Discretization-Optimization Methods for Optimal Control Problems

Transcription

1 Proceedigs of the 5th WSEAS It Cof o SIMULAION MODELING AND OPIMIZAION Corfu Greece August (pp399-46) Discretizatio-Optiizatio Methods for Optia Cotro Probes ION CHRYSSOVERGHI Departet of Matheatics Schoo of Appied Matheatics ad Physics Natioa echica Uiversity of Athes Zografou Capus 578 Athes GREECE Abstract: - We cosider a optia cotro probe described by oiear ordiary differetia equatios with cotro ad state costraits Sice this probe ay have o cassica soutios it is aso foruated i reaxed for he cassica cotro probe is the discretized by usig the ipicit idpoit schee for state approxiatio whie the cotros are approxiated by piecewise costat cassica oes We first study the behavior i the iit of properties of discrete optiaity ad of discrete adissibiity ad extreaity We the appy a peaized gradiet proectio ethod to each discrete cassica probe ad aso a correspodig progressivey refiig discretizatio-optiizatio ethod to the cotiuous cassica probe thus reducig coputig tie ad eory We show that accuuatio poits of sequeces geerated by these ethods are adissibe ad extrea for the correspodig discrete or cotiuous cassica or reaxed probe For ocovex probes whose soutios are o-cassica we show that we ca appy the above ethods to the probe foruated i the Gareidze for Fiay uerica exapes are give Key-Words: - Optia cotro discretizatio idpoit schee piecewise costat cotros peaized gradiet proectio ethod reaxed cotros Itroductio I this paper we propose discretizatio-optiizatio ethods geeratig cassica cotros istead of reaxed cotros (see [] [] [5]) for sovig optia cotro probes ad study their behavior i the iit i the fraewors of cassica ad reaxatio theories We cosider a optia cotro probe described by oiear ordiary differetia equatios with cotro ad ed-poit state costraits ad ed-poit cost Sice this probe ay have o cassica soutios it is aso foruated i reaxed for he cassica cotro probe is the discretized by usig ipicit idpoit schees for state ad adoit approxiatio the idpoit itegratio rue for approxiatio of the itegras ivoved i the derivatives of fuctioas whie the cotros are approxiated by piecewise costat cassica oes We first give various ecessary coditios for optiaity for the cotiuous cassica ad reaxed probes ad for the discrete probe Next we show that strog accuuatio poits i L of sequeces of optia (resp adissibe ad extrea) discrete cotros are optia (resp adissibe ad weay extrea) for the cotiuous cassica probe ad that reaxed accuuatio poits of sequeces of optia (resp adissibe ad extrea) discrete cotros are optia (resp adissibe ad weay extrea) for the cotiuous reaxed probe We the appy a peaized gradiet proectio ethod to each discrete cassica probe ad aso a correspodig ixed discretizatiooptiizatio ethod to the cotiuous cassica probe that progressivey refies the discretizatio durig the iteratios thus reducig coputig tie ad eory especiay for arge systes We prove that accuuatio poits of sequeces geerated by the fixed discretizatio ethod are adissibe ad extrea for each discrete probe ad that strog cassica (resp reaxed) accuuatio poits of sequeces of discrete cotros geerated by the progressivey refiig ethod are adissibe ad weay extrea for the cotiuous cassica (resp reaxed) probe For ocovex probes whose soutios are o-cassica we show that we ca appy the above ethods to the probe foruated i the Gareidze for Usig a stadard procedure the coputed Gareidze cotros ca the be approxiated by cassica oes Fiay uerica exapes are give Probes ivovig poitwise state costraits have bee studied i [4] [5] For various discretizatio ad optiizatio ethods i optia cotro see [-7] [9] [] ad refereces there

