Backs epping s abiliza ion o an unde ac ua ed 3 ×3 linea hype bolic
sys em o luid low equa ions
Flo en Di Meglio, Ra ael Vazquez, Mi osla K s ic, Nicolas Pe i
Abs ac — We in es iga e he bounda y s abiliza ion o a
pa icula subse o 3×3linea hype bolic sys ems wi h a ying
coe icien s on a bounded domain. The sys em is unde ac ua ed
since only one o he h ee hype bolic PDEs is ac ua ed a
he bounda y. The se up conside ed in he pape occu s in
con ol o mul iphase lows on oil p oduc ion sys ems. We use
a backs epping app oach o design a ull-s a e eedback law
yielding exponen ial s abili y o he o igin.
I. In oduc ion
We conside he p oblem o s abilizing a ce ain ype
o 3 ×3 linea hype bolic sys ems o anspo equa ions
wi h spa ially a ying coe icien s. We conside he case
whe e wo uncon olled PDE s a es ha e s ic ly posi i e
anspo speeds, whe eas he con olled PDE s a e has a
s ic ly nega i e speed. In addi ion, we assume ha one o
he uncon olled s a es is a Riemann in a ian , i.e. i sa is ies
a pu e anspo equa ion wi h a ying speed.
The s abili y o hype bolic sys ems has been p e iously
in es iga ed. In [4] using a Lyapuno app oach, he au ho s
conside he s abiliza ion by s a ic ou pu eedback o 2 ×2
linea (wi h cons an and a ying coe icien s) and quasilinea
hype bolic sys ems. Simila ly, su icien condi ions o he
s abili y o 2 ×2 linea sys ems wi h a ying coe icien s
and n×nquasilinea sys ems a e de i ed in [1] and [5], e-
spec i ely. Howe e , when hese condi ions a e no sa is ied,
mo e ad anced eedback laws a e needed.
The s abili y o hese sys ems is also in es iga ed in [11],
linking he Lyapuno and equency domain app oaches, as
well as in [16], ollowing a Lyapuno app oach. Besides,
in [10], su icien condi ions o he exac con ollabili y
o quasilinea hype bolic sys ems a e gi en. In e es ingly,
hey do no apply o he sys em conside ed in his pape .
A su icien condi ion is ha he numbe o con olled s a es
is la ge han he numbe o uncon olled ones, which is no
he case he e.
In [14], an ou pu eedback law is in oduced, which
allows s abiliza ion o 2 ×2 linea he e odi ec ional1hype -
bolic sys ems. A backs epping obse e -con olle s uc u e
is de i ed, yielding exponen ial s abili y in he L2-no m o
F. Di Meglio is wi h he Depa men o Mechanical and Ae ospace
Enginee ing, Uni e si y o Cali o nia San Diego, La Jolla, CA 92093-0411,
USA.
R. Vazquez is wi h he Depa men o Ae ospace Enginee ing, Uni e si-
dad de Se illa, Camino de los Descub imien o s.n., 41092 Se illa, Spain.
M. K s ic is wi h he Depa men o Mechanical and Ae ospace Enginee -
ing, Uni e si y o Cali o nia San Diego, La Jolla, CA 92093-0411, USA.
N. Pe i is P o esso a MINES Pa isTech, 60, Bd S -Michel, 75272 Pa is,
Cedex 06, F ance.
1i.e., whe e he wo s a es ha e anspo speeds o opposi e signs
he o igin o he conside ed sys em. The esul is ex ended
o he quasilinea case in [15].
In his pape , we ex end he s a e eedback in [14] o a
