Global Optimization for Solving Linear Non-Quadratic Optimal Control Problems

Transcription

1 Joural of Appled Matheatcs ad Physcs ISSN Ole: ISSN Prt: Global Optzato for Solvg Lear No-Quadratc Optal Cotrol Probles Jghao Zhu Departet of Appled Matheatcs ogj Uversty Shagha Cha How to cte ths paper: Zhu JH (06) Global Optzato for Solvg Lear No-Quadratc Optal Cotrol Probles Joural of Appled Matheatcs ad Physcs Receved: Septeber 5 06 Accepted: October 0 06 Publshed: October 3 06 Abstract hs paper presets a global optzato approach to solvg lear o-quadratc optal cotrol probles he a work s to costruct a dfferetal flow for fdg a global zer of the Haltoa fucto over a Eucld space Wth the Potryag prcple the optal cotrol s characterzed by a fucto of the adjot varable ad s obtaed by solvg a Haltoa dfferetal boudary value proble For coputg a optal cotrol a algorth for uercal practce s gve wth the descrpto of a eaple Copyrght 06 by author ad Scetfc Research Publshg Ic hs work s lcesed uder the Creatve Coos Attrbuto Iteratoal Lcese (CC BY 40) Ope Access Keywords Lear No-Quadratc Optal Cotrol Potryag Prcple Global Optzato Haltoa Dfferetal Boudary Value Proble Pral Proble I ths paper the otato represets a or for the specfed space cocered he pral goal of ths paper s to preset a soluto to the followg optal cotrol proble (pral proble ( ) short) where ( ) + F P u d t () 0 [ ] st 0 0 = A + Bu = a t R u R () R F s twce cotuously dfferetable o F 0 R P u s twce cotuously dfferetable o R P( u) > 0 u R I the cotrol syste AB are gve atrces R ad R respectvely ad α stads for a gve vector R We assue that P( u) l f > 0 (3) u u DOI: 0436/jap October 3 06

2 If P( u ) s a postve defte quadratc for wth respect to u ad F s a pos- tve se-defte quadratc for wth respect to the the proble ( ) s a classcal lear-quadratc optal cotrol proble [] he rest of the paper s orgazed as follows I Secto we focus o Potryag prcple to yeld a faly of global optzatos o the adjot varable I Secto 3 we deal wth the global optzato for the Haltoa fucto I Secto 4 we show that there ests a optal cotrol to the pral ( ) ad preset a atheatcal prograg I Secto 5 ad 6 we dscuss how to copute the global zer by a dfferetal flow ad preset a algorth for the uercal practce wth the descrpto of a eaple Potryag Prcple Assocated wth the optal cotrol proble ( ) let s troduce the Haltoa fucto wth the state ad adjot systes ( ) H u = A + Bu + F + P u () ( ) = H u = A + Bu 0 = a () = H u =A F = 0 (3) We kow fro Potryag prcple [] that f u ˆ () s a optal cotrol to the proble ( ) the t s a etreal cotrol Assocated wth the state varable ˆ () ad the adjot varable ˆ ( ) we have ( ˆ ) ( 0 ) ( ˆ ˆ ) ( ˆ) ˆ = H ˆ uˆ = Aˆ + Buˆ ˆ = a (4) ˆ = H u ˆ =A ˆ F ˆ = 0 (5) ad ( ˆ ˆ ˆ ) = ˆ ˆ H t u t t H t u t ( t) ( A ( t) Bu) F ( ( t) ) P ( u) a e t [ ] ˆ = ˆ + + ˆ + 0 (6) Sce (6) the global optzato s processed o the varable u over R for a gve t t s equvalet to deal wth the optzato (for obtag a global zer): ˆ P u + ( t) Bu (7) herefore we tur to cosder the followg optzato wth respect to a gve paraeter vector R P u + Bu (8) I ths paper for a gve adjot varable we solve the optzato (8) to create a u = h he Haltoa boudary proble () (3) we replace the fucto varable u wth the fucto h( ) ad solve the followg equato 860

