1227948

Contrôle de réacteurs de polymérisation, Observateur et
Invariance
Nasradine Aghannan
To cite this version:
Nasradine Aghannan. Contrôle de réacteurs de polymérisation, Observateur et Invariance. Automatique / Robotique. École Nationale Supérieure des Mines de Paris, 2003. Français. �tel-00006598�
HAL Id: tel-00006598
https://pastel.archives-ouvertes.fr/tel-00006598
Submitted on 27 Jul 2004
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ECOLE DES MINES
DE PARIS
N° attribué par la bibliothèque
|__|__|__|__|__|__|__|__|__|__|
THESE
pour obtenir le grade de
Docteur de l’Ecole des Mines de Paris
Spécialité “Mathématiques et Automatique”
présentée et soutenue publiquement par
Nasradine AGHANNAN
le 13 novembre 2003
CONTRÔLE de REACTEURS DE POLYMERISATION,
OBSERVATEUR ET INVARIANCE
Directeur de thèse : Pierre ROUCHON
Jury
M. Hassan HAMMOURI
M. Georges BASTIN
M. Witold RESPONDEK
M. Emmanuel TRELAT
M. Marc WEINBERG
Président
Rapporteur
Rapporteur
Examinateur
Examinateur
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. ) 0 -" ) &= 3= 4 ) = /# P + ) 9
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S T 8& " ) )
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-" H & / & 5 ) S T : # [email protected] -" 5 P / + . & + ) & + # ) 0 & & . , # + & X + ./& & X 6 , # , & /& : 8 & '
: # ! ) 7
W x1 x2 7 + ) [email protected]
-"
W y1 y2 7 & ) [email protected] -"
W P = P1 + P2 7 ( & 0 ( + P1 P2 [email protected] -"
! 0 7
W x1 x2 & ) ( F C31 F C32 ) [email protected] -"
W
y2 & ) ( F H21 F H22 /& [email protected] -"
W P & ( F Cata [email protected] ) ) = y1
+.# .#
( & Y 3 S T4 / + & / ) ( 0 7 ( 8 & 9 . / ( & , / / ) 7 ( 0 F # F /) . ( O + 0 ) & ) ;
- '
) ( @ & < ) 0 X
& ( / & ( . ( / 1 MP i
MPP i
MH2Ri
MCatai
Mtoti
F C3i
F H2i
Fi,sortie
F Cata
F H20i
xi
yi
Pi
P
Ai
qi
Ji
Q̇i
Ti
Te
∆Ti
V
ρP
ρPP
∆H
CpP
CpPP
CpH20
hP
hPP
Ri
Ri
/& Ri
Ri
Ri
1 ( Ri
1 ( /& Ri
1 ( 0 Ri
1 ( 1 ( / 5 Ri
$ Ri
$ /& Ri
2 Ri
2 ) Ri
0 /& Ri
Ri
& ) 5 Ri
2 Ri
2 / 1F / + / 5 I 1 1 : 8 8 8 /
: : W 7
d
MPP 1
dt
d
MP 1
dt
d
MPP 2
dt
d
MP 2
dt
= P1 − F1,sortie x1
[email protected]
= F C31 − F1,sortie (1 − x1 ) − P1
3"4
= P2 + x1 F1,sortie − F2,sortie x2
3?4
= F C32 + (1 − x1 )F1,sortie − F2,sortie (1 − x2 ) − P2
3K4
W 7
d
MCata1
MCata1 = F Cata −
F1,sortie
dt
Mtot1
d
MCata1
MCata2
MCata2 =
F1,sortie −
F2,sortie
dt
Mtot1
Mtot2
3!4
34
W /& 7
d
MH2R1 = F H21 −
dt
d
MH2R2 = F H22 +
dt
MH2R1
F1,sortie
Mtot1
MH2R1
MH2R2
F1,sortie −
F2,sortie
Mtot1
Mtot2
34
3J4
/ ( 0 # 0 / ( F C31 F C32 ( F1,sortie F2,sortie # + ) # # ( 7
V =
MP 1 MPP 1
+
ρP
ρPP
3L4
) && /& : ) 3L4
( ( 7
V̇ = 0
˙
M˙P 1 MPP
1
+
ρP
ρPP
F C31 − F1,sortie (1 − x1 ) − P1 P1 − F1,sortie x1
=
+
ρP
ρPP
=
, O & & 7
F1,sortie =
F C31
+ P1
ρP
1
ρPP
1
−
ρP
1
1
ρP
+ x1
1
ρPP
−
1
ρP
.
) 9 ( ( 7
F2,sortie
F C32 + F1,sortie
=
+ (P2 + x1 F1,sortie )
ρP
1
ρPP
1
−
ρP
1
ρP
1
+ x2
1
ρPP
−
1
ρP
.
# P1 P2 ( ( & 5 /) ) . ( / & / 7
1 9'!
d
(MP 1 hP (T1 ) + MPP 1 hPP (T1 )) = F C31 hP (T e)
dt
− F1,sortie (1 − x1 )hP (T1 )
− F1,sortie x1 hPP (T1 ) + J1 − Q̇1 .
&& / & / & '
d
d
d
(MP 1 hP (T1 ) + MPP 1 hPP (T1 )) = hP (T1 ) MP 1 + hPP (T1 ) MPP 1 .
dt
dt
dt
( [email protected] 3"4
(F C31 − P1 − (1 − x1 )F1,sortie )hP (T1 ) + (P1 − x1 F1,sortie )hPP (T1 )
= F C31 hP (T e) − F1,sortie (1 − x1 )hP (T1 ) − F1,sortie x1 hPP (T1 )
+ J1 − Q̇1 .
8 0
P1 (hPP (T1 ) − hP (T1 )) = F C31 (hP (T e) − hP (T 1)) + J1 − Q̇1 .
( /# P1 =
P1
7
F C31 CpP (T1 − Te ) + Q̇1 − J1
∆H
[email protected]
M & 0
P2 =
F C32 CpP (T2 − Te ) + F1,sortie (x1 CpPP + (1 − x1 )CpP )(T2 − T1 ) + Q̇2 − J2
∆H
[email protected]@4
( x1! x2! y1 y2 > . x1 x2 y1 y2 & ( # x1 O ) 0 8 7
3 .# )
MPP 1 = x1 Mtot1
Mtot1 = MP 1 + MPP 1
. /& ( 7
&&( ) d
d
MPP 1 = x1 Mtot1 + Mtot1 ẋ1
dt
dt
P1 − F1,sortie x1 = x1 (F C31 − F1,sortie ) + Mtot1 ẋ1
/P
1
ẋ1 =
V
1
+ x1
ρP
1
ρPP
1
−
ρP
− F C31 x1 + P1
Mtot1 =
P >
1
ρP
V
+ x1
1
ρPP
−
1
ρP
.
( ) 8 ) 7
P1 = A1 MCata1
P2 = A2 MCata2
P /) A1 A2 0 ( / 0 7
P1 P A1 =
A2 = 2
.
MCata1
MCata2
( P = P1 +P2 ) && /) ) 3!4 34 7
d
d
MCata1 + A2 MCata2
dt
dt
MCata1
MCata2
MCata1
= A1 F Cata −
F1,sortie + A2
F1,sortie −
F2,sortie .
Mtot1
Mtot1
Mtot2
Ṗ = A1
" >
ẋ1
ẋ2
ẏ1
ẏ2
d
MCata1
dt
d
MCata2
dt
Ṗ
$ ) 7
1
=
[email protected]"4
− F C31 x1 + P1
Mtot1
1
=
[email protected]?4
− (F C32 + F1,sortie )x2 + P2 + x1 F1,sortie
Mtot2
1
=
[email protected]
F H21 − F C31 y1
Mtot1
1
=
[email protected]!4
F H22 − F C32 y2 + F1,sortie (y1 − y2 )
Mtot2
MCata1
= F Cata −
F1,sortie
[email protected]
Mtot1
MCata1
MCata2
=
F1,sortie −
F2,sortie
[email protected] 4
Mtot1
Mtot2
MCata1
MCata2
MCata1
= A1 F Cata −
F1,sortie + A2
F1,sortie −
F2,sortie .
Mtot1
Mtot1
Mtot2
[email protected]
. )( 8 7
Mtot1 =
Mtot2 =
F1,sortie =
1
ρP
1
ρP
+ x1
ρPP
V
+ x2
1
1
ρPP
−
1
ρP
−
1
ρP
1
+ P1 ρPP
− ρ1P
1
1
+ x1 ρPP − ρP
F C31
ρP
1
ρP
F2,sortie
V
F C32 + F1,sortie
=
+ (P2 + x1 F1,sortie )
ρP
1
ρPP
1
−
ρP
1
ρP
1
+ x2
1
ρPP
−
1
ρP
.
! " #$ % &'
) : G1 2'$, / ) 7 ! <2 K <2
"@ . / ( ( 3 14
P / . F ) " ) 7
W / W . + ) 7 /( W , /& / 8 / /(5+ 5 3 & % 4 ) )
() % ( )( 8 + 7 # E /0 5 / & 9 7 ( ) & 0 3) $& "4 / 9 ( /() / & " : ( / % 0 " + 9
) + + 1 "@ ) / "" )( + ( % ? ( 3
# ''
./ : 0 ) G1 0 / " / /# # ) F + . /( K 5 % ) / /# & 1 #
7 #
. $& " + . / / ( ( + " G" ) K 6 &= 3'F *=4
9 : # 5 P / + 9#'
. & /( / /# ) + ) '
: $ () *
& + # ) 0 & & . , # + + / /# : 8 & # K ) 7
W x 7 # + W yC2 yC6 7 + ) / /# W P 7 ( K 0 7
W x & ( /( F C4
W yC2 yC6 & ) (
/# W P & ( F Cata
+.# F C2
F C6
/ .#
( & / + & / ) ( 0 7
( 8 & 9 . / ( & , / / ) 7 ( 0 F # F / /) . ( O + 0 ) & ) ;
- '
) ( @ & < 1 ) 0 X
& x yC2 yC6 P x xC2 xC6 P ( / &
. ( / MC2
MC4
MC6
MP E
MCata
Mtot
Fe
F C2
F C4
F C6
F Cata
Fsortie
F H20
x
xC2
xC4
xC6
yC2
yC6
P
PC2
PC6
A
J
Q̇
T
Te
∆T
V
ρM el
ρP E
∆H
CpM el
CpP E
CpH20
hC2
hC4
hC6
hP E
/ /( /# 1 ( 0 / 1 ( / 1 ( /( 1 ( /# 1 ( 1 ( 0 1 ( / 5 $ $ / $ /( $ /# $ / $ /# 2 2 / 2 /# ) & ) 5 2 2 / + ) 1F / + / 5 I 1 1 : 8 8 8 /
: / : /( : /# : $ *
W + ) 7
d
MP E
dt
d
MC4
dt
d
MC2
dt
d
MC6
dt
W 7
= P − xFsortie
[email protected]
= F C4 − xC4 Fsortie
3"4
= F C2 − xC2 Fsortie − PC2
3"@4
= F C6 − xC6 Fsortie − PC6
3""4
d
MCata
MCata = F Cata −
Fsortie
dt
Mtot
3"?4
( Fsortie # + ) && & / ρM el / ) /# +H ) 7
V =
: ) 3"K4 ( MM el MP E
+
ρM el
ρP E
3"K4
( 7
V̇ = 0
˙
MM
M˙P E
el
+
ρM el
ρP E
F e − (1 − x)Fsortie − P
P − xFsortie
=
+
ρM el
ρP E
=
, O & & 7
Fsortie =
Fe
+P
ρM el
1
ρP E
−
1
ρM el
1
ρM el
+x
1
1
ρP E
−
1
.
ρM el
# P (
( & 5 1 9'!
# P >
. ( / & / 7
d
MC2 hC2 (T ) + MC4 hC4 (T ) + MC6 hC6 (T ) + MP E hP E (T )
dt
= F C2 hC2 (Te ) + F C4 hC4 (Te ) + F C6 hC6 (Te )
− xC2 Fsortie hC2 (T ) − xC4 Fsortie hC4 (T ) − xC6 Fsortie hC6 (T )
− x Fsortie hP E (T ) + J − F H2O CpH20 ∆T
&& / & / & d
MC2 hC2 (T ) + MC4 hC4 (T ) + MC6 hC6 (T ) + MP E hP E (T )
dt
d
d
d
d
= hC2 (T ) MC2 + hC4 (T ) MC4 + hC6 (T ) MC6 + hP E (T ) MP E
dt
dt
dt
dt
'
( [email protected] 0 3""4 7
F C2 − xC2 Fsortie − PC2 hC2 (T ) + F C4 − xC4 Fsortie hC4 (T )
+ F C6 − xC6 Fsortie − PC6 hC6 (T ) + P − xFsortie hP E (T )
= F C2 hC2 (Te ) + F C4 hC4 (Te ) + F C6 hC6 (Te )
− xC2 Fsortie hC2 (T ) − xC4 Fsortie hC4 (T ) − xC6 Fsortie hC6 (T )
− x Fsortie hP E (T ) + J − F H2O CpH20 ∆T
8 0
P hP E (T ) − PC2 hC2 (T ) − PC6 hC6 (T )
= F C2 hC2 (Te ) − hC2 (T ) + F C4 hC4 (Te ) − hC4 (T )
+ F C6 hC6 (Te ) − hC6 (T )
+ J − F H2O CpH20 ∆T
−P ∆H = F C2 CpC2 + F C4 CpC4 + F C6 CpC6 (Te − T )
+ J − F H2O CpH20 ∆T
1/P /# 7
F C2 CpC2 + F C4 CpC4 + F C6 CpC6 (Te − T ) + F H2O CpH20 ∆T − J
P =
∆H
PC2
PC6 > PC2 PC6 )
# # ( # / /# # P # ) +
7
PC2 = S
) "T
PC6 = S
) T
xmol
C2
mol
mol
xC2 + xmol
C4 + xC6
xmol
C6
MCata mol
mol
xC2 + xmol
C4 + xC6
MCata
) 7 ( & # O 7 /) 0 ) ./# ST ) ) + ) / ) 3PC2 PC64 7
P = PC2 + PC6
S ) T xmol
C6
PC6 = PC2
S ) "T xmol
C2
( 7
xC2
P
xC2 + kxC6
kxC6
=
P
xC2 + kxC6
PC2 =
PC6
P k 7
S ) T " 1 S )
k=
=
S ) "T 3 S )
) O k > 1 /# + +
T
"T
+ ( / & 3 .# x! x x
> . x xC2 xC6 & ( # x O ) 0 8 7
C2
C6
MP E = x Mtot
. )
Mtot = MP E + MC2 + MC4 + MC6
&&( ) ( 7
d
d
MP E = x Mtot + Mtot ẋ
dt
dt
P − xFsortie = x(F e − Fsortie ) + Mtot ẋ
/P
)
1
ẋ =
Mtot
Mtot =
P >
1
ρM el
− xF e + P
+x
V
1
ρP E
−
1
.
ρM el
( ) 8 ) 7
P = A MCata
P /) A 0 ( / 0 7
P A=
MCata
.
( P ) && /) ) 3"?4 7
d
Ṗ = A MCata
dt
MCata
= A F Cata −
Fsortie .
Mtot
" >
$ ) 7
1
− xF e + P
Mtot
1
xC2
ẋC2 =
P
− xC2 F e + F C2 −
Mtot
xC2 + kxC6
1
kxC6
ẋC6 =
P
− xC6 F e + F C6 −
Mtot
xC2 + kxC6
d
MCata
MCata = F Cata −
Fsortie
dt
Mtot
MCata
Ṗ = A F Cata −
Fsortie
Mtot
xC2
yC2 =
1−x
xC6
yC6 =
1−x
ẋ =
3"!4
3"4
3" 4
3"J4
3"L4
3?4
[email protected]
. )( 8 7
Mtot =
1
ρM el
Fsortie =
+x
Fe
ρM el
+P
1
+x
ρM el
V
1
ρP E
1
ρP E
1
ρP E
−
−
−
1
ρM el
1
ρM el
1
ρM el
F C2 CpC2 + F C4 CpC4 + F C6 CpC6 (Te − T ) + F H2O CpH20 ∆T − J
P =
A=
P
MCata
∆H
.
1 ) /() % ) )
" : ./() / + 7 " : % & ( & 7 # ) O ( ) " : [email protected] # ) /() % ?" ??
3#/ -
& F " : ) ( . ) / & / ) # 0 & (F ) F F & ( 0 ) ( F # F ) 0 +
# 0 / # #
, ( ) 0 ) ( , O ( ) # ( / ) O ) /) 0 #
# W ( W W ( 5 ) () & F ( 7 # " : # ) B: C
$ > " ) . / / & 7
= -
1
ẋ1 =
− F C31 x1 + P1
Mtot1
1
ẏ1 =
F H21 − F C31 y1
Mtot1
d
MCata1
MCata1 = F Cata −
F1,sortie
dt
Mtot1
)
Mtot1 =
F1,sortie =
P1
1
ρP
V
+ x1
ρPP
−
1
1
ρP
1
ρP
+ P1 ρPP −
.
1
+ x1 ρPP
− ρ1P
F C31
ρP
1
ρP
1
& )(
) # x1 7
ξ1 = log
1
+ x1
ρP
1
ρPP
1
−
ρP
. 7
1
1
˙ξ1 = 1 − F C31 exp(ξ1 ) − 1 + P1
−
V
ρP
ρPP
ρP
exp(ξ1 )
ẏ1 =
F H21 − F C31 y1
V
1
d
MCata1 F C31
1
+ P1
−
MCata1 = F Cata −
dt
V
ρP
ρPP
ρP
x1 =
exp(ξ1 ) −
1
ρPP
−
1
ρP
1
ρP
Mtot1 = V exp(−ξ1 )
F1,sortie = exp(−ξ1 )
F C31
+ P1
ρP
1
1
−
.
ρPP
ρP
. E( ) 7

