ionic teresa š RECENT MATHEMATICAL INDY Nie
IN OPTIMIZATION AND CONTROL Collection : Sciences mathšnhatiques et wee ue
2
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INFORMATION PROCESSING:
Trang 3“8e de Mhes de Đan dần Raoifercrrenn.ifreee
‘te Sn Mig saan đau (CGI Nhhớigh LW Cae Nas)
Trang 4INFORMATION PROCESSING:
RECENT MATHEMATICAL ADVANCES IN OPTIMIZATION AND CONTROL Articles from the IFIP Conference July 21st-25th, 2003
Ecole des Mines de Paris Sophia Antipolis, France
Trang 7Vi ASTHEMATICAL ADWANCES IN OPTISZATION AND CONTROL VI Differentiabiity of the L1-Tracking Funetional Linked to tho Robin nverse Problem “
8 Checbune, 2, eric, K Kinin 1 Pasi nity Propo af Difcrortiabiey Property of the Maroing er #— PhDabadenofthe Pune ` + “
VIL Collision Detection Between Virtual Objects Uei ‘mization ‘Techniques (Charel Bars, Yelander Herman
1 Previots Wark m
2 Mybeht Callow Detection Alta 3 Siuulation Result a a 1 CConshsons std Future Work bì ‘VIL Optimality and Sensitivity Properties of Bang-Bang Controls for Linear Systems a rane egenhaner 1 Paolo, Optinality Codi 8 2 Thu Opeinal Co 3 Esatple, Sensi Analy oy Shooting 2 IX Optinal Vortices Control in Navier-Stokes Flows J Behe, $ Chanhane, K Kunisch 101
1 Adjoiae Based Optinal Coutol tại 2 The Opti Conte Prem, 3 FisrOnlerOpdunliy tetem H6 tì X, Production Planning In the Metal Industry us
1 Probie Do - ue
2 Onin ofthe Pte Plating Anpmoeeh Cancion and Outlook us tà XI, Tracking Autopilot for Underwater Robotie Vehicle 127 Jersy Caras, Zygmunt Bitowshs
Po Tatoation Ly
2 Dytmnic ees Dyn Conte uf Voie ln Sis Degas uf Uurater Vice of Freon 13 Ds 1 Cooruate Syetnos and Tracking Cota Station Rost re la
NÓ Condviem Ly
Trang 8Contes vú XIL The P-Regularity ‘Theory: Constructive Analysis of Nonlinear Optimization Problems lân (Die Brescia, ir Szccpemi Aes The go I Peegelar Mappings acl» Generalization ofthe Lyin
2 Shatarey aulEemul Xonlueir Tổ TP Em Piniple, Nessa and Sale Con
£ The P Factor Method Per Solving Sager Nonlinear Equi oT 5A’ Mextund fr Soteinz Nouoglar Equlty-Coneteinod Opt Ta ti Probie
(Gn Rall of laity Constrained Optimization Prk ——— 1a ee ee ee ee ee ee ee 1 Optimal Cort of eral Equations with Ea Linite
2 Opeinnton Slow Vntge Capita aT Ts TC En Two Sector Ramsey Vintnge Capital Nadel Tot {Beatin Capital Lene ase Mar Benois TT
XIV Difforeutition, Sousitivity Amalysis and Identification ‘of Hybrid Models ia John Mare, Tier Contos
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vil ATHEMATICAL ADWANCES IN OPTISRZATION AND CONTROL XVI Numerieal Calibration of Local Volatility Surfaces 205, Ai Bev Herdne Busca, Mona R Naguenee i rs Prt fe th Lo Say Sua am Fe bee taal Vans {SRS pete ae Bd oo Mth đã ‘Situation ‘no bug 22L 2mm Kiundi 1 etic Mỹ 3 a ne tow Bi {shaun Stat =
XVII A Study of Some Inverse Problems for Disteibuted Parameter Systeme by Optimal Control Theany 38 2 hain? Rte Smet to Stshe Pobin 3 3
5 — 0,
XIX, Static Task Assignment in Distsibutod Compating Systeme au
1 Xian Prt Dein out {Prion Renal mo ÔN nu mm XX Distance Minimization in Public Transportation
‘Networks with Elastic Demand 230 Yon, A Quilt Duhamel 1" Preis sein 20 2 The Pivewe Linear Model $ A TaNieohaniuie >a âm 4 — hageAHmeiMr tui Wedsle Bdhaue 3m
Trang 10Foreword
This wlune comprises sletod papers rom the 2st IFIP Conference ‘on Systems Mevdling Pentagon building ofthe Ecol ex Mines de Pati, in Sophia Autipols, od Optimization, Ths conference took plane a the
France, fous ly 2st 6 duly 25th 200%
The articles i this volome pret ew aden in applied mate matitsn catnputatismal sienfe la turn, now algoritis and met can bo use in dustrial applic Ince produetion planaing ia the metal soduste va, Examples of such applicstias ton pring, ae sstens to prevent ship collisions, ew autorploting systons for sul marines, optinizatin of pbc trammportations ays