pa icula ype o 3 ×3 linea sys ems, a ising in modelling
o mul iphase low. In [7], a model o gas-liquid low
in oil p oduc ion pipes is p oposed, unde he o m o a
3×3 quasilinea hype bolic sys em. The model ep oduces
an undesi able phenomenon, called slugging, ea u ing la ge
oscilla ions o he p essu e, low a es and mass hold-ups
e e ywhe e inside he pipes. The occu ence o slugging
co esponds o he ins abili y o he equilib ium o he model,
which can be s abilized by eedback ac ua ion o a al e
loca ed a he ou le o he pipes. Rema kably, one o he
s a es o he model, namely he mass ac ion o gas, is
a Riemann in a ian (see e.g. [6]). The linea iza ion o
he model p oposed in [7] leads o he sys em conside ed
h oughou his a icle.
The pape is o ganized as ollows. In Sec ion II, we
desc ibe he sys em unde conside a ion. In Sec ion III, we
p opose a backs epping ans o ma ion. The exis ence o he
ke nels di ec ly s ems om he esul s p esen ed in [14]. In
Sec ion IV, we p o e exponen ial s abili y in he L2-no m
o he a ge and o iginal sys ems. Finally, we discuss he
ob ained esul s and pe spec i es o u u e imp o emen in
Sec ion V.
II. Sys em de ini ion
Conside he pipe schema ically depic ed in Figu e 1. I is
illed wi h gas and oil coming om a ese oi o a mani old.
The geome ic dis ibu ion o bo h phases inside he pipes,
e e ed o as low egime, depends, among o he hings,
on he ma u a ion o he oil ield, he geome y o he pipe
and he na u e o he gas-liquid mix u e. One o hese low
egimes, called slugging, ea u es pe iodic oscilla ions o all
he physical quan i ies (such as p essu e, low a es, hold-
ups) inside he pipe. These oscilla ions a e a he bi h o
p oduc ion losses and may damage he acili ies. A possible
solu ion o supp ess hem is o use eedback con ol o he
ou le al e, which can be emo ely ac ua ed o s abilize
he low. The model p oposed in [7], which is o d i - lux
ype [2], [3] akes he ollowing o m
∂ζ
∂ ( ,x)+A(ζ)∂ζ
∂x( ,x)=S(x) (1)
on he spa ial domain [0,1]. The h ee dis ibu ed s a e
a iables a e he gas mass ac ion, p essu e and gas eloci y.
The exp essions o he Aand Sma ices a e gi en in [7].
Liquid
Gas
Ou le Val e
Inle
θ
Fig. 1. Schema ic iew o a pipe con eying oil and gas om a ese oi .
The bounda y condi ions a e exp essed as ollows
hl1(ζ( ,0))
hl2(ζ( ,0))!= 0
0!h (ζ( ,1),Z( ))=0 (2)
whe e Z( ) is he opening o he ou le al e, which is
he con ol inpu . Conside ing small a ia ions a ound an
equilib ium p o ile ¯
ζ(x) (co esponding o a gi en al e
opening ¯
Z) yields he ollowing linea sys em wi h a ying
coe icien s (see Appendix A o he comple e linea iza ion)
u1 ( ,x)+λ1(x)u1x( ,x)=0 (3)
u2 ( ,x)+λ2(x)u2x( ,x)+σ21(x)u1( ,x)+σ23(x) ( ,x)=0
(4)
( ,x)−µ(x) x( ,x)+σ31(x)u1( ,x)+σ32(x)u2( ,x)=0
(5)
on he domain ( ,x)∈R×[0,1]. The anspo speeds a e C1
unc ions o space sa is ying he ollowing inequali ies
∀x∈[0,1] −µ(x)<0< λ1(x)< λ2(x)
and we deno e
Λ(x)=









