1 r1 (x∗i − xki ) + c2 r2 (x∗ − xki ).. r1 , r2 ∈ [0, 1]. c1. . c2. ω. α. v, v¯. i = 1, . . . , p [v, v¯]. ω. 0. x, x ¯. x0i ∈ [x, x ¯]. vi0 ∈. k=0. ω k = αω k−1. x∗i. x∗. i. #. vik = ω k vik−1 + c1 r1 (x∗i − xki ) + c2 r2 (x∗ − xki ).. `± ]`bsU*x¶¨+¸bs}o!xpyxbsnwyb ³xpy]`xtpy]`b
(178) ®:}ut`wrxbsns»o}py]b wr³xnybgwy}~qbsp^pr]`b aË}t }w ± m±dI`prb py]`b
(179) }o pyx}t}¨py]`b
(180) w«prx b8n #. v. v¯. ` ± }w ]`bs± Ux¨}o pyx}tn }¨wypyx{!bsn2wrbg³xpy]xtpy]`b ®:}u`t1`wrxb8n »O}pr]`b wr³x{n«b`wr}~qbs!pÃpr]`b a¾}Tt `± TÀ¨prb wra,xtpyx}tÁ!wrx¶prb wrx}TtwrbsT]`bsUpy]`bst§nqpr}»`}py]`bswy³x{nybnyb!p t1*T}py}n«pybsÁ`± = xki + vik xk+1 i. #. x ¯. k := k + 1. Ö. efe g/h,h,i,j. x.
(181) 8 (0 ! (0 & ! !/- !K ,+, ' (, +
(182) ! !K' f!!/ ! \^-]oxx nIb wrb TtO}p2wrx¶`pr]`wy}oa bsxnrn2n«³2}Twsb ±¹¶¼ãgnyu`t`x¶prl'bspr*]`¨©be}w^T}T®wr1®1b b8nqx¬8p21prx}O}Tn«txpyxny}xtt bgb8]]npypy}-w®:~qb8bg py}T}wya,la- u}u`t`{x*s®:prbsb
(183) !}T®:a'b!pq³2uob pybsb8t*py}T]`t b wy}o!b8nyny}wns± \^\^]]xbng¸}Tu`wypya,x{!bb3 }nyt³^n«pywrwa xtO}Tp¹opytxa,'xpy¬8]bprx}T®1tL}?·!}T}Ttwyn«xpypyw]`a xtTpx{ngn¹u1wrn«b^b8Ápywrpybs}Ápyn«bs}T'¸T®mb
(184) l,py]`b3o}Txt`opyxnya,u`xx¬8prpy®x}bt:`b t1wr}®`pqlb a pybswyMa? n ± pr}'py]`b
(185) }T®o~qbs!pyx¸b¨©u`t!pyx}tJ»`xQ± b± . .
(186) . . . . . . . x∗. F -G. X. J(q) = FD (q) +. µi max{0, ci (q)} +. X. νj cj (q)2 ,. b8³ zO]ub wrb3x¶pqprl]`!b,}xt1t1nqoprwrb!·ªxtOn«prb n2p ®m}l ¨ pr]`± b,TãxtÁtbs}TzOu`uw xTpqn«lLbg!py}T]`tb
(187) nqpr}Twr®o~qxbstO!ppyn x¸b¨©u`t!pyx}txn-owrb bst`}pyn2b8TLn ®ml tLpy]`bUnyb!p
(188) }¨ ¨©}Tw1bstpqlUwra,b!pybswrn t ± !
(189) Ã *s !
