A non standard optimal control problem arising in an economic application

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Pr?:-quk;8 O p ~ n m l (701 3) X3.7(' )' F;3 71 4201 3 Brazilian Operabons Research Souely

Pnnted verslon ISSN 0101 -7438 1 Onllk veraon ISSN 1678-5'42

www w ~ c h brlpopc

A kON-STANDAIW

OPTIMAL

COKTROL

PROBLEM

ARISING

IN

l$Y

ECOKOklICS

APPLIC:\rl'IOTI'

Alan

**Zinobcr* and**

Suliadi Sufahani

I2rxcir.d Mq 3; 3 1 1 1 : Acc-lytllrl July 1OZ 201:

ABSTRACT. A rcccrlt optimal mntrol problcm it1 the arc3 of economics has mathmatical proptrtics that do not la11 into thc standard optimal control problem formulation. [n our prohlcn~ thc slate value at the titla1 time rhe sme, Y ( I - ) =

=:

is Cree i~nd unknown, and additionally the hgwngian intgrand in the f~tncrio~~ai 1s a p 1 w ~ 7 h . i ~ ~ consml firncl~mi nf lhc nnknt~wn valuc j i 1.1. 'I his Ir no1 a sundanl opl~mal eoncrol pn~blcni anti cilnnot ht. solrul u\lng l'oniyagin's Minimum I'nnciplc with the standad h o & q ctmctiiiins at thc final tlnw. 111 the stanchrtl pnbhlcm r ficc final state j - t f ' ~ yield\ a n ~ m x i i q houndarj condititrn p(7 ) 0, when ptl) 1s rhc cosmtc. Rccausc the integrand is a function of y~ T b , thc fkw ncccrvary condition IS that

r1T) should bc equal to a certain iatgral that is a continuous liuilction of T I . We iniroducc a continuous approximation of rbe piecewise cunslanl i n t g n ~ i d halction by using a hyperbolic rangen1 approach and ~ 4 v e ijn wdniple u s i q a C - ' dlooting dguriQm with Kwtunilrmion for sulvilig 1he'lic.u I'oint b u d ~ r y Value tbohlcm ( I'i'liVtJ). I'hc ni~n~nii\ing Tru. valr~c y( I'l is calculal~~l In :m outcr loop ~lcwlitm using the

(iulrlcn S L T ~ I ~ I or ljr~mt algurithm I:~)~npuratrvu 11o1dinc;lr prt~granmii~ig (h P) discrctc-timc rtsults arc r h ) pr~zmtrd

Keywords: optimal ammol, non-standard opiimal cnntn~l, piixiwisc constant ilitcgrand, ~wonomics, u)m- parativc ~bonlit~car progm~btning rwults.

Cdculus of Variatio~s (CoV) pnlvidcs thc ~lathcmalieal lkcory to solve extrcmixing iuneriotlal problems f i r ~ l i i c h a given iiinctional has tl smtionary value eitlier nlinirnunl or maxin~um [131.

Optimal control i s an ertensirln nf C:oV and it is a ma~hematical r)ptirni~atian mcihod Tur deriving optimal control politics. Optinla1 control c l ~ n n c l s thc paths of thc control variables t o optimiirc thc ~ 1 s t functitmal whilst satisfying (in this paper) ordinary diKe~cntial equations. I:cnnornics is

a sc~urcc of intcnsting applications o f Ihc lhcory of C'oV :md clplimal control 1'71. A

fns

classic:d manlpIt:b tllat m k t the use of optimal control am the drug bust hlrdtegy, topimil pndi~ctiun, optimal conmd in diwrclc mc~hanics, pllicy arangcmcnl and thc myalty payment pnlhlcm 15.7-91.

*Comspiw~ding aulhor

khtlcll ctr M n ~ k m a d c s and Stidslim, Ilnivmily of Shctlicld. Shcllicid Pi3 7K11. lJnillrl K i n ~ d t m ~ .

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64

A NON-STANDARD OPTJMAL CONTROL PROBLEM ARISING IN AN ECONOMICS APPLICATION

In a recently studied wonomics prohlum a firm struggling will1 inttnsivcly low demand imf their product w i l l purposely incmsc t hc marginal cost by rcdreing t hc production o r the product and incrmscthc sclling m c c

I

I 11. I'hcmlimrc the wquircmcnt 10 pay a flat-rate n~yalty on wlc has the c f f w ofincm~sing the marginal cost and rhcrcby Qccrcasing the olitpul whilc simultaneously incm5ing thc pricc. 'I'hc cr~c?ct of permitling a nonlinear n)yalty I c d s ti) u n o n - s ~ a n d d I'oV prublcm not previously considcrcd.