λ1(x) 0 0
0λ2(x) 0
0 0 −µ(x)










Σ(x)=









000
σ21(x) 0 σ23(x)
σ31(x)σ32(x) 0










The linea ized bounda y condi ions ead










u1( ,0)
u2( ,0)
( ,0)










=Q0









u1( ,0)
u2( ,0)
( ,0)










=









0 0 q1
0 0 q2
0 0 1




















u1( ,0)
u2( ,0)
( ,0)










(6)
( ,1) =U( ) (7)
whe e U( ) is he new con ol inpu , and q1and q2a e non-
ze o. Sys em (3)–(5) wi h bounda y condi ions (6), (7) o ms,
along wi h an app op ia e ini ial condi ion, a well-posed
p oblem. Howe e , as demons a ed in [7], he equilib ium
αβ
T anspo di ec ions
Sou ce e ms
Bounda y condi ions
O iginal sys em
Ta ge sys em
Fig. 2. Schema ic iew o he con ol design o 2 ×2 he e odi ec ional
sys ems. The backs epping ans o ma ion comple ely emo es he in e nal
coupling be ween bo h s a es. The esul ing a ge sys em is exponen ially
s able.
u1≡u2≡ ≡0 may be uns able, especially o la ge alues
o ¯
Z. In he nex sec ions, we p opose a s abilizing eedback
law ollowing a backs epping app oach.
III. Backs epping ans o ma ion
A. Analogy wi h 2×2he e odi ec ional sys ems
Sys em (3)-(4)-(5) is no he mos gene al o m o 3 ×3
hype bolic sys ems. Because he i s line o ma ix Σ(x) is
illed wi h ze os ( he i s s a e is a Riemann in a ian ), i s
s uc u e esembles ha o 2 ×2 he e odi ec ional sys ems.
In [14], a s abilizing eedback con ol law is p oposed o
hese sys ems, along wi h obse e s o bo h he colloca ed
and non colloca ed cases. We p opose o exploi he esem-
blance wi h 2 ×2 he e odi ec ional sys ems o design an
con olle s uc u e o ou 3×3 (2×2+a Riemann in a ian )
sys em.
In pa icula , when designing a backs epping ans o ma-
ion, he ke nel equa ions ha allow o supp ess he in e nal
coupling be ween he las wo s a es (u2and ) a e exac ly
he same as he ke nel equa ions in he 2 ×2 case. The
coupling be ween he i s s a e (u1, he Riemann in a ian )
and he las s a e ( , he con olled s a e) can be supp essed
by he backs epping design as well. Impo an ly, he coupling
be ween he wo homodi ec ional s a es (u1and u2)does no
need o be supp essed, since i does no a ec he s abili y o
he a ge sys em. This idea can be summa ized by Figu es 2
and 3. This sugges s ha a gene aliza ion o his esul o
(n+1)×(n+1) sys ems, wi h ns a es wi h posi i e speeds and
one con olled s a e wi h a nega i e speed may be possible.
We discuss his ma e in mo e de ail in Sec ion V.
α
α
β
σ
σ
σ
σ
T anspo di ec ions
Sou ce e ms
Bounda y condi ions
O iginal sys em
Ta ge sys em
Fig. 3. Schema ic iew o he con ol design o a pa icula 3 ×3 sys em
(2 ×2+a Riemann in a ian ). The backs epping ans o ma ion comple ely
emo es he in e nal coupling be ween he α2and βs a es, and be ween α1
and β. The coupling be ween α1and α2(u1and u2in he o iginal sys em)
is sligh ly modi ied, as he e is now an in eg al sou ce e m p opo ional
o α1in he p opaga ion equa ion o α2. The esul ing a ge sys em is s ill
exponen ially s able.
B. Ta ge sys em
We wan o map he o iginal sys em (3)–(5) o he ollow-
ing a ge sys em
α1 ( ,x)+λ1(x)α1x( ,x)=0 (8)
α2 ( ,x)+λ2(x)α2x( ,x)
+σ21(x)α1( ,x)+Zx
0
c(x,s)α1( ,s)ds =0 (9)
β ( ,x)−µ(x)βx( ,x)=0 (10)
whe e cis a C0 unc ion o be de e mined de ined on he
iangula domain
T={(x, ξ) : 0 ≤ξ≤x≤1},(11)
along wi h bounda y condi ions
α( ,0) =Q0α( ,0) =









0 0 q1
0 0 q2
0 0 1










α( ,0) β( ,1) =0 (12)
The L2-s abili y o sys em (8)-(9)-(10) is in es iga ed in
Sec ion IV. We now p opose a s a e ans o ma ion ha maps
he o iginal sys em (3)-(4)-(5) o he a ge sys em.
C. Backs epping ans o ma ion
To ans o m sys em (3)-(4)-(5) in o he a ge sys em (8)-
(9)-(10), we conside a backs epping ans o ma ion o he
ollowing o m.
γ( ,x)=w( ,x)−Zx
0
K(x, ξ)w( , ξ)dξ(13)
whe e w=(u1,u2, )Tand γ=(α1, α2, β)Tand he gains o
he ke nel ma ix
K=