1)
−F C31 exp(ξ
0
0
V
 exp(ξ )
1
1)

0
F H21 − F C31 y1
−F C31 exp(ξ
 V
V
J =


1
1
+
P
0
0
− V1 FρC3
1 ρPP −
P
) 7

1)
−F C31 exp(ξ
V


 exp(ξ1 )

F H21 − F C31 y1
S =  2V



0
exp(ξ1 )
2V

1
ρP







F H21 − F C31 y1
0
1)
−F C31 exp(ξ
V
0
0
− V1
F C31
ρP
+ P1
1
ρPP
−
1
ρP








: S 8 &) ) + 7

1)
−F C31 exp(ξ
V

D = det 
 exp(ξ1 )
2V
F H21 − F C31 y1
'
D=
exp(ξ1 )
2V
exp(2ξ1 )
(F C31 )2
V2
1
1−
4

F H21 − F C31 y1 


exp(ξ1 )
−F C31 V
F H21
− y1
F C31
2 1 D > 0 0 ≤ FF H2
≤ 1 0 ≤ y1 ≤ 1 C3
R1 . 8 7
1
1
ds2 = (dξ1 )2 + (dy1 )2 + (dMCata1 )2
2
(dx1 )2
1
1
=
−
ρPP
ρP
1
1
+
x
1 ρPP −
ρP
% > :
1
ρP
2
2
2 + (dy1 ) + (dMCata1 )
/ / 3"!4 0 3"J4 7
& . 1
1
1
1
ẋ =
+x
−
− xF e + P
V ρM el
ρP E ρM el
1
1
1
1
xC2
ẋC2 =
+x
−
P
− xC2 F e + F C2 −
V ρM el
ρP E ρM el
xC2 + kxC6
1
1
1
1
kxC6
ẋC6 =
+x
−
P
− xC6 F e + F C6 −
V ρM el
ρP E ρM el
xC2 + kxC6
1
d
MCata F e
1
+P
−
MCata = F Cata −
dt
V
ρM el
ρP E
ρM el
& )( ) 7
ξ = log
1
ρM el
ψ = xC2 + xC6
xC6 = xC6
MCata = MCata
+x
1
ρP E
−
1
ρM el
. ) 7
1
1
˙ξ = 1 − F e exp(ξ) − 1
−
+P
V
ρM el
ρP E
ρM el
exp(ξ)
ψ̇ =
− ψF e + (F C2 + F C6) − P
V
exp(ξ)
kxC6
ẋC6 =
P
− xC6 F e + F C6 −
V
ψ + (k − 1)xC6
1
d
MCata F e
1
+P
−
MCata = F Cata −
dt
V
ρM el
ρP E ρM el
. E( 
−F e exp(ξ)
0
V

−F e exp(ξ)
α1

V


J =
α3
α2


0
0
)
0
0
exp(ξ)
V
/ 7
− Fe −
kψ
(ψ+(k−1)xC6 )2
0

0
0
P
0
− V1
Fe
ρM el
+P
1
ρP E
−
1
ρM el
exp(ξ)
α1 =
− ψF e + (F C2 + F C6) − P
V
exp(ξ)
kxC6
P
α2 =
− xC6 F e + F C6 −
V
ψ + (k − 1)xC6
kxC6
exp(ξ)
P
α3 =
V (ψ + (k − 1)xC6 )2
E( & ) & # &+ 7 + 5 &) & (α1 , α2 , α3 )
( + ' ) M / J T M + M J 8 &) AT & A







