Optimal control, contact dynamics and Herglotz variational problem

Jou nal o Nonlinea Science (2023) 33:9 h ps://doi.o g/10.1007/s00332-022-09861-2 Op imal Con ol, Con ac Dynamics and He glo z Va ia ional P oblem Manuel de León1,2 ·Manuel Lainz1·Miguel C. Muñoz-Lecanda3 Recei ed: 25 June 2020 / Accep ed: 7 Oc obe 2022 / Published online: 11 No embe 2022 © The Au ho (s) 2022 Abs ac In his pape , we combine wo main opics in mechanics and op imal con ol heo y: con ac Hamil onian sys ems and Pon yagin maximum p inciple. As an impo an esul , among o he s, we de elop a con ac Pon yagin maximum p inciple ha pe mi s o deal wi h op imal con ol p oblems wi h dissipa ion. We also conside he He glo z op imal con ol p oblem, which is simul aneously a gene aliza ion o he He glo z a ia ional p inciple and an op imal con ol p oblem. An applica ion o he s udy o a he modynamic sys em is p o ided. Keywo ds Con ac Hamiol onian sys ems ·Op imal con ol ·He glo z p inciple · P esymplec ic sys ems ·Pon yagin maximum p inciple Ma hema ics Subjec Classi ica ion 37J55 ·49S05 ·70Q05 ·34H05 ·49K15 ·49K20 · 93C15 Con en s 1 In oduc ion ............................................. 2 2 P econ ac Hamil onian Sys ems ................................... 4 Communica ed by Alain Go iely. BManuel de León [email p o ec ed] BManuel Lainz [email p o ec ed] Miguel C. Muñoz-Lecanda [email p o ec ed] 1Ins i u o de Ciencias Ma emá icas (CSIC-UAM-UC3M-UCM), Mad id, Spain 2Real Academia de Ciencias Exac as, Físicas y Na u ales, Mad id, Spain 3Depa men o Ma hema ics, Uni e si a Poli ècnica de Ca alunya, Ba celona, Spain 123 9Page 2 o 46 Jou nal o Nonlinea Science (2023) 33 :9 2.1 Con ac Mani olds and Hamil onian Sys ems .......................... 5 2.2 P econ ac Mani olds and Hamil onian Sys ems ........................ 6 2.2.1 P econ ac Hamil onian Sys ems and he Cons ain Algo i hm ............. 7 2.2.2 Mo phisms o P econ ac Hamil onian Sys ems ..................... 8 2.3 The Lag angian Fo malism ................................... 9 2.3.1 The He glo z Va ia ional P inciple ............................ 10 3 A Quick Su ey on Op imal Con ol and Pon yagin Maximum P inciple ............. 11 3.1 The Op imal Con ol P oblem .................................. 11 3.1.1 S a emen o he P oblem ................................. 11 3.1.2 The Ex ended Op imal Con ol P oblem ......................... 12 3.2 The Pon yagin Maximum P inciple .............................. 13 3.3 The P esymplec ic App oach o PMP .............................. 14 4 Dynamics o Vec o Fields as Con ac Dynamics .......................... 16 4.1 The Gene al Case ........................................ 16 4.2 The Case M=R×Mo..................................... 16 4.2.1 The Symplec ic Case ................................... 17 4.2.2 The Rela ion wi h Con ac Dynamics .......................... 18 5 The Con ac Dynamics App oach o Pon yagin Maximum P inciple ............... 19 5.1 S a emen o he P oblem .................................... 20 5.2 No mal Solu ions: po=λo=/0................................ 21 5.3 Abno mal Solu ions: po=λo=0............................... 21 6 He glo z Va ia ional P oblem as an Op imal Con ol P oblem ................... 23 6.1 S a emen o he P oblem .................................... 23 6.2 Op imal Con ol App oach o he He glo z Va ia ional P oblem ................ 24 6.3 Applica ion o he P esymplec ic Fo m o he Pon yagin Maximum P inciple ........ 26 6.3.1 The Ex ended P oblem .................................. 26 6.3.2 Solu ion o he Ex ended P oblem wi h he P esymplec ic Fo m o he Pon yagin Max- imum P inciple ...................................... 27 6.4 The Final Resul s ........................................ 29 7 He glo z Op imal Con ol P oblem ................................. 30 7.1 S a emen o he P oblem .................................... 30 7.2 Solu ion o He glo z Op imal Con ol P oblem ......................... 31 7.3 Con ac Fo mula ion o he No mal Solu ions ......................... 33 7.4 Reduc ion o he P oblem .................................... 35 8 Applica ion: Op imal Con ol on The modynamic Sys ems ..................... 37 8.1 Homogeneous Hamil onian Sys ems and Con ac Sys ems ................... 37 8.2 Con ol o Con ac Sys ems ................................... 38 8.3 Applica ion o The modynamic Sys ems ............................ 39 8.4 Example: Gas–Pis on–Dampe Sys em ............................. 40 9 Conclusions and Fu u e Wo k .................................... 43 Re e ences ................................................ 44 1 In oduc ion This pape ies o combine wo impo an opics in mechanics and con ol heo y: Hamil onian con ac sys ems and Pon yagin maximum p inciple in op imal con ol. On he one hand, Hamil onian con ac sys ems a e ge ing a g ea popula i y in ecen imes because hey allow o desc ibe dissipa ion dynamics, and se e al o he ypes o physical sys ems in he modynamics, quan um mechanics, ci cui heo y, con ol heo y, e c. (see o ins ance B a e i 2019;Go o2016; Kholodenko 2013; Rami ez e al. 2017; de León and Sa dón 2017; Gase e al. 2020b; Simoes e al. 2020; Sussmann 1999). Recen ly, a gene aliza ion o con ac geome y has been de eloped o desc ibe ield heo ies wi h dissipa ion (Gase e al. 2020a,c). In ac , he Hamil onian 123 Jou nal o Nonlinea Science (2023) 33 :9 Page 3 o 46 9 o mula ion in he scena io o con ac s uc u es exhibi s e y di e en cha ac e is ics o i s coun e pa in symplec ic mani olds. Indeed, hese di e ences a e based on he ac ha in he con ac case hey a e Jacobi s uc u es, mo e gene al han hose o Poisson ela ed o he symplec ic ones. In a ia ional e ms, one can show ha con ac Hamil onian equa ions can be de i ed om he so-called He glo z p inciple, which includes as a pa icula case he classical Hamil on p inciple. On he o he hand, he Pon yagin maximum p inciple (PMP), see Pon yagin e al. (1962), Ba be o-Liñan and Muñoz-Lecanda (2009) and e e ences he ein, is he mos use ul ins umen o inding solu ions o an op imal con ol p oblem. In ac , he PMP is he pa adigm in he heo y o op imal con ol, and since i s o mula ion has ne e ceased esea ch on i s inc edible p ope ies, om e y di e en poin s o iew, al hough we will ocus he e on i s mo e geome ic aspec s. An immedia e issue a ising om possible applica ions is ha o s udying p oblems o op imal con ol om he poin o iew o Hamil onian con ac sys ems, and he e o e o sys ems wi h dissipa i e p ope ies among many o he s. And, hen, i seems e y na u al o ask whe he a Pon yagin maximum p inciple could be de eloped o deal wi h a con ac con ol p oblem. To ou knowledge he ela ionship be ween con ac Hamil onian sys ems and he Pon yagin maximum p inciple was i s no iced in Ohsawa (2015) and de eloped in Jó´zwikowski and Respondek (2016). T ying o look o bo h opics wi h a common iewpoin , we conside wea he he solu ion cu es o he Pon yagin maximum p inciple admi a o mula ion in e ms o Hamil onian con ac sys ems in an adequa e mani old. Con e sely, we examine i he He glo z a ia ional p oblems can be unde s ood as a pa icula class o op imal con ol p oblems. Wi h all his in mind, he pape is s uc u ed as ollows. Sec ions 2and 3a e dedica ed o e iew he elemen s o Hamil onian con ac sys ems and Pon yagin maximum p inciple, bo h necessa y o unde s and he objec o he manusc ip . So, Sec . 