3 3 Global Optzato ˆ = H u = A + Bh 0 = a (9) = H u =A F = 0 (0) I ths secto for a gve paraeter vector optzato proble [3] [4] P u to create a fucto ( u h C R C ) u = R we deal wth the followg global + Bu It follows fro (3) that there est postve ubers β ad r such that (3) P u β u u > r (3) It follows fro (3) that there est postve ubers β ad r such that whe > r P u β u (33) { } Wthout loss of geeralzato we assue that β r P Lea 3 For gve R let equvalet to the the followg global proble proof: Let < B α = the the global proble s β : P u + Bu u α { } R be gve Sce α > P 0 + ad α > r he whe u > α we have r P 0 + > β > 0 t s clear that ( ) β ( β ) P u + B u u B u = u B u > u O the other had for u α ( ) P u + B u P u + Bu u α + > whe u > α we have But sce P( 0) P( 0) P( 0) P( u) (34) P u + B u u > α > P 0 + > P 0 = P 0 + B 0 P u + Bu Sce we have show above that for all u P( u) + ( B ) u P u ( u) + Bu α R u α otg that { } P ( u) + Bu = P ( u) + Bu u R we have α u α he lea has bee proved Cosequetly by Lea 3 we coclude the followg lea Lea 3 Let u be a zer of P u ( u) + Bu α he u P u + Bu over P u + B = ad zer of R Moreover 0 s a - 86

4 u + B β Reark 3 Sce P( u) P ( u) + Bu over u α zer of ato P u > 0 + Bu over u R u s the uque zer of he t follows by Lea 3 that u s also the uque R herefor u s uquely detered by the equ- ( P u Bu) P( u) B 0 + = + = By eleetary calculus [5] the above equato defes a plct fucto of the varable deoted by ( ) : paraeter 4 Haltoa Boudary Value Proble u = h = u whch s cotuously depedet of the I ths secto we solve the followg Haltoa boudary value proble: = H u = A + Bh 0 = a (4) = H u =A F = 0 (4) Equato (4) ca be rewrtte by the tegral for ( t) A ( ν t ) Substtutg t to Equato (4) we have ( ) = e F ν d ν (43) t ( A ν t ( ( ν) ) ν) t 0 t = A t + Bh e F d 0 = (44) I the followg we show that Equato (44) has a soluto the together wth (43) we obta a soluto to Haltoa boudary value proble (4) (4) ad Sce F( ) s twce cotuously dfferetable o K = a F( ) Let a B A M = A ( a + ) + + B e K β ([ ] ) X = C 0 R R we ay defe Cosder the ball cetered at a X (regardg a as a fucto costatly equal to the vector a): G: { X ( t) a t [ ] } Ω= : 0 For a real uber such that 0 < < defe a operator M Ω X whch acts o each eleet Ω to produce a age G satsfyg (otg that the tegral (44) eeds the forato of ( ν ) o the whole terval 86

5 [ 0 ] ) for [ 0 ] t whle for t [ ] ( ( ) 0 ) t A ν s ν ν s G t : = a + A s + Bh e F d d s ( ( ) ) A ν s ν ν s G t : a + A s + Bh e F d d s 0 By a eleetary estato we have for t [ 0 ] (45) A ( s) t + B e K G t a A s + B ds M ( t 0) M 0 β (46) whle for t [ ] A ( s) + B e K G t a A s + B ds M 0 β whch ples that G Ω It s also clear that G s a cotuous ad copact appg he by Schauder fed-pot theore there s a eleet ˆ Ω such that Gˆ = ˆ It follows that ˆ () s a soluto to (44) for t [ 0 ] For t [ 0 ] ( t) ( ˆ ) ˆ A ( ν t ) = e F ν d ν t let (47) By a tradtoal approach the classcal theory of ordary dfferetal equato we see that the soluto ( ˆ ( t) ( t) ) t [ 0 ] ca be eteded to [ ] (43) we see that ˆ ( ) ˆ ( ) 0 he by (44) s a soluto to Haltoa boudary value proble (4) (4) We coclude the followg result heore 4 here ests a soluto par ˆ ( ) ˆ ( ) value proble (4) (4) Let h( ) : = ad ˆ ( ) ˆ ( ) u to Haltoa boudary be a soluto of the Haltoa boudary value proble (4) (4) he by the defto of the Potryag etreal cotrol we uˆ t = h ˆ t t 0 s a etreal cotrol to the pral proble coclude that ( ) [ ] Reark 4 Moreover otg that F( ) 0( R ) P( u) 0( u R ) ad > by () we see that the Haltoa fucto s cove o the state ad cotrol varables respectvely Meawhle otg that u h( ) = does ot deped o the state varable by tradtoal optal cotrol theory we kow that the û t = h ˆ t s also a optal cotrol to the optal cotrol etreal cotrol proble ( ) I other words the practce for solvg ( ) we oly eed to copute a soluto of the followg dfferetal boudary value proble: = A + Bh ( ) (48) =A F( ) (49) 863