(190) ! " i∈I. j∈E. ci (q) ≤ 0. I. E. J(q) = FD (q) + µ max{0, −λmin(q)} + ν(V (q) − V ∗ )2 µ. ν. (4 :<.<> > 78:*@ 4<@ ,%3 , 0@ ,D4 :. #"%$. ãt wyxa§ wr}lLt !u`wrwyx¶prwrb b wrtOxp}Tt }oprxa,x¬stªpyx§}tªn«b8`!wr}T}t®``b awrlL³ wyb,xpy³^bswyx}tOtp
(191) py}§}T± o¦pyxa,bUx³^¬sb,tOpq³ p
(192) }§py}L!wrn«xpypyu1b owrx{lx³¨]³ b bUwrbs³ tªbxnya'b bswy}p-¸Tb pr]``b§`wrn«}Tb8T!}T]Lt`³]`wrl´x{]ª!wrxxnpyb `wrb x}¸t bs}T³1xb8py]L}xuot pUV o1 X b pyb t1wrx°}wnyxa-pyxt`u`{prpyb,]`bÁ§`cgwrxa,ny]ªwyTl a,}Tt`b,b³py]`}m}ªbswyb,a-}ut]4b± `¦?lTbb w ¨©}uopr}x³¿x¬sbspyn]`b prx{nyng]`u`b
(193) ®®1nyb8nq}OpT!Ib xpr¸}bst§}®opyp]`b,xtÁn«pypy,}3wcg}pyobsprnyxlÁ]a,}b8x¬ zO¨¹b u`prx]`xb'®`}wyxpytu`]UaËb wpy1]``bg}T?x}tOlTpyp b ]w b w2V X}Q±
(194) t`bg\^x¶]`pn2b' }¨©}}oa,wprx³`a]`b x{a,]n«b pytOwpyp ]`prbb n«pyxpywbs}3npr}b owrTprb-lUxa,o}bsx¨+¬ t`}b }t`prb
(195) bsL± ®O?TÀl p^lTb xnw ¨©}Tt}³n »owyb8n«:bs!pyx¸b l±¹\^]`b
(196) }wrwyb8n«:}t`xt`'!}Tt!u`wrwrb tOp }Topyxa,x¬8pyx}t`wr}®`b a x{n2py]b tÁ³wyxp«prb tÁ@ n ³]b wrb x{n'¸bs!py}Tw2}¨b tpy] ± prxgt]`n4x{pyn]`obn«b8prn«nyx`w«prt-x¶xpytpy1x}Tt`1x,tO}Tp+xatO³ pb2pypywr}e!x}· nyt1}n«¸x{ob b w@ ¸0³2wrxbà py}x}ttnyn4x`³b x¶w0prpr]-]`wrbÃbs}Tny1ob8pyxa-p4u`pr}ea }Tw«pr]`}}¨mTpy}]`tbÃprowrxwyb8b p0py¨©xu`}tt1!n pyx}t»nyu` a,a,=± ^`wywrx¬ }b8a nyu`®wynyx{prTx!}Tb tn wybs®ml pybspy}'py]`b nyu`®ny±\^ ]`b b
(197) ¸b8wrpybg}Tw obst`x{}n2prpybs]`bstÁ®Ol,³prwy]`xp«b
(198) prb !t§}mb Un xbstTpn tUpy]`}On«bgpy} T. JA. JB. U ⊂ Rn. JA. W. JB. q = (¯ u, w) ¯. u ¯. w ¯. minu∈U s.t.. c. F G. JA (u, w) ¯ c(u, w) ¯ =0. minw∈W s.t.. nc. ∗ qA. S = (ω1 , . . . , ωn ).. . U. W. w1 , . . . , wn−p. F G. JB (¯ u, w) no constraint. JA. ωi. . u 1 , . . . , up. q. ∗ q = qA +S. . u w. . Õ eÕ HL. NM.
(199) T.