2 Proceedigs of the 5th WSEAS It Cof o SIMULAION MODELING AND OPIMIZAION Corfu Greece August (pp399-46) he cotiuous probes Cosider the foowig optia cotro probe he state equatio is give by the differetia syste y'( t) = f( t y( t) w( t)) for aa t I : = [ ] d y() = y yt ( ) the costraits o the cotro w are wt () U for aa t I ' where U is a copact subset of d the costraits o the state y: = yw are G( w): = g( y( )) = G( w): = g( y( )) where the vector fuctios g g tae vaues i respectivey ad the cost fuctioa to be iiized is G( w) : = g( y( )) We defie the set of cassica cotros by d ' W : = { w: I U w easurabe} L ( I ) ad the set of reaxed cotros (for the reevat theory see [] ad [8]) by R: = { r: I M( U) r weay easurabe} Lw ( I M( U)) L( I C( U))* where M ( U ) (resp M( U )) is the set of Rado (resp probabiity) easures o U he set W (resp R ) is edowed with the reative strog (resp wea star) topoogy ad R is covex etrizabe ad copact If each cassica cotro w( ) is idetified with its associated Dirac reaxed cotro r(): = δ w() the W ay be cosidered as a subset of R ad W is thus dese i R For a give fuctio φ BIU ( ; ) where B is the set of Caratheodory fuctios i the sese of Warga [] ad r R we use the sipified otatio φ( trt ()): = φ( turt ) ()( du) U We ca ow defie the reaxed probe he state equatio is y'( t) = f( t y( t) r( t)) for aa t I y() = y with y: = yr the cotro costrait is r R ad the state costraits ad cost are defied as i the cassica probe but with w repaced by r accordig to the above otatio We deote by the Eucidea or i We suppose i the seque that the fuctio f d is defied o I U easurabe for yu fixed cotiuous for t fixed ad satisfies f ( t y u) ψ() t + β y d for every ( t y u) I U with ψ L ( I) β f ( t y u) f( t y u) L y y for every ( t y y u) I d U heore For every reaxed (or cassica as W R) cotro r R the state equatio has a uique absoutey cotiuous soutio y: = yr Moreover there exists a costat b such that y b for every cotro r R r Let B deote the cosed ba i d with ceter ad radius b defied i heore We suppose ow i additio that the fuctios g = are cotiuous o B heore he appigs G : W(resp R) = are cotiuous o W (resp R ) If the reaxed probe is feasibe the it has a soutio Note that i the cassica probe we have y'( t) f( t y( t) U) (veocity set) whie i the reaxed oe y'( t) co[ f( t y( t) U)] he cassica probe ay have o cassica soutio ad because W R we have i geera c : = i G ( r) if G ( w) : = c R costraits o r costraits o w where the equaity hods i particuar if there are o state costraits sice W is dese i R Usuay uerica ethods sighty vioate the state costraits; so approxiatig a optia reaxed cotro by a reaxed or a cassica oe hece the reaxed optia cost c R is ot a drawbac i practice (see [] p 48) Note aso that approxiatig sequeces of cassica cotros ay coverge to reaxed oes I order to state the various ecessary coditios for optiaity we suppose i additio that the fuctios f f y f u are defied o I B' U' where B ' (resp U ' ) is a ope set cotaiig B (resp U ) easurabe o I for fixed ( yu ) B U cotiuous o B U for fixed t I ad such that f ( t y u) ξ () t f ( tyu ) η() t y for every ( t y u) I B U with ξη L( I) ad that the fuctios g g y = are defied o B ' ad cotiuous o B u W

3 Proceedigs of the 5th WSEAS It Cof o SIMULAION MODELING AND OPIMIZAION Corfu Greece August (pp399-46) heore 3 (i) If U is covex the for ww ' W the directioa derivative of the appig G = defied o W is give by G ( w+ α( w' w)) G ( w) DG ( ' ): i w w w = + α α = z ( t ) f ( t y ( t u ) w ( t ))[ w '( t ) w ( t )] dt where y: = yw ad the adoit state z : = z a row w vector fuctio ( = ) or a atrix fuctio ( = ) is defied by the iear adoit equatio z '( t) = z ( t) f ( t y( t) w( t)) for aa t I y z( ) = gy( y( )) with y: = yw where the cotros are cosidered as cassica oes (ii) For rr ' R the directioa derivative of the appig G = defied o R is give by G ( r+ α( r' r)) G ( r) DG ( ' ): i r r r = + α α = z () t f ( t y () t r '() t r ()) t dt where y: = yr ad the reaxed adoit z : = z is r defied by the iear reaxed adoit equatio z '( t) = z( t) fy( t y( t) r( t)) for aa t I