2is de o ed jus o ecall he main no ions and esul s abou con ac Hamil onian sys ems, including he so-called He glo z p inciple, a na u al ex ension o he well-known Hamil on p inciple. As we said abo e, his sec ion will acili a e a be e unde s anding o he es o he pape . Sec ion 3is dedica ed o he Pon yagin maximum p inciple in se e al o mula- ions. We in oduce he classical op imal con ol p oblem, he associa ed ex ended sys em, he classical Pon yagin maximum p inciple and i s ans o ma ion in o he symplec ic and p esymplec ic o mula ions. This las one is used in se e al sec ions o he a icle. In Sec . 4we discuss an in e es ing pa icula case o Hamil onian dynamics; indeed, gi en a ec o ield Xon a mani old M, one can de ine he comple e li o X o he co angen bundle T∗Mwhich is jus he Hamil onian ec o ield co esponding o he Hamil onian unc ion de e mined by X: jus i s e alua ion on he poin s o he co angen bundle. Hence he dynamics o a gene al ec o ield is desc ibed as he co esponding o a Hamil onian ec o ield in a symplec ic mani old. Bu his dynamics on T∗Mis iche han one could expec . In ac , i he mani old Mdecomposes as M=R×Mo, and he ec o ield has a pa icula symme y p ope y, one has a e y na u al se ing o iden i y wo di e en geome ic beha iou s acco ding o he alue o he momen un po 123 9Page 4 o 46 Jou nal o Nonlinea Science (2023) 33 :9 co esponding o he global coo dina e xo. Indeed, one is a (p e) symplec ic geome y, when po=0, and he second one, a con ac geome y, when po=/0. Sec ions 5,6and 7a e he bulk o he pape . Sec ion 5is in a b oade sense a di ec applica ion o Sec . 3. We conside an op imal con ol sys em gi en by (M,U,X,I,xa,xb)whe e M=R×Mo, ha is, we s udy he so-called ex ended sys em associa ed o an op imal con ol p oblem de ined by a ec o ield depending on con ols, X(x,u), and a cos unc ion F. Applying Theo em 5in Sec . 3, we know ha his p oblem is equi alen o sol ing he dynamics o he p esymplec ic sys em (T∗M×U,ω,H), whe e X=F∂ ∂xo+Xi∂ ∂xi,H=Fpo+Xipiis he linea Hamil- onian gi en by X, and ωis he p esymplec ic o m ob ained by li ing he canonical symplec ic o m, ωM∈2(T∗M), oT∗M×U. He e, U ep esen s ob iously he space o con ols. The co esponding p esymplec ic algo i hm p o ides he solu ions, and we can dis inguish wo cases: he egula one, when he con ols can be ob ained as unc ions o he es o a iables, o he singula one, ha p oduces highe o de condi ions. Again, he e olu ion o he momen um pois cons an , and his pe mi s, as abo e, o discuss he cases whe e po=0o po= 0. Wi h his in mind, we a e able o s a e he Con ac Pon yagin maximum p inciple (Theo em 4). Sec ion 6is jus de o ed o in e p e he He glo z p inciple as an Op imal Con ol P oblem, and de i e he He glo z equa ions o mo ion using he co esponding Pon- yagin p inciple. In Sec . 7we s a e he He glo z Op imal Con ol P oblem and ind he solu ion equa ions. In his si ua ion, he ex emal condi ion, gi en as an in eg al o he cos unc ion in he classical op imal con ol p oblems, is changed in o an ex emal condi ion on he solu ions o a di e en ial equa ion on a new a iable o be maximized. This p oblem is a gene aliza ion o he classical op imal con ol sys ems in he sense ha we ob ain he classical equa ions i he cos unc ion and he ex emal condi ion is like in he classical si ua ion. Finally, in Sec . 8we apply he abo e esul s o an example coming om The modynamics. Being awa e ha in p ac ical applica ions o op imal con ol i is necessa y o use mo e gene al classes o unc ions and mappings, as i is usual in his kind o heo e ical app oaches, all he mani olds and mappings a e conside ed as o C∞-class. The usual Eins ein con en ion o summa ion indices will be unde s ood unless indica ed. As gene al e e ences o no a ions and basic esul s on geome y, mechanics and con ol we use (Ab aham and Ma sden 1978; Bullo and Lewis 2005; Bloch 2015). 2 P econ ac Hamil onian Sys ems In his sec ion we e iew he necessa y heo y o con ac mani olds, con ac and p econ ac dynamical sys ems, in bo h Hamil onian and Lag angian o mula ions, and He glo z a ia ional p inciple and i s gene alized Eule –Lag ange equa ions. See A nold (1978), B a e i (2017), B a e i e al. (2017), de León and Lainz-Valcáza (2019),Gase e al.(2020a), Geiges (2008), Guen he e al. (1996), Lainz-Valcáza and de León (2019) and Liu e al. (2018) o de ails. 123 Jou nal o Nonlinea Science (2023) 33 :9 Page 5 o 46 9 2.1 Con ac Mani olds and Hamil onian Sys ems Acon ac mani old (M,η)is a (2n+1)-dimensional mani old equipped wi h a con ac o m η, ha is a 1- o m sa is ying η∧(dη)n= 0. Then, he e exis a unique ec o ield R, called he Reeb ec o ield, such ha iRdη=0,iRη=1.(1) Gi en (M,η), he e is a Da boux heo em o con ac mani olds: a ound each poin in Mone can ind local Da boux coo dina es (qi,pi,z)such ha η=dz−pidqi,R=∂ ∂z.(2) As an example, and a na u al model, we ha e he ex ended co angen bundle T ∗Q× Ro an n-dimensional mani old Q, which ca ies a na u al con ac o m ηQ=dz−θQ,(3) whe e θQis he pullback o he Liou ille 1- o m o T∗Q,θQ=pidqi, being (qi,pi,z) he na u al bundle coo dina es o T∗Q×R. I (M,η)is a con ac mani old, he map: ¯ :TM →T∗M, → ι dη+η( )η. is a ec o bundle isomo phism o e M. Gi en a Hamil onian unc ion H:M→R, we can de ine a dynamical sys em. The iple (M,η,H)is called a con ac Hamil onian sys em. The associa ed Hamil onian ec o ield XHis he solu ion o he ollowing equa ion ¯ (XH)=dH−(R(H)+H)η. (4) In Da boux coo dina es, XHhas he local exp ession XH=∂H ∂pi ∂ ∂qi−∂H ∂qi+pi ∂H ∂z∂ ∂pi +pi ∂H ∂pi −H∂ ∂z.(5) The e o e, an in eg al cu e (qi( ), pi( ), z( )) o XHsa is ies he di e en ial equa- ions dqi d =∂H ∂pi , dpi d =− ∂H ∂qi−pi ∂H ∂z, 123 9Page 6 o 46 Jou nal o Nonlinea Science (2023) 33 :9 dz d =pi ∂H ∂pi −H. 2.2 P econ ac Mani olds and Hamil onian Sys ems Le ηbe a 1- o m on an m-dimensional mani old M. We de ine he cha ac e is ic dis ibu ion o ηas C=ke η∩ke dη⊆TM,(7) which we suppose o be egula , ha is, o cons an ank. We say ha ηis a 1- o m o class c i he ank o he dis ibu ion Cis m−c. The e exis some cha ac e iza ions o his no ion o a 1- o m gi en in he ollowing (Godbillon 1969). P oposi ion 1 Le ηbe a one- o m on an m-dimensional mani old M. Then, he ol- lowing s a emen s a e equi alen : 1. The o m ηis o class 2 +1. 