(200) !"!$#&%
(201) '() *' %! +,-
(202) %. !/(0 . t wrbsny:bsprx¸Tb l± ±\^\^]]bb!owrb8un«!xx{t 0:¸}xwrtOxp^®`x{nb8n-t`}wr³Xb {pypr]`bsb´]py} }x{!b }¨wrpyb]`bt`}ny`³ x¶pypyxt`apywrx· t´¨©}Ttw *py]`b3pro]`bsb l xnyx}Twrt b xtÁ³]`x{]³^?l,py}Uny`x¶pxtOpy} t ± pr®1g]`neb$nybsaUwy}Tbs}T1nrtUny}Oxn«prb8]`t´Ábeax¨©t}TpyV }1Ywr³xX+· ³2xt`b-3p3³^x{pro]`b8tOb`pe± }pr¦o}prb
(203) xa3]` u`}m}a}Ttnynyb
(204) x`prb ]`w b3}py¨]`ny`b py]`x¶b!pybÁ·opyx}Tt`®ot~qbsnyax!}TpytUxpy¸wr}bUx¶¨·§¨©pru]`tsb !pr}`xwwr}Toxtaxt` wrl'pr}U¨©u`± pyt]\^!b-py]`xb }x{xn'tbstm`¸`bswr!}TpyT}T®:wr] n}uo}xpn¨ xprn}oprxa,x¬ bs¸u`b xtpy]boxwrbs!pyx}t}¨¹'u`t`xp¸bs!py}Tw ± ∗ q = qA + u1 ω1 + . . . + up ωp + w1 ωp+1 + . . . + wn−p ωn. JA. u1 , . . . , u p. JB. w1 , . . . wn−p. S. u. w. ∗ qA. JA. JA. ω ∈ Rn. ∗ qA. ε2 T ∗ ω HA ω + O(ε3 ), 2. 1¨©®³}T]w^nyb xµ`wrne·ob }bs¨¹¸n«b8apy}Twr¼Qn bst`}unyu±] \^]Á]»prbeT]n}pg®`n«a~qpybs]`b-prxpr¸TTben^x x{1n}Oprny}-nyx`®wr}bT¸T:±¹bs!}Tjopynyx}bgtw 3nb n«wybx p«Tprnyxt`ny}o'!x{nrpyn«b8}o x³×pyx¶¨©bspr}U]w ³¸x¶prwr]xpypy]`x2}b t³2n^b ]`pr}T]]x? ¸pbeb }¨4wrb » á `tbsnyxT³2tbgnywy}T 1b}O¨©n«}bew py}3 ± }tnyx{ob w py]`b bsxTb tm¸b8py}TwrnÃnrn«}o!x{prbs,pr}pr]`b
(205) n«a bsn«p b xbstm¸?ubsn }¨ n TÀ¨¹!}Ttn«pywxtTpn^`:bsw2³2bnrnyu`a,bgxtbswxtobs1bst`t!bt. Umt`}³bs`b }¨py]`b
(206) TwrToxbstOp2xt pprr^}']`u`}wypytpy]`]`}bbswra,wy a,}t}`n«wr®pybwT»¹n«x®mx{tOn l´p^]m lm:b wrwy¼Àa'nyu`¼ãwyjo¨á]!a'b8}n x{¨0mpyp,]`b}wyn«pyu`]`®1}Tn«}TTt!bxT¬8b t`prbsx}Twrt» pyb8`,wy}o®O l3bsnrpyn-]b ³2aÁb»m³^xJ± btO±p'py]pybg}ªny!u`}T®tnynqprwyu1 bpprt`t bs}TtTw«p¼ ∗ ∗ ∗ T JA (qA + εω) = JA (qA ) + ε(∇JA ) ω+. ∗ ∗ HA = HA (qA ) j ω. n−p. ∗ T (∇JA ) ωi = 0. ε. F. i > nc. G. F AG. ε2 T ∗ ω HA ω + O(ε3 ) 2. ∗ ∗ JA (qA + εω) − JA (qA )=. ∗ HA. JB. ∗ qA. ∗ ∇c∗i = ∇ci (qA ),. i = 1, . . . , nc .. {ω1 , . . . , ωnc }. ci = 0, i = 1, . . . , nc ω1 =. ∇c∗1 , k∇c∗1 k. ω ˜ 2 = ∇c∗2 − (∇c∗2 , ω1 )ω1 , ω ˜ i = ∇c∗i −. i−1 X. k j ωj ,. kj. s.t. ω ˜ i ⊥ ωj. ω2 = ω ˜i =. ω ˜2 , k˜ ω2 k. ω ˜i , k˜ ωi k. i = 3, . . . , nc .. T\^x]¸Tb§b t}®Owypyl*]`py}T]`b
(207) }t1a2py`wrx¶wy· }~qbs!»`py`x}bst nr!}wrxtO®:pybs}°Ápy]n b¨©}Tn«u`®1}³n«n T» !bprt`bstOp'pr}ªpy]`b }tn«pywxtOp']mlm:b wy¼Ànyu`wy¨á!b8n'xn j=1. P. P =I−. nc X i=0. Ö. efe g/h,h,i,j. ωi ωiT ..