) ) () " : ' ( W " (x̂1, x̂2 , ŷ1 , ŷ2 , M̂Cata1 , M̂Cata2 ) / / 7
3 .# 9 #
1
− F C31 x̂1 + P1
Mtot1
1
x̂˙ 2 =
− (F C32 + F1,sortie )x̂2 + P2 + x̂1 F1,sortie
Mtot2
˙ŷ1 = 1
F H21 − F C31 ŷ1
Mtot1
˙ŷ2 = 1
F H22 − F C32 ŷ2 + F1,sortie (ŷ1 − ŷ2 )
Mtot2
x̂˙ 1 =
M̂Cata1
d
M̂Cata1 = F Cata −
F1,sortie
dt
Mtot1
M̂Cata1
M̂Cata2
d
F1,sortie −
F2,sortie
M̂Cata2 =
dt
Mtot1
Mtot2
)
Mtot1 =
Mtot2 =
F1,sortie =
1
ρP
1
ρP
+ x̂1
ρPP
V
+ x̂2
1
1
ρPP
−
1
ρP
−
1
ρP
1
+ P1 ρPP
− ρ1P
1
1
+ x̂1 ρPP − ρP
F C31
ρP
1
ρP
F2,sortie
V
F C32 + F1,sortie
=
+ (P2 + x̂1 F1,sortie )
ρP
W : 1
ρPP
1
−
ρP
1
ρP
1
+ x̂2
(x̂, x̂C2 , x̂C6 , M̂Cata , ŷC2 , ŷC6 ) / / 7
1
ρPP
−
1
ρP
.
1
− x̂F e + P
Mtot
1
x̂C2
=
P
− x̂C2 F e + F C2 −
Mtot
x̂C2 + kx̂C6
1
kx̂C6
=
P
− x̂C6 F e + F C6 −
Mtot
x̂C2 + kx̂C6
x̂˙ =
x̂˙ C2
x̂˙ C6
M̂Cata
d
M̂Cata = F Cata −
Fsortie
dt
Mtot
x̂C2
ŷC2 =
1 − x̂
x̂C6
ŷC6 =
1 − x̂
)
Mtot =
1
ρM el
Fsortie =
$' # (
# ! + x̂
Fe
ρM el
+P
1
+ x̂
ρM el
V
1
ρP E
1
ρP E
1
ρP E
−
−
−
(( (+
1
ρM el
1
ρM el
1
ρM el
() # Densité R1
mesure
estimation
Densité R2
mesure
estimation
$+ , - . /
Offgaz C2
yC2 mesuré
yC2 estimé
Offgaz C6
yC6 mesuré
yC6 estimé
$+ , 0&1 $ - 2 /
1 $& @ / " & /# / ( ( / ' 9 / ) 7 0 & / ,1 1 $& " ( / / )
) ) 0 /F&= . + 0 / & 7 ( #
. ) 8 % )& ,#
. # & # % : /# " &
) F 9 + 7
ẋ1
ẋ2
ẏ1
ẏ2
Ṗ
=
=
=
=
=
fx1 (X)
fx2 (X, F C31 )
fy1 (X, F C31 )
fy2 (X, F C31 , F C32 )
fP (X, F C31 , F C32 )
+
+
+
+
+
gx1 (X)F C31
gx2 (X)F C32
gy1 (X)F H21
gy2 (X)F H22
gP (X)F Cata.
) X = (x1 , x2 , y1 , y2 , P ) . + g + / ' # 3) B5C4 ξ ∈ {x1 , x2 , y1 , y2 , P } O 0 uξ 7
ξ˙ = fξ + gξ uξ
7
1
uξ =
gξ
− fξ + Kp (ξr − ξ) + Ki
(ξr − ξ)
+ gξ / ξr & & 3 + 4 0 ) ξ (Kp, Ki ) & & > . ) #
x1 x2 ) /() .
D & F # X ) 7
D=
1
ρP
+X
1
.
1
ρPP
−
1
ρP
$& @ # ( ) 0 + # 0 ) ) & # x ∈ {x1, x2 } 7
1
ux =
gx
− fx + Kp (xr − x ) + Ki
(xr − x ) .
' 8 ) & x 7 & 8 ( + + 3 ) 4 . & % / )& x ) xr % & /) ( 1 ) + /() $' # ,#
(( (
() % " ./ % 0 ) / 9 +
Densité R1 et R2
x1
x1r
x2
x2r
FC3 R1 et FC3 R2
R1
R2
$+ & - /
> 1 $& ?
# & # 5 5 & 8#
: 8 ) /O % & O (( ( Production
P
Pr
FCata
$+ & - 3 /
> 1 $& K
# & 5 ' () & & 0 & # 7 ( + 0 = /# 1 $& ! & ((
( / ? / + # ) &" > 1 $& # & /& / 5 0 3 & 4 . & & + &) + : F
Densité R1
x1
x1r
FC3 R1
$+ , - 3 /
/& X /) 7 ( )
Hydrogène R1
y1
y1r
FH2 R1
$+ & ( & - . /
1 # > '
? @*%<
) # 6 0 # (5+ % 7 /) F 0 / (
6 ) 8 0 ) / & / ( & & & $ & 5 / & ( + N / )
# % # -: # / / 0 # )( 0 7 ) # / # & / : ) ) / + # ./ / & & # / ' " > 8 / [email protected]"4 # " ) / # ) / ψP ψPP
R1 ' /( 0 & & 7
ψPP
) αP αPP # +
. ( / ) )( 7
ψP = αP MP
= αPP MPP
d
ψP = FP − (1 − χ)Fsortie − PP
dt
d
ψPP = PPP − χFsortie .
dt
Fsortie & ) ( / # 0 ψP ψP P . + χ 8 7
FP
χ=
ψPP
.
ψP + ψPP
PP P + : @ ( ) +
χ 7
PP
1
χ̇ =
V
1
+χ
µP
1
1
−
µPP
µP
(−χ(FP + PPP − PP ) + PPP )
3?"4
) µP = αP ρP µPP = αPP ρPP ( / 8 1 P / ) ) & ) ( ) / [email protected]"4 / αP = αPP = 1 PPP = PP 1 / 8
./ 3?"4 ) / & (R+∗ , ×)
. / + & & )( ψP ψPP F ) 3 (4 ( # 9 ) 5 & > ) / ) & 0 + 0 +
& : F MPP
x=
MP + MPP
ψP = αP MP
ψPP = αPP MPP
χ=
ψPP
αPP MPP
=
=
ψP + ψPP
αP MP + αPP MPP
x+
x
.
(1 − x)
αP
αPP
/ / () + / (x− x̂) / x̂ x
/ ) 3?"4 : F (x− x̂) & / ) 7
χ+
(χ − χ̂)
χ
−
(1 − χ) χ̂ +
αPP
αP
8 χ̂ 7
χ̂ =
x̂ +
χ̂
(1 − χ̂)
αPP
αP
x̂
.
(1 − x̂)
αP
αPP
/()
+ ) 7
1
χ̂˙ =
V
1
1
1
+ χ̂
−
µP
µPP
µP
1 − χ̂ χ
−χ̂(FP + PPP − PP ) + PPP − k H
χ̂ 1 − χ
3??4
& /() G + P k & 3C 1 # 4 / @ χ & / χ () ) / / 3?"4 / & ' F / ) # < & ) BC 7
1 − x̂ x
1 − χ̂ χ
=
.
χ̂ 1 − χ
x̂ 1 − x
& / & k + ( () )& , /() ) 7
1
1 − χ̂ χ
1
k
1
+ χ̂
−
2 = −
H
.
V
µP
µPP
µP
1
+ χ̂
µP
1
1
−
µPP
µP
χ̂
1−χ
' > () )& 3) 4 7 χ 3??4 ( )& & )(
ξˆ = log
0 1
˙
ξˆ =
V
E(
J
1
1
1
ˆ
−
(FP + PPP − PP ) + PPP
− exp(ξ) −
µP
µPP
µP
1
− µ1P
χ
1
1
µPP
−1
−
+ kH
ˆ − 1
1−χ
µPP
µP
exp(ξ)
7
µP
2
1−χ̂ χ
H
χ̂ 1−χ
1
χ
1
1
ˆ
J = − exp(ξ) (FP + PPP − PP ) + k −
.
2
V
µPP
µP
1−χ
1
ˆ
exp(ξ) − µP
, &+ W (FP + PPP − PP ) + 7 FP − PP = αP (Fmassique − Pmassique ) > 0 3 + 4 PP P ≥ 0
W O & k > 0
W ) H + H )
) /() 3??4
'2/
# # ' 9# 9# ) 9! # ' ' %%: A 9 / #
' '
#
/
1 / / ) ) / + F # ) ) ) () ) ) 0 ) 9 . ) = & /( K ) / & + 7 & )
: ! .& & > ẋ = f (x) )
y = h(x) ) / & + G 8 () ) + x̂˙ = fˆ(x̂, y) ) y = h(x) x̂ / x M 8 ) & + & / y 0 ) & / ) /() x̂˙ = fˆ(x̂, y) 7 ( ) ) + / / ) ) ( . )& /() & ( ) )& /() ) ) / D & 0 / G E( ∂ fˆ/∂ x̂ 8 &) 3 4
" " ( > 1 ! ) /() 7 .& & 3 8& 4 ( / () / 3 )4 ) 0
& 8& . & () /: .& & . / - 8 / & / 8
& ) )& E( /() 8 &) 3 4 8 / 0 / & /() 1/
) () 9 # 8& ) ( / F # ( - ) 9 & F ( / & 0 / " M # ( 5 &) /O / # ( + && . + /(5 / BC
%# 1 # / / () . ( / / / ) ) ( / & # )& () ) ) ) )& &( /( /# / + & ) V
) # c1 c2 y 7
d
c1 = F (cin
1 − c1 )
dt
d
V c2 = F (cin
2 − c2 )
dt
c1
y=
.
c1 + c2
V
3?K4
in
( / (cin
1 , c2 ) ) / & ( F
+ (c1 , c2 ) # g/L mol/L & 8 ) M1 M2 0 / C1 = M1c1 C2 = M2 c2 F
C1in = M1 cin
1
C2in = M2cin
2 & 7
d
C1 = F (C1in − C1 )
dt
d
V C2 = F (C2in − C2 ).
dt
y
V
0 /
7
Y = C1 /(C1 + C2 ) = y/(y +
0
+ &
M2
(1 − y)).
M1
/() ) P ĉ1 ĉ2 ŷ ) c1 c2 y
k & /() 7
ŷ 1 − y
d
in
V ĉ1 = F (c1 − ĉ1 ) − kĉ1 log