2. A e e y poin o M, η∧(dη) =/0,η∧(dη) +1=0.(8) 3. A ound any poin o M, he e exis local Da boux coo dina es x1,...x ,y 1,...y , z, u1,...us, whe e 2 +s+1=m, such ha η=dz−  i=1 yidxi.(9) In hese Da boux coo dina es, he cha ac e is ic dis ibu ion o ηis gi en by C=∂ ∂uaa=1,...,s.(10) A pai (M,η) o a mani old Mequipped wi h a o m ηas abo e will be called ap econ ac mani old (see Godbillon 1969). The o m ηwill be called a p econ ac o m. Rema k 1 The dis ibu ion Cis in olu i e and i gi es ise o a olia ion o M.I he quo ien π:M→M/Chas a mani old s uc u e, hen he e is a unique 1- o m ˜ηsuch ha π∗˜η=η. F om a di ec compu a ion, ˜ηis a con ac o m on M/C. This jus i ies he name o p econ ac o m. Gi en (M,η), he ollowing map :TM →T∗M → ι dη+η( )η, (11) 123 Jou nal o Nonlinea Science (2023) 33 :9 Page 7 o 46 9 is a mo phism o ec o bundles o e Mand i s ke nel is C. AReeb ec o ield o (M,η)is a ec o ield Ron Msuch ha ιRdη=0,η(R)=1,(12) o , equi alen ly (R)=η. We no e ha he e exis Reeb ec o ields in e e y p econ ac mani old. Indeed we can de ine local ec o ields R=∂ ∂zin Da boux coo dina es and can ex end i using pa i ions o uni y. Howe e , unlike on con ac mani olds, hey a e no unique. In ac , gi en a Reeb ec o ield Rand any sec ion Co C, we ha e ha R=R+Cis ano he Reeb ec o ield. 2.2.1 P econ ac Hamil onian Sys ems and he Cons ain Algo i hm Ap econ ac Hamil onian sys em is a p econ ac mani old (M,η) wi h a smoo h unc ion H:M→Rcalled he Hamil onian. We deno e i by (M,η,H). Fo a p econ ac Hamil onian sys em (M,η,H), gi en a submani old M⊂M, aHamil onian ec o ield along Mis a ec o ield X∈X(M), such ha X|M∈ X(M)and solu ion o he equa ion (X)=dH−(H+R(H))η, (13) a he poin s o M, and being Rany Reeb ec o ield. I can be seen ha , i his equa ion holds o one Reeb ec o ield, i will hold o all o hem. No ice ha , since is no an isomo phism, hen (13) migh no ha e solu ions a e e y poin o he mani old M. Fu he mo e, solu ions, i hey exis s, a eno necessa ily unique. Indeed, adding a sec ion Co C o a solu ion Xgi es ise o a new solu ion X=X+C. In o de o ob ain he maximal submani old along which Hamil onian ec o ields a e de ined, we can de elop a cons ain algo i hm.Todoso,le γH= dH−(H+R(H))η ∈1(M)and de ine induc i ely M0=M, and o any posi i e in ege i, Mi={p∈Mi|(γH)p∈(TpMi−1)},(14) whe e we assume ha all Mia e mani olds. The algo i hm will e en ually s op, ha is, we will ind a posi i e in ege isuch ha Mi=Mi−1. We call his submani old he inal cons ain submani old M .I M has posi i e dimension, he e will exis Hamil onian ec o ields along M .The pai (M ,X)will be called a Hamil onian ec o ield solu ion o he Hamil onian p econ ac sys em (M,η,H). A use ul cha ac e iza ion o such pai s is gi en by he ollowing P oposi ion 2 X is a Hamil onian ec o ield along M o (M,η,H)i and only i , a he poin s o M, η(X)=−H,(15a) 123 9Page 8 o 46 Jou nal o Nonlinea Science (2023) 33 :9 LXη=gη, (15b) whe e g :M→R. Mo eo e , i his holds, hen g =−R(H) o any Reeb ec o ield R. P oo Le Xbe a Hamil onian ec o ield along M. By he de ini ion o , equa ion (13), a he poin s o M, becomes ιXdη+η(X)η =dH−(H+R(H))η, (16) and, by con ac ion wi h R, we ob ain η(X)=−H.(17) Combining (16) and (17), we deduce ιXdη+dιXη=−R(H)η, (18) bu he le -hand side o his equa ion equals LXηby Ca an’s o mula, hence X ul ills (15) a he poin s o M. Now assume ha Xsa is ies (15) on he poin s o M. Once again, by con ac ion o (15b) wi h a Reeb ec o ield R,weha e g=ιRLX(η) =ιR(ιXdη+d(η(X))) =−ιR(dH)=−R(H). (19) Combining his wi h (15), we can easily e ie e (16).  2.2.2 Mo phisms o P econ ac Hamil onian Sys ems Le (M,η,H)and (¯ M,¯η, ¯ H)be p econ ac Hamil onian sys ems. A map F:M→¯ M is said o be a con o mal mo phism o p econ ac sys ems i F∗¯η= ηand F∗¯ H= H o some non- anishing unc ion :M→R.I =1, we say ha Fis a s ic mo phism o p econ ac sys ems. Theo em 1 Le F :M→¯ M be a con o mal mo phism o p econ ac sys ems. Assume ha X,¯ X a e F- ela ed ec o ields de ined along submani olds M⊆M and ¯ M= F(M)⊆¯ M, espec i ely. The e o e, i ¯ X is a Hamil onian ec o ield along ¯ M, hen X is also a Hamil onian ec o ield along M. P oo Since ¯ Xis a Hamil onian ec o ield, i s sa is ies (15) along ¯ M ¯η( ¯ X)=−¯ H,(20a) L¯ X¯η=¯g¯η. (20b) Pulling back by F, we ob ain η(X)=− H,(21a) 123 Jou nal o Nonlinea Science (2023) 33 :9 Page 9 o 46 9 LX( η) =(¯g◦F) η. (21b) F om his exp ession, we ob ain η(X)=−H,(22a) LX(η) =gη, (22b) whe e g=¯g◦F−(LX )/ . Hence Xis a Hamil onian ec o ield.  Obse e ha i Fis a di eomo phism, hen we ha e a bijec i e co espondence be ween pai s o Hamil onian ec o ields along submani olds. 2.3 The Lag angian Fo malism Unlike T∗Q×R, he mani old TQ×Rdoes no ha e a canonical con ac s uc u e. Howe e , gi en a Lag angian unc ion L :TQ ×R→Rone can cons uc he 1- o m ηL=dz−θL,(23) whe e θLis he associa ed Lag angian 1- o m, which in bundle coo dina es (qi, i,z) is w i en as θL=∂L ∂ idqi.(24) The Lag angian Lis said o be egula i i s Hessian ma ix wi h espec o he eloci ies, (Wij)=∂2L ∂ i∂ j,(25) is egula . One can see ha ηLis con ac o m when Lis egula . Fu he mo e, ηLis a p e- con ac o m when (Wij)has cons an ank (see de León and Lainz-Valcáza 2019, Sec ion). The ene gy o he Lag angian is EL=(L)−Lwhe e is he canonical Liou ille ec o ield on TQ,= i∂ ∂ i, ex ended in he usual way o TQ×Rwi h he same local exp ession. Hence, p o ided Lis such ha (Wij)has ull ( esp. cons an ) ank we ha e ha (TQ×R,η L,EL)is a con ac ( esp. p econ ac ) Hamil onian sys em. Le ξLbe a Hamil onian ec o ield o his con ac o p econ ac sys em. F om a di ec compu- a ion one can see ha e e y in eg al cu e (qi( ), i( ), z( )) o ξLis a solu ion o he He glo z equa ions: d d ∂L ∂ i−∂L ∂qi=∂L ∂ i ∂L ∂z,(26) 123 9Page 16 o 46 Jou nal o Nonlinea Science (2023) 33 :9 na u al p ojec ion M×U→M, conside ing he con ols as he elemen s o he ib es. The local equa ions a e he same ha we ha e ob ained in he i ial case o he con ols. 4 Dynamics o Vec o Fields as Con ac Dynamics I is well known ha he in eg al cu es o a ec o ield in a mani old Mcan be ob ained as p ojec ion o in eg al cu es o a Hamil onian ec o ield in he co angen bundle. We can ex end his dynamics o he con ac associa ed mani old TM×R,as in Eq.(3), wha gi es he addi ional equa ion ˙z=0, ha is in a i ial way. We wan o ob ain a non- i ial ex ension. In his sec ion we s udy how o ob ain hese in eg al cu es as solu ions o a con ac dynamical sys em in an adequa e con ac mani old, a leas in he case ha he o iginal ec o ield has some symme y p ope ies. He e we eco e a simila si ua ion we had in he Pon yagin maximum p inciple in i s symplec ic app oach. See Sec . 3. 