HE PUBLISHING HOUSE PROCEEDINGS OF HE ROMANIAN ACADEMY, Series A, OF HE ROMANIAN ACADEMY Volume, Number 4/200, pp 287 293 SUFFICIEN CONDIIONS FOR EXISENCE SOLUION OF LINEAR WO-POIN BOUNDARY PROBLEM IN MINIMIZAION OF QUADRAIC FUNCIONAL Mihai POPESCU Insiue o Mahemaical Saisics and Applied Mahemaics o he Romanian Academy PO Box -24, RO-0045, Buchares, Romania, E-mail: ima_popescu@yahoocom he quadraic uncional minimizaion wih dierenial resricions represened by he command linear sysems is considered Deerminaion o he opimal soluion implies he solving o a linear problem wih wo poins boundary values he proposed mehod consiss in he consrucion o a undamenal soluion S ( a n n symmeric marix From he exremum necessary condiions i is obained he Ricai marix dierenial equaion having he S ( as unknown undamenal soluion is obained he paper analyzes he exisence o he Ricai equaion soluion S ( and deermine he opimal soluion o he proposed opimum problem Key words: Quadraic uncional minimizaion, Suicien condiions, Dierenial resricions, Linear sysem, Opimal soluion, wo poins boundary values INRODUCION In he conrol heory a special imporance is accorded o he quadraic linear problem he ineres is jusiied by he grea number o is pracical applicaions A represenaive model is oered by he linear regulaor problem corresponding o he quadraic uncional minimizaion, wih dierenial resricions, deined by linear command sysems [2, 3, 4, 5] Uilizing he Bellman equaion, he Ricai dierenial equaion associaed o he proposed opimum problem is obained he opimal conrol eedback and he minimum value o he perormance index is expressed as a uncion o he Riccai [2, 4, 5] hereore, deermining he exisence condiions or he soluion o Riccai equaion becomes a necessiy Also, aking ino accoun he hypohesis o he nilpoen srucure o he bilinear sysems he opimal conrol or he quadraic uncionals class [7, 8, 9] is obained he conrol on he neighbouring exremal uilizes he ransiion marices and heir symplecic properies he admissible opimal neighbouring rajecory is obained by he inegraion o he variaional Hamilonian sysem wih boundary condiions obained by he cancellaion o he exremised uncional his approach is a linear problem wih wo poin boundary values he exisen resuls in he quadraic linear problem can be exended o dierenial resricions having he orm o command sysems wih a ree, perurbing erm he presen sudy is based on his approach 2 HE OPIMAL CONROL PROBLEM Le s consider he ollowing conrolled linear dierenial sysems, o order n : n x = A () x+ B () u+ F (), x(0) = x R, [0, ] () he elemens o he marices A and B and he componens o he vecor F ( are coninuous real uncions deined wihin [0, ] 0
288 Suicien condiions or exisence soluion o linear wo-poin boundary problem in minimizaion o quadraic uncional 2 n m E = R is he sae space and U = R represens he parameers space A is an n n marix and B, x, F and u are, o he n m, n, n, m dimension, respecively Le s assume ha he quadraic perormance index is expressed by he ollowing uncionals: = 2 + + Φ 2 0 J xqx () uru () d x( ( x, (2) where Q, Φ are n n symmerical nonnegaive marices and R is a m m posiive deined marix he proposed opimum problem is equivalen wih he deerminaion o he conrol vecor which minimizes he perormance index (2) under he resricions () hen, he Hamilonian H can be wrien: u U H( x, λ, u, = x Q( x+ u R( u+λ ( [ A(, x+ B( 2 2 u+ F( ], (3) where λ () is adjoin variable he opimal conrol u is obained rom: where Changing ( x, λ, u, = 0 H u (4) u R B u in (3), he opimal