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
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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
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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
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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.
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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,
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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)
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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)
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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)
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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)
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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
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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.
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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
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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
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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.
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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,
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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.
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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.
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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
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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).
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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.
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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.
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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
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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)
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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.
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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
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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
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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)
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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,
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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)
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