4.1 The Gene al Case Le Mbe a mani old and X∈X(M)a ec o ield. Le ˆ X:T∗M→R he na u al unc ion de ined by ˆ X(α) =α(X)=<α, X>. In a canonical coo dina e sys em (xi,pi)in T∗M, we ha e ha ˆ X(x,p)=piXi. As i is well known, i ωM=−dθMis he symplec ic canonical 2- o m in T∗M, we can conside he Hamil onian symplec ic sys em (T∗M,ωM,ˆ X). Then he Hamil- onian ec o ield Yˆ X∈X(T∗M), de ined by i(Yˆ X)ωM=dˆ X, has local exp ession X=Xi∂ ∂xi,⇒Yˆ X=Xi∂ ∂xi−pj ∂Xj ∂xi ∂ ∂pi i (xi)and (xi,pi)a e coo dina es o Mand T∗M espec i ely. By his local exp ession we ha e ha Yˆ X=X∗, whe e X∗is he so-called canonical li ing o X∈X(M) o T∗M. The in eg al cu es o Yˆ Xp ojec ed o Ma e he in eg al cu es o Xas we can see by di ec obse a ion o he abo e local exp ession. Wi h his me hod, we ha e ans o med any ec o ield in a Hamil onian one bu doubling he dimension. Fo de ails abou hese cons uc ions we e e o de León and Rod igues (1989), Yano and Ishiha a (1973). Obse e ha he Hamil onian ˆ Xdepends linea ly on he momen a. 4.2 The Case M=R×Mo In his Sec ion we analyze he speci ic case whe e in he mani old M he e is a pa icula “di ec ion”, ha is M=R×M0. This si ua ion allows us o spli up he in eg al cu es o a ec o ield, wi h a symme y p ope y, in o wo di e en classes: one ollowing a symplec ic geome y and he o he class unde a con ac geome y. This s udy is 123 Jou nal o Nonlinea Science (2023) 33 :9 Page 17 o 46 9 connec ed wi h he ideas de eloped in Ohsawa (2015) whe e he con ac s uc u e is associa ed o he p ojec i e mani old associa ed o T∗M. 4.2.1 The Symplec ic Case Suppose now ha we ha e one di ec ion specially iden i ied in he angen bundle o he mani old, ha is M=R×Mo. When necessa y we deno e by (xo,xi)a coo dina e sys em in Mand (xo,xi,po,pi)i s na u al ex ension o T∗M. Le X∈X(M)and suppose ha ∂ ∂xo,X=0. In coo dina es his means ha , i X=Xo∂ ∂xo+Xi∂ ∂xi, hen he coo dina es Xoand Xio he ec o ield Xdo no depend on xo. In pa icula his implies ha Xis p ojec able o Mo. Rema k 4 Wha is he meaning o his si ua ion? Suppose we ha e wo ec o ields Xo,X∈X(M)wi h [Xo,X]=0. Then a ound any egula poin o Xowe can choose a local coo dina e sys em (U,xo,xi), wi h i=1,...,n,i dimM=1+n, and U⊂Man open se , wi h Xo|U=∂/∂xo. Hence we ha e he abo e si ua ion bu locally. In his case he local decomposi ion {xo}×{xi}is no unique. This is wha we called abo e “pa icula symme y p ope y” o he ec o ield X. We can obse e ha i is a common si ua ion a leas locally. This is a si ua ion we a e going o ackle when ying o ela e con ac s uc u es and op imal con ol. The a iable xowill co espond o he cos unc ion Fas we ha e seen in Sec . 3in ou e iew o he Pon yagin maximum p inciple. I we p oceed in his case as abo e in he gene al si ua ion, wi h i(X∗)ωM=dH, whe e he Hamil onian unc ion His de ined by H=ˆ X=poXo+piXi hen he co esponding Hamil onian ec o , using [∂/∂xo,X]=0, is gi en by X∗=Xo∂ ∂xo+Xi∂ ∂xi−0∂ ∂po −po ∂Xo ∂xi+pj ∂Xj ∂xi∂ ∂pi . The associa ed sys em o di e en ial equa ions is: ˙xo=Xo,˙po=0,˙xi=Xi,˙pi=−po ∂Xo ∂xi−pj ∂Xj ∂xi This is he desc ip ion o he Hamil onian sys em (T∗M,ωM,H)wi h H=ˆ X. 123 9Page 18 o 46 Jou nal o Nonlinea Science (2023) 33 :9 4.2.2 The Rela ion wi h Con ac Dynamics Obse e ha he ec o ield X∗is angen o he submani old de ined by po= cons an , hence we can educe he p oblem o hose hype su aces o T∗M.Weha e wo di e en si ua ions and, by compa ison wi h he si ua ion o he op imal con ol and he symplec ic Pon yagin maximum p inciple, we will call no mal and abno mal si ua ions. (a) The no mal si ua ion po=/0 Fo λo∈R,λo=/0, le N⊂T∗Mbe he submani old de ined by po=λoand le j:N→T∗Mbe he na u al inclusion. Ob iously he dimension o Nis odd, hence i canno be a symplec ic mani old. We deno e by (xo,xi,pi) he coo dina es induced in Nby he coo dina es we ha e in T∗M. Conside now he canonical 1- o m θM∈1(T∗M)and le η=−j∗θM, hen we ha e he ollowing esul Lemma 1 (N,η)is a con ac mani old. The Reeb ec o ield is R =−1 λo ∂ ∂xo. The p oo is di ec using i s local exp ession, η=−λodxo−pidxi. The minus sign comes om a con en ion in he de ini ion o he symplec ic o m in T∗Mand he 1- o m and 2- o m in a con ac mani old. Le HN=j∗Hbe he es ic ion o H o N. We ha e ha , locally, HN= λoXo+piXiand we ha e a Hamil onian con ac sys em gi en by (N,η,HN).Le XN∈X(N)be he co esponding con ac Hamil onian ec o ield, ha is: i(XN)η =−HN,i(XN)dη=dHN−(L(R)HN)η whose local exp ession is XN=Xo∂ ∂xo+Xi∂ ∂xi−λo ∂Xo ∂xi+pj ∂Xj ∂xi∂ ∂pi ,(34) wi h he usual no a ion con using he unc ions on T∗Mand hei es ic ions o N. Wi h his in mind, we ha e ha : Theo em 6 The ec o ield X∗∈X(T∗M)is angen o N and, on he poin s o N, i is equal o XN. Hence he no mal in eg al cu es o he ec o ield X∗a e solu ions o a Hamil- onian con ac dynamics on a co esponding con ac mani old. The con ac sys em is (N,η,HN). Commen : A li le calculus He e we gi e he co esponding calculus o ob ain he exp ession in (34). We ha e ha HN=λoXo+piXiand η=−λodxo−pidxi. Deno ing XNby XN=ao∂ ∂xo+ai∂ ∂xi+bi ∂ ∂pi 123 Jou nal o Nonlinea Science (2023) 33 :9 Page 19 o 46 9 he i s con ac dynamical equa ion is: i(XN)η =−HN⇒−λoao−aipi=−λoXo−piXi and he second one i(XN)dη=dHN−(L(R)HN)η ⇒ −bidxi+aidpi=λo ∂Xo ∂xidxi+Xidpi+pj ∂Xj ∂xidxi. Hence ai=Xi,bi=−λo ∂Xo ∂xi−pj ∂Xj ∂xi,ao=Xo as we wan ed. (b) The abno mal si ua ion po=0 This case co esponds o λo=0 and he submani old No⊂T∗Mde ined by po=0. Le jo:No→T∗Mbe he na u al inclusion and ηo=j∗ oθM. Obse e ha ηo=−pidxiis no a con ac o m. In ac , as mo=dim Mo,we ha e ha ηo∧(dηo)mo−1=/0, bu ηo∧(dηo)mo=0. We can conside he 2- o m ωo=dηo, he Hamil onian Ho=j∗ oHand he p esymplec ic mani old (No,ω o,Ho). Obse e ha ke ωo={ ∂ ∂xo}. The Hamil onian p esymplec ic equa ion i(Xo)ωo=dHo gi es he solu ion Xo=Xi∂ ∂xi−pj ∂Xj ∂xi ∂ ∂pi +A∂ ∂xo, whe e Ais a bi a y and co esponds o ke ωo. In ac we ha e ha ˙xo=A. I does no exis any cons ain because he ec o ield Xois de ined on he whole mani old No. This is because he only cons ain is gi en by LTHo=0 wi h T∈ ke ωoand his is ul illed globally on No. Commen : Obse e ha T∗M=λ∈RNλ, hence wi h his decomposi ion we ob ain all he solu ions o he ini ial Hamil onian p oblem on T∗Mgi en by he Hamil onian H. 5 The Con ac Dynamics App oach o Pon yagin Maximum P inciple Following he ideas o he p e ious sec ions, we s udy a con ac app oach o he Pon yagin maximum p inciple, in pa icula o he so-called no mal solu ions o he op imal con ol p oblem. In pa icula we will ob ain he no mal solu ions o an op imal 123 9Page 20 o 46 Jou nal o Nonlinea Science (2023) 33 :9 con ol p oblem as p ojec ion o he in eg al cu es o a Hamil onian con ac sys em in adequa e mani olds. The abno mal solu ion can be ob ained wi h ano he di e en app oach gi en a he end o his sec ion. 5.1 S a emen o he P oblem Le (M,U,X,I,xa,xb)be an op imal con ol p oblem. We know by Theo em 5 ha o sol e his p oblem we need o s udy he associa ed Hamil onian p esymplec ic sys em (T∗M×U,ω,H), ha is o ob ain an in eg al cu e o he ec o ield XH solu ion o he equa ion i(XH)ω =dH, whe e ω=π∗ 1ωo=dxo∧dpo+dxi∧dpi,H=ˆ X=poF+piXi and π1:T∗M×U→T∗M. Recall ha ke ω={∂/∂ua}. The solu ion o he equa ion i(XH)ω =dHis gi en by: XH=F∂ ∂xo+Xi∂ ∂xi−λo ∂F ∂xi+pj ∂Xj ∂xi∂ ∂pi +Aa∂ ∂ua.