dt
1 − ŷ y
1 − ŷ y
d
in
V ĉ2 = F (c2 − ĉ2 ) − kĉ2 log
dt
ŷ 1 − y
ĉ1
ŷ =
.
ĉ1 + ĉ2
3?!4
M ) / 9
& / / ) Ĉ1 C2
1−Y
ĉ1 c2
ŷ 1 − y
=
.
=
=
ĉ2 c1
1 − ŷ y
Ĉ2 C1
1 − Ŷ Y
Ŷ
ŷ 1−y
0 log 1−ŷ
/ + (ŷ − y) y
/ ) () ) : F / (ŷ − y)
/ # + 7
/ ) ) / 3 B6C4 3 BC4 / (ŷ − y) ) / /
() ) / ) ) K? ( # 3?K4 () &( )& k + 7 ) )
/() ds2 = (dc
+
(c )
(dc )
BC
(c )
1
2
2
2
2
1
2
2
& 1 [email protected] & + 8 () ) () / / ) G ) ) 9
K" ) ( / # ) # /(
) ) 7 K? KK K!
M / ) / # P / ( / 6 ) .
& /() ) ) ) / & + 7 F K
/ +H BC P ( < &
) # &# & + BC ) 8 / & ) / & +
8 /() & ) /
& /5+ S & T S T ./(5+ ) ) /() , ( 8 ) & +( 3C k , k ∈ N # 4
' ( X 9 7 # ( 8 Rn ( Rn ) ' / 0 / ( G(X ) (5) X 9 7 + ( (G(X ), o) & ' G(X ) & / ( X / ( X /) 0 & G(X ) # X = Rn & + )( X 9
&# ) *
> ! " # $ ! B 9# !# C
! X
ϕ : G −→ G(X )
% ! % % (G, ·) & X ' ( $% g G X ϕg ϕg1 ·g2 (x) = ϕg1 ◦ ϕg2 (x) = ϕg1 ϕg2 (x) ∀g1 , g2 ∈ G,
' + 7
∀x ∈ X .
) / & ∀x ∈ X , ∀g ∈ G, ϕg (x) = g.x = w(g, x).
. # # K? KK / 9 8 M & G(X ) F + 0 +
/ ( 8 G & P 8 & G(X ) ' & 8 3 .#
GD
& ẋ = f (x)
P / x 0 ( ) X Rn G
& X X = ϕg (x),
P
3?4
& +
g ∈ G,
F 1 G & + X & 8 x ∈ X / ϕg (x) /9 8 g ∈ G / : 0 g & r & G
ϕg
) $
B .#
g ∈ G GD ∀x ∈ X
> ) % 3?4 C
f (ϕg (x)) =
∂ϕg
(x) · f (x).
∂x
G
' G & . 8 ) 0 g ∈ G ) Ẋ = f (X) / 0 & / & + 3?K4 & + G r = 2 #
+ M1 > 0 M2 > 0 (C1 , C2 ) = ϕ(M1 ,M2 ) (c1 , c2 ) = (M1 c1 , M2 c2 )
P ) ϕ(M ,M ) / G # (M1 , M2 ) in
in
in
in
in
(cin
1 , c2 ) + (C1 , C2 ) = (M1 c1 , M2 c2 ) in
) 0 / 5 (cin
# 1 , c2 ) 7 /
d in
d in
dt c1 = 0 dt c2 = 0 /() % ( + 8 & + & / ) 8 "
1
2
GD & h : x → y = h(x) X . y 8 Rm 3dim y = m4
#
) +
( ) Y GD C > % G
y = h(x) * G +( )&
B #
%
ẋ = f (x)
G x d
x̂ = fˆ(x̂, h(x))
dt
g ∈ G
x̂ ∂ϕg
(x̂) · fˆ(x̂, h(x)) = fˆ(ϕg (x̂), h(ϕg (x))).
∂x
8 & 8 /() & ) X̂ = ϕg (x̂) 3 #
d
X̂ = fˆ(X̂, h(X))
dt
x̂ ∈ X 4 X = ϕg (x) ./()
/() G ) 1 3?!4 /() & 8 lim x̂(t) = x(t).
G, -,
t−→+∞
M 0 # & 7
W # ( p
) wi (x) i = 1, ..., p 8 / / ) ) 0 ) G7
∂ϕg
(x) · wi (x).
∂x
+ Ji(x̂, h(x)) i ∈ {1, ..., p}
∀g ∈ G, ∀i ∈ {1, ..., p} wi (ϕg (x)) =
W
) 8 g ∈ G x̂ x
Ji (ϕg (x̂), h(ϕg (x))) = Ji (x̂, h(x)),
i ∈ {1, ..., p}.
) d
x̂ = f (x̂) +
(Ji (x̂, y) − Ji (x̂, ŷ))wi (x̂)
dt
i=1
p
3? 4
() ) / 8 /) # )& x̂ ) x ) / ) +# 9 ) + ) ) ( + / / ) / & + ) .
> - % % !. ẋ = f (x) G ( y = h(x) m( & m " !. & x̂ y I(x̂, y) = (I1 (x̂, y), ..., Im (x̂, y)) % @ g ∈ G x̂ x
B+# 9 C
I(ϕg (x̂), h(ϕg (x))) = I(x̂, h(x))
" x̂ & y → I(x̂, y) # / $ & %
h(x) = h(x̂) I(x̂, h(x̂)) = 0.
I + ) I ) /( ( 8 ) ) / B" GD C > - % % !. ẋ = f (x)
G ( 0 Y y = h(x) G ) & G x
# Y & 1 g ∈ G 2 " g Y % h ◦ ϕg = g ◦ h(
X = ϕg (x) / Y = h(X) G ) & 8 Y
0 y . y 3?K4 G ϕ(M ,M ) 0
1
Y =
C1
M 1 c1
=
=
C1 + C2
M 1 c1 + M 2 c2
y+
2
y
M2
(1
M1
− y)
8 / G / 8 & 8 / G
/ / / h ) 9 ( h + G ) 7 / # 3?K4 / h(c1 , c2 ) = c1 + c2 8 G ) : F Y = C1 + C2 8 + y = c1 + c2 3 + P M1 = M2 4
+ y / (c1 , c2 ) 1/ ) /
& / ) / 3 Y 4 0 ) 3 y4
# " *
B+/ 9# 9 C
% G ẋ = f (x) - % x0 & !.
G
y = h(x)(
G g → ϕg (x)
! r = dim G g = r ≤ n = dim x( 0 x0 2 m = dim y " !. Ii (x̂, y) i = 1 . . . m % & (
. / G G & F) 3/ 0 & / ) 4 .
& . G & # )
h /
(# # + Ii '-
( / # ) 1 ) Σ 0 / 8 (x̂, y) ξ # (x̂, y) ./ G ( 8 y G ) g ∈ G
+ ) (x̂, y) → (ϕg (x̂), g (y)).
./ & G / & & (x0 , y) . 8( ξf ) 9 + 0
/ ( x̂ a = ψf (x̂, Ξf ) & 8 + Y = (y, a) 0 g ∈ G a / Y = (y, ψf (x̂, Ξf )) := Υ(y, x̂, Ξf ).
, ) Ξ0f m + Υ(y, x̂, Ξ0f )
) 1 & Υ
0 y # & 0 m '
I(y, x̂) = Υ(y, x̂, Ξ0f ) − Υ(ŷ, x̂, Ξ0f )
( / ) /# 3?K4 / / ) I(x̂, y) = log
+/ =
'
ŷ 1 − y
1 − ŷ y
#
.
.#
+ & ) V
) n (c1 , . . . , cn ) 7
-
V
d
ci = F (cin
i − ci ) + ri (c1 , . . . , cn )
dt
ci
, i = 1, ..., n
yi = n
h=1 ch
!
in
( / (cin
1 , . . . , cn ) / . n & ri (c1, . . . , cn ) 7 n + 9 & & + yi 0
/ (c1 , . . . , cn )
./# 3?K4 0 7 ) n = 2 r ≡ 0
F
: & / x = (c1, . . . , cn ) 8 X = (R+ )n ⊂ Rn . y = (y1 , . . . , yn ) ) Y = [0, 1]n ./ & & / 8 7
&# ϕ:
n
G = ((R+
−→ G(X )
∗ ) , .)
ϕg
g = (M1 , . . . , Mn ) −→
Z 0 0 Y ρg :
ϕg :
X
−→
X
(c1 , . . . , cn ) −→ (M1 c1 , . . . , Mn cn ).
7
Y
−→ Y
M n yn
nM1 y1
n
.
(y1 , . . . , yn ) −→
,
.
.
.
,
M i yi
M i yi
i=1
i=1
/ ) 8 " 9 ) 8 ) / & cin
i ri ) # ) / ) / & 8 & /() + ) & + & / ((ci), (yj )) . / ) 7
* Cio = Mi ci ,
i = 1, ..., n
Cio
,
ci
i = 1, ..., n
Mi =
P (Cio) + / & / Mi yi
Yi = n
,
h=1 Mh yh
i = 1, ..., n
( + ) ) 7
Hi (
c, y) =
Cio
y
ci i
n Cho ,
y
h=1 ch h
i = 1, ..., n
P c = (ci) y = (yi) M ( + ) #
) 3 4 7
Iij (ĉ, y) = log
yi yj
yj yi
/ / ) i j {1 . . . n}
= # /() / / ) Iij n & 8 # /
& & / / 7
vi (c1 , . . . , cn ) = ci
∂
,
∂ci
∀i ∈ {1, . . . , n}
) ) ( /() ) 3k > 0 &&4 7
d
yj
V ci =F (cin
i ) + ci ri (( )1≤j≤n ) − k ci log
i −c
dt
yi
ci
yi = n
h=1 ch
,
n
yi yh
yh yi
h=1,h=i
3?J4
∀i ∈ {1, . . . , n}
) / & ĉi = ci
) & dtd ci = dtd ci ) # /& ri 7
ri (c1 , . . . , cn ) = ci ri
c1
cn
,...,
ci
ci
! 9 #
& 7
ci )
ξi = log(
= ci ri
y1
yn
,...,
yi
yi
.
) )& i = 1, . . . , n
. /() ) 7
d
yj
V ξi =F (cin
)1≤j≤n ) − k
i exp(−ξi ) − 1) + ri ((
dt
yi
n
n
yh
(ξi − ξh ) + log(
)
yi
h=1,h=i
h=1,h=i
! " . E( 3 0 ξi 4 & 0 7