Hamilonian = λ (5) H can be wrien: H = x Qx +λ Ax λ BR B λ+λ F (6) 2 2 he deerminaion o he opimal soluion resuls rom he inegraion o he Hamilonian sysem: wih he boundary-condiions H x = λ H λ= x = x0 a) x 0, b) λ ( ) =Φ( ) x( ) Solving o he equaion (7) under he above condiions (8) deines he wo-poin linear boundary value problem, (7) (8) 3 SOLVING MEHOD FOR WO-POIN LINEAR BOUNDARY VALUE PROBLEMS he sysem (7) become λ= Qx A λ x = Ax BR B λ+ F (9) Using he symplecic properies o he ransiion marices, he case F = 0 has been discussed in [6] and [] We aim obained a undamenal soluion or he linear wo-poin boundary value problem represened by he inhomogeneous dierenial sysem (9) wih he boundary condiions (8)
3 Mihai Popescu 289 Le s consider an S ( square marix o order n and an h ( vecor o dimension n, which will be deerminaed so ha he soluion x ( and adjoin variable λ ( o he sysem (9) wih he inal soluion 8(b) saisies he relaion λ ( ) = S( x( + h( (0) he dierenial equaions or S ( and h ( are chosen so ha o have d d or any soluion o he equaion (9) From () i ollows [ Sx h ] () () + () λ () = 0, () S x + S x + h λ = 0 (2) Replacing he adjoin variable (0) in (9) and opimal conrol (5) i resul a) x = ( ABR B S) x BR B h+ F b) λ= Q+ A S xa h Considering (3) he dierenial sysem (2) becomes + + + + + + + = 0 (4) S Q SA A S SBR B S x h SBR B S A h SF Relaion (4) is saisied or any x i S ( and h ( are deermined such ha we ge a) b) (3) S + Q + SA + A S SBR B S = 0, (5) S( ) = Φ ( ) = Φ, (6) respecively h + SBR B S + A h+ SF = 0, (7) he boundary condiions (6) and (8) resul rom (0) and (8b) h ( ) = 0 (8) 4 ANALYZE O DEERMINE HE EXISENCE OF A SOLUION DIFFERENIAL EQUAION Uilizing he HGMoyer s resuls [], he suicien condiions or he exisence o a soluion o he Riccai marix dierenial equaion (5) are esablished heorem he suicien condiions or he exisence o symmeric marix S ( where [ 0, ] saisying equaion or which are S = Q+ SA+ A S C+ B S R C+ B S (9) S( ) = Φ (20)
290 Suicien condiions or exisence soluion o linear wo-poin boundary problem in minimizaion o quadraic uncional 4 By reining he equaion (9) using he noaions QC R C 0 [0, ], (2) R > 0 [0, ] (22) Φ 0 (23) Q Q C R C =, (24) A = A BR C (25) i is obained he equaion (5); hereore he problem o he exisence o a soluion or equaion (9) is reduced o inding o a soluion or he equaion (5) heorem 2 he suicien condiion or he exisence o S ( where 0, ] saisying he equaions [ S = Q+ SA+ A S SBR B S (26) S( ) = Φ (27) is he exisence o an n n symmeric marix P (, having ime coninuous diereniable uncions deined or [0, ] as elemens, such ha B P= 0, [0, ], (28) P + Q+ PA+ A P= M() 0, R > 0, [0, ], (29) Φ P ( ) = G 0 (30) Proo Le s consider P ( + S ( = P(, (3) where P ( and S ( are symmerical marices According o he hypohesis, a symmerical P ( exiss saisying (28), (29), and (30) We have (32) P= Q+ PA+ A PM Uilizing he hypohesis (28) we rewrie (32) as (33) P = Q + PA + A P M PBR B P Subsiuing he value o P rom (3) in (33) we ge P S = Q+ A P+ S + P+ S AM P+ S BR B P+ S = Q+ A P+ S + + P + S A M SB R B S SBR B P PBR B S PBR B P Furher subsiuing S = P P (35) and considering (28) he equaion (34) becomes (34)