6 [ ] 0 = a = 0 t 0 (40) We preset a uercal ethod to deal wth the dfferetal boudary value Equato (49) Equato (40) as follows Defe a esh by dvdg the te terval [ 0 ] evely 0 < t < < t < t = 0 N t t = : 0 N N = = + Cosder solvg for 0 N wth t = (due to the boudary codto For the requreet o the adjot varable ( t N ) 0 N the teded approato of of the dfferetal boudary value Equato (49) Equato (40)) we cosder the followg dfferece equato: + = A + Bz ; + =A F ; z = h( ); 0 = a; N = 0; = 0 N Solvg the dfferece equato above we ca get the valyue 0 Accordg to classcal uercal aalyss theory the soluto of above dfferece equato wll coverge to the soluto of dfferetal boudary value proble (48) - (40) Apparetly we eed to copute h( ) uercally It wll be gve et secto 5 Coputg h() by a Dfferetal Flow I ths secto we study how to copute h( ) = R we solve the followg global optzato proble P u u + Bu For a gve paraeter vector (5) to create a fucto u = h( ) I the followg we wll detere the value of h( ) by a dfferetal flow Sce the Hesse atr fucto of P( u ) s postve defte by the classcal theory of ordary dfferetal equato for gve R the followg Cauchy tal value proble [3] [6] creates a uque flow ξ ( ρ) ρ 0: dξ + P( ξ) + ρi ξ = 0 ξ( 0 ) = u ρ [ 0 ) dρ (5) such that ( ξ ( ρ) ) B ρξ ( ρ) P + + = 0 (53) otg that P( u ) + B = 0 sce u s the zer of P ( u) (Lea 3) o epla the uqueess of ξ + Bu over ρ we refer to the fact that R 864

7 P( u) + ρi > 0 u R for ρ [ 0 ) hus cobg (53) ξ soluto of the equato P( u) + B + ρu = 0 I what follows we choose a real uber ρ > such that o P u B ρ u By Brouwer Fed-Pot theore ([7]) there s a pot u { u } P( u ) B Moreover we have u ρ Cρ ρ s the uque such that = u (54) (55) where the postve costat C s oly depedet of the paraeters I the followg there are several tes of appearg the character C whch ay deote dfferet postve costats oly depedet of the paraeters It follows fro (54) that P u + B + ρ u = (56) By (53) ad the uqueess of the flow ξ ( ρ) ρ 0 we see that u 0 ξ ρ = (57) ad that the flow ξ ( ρ) ρ 0 ca also be got by the followg Cauchy tal value proble Certaly dξ dρ ( ξ) ρ ξ ξ( ρ ) ρ [ ) + + = = P I 0 u 0 ξ 0 = u Although t s hard to get u ad pute uercally aother vector stead of u by the followg result (58) u eactly we ca co- heore 5 Let the flow ξ ( ρ ) be got by the followg backward dfferetal equato he dξ dρ + P ξ + ρi ξ = u ( P I) ( P B ) [ ) 0 (59) ξ ρ = = 0 + ρ 0 + ρ 0 (50) u u Cρ (5) ( ) P ξ + B Cρ ξ u Cρ (5) 0 0 where the postve costat C s oly depedet of the paraeters ad selected to be suffcetly large satsfyg (54) (55) Proof: Whe of the org by (54) we have ρ > s suffcetly large ρ s u s ear the org I a eghborhood 865

8 ad Notg P P 0 + P 0 u + O u = P u =ρ u B ( P + ρ I) u = P B + O( u ) > 0 cosequetly we have ( ρ ) ( ) u = P 0 + I P 0 + B + O u Let we have hus by (55) ( ρ ) ( ) u : = P 0 + I P 0 + B u u = O u u u Cρ (53) where the postve costat C s oly depedet of the paraeters By the way we deduce that u u u u C + + otg that I the followg we eed to keep d that By (53) (54) for suffcetly large ρ > P u + ρ u + B = (54) P u ρ u B + + ρ we have 0 ρ ( ρ ) = P u + u P u + u u u a P( u ) u u ρ + u + C ρ ρ + C Cu u Cρ + Cρ Cρ (55) otg that the equalty process the value of the costat C has bee chaged several tes but oly depedet of gve forato lke PB Sce P ξ ( ρ) + ρξ ρ s a costat alog the flow (50) we have Cosequetly for ρ = 0 we have ( ξ ( ρ) ) ρξ ( ρ) ρ P + = P u + u ( ξ ) ρ P 0 = P u + u ξ ρ otg that (59) hus by (55) ( ξ ) ρ P 0 + B < C (56) Further otg that P( u ) B 0 + = we have 866