−F cin
1 exp(−ξ1 )
1 
A= 
0
V



 1 
+ 
0
V 

n )
−F cin
exp(−
ξ
n
−(n − 1)k
k
k
k
···
k
···
k
k
−(n − 1)k






. &) 7 −nk ) n − 1 ) ) & 0 @
. &) E / ( {ß1, . . . , ir } n
k ∈ J, cin
k = 0 B = (e1 , . . . , en ) ( R 8 &) & {ek , k ∈ J} ) A 8 &) /() &( 3 BC4 ) )& &( ./(
) 3?J4 () &( ) 7 ' /() (ξi ) ) : # E( , ) /() 3?J4 ) 0 ) 7
2
ds =
2
n dci
i=1
ci
=
n
(dξi )2
i=1
/ 0 D & & + & / / +/ E# D
' /# ) & 7
-
ẋ = u cos(θ)
u ) ẏ = u sin(θ)
u
θ̇ = tan(ϕ).
L
$& KK P = (x, y) ) ϕ / & ) /
( 1 ϕ
θ
L
y
x
# - $ ) 7
˙
P =uτ
u
τ˙ = tan(ϕ) ν
L
)
τ=
cos θ
sin θ
ν=
− sin θ
cos θ
.
&# ./ (x, y, θ) 8 /
X = R2 × S1 ' u ϕ , ) 7 / / & :3"4 / +H & / ./ & :3"4 / 7
ψ:
G = SE(2) −→ G(X )
ψg
g = (rotα , Ta ) −→
)
ψg :
X
−→
X
(P , τ ) −→ (rotα (P + a), rotα (τ )).
#
!
& / & α /& Ta ) a , /& / & + 0 ) 0 & ./ ) ( # $ rotα
= # ) /() ) BC P = (x, y)
0 τ θ . B7C () 8 D / 0 ) () / ) / & :3"4
I 0 /() BC , / ) 7
dˆ
tan ϕ ˆ u ˆ 1
ν + (λ − (P − Pr ))
λ=u
dt
L
a
a
1
ˆ
τˆ = λ − (P − Pr )
a
P νˆ ) 0 τˆ
, ( )( # λ = τ + a1 P P Pr
8 + . /() )& a
&+ () ( ) 7
W ./ ) # (P − Pr )
W ./ ) # + / ) $ 0 /# K? 5 / /() λ ) / & :3"4 λˆ / +/ -
# & ' ) A -: '1:D
-
7
' ẋ = Ax
y = Cx
) x ∈ Rn y ∈ Rp A ∈ Rn × Rn C ∈ Rp × Rn
./ x 8 X = Rn . y 8 Y = Rp ' / & + ) 7
&# ϕ : G = (R∗ , .) −→ G(X )
g=λ
−→
ϕg
Z 0 0 Y ϕg : X −→ X
x −→ λx.
7
ρg : Y −→ Y
y −→ λy.
/ / ) / & + & / X × Y . / ) 7
+# xo = λx̂,
x̂ ∈ X
y o = λŷ,
y∈Y
) (xo, yo ) ∈ X × Y + ( . λ = xx̂ i {1, · · · , n} : 5 ( + ) + ) 7
o
i
Hij (x̂, y) =
( / ) ) 7
ei,j (y, x̂) =
P ŷ 8 ŷ = C x̂
yj
x̂i
( yj
ŷ j
−
x̂i
x̂i
$ ' () ) n ) ) ) 7
= # vi(x̂) = x̂i
' ( /() 7
x̂˙ = Ax̂ +
∂
,
∂ x̂i
∀i ∈ {1, · · · , n}.
p
n
i=1
kij ei,j (y, x̂) vi (x̂)
j=1
ŷ = C x̂
7
x̂˙ = Ax̂ + K(y − ŷ)
ŷ = C x̂
) K = (kij ) ∈ Rn ×Rp & (A, C) ()( 3 & D 4 K 0 ( () )& / & & + ) / / () 6
- F' 9 . ( + / / ) . & /() ) /) ) [email protected] / ) G ) . & + O BC BC # 1 BC
) 7
6 #
# " $
GD
B- F' 9 # #
#C
- G ! " 2 ! X ( $ f # X G * $ ) + [v, f ] = 0
! # v ! (
F () ) # () 9
0 / : & /() + 7
6 - F' 9 # # #
ẋ
x̂˙
)
F1 (x, x̂) =
= F (x, x̂) = F1 (x, x̂) + F2 (x, x̂)
f (x)
f (x̂)
F2 (x, x̂) =
0
W (h(x), x̂)
.
' / & + G / X × X P 8 / (x, x̂) 7
w̃ : G × (X × X ) −→
X ×X
(g, x, x̂)
−→ (w(g, x), w(g, x̂))
P w & / 0 / G X . &
# / G X × X + 7
V (x, x̂) =
v(x)
v(x̂)
8 P v & 8 / G X . / ) /() / 7 & 8 V / G X × X ) F 8 /() ) 8
[V, F ] = 0
1 ) .
[V, F ] = [V, F1 ] + [V, F2 ].
∂
∂
+ v(x̂)
∂x
∂ x̂
∂
∂
+ f (x̂)
F1 = f (x)
∂x
∂ x̂
∂
F2 = W (h(x), x̂)
∂ x̂
V = v(x)
( ∂v(x) ∂
∂f (x)
− f (x)
[V, F1 ] = v(x)
∂x
∂x
∂x
∂v(x) ∂
∂f (x)
− f (x̂)
+ v(x̂)
∂ x̂
∂ x̂
∂x
∂v(x̂) ∂
∂f (x̂)
− f (x)
+ v(x)
∂x
∂x
∂ x̂
∂v(x̂) ∂
∂f (x̂)
− f (x̂)
.
+ v(x̂)
∂ x̂
∂ x̂
∂ x̂
' [V, F1 ] = 0 7 3 ) f / G X 4 # 3x x̂ # )( 4 1 [V, F ] = 0 0 [V, F2 ] = 0 ' ∂v(x̂) ∂
∂W (h(x), x̂)
− W (h(x), x̂)
[V, F2 ] = v(x̂)
∂ x̂
∂ x̂
∂ x̂
∂v(x) ∂
∂W (h(x), x̂) ∂
− W (h(x), x̂)
+ v(x)
∂x
∂ x̂
∂ x̂ ∂x
0 ) 7 3 8
/() ) 4
*
B- 9 # # #C
& ! G X × X ! # v & G X [v(x̂), W (h(x), x̂)]|x +
3 Lv h(x) ) . x (
h
v
∂W
Lv h(x) = 0
∂y
3
[, ]|x
!# % $ # & + & / () ( / / /() ' + ( ( 8 ( 9 () ) / ) / ) ( . )& ( ) 7 () ) )& ) ) ( ) )& &( & M ( # )& / () ) + / 8 D & / 0 + & . + /(5 / ( BC
%# . () & & 3) # B: C4 % 5 % 3) B8: : 7C4 1 # S [
3 /) 50 + K4 /() 7 .& & . /: .& &
F 7 # # / ( / 8& S [ & / ) 50 %
3+ & B65C4 / ( % 3+ & B5C4 ./() 0 ) ) 2 /
B5C / - / 8& 3 & ) ) ( ) B5: 66C4 8 / & 1 ) # %
/ 50 & - ) # %( B: 8C 8 5 B8: 7: C %
B: C ( B5: : : C
. ) )& ( # 7
# W ) ) )
/() 0 / E( ( W B: C ( / # / - 8 / 8& # & /() 0 <
B6: 6C
8 /# / 0 & 3/ 0 / &
+ # 4
1 8& : 3 ( 4 / & / q̈ = 0 P q : : & / / ∇q̇ q̇ = 0
P ∇ & - ' 8& q ' 0 ( q̂ v̂ q )
q̇ = v . q̈ = 0 , O /() . (&
) q̂˙ = v̂ − α(q̂ − q),
v̂˙ = −β(q̂ − q)
) α β # ) )& # 1 P ∇q̇ q̇ = 0 / q̂ − q / 7 & &q̂ F P F (q̂, q) & q̂ q ' 0 /
( ∇q̇ q̇ = 0 9
/() ) q̂˙ = v̂ − α &q̂ F, ∇q̂˙ v̂ = −β &q̂ F.
, ) 0 & q () ( 8 q̂ q &q̂ F 0 / & 0 q̂ ) v̂ 8 & ( t → q̂(t) ) ) ∇q̂˙ v̂
& ( 8 & ( / )& α β + , ( ( &) ( # 6 & 3) & & / ) ) ( &) B6C4 / ( / / # () )& F / /
( # &q̂ F, ∇q̂˙ v̂ = −β &q̂ F + R(v̂, &q̂ F )v̂
P R ( - R(v̂, &q̂ F )v̂ 0 &q̂ F 0 v̂ ( && & + ) 1 & 0 )& # F ( 3 & 4
' ) 0 / /& B: C α β +
() # )& 5
t → q(t) : F q̂ q ) /() / 7
q̂˙ = v̂ − α
∇q̂˙ ξ = ζ − αξ,
∇q̂˙ ζ = −βξ
P ξ = q̂ − q # 0 / : 3q̈ = 04 7
q̃˙ = ṽ − αq̃,
ṽ˙ = −β q̃
) q̃ = q̂ − q ṽ = v̂ − v . R = 0 ) ) ∇q̂˙ \ )
/ d/dt : ./5 / + # 8 & /() / 3) 3K44
& . [email protected] ) # 8 & 1 !" ) /() & / ) # + # 1 !? ) (
# 3 4 5 3 )& 4 /O ) !K 1 / # # " -
! ! '.# $ ' .& & ) ) 8& n / - 3) B5C4 . q ∈ M (q i )i=1...n . .& & 7
1
L(q, q̇) = gij (q)q̇ i q̇ j − U (q)
2
# P 8 ) g(q) = (gij (q))i=1...n,j=1...n 8 3 / 4 + U (q) / & . /: .& & d
dt
∂
∂
L = i L + ui (q, t),
i
∂ q̇
∂q
i = 1, ..., n
P u(q, t) = (ui (q, t))i=1,...,n + t q & 0
. + - /: .& & ∇q̇ q̇ = −&q U (q) + g(q)−1 u(q, t) ≡ S(q, t)
P ∇ &q g(q)−1 ) .)) / &
0 - / ) g(q)
q S(q, t) ) M 1 /: .& &
/ ) 7
3?L4
q̈ i = −Γijk (q)q̇ j q̇ k + S i (q, t)
P Γijk 3( F4 Γijk
1
= g il
2
∂glk ∂gjl ∂gjk
+ k −
∂q j
∂q
∂q l
) gil g−1 ) 7
X 0 + # & 8 9 +
- ∇q̇ v ) ) ) v & q̇ 1 ( P { }i & 8 i
{∇q̇ v}i = v̇ i + Γijk (q)v j q̇ k
= # -.#
1 8 () .& & 8 /: .& & 3 4 ) /() ) (5+ ) # ) 8 /() ) 8 #
( # . )& () !?
' / 3
/ 0 q 4 / ) 3 / 0 q̇i = vi 4 q̂ v̂ q ) v : 8 ) # / 7
∇q̂˙ q̂˙ + α &q̂ F (q̂, q) = T//q→q̂ S(q, t) − β &q̂ F (q̂, q)
3 .# 9 #
i
+ R (q̂˙ + α
&q̂ F (q̂, q)), &q̂ F (q̂, q)
(q̂˙ + α
&q̂ F (q̂, q))
/ P
W
W
q̂˙ = v̂ − α &q̂ F (q̂, q)
∇q̂˙ v̂ = T//q→q̂ S(q, t) − β &q̂ F (q̂, q) + R(v̂, &q̂ F (q̂, q))v̂
3K4
α β F (q̂, q) +
& q q̂ + ( 8 & q̂ q O W T//q→q̂ q ) q̂ & & 5& q 0 q̂
/ / & 0 q ) / & 0 q̂ F ( 8 q q̂ O W R ( - 1 (qi) /() / q̂˙i = v̂ i − α
&q̂ F (q̂, q)
i
v̂˙ i = −Γijk (q̂)v̂ j q̂˙k + T//q→q̂ S(q, t) − β
i
&q̂ F (q̂, q) + R(v̂, &q̂ F (q̂, q))v̂ .
() / # q 7 ∇ + F / T//q→q̂ R (5 0 - M . $& @ /() 3K4 / 8& M - ) 0 & F M ( + : : F ) ( # S(q, t) & 4 q *
& 4 q̂ & &' γ 7 / 5 T//q→q̂ S(q, t) 0 / & S(q, t) 0 Tq M 9 ( ) ∇q̂˙ v̂ ∈ Tq̂ M H / ) (q̂ − q) &q̂ F (q̂, q) ( 5 q q̂ ) . −β &q̂ F (q̂, q) 9 / : . R(v̂, &q̂ F (q̂, q))v̂ ) 0 ) ( F / ( 0 ( 3) B6C4 1 $& @ / & & γ 5 q 0 q̂ ) M ' ) # / & S(q, t) &q F (q̂, q) 9 T//q→q̂ S(q, t) T//q→q̂ &q F (q̂, q) = −&q̂ F (q̂, q)