(35) Obse e ha his solu ion exis s all o e he mani old T∗M×Uand ha pois cons an o e e y cu e solu ion o he p oblem. The las e m co esponds o he elemen s o ke ω. The minimali y, compa ibili y, condi ions a e ∂H ∂ua=0 o e e y a, a e used o de e mine he con ols. As we said in Sec . 3, i he compa ibili y equa ions allows us o de e mine he con ols u1,...,uk, ha is we can ob ain ua=ψ(xo,xi,po,pi), hen we say ha he op imal con ol p oblem is egula , o he wise i is called singula . In he singula case, i is necessa y o apply an algo i hm o cons ain s, ha is o go o highe o de condi ions, o ob ain he con ols pe haps on a submani old o T∗M×U. Suppose ha we a e in he egula si ua ion, hence we ha e de e mined he con ols by he compa ibili y condi ions. Wi h he egula i y assump ion as he con ols uahas been de e mined, we ha e ha XHis p ojec ed o he mani old T∗Mand has componen s only in (xo,xi,po,pi).Then we ha e: XH=F∂ ∂xo+Xi∂ ∂xi−λo ∂F ∂xi+pj ∂Xj ∂xi∂ ∂pi (36) because we a e in he symplec ic case. We know ha , o all he solu ions o he associa ed p esymplec ic o mula ion, we ha e ha he momen po( )is a cons an . Following he p e ious sec ion, we will y o classi y he solu ions acco ding o he eal alue o po. Hence we de ine and s udy (a) No mal solu ions: hose wi h po=λo=/0. (b) Abno mal solu ions: hose wi h po=λo=0. 123 Jou nal o Nonlinea Science (2023) 33 :9 Page 21 o 46 9 5.2 No mal Solu ions: po=o=/0 Le N⊂T∗Mbe he submani old de ined by po=λoand j:N→T∗Mbe he na u al inclusion. We deno e by (xo,xi,pi) he coo dina es induced in Nby he coo dina es we ha e in T∗M. Conside now he canonical 1- o m θM∈1(T∗M)and le η=−(j)∗θM, hen we ha e ha Lemma 2 (N,η)is a con ac mani old. The Reeb ec o ield is R =−1 λo ∂ ∂xo. Le HN=(j)∗H he es ic ion o H o N, hen HN=λo(X)o+pi(X)iand we ha e a Hamil onian con ac sys em gi en by (N,η,HN).Le XN∈X(N)be he co esponding con ac Hamil onian ec o ield, ha is he solu ion o he equa ions i(XN)η =−HN,i(XN)dη=dHn−(L(R)HN)η whose local exp ession is XN=Xo∂ ∂xo+Xi∂ ∂xi−λo ∂Xo ∂xi+pj ∂Xj ∂xi∂ ∂pi .(37) Wi h he usual no a ion deno ing by he same names he unc ions on T∗Mand hei es ic ions o N. Wi h his in mind and ollowing Sec . 4.2.1,weha e ha : P oposi ion 3 The ec o ield XH∈X(T∗M)is angen o N and, on he poin s o N, i is equal o X N. Hence, o e e y u∈U, all he no mal solu ions o he op imal con ol p oblem a e solu ions o a con ac Hamil onian p oblem. 5.3 Abno mal Solu ions: po=o=0 Le No⊂T∗M he submani old de ined by po=0. Le jo:No→T∗Mbe he na u al inclusion and ηo=j∗ oθM. As abo e, ηo=−pidxiis no a con ac o m and we ha e ha ηo∧(dηo)mo=0. We can conside he 2- o m ωo=dηo, he Hamil onian Ho=j∗ oHand he p esymplec ic mani old (No,ω o,Ho). Obse e ha ke ωo={ ∂ ∂xo}. The Hamil onian p esymplec ic equa ion i(Xo)ωo=dHo gi es he solu ion Xo=Xi∂ ∂xi−pj ∂Xj ∂xi ∂ ∂pi +Aa∂ ∂ua, 123 9Page 22 o 46 Jou nal o Nonlinea Science (2023) 33 :9 whe e Aaa e a bi a y and co espond o ke ωo. And i does no exis any cons ain because he ec o ield Xois de ined on he whole mani old No. No e: We can also sol e he p econ ac p oblem gi en by (Nu o,η u o,Hu o). Commen : Obse e ha T∗M=λ∈RNλ, hence wi h his decomposi ion we ob ain all he solu ions o he Hamil onian p oblem on T∗Mgi en by he Hamil onian H. Some o hem, he no mal solu ions, as con ac p oblems, and he abno mal solu ions as symplec ic ones. Wi h all his in mind, we ha e p o ed he ollowing Theo em 7 (Con ac Pon yagin maximum p inciple) Conside he op imal con ol p oblem (M,U,X,I,xa,xb), wi h U ⊂Rkan open se . Le ˆσ:I→T∗M×U= T∗R×T∗Mo×U, ˆσ=(σT∗M,σ U), be a solu ion o he p esymplec ic Pon yagin maximum p inciple o such p oblem and suppose we a e in he egula case, ha is he minimali y condi ions (∂ H/∂u)=0, o e e y u ∈U and o e e y ∈I allows o de e mine he con ols. Then (a) i ˆσis a no mal solu ion wi h po=λo=/0, hen s i is an in eg al cu e o he con ac Hamil onian sys em (N,η,HN), as desc ibed abo e, wi h HN= λoF+piXi. (b) i ˆσis an abno mal solu ion, hen i is an in eg al cu e o he p esymplec ic Hamil onian sys em (No,ω o,Ho), as desc ibed abo e, wi h Ho=piXi. Fo he no mal solu ions, hey sa is y he di e en ial equa ions: ˙xo=∂H ∂po =F,˙po=∂H ∂xo=0,(⇒po=c ) ˙xi=∂H ∂pi =Xi,˙pi=− ∂H ∂xi=−po ∂F ∂xi−pj ∂Xj ∂xi ∂H ∂u1=0,..., ∂H ∂uk=0 whe e H=λoF+piXiwi h λo=/0. Fo he abno mal solu ions, he co esponding di e en ial equa ions a e ˙xi=∂H ∂pi =Xi,˙pi=− ∂H ∂xi=−pj ∂Xj ∂xi ∂H ∂u1=0,..., ∂H ∂uk=0 whe e H=piXi. Commen : We no e ha in Ohsawa (2015), an app oach o Pon yagin maximum p inciple is gi en in e ms o con ac sys ems. Indeed, he au ho wo ks in he p ojec- i iza ion o he co angen bundle, PT∗(M0×R). In ha app oach, he no mal and abno mal solu ions a e uni ied, and he abno mal ones co espond o he hype plane a in ini y. Howe e his mani old is no a con ac mani old in he sense we a e using, so we a e o ced o emo e he hype plane a in ini y and ob ain T∗M0×R. 123 Jou nal o Nonlinea Science (2023) 33 :9 Page 23 o 46 9 This can be seen as a di e en o mula ion o Theo em 7in ou pape , whe e we ea sepa a ely bo h kind o solu ions, which co espond o di e en geome ies (con ac and p esymplec ic). In addi ion, in ha e e ence he au ho does no s udy he ela ionship be ween he He glo z a ia ional p inciple and op imal con ol, which is he ocus o ou pape . 6 He glo z Va ia ional P oblem as an Op imal Con ol P oblem In Sec . 2.3.1 we ha e s udied he He glo z a ia ional p inciple; he e we ob ained he con ac equa ions o a Hamil onian con ac sys em as solu ion o a a ia ional p oblem bu wi h a gene aliza ion o he Hamil on a ia ional p inciple. This mo e gene al p inciple was s a ed and sol ed in 1930 by Gus a He glo z, see He glo z (1930) and Guen he e al. (1996). The idea was o change he in eg al s a emen on he cu es solu ion o he p oblem by a di e en ial equa ion de ined p ecisely by he Lag angian unc ion. In e es in his app oach has been inc easing since he las e e ed publica ion and i s ela ion wi h con ac dynamics and dissipa ion sys ems, see o example Geo gie a and Guen he (2002) and B a e i e al. (2017) and e e ences he ein. In his sec ion we app oach He glo z p inciple as an op imal con ol p oblem and ind he co esponding di e en ial equa ions, he gene alized Eule –Lag ange equa ions, wi h a new p oo h ough he Pon yagin maximum p inciple. 6.1 S a emen o he P oblem We begin ecalling he s a emen o he He glo z a ia ional p oblem as we did in Sec . 