The onganizers would like to thank Yee Labourenr, head of the opin Antipolis division ofthe coke des Mines and Michel Comaed, hod of INRIA Sophia Antipolis who made
this conference
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CHAPTER I
PROBLEM OF ADAPTIVE MINIMAX CONTROL FOR PURSUIT-EVASION PROCESS Anne F Sho
nical system objects Pant 17 ( that consist of €
respectively) are deseibed by linear ad conver discrete-time recurteat
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3 NHTHEVATICM ADVIACES X GPHIMHZATION AND CONTROL srnlel by n dEcntetime vector equation, whieh depends arly on the nhược eters of abjece 17 at, vit the transformation matrix, om Une pase yoctors of object J Tes ansed that the sts
lug all priors defined system paraunetersaze law aad ate cones, lee said bounced polshedns [vitb & fate mnber of verte) i the corresponding Exetean verte spaces Under these assmiptions te formule and sole the problem of aalaptive maha contol foe the Pursuit-evasion process the dicestestime dvoantical omplete formation,
‘To organi the minimax contro) by the purit a & chosen class of the advise ndaptive contol strates we propose & recurrent Troonlume, ench stop of whi i based om realiat Ininitns nonlinear filtering process (oe [5 [6}) and on solve linear ‘and conges programming problems "The resus obtained i this port are based om aja aa be weed for compater sinulatina of av actual dynamical processes and for desig ing wf optics digital contri and navigation syste or haolof
ical nad transportation nystenis Mathinatient mode had considered, for example, in (I ss with ite
1 Deseription of the Problem
On given ineegervalued tine iatoerad TT = Wks esate of sth tio controled objects ~ abject J, coutrolel hệ the purener P aud ob (T 23 0) Be consider a anllistep dynaaical system which ©
ject IT, conteoled by the euler B The motion Ty the linear diverece-time recurrent vector equation of object I derived ít 1)= A)g0) + B40), @ sud the motion of objet 17 is described by the nontineae discrete-time HED) = Hi.) ®
Men € 07T C1: g € RF anil = € RY are the phase voetom nf object 1 sud 1, eapoetvnly (7,9 € Ny hore N Bethe et ofall watoral nares for n € N, R” is the nedinensional Eucldenn vector space) u() © RP ad (2) € RY age the contools of parsuer P aad evadr £, respectively ‘onstvained by the sate
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Probie of Aeptve Animas Cont for Purl-Ehasem Pmrres
OTST x Re RE RY for all TET is in the collection of variables (2,2): forall ‘the object 1 Tore due ea (2), (2) sation @ TF the wedor Red Fonts contin phase vector s7) € R* Me coustralne Wer cr “
ai the sen UỤ, Vp, and 2° are somves, close, at bontded pole drone (with afte ine of vertices) ithe ses RP RY, a respectively fir eal the eilistion (2) € 0.11 2° whee the ek 2 aces, The (2À ) — ỰVb SnỦ, 2€ Z2 6 VỊ) C TR {Let us describe the nfometonal possibilities of the purser Inthe pcul-evsion proces (the aapive mua conta ly the proces Ue puso of objec FF By beet Ð) ‘We msume the Er suy ở € TTT nd any ager ne Hater
ne by the purser P90) ~ w hie he intial phase sate of objet Zw) — {uleeramr whe ‘ee the mat cenlskdoae of the cong of the purse Pon the terval đế: <Ó) — (26) (0 € R® ad m € ÁN m < 3, vhích Uhe ast realizations of the sguat on ce eral, howe ve cự) (0) — ca Bed) ae geratel for al €€ Up by the dveeectine ing quantities ago measued and «)= EQtl)s0] + PA), 6ì ‘where E(t) the measurement error satsng he constrain eS, CR! VEN) t6