In

this papcr wc will dcmcmstratc somc orsolving Lllc p r o h l ~ m without discussing thc prccisc c~onomics dcuils.

Lct us start by considering a simplc problcm. Wc wish to dclcmiinc thc control runc.tion r r ( t )

whcru L ( 10, 7'

1

that maximiws the integral fiunctiowal

sllhjcct sdisrying ~ h c ~ ~ : i l c oniinary difknlnlial cqudion

$

N O ) with ~ ' ( 0 ) known and thc endpoint state valuc y ( T ) at timc r = T unknown. Tllc Lagran2ian intcgratid dcpcnds on the unknt~wn linal ralucy(7') and rhis is nit a skmJard optimal ctmln~l prohlcm bwausc l c lirnaion dopcnds on ?(TI. A standard optimal control often solves a problem with thc final state ,FIT) rrw and [hen thc boundary mndition p(T') is ~ c r o

1

121. 'l+c i Iamiltonian f i ~ r opiirnal contrid thcory arises wllcn using Potitryagin's hiIinimtjn1 Principle (PMP) [13]. It is uscd to find the optimal a)nln>l Tor taking a dynamicdl system Iixrrn one s ~ t c LO xnothcr a ~ h j c c t to satisrying thc undcrl ying ordinary dift'crential equations [12].

Wc

introduce a continuuus approximation of thc picucwiw conslant intcgrdnd runction hy using a hypcshc>lic Langcnt appnuch. 'I'he prcxanl piper will indicatc how such problctns can be sol\rcd. In thc acxt section

wc

wiil discuss thc theoretical q p n w o h in o d c r t o solve the problem. IVc will fi,llow tho work oTMalinowska mind lilm 161, in proving thc boundary condition for CoV [2]. Then wc will dctnonstratc a numerical cxamplc

and

wo will IISL' (: I I Lo solve il. We also cornparu 1hc rc$ulk wilh a nonlinear pn)gramming(NI') discrctc-time appmqch and then prcscnt somc conclusions.

2 NUN-STANDARD OPTiMAL CONTROL

In classical optimal control problcnls. the function (1) docs not depend specifically on tlic frcc value -v( 1'). I lowover, in our caw, lhc I agrmgidm inirxrdnd runction 1 d~qwnds on ~ ( 7 ) . We prcsent a rcccnt thcorem that

indicates

that y ( T ) is cqual to a certain intcgnrl at timc T. Ma-

linowska and li~ms 161 harc proven rhis hounda~ condititm for C:oV on limc scales 121. I'hc tbllowing t h t w m c m bc applicd to our problem.

'fhcnrcm: 16.71

If j c .) is thc solution of thc following problcm

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ALAEI ZINOBER and SUUADI SUTAI IAN1

65

4hcn

d -

F [ f .

,E(/),

j(&),

: ~ ( l j ) = f;:(f. j(/), j [ f 1. j ( 1 . ) )

dl: .P

fur all 1 E 10. 1.1. M o ~ ) \ w

1;rom an ovlimal wnln)l penpcxtlvconc

1

12. 13

1

ha% thccc~talo p ( ? ' ) f < { t . ji~). ; ( I ] . y(?'l)- Hcncc

p ( T ) =

-

I'

fi

( r .

~ ( t ) .

&r).

j t l-l)'/r

(6) where p(f is the I lamiltt~nian mul~iplim or ciwtale runcticm. Ihctmm 1 shows lhal lhc nCLiC'ssary optimal condition docs ~iot havc p l T ) = O as is thc case for thc standard classical optimal coritrol pnlblcm.

Suppose that wc wish Lo considcr a pia..cwisc consrant functiorl f' in (2). For cxamplc, considcr

w hcm : .vi 7) and

and

a for J.

.:

0.5:

?+' =

h for 1 > 0.5,-

with r r and

t~

rcal nuniberr. We have a discontinuous intwgand thal cannot hc difrercntiatcd ax mluircd in ihc final bour~day condition (71. WC introduce a conti~luous approxin~wlion of thc piwcwisc ujnstanl int~grdnd funclion by using a hyperbolic tangent appwdch in c d e r to have a continuous smooth li~nc~ion. Illl~cf appwximaiions haw been lricd but thcy rquircd hundrcds of terms, fir insmcc in a Fourier strius approximation. 'Ihu tanh(kj) q r c ~ x i m a ~ i o n used hen: is vcry accutatc. A numerical cxsmplo will hc prcscntud in thc next swlion.