2.3.1. Le Qbe a smoo h mani old and F:TQ×R→Ra smoo h unc ion and conside he ollowing p oblem: He glo z a ia ional p oblem: Find cu es =(γ, ζ ) :I=[a,b]→Q×R, such ha (1) end poin s condi ions: γ(a)=qa,γ(b)=qb,ζ(a)=0, (2) is an in eg al cu e o ˙z=F(q, ,z):˙ ζ=F(γ ( ), ˙γ( ), ζ( )), o e e y ∈I, and (3) ex eme condi ion: ζ(b)is maximum o e all cu es sa is ying (1) and (2). Obse e ha we ha e conside ed he di e en ial equa ion ˙z=F(q, ,z)depending on he cu es γ. In he case ha he unc ion Fdoes no depend on he a iable z, ha is F:TQ→R, hen he di e en ial equa ions is ˙z=F(γ, ˙γ), hence by in eg a ion, he p oblem is he classical a ia ional one de ined by: ind he cu es γ( )minimizing S[γ]=b a F(γ ( ), ˙γ( )d wi h ini ial condi ions γ(a)=qa,γ(b)=qb. 123 9Page 24 o 46 Jou nal o Nonlinea Science (2023) 33 :9 As we know, He glo z ob ained ha he cu es γsolu ion o his p oblem sa is y he so-called gene alized Eule –Lag ange equa ions d d ∂F ∂ iγ −∂F ∂qi−∂F ∂z ∂F ∂ i=0. In his sec ion we will ob ain hese di e en ial equa ions as an applica ion o he Pon yagin maximum p inciple o a sui able op imal con ol p oblem associa ed o he He glo z a ia ional p oblem. To do so, we begin by gi ing a geome ic s a emen o he He glo z p oblem. Gi en he unc ion F:TQ ×R:→ R, conside he igh up iangle o he ollowing diag am TR τo TQ×R τQ×IR π2 Z R I ˆ  Q×R whe e Z∈X(R,π 2)is he ec o ield on Ralong he p ojec ion π2de ined by Z=F∂ ∂z. Now aking he ull diag am, we ha e he ollowing p oblem associa ed wi h he ec o ield Z Geome ic He glo z a ia ional p oblem: Find cu es :I=[a,b]→Q×R, =(γ, ζ), such ha (1) end poin s condi ions: (a)=(qa,0), γ (b)=qb, (2) is an in eg al cu e o Z:˙ ζ=F(˜ ( )), o e e y ∈I, whe e ˜ =(γ = (γ, ˙γ),ζ), and (3) ex eme condi ion: ζ(b)is maximum o e all cu es sa is ying (1) and (2). Ob iously he wo abo e p oblems a e equi alen . The di e ence is only in he lan- guage used o s a e hem. 6.2 Op imal Con ol App oach o he He glo z Va ia ional P oblem Associa ed o he unc ion F:TQ×R:→ R, conside he ollowing diag am T(Q×R) τQ×R TQ×RτQ×IR Y Q×R 123 Jou nal o Nonlinea Science (2023) 33 :9 Page 25 o 46 9 whe e Yis he ec o ield on Q×Ralong he p ojec ion τQ×IR:TQ×R→Q×R de ined by Y((q, ),z)=((q,z), , F), which in local coo dina es Yis gi en by Y= i∂ ∂qi+F∂ ∂z. This ec o ield co esponds o he sys em o o dina y di e en ial equa ions: ˙qi= i,˙z=F(qi, i,z). Obse e ha he i s sumand o he ec o ield Yis a canonical ec o ield along he p ojec ion τQ:TQ →Q, i co esponds o he iden i y map ITQ :TQ →TQ. Hence he ec o ield Yis associa ed in a na u al way o he unc ion F. These elemen s de ine a con ol sys em wi h ec o ield Y ∈X(Q×R,τQ×IR), on he s a e space Q ×R, and wi h he ib es o TQas he se o con ols; ha is o e e y s a e (q,z)∈Q×R, he con ols a e he elemen s ∈TqQ. On his con ol sys em we s a e he ollowing op imal con ol p oblem: Conside he diag am T(Q×R) τQ×R TQ×RτQ×IR Y Q×R I  ˜   whe e, i =(γ, ζ), hen ˜ =(γ ,ζ)=((γ, ˙γ),ζ). Fo a cu e :I→Q×R, we ake i s canonical li ing o he angen bundle, :I→T(Q×R), ha is: i =(γ, ζ) hen =((γ, ζ ), ( ˙γ, ˙ ζ)). We say ha a cu e is an in eg al cu e o he ec o ield Yi : =Y◦˜ , ( ˙γ,˙ ζ) =Y(γ, ˙γ,ζ)=( i(γ, ˙γ),F(γ, ˙γ,ζ)), which, in local coo dina es, is a solu ion o he abo e sys em o di e en ial equa ions: ˙qi= i,˙z=F(qi, i,z). Hence we ha e he op imal con ol p oblem gi en by: Op imal con ol p oblem associa ed o He glo z a ia ional p oblem: Find cu es :I=[a,b]→Q×R,=(γ, ζ ), such ha (1) end poin s condi ions: (a)=(qa,0), γ (b)=qb, (2) is an in eg al cu e o Y:( )=Y(˜ ( )), o e e y ∈Iand (3) op imal condi ion: ζ(b)is maximum o e all cu es sa is ying (1) and (2). 123 9Page 32 o 46 Jou nal o Nonlinea Science (2023) 33 :9 whe e Z=X+Y, ha is Z=Xi∂ ∂xi+F∂ ∂z, locally. And he cu es a e =(γM,γ z), γ=(, γU)=(γM,γ z,γ U). Then we ha e ano he equi alen s a emen : He glo z op imal con ol p oblem: Find cu es γ:I=[a,b]→M×R×U, γ=(γM,γ z,γ U),=(γm,γ z)such ha (1) end poin s condi ions: γM(a)=xa,γM(b)=xb,γ z(a)=0, (2) Mis an in eg al cu e o Z:˙ M=Z◦γ, and (3) op imal condi ion: γz(b)is maximum o e all cu es sa is ying (1), and (2). Condi ion (2) is w i en as (˙γ,˙γz)=Z◦γ, ha is ˙xi=Xi(x,z,u), ˙z=F(x,z,u). (45) which a e he same se o di e en ial equa ions as Eq. (44). Hence bo h p oblems a e equi alen . In he sequel we e e o his second o m. Obse e ha wi h his app oach, we ha e a classical op imal con ol p oblem and we can ind i s solu ion ollowing he me hod o Sec . 3, in pa icula by applying he weak p esymplec ic o m o he Pon yagin maximum p inciple, Theo em 5.In his case, he unc ion o op imize is one o he di ec ions o s a e space which is gi en by z. We begin, as usual, by ex ending he ec o ield, hence ob aining he ex ended sys em adding a new a iable xo o he a iable z o maximize. The new ec o ield is X=F∂ ∂xo+Xi∂ ∂xi+F∂ ∂z∈X(R×M×R). Then he associa ed Hamil onian is H(xo,po,xi,pi,z,pz,u)=poF+piXi+pzF, de ined on he mani old T∗(R×M×R)×U. The p esymplec ic o m is ω= dxo∧dpo+dxi∧dpi+dz∧dpz, wi h ke nel gi en by he angen ec o ields o U, and he Hamil onian ec o ield XH, solu ion o he equa ion iXHω=dH, is locally gi en by XH=F∂ ∂xo+0∂ ∂po +Xi∂ ∂xi+F∂ ∂z −po ∂F ∂xi+pj ∂Xj ∂xi+pz ∂F ∂xi∂ ∂pi −po ∂F ∂z+pi ∂Xi ∂z+pz ∂F ∂z∂ ∂pz +Aa∂ ∂ua, whe e he las e m co esponds o he ke nel o ω. Obse e ha his solu ion exis s all o e he mani old T∗(R×M×R)×Uand ha pois cons an o e e y cu e solu ion o he p oblem. 123 Jou nal o Nonlinea Science (2023) 33 :9 Page 33 o 46 9 Being a p esymplec ic sys em, he compa ibili y equa ions a e gi en by i(Z)dH= 0 o e e y Z∈ke ω, ha is equa ions ∂H ∂u1=0,..., ∂H ∂uk=0 (46) which, oge he wi h he equa ions coming om he ec o ield XH,gi eusase o equa ions o sol e he op imal con ol p oblem. Recall ha hese compa ibili y condi ions a e he same ha he op imali y ones. As in o dina y op imal con ol p oblems, suppose ha he compa ibili y equa- ions allow us o de e mine he con ols u1,...,uk, ha is we can ob ain ua= ψa(xo,xi,po,pi), hen we say ha he op imal con ol p oblem is egula ,o h- e wise i is called singula . In he singula case, i is necessa y o apply an algo i hm o cons ain s, ha is o go o highe o de condi ions, o ob ain he con ols pe haps on a submani old o T∗(R×M×R)×U. The di e en ial equa ions associa ed wi h he abo e ec o ield XH, oge he wi h equa ions (46) a e he solu ion equa ions o he He glo z op imal con ol p oblem. Rema k 5 To unde s and he signi icance o hese equa ions, we can compa e he abo e se o equa ions wi h he co esponding ones o a classical op imal con ol sys em. Apa om he compa ibili y condi ions, which a e he same, he ec o ield solu ion, see Theo em 5, was gi en by XN=F∂ ∂xo+Xi∂ ∂xi−λo ∂F ∂xi+pj ∂Xj ∂xi∂ ∂pi .