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RECENT MATHEMATICAL INDY Nie

IN OPTIMIZATION AND CONTROL

Collection : Sciences mathšnhatiques et wee ue

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ECOLEDES MINES DEPARIS

Les Presses `

INFORMATION PROCESSING:

RECENT MATHEMATICAL ADVANCES

IN OPTIMIZATION AND CONTROL

“8e de Mhes de Đan dần

Raoifercrrenn.ifreee

‘te Sn Mig saan đau (CGI Nhhớigh LW Cae Nas)

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INFORMATION 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

OL

iy NHƯHEVHTICM ADVIACES IS GPHIMHZATION AND CONIMOL

Vi 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

VIL Collision Detection Between Virtual Objects Uei ‘mization ‘Techniques

(Charel Bars, Yelander Herman

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

2 Onin ofthe Pte Plating Anpmoeeh Cancion and Outlook us tà

XI, Tracking Autopilot for Underwater Robotie Vehicle 127 Jersy Caras, Zygmunt Bitowshs

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

Anpenlse ROV Siaplitiot 8 1 Paranetine us

Contes 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

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

vil ATHEMATICAL ADWANCES IN OPTISRZATION AND CONTROL XVI Numerieal Calibration of Local Volatility Surfaces 205,

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

XIX, Static Task Assignment in Distsibutod Compating Systeme au

2 The Pivewe Linear Model $ A TaNieohaniuie >a âm

4 — hageAHmeiMr tui Wedsle Bdhaue 3m

Foreword

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

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CHAPTER I

PROBLEM OF ADAPTIVE MINIMAX

CONTROL FOR PURSUIT-EVASION

nical system objects Pant 17 ( that consist of €

respectively) are deseibed by linear ad conver discrete-time recurteat

sector equations, epson Phe sat of the signal with enor fe ge

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

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

ad (2) € RY age the contools of parsuer P aad evadr £, respectively ‘onstvained by the sate

vớ) € Uy C R®, r0 € VE CRY (ng EN} @ A(e) and BÍ) are re matviow of ores (Fe) anu (2p), respeeivey sd or all 6 O° TT ta nti Aft bs msn be verte the

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

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

“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

ia omen, chee, and

1 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

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

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 =),

mo €L) = (€Q)]yerrrg ofthe signal on the time interval 5,0, espe sels

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

(): 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

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

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

20) = BNF A tr) 4.0)

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

‘Theo, for estimating thơ quaBy of the pursuit

dynatcal aster (1)-(6) cn the tne luterval 2 C ĐT hy te pursue£ P2 Me deRae the funeteaal 9 = W(x) x (FO) x 5,0) — FU in

fc a way that Gor ny auall «> Qa yeubation 1001/40) € HPT) 0G 8) 3070) wie fs deine By te rela

evasion proosse in

Probie of Aeptve Animas Cont for Purl-Ehasem Pmrres 7

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

1) of the eouteol

SE) of the purser Pan the sigual sỸ

mộ-Ÿ]()), the -podlion 11), mac

8 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

(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 afer — 1,4,9 TWiT He lT— N.a(T)) — 2), 289)

where the (T= 1}-postion ofthe pucsuor P

Prati of tt Aidner Chi đọc Đơn Eiamon P

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

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

SHO COE AY) — JO) = Hwan aK

whore p and 2 denote respectively the diameter and the center ofthe Ile, wea Red domsin containing the origin av fia pesitive Function tong 09 2en0 with p ‘Therefore, to mininze Hie erterion J, we hae

12 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

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

4 given o € £40) whose support does wot contain the center 9 oft hols wo consider» ei uy sting to the probe

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

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

"`

<Fiyherves= 0 Yer t0 + 0 diferentiahte frnction J, 7

We asinme that there existe fet wy © Y vated aja state station

< DFulw)est vey = —Diwele EV ) hat ave known four veal munbers Spy, Apa, Ay4 and 63 0) ending tn sero with such that

is)

< Ri0g)— Falta) — DEW(Wo}(y —ehen > = Fler FOL} o

Slip) — leh = Sedban bolo) ny Jal ~ tang) = Day(uallny — te) = Kedan + alfa (yy Then,

fp) ~ Heo) = HON Be + 6634815 +8) FOLIO

3 Main Results

sven, we nRrodue the nolation

The 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

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

HH NMHỢNHĐMAICAE ADVECES IN OPTIMIZATION AND CONTROL functions, uation (1), nụ tớ nem that ey — Wand we Foous on ys Dy

ag af ll one solution satan

Por ait open and boon subset O or R8, than exits a constont 0 auch that forall 9 sudReienty smal ud all © HO),

AMR O02 iy Slewelioy:

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

Approximation of the Perturbed Solution Thc fundamental olution in dimension 2

sna rhe remind

Is NHỢNHĐMAICAE ADEECES IN OPTIMIZATION AND CONTROI Asymptotic Expansion of the Cost Punction, - No bao

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,

< Re,)— DRu,l0lees > — nlltlussim)

The 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

si=0(!), =a) mạn

3N NHAHENTICAM, ADVANCES IN GPTIMIZATION AND CONTOL

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

LETTEIITTLDIA = TOT =E:

aa Et Tote vàn [REDON ave =O [Boe Wey =e

The Topological Asompotic for Nonknrve Systeme ma

[ 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

“MS Bmlemeialslelnn ơi net A AD}

3.6 Particular Cost Funetions

TUEOREAC 7 For the following cost fmetions and on operator Sut ‘sfiing Hypothesis 6 unde the lnpotheses of Theorem 3 oD (resp Theorem 3 en JD), Equetions (10) and (11) hold eth F() ~ @ (resp Jia) 1) np) ana the corresponating voles of 8s, and Bi

38 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

here ue EIQ) WEB RYP p> Ne >,

đi" ant b= [PROM 9

‘The Total Asst fr Nontiear Sites a

At the point ay

gữn) — man) 1a the expevinone, we have taken Vion — I0, ghích corepondk tỏ n Reynolis nue ealeulated with respect to the Tole section equal to

40 The topological gealioat i repeeseate i Figuer 1 Lara 16 be quite ge sl oud ews lion hel

21 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

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