000
0k22 k23
k31 k32 k33









a e ye o be de e mined. Di e en ia ing (13) wi h espec o
space yields
γx( ,x)=wx( ,x)−K(x,x)w( ,x)−Zx
0
Kx(x, ξ)w( , ξ)dξ
while di e en ia ing (13) wi h espec o ime, and in eg a ing
by pa s yields
γ ( ,x)=−Λ(x)wx( ,x)−Σ(x)w( ,x)
+K(x,x)Λ(x)w( ,x)−K(x,0)Λ(0)w( ,0)
−Zx
0hKξ(x, ξ)Λ(ξ)+K(x, ξ)Λ0(ξ)−K(ξ)Σ(ξ)iw( , ξ)dξ
Using hese exp essions in o he a ge sys em equa ions, and
using he plan equa ions and he ac ha w1( ,x)=α1( ,x)
yields
0=hΣ0(x)−Σ(x)+K(x,x)Λ(x)−Λ(x)K(x,x)iw( ,x)
−K(x,0)Λ(0)w( ,0) −Zx
0hKx(x, ξ)Λ(x)+Kξ(x, ξ)Λ(ξ)
+K(x, ξ)Λ0(ξ)−K(x, ξ)Σ(ξ)+C(x, ξ)w( , ξ)dξ
whe e C(x, ξ)=









0 0 0
c(x, ξ)00
0 0 0










and Σ0(x)=










0 0 0
σ21(x)00
0 0 0










. Since his has o be e i ied o all
w( ,x), his yields he ollowing equa ions o be sol ed
0=K(x,0)Λ(0)Q0(14)
0= Σ0(x)−Σ(x)+K(x,x)Λ(x)−Λ(x)K(x,x) (15)
0= Λ(x)Kx(x, ξ)+Kξ(x, ξ)Λ(ξ)
+K(x, ξ)Λ0(ξ)+ Σ0(x)K(x, ξ)−K(x, ξ)Σ(ξ)+C(x, ξ)
(16)
on he iangula domain Tde ined by (11). De eloping he
equa ion (16) yields he ollowing 5 PDE
λ2(x)k22
x+λ2(ξ)k22
ξ=−λ0
2(ξ)k22 +σ32(ξ)k23 (17)
λ2(x)k23
x−µ(ξ)k23
ξ=σ23(ξ)k22 +µ0(ξ)k23 (18)
−µ(x)k31
x+λ1(ξ)k31
ξ=σ21(ξ)k32 +σ31(ξ)k33 −λ0
1(ξ)k31
(19)
−µ(x)k32
x+λ2(ξ)k32
ξ=−λ0
2(ξ)k32 +σ32(ξ)k33 (20)
−µ(x)k33
x−µ(ξ)k33
ξ=σ23(ξ)k32 +µ0(ξ)k33 (21)
whe e, o no a ional con enience, ki j(x, ξ) was abb e ia ed
o ki j. This also yields he ollowing exp ession o c(x, ξ)
c(x, ξ)=k22(x, ξ)σ21(ξ)+k23(x, ξ)σ31(ξ) (22)
Equa ions (14) and (15) gi e he bounda y condi ions o
equa ions (17)-(21) as ollows
q2λ2(0)k22(x,0) =µ(0)k23(x,0) (23)
k23(x,x)=−σ23(x)
λ2(x)+µ(x)(24)
k31(x,x)=σ31(x)
λ1(x)+µ(x)(25)
k32(x,x)=σ32(x)
λ2(x)+µ(x)(26)
µ(0)k33(x,0) =q1λ1(0)k31(x,0) +q2λ2k32(x,0) (27)
A di ec applica ion o [14, Theo em 4] wi h, on he one
hand
F1=k22 F2=k23
and, on he o he hand
F1=k33 F2=k32 F3=k31
yields he ollowing lemma.
Lemma 3.1: Conside he hype bolic sys em (17)-(21)
wi h bounda y condi ions (23)-(27). Unde he assump ions
λ1, λ2, µ ∈ C1([0,1]),∀i,jσi j ∈ C0([0,1])
he e exis s a unique con inuous solu ion K.
In pa icula , c(x, ξ)=k22(x, ξ)σ21(ξ)+k23(x, ξ)σ31(ξ) is a
con inuous unc ion on T, and he e o e i is bounded on T
which is c i ical in he p oo o s abili y o Sec ion IV.
D. In e se ans o ma ion
The in e ibili y o ans o ma ion (13) is p o ed in [12],
along wi h he boundedness o he in e se ans o ma ion
ope a o . The in e se ans o ma ion eads
w( ,x)=γ( ,x)+Zx
0
L(x, ξ)γ(x, ξ)dξ(28)
whe e he in e se ke nel Lis implici ly de ined on Tby he
ollowing in eg al equa ion
L(x, ξ)=K(x, ξ)−Zx
ξ
K(x,s)L(s, ξ)ds (29)
Gi en he pa icula o m o K,Lhas he ollowing o m
L=