Tq̂ M , /& / +
& α β / 3K4 8 () #
/ ) 8& M % # # / ' !!.#
# > '
( .& & 7
0
&
1
1
L(q, q̇) = q̇ 2 − q 2
2
2
/ ) q ∈ R 7 q̈ = −q / 8& : / /() 3K4 / 7
q̂˙ = v̂ − α(q̂ − q)
v̂˙ = −q − β(q̂ − q)
[email protected]
& α β + )& q̂ v̂ ) ) q q̇
> & ) r = exp(q) . .& & ) L(r, ṙ) =
/ 7
1 ṙ2 1
− (ln r)2
2 r2 2
ṙ = w
ẇ =
w2
− r ln r
r
r ∈]0, +∞[.
:) /() 3K4 7
r̂˙ = ŵ − α &r̂ F (r, r̂)
ŵr̂˙
+ T//r→r̂ (−r ln r) − β
ŵ˙ =
r̂
&r̂ F (r, r̂) + R(ŵ, &r̂ F (r̂, r))ŵ.
3K"4
. g = g11 ) g11 (r) = r1 . ( F ) Γ111 (r) = − 1r ./ & 5 r1 r2 2
r2
γ(s) = exp ln
s + ln r1
r1
) γ(0) = r1 γ(1) = r2 & r1 r2 ) dg (r1 , r2 ) =
0
1
g(γ(s))(γ (s))2 ds = | ln r2 − ln r1 |
# P & 8 d/ds . F (r̂, r) 1
F (r, r̂) = (ln r − ln r̂)2
2
& (ln r̂ − ln r)
= r̂(ln r̂ − ln r).
&r̂ F (r, r̂) = r̂2
r̂
./ & & 5 r s=0
r̂ r̂
s → γ(s) = exp ln
s + ln r
r
s = 1 / 7
1 γ (s)u = 0
u −
γ(s)
u(0) = −r ln r
7
u(s) = −r ln r exp ((ln r̂ − ln r)s) .
) 3K4 T//r→r̂ (−r ln r) = u(1) = −r̂ ln r.
r 7
r̂˙ = ŵ − α r̂(ln r̂ − ln r)
ŵr̂˙
− r̂ ln r − β r̂(ln r̂ − ln r)
ŵ˙ =
r̂
( #
/ 8& + # ( F
1 /# /() /
) 7 / r̂(ln r̂ − ln r) F / ( r̂ − r )& / ) 8 / /&
5 /# [email protected] r . g ) g11 (q) = 1 ) F
&q̂ F (q, q̂) = q̂ − q
T//q→q̂ S(q, t) = T//q→q̂ (−q) = −q
R(v̂, &q̂ F (q̂, q))v̂ = 0.
# () [email protected] 3K"4 # # 9 ()
# F / # F T//q→q̂ # : & /) + ( # R(v̂, &q̂ F (q̂, q))v̂
i
i
= Rjkl
v̂ k {&q̂ F (q̂, q)}j v̂ l
i
) Rjkl
( 7
∂Γikl ∂Γijl
=
−
+ Γipj Γpkl − Γipk Γpjl .
j
k
∂q
∂q
q F T//q→q̂ i
Rjkl
q̂ # ) 2F = gij (q)(q̂ i − q i )(q̂ j − q j ) + O( q̂ − q 3 )
3K?4
{&q̂ F }i = q̂ i − q i + O( q̂ − q 2 )
3KK4
{T//q→q̂ w}i = wi − Γijl (q)wj (q̂ l − q l ) + O( q̂ − q 2 )
3K!4
w 0 / & q 0 M . & ) 8 & . ) 8 & + . ) /# 3) B66: 6C
4 && & + & F (
) # # 0 / " 3K4 9 & 7
q̂˙i = v̂ i − α(q̂ i − q i )
v̂˙ i = −Γijk (q̂)v̂ j q̂˙k + S i (q, t) − Γijl (q)S j (q, t)(q̂ l − q l )
i
−β(q̂ i − q i ) + Rjkl
(q)v̂ k (q̂ j − q j )v̂ l .
3K4
1 Γijk (q̂)v̂j q̂˙k Γijk (q̂) Γijk (q) / ) ) v̂ 0 q̂˙ i
i
Γijl (q)S j (q, t)(q̂ l −q l ) Rjkl
(q)v̂ k (q̂ j −q j )v̂ l ) Γijk (q̂) Rjkl
(q̂) /
/& ( ./ 9 3K4 # 7
W . & # ) 9 ) / (gij ) ) 0 q 5 /0 / "
W ) )& 3K4 α β +
# ! 9 #
. /() 3K4 3q̂ ≈ q4 8 B: C 7 # / #
3?L4 3K4 ) )& ) 7
# " + > . # % 3?L4 # ! T M ( K T M 2 . " α β * ! & +(
0 2 ε > 0 *" K α β + µ > 0 % 4 G T M *" α β + % 3?L4
K [0, T [ t → X(t) = (q(t), q̇(t)) ∈ K
T ≤ +∞ 3K4 t → X̂(t) = (q̂(t), v̂(t)) X̂(0) # dG (X̂(0), X(0))
≤ ε # t ∈ [0, T [ ∀t ∈ [0, T [
dG (X̂(t), X(t)) ≤ dG (X̂(0), X(0)) exp(−µt).
dG ! ! % 1 %
G
T M (
. G + ) 8 < B6C / 0 /# 8 / & 0 T M . & /() α
β 8 G ( /( / )& + & dG (X̂(t), X(t)) + t
' #/ ' W 7 q 3 ) & 4 ) 0 q̂ v̂ 3K4
(q̂ + δq̂, v̂ + δv̂)
8 ) & (q̂, v̂) 0 + (δq̂, δv̂)
W 7 + G 8( & T M /() 3K4 q̂ q
# (- ' - .# 9 #
( / E( B5C
/ 0 ) / & ∇q̇ q̇ = 0 3) # :4
2 + (qi)
+ 8 ) δv̂ /
8 ) ζ 0
/ & q̂ 0 M ) (ξ = δ q̂, ζ) 0 (δ q̂, δv̂)
5(((( ) .% ξ ζ > 5 3S [4 7
ξ i = δ q̂ i
ζ i = δv̂ i + Γijk (q̂)δ q̂ k v̂ j ,
i = 1, ..., n
3K 4
'
) 8 ξ i ζ i # # ) ξ ζ 9 / & q̂ 0 M : F ) && 7
W ξ = γ̇(0) > 0 ) γ & 5 γ(0) = q̂
γ() = q̂ + δq̂ W ζ = T//q̂+δq̂→q̂ (v̂ + δv̂) − v̂
. ) & ξ ζ 8 & ( t → q̂(t) ) ) ) ∇q̂˙ ξ ∇q̂˙ ζ 0 / & q̂ 0 M ) 3K4 7
i
d i
i
δ q̂ = δv̂ − α δ &q̂ F (q̂, q)
dt
d i
3KJ4
δv̂ = − δ Γijk (q̂)v̂ j q̂˙k
dt
i
+ δ T//q→q̂ S(q, t) − β δ &q̂ F (q̂, q) + δ R(v̂, &q̂ F (q̂, q))v̂
5(((( ∇q̂˙ ξ
i
∇q̂˙ ξ # > 1 ) i = 1, ..., n 7
= ξ˙i + Γijk (q̂)q̂˙k ξ j
i
= δv̂ i − α δ &q̂ F (q̂, q)
+ Γijk (q̂) v̂ k − α {&q̂ F (q̂, q)}k ξ j
i
i
i
k j
i
k j
= δv̂ + Γjk (q̂)v̂ ξ − α δ &q̂ F (q̂, q)
+ Γjk (q̂){&q̂ F (q̂, q)} ξ .
( 7
∇q̂˙ ξ = ζ − α ∇ξ &q̂ F (q̂, q)
)
∇ξ &q̂ F (q̂, q)
5(((( ∇q̂˙ ζ i
i
= δ &q̂ F (q̂, q)
+ Γijk (q̂){&q̂ F (q̂, q)}k ξ j .
> ) ) ζ ∇q̂˙ ζ
i
7
3KL4
= ζ̇ i + Γijk (q̂)q̂˙k ζ j .
1/ 3KJ4 3K 4 ) d i d i
δv̂ +
Γjk (q̂)v̂ j ξ k
dt dt = − δ Γi (q̂)v̂ j q̂˙k
ζ̇ i =
jk
+ δ T//q→q̂ S(q, t) − β
+
d i
Γjk (q̂)v̂ j ξ k
dt
&q̂ F (q̂, q) + R(v̂, &q̂ F (q̂, q))v̂
i
# δ
) Γijk (q̂)v̂ j q̂˙k
δ q̂
δv̂ 7
∂Γijk (q̂) c j k
ξ v̂ q̂˙ + Γijk (q̂)δv̂ j q̂˙k + Γijk (q̂)v̂ j ξ˙k
=
c
∂ q̂
i
∂Γjk (q̂) c j k
=
ξ v̂ q̂˙
∂ q̂ c
+ Γijk (q̂)ζ j q̂˙k − Γijk (q̂)Γjpc (q̂)v̂ p ξ c q̂˙k
+ Γi (q̂)v̂ j ξ˙k
jk
∂Γi (q̂)
d i
jk
j k
Γjk (q̂)v̂ ξ =
q̂˙c ξ k v̂ j + Γijk (q̂)ξ k v̂˙ j + Γijk (q̂)ξ˙k v̂ j
c
dt
∂ q̂
∂Γijk (q̂) c k j
=
q̂˙ ξ v̂
∂ q̂ c
− Γijk (q̂)Γjpc (q̂)ξ k q̂˙c v̂ p
j
+ Γijk (q̂)ξ k T//q→q̂ S(q, t) − β &q̂ F (q̂, q) + R(v̂, &q̂ F (q̂, q))v̂
+ Γijk (q̂)ξ˙k v̂ j
' ( # 7
d i
j k
i
j ˙k
Γ (q̂)v̂ ξ − δ Γjk (q̂)v̂ q̂
dt jk
= −Γijk (q̂)ζ j q̂˙k
j
+ Γijk (q̂)ξ k T//q→q̂ S(q, t) − β &q̂ F (q̂, q) + R(v̂, &q̂ F (q̂, q))v̂
i
i
(q̂)
∂Γ
∂Γ
(q̂)
pc
pk
+ ξ k q̂˙c v̂ p
−
+ Γijc (q̂)Γjpk (q̂) − Γijk (q̂)Γjpc (q̂)
∂ q̂ c
∂ q̂ k
# ' ( ζ̇ i 7
ζ̇ i = − Γijk (q̂)ζ j q̂˙k
+ δ T//q→q̂ S(q, t) − β
&q̂ F (q̂, q) + R(v̂, &q̂ F (q̂, q))v̂
+
Γijk (q̂)ξ k
T//q→q̂ S(q, t) − β
i
&q̂ F (q̂, q) + R(v̂, &q̂ F (q̂, q))v̂
j
i
− ξ k q̂˙c v̂ p Rkcp
(q̂)
/# S [ ) 7
˙
∇q̂˙ ζ = −R(q̂, ξ)v̂ + ∇ξ T//q→q̂ S(q, t) − β∇ξ &q̂ F (q̂, q) + ∇ξ R(v̂, &q̂ F (q̂, q))v̂
P
i
j
i ∇ξ T//q→q̂ S(q, t)
= δ T//q→q̂ S(q, t)
+ Γijk (q̂)ξ k T//q→q̂ S(q, t)
∇ξ
R(v̂, &q̂ F (q̂, q))v̂
i
i
j
= δ R(v̂, &q̂ F (q̂, q))v̂
+ Γijk (q̂)ξ k R(v̂, &q̂ F (q̂, q))v̂
( ∇ξ ∇ξ / /& / ) ) ) 0 ξ 7 ) R(v̂, &q̂ F (q̂, q))v̂ 8 & ( t → q̂(t) 8 ) ) ξ / 0 q̂˙
:# ∇ξ δv̂ i = ζ i − Γijk (q̂)δ q̂ k v̂ j
i
i
δ &q̂ F (q̂, q)
= ∇ξ &q̂ F (q̂, q) − Γijk (q̂)δ q̂ k {&q̂ F (q̂, q)}j
+ /# ) 7
∇ξ
R(v̂, &q̂ F (q̂, q))v̂ = (∇ξ R)(v̂, &q̂ F (q̂, q))v̂ + R(v̂, ∇ξ &q̂ F (q̂, q))v̂
+ R(ζ, &q̂ F (q̂, q))v̂ + R(v̂, &q̂ F (q̂, q))ζ.
P ∇ξ R & ) ) ( & ξ # $ ( + ) )
3K4 0 q̂ v̂ 7
∇q̂˙ ξ = ζ − α∇ξ &q̂ F (q̂, q)
˙ ξ)v̂ + ∇ξ T//q→q̂ S(q, t) − β∇ξ & F (q̂, q)
∇q̂˙ ζ = −R(q̂,
q̂
+(∇ξ R)(v̂, &q̂ F (q̂, q))v̂ + R(v̂, ∇ξ &q̂ F (q̂, q))v̂
+R(ζ, &q̂ F (q̂, q))v̂ + R(v̂, &q̂ F (q̂, q))ζ
P ξ 0 ) q̂ Z S = 0 F
E( 7
= 0 3!4
) / D2 ξ
˙ ξ)q̂˙
= −R(q̂,
Dt2
) D/Dt = ∇q̂˙ q̂ q 3
5(((6( . q̂ $ q >
v̂ v4 / ) ) ( & q̂ − q 7
∇ξ &q̂ F (q̂, q) = ξ
&q̂ F (q̂, q) = 0,
1 ∇ξ T//q→q̂ S(q, t)
i
q̂˙ = v̂.
i
= δ T//q→q̂ S(q, t)