(208) ?. (0 ! (0 & !
(209) !/- !K,+, '(, +
(210) ! !K'f!!/ !. ¦ b py]b t§ob!µ1t`b py]`b
(211) apywrx¶· o» ³]`x{]Áxn^wrbstn«lma,a,b!prwyx{» tmtu`Uonybst` }bprbg}x¨ prn b xtbstm¸?pr]`}Tunybsb n}®m¨ l +± ÃzOupyx}t á ±¹^\^]]`bst`nybTbsxtn py} u1obpr]`bg¸Tbs!py}wn¹pr]p^®:b }t`-py}3py]`b 0 HA. 0 ∗ HA = P HA P. P. µ0j , j = 1, . . . , n F G 0 HA. ε2 µ0j + O(ε3 ). 2. ^\nybs] b3}tUwrwyl
(212) x}¨©wru`x4t®:!bspyn«xp}t`b!µx{t`nxprpy]`x}b tLt,}pr]`¨ bg nyu`®ny¼À `bxa,n«1b t1t`n«xt`}b8t,®mnyl-u`pr®]`nyb b-nb x*bsn«tOpyw¸Tbspr!b pyT}l*wn¹pr}*nrnya,}mx txxa,pyb8x-¬ b3³pyx]`py] b pr]`bnya,bsn«pt`}t¬ b wr},®ny}uopyb ¸u`b8n ± ∗ ∗ JA (qA + εωj ) − JA (qA )=. n−p. n−p. |µ0j |. #"!. 2/,%.10@ ,D4 : @-4 '*)6> 4#6#):<03 ,/. <>
(213) 4 B2%783. rpprTãt]`]`b*b§pr]`n«nqprbp`®`®`wrxxb xx¶¸mpqpqll´x}»+uTwynn«bs:`wrbswrbs!bsnyp'bsnyb x{tOwrn,pyb8] }L¨eVÑY ®mTXQwyl»1b8}pyt`]`p'b*lUxa,nypra]`:b}wyoprb8wnqt1p3!bÁb ]xTb nttm ®:¸b ³ b u`tbÁbT ³2»4}³^}tu`ny{x{n ´ob pr®:wywrb8bsbÁpyxtOb8pynbs}Twyb8®onnq~qprbs bs!} pytxn«¸xpytbw¨©pyu`x]`tOtbÁps!±$py®:x}abstn«}p,»:³2³¸b ¸T]`b xu`w8bb » ³]mlo]oxwr]*}oo lm}tu`{a,®:x{gbn«Tpr]`®`xxb x¸pqb8l*0!±+}T\^u`]`xn ®:x{b
(214) nÃbT±u1±Twrt`b8tO!pyb8b nyb8nr0±wrl-pr}3ob8!x{obe³]b!py]b w2¨©}w^
(215) x¸bstUxto²}³ ¸bs}o x¶pql Tãt,eµwrn«p¹nqprb '³ b!}Ttnyxobsw4pq³2}}oprxa,x¬sprx}Tt3wy}T®`bsan »}Tt`b ³xpy]3py]b^`wrTn}T®o~qbs!pyx¸b^t-}t`b ³nyxa-xpy]Uu`pypr]`bgt`b b }Txub n«tml¸t1u`b^¨©n«u`prt1w«pyp