For all €€ ĐT am the phase vectors g) € RP of chiet ƒ, ve ae sane hat yt) and PUY) ae the rod (on ] ml ly x1} nhe, respectively, nad for all y() the rank of the masrix 2(y()) ie equal to the dimeasion m of ehe vwetor a(t onmlel polyholron (with finite munaber of vertion) ia the space Re Davis ee purst-evasion process the parses Palo kaw the set 2(0) = Z SZ" RE af allowed ital states of object II, whích sc Consistent (Se 2) with the atl signal ay Equations and constraints (1-65) sre al eset i to
The pursuit-ovasion paoeess performance is estimated by the mao tude af the distance bạReeen HENGK Er: be) at the time T nbhrte ƒ and [1 inthe space RE (whe “Then, fo te aysteus(1)-(G), the al of the adaptive enim canto for the pursit-erasion ress can be tated fan the viewpoint of the yeu Pas falls on given te etorval the pave P shold
Trang 151 MATHEMATICS ADVANCES IN OPTIANZATION AND CONTROL
rn the contol 9) = {oar Gr all #€ T=, wt U) in the poison mode os salto of the laptit múuimax strategs
| the chosen elas of admissible adaptive atategles), wang the ada sible (by vite of (1)-(6)) wealization of the sigual x) — (e() herr Fogtioe wth all walla informuston ahont his process, in such h twny thatthe minal magnitud of the đồg—tmte beteeen realizations the vectors [aT He = (yulF) iT ene 204) E RẺ sa [4T)]»
(UT) 2AT) s[T)) € RẺ ÌN bệ nai (where y(T) isthe eal- lantion of the pase vector of te cbjact 7 at the hủ Ï comiesiouding te the oantrl u() and (7) is ads reaizatoa oŸ the phưec sectar ff object ah the tne T which cin bestimmt in hs pres only the information sot (se 3| J0) that the evader ean have complete information about pacamcters of system (1}-(6) ou the tape Sateeval OT anu bie alan in Tee assume te pursuit-evanion pactan is diametrically auposal lo the âm d the horse P
2 Formulation of the Problem
For a strict mathematical farmolation ofthe problem of the slate tninim contol for the pursit-eeson procs in screens eal system (1) (6), wn intone some definite, Yo Nand Ta COT (7 < we denote by S77) thế nets space of Tanctione 2 FD — "of the integer argument where te mete is deine by tele el) = sa e100) — 228 I este) & 8a
and by complS,(F0 we di (7) that are aonempty and compact in the sense of this mete ge [jy the Eleanor in RO
Using the constraint (8) we define the set Z1) € aimnp(S,lT, 0= TỊ)
fsảnieietonteok tổ) — {0()1„zxcy sắn phner Pn lùn 7 x§, the set of all subsets of the spare =), "”" OA) — (we): ale
(FIT, VEE TTT, aft) €Ui} Ving constraluts (1) aml (6), we define similcly ee sets VF} € sonp(Sum Ð= DJ xá ức) & compl SEF 1.0) of adobe con trob vt} — {el }genger of the vader Ế aml mhuExiMle tneereneut
mo €L) = (€Q)]yerrrg ofthe signal on the time interval 5,0, espe sels
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Probie of Aeptve Animas Cont for Purl-Ehasem Pmrres 5
We denote by (FB) C SFT 1) the set of abs, by virtn of (1-18), ealzations we) ~ {U8 herery of he sigonl — on thc tìme
We eal th olbotion x (7}.2(7)) € OT RY x compl) (vhore Z[) tế sot of adminsble phosn states 2(7) € RE of 6 Tí ạt thế time moment 7 vít) = wụ — (Octo dZo}ou ~ (O40 Z0)) the r-postiou of the punsber P ln the discetetime synamilea stony (1-6) whee the nowenipty st js defined by
3€ 5i, a4 = Elwes + FOO}
For llr €O7T we lo define the set ofall amie r-positons of Ae ponte P by Wa) — (rp RE oem) for rt WE) WEG) — 0) — (pnd (OJ HR" x compl) Now for» fixed time interval rod ĐỊT Lz < #), adi tions, by setae of (1)-(0), the rpeition (7) the ote a
we dnote by fAraPsr(out}C) the se ofall pis (207) 20)) € Jx V9) cooeeM (so oth hs aca on the tne inter Fo
AG Beir) we} oe) — {EGF EOI AEN € ZO VED, Y£€TTT.Đ, 340) € Eụ, e0) = Eg0) (): S) = s9,
"ve (5.0, lz),0)) n4 3/5, 5z) 8C] denote te set
Tioneofohjets I and Hof the tne interea at the time £€ 7T, respectively: By virtue of (1) and (2), the motioue of objects and 17 ane generated, respectively, by the pats (y(r}sut-} aud (2(7},8()) We call the set " re mh of mo (0) = wŒS,với ATA wtr jm) = (2%) S01 eR, 90) = su, ivl,eO),