z( ) = gy( y( )) with y: = yr (iii) he appigs ( ww ') a DG ( ww ' w) (resp( rr') a DG ( rr' r) ) = are cotiuous o W W (resp R R ) heore 4 (i) If U is covex ad the cotro w W is optia for the cassica probe the w is weay extrea cassica ie there exist utipiers λ λ λ with λ λ () = = λ DG ( w w' w) = u λ = such that = λ z() t f ( t yt () wt ())[ w'() t wt ()] dt for every w' W () λ G( w) = (trasversaity coditio) he coditio () is equivaet to the poitwise wea cassica iiu pricipe λ z () t f ( t y() t w()) t w() t = u = i λz( t) fu( t y( t) w( t)) u for aa t I u U = (ii) If the cotro r R is optia for either the reaxed or the cassica probe the r is strogy extrea reaxed ie there exist utipiers as i (i) such that (3) λ DG ( r r' r) = = = λ z () t f( t y() t r'() t r()) t dt for every r' R (4) λ G() r = he coditio (3) is equivaet to the poitwise strog reaxed iiu pricipe λ z () t f( t y() t r()) t = = i λz( t) f( t y( t) u) for aa t I u U = If U is covex the this pricipe ipies the poitwise wea reaxed iiu pricipe λ z() t f( tyt () rt ()) rt () = u = i λz( t) fu( t y( t) r( t)) φ( t r( t)) = φ = for aa t I where the iiu is tae over the set B( IUU ; ) of Caratheodory fuctios φ : I U U which i tur ipies the goba wea reaxed coditio λ z () t f ( t y() t r())[ t φ( t r()) t r()] t dt u for every φ B( IUU ; ) A cotro r satisfyig this coditio ad (4) is caed weay extrea reaxed 3 he discrete probes I the seque we suppose that the fuctios f fy f u ad g gy g u = are cotiuous i a their arguets Let ( N ) be a icreasig sequece of positive itegers such that N We set N : = N N ': = N ' h : = / N ti : = ih i= N Ii : = [ ti ti ) i= N IN : = [ tn tn] We defie the set of discrete cassica cotros W : = { w W w ( t): = wi U i Ii i= N} o

4 Proceedigs of the 5th WSEAS It Cof o SIMULAION MODELING AND OPIMIZAION Corfu Greece August (pp399-46) For a give discrete cotro w W the discrete state y : = y = ( y yn ) is the soutio of the w ipicit idpoit schee yi = yi + h f( ti yi wi ) i= N y = y : with yi : = ( yi + yi )/ ti : = ( ti + ti )/ heore 5 If h < / L the the discrete state y is uiquey defied ad there exists a costat b ' such that W yi b' i= N for every ad w Fro ow o we suppose that h < / L he discrete state equatio ca the be soved uericay for each i= N by the stadard predictor-corrector ethod he discrete cotro costrait is w W Defie the discrete appigs G ( w ): = g( yn) = he discrete state costraits are either of the two foowig oes Case (a) G ( w ) ε Case (b) G ( w ) = ε ad G( w ) ε where the adissibiity perturbatios ε are appropriate positive ubers or vectors covergig to zero to be defied ater he discrete cost to be iiized is ( G ) w heore 6 he appigs w y w G ( w ) are cotiuous o W If ay of the two above discrete probes is feasibe the it has a soutio heore 7 If U is covex the for w w' W the directioa derivative of the appig G = defied o W is give by DG ( w w' w ) N = h zi fu ( ti yi wi )( w' i wi ) i= where the adoit state z is give by the iear ipicit schee z = i z + i h zi fy ( ti yi wi ) i= N zn = gy ( yn ) with y : = y w he appigs ( w w' ) a DG ( ' w w w ) for = are cotiuous o W W We ow state the discrete ecessary coditios for optiaity heore 8 If U is covex ad w is optia for the discrete probe with state costraits Case (b) the w is discrete extrea cassica ie there exist utipiers λ λ λ with λ λ (5) = = λ = such that λ DG ( w w' w ) for every w' W (6) λ [ G ( w ) ε ] = he coditio (5) is equivaet to the discrete poitwise wea cassica iiu pricipe = λ z f ( t y w ) w i u i i i i = i λ zi fu( ti yi wi ) u i= N u U = 4 Behavior i the iit I this sectio we study the behavior i the iit of properties of discrete optiaity ad of discrete adissibiity ad extreaity Defie the piecewise costat fuctios y (): t = ( yi + yi )/ t I i i= N ad the piecewise iear fuctios yˆ (): t = yi + ( t ti ) f( ti yi wi ) t I i i= N heore 9 (Cosistecy of states ad fuctioas) (i) Let ( w W ) be a sequece such that w w i L strogy he w W yˆ y y y uifory where y: = yw ad G ( w ) G( w) = (ii) Let ( w W R) be a sequece such that w r i R he yˆ y y y uifory where y: = yr ad G ( w ) G ( r) = heore (Cosistecy of adoits ad derivatives of fuctioas) (i) If ( w W ) is a sequece such that w w i L strogy (resp w r i R ) the zˆ z z z uifory where z : = z (resp z : = z ) w r