(216) x}tTna,}Tb'®o~q®1bsb !pqpy³ x¸bsbTb f±tTãt*py]`b't`pq³2b!·m}Áp2}n«¨ÃpybsprU]`b ³2a¯b³2py}§tOopb!pypr}-b wr a,}xtt`nybx{ocgb w ny®1]}pyb8]UzTu`xwr}x®®wyxbsu`a,a n :xpy}bsxwrtOprpynsx}±tenÃ:¨©} w2pr ]`u`b a'pyxb }prtn¹a'}owyobbs:J±¹b wy\^¨©}]`wrb a,wrbsbs3nyu`wyuprt`n2t`xwrt`bg{ pr| b w^j! }a,}Twywyb8xpyU]a³xpy³]*x¶prpy]§]`b 8}gwrx1xtwypyx{:!|Ãbsºgnta,}osobsTJ ± \^]bow-a'xt`xa,x¬8pyx}t`wr}®`b a ? Ãxn2t`}³æ¨©}Twya3u`pyb8³x¶pr]`}u`p pr]`b
(217) nqp®`x¶pql }tn«pywxtOp ³]`xb UTb bs`xt3pr]`b ¸}Tu`a,b
(218) t1*pr]`b
(219) ®:}?·!}TtnqprwrxtOp8± =F G. minq s.t.. FD (q) V (q) = V ∗ q ≤ q ≤ q¯. \^]b}³2b wtu`1bsw®1}?·Á!}t1nqprwrxtOprn^wybxtÁ}Tu`w`x{ pyx}tx¸b t®ml {joxprtx }Tb
(220) tn^py]³2b3bn«b pr·o®1xb8x¶pqplÁpy}'!}Tµtt1n«pyw,xtTnypgxa,xnex t`w}pgwyb8Tn«upr¶xp8¸T± b-Tãtpob prb8]`*b-³2}Tob
(221) py]xa-?¸Tub a ³2b
(222) ¨©}Tu`t§xt§py]b`wrb ¸mx}Tunes{!uo¼ q = (0.03, 0.03, −1.0, 0.5, 0.5),. ³]x]lmxbs`n. q¯ = (0.08, 0.08, −0.5, 0.8, 0.8).. q ∗ = (0.08, 0.030277, −1, 0.8, 0.64325). FD = 5.35967,. Re(λmin ) = 0.0778279,. V = 0.00197193.. Õ eÕ HL. NM.
(223) 1.
(224) !"!$#&%
(225) '() *' %! +,-
(226) %. !/(0 . }a,wyx{n«}TtU³xpy]Ápy]b}Twyxxt:a,}oobs4n«]}³n3T}m}mTwybsb a,b tOp^}¨py]`b
(227) wrbsnyu`prn x\^t]^b xµ11wr±n«?p± !}a,:}t`bstOp+}¨pr]`bb xbsto¨©u`t!pyx}t3¨©}w+py]b^²}³ wr}ut
(228) pr]`b}oprxa`obsnyxTt x{n¹`}pypyb8 FD = 5.36048,. Re(λmin ) = 0.0777728,. V = 0.00197193.. q∗. xu`wrbd? +xbsto¨©u`t!pyx}t§nrn«}o xpybsUpy},pr]`bn«absn«pb xb tm¸u`b¨©}w`bsnyxTt pr\^]`]b
(229) b't`bswyT}T®`prxbs¸Tabny}a¨ py]`b8bnqpabsxT·ob xa,tm¸x¬8u`pybx}± t}¨ py]`bUnyab8nqp b xbstO¸u`b'x{n ³wyxp«prb tªn a,xtxa,x¬sprx}Tt}¨ ^.