(eG) 00) € RB alr) al) s) tho information set of the pursuor P (see [J-5)} of at posteior’ mine max filtering proces for the dieseto-tine dynamical system (1)-(6) fn the Unie interval 9 eoeresponting tothe instant aad collection (e],MC).2(1) € WUỢO x Tim H x ĐỊ 9) Note that He the set of sdniedLlc malzaiobe of thơ phase vector of oiject UF atthe tine
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6 AWTHEMATICAL ADWANCES LN OPTISZATION AND CONTROL ‘9 whieh are consistant wth al information about this sytoen Koma to thế purser P on the sme interval 7,2 ar « fed thúc tcerval 7,0 © IE (r < 9), position wr)
(che Z(r)] € Wx) of the pussner P an his consol a) € CGI) se defi the footings MF Am(r}.u = (oO): we 80/71, VEETTTTS +9 = E0l9)s(9 1 Fe) 6)= wỆS0,w).n), s0) = 503071513), go Wf,efr).0,10)) = 40): z(0)€ P0) x40) = {0.000).2(0)], g0) (().06)) € ZU x VED) o I yr} 00) 20) = BNF A tr) 4.0)
2b) € UAT or) ty)
nud ee call them the et of adit sigaals on the time interval 7 nt these of ade s positions ofthe pmesuer P, respectively hich ‘ormespon t0 the 7 psiton {7} and the sono] a) at Pog: Wer) x UGA) «00
sesgulng to any collection (7), u-)(9)) € We) x UE) x29) a D-position wa) = 4 yD) 210} AEM, cane T) —+ Wd} be a mnlistep maprin 9) = 19, 9'°(9) 2°00} = Ezsfelz).lf3,5l3) —- (U) wh AV) = oF yled, ate, ZCI FA wtr) wl) (02) = faint), 200)}, l0) = vội 0) ‘Theo, for estimating thơ quaBy of the pursuit
Trang 18Probie of Aeptve Animas Cont for Purl-Ehasem Pmrres 7
(2490) = (LEV, mtr) 9,06 bd tủ
Hore und below, for a € Nok © Take = (eigzays ony)! RP ad XC RR, the expressions (#4) — (riage tay) © RE and (Xe đige RSSg— [rluz X] C RẾ ae the bprojection ofthe
fad the A-peo jection of the set X, eepetitlgi MỆT] = {7 We) (wit) =
Now, ited some definitions An asks adaptive control dinlegy UT of the purser P for the pursut-evnston proves (1)-(6) em th time inlervn TT thề mappiuz
UIT —— Us, ghính aeolalee tà every tme monen€
sai say pelhleteeleatlen of the >epeeldin w(x) = fe 4) 20)) € EJ(6{0) — sọ] the sec U(w{9)) C Uh of the eomtaole u(t) € 0 of the pusser P- We denote the set of al adnsible adaptive control Stelogies nf the pueser For any small > O we deine the alaptiveeensimas ctrl stra P for this prooess ty U sấy f the pưeul-ewsdon process (1)-(8) as &wealiation of « speci adaptive coutrol strategy US — UL w(r)) 11) 20(0) — ap) foun the cles of aes adap coe € Ur (r eM T— Tale) & UY, vhich is formally sesribed by the followlugeebationsipe 1) for ally €,T—T and r-positions wl") = fr gfe), 2%) € WO my 7 WE) el) — nh ee vile#U )~ 0x20 €U tủ 2 v0, et 2) for oll 7 © TST ond T0) A00006.0)0)] te ) = th) Were) =U dị Here, wf = (0.0 Mis: for aduissile past eeallzations on the 3 = Db where tral) 2840)} forma 1) of the eouteol
SE) of the purser Pan the sigual sỸ
Trang 198 AWTHEMATICAL ADWANCES LN OPTISZATION AND CONTROL E7TT,vf90) 1# (2,c È DỊ š (0) = EFT T wher ul 06) defined by the eelatonship (18) and the nha QFN +6 (0%! where the fuaettonal
ber imal genrnto ranh corepondin to realization of the adaptive minimax eon sayy UE) ¢ U° of he rgnser P on th me tư 77-71 md duoymlael tp fibing Rhhlimsldp 1) = nụ FETT wiry ate ole + UNE
(alr) = ATT Laer aed,
sy nssurnptions for the dicrte-tmw dyna system 1 the functional >, follows that forall si moments 7 € 0,T=T aust rpositions w(t) € W(7) (0) = ay), the
sdx UỆ (7l) ac nonetnpty
t thế toaliadioes of the contool a!) of the pursnar Pant the mái 2
ST, sỹ sŸ”1()) ae thế mse of sing the adapive minimax
trol strategy Ue) € U" on the time inter and the gai 9T) € ĐŒT TT, xố TT — Đ,Ệ Ì(T — Đ)) satisie the following relatonsbip
ST afer — 1,4,9 TWiT He lT— N.a(T)) — 2), 289)
Trang 20Prati of tt Aidner Chi đọc Đơn Eiamon P "`" an st OTT T = AT ATT wlcler— 1,7 19, Thea, we call the suber LOT) = AO, ws.)