5 Proceedigs of the 5th WSEAS It Cof o SIMULAION MODELING AND OPIMIZAION Corfu Greece August (pp399-46) (ii) If ( w W ) ( w' W' ) are sequeces such that w w w' w' i L strogy the DG ( w w' w ) DG ( w w' w) heore (Cotro approxiatio) (i) For every w W there exists a sequece ( w W ) that coverges to w i L strogy (ii) For every r R there exists a sequece ( w W R) that coverges to r i R We suppose i the seque that each cosidered cotiuous cassica or reaxed probe is feasibe he foowig theore addresses the behavior i the iit of optia discrete cotros heore If there are state costraits we suppose that the sequeces ( ε ) i the discrete state costraits Case (a) coverge to zero ad satisfy G ( w ) ε ( ) G w% ε ε % for every where ( w% W ) is soe sequece covergig i L (resp i R ) to a optia cotro if it exists (resp which exists) w% W (resp r% R) of the cassica (resp reaxed) probe For each et w be optia for the discrete probe Case (a) he every accuuatio poit of ( w ) i L (resp R ) is optia for the cotiuous cassica (resp reaxed) probe Next we cosider the discrete probes with state costraits Case (b) We first costruct sequeces of perturbatios ( ε ) covergig to zero ad such that the discrete probe is feasibe for every as foows For each et w' W be a soutio of the foowig auxiiary iiizatio probe without state costraits = + w W c i{ [ G ( w )] [ax( G ( w ))] } he set ε : = G( w' ) = ε : = ax( G( w' )) = Usig our assuptios it ca be show that c hece ε ad ε he foowig theore addresses the behavior i the iit of adissibe ad extrea discrete cotros heore 3 For each et w be adissibe ad extrea for the discrete probe Case (b) with the perturbatios ( ε ) costructed as above he every accuuatio poit of ( w ) i L (if it exists) is adissibe ad weay extrea cassica for the cotiuous cassica probe ad every accuuatio poit i R (which aways exists) is adissibe ad weay extrea reaxed for the cotiuous reaxed probe 5 Discretizatio-optiizatio ethods We suppose here that U is covex Let ( M ) ( M ) be oegative icreasig sequeces such that M as ad defie the peaized discrete fuctioas G ( w ): = G( w ) + { M G ( w ) + M [ax( G ( w ))] Let γ bc () ad et ( β ) ( ζ ) with ζ be positive decreasig sequeces that coverge to zero he agorith described beow cotais various optios I the case of the progressivey refiig versio we suppose that either N( + ) = N( ) or N( + ) = µ N( ) for soe iteger µ I this case we have W W + ad thus a cotro w W ay be cosidered aso as beogig to W + ad therefore the coputatio of states adoits ad derivatives of fuctioas for this cotro but with the possiby fier discretizatio + aes sese Agorith Step Set : = : = choose a vaue of ad a iitia cotro w W Step Fid v W such that L i [ ( ' DG w ) v w v' W ( γ /) v' w ] L d = DG w v w e β set w : = w v : v e : = DG ( w v w ) + ( γ /) v w = + ad set : ( ) Step 3 If = e : = e d : = d : = + [ : = + ] ad go to Step Step 4 (Ario step search) Fid the owest iteger s vaue s say s such that α = c ζ (] ad α satisfies the iequaity