+ Γijk (q̂){T//q→q̂ S(q, t)}k ξ j
i
T//q→q̂ S(q, t) = S(q, t)i − Γijk (q)(q̂ j − q j )S(q, t)k
q q̂ ) 7
i
δ T//q→q̂ S(q, t)
= −Γijk (q)ξ j S(q, t)k
Γijk (q̂){T//q→q̂ S(q, t)}k ξ j = Γijk (q)ξ j S(q, t)k
0 (q̂ − q)
∇ξ T//q→q̂ S(q, t) = 0.
# 1 q = q̂ 3!4 ) ∇q̂˙ ξ = ζ − α ξ
∇q̂˙ ζ = −βξ
[email protected]
1/ ) : ) /() . (& ( [email protected]
/( /() q̂ = q 1 & & q̂ q 7 # 0 /() ) /( 8 / / /() / 0
8( & T M 5(((( - 4 # ! T M > & (
) α β + " ' ''
−α 1
A=
−β 0
G ;= #
8 ) Q At Q + QA = −I
a c
Q=
c b
a
b
V (ξ, ζ) = ξ, ξ + cξ, ζ + ζ, ζ
2
2
3!"4
P , & 0 g 0 / &
T M / Q 8 ) 7 (qi, vi ) & ) (δqi, δvi ) & (q, v) 0
# TM
i
V δq , (δv +
Γikl (q)v k δq l )i=1...n
a
gij δq i δq j
2
+ c gij (δv i + Γikl (q)v k δq l ) δq j
=
b
gij (δv i + Γikl (q)v k δq l ) (δv j + Γjkl (q)v k δq l )
2
1 (qi, vi ) 2n × 2n + q v : X = (q, v) & G(X)
8 T M , /& / ) & 8 < T M 3) B6: 6C4 7 ( < )
(a, b, c) = (2, 2, 0)
+
5(((( 0 ! > . 0
#
ξ
ζ
) 8 [email protected]
dV
= a∇q̂˙ ξ, ξ + c∇q̂˙ ξ, ζ + c∇q̂˙ ζ, ξ + b∇q̂˙ ζ, ζ
dt
1
= − (ξ, ξ + ζ, ζ).
2
λ > 0 dV
≤ −λV.
dt
& 8 /() 3K4 G(X) q̂ = q ) v̂
8 + X̂˙ = Υ(X, X̂) /() 3K4 Ẋ =
Υ(X, X) 0 ) 3?L4 ./ & dV
≤ −λV & 8 dt
/ & ) ) 8 7
t
∂G ∂Υ ∂Υ Υ(X, X̂) +
G(X̂) + G(X̂)
≤ −λG(X̂)
∂X X̂
∂ X̂ (X,X̂)
∂ X̂ (X,X̂)
3!?4
X = (q, v) X̂ = (q, v̂) ) q v v̂ ( G 8 ) 3!?4 0 X X̂ & 0 < ρ < λ # ε > 0 X K X̂ ) 8 dG (X, X̂) ≤ ε
) t
∂G ∂Υ ∂Υ Υ(X, X̂) +
G(X̂) + G(X̂)
≤ −ρG(X̂).
∂X X̂
∂ X̂ (X,X̂)
∂ X̂ (X,X̂)
# X̂(0) O X(0) / 0 dG (X(0), X̂(0)) ≤ ε
1/ / # ) t > 0 ρ
dG (X̂(t), X(t)) ≤ dG (X̂(0), X(0)) exp(− t).
2
$& " t ∈ [0, T [ dG (X̂(t), X(t)) ≤ ε
X̂(t) & ) )& # ) µ = ρ/2 ) ? ) , +/
) /# B8C ( / &)
. ( / 9 /() 7 F & 8 5 ( F &+ 0 ( #
! % %
3 .# # - ) ( ( 3( 4 $& ? ) r ( θ / & ) /= M u % ./ & 7
T =
/ & &)
1 2
ṙ + (1 + r2 )θ̇2
2
7
U = r sin θ.
( ) 7
ṙ = vr
θ̇ = vθ
v̇r = rvθ 2 − sin θ
−2r
u − r cos θ
v̇θ =
v
v
+
r
θ
1 + r2
1 + r2
9 # -.#
% >
3!K4
. g 8 / & / 7
1
0
0 1 + r2
. ( F 7
Γ212
Γ122 = −r
r
= Γ212 =
1 + r2
. ( -
# 7
1
1 + r2
1
=+
1 + r2
1
=+
(1 + r2 )2
1
=−
(1 + r2 )2
1
R122
=−
1
R212
2
R121
2
R211
. ( 7
p
Rs = g li Rpli
=
−2
(1 + r2 )2
. ( ( ( &)
%0 & > /() #
3K4 7
r̂˙ = v̂r − α(r̂ − r)
˙
θ̂ = v̂θ − α(θ̂ − θ)
u − r cos θ
˙
˙v̂r = r̂ θ̂v̂
θ + − sin θ + r̂(θ̂ − θ)
1 + r2
−1
1
2
− β(r̂ − r) +
v̂r v̂θ (θ̂ − θ) +
v̂ (r̂ − r)
1 + r̂2
1 + r̂2 θ
v̂˙ θ =
−r̂ ˙
˙
(r̂v̂θ + v̂r θ̂)
1+ r̂2
r̂
u − r cos θ
u − r cos θ
−
+ (θ̂ − θ)(− sin θ)
(r̂ − r)
+
1 + r2
1 + r̂2
1 + r2
1
−1
2
v̂ (θ̂ − θ) +
v̂r v̂θ (r̂ − r)
− β(θ̂ − θ) +
(1 + r̂2 )2 r
(1 + r̂2 )2
3!!4
#
! a) r
b) theta
1.5
1.5
real
int.obs.
simple
1
1
0.5
0.5
0
−0.5
0
−1
−0.5
0
5
10
15
−1.5
0
5
c) r
10
15
10
15
d) theta
1.5
1.5
1
1
0.5
0.5
0
0
−0.5
−0.5
0
5
10
15
−1
0
e) d(r)/dt
5
f) d(theta)/dt
1.5
4
3
1
2
1
0.5
0
−1
0
−2
−0.5
0
5
10
15
−3
0
5
10
,* , , - = * 56 , = ,* ' 55 = ,* 52
15
# ) $& K
% ( / (
( (r = 0, θ = 0, vr = 0, vθ = 0) 7 u = −20 θ + 101 r r ( & ) −2 1 ) 5 & #
+ br bθ ) # r θ + + && / ( ) /() 3!!4 ) ) 7
"# #'.#
r̂˙ = v̂r − α(r̂ − r)
˙
θ̂ = v̂θ − α(θ̂ − θ)
v̂˙ r = r̂v̂ 2 − sin θ − β(r̂ − r)
3!4
θ
−2r̂
r cos θ − u
v̂r v̂θ −
− β(θ̂ − θ)
v̂˙ θ =
2
1 + r̂
1 + r2
/
() )
/ 5 r θ , )& & O & ) (
() O ( ) P & & . 7
I '() 3!!4 3!4
r
"
"@
vr
!
!"!
θ
"
vθ
"
& α β O & () 3!!4 3!4 # )& & # + br bθ 8 $& K ) ) ) & 7
1√
2
2
α = 2ωo
ωo =
β = ωo2
/ ) ( ( # ) 2
## 1 $& K & & ( P r θ & # + br bθ ) ) /() 3!4 )& 7 β / = & F ( &)
/() 3!!4 )& / ( # ) /# /() " -
! !
# 0 ( BC
+ 0 7
3 .# 9 # -.#
.& & 0 [email protected] 5 q̇ = v
∇q̇ v = S(q, t) − A(q)q̇.
8 ) / ) 8& 5 8& q 0
) q̇ /() ) 7
q̂˙ = v̂ − α &q̂ F (q̂, q)
3! 4
∇q̂˙ v̂ = T//q→q̂ S(q, t) − T//q→q̂ A(q)T//q̂→q q̂˙
−β &q̂ F (q̂, q) + R(v̂, &q̂ F (q̂, q))v̂
A(q)
(- 9'.# 9 # B5C : + ) 3! 4
/ ∇q̂˙ ξ = ζ − α∇ξ &q̂ F (q̂, q)
˙ ξ)v̂ + ∇ξ T//q→q̂ S(q, t) − ∇ξ T//q→q̂ A(q)T//q̂→q q̂˙
∇ ˙ ζ = −R(q̂,
q̂
−β∇ξ &q̂ F (q̂, q) + (∇ξ R)(v̂, &q̂ F (q̂, q))v̂ + R(v̂, ∇ξ &q̂ F (q̂, q))v̂
+R(ζ, &q̂ F (q̂, q))v̂ + R(v̂, &q̂ F (q̂, q))ζ
3!J4
)
# i i
∇ξ T//q→q̂ A(q)T//q̂→q q̂˙
= δ T//q→q̂ A(q)T//q̂→q q̂˙
j
+ Γijk (q̂)ξ k T//q→q̂ A(q)T//q̂→q q̂˙
.
. q = q̂ > )& () 7
@ /( /() 3! 4 q̂ = q
" & ) ) q̂ q
/# ) 3!J4 q = q̂ ,
0 /# ∇ξ T//q→q̂ A(q)T//q̂→q q̂˙ .
%0 T
A(q)T//q̂→q q̂˙ >
i
k
i i
j
j
˙
˙
˙
T//q→q̂ A(q)T//q̂→q q̂
= A(q)T//q̂→q q̂ − Γjk (q)(q̂ − q ) A(q)T//q̂→q q̂
/P
/
//q→q̂
i
T//q→q̂ A(q)T//q̂→q q̂˙
=A(q)ij q̂˙j + Γjmc (q)(q̂ m − q m )q̂˙c
i
j
j
k ˙h
h
m
m ˙c
− Γjk (q)(q̂ − q ) A(q)h q̂ + Γmc (q)(q̂ − q )q̂ .
) T//q→q̂ A(q)T//q̂→q q̂˙
i
q̂ − q
=A(q)ij q̂˙j + A(q)ij Γjmc (q)(q̂ m − q m )q̂˙c − Γijk (q)(q̂ j − q j )A(q)kh q̂˙h .
δ T//q→q̂ A(q)T//q̂→q q̂˙
,
i
δ T//q→q̂ A(q)T//q̂→q q̂˙
= δ A(q)ij q̂˙j + A(q)ij Γjmc (q)(q̂ m − q m )q̂˙c
− Γi (q)(q̂ j − q j )A(q)k q̂˙h
jk
/P
h
i
˙
δ T//q→q̂ A(q)T//q̂→q q̂
=A(q)ij ξ˙j
+ A(q)ij Γjmc (q)ξ m q̂˙c + A(q)ij Γjmc (q)(q̂ m − q m )ξ˙c
− Γi (q)ξ j A(q)k q̂˙h − Γi (q)(q̂ j − q j )A(q)k ξ˙h .
jk
h
jk
h
## i
ξ i = ∇q̂˙ ξ − Γijk (q̂)ξ j q̂˙k
ξ (q̂ − q) 7
i
ξ i = ∇q̂˙ ξ − Γijk (q)ξ j q̂˙k .
' ( ) ξ
j
i
δ T//q→q̂ A(q)T//q̂→q q̂˙
=A(q)ij ∇q̂˙ ξ − A(q)ij Γjmc (q)ξ m q̂˙c
(q̂ − q) 7
c
+ A(q)ij Γjmc (q)ξ m q̂˙c + A(q)ij Γjmc (q)(q̂ m − q m ) ∇q̂˙ ξ
h
− Γijk (q)ξ j A(q)kh q̂˙h − Γijk (q)(q̂ j − q j )A(q)kh ∇q̂˙ ξ
8 7
i
i ˙
δ T//q→q̂ A(q)T//q̂→q q̂
= T//q→q̂ A(q)T//q̂→q ∇q̂˙ ξ
# ξ
q − q̂
7
− Γijk (q)ξ j A(q)kh q̂˙h .
j
Γijk (q̂)ξ k T//q→q̂ A(q)T//q̂→q q̂˙
,
j
Γijk (q̂)ξ k T//q→q̂ A(q)T//q̂→q q̂˙
= Γijk (q̂)ξ k A(q)jh q̂˙h
∇ξ T//q→q̂ A(q)T//q̂→q q̂˙
q
q̂ &1 &0 ( 3!L4 34 ( 7
∇ξ
3!L4
34
> : ˙
T//q→q̂ A(q)T//q̂→q q̂ = T//q→q̂ A(q)T//q̂→q ∇q̂˙ ξ .
. q̂ = q ) T//q→q̂ = IdT (M ) = T//q̂→q 7
∇ξ
q
T//q→q̂ A(q)T//q̂→q q̂˙ = A(q̂)∇q̂˙ ξ.
./ )
3! 4 / q̂ = q
∇q̂˙ ξ = ζ − α ξ
∇q̂˙ ζ = −βξ − A(q̂)(ζ − α ξ)
[email protected]
# :# [email protected] , & )(
''
ψ = ζ − αξ.
./ [email protected] (ξ, ψ)
∇q̂˙ ξ = ψ
∇q̂˙ ψ = −βξ − (A(q̂) + αIn )ψ
3"4
P In & n
α β + + . ) ) 7
V (ξ, ψ) =
) ) 7
β
1
ξ, ξg(q̂) + ψ, ψg(q̂) .
2
2
V̇ = −ψ, (A(q̂) + αIn )ψg(q̂) ≤ 0.
. ) V̇ 8 &) /( 7 + V ) A F ) 9 α : F −ψ, A(q̂)ψg(q̂) &+
/ / + , ( ( / #
W 3 q̂4 ) Ẇ 8 &) 0 5 A /) / ) 0 )& 3! 4
# . ( 3 /# ( ( ) !K4 0 & )& () 3K4 = & 9 + ) (
q̂ q 9 / + / ) 1 )& /) # , ( / # P M & . / ) 0 )& + ./() 3K4 # + 9 0