the éoptimal guarantee! result corespouting 10 the reallaatou of the adaptive covinimas coucol strategy Uy € U" of the pnesuer P om the nv latrval 0T, In vow of the at sfthe hdVe c-ninlatax coutro ainertttioe dyuAamieal set (1) (6) 1 efintons, sea frou the asain problem Tor the pmisuit-esasiou proce lu the
Problem 1 For the inital position (0) = wn = (0.40 Zo} € Wy of the psuer P ia the dlsenete-thuw dyuaaical system {1)-(@), and for any snall¢ > 0 it roquierl th determine his adaptive enna contro strategy UL! € U* and the eoptimal gnaranted xu (0T) ‘conning to the rsleaton of ths strategy om th time ater, the sequotco of one fort pursnit-eason process asthe react ay hp oi — Conehu
‘To organize the adaptive emininas control of the pursuit as eel Tanto af a specie adaptive coutiol steatagy Ứ] € U", we propose a recereat arity, which seduces the aalkstop probe sell tion of the seqwenne af one-step optinantion prublons, wad each step of which is based on realiation of pesteroes mins korn proces (oor |] (5) ato solving of linvar and conve programing problems
References
Lu) 6, A Boson and S Bước Diese and opt hic sat TEE Tne Auta Cant 0,86 B90, [a] NX mwah Theor of Conte of ation Nana Mow, 196 in Rein BLINN Kiwoki an AL si Co
Trang 21
10 NHITHENATICAM abe
LACES IN OPTIMIZATION AND CONTROL [PLA Shore Mina str or state thio of boaloe đit tm etition ml ete Cmte S24} 81,10, IB) AB: Shrihn, Muir Batiaton med Cota Dice Tne Davee Sytem Url Sate Univer Pair Elana HW Raia
Trang 22CHAPTER IL THE TOPOLOGICAL ASYMPTOTIC FOR NONLINEAR SYSTEMS Application to the Navier-Stokes Equations Sanuel Amslutz ` MHP Intreduetion
The topological sensitivity analysis consists bn providing an aay totie expansion of a shape function with cespoct to the ser ofa smal hole imemhel nojd the ounin, Generally, this expansion sf the For
SHO COE AY) — JO) = Hwan aK
Trang 2312 MATHĐMAICAE ADLECES IN OPTIMIZATION AND CONTROI Iotest to sort hoi where the
renuukleale 1o nũrief Torntise si & doecnit đưct
inl gratont 9 is wegation Thác thm band on the tse of ke
The topological asyinptotie as boon instigated by the works of A Setinmacter |, 3 Sokolewski and A Zachoandd lR, Thea, ẤM, Mac mona [5 Intzoduoot a genoeaization of the adjoiat wlethod to compte 2 eapily for large clans of com fytctions This appeaach, combined
sith a truncation technsque, as permitted to determiue the tupologiea teuiont for vasious problems = Uearolmtieity [2], Dirichlet |, Stok HỗI aml Heats 6) equations
For nonlinear systems the methods proviowsly vse ave to be widely inode, First, to avoid some extent dificult, the truneation i bandnnel Seconil: thiet of the noalipariey ha to bế crlhiuee For a costly class of operncors aan! a Dist coiton on the bowl
ney of the Hole, we show
hat his contusion fh of ancon order At XaKake of sppledfbie me wehnew the brown Toslis suncordag tbe
previous liar systems and we obtain the topokagral sensitivity ofthe NavierStolvs equations In this paper, all the prooGs are omitted, ‘The Inater rosie i Museen by & númerleal experipent
1, Presentation of the Problem 11 — TheTmi
| Problem
Trang 24
The iyshgiedl Asơnplisie fac Nodicr Sgdens ry where Y= HO) ad Fy bs the nay dete by
1.2 The Perturbed Problem
TL sẽ be am open and bounded subset of RY containing the origin swith mot and comet
Daramctor p>, we conser the perforated đunuiu Ap = 4p where Gp ty bps Bor omvenienon, we wll nestane Tlhv perturbed feld n, is supposed to he £1? jn weighborkood in the soquel ht o ~ 0 sail be solution to the PDE { Arpt Blu) = om Bp ep = Doon T đi We define ty mg Av vy
SAlvsuy CS cRúhrsewL[ nay 8 Dặc tạ thề Gisen fonnla uy sui-fis
oH
Herp extended by aon inside sys Tht convention wil be ja Pliitoly need thozhone thie pape
The Dirichlet cmaition on PF cold be enced by wn boundary on ition making Problems {1) and (8) #elkpesed Ín elevnat funesional
18
Trang 25
HÀ: NHỢNĐMAICAE ADEECES IN OPTIMIZATION AND CONTROL 2 An Adjoint Method
‘The tupaloseal ssymptotie Bs given Dy the following theörcm, ph vine that sone hypotheses Tuner which conditions those hypotheses hol are satised In Sortion IS, wo il explain
