P oceedings o he Es onian Academy o Sciences,
2014, 63, 1, 26–32
doi: 10.3176/p oc.2014.1.05
A ailable online a www.eap.ee/p oceedings
Connec ions in con ol s a egy
Maido Rahulaa∗and Pe Vaˇ
s´
ıkb
aIns i u e o Ma hema ics, Uni e si y o Ta u, J. Lii i 2, 50409 Ta u, Es onia
bIns i u e o Ma hema ics, B no Uni e si y o Technology, Facul y o Mechanical Enginee ing, Technick´
a 2, 616 69 B no,
Czech Republic; [email p o ec ed].cz
Recei ed 15 Augus 2013, e ised 20 Sep embe 2013, accep ed 14 Oc obe 2013, a ailable online 14 Ma ch 2014
Abs ac . We p esen an in ini esimal in e p e a ion o he con ol heo y, pa icula ly o he pa conce ning dynamic sys ems. We
use he o iginal concep o a bundle connec ion, which lies in he idea o ib e anspo a ion along a pa h on he base mani old. The
con ol o a p ocess leads also o he anspo a ion o ib es, and he con ol s a egy, i.e. he choice o a sui able sys em con ol
in o de o op imize he p ocess co esponds o he choice o a pa h on he base mani old. The iple o c ucial e ms o con ol,
aim–con ol–s a egy, ansla es in he e ms o connec ions as ib e–connec ion–cu e. Such a scheme is qui e con incing, bu i
also wo ks well in dynamic sys ems analysis.
Key wo ds: con ol heo y, connec ion.
1. INTRODUCTION
When con olling a sys em, we no only apply one con ol model bu also y o ind a mo e sui able con ol
model among he possible ones, i.e. we sea ch he con ol s a egy. We dis inguish he ollowing s ages:
con olled p ocess – con ol co ec ion – s a egy choice.
Le us desc ibe he ma hema ical se ing. Le X,Y, and Zbe h ee ec o ields and le us deno e by
a =exp X,b
σ
=exp
σ
Y,c
τ
=exp
τ
Z
he app op ia e lows. I we unde s and he low as a mo ion, he ec o ield can be seen as s opping he
mo ion a he p ecise momen (s op-scene). Sho ly, a ec o ield is an in ini esimal ep esen a ion o he
low.
The low c
τ
o he ec o ield Z ep esen s he con olled p ocess (i is also possible o eplace i by a
anspo o an a bi a y enso ield along he low c
τ
). Fu he mo e, he low b
σ
ac s on he ec o ield Z
low c
τ
as ollows; see [1,3]:
c
τ
Ãb
σ
c
τ
b−1
σ
,ZÃZ
σ
.
This co esponds o he change o con ol. I in addi ion he low a ac s on b
σ
, we ha e a con ol s a egy
b
σ
Ãa b
σ
a−1
,YÃY .
∗Co esponding au ho , [email p o ec ed]
M. Rahula and P. Vaˇs´
ık: Connec ions in con ol s a egy 27
Conce ning he s a egy {XÃ{YÃZ}}, i is ob ained as a composi ion o he ac ion o b
σ
on c
τ
and he
ac ion o a on b
σ
,
c
τ
Ãb
σ
c
τ
b−1
σ
Ã(a b
σ
a−1
)c
τ
(a b
σ
a−1
)−1
.
The ec o ield Yplays he ole o he one con olled by he ec o ield Xand he ole o he con olling
ield o e Z.
The main goal o he pape is o desc ibe he ans o ma ion o he pa ame e s when he p ocess Zis
changed acco ding o he s a egy app op ia e o he ec o ield Xunde he ac ion o he ec o ield Y.
2. VECTOR FIELDS
Le Mbe a smoo h mani old. The de i a i es o a unc ion :M→Ralong he ec o ields X,Y, and Z
a e de ined by
X .
= ( ◦a )0
=0,Y .
= ( ◦b
σ
)0
σ
=0,Z .
= ( ◦c
τ
)0
τ
=0.
The ec o ield Yis anspo ed along he low o X, which can be unde s ood as an in ini esimal
in e p e a ion o such anspo a ion (s op-scene) – he b acke o ec o ields, i.e. Lie de i a i e LXY=
[X,Y].