000
l2,1l2,2l2,3
l3,1l3,2l3,3










whe e he li j a e con inuous unc ions on T.
IV. Con ol law and main esul
A. S abili y o he a ge sys em
Be o e s a ing he main esul , we p o e exponen ial
s abili y o he a ge sys em in he ollowing Lemma.
Lemma 4.1: Conside sys em (8)-(9)-(10) wi h bounda y
condi ions (12) and ini ial condi ions u0
1,u0
2and 0. Unde
he assump ions
λ1, λ2, µ ∈ C1([0,1]), σ21 ∈ C0([0,1]),c∈ C0(T),
u0
1,u0
2, 0∈ L2([0,1]),(30)
he o igin is exponen ially s able in he L2-no m.
P oo Conside he ollowing candida e Lyapuno unc ion
V( )=Z1
0
pe−δxhα1( ,x)2+α2( ,x)2i+e−δxβ( ,x)2dx (31)
whe e pand δa e s ic ly posi i e eal numbe s o be
de e mined. Di e en ia ing Vwi h espec o ime yields
˙
V( )=
Z1
0h−pe−δx(λ1(x)α1( ,x)α1x( ,x)+λ2(x)α2( ,x)α2x( ,x))
−pe−δxσ21(x)α2( ,x)α1( ,x)+eδxµ(x)β( ,x)βx( ,x)
−pe−δxα2( ,x)Zx
0
c(x,s)α1( ,s)ds#dx
=h−pe−δxλ1(x)α1( ,x)2−pe−δxλ2(x)α2( ,x)2
+eδxµ(x)β( ,x)2i1
0
+Z1
0hp(λ0
1(x)−δλ1(x))e−δxα1( ,x)2
+p(λ0
2(x)−δλ2(x))e−δxα2( ,x)2−(µ0(x)+δµ(x))eδxβ( ,x)2
−2pe−δxσ21(x)α2( ,x)α1( ,x)idx
−Z1
0Zx
0
2pe−δxα2( ,x)c(x,s)α1( ,s)dsdx (32)
Deno ing
kck∞=max
(x,s)∈T
|c(x,s)|,
he las e m can be uppe -bounded as ollows
−Z1
0Zx
0
2pe−δxα2( ,x)c(x,s)α1( ,s)dsdx
≤pkck∞ Z1
0Zx
0
e−δxα2
1( ,s)dsdx
+Z1
0Zx
0
e−δxα2( ,x)2dsdx!
=pkck∞ Z1
0
α2
1( ,s)Z1
s
e−δxdxds
+Z1
0Zx
0
e−δxα2( ,x)2dsdx!
=pkck∞ Z1
0
α1( ,x)2e−δx−e−δ
δdx
+Z1
0Zx
0
e−δxα2( ,x)2dsdx!
≤pkck∞ Z1
0
α1( ,x)2e−δx
δdx +Z1
0Z1
0
e−δxα2( ,x)2dsdx!
=pkck∞ Z1
0
α1( ,x)2e−δx
δdx +Z1
0
e−δxα2( ,x)2dsdx!
(33)
Plugging (33) in o (32) and using he bounda y condi ions
yields
˙
V( )≤hpλ1(0)q2
1+pλ2(0)q2
2−µ(0)iβ2( ,0)
+Z1
0 α1( ,x)
α2( ,x)!T
P(x) α1( ,x)
α2( ,x)!e−δxdx +Z1
0
ρ(x)eδxβ( ,x)2dx
(34)
wi h
P(x)= p(λ0
1(x)−δλ1(x)) +pkck∞
δ−pσ21(x)
−pσ21(x)p(λ0
2(x)−δλ2(x)) +pkck∞!
(35)
and
ρ(x)=(µ0(x)+δµ(x)) (36)
We now seek 2 pa ame e s pand δsuch ha he ollowing
inequali ies a e sa is ied
phλ1(0)q2
1+λ2(0)q2
2i−µ(0) <0 (37)
and, o all x∈[0,1]
λ0
1(x)−δλ1(x)+kck∞
δ<0 (38)
λ0
2(x)−δλ2(x)+kck∞<0 (39)
(µ0(x)+δµ(x)) <0 (40)
"λ0
1(x)−δλ1(x)+kck∞
δ#λ0
2(x)−δλ2(x)+kck∞
−σ21(x)2>0
(41)
Fi s , we pick
0<p<µ(0)
λ1(0)q2
1+λ2(0)q2
2
so ha (37) is sa is ied. Besides, inequali ies (38)-(39)-(40)-
(41) ew i e
λ1(x)δ2−λ0
1(x)δ− kck∞>0 (42)
λ2(x)δ−λ0
2(x)− kck∞>0 (43)
µ(x)δ+µ0(x)>0 (44)
λ1(x)λ2(x)δ3−λ1(x)λ0
2(x)+kck∞λ1(x)+λ0
1(x)λ2(x)δ2
+hkck∞λ0
1(x)− kck∞λ2(x)+λ0
1(x)λ0
2(x)−σ21(x)2iδ
+kck∞λ0
2(x)+kck∞>0.(45)