THrohEu | Let V bea te oct, For all pe Re, 10660 fe differntionte map Ey: V — "` <Fiyherves= 0 Yer t0 + 0 diferentiahte frnction J, 7
Trang 26The iyshgiedl Asơnplisie fac Nodicr Sgdens 6 3.1 Topological Sensitivity in 3D
Approximation of the Perturbed Solution, Fire opproviososion, We split int tp = We thy ty where hy is solution to [%1 ns
ad ste eetaalnder The dousinant part of ty — be hg
Second approsimation, Wo st Hy) = hg) and we 4) ute Hy Wy 1D solution to (3 ụ ¬ ` H = ~w(0) on da ‘The fuvetion H can be expliited with the belp of « single Ine tt |
He) = -a) fee vito)
where the fundamental solution of the Laplace operator iv 3D i Fe) ire
and 3) © A174) Is the unique soltion to
Asymptotic Expansion of the Cast Fumetion Iu order to apply ‘Theorem 1, ww nel to determine fp) Bets Bray 83 and đụa, Thế valine of 6 sud 8 ate given in Seetion IL
Trang 27HH NMHỢNHĐMAICAE ADVECES IN OPTIMIZATION AND CONTROL functions, uation (1), nụ tớ nem that ey — Wand we Foous on ys Dy < Fal — Big) ta » te
plac tp yn y+ ep aling change of varie al app inating suceesieey Hy hy HF ana ra(ar) by a0), we obtain
1 can be proved that £{p) ~ o(9) Next da tothe jump relation of the gle ayer potential, we have 3yHf = —uo( Equation (8) Los wit 0} ou dy We deduce that
flp)=@ and dp, = PewlO)ra(0},
re [im a
Thus, uudoe the fllowing hypothesis, the asymptotic expansion ofthe ost Tauetion is provided by Theorent Ủ
here
Hyporunsis 21 There erste N€ R and some constant e > 0 sch that for any Fe HMO), Â â HEE) and w= HCC) with Talsg, < As the poten
{ Bey Hale) = fin Op eS pont b= 0 om Oy ag af ll one solution satan 2 Theve eels consent ¢ > 0 auch that forall lieve scr ntty snot YO) with KRatÐla < san t2)
Honma the eqs the diet ten bs considered a od
Por ait open and boon subset O or R8, than exits a constont 0 auch that forall 9 sudReienty smal ud all © HO),
Trang 28
The iyshgiedl Asơnplisie fac Nodicr Sgdens " 4 When oly ted ta se,
< Real) ~ ĐR, (me > = allege + lil
Some extpls: of such finetions # ave given in Section Tks As a ve itn the differential operators of order 2 whic, bọt sHXy the hind assumption
counterexam paricular.d
‘Tueowea 3 (raroLOGICAL seaxstrivery 1D) Soppwsing that 1 the fonction & satisfies Uypothesis 2 on la lụa < AI
1 the cost function satgfies Equations (10) ond (21) with Sn) = the aint Bquation (7) tox af fast one sofution ey € 14}, the dict and adjoint stator oy ad 2 are of close C2 in a neg 3 + the weBeont P à daÑns bự (1,
he su function das the following asymptotic expension
l6) —JN) = øPAelln) 4đ +altrelg) —— 8) 3.2 Topological Sensiti ity 2D
Trang 29Is NHỢNHĐMAICAE ADEECES IN OPTIMIZATION AND CONTROI Asymptotic Expansion of the Cost Punction, - No bao — Ea (I sai} “ED “The latter equality comes steaightoweurdly from the fact that E is prowl that £(p) = of 1/ lp} This < Ful oh) wih to), fuudenental solution, Te enn Tu) top am ẩn (9)840) ‘Wie mem again in hú che that Bry — We deuce the topologien semiotic given ia Theorem 3, Kor tie prof, the fllowing hypoth ie equiv,
Hyvornssis ¢ 1 There existe» €t2| and elle t2e| auch that & fam le extended io a map, all denoted by ®, chick ie rontinwous fram WMO) sate LO} for eny open ead bounded sabaet O of e 2 These extla NER and some constant > 0 much that for any FE HH) e € LAE) and w WED) with hyve) © the protien gone Bet Rae) ~ fin Op b= 0 om
au ai fast one solution tating
elu, Sf lean, believer
¥ There evils 4 constant o> D auch that for any open set OC and Jor all uu € WER(O) with Mlhpego) < À em [ey suffice ntly sal,
We Wenoy < ells iey When Ith» tendo to zero,
Trang 30The Topological Assopotic for Nonkinve Systm " Tueoneat 5 (TOPOL06ICAL SENSITIVITY HW 2D) Sapposing the
1 the function & satnfies Hypothesis 4 ae |sallyssjm < À the cont faction satnfos Batons (10) and (11) with (0) 1/lap the adjoint Bguation (7) bow af lust one sotuion sọ € HỆ, fhe dirt and atin sates ay al sy are of lass Con weigh he cost faction hos the fallawing asymptotic ezpension
ie) — 0) = = armteyralO) + B+ Boal + 0 16) HO) HO) = GE BrwtOhot0) + 8 Bl 08) 3.8 Generalization