Rema k 1. One can ob ain he b acke o wo ec o ields [X,Y] = XY −YX by double di e en ia ion o a
unc ion along he ec o ield low a b
σ
a−1
w. . .
σ
and hen w. . . :
◦(a b
σ
a−1
)−1(.)0
σ
=0
−→ −¡Y( ◦a )¢◦a−1
(.)0
=0
−→ (XY −Y X) .
Nex , he anspo o an a bi a y smoo h enso ield along a ec o ield low is de ined by he Lie–
Maclau in se ies. Fo example, he anspo o a ec o ield Zalong he low b
σ
is de ined by
ZÃZ
σ
=Z+Z0
σ
+Z00
σ
2
2+... =
∞
∑
k=0
Z(k)
σ
(k)
k!,
whe e he coe icien s Z(k)=LYZ(k−1)
,k=1,2,..., a e Lie de i a i es o Zwi h espec o Y.
In ou si ua ion, he ec o ield Xplays h ee oles:
1. The ec o ield Xi sel causes he p ocess as a mo ion in i s low a .
2. The ope a o LX ans o ms he con ol YÃ[X,Y].
3. The ope a o LLXde ines he con ol s a egy – con ol o con ol LYÃ[LX,LY] = L[X,Y]. No e ha
he abo e equali y can be ob ained om he Jacobi iden i y:
£[X,Y],Z¤+£[Y,Z],X¤+£[Z,X],Y¤=0
o equi alen ly
£[X,Y],Z¤=£X,[Y,Z]¤−£Y,[X,Z]¤.
No e ha he p ocess can be in luenced only by some ou e p ocess, no by i i sel . Indeed, i we admi
ha he ope a o LXac s on he con ol by he p ocess X, we ob ain:
LXX= [X,X] = 0.
We assign he ollowing ope a o s o he ec o ields X,Y, and Z:
1. he ope a o Zimplemen ing he p ocess (mo ion in he low c
τ
);
2. he ope a o LYimplemen ing he con ol o he p ocess ZÃ[Y,Z](mo ion o he low c
τ
in he low b
σ
);
3. he ope a o LLXde ining he con ol s a egy LY(mo ion in he low a o a mo ion o he low c
τ
in he
low b
σ
).
28 P oceedings o he Es onian Academy o Sciences, 2014, 63, 1, 26–32
3. CONTROL AND CONNECTION
We will ollow he no a ions app op ia e o he heo y o connec ions on ib ed mani olds; see, e.g., [1,4].
Le us conside a ec o bundle
π
:M1→Mwi h n-dimensional base mani old Mand -dimensional
ib es. The s anda d ib e is isomo phic o R . On a neighbou hood U⊂M1we ha e local coo dina es
(ui
,u
α
), whe e (ui)deno es he base coo dina es and (u
α
) he ib e coo dina es. P ecisely, ui=¯ui◦
π
, whe e
¯uideno es he local coo dina es on he neighbou hood
π
(U)⊂M. The coo dina es (u
α
)a e he coo dina es
o R
.La in indices i,j, . . . ange om 1 o n, G eek indices
α
,
β
, . . . ange om n+1 o n+ .
We de ine wo ec o ields:
Y=y
α∂α
and Z=z
α∂α
.
He e z
α
a e he unc ions depending on he ib e coo dina es u
α
only, while y
α
a e he unc ions o all
coo dina es (ui
,u
α
). The low b
σ
=exp
σ
Yis de ined on he neighbou hood Uby a sys em o ODEs
du
α
d
σ
=y
α
(ui
,u
β
).(1)
Indeed, now we can see he connec ion be ween dynamic sys ems, see [2], and he con olling pa ame e s
(ui). As men ioned abo e, hese pa ame e s a e li ed om he base
π
(U)⊂M o he neighbou hood
U⊂M1, i.e. ui=¯ui◦
π
.
On e e y ib e, he ec o ield Yinduces a amily o ajec o ies – phase po ai . When he ib e is
changed, he ec o ield Ychanges oo and so does he phase po ai , i.e. he con ol {YÃZ}. A ques ion
a ises: how do he pa ame e s (ui)a ec he con olling p ocess?
Le us conside he coo dina e map
Φ:(ui
,u
α
)Ã(ui
,s,I
κ
),k=n+2,...,n+ ,
whe e sis a canonical pa ame e , i.e. LYs=0, and I
κ
is a sys em o −1 independen in a ian s o he
ec o ield Y. The coo dina es (ui
,I
κ
) o m a comple e sys em o local in a ian s o Yon he mani old M1.
