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