(47) Compa ing his ec o ield XNwi h he abo e XH, in his las we ha e a new a i- able, z, hence wo new e ms, one o ˙zand he o he o ˙pz. Mo eo e , he e m co esponding o pihas changed. Bu i he cos unc ion Fdoes no depend on z, hen we ha e ha ˙pz=0, hence pz=cons an , and bo h equa ions, he classical and he He glo z op imal con ol, a e he same. In ac in his las case, we can change he di e en ial equa ion ˙z=F(x,z,u) and he op imali y condi ion by he in eg al o be op imized b a F(x,u)d and we ob ain exac ly he classical p oblem. Hence, as we p oposed a he beginning o he sec ion, we ac ually ha e a gene al- iza ion o he classical op imal con ol p oblem om he poin o iew o he equa ions sol ing he p oblem. 7.3 Con ac Fo mula ion o he No mal Solu ions We can analyze he se o no mal solu ions, ha is po=/0, in he aim o Sec . 4.2.2 and ob ain hese solu ions as in eg al cu es o con ac dynamical sys ems. 123 9Page 34 o 46 Jou nal o Nonlinea Science (2023) 33 :9 To p oceed suppose we a e in he egula si ua ion, ha is he maximali y condi ions allows us o de e mine he con ols. To s udy his si ua ion we can ix he con ols, hey a e de e mined by he las equa ions solu ion o he p oblem, and analyze he o he equa ions as solu ions o a symplec ic p oblem. Then, once ixed u=uo, ou mani old is T∗(R×M×R). In his mani old we can analyze he p oblem as a con ac dynamical sys em. Fo a gi en λo∈R,λo=/0, conside he submani old Nλo⊂T∗(R×M×R), gi en by po=λoand he na u al injec ion jλo:Nλo→T∗(R×M×R).Le η=−j∗ λoθ∈1(Nλo). Then we ha e Lemma 4 Fo e e y ixed u ∈U, he mani old (Nλo,η)is a con ac mani old. I s Reeb ec o ield is gi en by Rλo=−1 λo ∂ ∂xo The p oo is s aigh o wa d using he local exp ession o η η=−λodxo−pidxi−pzdz. Le HNλo=j∗ λoHand conside he Hamil onian con ac sys em gi en by (Nλo,η,HNλo).Le Z∈X(Nλo) he co esponding Hamil onian ec o ield, ha is he solu ion o he con ac equa ions i(Z)η =−HNλo,i(Z)dη=dHNλo−(L(Rλo)HNλo)η whose local exp ession is XH=F∂ ∂xo+0∂ ∂po +Xi∂ ∂xi+F∂ ∂z −po ∂F ∂xi+pj ∂Xj ∂xi+pz ∂F ∂xi∂ ∂pi −po ∂F ∂z+pi ∂Xi ∂z+pz ∂F ∂z∂ ∂pz . Wi h he abo e exp essions and commen s we ha e p o en he Theo em 11 The no mal solu ions o he p oblem 7.2 co esponding o po=λo=/0 a e he p ojec ions o R×M×R×U o he cu es solu ion o he con ac Hamil onian p oblem gi en by (Nλo,η,HNλo). The co esponding di e en ial equa ions o he cu es solu ion o his Hamil onian con ac p oblem a e : ˙xo=F ˙xi=Xi ˙z=F 123 Jou nal o Nonlinea Science (2023) 33 :9 Page 35 o 46 9 ˙pi=−po ∂F ∂xi−pj ∂Xj ∂xi−pz ∂F ∂xi ˙pz=−po ∂F ∂z−pi ∂Xi ∂z−pz ∂F ∂z Toge he wi h he maximiza ion condi ion, ha is he cons ain s ob ained om he compa ibili y o he p esymplec ic equa ion ∂H ∂u1=0,......, ∂H ∂uk=0 7.4 Reduc ion o he P oblem We ema k ha his p oblem is a gene aliza ion o He glo z a ia ional p inciple. On he p e ious sec ion, we showed ha he equa ions ob ained h ough he Pon yagin maximum p inciple could be educed o ob ain he He glo z equa ion. In his sec ion, we show ha a simila educ ion can be applied in his mo e gene al case. We see om he di e en ial equa ions abo e ha , aking he same ini ial condi ion o bo h a iables, we will ha e xo=z o he solu ions o p oblem 7.2. Then one o hem is i ele an o he p oblem, we can elimina e i . As he momen um co esponding o xois cons an , we can elimina e he pai (xo,po). Obse e ha , in ac , pzis also i ele an o he p oblem. Indeed, we can educe he dimension o he s a e space o he p oblem; his new mani old is wha we will now cons uc . Conside he Hamil onian H0:W0=T∗M×R×U→R, (xi,pi,z,ua)→ piXi(xi,z,ua)−p ∗ 1F(x,z,u), (48) whe e p 1:T∗M×R×U→M×R×Uis he na u al p ojec ion. Also conside he canonical con ac o m on T∗M×R η0=dz−pidxi.(49) Theo em 12 The no mal solu ions o he p oblem 7.2 co esponding o po=λoa e he p ojec ions o R×M o he cu es solu ion o he con ac Hamil onian p oblem gi en by (T∗M×R,η 0,H0). The equa ions o mo ion o he a o emen ioned Hamil onian p oblem a e ˙xi=Xi,(50a) ˙pi=pi ∂F ∂z−pj ∂Xj ∂xi+∂F ∂xi−∂Xj ∂zpipj,(50b) ˙z=F(50c) 123 9Page 36 o 46 Jou nal o Nonlinea Science (2023) 33 :9 subjec ed o he cons ain s ∂H ∂ua=∂F ∂ua−pj ∂Xj ∂ua=0.(50d) Rema k 6 In he case ha he p oblem is singula , one would wo k ins ead wi h he p econ ac sys em (T∗M×R×U,η 0,H0), applying he app op ia e cons ain algo- i hm. P oo Le γbe a solu ion o he He glo z op imal con ol p oblem. By Theo em 11, we know ha he e exis s a solu ion cu e σo he co esponding con ac sys em on Nλ0. In o de o p o e his heo em, we will p ojec σon o a solu ion o he sys em (T∗M×R,η 0,H0). Fi s o all, no ice ha he solu ions sa is y x0=z, hence σwill lie on he subman- i old j:˜ Nλ0→Nλ0de ined by x0=z. The dynamical ec o ield Xo he p econ ac sys em (Nλ0,η λ0,Hλ0)is angen o he submani old ˜ Nλ0. Indeed, he es ic ion o X o ˜ Nλ0a e jus he equa ions o mo ion ˜ Xo he induced p econ ac sys em (˜ Nλ0,˜ηλ0=j∗ηλ0,˜ Hλ0=j∗Hλ0).In coo dina es ˜ηλ0=(−λ0−pz)dz−pidxi,(51a) ˜ Hλ0=(λ0+pz)F+piXi(51b) Conside he ollowing commu a i e diag am, ˜ Nλ0 W0R×M×R M×R τ λ0 τ0π1 (52) whe e λ0xi,z,pi,pz=xi,−(λo+pz)pi,z.(53) No ice ha λ0is a subme sion and a con o mal equi alence o p econ ac sys ems: ∗ λ0η0=−(λo+pz)˜ηλ0,(54a) ∗ λ0H0=−(λo+pz)˜ Hλ0,(54b) By Theo em 1p ojec ions o he solu ion cu es o he p econ ac sys em on ˜ Nλ0a e solu ion cu es o he con ac sys em on TM×R. 123 Jou nal o Nonlinea Science (2023) 33 :9 Page 37 o 46 9 As a consequence o his heo em, we can ob ain again he He glo z equa ions. Conside he He glo z p oblem in Sec . 6.1 o a Lag angian L:TQ×R→R. No ice ha his p oblem is a pa icula case o he He glo z op imal con ol p oblem, whe e •Con ols a e he eloci ies ua= i. •The cos unc ion is he Lag angian F=L. •The con ol equa ion is X= i∂ ∂xi. The solu ions o his p oblem a e gi en by Theo em 12: ˙qi= i,(55) ˙pi=pi ∂L ∂z+∂L ∂qi(56) ˙z=L(57) wi h he cons ain s ∂L ∂ i=pi,(58) which a e p ecisely He glo z equa ions. 8 Applica ion: Op imal Con ol on The modynamic Sys ems One possible applica ion o his heo y is he s udy o he modynamic p ocesses which minimize o maximize some he modynamic po en ial. As an example, we apply ou o malism o he con ol sys ems conside ed in Van de Scha and Maschke (2017). The ela ion be ween symplec ic and con ac mani olds ia he symplec i ica ion p ocedu e has pe mi ed o go deepe in he geome ic desc ip ion o he modynamic sys ems. This way has been explo ed in Balian and Valen in (2001) (see also A nold 1978; Libe mann and Ma le 1987; Ibáñez e al. 1997). 