8 6 + (
( 8 B6: C # B6C
1 ) ) / ( /() F ) 0
() ) 9 ) K ) / & + ! .& & ) 0 & 8& ) # () ) / ) / ) 0 ) & )
& () / ( & & , 0 + # ) 9 ( + ( -+ ( + ) / ) ) 7 / 6 # ( / +
) # / # ( / 8 /& 7
/ 6 ) ( 8 #
& & ) & / / / ) 9 = & : ( ) () # ) + ) ) K /() ) , /& ( 1( # 7
( + J BC K
= / R3
./ ( ) ) Σ 0 R3 & + G & Σ # ( . / ) ( & & + ) 8 5
(ξ1 , ξ2 , ξ3 ) $& @ ξf = (ξ1 , ξ2 )
( ) ξb = ξ3 ( ) ) # P S 9
( / 0 / #
g G S = ϕg (P ) O ) 8
# 9 ξ3 7 ξ3 + ) 2 + ) + ξ3 5 (p1 , p2 , p3 ) 3 ) (s1 , s2 , s3 ) (r1 , r2 , r3 )4 / P 3 ) S R4 + P S 0 9 ( #
R
P
S
g
ξ3
ξ2
ξ
O
1
!, & / G & Σ 7
g ∈ G 3 4 7
s1 = (ϕg (P ))1
s2 = (ϕg (P ))2
s3 = (ϕg (P ))3
. ( S ) 0 ) / 5 & ) + ) ) # 0 g R 9 (
+ ' / g(P ) & ) 8 R = ϕg(P ) (P ) / ) 3 4 7
r1 = (ϕg (P ))1
r2 = (ϕg (P ))2
+ / 3 & #
# 0 G 9 24 ) 7
g(P ) = γ(P, r1 , r2 )
. + ) 7
I(P ) = (ϕγ(P,r1 ,r2 ) (P ))3
: F I(P ) = r3 R S 9 ( ) 7 I(S) =
r3 = I(P ) ϕγ(S,r ,r ) (S) = R
1
2
!'' & G & & ) Σ σ )
( r r < σ 1 ξ Σ Ξ = ϕ(ξ, a)
+ 0 / G a /(
3 ( / ) + #4 ( a + ξ = (ξb , ξf ) ) ( ξb σ − r 8( ξf r . +
Ξ = ϕ(ξ, a) Ξb = ϕb (ξ, a)
Ξf = ϕf (ξ, a)
) a → ϕf (ξ, a) )( ξ ψf (ξ, Ξf ) / ) 3ϕf (ξ, ψf (ξ, Ξf )) ≡ Ξf 4 ( Ξb = ϕb (ξ, ψf (ξ, Ξf )) := ψb (ξ, Ξf ).
ζ ξ 9 ( 9 (
& 8 / + Ξf 8# 0 Ξ0f # 3 4
ψb (ξ, Ξ0f ) = ψb (ζ, Ξ0f ).
: / σ − r + ξ
+ Ξ = ϕ(ξ, a) ) ψb (ξ, Ξ0f ) ≡ ψb (ϕ(ξ, a), Ξ0f ).
→ ψb (ξ, Ξ0f )
)
. B: C ) ẋ = f (x, t)
9 # ) & & 6
) 2 B " C > - . % ẋ = f (x, t) !
*C 1 2 + # M !. ( - g % M ( - U ⊂ M M ( ) % f U
% g % 7 # ! & 1 & 2 λ > 0 % x U t
∂g
∂f T
∂f
+
f (x, t) ≤ −λg(x).
g(x) + g(x)
∂x
∂x ∂x
) ) 5 8 8 &
# " . > - .
M
% ẋ = f (x, t) ! # !. ( - g(x) % M ( - X(x, t) 8 1 f
d
X(x, t) = f (X(x, t), t)
dt
X(x, 0) = x
∀t ∈ [0, T [
T ≤ +∞.
2 x0 x1 M ! % γ(s) % 9 x0 = γ(0) x1 = γ(1)( -
: f U ⊂ M λ # ;
: X(γ(s), t) 1 U s ∈ [0, 1] t ∈ [0, T [ (
λ
3 dg dg (X(x0 , t), X(x1 , t)) ≤ e− 2 t dg (x0 , x1 ) ∀t ∈ [0, T [
! % 1 % g(
) / + . 3 ) 4 l(t) & ( (X(γ(s), t), s ∈ [0, 1]) g
(#
1
dX(γ(s), t) T
dX(γ(s), t)
ds.
g(X(γ(s), t))
ds
ds
l(t) =
) 0
d
l(t) =
dt
d
1
dt
0
2
dX(γ(s),t) T
g(X(γ(s), t)) dX(γ(s),t)
ds
ds
dX(γ(s),t) T
g(X(γ(s), t)) dX(γ(s),t)
ds
ds
ds.
d
∂
d
d dX(γ(s), t)
= f (X(γ(s), t), t) =
f (X(γ(s), t), t) X(γ(s), t)
dt
ds
ds
∂X
ds
( d
dt
dX(γ(s), t) T
dX(γ(s), t)
g(X(γ(s), t))
ds
ds
)
P (s, t) =
f dX(γ(s), t) T
dX(γ(s), t)
=
P (s, t)
ds
ds
∂f (X) T
∂f (X) ∂g(X)
g(X) + g(X)
+
f (X, t)
∂X
∂X
∂X
X = X(γ(s), t).
U # λ > 0 P (s, t) ≤ −λg(X(γ(s), t)).
' ( / & ) ) 0
5 d
l(t)
dt
λ
d
l(t) ≤ − l(t)
dt
2
λ
l(t) ≤ l(0)e− 2 t ∀t ∈ [0, T [.
dg (X(x0, t), X(x1 , t)) ≤ l(t) l(0) = dg (x0, x1 ) 3
# x0 x1 4 F γ & ( / ) K ( / 6 &
) / ) + # & ) 0 & %# . 6 ) + /(5 ( # , ( / ) )
# & ) ) / 7 (x, y) /
# & t / ) ∂I(x, y, t) ∂I(x, y, t)
∂I(x, y, t)
dI
=
+
u+
v=0
dt
∂t
∂x
∂y
& ) I + & 8 I(x + u, y + v, t + 1) = I(x, y, t)
3?4
P (u, v) / . 3?4 + / ) 0 # ) u v 7 F + ∂I ∂I
) & / ∇I = ( ∂x
, ∂y ) ) 6 7 & ) . + 7
W 1 F B6: 5C + / 8 0
# 7 3?4 )
G( (u, v) = &
2
∂I
u
.∇I dxdy + λ||v||2
+
v
∂t
W 1 / 8
3K4
( ( ) 7
2
I(x + u, y + v, t + 1) − I(x, y, t) dxdy
3!4
+ 9 &( 8 P 6 9 W 1 + B: C / + ./ B7C @LL ) # ) F ) 0 1 # ) / 6
( ) 8 0 ./ 8 / ) # & : F / I / 6 3?4 J = φ(I) φ F 8 & ( # ) # # & ) K & + & F 8 & 1 @ # # &
/ 1 " 0 / 8 ( 6
= # ' ) /& ( / t # (x, y) I(x, y, t) ./ 6 3?4 /# / / (u, v) = (u(x, y), v(x, y)) ) ) ) # (x, y) ) ) % 9'.# # 2 .#
I(x + u, y + v, t + 1) = I(x, y, t)
/# dI
∂I
u
=
+
.∇I = 0
v
dt
∂t
34
) ∇I & / 6 ) # & : F I 34 F φ & φ(I) : F ) ∂φ dI
dφ(I)
=
= 0.
dt
∂I dt
. ) 0 & 7
W . F 3K4 / ) / & + I ) φ(I) + X -
9
.#
2
∂I
u
+
.∇I dxdy
v
∂t
E( F φ , / & λ 8 W 9 3!4 / ) : F F φ 8 / 0 # (x, y) + & E( φ ./ 8 / ) # & & ( / + .
6 8 , ( ( ) 0 ) + Σ(I) ⊂ Z2 / ( 8 8 # /& I + σ≤I (a) σ≥I (a) 8 ) (x,
y)
∈
Σ(I)
)
8
I(x,
y)
≤
a
I
, ∀a ∈ R
σ≤
(a) =
Σ(I)
(x, y) ∈ Σ(I) ) 8 I(x, y) ≥ a
I
, ∀a ∈ R.
σ≥ (a) =
Σ(I)
= 9# + / ) I
Ir
r
EIr (I1 (x1 , y1 ), I2 (x2 , y2 )) = σ≤
(I2 (x2 , y2 )) − σ≤
(I1 (x1 , y1 ))
+σ Ir (I2 (x2 , y2 )) − σ Ir (I1 (x1 , y1 ))
≥
≥
/ # (x1, y1 ) & + / I1 ) / # (x2, y2 ) &
) + / I2 Ir & ) + ' ) 8 # 8 / ) 8 / 8 φ
F 8 & ) φ(I)
σ≤ (φ(J)(xo , yo ))
) 8 φ(I)(x, y) ≤ φ(J)(xo , yo )
Σ(I)
(x, y) ∈ Σ(I)
) 8 φ(I)(x, y) ≥ φ(J)(xo , yo )
Σ(I)
=
=
φ(I)
σ≥ (φ(J)(xo , yo ))
(x, y) ∈ Σ(I)
( # F ) φ 7
W φ + φ(I)
I
σ≤ (φ(J)(xo , yo )) = σ≤
(J(xo , yo ))
φ(I)
I
σ≥ (φ(J)(xo , yo )) = σ≥
(J(xo , yo ))
W ( φ(I)
I
σ≤ (φ(J)(xo , yo )) = σ≥
(J(xo , yo ))
φ(I)
I
σ≥ (φ(J)(xo , yo )) = σ≤
(J(xo , yo )).
/ ) 7
Eφ(Ir ) (φ(I1 )(x1 , y1 ), φ(I2 )(x2 , y2 )) = EIr (I1 (x1 , y1 ), I2 (x2 , y2 )).
.
#
# # 2 .#
/ 8 ( 6 #
& ) ) F ( 7 & ) / & 0 / # P / 6 /& . ( 5 / ) / /
6 # (xo, yo ) / ) 7
0'
9G
J(u, v) =
W (xo ,yo ,t)
# 9# 2
EIr I(x, y, t), I(x + u, y + v, t + 1) dxdy
P W (xo , yo , t) + 9 # (xo, yo ) /& 0 / t ) 7 & ./& Ir + & ( 7 ) # Ir = I(·, ·, t) Ir = I(x, y, t)|(x,y)∈W P Wr
+ 9 (xo, yo ) /& 3/ /) + 0 /& / / + /&4
. ) ) (u, v) + 9 W (xo, yo ) & u(t + 1) = λu(t) + α &x J(u, v)
v(t + 1) = λv(t) + α &y J(u, v)
P λ + α & & ./) & / & /& 8 r
) ) 7 S#[ $& @ ST $& " S ([ $& ? ' + 9 / /( / ( / ) 7 # ) F ( / 6 S2#[ 0 +H \) / & 1 $& K ! 7 # ) ) ) # & F $' # # # +H 0 / ) 0 ) 7 ()
/) / ) ( 6 # ) W (xo , yo , t) + 9 8 × 8
# . + 9 + Wr 72 × 72 # (xo, yo ) Wr / & ) # ./ 9 # ) K ) / 0 ) 7 # # K /& . ( ) / 0 8 / ' 8 +9 .:; × 52 < '
' 8 9 == × 52 < '
' 8 ,9 6; × 52 < '
' 8)'9 .53 × ==2 < ' - ' 8)'9 .53 × ==2 < ' - .; &
1 # ) ) # / + , ) ) 8 & B5C B66C # + * ( q(t) t 8 ) -
M ' ) ξ 8 & ( q(t) M ' / 0 ) ξ(t) = ξ(q(t)) ∈ Tq M 7 Tq M & / & 0 M q
. ) /( # / & ,
+ & 0 ) 9 & / h = 0 7
∆(h) =
1
T//q(t+h)−→q(t) (ξ(t + h)) − ξ(t)
h
1
(ξ(t + h) − ξ(t))
h
0 Tq(t+h) M = Tq(t) M 3) $& @4 T//q(t+h)−→q(t) /
Tq(t+h) M ) Tq(t) M 7 ∆(h) =
ξ(t + h) ) F ) 0 ) X / 0 h
: 3K!4 ( /# ) h 7
(
(
1 i
ξ (t + h) − Γij,k (q(t))(q j (t) − q j (t + h))ξ k (t + h) − ξ i (t)
h
q j (t + h) − q j (t) k
ξ i (t + h) − ξ i (t)
+ Γij,k (q(t))
ξ (t + h)
=
h
h
0 h
lim {∆(h)}i = ξ˙i + Γi (q)q̇ j ξ k
{∆(h)}i =
' ( j,k
h→0
lim ∆(h) = ∇q̇ ξ.
h→0
) ) ∇q̇ ) ξ 8 & (
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