Now we can de ine he subme sion o he mani old M1on o he ib e R
,
ϕ
:M1→R :(ui
,u
α
)Ã(s,I
κ
).
A ib e o he subme sion
ϕ
has he dimension nand o ms he amily o he in eg al su aces which de ine a
ho izon al dis ibu ion 4h.Thus on he ib a ion
π
, a ze o o sion connec ion s uc u e 4h⊕4 is de ined.
Le us conside he adap ed basis
(XiX
α
) = µ
∂
∂
uj
∂
∂
u
β
¶·Ã
δ
j
i0
Γ
β
i
δβ
α
!,µ
ω
i
ωα
¶=µ
δ
i
j0
−Γ
α
j
δα
β
¶.µduj
du
β
¶,
whe e he ec o ields
Xi=
∂
∂
ui+Γ
α
i
∂
∂
u
α
o m a base o he dis ibu ion ∆hand he o ms
ωα
=du
α
−Γ
α
idui
anish on he dis ibu ion ∆h. The numbe o pa ame e s Γ
α
iequals n and hey de ine he dis ibu ion
∆huniquely. On he o he hand, he pa ame e s Γ
α
ia e de e mined by se ing he unc ions
ϕα
equal o a
cons an on he ib es o he subme sion
ϕ
, mo e p ecisely by hei di e en ials:
d
ϕα
=
ϕα
idui+
ϕα
β
du
β
=
ϕα
β
(du
β
+¯
ϕβ
γϕγ
idui) =
ϕα
βωβ
=⇒Γ
α
i=¯
ϕα
γϕγ
i,
whe e he coe icien s o d
ϕα
a e he pa ial de i a i es o
ϕα
.The ma ix (
ϕα
β
)is he in eg a ing ma ix
wi h espec o he o ms
ωα
and i s in e se is (¯
ϕβ
α
).
M. Rahula and P. Vaˇs´
ık: Connec ions in con ol s a egy 29
Theo em 1. The ec o ield Y is p ojec ed by he subme sion
ϕ
:M1→R on o he ec o ield T
ϕ
Y on
he s anda d ib e R . In he coo dina es (s,I),whe e s deno es he canonical pa ame e and I is a sys em
o he base in a ian s, he ec o ield T
ϕ
Y ep esen s he ope a o
∂
s
.
=
∂
∂
s. The ec o ield Z is exp essed
uniquely in he basis (
∂
s,
∂
I)and he p ocess con olled by Z is,in he coo dina e sys em (s,I),desc ibed by
he unc ions s◦
ϕ
and I ◦
ϕ
. These unc ions depend on he pa ame e s u
α
and he con olling pa ame e s ui.
P oo . A amily o he ib es co esponding o he subme sion
ϕ
is de ined by he solu ion o he sys em
o di e en ial equa ions (u
α
)
σ
=
ϕα
(
σ
,ui
,u
β
); see sys em (1). Fu he mo e, an a bi a y sec ion o he
ib a ion
π
can be ex ended in o he sys em o imp imi i i y app op ia e o he low bs, i.e. he amily
o he ib es co esponding o he subme sion
ϕ
. The ec o ield Yis
ϕ
-p ojec ed on he ib e R . An
in eg able dis ibu ion ∆h=Ke T
ϕ
in he ib a ion
π
de ines a ze o cu a u e connec ion and hus on he
neighbou hood U he basis and he co-basis o he dis ibu ion ∆his de ined as ollows:
Xi=
∂
i+Γ
α
i
∂α
,
ωα
=du
α
−Γ
α
idui
.
Le us ecall ha an a bi a y ec o ield ¯
Xon he base mani old Mcan be li ed om M o he ho izon al
dis ibu ion ∆h:
¯
X=¯xi¯
∂
iÃX=xiXi,whe e xi=¯xi◦
ϕ
.
In ou no a ions, he basis Xi ep esen s he ope a o s ¯
∂
i om he neighbou hood
π
(U)li ed o he
dis ibu ion ∆h.