5 Mihai Popescu 29 We chose P S = Q+ A P+ S + P+ S AM SBR B S + PBR B P = + + + P M A P PA PBR B P (36) (37) P G =, (38) ha can be wrien ( P ) M A ( P) ( P) A ( P) BR B ( P) = + +, (39) P = G (40) From (39) and (40), i appears ha he uncion ( P) saisies he Ricai equaion or which he condiions o heorem represened by M () 0 [0, ], (4) R () > 0 [0, ], (42) G 0 (43) are me hereore ( P) exiss or any [0, ] Replacing he expression (37) in (36) and he boundary consrain (38) in (30) we ge S = Q + SA + A S SBR B S, (44) respecively Φ P ) S ( ) = G = P( ) (45) ( or S ( ) = Φ (46) Because (44), (46) are idenical o (26), (27) and P ( and P ( exis or any [0, ], using (3) i ollows ha S ( = S( exiss or any [0, ] hus heorem 2 is proved he relaion beween he suicien condiions or he exisence o he soluion o he Ricai equaion ormulaed in he previous heorems is esablished by heorem 2 heorem 3 he condiions in heorem 2 are weaker han hose in heorem Proo I replacing Q and A wih ( Q C R C) and ( A BR C), respecively, he condiions or he exisence o he soluion or he equaions (26), (27) expressed by heorem 2 come o he consrucion o a symmerical marix P (, [0, ] so ha B P= 0 [0, ], (47)
292 Suicien condiions or exisence soluion o linear wo-poin boundary problem in minimizaion o quadraic uncional 6 P + Q C R C+ P A BR C + A BR C P= M() 0 [0, ], (48) Φ P ( ) = G 0 (49) I he condiions rom heorem are veriied, hen he condiions (47), (48), (49) are saisied by P = 0, and hus heorem 3 is proved where 5 HE OPIMAL SOLUION Inegraing he linear dierenial equaion (7) wih he boundary condiions (8) we obain τ h = exp K( τ)dτ exp K( s)d s S( τ) F( τ) d τ, 0 0 0 (50) K SBR B S A = + (5) Cauchy s problem soluion or he dierenial equaion (3) wih he iniial condiion x 0 is where we have noed τ x = exp X( τ)dτ x0 + exp X( s)d s Y( τ)d τ, 0 0 0 (52) X A BR B S = (53) Y BR B h F = + (54) For he values o h ( and x ( resuled rom (50) and (52) he opimal conrol becomes u () = R B P() x() + h() (54) [ ] 6 CONCLUSIONS he presen sudy proposes a mehod or he solving o he linear wo-poin boundary value problem his is equivalen o he inding o he opimal soluion or he saic quadraic uncionals wih dierenial resricions represened by he inhomogeneous linear conrol sysem I he adjoin variables are expressed as uncions o he sae variables, rom he necessary exremum condiions, he Ricai marix dierenial equaion associaed o he opimum problem is obained he suicien condiions or he exisence o he soluion o he Ricai equaion ha ensure a local weak minimum in he analyzed opimal non-singular conrol are obained REFERENCES MOYER, H G, Suicien condiions or a srong minimum in singular conrol problems, SIAM J Conrol,, 3, pp 620 636, 973 2 BELLMAN, R, Dynamic Programming, Princeon Universiy Press, 977 3 COURAIN, R F, PRICHARD, A J, Ininie dimensional linear sysems heory, Lecure Noes in Conrol and Inormaion Sciences, Springer Verlag, New York, 978 4 RODMAN, L, On exernal soluion o he algebraic Ricai equaion, Lecures in Appl Mahemaics, 8, pp 3 327, 980
7 Mihai Popescu 293 5 ZABCZYK, J, he linear regulaor problem and he Ricai equaion, Mahemaical Conrol heory,, ICP, pp 44 55, 2004 6 HULL, G D, Opimal conrol heory or applicaions, Springer Verlag, New York, 2003 7 POPESCU, M, On minimum quadraic uncional conrol o aine nonlinear conrol, Nonlinear Analysis, 56, pp 65 73, 2004 8 POPESCU, M, Conrol o aine nonlinear sysems wih nilpoen srucure in singular problems, Journal o Opimizaion heory and Applicaions, 24, 5-7, pp 455 466, 2005 9 POPESCU, M, PELLEIER, F, Courbes opimales pour une disribuion aine, Bull Sci Mah 29, pp 70 725, 2005 0 POPESCU, M, Sweep mehod in analysis opimal conrol or rendez-vous problems, J Appl Mah & Compu 23, 2, pp 243 256, 2007 POPESCU, M, Variaional ransiory processes, nonlinear analysis in opimal conrol, Edi ehnică, Buchares, 2007 Received May 2, 200