‘The wosulte of Theorems Sand 5 can easly be goucrllzd to the ens seloge the Laplace operstor & b the following properties placed yaa operator 3 sity Hivvorusis 6 For any ajen and bounded wet OC RN, A ix defied hy YO) = VOY Asa ditavey, (0) in else subspace of HOY, "` Ais atensor of ender 4 such that A20 340214 WE Mya
the fundamental matris of A satisfies, in the sense of the uniform ‘norm with respect fo the angular coordinate
Trang 313N NHAHENTICAM, ADVANCES IN GPTIMIZATION AND CONTOL ae 0) in 2D, Bey -BEQ ro 2p ` For su «wetor operat, the sealer P inst be rept by the 3m tnt of te Hear ap € Rm Pa fs eg the vuÖqt Slatin "na an ‘Thon, under the rospetioe hypotheses replving A by 3, 174} by YO) and HO) by Yo(O), we baw the uf Theurems 3 and 5 satin ly topological asymplotics Ho) = HO} = pl Pach).y(0} +8044 52a)+0l0) lap, ĐẠI 1 1 Jo) H0) = GE Larry) 4b bdr] +E) BD t9) 3.4 Examples
We give i Table some examphs of diferontinl operators & satisfying Hypothesis @ and of wonlinear perturbations © verifying Hypothesis 2 in dineasion 3.and Hypothesis ia dimesiou 2 The checking of these Hype ‘except for the Navie-Stales problem where in wery technical and ont of the nope of this paper We give in Table 2 the erespontng matrix Q iu 2D fa clasts the pin stain cose rvonted, For plain strss, ẤT = 3y/(À £ 2u) mat lờ enbstRutel Thế — = Ber 3m |
LETTEIITTLDIA = TOT =E: aa Et Tote vàn [REDON ave =O [Boe Wey =e
Trang 32
The Topological Asompotic for Nonknrve Systeme ma Seton | uc TH SA 3.5 Spherical Hole (3D)
We supyse here thar = B(0, 1) and tht the operator 3 i isotropic, that is, Ele) = E(r), In this caso, the doasity solution to (17) constant and thee exists Af 21B) suất thác [ Be-witay) a vee Thus, y= Maat Pa Ma, We deduce “Table 3 gations the values of A cornsonding tothe operators prsbeuted in Tale | (ae x
Eleskay | as Em) - ĐT TÔ “Ex
Seo | ete) Seen M
“MS Bmlemeialslelnn ơi net A AD}
3.6 Particular Cost Funetions
Trang 3338 NHTHEVATICM ADVIACES IX GPHIMHZATION AND CONTROL 1 IP the coat function is ofthe form
Ile) = Jp
whore > 0 and J is ifferentible om Va(2\ BETH), then 2 For the cost fanetion ss = fete shore uy € L2(0)" 9 L9BIO 2)" 9 > NH > 0, Sn =82=0 9 For the eos function `
here ue EIQ) WEB RYP p> Ne >, đi" ant b= [PROM 9
Qn) (0) 020
4 A Numerical Bxample
We illustrate the use of the topological asytoptotic an aay example Laka foun f] where cs atid i the context ofthe Stakes equations
1 consists in inserting some stall stacks ia Lan fll with aa incotnnseiĐle aid in ander ta apprasimate a target How ay Here 8 te 4 NavierStokos mod, ‘The geometry and the howndary eonitons co givan in Figure L
he cost fonction to be asinine i defined by
sey — fsa =1
Trang 34‘The Total Asst fr Nontiear Sites a
Pape 1 Ov the 46 te bay he A he eb
‘elon Van f the average velocity of the inom Auk “his mona that we wish to obtain w pastbolie velocity profile with-» masini esd at the cop of the tank For dae fueton, the topological gradient At the point ay
gữn) — man)
Trang 3521 NHHTHEVHTICM ab
LACES IN OPTIMIZATION AND CONTROL FF A Sener Topsoieptiserny tom Bastien unter Verwendeng {on ojahpotmerenircterien PRD tha, Unstone
Sewn, 116,
IH 2 Soni al A acini On the oy dentin eine
Trang 37
2E NHTHEVATICM ADVIACES IX GPHIMHZATION AND CONTROL Introduction
Torta devetopments ave ten place In the eld of dterration of inbreel cũnctte grieturee drag Hn Ist 20 soars sow m3 Methods nf modeling of chierhinion of enforced concrete nce
a be dived at the bad Han
Level; Scientific base ew Lowel 2 Bader Ie
Level 1 Techie
130m advanced scinntific level based on an exnet modelling oft dbeterioeation profile (deteriopaion tute ax function of the Tite) Ai ‘acs information of the sonceote microstructure is weed 0 a
Ue liso of ehloride fons lat the couerete Further, advanced Infor adidn an the evieohtaeital oadiug of the structure fs used (2 model 8 Ube time dependency ofthe eller cod entratin un the xui dể Hinestmactr ‘Three los! 3 onteling nf the eterno iene se single structures Level 3 anodes ar base on a full probabilistic de serption ofthe sarlous quantition which affet the deterioration of the