8.1 Homogeneous Hamil onian Sys ems and Con ac Sys ems The e is a close ela ionship be ween homogeneous symplec ic and con ac sys ems, see o example Van de Scha and Maschke (2017) whe e his ela ion is s udied. He e we b ie ly ecall he ideas we need o ollow he example. In he gene al case, i π:M→Bis a ec o bundle, a unc ion F:M→R is homogeneous i , o any ep∈Mp=π−1(p)wi h π(ep)=p∈B,weha e F(λep)=λF(ep). In his si ua ion he unc ion Fcan be p ojec ed o he p ojec i e bundle P(M)o e Bob ained by p ojec i iza ion on e e y ib e. We a e in e es ed in he case ha M=T∗(Q×R)→Q×R, wi h na u al coo dina es (qi,z,Pi,Pz) Le Hbe an homogeneous Hamil onian unc ion on T∗(Q×R). Locally, we ha e ha H(qi,z,λPi,λPz)=λH(qi,z,Pi,Pz), o all λ∈R. Equi alen ly, one can 123 9Page 38 o 46 Jou nal o Nonlinea Science (2023) 33 :9 w i e Hqi,z,Pi,Pz=−Pzhqi,−Pi/Pz,z,(59) o Pz=/0, whe e h:T∗Q×R→R,h(qi,pi,z)=H(qi,z,−pi,−1)is well de ined. Wi h he abo e changes, we ha e iden i ied he mani old T∗Q×Ras he p ojec i e bundle P(T∗(Q×R)) o he co angen bundle T∗(Q×R) aking ou he poin s a in ini y, ha is he subse de ined by {Pz=0}. Following Van de Scha and Maschke (2017, Sec ion 4.1), he map :T∗(Q×R) {pz=0}→T∗Q×R (qi,z,Pi,Pz)→(qi,Pi/Pz,z)=(qi,pi,z), (60) sends he Hamil onian symplec ic sys em (T∗(Q×R) {pz=0},ωQ×R,H)on o he Hamil onian con ac sys em (T∗Q×R,ηQ,h), whe e ωQ×R=dqi∧dPi+dz∧dPz and ηQ=dz−pidqi. Obse e ha he na u al coo dina es o T∗Q×R, deno ed by (qi,pi,z), co espond o he homogeneous coo dina es in he p ojec i e bundle. In ac , he map is he p ojec i iza ion; i.e., he map ha sends each poin in he ibe s o T∗(Q×R) o he line ha passes h ough i and he o igin. I can be shown ha p o ides a bijec ion be ween con o mal con ac omo - phisms and homogeneous symplec omo phisms. Mo eo e , maps homogeneous Lag angian submani olds L⊆T∗(Q×R)on o Legend ian submani olds L= φ(L)⊆T∗Q×R. See Van de Scha and Maschke (2017) and Sec . 8.3 o mo e de ails on his opics. 8.2 Con ol o Con ac Sys ems On he con ac na u al mani old T∗Q×R, wi h coo dina es (qi,pi,z), assume ha we a e gi en a pa ame ized amily o Hamil onians h:T∗Q×R×U→R,U⊂Rk, wi h Hamil onian con ac ec o ields Xhu, whe e hu(qi,z,pi)=h(qi,z,pi,u). Then we can de ine he con ol sys em Z(q,p,z,u)=Xhu(q,p,z), whe e he ollowing diag am is commu a i e: T(T∗Q×R) τT∗Q×R T∗Q×R×U Z πT∗Q×R I γ 123 Jou nal o Nonlinea Science (2023) 33 :9 Page 39 o 46 9 A cu e γ:I→T∗Q×R×Uis an in eg al cu e o Z, ha is =Z◦γ,i in local coo dina es sa is ies he di e en ial equa ions dqi d =∂hu ∂pi , dpi d =− ∂hu ∂qi−pi ∂hu ∂z, dz d =pi ∂hu ∂pi −hu. One can conside he He glo z op imal con ol p oblem gi en by Z, as we s a ed in Sec . 7.2. Then, by Theo em 12, we know ha he no mal solu ions a e he p ojec ions o he solu ions o he con ac sys em (T∗(T∗Q)×R,η T∗Q,H), whe e H=pqi ∂hu ∂pi −ppi ∂hu ∂qi−pi ∂hu ∂z−pi ∂hu ∂pi +hu.(61) 8.3 Applica ion o The modynamic Sys ems We conside he modynamic sys ems in he so-called en opy ep esen a ion. Hence he he modynamic phase space, ep esen ing he ex ensi e a iables, is he mani old T∗Q×R, equipped wi h i s canonical con ac o m ηQ=dS−pidqi.(62) The local coo dina es on he con igu a ion mani old Q×Ra e (qi,S), whe e Sis he o al en opy and qi’s deno e he es o ex ensi e a iables. O he a iables, such as he in e nal ene gy, may be chosen ins ead o he en opy, by means o a Legend e ans o ma ion. The s a e o a he modynamic sys em always lies on he equilib ium submani old L⊆T∗Q×R, which is a Legend ian submani old, ha is, η|TL=0 and dim L= dim Q=n. The pai (T∗Q×R,L)is a he modynamic sys em. The equa ions (locally) de ining La e called he s a e equa ions o he sys em. On a he modynamic sys em (T∗Q×R,L), one can conside he dynamics gene - a ed by a Hamil onian ec o ield XHassocia ed o a Hamil onian h. I his dynamics ep esen s quasis a ic p ocesses, meaning ha a e e y ime he sys em is in equilib- ium, ha is, i s e olu ion s a es emain in he submani old L, i is equi ed o he con ac Hamil onian ec o ield Xh o be angen o L. This happens i and only i h anishes on L. Equi alen ly, by Sec . 8.1, one can conside he ex ended he modynamic phase space T ∗(Q×R)wi h i s canonical symplec ic o m ωQ×R=dqi∧dPi+dS∧dPS.(63) 123 9Page 40 o 46 Jou nal o Nonlinea Science (2023) 33 :9 In his o mula ion, a he modynamic sys em is a uple (T∗(Q×R), L)), whe e L is a homogeneous Lag angian submani old. Dynamics a e gi en by a homogeneous Hamil onian K. See Van de Scha and Maschke (2017) o de ails and ecall we ha e iden i ied, in Sec . 8.1, he bundle T∗Q×Rwi h he p ojec i e bundle P(T∗(Q×R)). Po - he modynamic sys ems we e in oduced in Van de Scha and Maschke (2017), bu in a homogeneous symplec ic o malism. De ini ion 1 (Po - he modynamic sys em)Apo - he modynamic sys em on T∗(Q× R)is de ined as a pai (L,K), whe e he homogeneous Lag angian submani old L⊂ T∗(Q×R)speci ies he s a e p ope ies. The dynamics is gi en by he homogeneous Hamil onian dynamics wi h pa ame ized homogeneous Hamil onian K:= Ka+ Kc αuα:T∗(Q×R)→R,u∈Rk,Kc:T∗(Q×R)→Rk, wi h Ka,Kcbo h equal o ze o on he poin s o L, and Kaas he in e nal Hamil onian. One need he addi ional condi ion ∂K ∂S|L≥0,(64) so ha he second law o he modynamics holds. Using he esul s o Sec . 8.1, we could ins ead conside he ollowing con ac o mula ion. De ini ion 2 (Po - he modynamic sys em, con ac o malism) A po - he modynamic sys em on (T∗Q×R,ηQ)is de ined as a pai (L,h), whe e he Legend ian submani old L⊂T∗Q×Rspeci ies he s a e p ope ies. The dynamics is gi en by he con ac Hamil onian dynamics wi h pa ame ized con ac Hamil onian h=ha+hc αuα: T∗Q×R→R,u∈Rm,hc:T∗Q×R→Rk, wi h ha,hcze o on L, and he in e nal Hamil onian hasa is ying ∂h ∂S|L≥0,(65) so ha he second law o he modynamics holds. Ou heo y p o ides ools o unde s and which o he a ailable he modynamic p ocesses minimize he en opy p oduc ion o he sys em. Obse e ha we can conside p ocesses ha maximize o minimize o he he modynamic a iables, such as he ene gy, ia a Legend e ans o m. 8.4 Example: Gas–Pis on–Dampe Sys em We end his sec ion wi h an explici example which can be ound in Van de Scha and Maschke (2017). Conside an adiaba ically isola ed cylinde closed by a pis on con aining a gas wi h in e nal ene gy U(V,S). The ex ended phase space has he ollowing ex ensi e a iables • he momen um o he pis on π, 123 Jou nal o Nonlinea Science (2023) 33 :9 Page 41 o 46 9 • he olume o he gas V, • he ene gy E, • he en opy S. They co espond o Q×Rwi h local coo dina es (V,π,E,S). The Legend ian sub- mani old is gi en by L=(V,π,E,pV,pπ,pE,S)|E=π2 2m+U(S,V), pV=−pE ∂U ∂V, pπ=−pE π m,pE=1/∂U ∂S(66) The ene gy is hen gi en by h=pV π m+pπ−∂U ∂V−dπ m−d(π m)2 ∂U ∂S +pπ+pE π mu,(67) whe e dis he diame e o he pis on and mis i s mass. The Hamil onian ec o ield is gi en by Xh=π m ∂ ∂V+−πd m+u−∂U ∂V∂ ∂π +πu m ∂ ∂E +⎛ ⎝⎛ ⎝pπ ∂2U ∂V∂S−π2d∂2U ∂S2 m2∂U ∂S2⎞ ⎠pV+pπ ∂2U ∂V2−π2d∂2U ∂V∂S m2∂U ∂S2⎞ ⎠ ∂ ∂pV +⎛ ⎝⎛ ⎝pπ ∂2U ∂V∂S−π2d∂2U ∂S2 m2∂U ∂S2⎞ ⎠pπ+dp π m−pEu m−pV m+2πd m2∂U ∂S⎞ ⎠ ∂ ∂pπ +⎛ ⎝pπ ∂2U ∂V∂S−π2d∂2U ∂S2 m2∂U ∂S2⎞ ⎠pE ∂ ∂pE +π2d m2∂U ∂S∂ ∂S(68) We cons uc he con ac Hamil onian sys em (T∗(T∗Q)×R,η T∗Q,H)as in (61): H=− dp π m−pEu m−pV m+2πd m2∂U ∂SPπ −pπ ∂2 (∂V)2U(V,S)−π2d∂2 ∂V∂SU(V,S) m2∂U ∂S 2PV −πd m−u+∂ ∂VU(V,S)Ppπ+πPpEu m+πPpV m−π2d m2∂U ∂S , (69) 123