6 Proceedigs of the 5th WSEAS It Cof o SIMULAION MODELING AND OPIMIZAION Corfu Greece August (pp399-46) G ( w + α ( v w )) G ( w ) α be s ad the set α: = c ζ Step 5 Set w : ( ) + = w + α v w : = + ad go to Step I this Agorith we cosider two versios: Versio A : = + is sipped i Step 3: is a costat iteger chose i Step ie we choose a fixed discretizatio ad repace the discrete fuctioas G by the perturbed oes Versio B : = + is ot sipped i Step 3: i this case it ca be show that ie we have a progressivey refiig discrete ethod ad we ca tae = i Step hece = i the Agorith he progressivey refiig versio has the advatage of reducig coputig tie ad eory ad aso of avoidig the coputatio of iiu feasibiity perturbatios (see Sectio 4) It is ustified by the fact that fier discretizatios becoe progressivey ore efficiet as the iterate gets coser to a extrea cotro whie reativey coarser oes i the eary iteratios have ot uch ifuece o the fia resuts If γ > (peaized gradiet proectio ethod) oe ca see by copetig the square that Step reduces to fidig for each i the proectio of a vector oto U If γ = (peaized coditioa gradiet ethod) Step reduces to the iiizatio of a iear fuctio o U for each i A (cotiuous strogy or weay cassica or reaxed or a discrete) extrea cotro is caed abora if there exist utipiers as i the correspodig optiaity coditios with λ = (or λ = ) A cotro is adissibe ad abora extrea i exceptioa degeerate situatios Defie the sequeces of utipiers λ : = M G ( w ) λ : = M ax ( G( w )) where ax deotes a vector of ax vaues ad w is defied i Step 3 of the Agorith heore 4 (i) I Versio B et ( w ) be a subsequece (if it exists) of the sequece geerated by the Agorith i Step 3 that coverges to soe w W i L strogy as (hece ) If the sequeces ( λ ) are bouded the w is adissibe ad weay extrea cassica for the cotiuous cassica probe (ii) I Versio B et ( w ) be a subsequece of the sequece geerated by the Agorith i Step 3 that coverges to soe r i R as (hece ) If the sequeces ( λ ) are bouded the r is adissibe ad weay extrea reaxed for the cotiuous reaxed probe (iii) I Versio A et ( w W ) fixed be a subsequece geerated by the Agorith i Step 3 that coverges to soe w W as If the sequeces ( λ ) are bouded the w is adissibe ad extrea for the fixed discrete probe (iv) I ay of the above covergece cases (i) (ii) (iii) suppose that the (discrete or cotiuous) iit probe has o adissibe abora extrea cotros If the iit cotro is adissibe the the sequeces of utipiers are bouded ad this cotro is extrea as above I practice by choosig oderatey growig sequeces ( M ) ad a sequece ( β ) reativey fast covergig to zero the resutig sequeces of utipiers ( λ ) are ofte ept bouded Whe directy appied to ocovex optia cotro probes whose soutios are o-cassica reaxed cotros the cassica ethods ofte yied very poor covergece For this reaso we describe ow aother approach that uses the Gareidze foruatio of the probe For sipicity we cosider the case without state costraits We suppose that U is covex Cosider the reaxed probe with state equatio y'( t) = f( t y( t) r( t)) for aa t I y() = y cotro costrait r R ad cost fuctioa Gr (): = gy ( ( )) For each t I fixed the vector f ( t y( t) r( t )) d beogs to co[ f( t y( t) U)] hece d + f ( t y() t r()) t = v () t f( t y() t w ()) t with v ( t) [] d + v () t = ad by Fiippov s seectio theore (see []) we ca suppose that v w are easurabe herefore the cotro r yieds the sae state y as the Gareidze cotro d + r : = v ( t) δ Coversey G w () t every such a cotro r G is ceary a reaxed cotro r that yieds the sae state herefore the above reaxed cotro probe is equivaet to the foowig exteded cassica oe with state equatio

7 Proceedigs of the 5th WSEAS It Cof o SIMULAION MODELING AND OPIMIZAION Corfu Greece August (pp399-46) d + y'( t) = v ( t) f( t y( t) w ( t)) for aa t I y() = y cassica cotros v = ( ) w = ( ) cotro costraits d + v w v () t = v ( t) [] w ( t) U = d + ad cost Gvw ( ) = g( y ( )) Cosequety we ca appy the ethods described above to this probe he ai disadvatage of this approach is that the diesio of the cotro space is rapidy icreased It ca therefore be successfuy appied for reativey sa diesios dd ' he Gareidze reaxed cotros coputed thus ca the be approxiated by cassica cotros usig a stadard procedure (see []) If U is ot covex oe ca use ethods geeratig reaxed cotros to sove such strogy ocovex probes (see [] [] [5]) 6 Nuerica exapes Let I : = [] a) Defie the referece state yt (): = e