3 N I I % l R K l C A I ~ EXAM PI,E

'Jhc

cxamplc is a sin~plificd vcrsion ofthc propclscd monomics problcms bul has thc samc main t a t u m . I'he sutt: qualion (Olfl! system) i s dwribed by

We wish lo rnaxirnisc

. / / ( - ) I

=

I'

( { t , y t ~ ] . ~ ( b ) . :)dl

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i s a contintlous function

and

T

= 10. W{y) is a piecewise constaiit function where

I

I for .I. ( 0 . 5 ~ W (J) =

1.2 fur 3 =. 0.52

1% :ippmximatc W ( y ) using a hypcrbulic tangcni fi~nction

ant! w c sclc~1 A- s u i ~ b l y large. Ici 450 say. llilr larger values of

4-

wc obtain a smrwthcr approximation of tlic filnctiori F t ' ( j ) (scc Fig. 1). Thc known initial statc is y(0) = 0 and final s m c valuc : y i 7 j is lin. 'l'hcrclirrc ~ h c I lamilionian is dcscrihcd by H [ I . J., i l l p ) ₁p.lt

and

'I'hc wsmc saLislim

[image:4.595.50.462.229.708.2]

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-- -- MAN ZINOBER and SUUADl SUrAllANl

67

Ilquarion (6) holds a)

(

- ( ~ ) ? * { t ) ) ) d f W i y ) ! i f ) sin

4

RESULTS

I here are several necessary conditions that need LU be wtisfid, such the stare equiltiun (9) and thc costatc quation ( 16) along with lhc stationariiy condition (17). Also thc initial condition of y(i)) is given and we select ti guessed initial wlue of' p ( O ) . We a l w n d to make bure the boundary condition of ( 18) wH1 bc saiisficd :it the finit1 time. I-, 'This is a two point boundary value prublen~. We need to m b u r e that tlie ite:raliun ~ x l u e of r osed in (1 7) will be q u a 1 to L'I /') a1 the final time 7'. 'This is satisfied in our algorilhm only whcn tllc p(7') boundary condition co~iverges. I'hen we will have the optinltll sulution. 'llie r n i r ~ i ~ ~ ~ i s i n g h e value XI 1') is calculated in an imtcr loop iteralion using ihc Goldcn Sccrion (or Brcnt) maximising linc search algorithm We used the I \ . ~ w T I shouting met1:tllud with 1 w p u e d values 1:1 = ptO) and v2 =

pr

i'). We usc thc programming languagc C i I in urdcr lc) wlvc Lhc shooting nlcthvd and oihcr algorithms described in the NurnMical Keceipe librdry 181. We itmite the system with the m t e quation ~ ( t ). c o s ~ a c equation p(c), the inlcgral of q! t and Ihc cost function J l t ) .

An optimal solution ivzis u b ~ a i n d u i t l ~ high accuracy, Tlie msulls are

I:igurcs 2 4 shows ~ h c optimal cuwcs o r ihc skdlc? variahlc y(t ), adjoint varjablc / , ( I ) and the integml ciaIl) and the control variable t i ( ! 1.

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-G.B

0 2 4 6 S 1C

5nie

k3"lpre 3 - Castale v time slming algorithm.

3 C5

0 2 4 a 8 1U

[image:6.599.131.343.71.407.2]

tine

Figure 4 Control v tinie shooting algorithm.

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I:igurcu 5 7 show \:hc plot of optimal rcuulb for h o ~ h l c shooting and mnlincar p ~ ~ g r a m m i n g

{SP)

appmaclles fbr the state variable y (t 1, cosiate arrd c o n t r l variable ri ( I 1. The wsults are

ewcniidly the u m c . 'I'ke NI' rcuuku arc gtwd but the I lamillr~nian uhcn)ting appvach 1s a much more accurate approach.

In this paper we Iravve sliown ho>w to solrc a non-standard optimal ccmlrol pn>hl~m- Wc hare p ~ ~ c n l c d thc ncccssary conditions and Ihc wn~pulational proccdurcs in order to obtain optimal so)lulbn~. 'lhc ckplirna1 solcllitrn ol'a lust pwhlern has h~wn prcswtcd. A shtxliing algi~rithrn to- gcthcr with a n~wimising approach was used to obtain a highly accurdc solution, and con1p:urd wilh a discrc~c-time nonlinear pn~grdmming solution. Our ~cxhniquw can bc applied 10 the 'ac- h ~ a l mthcr morc wmplicalcd ccononlics problcnl whcrc thc Lagrangian inrcgrand is pic~czrise ctbnstani in many s ~ g a and depends upcm ! ( I ' ) which is a prk~ri unknown.

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[image:8.595.144.347.78.238.2]

70

A hW3KSTANDARD OPTIMAL CONTROL PROBLEM ARISING IN AN EGOhlOMlCSAPPLlCATlON

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KAIVA'V'III K . 20nX. Opliimal prr~duclian suhjiwl tc) pi~y:cwi.w mndnoous n~yally payment obligations, in~enlal repun.