(230) I. q∗. minq s.t.. −Re(λmin (q)) V (q) = V ∗ q ≤ q ≤ q¯.. g n^}oprxa3u`a¿³2b }®opxt ³]x]lmxbs`n xxcett`}*opÃx{ pyny]u`pywrxpe`t`wypyx{]ny b-xt`prpqTwr³2TlT}*o»?b!¨©T¼À}}}Tw¹*pr{]`n®:x{b!n¹pqwy³21b}Tb tbsxtO}t,pppy³2]`py}b^bp]pq³ ?¸}
(231) l§bT}TtTp{nswyT±fT}b at`w+x{}b nq³2prxxb bs¸T±tmb ¸?w8e»¨Ãpyu]!b2}Tbeu`xotwx n«:!bT}Tbs»1a-wyxbsp®`tx³2!tb8}nÃuprx}TÁwrt3bt`zO³}u`³xpyxpy]Ube®:b'nya{xtOwrpybsJbs»Owyw+b8n«nq]``pr}wrx³^t ¼ pr}}*py]`nybsb w
(232) b }py®o}*~qb8³]prx¸Tpeb3b pr·m}mpy}b tOa3peuxpg]x{n±*:\^}T]nrnyxnx®`x{nb opy.} }Tt`Ubb bsxÁt°pypr]`]`b-b,¸?¨©wrua,bb }³2¨¹}})w U§tb}}¨2®ocg~qb8nypr]ªx¸TTb
(233) ³a,x¶bpr]`ny}xa3uopgu`}Opyn«xx}t`t°x³t§]`pyx{]`] b ³^n^ob8ny wyx®:bsxt§jmb8pyx}t @`±T± Tãt'}uw nyb2³ b!}Ttnyxobswpy]`bow n`wrxa,wyl ¨©utprx}TtQ» prT]`nb
(234) nybsa,!}T}oto`x¶µ1bswyl¨©¨©u`utt!pyprxx}}TttJQ» ±jmxt1!b³2b
(235) ³2tOppy]`b¨©u`tprx}Tt» {npr}]?¸b
(236) :t}Tny3xpypyx]`¸bb
(237) ny¸au`bsb8nenq³ p¹b b x¨©}obs tmu¸?n}Tut b q ∗∗ = (0.08, 0.03, −0.660609, 0.8, 0.677202). FD = 5.41806 Re(λmin ) = 0.0783713,. E . V = 0.001972.. . JA = F D. ∗. JB = eβ(−Re(λmin )+Re(λmin )). Ö. efe g/h,h,i,j.
(238) *?. (0 ! (0 & !
(239) !/- !K,+, '(, +
(240) ! !K'f!!/ !. Q ae}³ bs¸bsws»0p ³2b']1?¸b Tpyx¸b,!}Ttn«pywxtTpn »Ita,b lÁpy]`b¸T}u`a,b' }tn«pywxtOpt1pr]`wybsb3®:}?· ³³^ }]ttOxn«pgpy] wpy}* xtOa,tprns}o]1±oxw¨©lÁo¨×pybslLpyw]`®1b3bwy¨©}}uwr~qnybsa-bsuprx}T¨©}Ttpyw-x}}ÁtLtLn«py`t]`Lb8xpn«b,T}ny¨2!ny}Tu`pytb a,n«wrpybwrwx¶pr}}xttTwrb-xpb8nµ`n ±pr·o]`b8ºgb §wru`b3®1b}?x{ngpy·L}Á} t`pr}]`tl§x{n«n
(241) py}wu`t`tob'x¨átO?ope¸Tb b!}T·ou`wy:bswb-bs®`!}py¨b,xt`¨©n«*wyxbspyprbsu]o}Tpyp ax}}t°prb ]`³ ¨×b psbw » µ`·obstÁ}`pyxa'x¬ b }¸bsw pr®:]`}?b
(242) ·§wr!b }at1nqprxt`wrxt`xtO-prn2¨©}Tpru`u`wwrt¸xwrt1x®`prbsx¸Tn^b
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