I is now clea ha he ec o ield Xbeha es wi h espec o he ec o ield Yas an in ini esimal
symme y, i.e. [X,Y] = 0, and hus he impac on he ec o ield Y anishes. In o he wo ds, he p ocess
app op ia e o he ec o ield Zis de ined on he ib e in he coo dina es (s,I), whe e he unc ions s◦
ϕ
and
I◦
ϕ
depend on he pa ame e s u
α
and he con olling pa ame e s ui. The ec o ield Xa ec s he ec o
ield Zindi ec ly by means o he in a ian s o he ec o ield Y.
Rema k 2. The componen s y
α
o he ec o ield Ydepend linea ly and homogeneously on he ib e
coo dina es. Thus he de ining sys em is desc ibed by he sys em o linea di e en ial equa ions
du
α
d
σ
=y
α
β
(ui)u
β
.
4. APPLICATION
On he bundle1
π
:R3→R:(u,x,y)Ã(u)
wi h he ib e coo dina es (x,y)and he con olling pa ame e (o base coo dina e) (u)we ha e he ec o
ield
Y=
∂
∂
x+ux
∂
∂
y.
We de ine i s low bs=expsY , he canonical pa ame e s, and he in a ian Io Yas ollows:
½˙x=1
˙y=ux ⇒½xs=x+s
ys=y+u(xs +s2
2),½s=x
I=y−ux2
2.
We check ha LYs=1,LYI=0. The ajec o ies on he ib es a e pa abolas depending on he pa ame e u.
1He e, o he sake o simplici y, we deno e he local coo dina es by (u,x,y)ins ead o (u1
,u2
,u3)bu no e ha he ib e
coo dina es (x,y)a e in no way ela ed o he componen s (xi
,y
α
)o he ec o ields Xand Y.
30 P oceedings o he Es onian Academy o Sciences, 2014, 63, 1, 26–32
The subme sion
ϕ
:R3→R:(u,x,y)Ã(s,I)p ojec s he space R3on o he plane sI. The angen
mapping T
ϕ
is de ined by he ollowing di e en ials and by he Jacobi ma ix:
½ds =dx
dI =−x2
2du −uxdx +dy,µ0 1 0
−x2
2−ux 1¶.
The ec o ield Ywi h he componen s (0,1,ux)is p ojec ed o he plane sI in which i o ms he ope a o
T
ϕ
Y=
∂
s(see Fig. 1).
Thus on he bundle
π
a ho izon al dis ibu ion
4h=Ke T
ϕ
is de ined. The co-basis on 4his o he o m
½
ω
2=ds =dx = (dx −Γ2
1du)
ω
3=uxds +dI =dy −x2
2du = (dy −Γ3
1du),
and he connec ion coe icien s a e
µΓ2
1
Γ3
1¶=µ0
x2
2¶.
The adap ed basis o he dis ibu ion 4his cha ac e ized by he ollowing:
X1=
∂
u+x2
2
∂
y,µ
ω
2
ω
3¶=µdx
dy ¶−µ0
x2
2¶·(du).
The ope a o X1commu es wi h he ec o ield Y, i.e. [X1,Y] = 0,and anishes unde he p ojec ion T
ϕ
,
i.e. T
ϕ
X1=0. The co-basis admi s an in eg a ing ma ix as ollows:
µ
ω
2
ω
3¶=µ1 0
us 1¶·µds
dI ¶⇒µ1 0
−ux 1¶·µ
ω
2
ω
3¶=µds
dI ¶.
H
H
H
H
H
H
H
H
H
H
H
H
C
C
C
C
C
C
C
C
C
CO
+
=
-
u
6
¤¤¤¤¤¤¤¤¤¤
¤º
¢¢¢¢¢¢¢¢¢
¢¸
¾
T
ϕ
∂
s
sI
Y
Fig. 1. Mapping T
ϕ
:Y→
∂
s.

M. Rahula and P. Vaˇs´
ık: Connec ions in con ol s a egy 31
The di ec impac o he pa ame e (u)on he ope a o Yis elimina ed. Indeed, because he p ojec ion
ϕ
a ge s on he ib e xy, i is possible o change he coo dina es unde he condi ion u=cons om (x,y)
o (s,I),
½x=s
y=us2
2+I,µ1 0
us 1¶.
Using he Jacobi ma ix ( he igh -hand side), we can change he basis o he new na u al one and we ob ain
he ollowing ames and co- ames:
³
∂
∂
x
∂
∂
y´=³
∂
∂
s
∂
∂
I´·µ1 0
−us 1¶,µdx
dy ¶=µ1 0
us 1¶·µds
dI ¶.