9 otg that P( u ) B 0 ξ ( 0) ξ ( 0) ( ) ( ( 0) ) ( ) ( ξ ( 0 )) ρ u C P P u = C P + B P u B ξ = C P + B C + = ad also otg that the equalty process the value of the costat C takes dfferet postve values whch are oly depedet of gve forato lke PB he theore has bee proved Reark 5 Coparg wth P( u ) B 0 ca solve the Cauchy tal proble (59) (50) stead of (58) to get ( 0) approato of u I what follows we gve a algorth to copute h + = the coputato practce we a dscrete soluto to Haltoa boudary value proble (4) (4) Algorth 5 ) Gve ) Gve R ; > ρ > ; 0 u ξ as a = uercally fdg 3) Get the flow ξ ( ρ ) by solvg the followg Cauchy tal proble dξ + P( ξ) + ρi ξ = 0 ξ( ρ ) = ( P( 0) + ρ I) ( P( 0 ) + B ) ρ 0 ρ ; dρ 4) f 5) ( ξ 0 ) ( ) ξ ( 0) P + B h = u stop; Otherwse go to 5); ρ = 0ρ go to 3) Reark 5 For the step 3) of above algorth we preset a uercal ethod to deal wth the Cauchy tal proble as follows Defe a esh by dvdg the te terval [ 0 ] evely 0 < ρ0 < < ρm < ρm = ρ ρ ρ j ρ = + j : s j 0 M M = = Cosder solvg for ξ0 ξ ξm wth j We deal wth the followg dfferece equato ξ j ξ j = P( ξ j) + ρj I ξ j j = M s 6 A Descrpto of a Eaple ξ the teded approato of ( j ) ( ) ξm = P 0 + ρ I P 0 + B Let s cosder to solve the followg optal cotrol proble uercaly: 0 4 ( + + ) u u dt st = + u 0 = a 4 R Let ξ ρ where state ad cotrol varables all take values P u = u + u We have 3 4 P ( u) = 4u + u P ( u) = u + H( u ) = ( + u) + + u + u We have the followg Haltoa boudary value proble ad a global optzato proble: 867

10 = + u 0 = a; ( ) t [ ] = + = 0 0 { } [ ] 4 u t = arg u + u + t u ae t 0 We eed to solve the followg dfferetal boudary value equato: ( ) = + h 0 = a; ( ) = + = 0 whch yelds the correspodg dfferece equato: = + h = = N 0 = a N = 0 Notg that P ( u) + ρ = u + + ρ ρ 0 ad P 0 = 0 by Algorth 5 ad Reark 5 gve postve tegers N M (properly large) ad postve real ubers (properly sall) If N ρ > ξ we cosder solvg the followg dfferece equato: + = + z; + ; = ( ) ( ) j ξ j ( ) ( ) = ξ j j s + + ξ j s ( ) ξ = + ρ ; M z = ξ ( ) ; 0 0 = a; N = 0; = 0 N j = M the the dscrete soluto of a optal cotrol uˆ z = 0 N Otherwse let above dfferece equato aga Refereces ρ = 0ρ ad N = N M = M ad do the [] Sotag ED (998) Matheatcal Cotrol heory: Deterstc Fte Desoal Systes d Edto Sprger New York [] Potryag LS (964) he Matheatcal heory of Optal Processes Pergao Press Oford UK [3] Zhu JH Wu D ad Gao D (0) Applyg the Caocal Dual heory Optal Cotrol Probles Joural of Global Optzato

11 [4] Zhu JH ad Zhag X (008) O Global Optzatos wth Polyoals Optzato Letters [5] Boothby WM (007) A Itroducto to Dfferetal Mafolds ad Reaa Geoetry Elsever Pte Ltd Sgapore [6] Zhu JH ad Lu GH (04) Soluto to Optal Cotrol by Caocal Dfferetal Equato Iteratoal Joural of Cotrol [7] Brow RF (988) Fed Pot heory ad Its Applcatos Aerca Matheatcal Socety New York Subt or recoed et auscrpt to SCIRP ad we wll provde best servce for you: Acceptg pre-subsso qures through Eal Facebook LkedI wtter etc A wde selecto of jourals (clusve of 9 subjects ore tha 00 jourals) Provdg 4-hour hgh-qualty servce User-fredly ole subsso syste Far ad swft peer-revew syste Effcet typesettg ad proofreadg procedure Dsplay of the result of dowloads ad vsts as well as the uber of cted artcles Mau dsseato of your research work Subt your auscrpt at: Or cotact jap@scrporg 869