Lecture 3

Bolzaprobkmdffeca.o.es,JEQEECxctb-fijutixeti.atDdt

sit . X. et) = f- Cfi Nfl , Oltl ) , Xfto) - Xo , OGIEt E Cfo it ,]

Alternatively , this is constrained optimization problem.

inf JCQ , IT = - -n

' Isubj to Kaya fctix cacti )

• Lagrange multipliers .

Mein IA) subj got -0 → Ca) t AGH.a LGA)

22£70 , 2×1=0'

• ZEO (Lagrange problem )

¥71 fff' Lcf, XHI , 041 ) dt subj to KCH - fltixltlioctl ) - O -

I = ftp.LCtixctiiectbdttfftoixctlfxltl-fcfixethoctijfdt .Fats : X - fctixco) ( Tx H)FM" "

* Lff, XHI , OH , ) - Ict) -Pxflfxctl.OHDTXIH-0.lt/a-7xCfft,XlthOAlYXlt1-HtixHkOetiD-Hct, natural, Ooty

Foch : L= ftjltctcxcthaltlcoetildt + JI' Actlxitidt .

To HCTIXHIIXHI , Oct ' ) = O

pontryagiismaximumprina.pl#

Define Hamiltonian H ' Cto .fi XIRDXIRDX → IR.

Hft , x, p , O ) a ptfctix.co) - LAX, 03

TheoremCPM#

Let 0£ be an optimal control , ¥ be its controlled trajectory .

Then , there exists PI : Gotti ] → IRA which is abs . cts . and

X.* a) = facial , Ctl ) = Fp Hct, ETH , PTH , HD e IlokXo

15*47 = - TxHtt, Cfb PTA , D , p*C a - Tx# CX*Ct,) ){HH, x' Al , PTH , OTH ) J Hct, GI , F'All O) for all Of ⑦

holds t- a.e .

P"

is called tae co -state - / adjoint .

Pnofoftheplup

Step I : conversion to Mayer Problem C ↳ 07

Define KOCH a Lctixltl , Oct ) ,4%7=0 ⇒ xht-fgth.cn - - 7 It .

Cx3x ) → 5 , Chef ) -1£ , IDEK #CX) x Xo .

⇒ iff B-Cxcti , ) subj E- faxed . Xcx -Xo.

Step 2 : Needle perturbation .

let Q"

be an optimal control .

Fix EE Hoste) , SE HO , ETO'

O

Define EE with ATotti.Oe a { S tf CE-e, TT s

Out.

Act) otherwise g€s t

Define X.ect) - fltixectl , D Xel - Xo

÷:*::Steps : Variational Equation

Define VA) = lim XeG3-X*# C- ETC, 't, ]E.→ Ot E .

On tfccit ,] , X*, Xa satisfies the same equation

X. = f-Ct,xc0Y

⇒ VA) =F×fHyx*Cf) , OTH ) VA) ,VCE ) - Vo .

VEI ' Vo - diff, { { ( Xotfotfttixdfhoeetlldt - Xo -fiftieth ④( TE x

↳e- diff, { If fctixodti , It - tefeefftsxtti . Idf }

.

'e'

fcf,x*kl, S ) - fff, x'Ce ) , Oye, ) ( tebefafffmd.tt ) .T

✓Cf) = If Cf , x' HI , ti ) VH) / VK)

is

×eSteps : Optimality condition .

VCTDT CHAD ) 70 .

in

depends my, s

⇒ Go to adjoint equation .

Tittle AHIVC't) adjoint a pitta - AARP .

If put = vtpt pair= VTGAT )p t PTA v ' = O ^

I'putt a constant in t IDefine co - State ,

or adjoint .( i¥¥: : : :*:¥¥i÷.. ...J p'

very = p*Ai5VHi) E O .

⇒ puttee,x' let , s p9e5' fee, x'to, )

(TT Tt¥¥¥ )

Step 5 : convert back to Bolz a problem

ii. out - o i *Mt" ? Essex, + xD( ⇒ p. .µ . ., a ,

= → . )J puff x'Khs) - Ha is € .

÷

jut . . r .

→ m*⇒⇐iii÷÷T=→IHCtixipiyn-sypptk-hltix.vn#HltixiP*I

Boblemwiuxtvaab6nm .

''

If -8C = fat'HGxHh0HDdtt§CxSubj to X=fGixcO) I tfcto.fi] c X Hok Xo

Xanax ,

prep.

x. * = 7pHHix7P7O ) x'Itoh Xo x-

Cti - X ,

P'*= - *Hit , PTO

"

) p9ti=-FIlx4{Htt,x9H , Petticoat ) 3 Hit , HI , F'Hiro ) ht OG

Example KEK 047

Myth f! Wtf COA) - i)2df I XHk§ , Hilal,

*

o 't ""' II sit: :I is global minimizer .

Lcf ,xD) = XZCO-D2 Hct, xp, = PO - X' CO- D2-

PMP : ITH a O*Cf) XM- 11=01 X'Cl) = I

154ft I 2x'G) Cl - OTH )2

OTH G arefmaxfptftlo - Hittite-02 )} .

0%1 af 9 '

off!? 15×07=0 htt .

XA) position of car at time # .

Exampled

,#>x.

ix.CHI OCH OCH C- E-Ill]

→Max10,041)

iff ICE] = -xx) + Jgtlzmaxfo, OHD' dt =x. = V X Col = O '

OCH E ⑦ a f-I , I]✓ a O '

VCO) = O-

⑧

solution using PMP .

Hct , x, V , Px , Pvc =

(Pyg )"

(tf) - ImaxCOOP'

= p*v + Pu O - Imax Coop-

PMP Equations.

x.* = V* XIN - O

j' a O* 4-907×0 .I er, ¥¥¥i :L .

⇒ PEH - I ft ⇒ PECH - T - t .

Hctix"

V'Al,

HI , Ritu, O ) = V'A) + OCT-t) - Imam0,032-

EG) a argvnax H o - T- t . -3..

Off'll] .

-

= mint-til ) H

T's ftT-f- I