t ad cotro t [ 5) wt (): = s ( s) t [5] with s= ( t 5) / 75 Cosider the foowig probe with state equatios y' = y + siy siy + w w y' = 5[( y y) + ( w w) ] y () = y () = cotro costrait set U = [ ] ad cost G( w) = y() Ceary the optia cotro ad state are w ad y he discrete gradiet proectio ethod without peaties was appied to this exape with γ = 5 N = 8 ad zero iitia cotro After 9 iteratios i we obtaied the cotro show i Fig ad the foowig resuts - G ( w ) = 898 e = η = 334 ζ = 864 where η is the discrete ax state error at the edpoits of the itervas I i ad ζ the discrete ax cotro error at the idpoits of these itervas b) With the costrait set U = [ 73] the cotro costraits beig ow stricty active for the ethod ad for the probe we obtaied after 9 iteratios i the cotro show i Fig ad the resuts - G ( w ) = e = 33 c) With the first state equatio repaced by y' = y+ w the costrait set U = [ 58] the additioa state costrait G ( w) = y ( ) 5= ad appyig here the discrete peaized gradiet proectio ethod we obtaied after 96 iteratios i the cotro show i Fig3 the state show i Fig4 ad the resuts G ( w ) = G ( w ) = 57 e = 6367 d) Cosider the foowig ocovex probe with state equatios y' = y+ w y' = 5( y y) w y () = y () = with yt () = e t cotro costrait set U = [ ] ad cost Gw ( ) = y () he uique optia reaxed cotro is ceary r*( t) = ( δ + δ)/ with optia state y* = y ad optia cost Gr ( *) = Note that the optia reaxed cost ca be approxiated as cosey as desired with a cassica cotro but caot be attaied for such a cotro Sice here the veocity set f ( t y U ) is a cotiuous arc i hece a coected set i the Gareidze foruatio ivoves oy three cotros v uw y' = y+ vu+ ( v) w y' = 5( y y) vu ( v) w y () = y () = with v [] ad uw [ ] Appyig without peaties the discrete coditioa gradiet ethod (ie with γ = ) which yieded a better covergece for this specia probe with N = 8 ad iitia cotros v : = 5+ 3t u : = 7 3t w : = 7+ 3t we obtaied after iteratios i the cotro v 5 with ax error 6 5 the cotros u = w = exacty the optia state with ax error e = 355 ad the cost G ( v u w ) = Fiay the progressivey refiig versio of the ethods where aso appied to the above probes with successive step sizes /3 /64 /8 i three eary equa periods ad yieded resuts of practicay siiar accuracy but required here about haf the coputig tie

8 Proceedigs of the 5th WSEAS It Cof o SIMULAION MODELING AND OPIMIZAION Corfu Greece August (pp399-46) W - W 3-7 W Y 5 t Fig Fig Fig3 5 t 5 t Refereces: [] I Chryssoverghi A Bacopouos B Koiis ad J Coetsos Mixed Fra-Wofe Peaty Method with Appicatios to Nocovex Optia Cotro Probes Joura of Optiizatio heory ad Appicatios Vo94 No 997 pp [] I Chryssoverghi J Coetsos ad B Koiis Discrete Reaxed Method for Seiiear Paraboic Optia Cotro Probes Cotro ad Cyberetics Vo8 No 999 pp [3] I Chryssoverghi Approxiate Gradiet Proectio Method with Geera Ruge-Kutta Schees ad Piecewise Poyoia Cotros for Optia Cotro Probes to appear i Cotro ad Cyberetics 5 [4] I Chryssoverghi J Coetsos ad B Koiis Discretizatio Methods for Optia Cotro Probes with State Costraits to appear i Joura of Coputatioa ad Appied Matheatics 5 [5] I Chryssoverghi J Coetsos ad B Koiis Discretizatio Methods for Nocovex Optia Cotro Probes with State Costraits to appear i Nuerica Fuctioa Aaysis ad Optiizatio 5 [6] I Chryssoverghi Discretizatio Methods for Seiiear Paraboic Optia Cotro Probes to appear i Iteratioa Joura of Nuerica Aaysis ad Modeig 5 [7] E Poa Optiizatio: Agoriths ad Cosistet Approxiatios Spriger 997 [8] Roubíče Reaxatio i Optiizatio heory ad Variatioa Cacuus Water de Gruyter 997 [9] V M Veiov O the ie-discretizatio of Cotro Systes SIAM Joura o Cotro ad Optiizatio Vo35 No5 997 pp [] A Schwartz ad E Poa Cosistet Approxiatios for Optia Cotro Probes Based o Ruge-Kutta Itegratio SIAM Joura o Cotro ad Optiizatio Vo 34 No pp [] J Warga Optia Cotro of Differetia ad Fuctioa Equatios Acadeic Press 97 5 t Fig4