Lew 2 isan advanced engincorig love basad on average material parneters sic ks the ffosion eoeicent Level 2 aeling t bas fou average Loaling parameters eg, the corde concentration on the soca su on simplifications regarting the ingress sch se using Picks Inet The deterioration modeling fs Tinto tm single r few đeteie ratio mechanisms like chlor vel 2 models ean be used for design of now structures, but also fox lussd rv sion of the minh assem deteviration cates svoupe of exstingstructtes
Trang 38Maiesing Corosion Crs
1, State of the Art
Tà túc paper only bore induced curtsion of the einficoment fe idee tho rate of chore pensteation ito concrete is eed ty irks lw of iflsioa, then it eat be shown that the tne Tore to inltrlen of reinforcement corrosion ix
cB lapt (Ge
me-ip(s(đ
cote the concrete cowie D i the dfn event, Car the ina elo cocentcain In he concrete, fb the ctr Satin Ail the yarunctene mentioned ahowe ar wld by secure variables m" After corosion fltation te ust products wil iniilly Ge por e arson the stoe/conerse suifuce caused bự the truinhinh Boat
paste to Me] nại caapprel/entriwxl ir voids aul then rel i a "pansion for the eonencte near the reiaforcmont, As ạ snl of this ttle stretoes neo Hntinted in the conerste, With inereasing corrsion he angle cesee il radl ertiealvalue Dring this process the volume ad crache vi bec cts a intial crade
Ing of the concrete Wy wil gceupy three volutes, namely the porous
one Wyo the expansion of the concrete de to rust pressure Wey ani the sce ofthe rorroed stool Wc With thi modeling al Some Ininor iplBentions i oan then he show that the ine frm corso Imitation tcc nition iss [2
1
Mew = Datos STIR
sce Di
non corrosion ete, pat the density of the sok mpage 8 the LAendty of the nest paduets, Te ein the derivation of (2) assum the diameter Dr(2) oF the eenforcement bar at the tine sm Đề
the diet i the rănfrsswienk ae pe he anual
Trang 39
28 MATHEMATICAL ADWANCES IN OPTISRZATION AND CONTROL Alter formation of the initia! crack the rohar eroseseotion is further feducnd du to the continued corres, al the erick Width Wong Ieeassd Espetbuents show that the function bathoen the eedactin of te rebar diameter AD and the cornesponding incense in crack width Bitom i give Lone inerval St ensured cn the surfece of he concrete specie ran be approximate bà e linear fet on = 7D 0 whore the factor ie of the onde 1.8 495, 1 flins fom (3) and (2) tint Met) = temest Tet) +4 Diol F (Tomes) 4 — Prat) 5) tên, ‘ et) lee the eritical crack width be By setting (Tal) = 1 ing expression is obtained for ring atthe He Toran th fol Testu Teva th) nc Fench) 208 te inital oracle width He the Se Tic
Several rescazthers have investigated the evolution of eortsion oaks iw reinforced eonerete beans experinentalle Attr foroation of the ini ‘ial crack tomsion, and the width ofthe erack tit Berens, In he experie the rebar crostini farther gece ds to th continued ‘ments ao impressed euttent age aormaly used to atc coreode tbo crack evaotion fs manaured by’ che nse af stain gages attached tthe Teawis The les of bar seetions is then mouitonel ant the crnespouding
surf of te beams hr st exp
anette aaction betwen the eo Auction ofthe robe dnnetor aa the wnat crack viíHh metsorsl J the surface of the conerote specimen ean be approximated by Hit sar Tupetion Based on this carlton she increased erack width with thine maybe atoll
Trang 40
Maiesing Corosion Crs Py
sand the correspond ed by tho uso of stxin gauges attachod to the surface of the beams In all four experiments the fnaction betieon the reductiou of the bát Sameer ant the nasdanvan cack wlth measured im the surface of the concrete specimen ean he approximated la nea fieton, se Bee
ork ewlition sm
To Ate be the ince ie eenck wide in the ine interval AC aad ke the contsponding loss uf rebar deter be ADtor Th Âm, ve ‘ikon Rhone is of the onder 1.5 Ii The factor depends on the erosescetioul data to 3 in the experiineuts rported ja 3, FEM Verification