Le us ocus on he ib e. No e ha he ac ion o he e ical ec o ield Zcan be unde s ood as an ac ion
on a enso ield. Conce ning he ac ion o he ope a o Yon he ec o ield Zin he o m
YÃZ=
µ∂
∂
x+
ν∂
∂
y,
wi h he componen s (
µ
,
ν
), we can see ha in new coo dina es i educes o he ac ion o he ope a o
∂
son
he ec o ield ˜
Zdepending on he pa ame e uonly:
∂
sØ
Z=
µ∂
s+
ν∂
I−u
µ
s
∂
I.
No e ha Zand ˜
Za e he same ec o ield, only exp essed in he coo dina es (x,y)and (s,I), espec i ely.
The ope a o s T
ϕ
Yand
∂
sa e he same ope a o s exp essed in di e en coo dina e sys ems.
Thus we change he con ol:
{YÃZ}Ã{
∂
sØ
Z}.
Rema k 3. As an example, le us conside he ope a o o o a ion
Z=−y
∂
∂
x+x
∂
∂
y.
In coo dina es (s,I),i can be w i en in he o m ˜
Z=−I
∂
s+s
∂
I+u(...), i.e. in such a o m ha some new
ope a o wi h coe icien uis added. Such a p ope y holds o an a bi a y linea dynamic sys em.
The con ol {YÃZ}is desc ibed in he coo dina es (x,y), while he con ol {
∂
sØ
Z}is exp essed in
he coo dina es (s,I). The pa ame e ua ec s he con olled ield ˜
Zdi ec ly.
5. CONCLUSION
The con ol o a dynamic sys em is iewed by means o di e en ial geome y as he ec o ield Yon he
bundle
π
:M1→Mwi h he s anda d ib e R and he base mani old M=Rn
.The subme sion
ϕ
is de ined
in such a way ha he ec o ield Yis p ojec ed o he ib e R
.The dis ibu ion 4h=Ke T
ϕ
gi es ise
o he possibili y o elimina ing he dependence o he ec o ield Yon he con olling pa ame e u. The
change o a iables o (s,I), whe e sis he canonical pa ame e and Iis he in a ian o he ield Y, changes
he con ol (Y→Z) o he con ol {
∂
sØ
Z}, whe e he ield
∂
sno longe depends on he pa ame e uwhile
he con olled ield ˜
Zdoes so.
32 P oceedings o he Es onian Academy o Sciences, 2014, 63, 1, 26–32
ACKNOWLEDGEMENTS
The i s au ho was suppo ed by Es onian Ta ge ed Financing P ojec SF0180039s08. The second au ho
was suppo ed by he p ojec NETME CENTRE PLUS (LO1202). The esul s o he p ojec NETME
CENTRE PLUS (LO1202) we e co- unded by he Minis y o Educa ion, You h and Spo s wi hin he
suppo p og amme “Na ional Sus ainabili y P og amme I”.
REFERENCES
1. A anasiu, Gh., Balan, V., B ˆ
ınzei, N., and Rahula, M. 2009. Di e en ial-Geome ic S uc u es. Tangen Bundles, Connec ions
in Fibe Bundles, Exponen ial Law and Je Spaces. Lib okom, Moscow (in Russian).
2. Pe ko, L. Di e en ial Equa ions and Dynamic Sys ems. Sp inge , 1991.
3. Rahula, M. New P oblems in Di e en ial Geome y. Wo ld Scien i ic Publishing Co. P e. L d., 1993.
4. Rahula, M. and Vaˇ
s´
ık, P. A no e on je and geome ic app oach o highe o de connec ions. Sp inge P oc. Ma h. S a is ics ( o
appea ).
Seos used juh imise eoo ias
Maido Rahula ja Pe Vaˇ
s´
ık
Juh imisel on kolm aspek i: juhi a p o sess, juh i p o sess ja juh imise alik/s a eegia. Ma emaa iliseks
mudeliks on kih kond, kus seos us m˜
oju ab oimu a kihil, ja baasipa amee id, milles s˜
ol ub seos us. Need
m¨
a¨
a a ad juh imise s a eegia. Osu ub, e baasipa amee eid ˜
oib seos use abil o sekohe kihile suuna a, s
juh i as p o sessis juhi a asse.