Inequali ies (42)-(43)-(44)-(45) a e sa is ied o a su icien ly
la ge δ. Indeed, since assump ions (30) hold, all he anspo
speeds, hei de i a i es, and he sou ce e ms a e uppe -
bounded in absolu e alue, and he e exis s such ha
∀x∈[0,1] µ, λi(x)>  > 0i=1,2
Thus, o all x∈[0,1], P(x) in (35) is posi i e de ini e, ρ(x)
in (36) is s ic ly posi i e and he e exis s such ha (34)
yields
˙
V≤ −V( ) (46)
B. Con ol law and main esul
F om ans o ma ion (13) e alua ed a x=1, one ge s
U( )= ( ,1)
−Z1
0
k31(1, ξ)u1( , ξ)+k33(1, ξ)u2( , ξ)+k33(1, ξ) ( , ξ)dξ
(47)
We now s a e he main esul o he pape
Theo em 4.2: Conside sys em (3)-(4)-(5) wi h bounda y
condi ions (6)-(7), ini ial condi ions u0
1,u0
2, 0, and he con ol
law de ined by (47). Unde he ollowing assump ions
λ1, λ2, µ ∈ C1([0,1]),∀i,jσi j ∈ C0([0,1]),
u0
1,u0
2, 0∈ L2([0,1])
he o igin is exponen ially s able in he L2sense.
P oo F om he con inui y o he in e se backs epping
ans o ma ion, we ha e he ollowing uppe bound (see,
e.g., [13])
kw( ,·)k2
L2([0,1]) ≤(1+kLk∞)kγ( ,·)k2
L2([0,1])
Besides, om (46), one ge s
kγ( ,·)k2
L2([0,1]) ≤e− kγ(0,·)k2
L2([0,1])
which concludes he p oo .
V. Discussion and pe spec i es
We ha e p esen ed a backs epping con ol design o
a pa icula 3 ×3 linea hype bolic sys em wi h a ying
coe icien s, yielding exponen ial s abili y o he o igin in
he L2sense. As explained in Sec ion III-A, he esul
may be gene alized o (n+1) ×(n+1) sys ems whe e he
con olled s a e has a nega i e anspo speed and he no he
s a es ha e a posi i e speed. Such a esul would exploi he
ac ha , no ma e how s ong he coupling is,2an n×n
homodi ec ional sys em wi h 0 inpu a he inle bounda y is
always s able. This can be seen by conside ing a Lyapuno
unc ion o he o m V( )=R1
0
n
P
i=1
e−δxpiu( ,x)2dx, wi h
su icien ly small piand su icien ly la ge δ. The coupling
be ween he con olled s a es and he nuncon olled s a es
would be supp essed by he backs epping ans o ma ion.
This is a di ec ion o u u e wo k.
Impo an ly, he p oposed eedback law equi es ull-s a e
measu emen . This assump ion is no ealis ic o he consid-
e ed applica ion, whe e senso s a e expensi e and di icul
o ins all, and e en mo e di icul o main ain. Bounda y
measu emen is a much mo e likely scena io. Usually, oil
p oduc ion acili ies a e ela i ely well equipped a hei ou -
le , whe e p essu e, low and densi y measu emen s may be
a ailable. When bo om p essu e senso s a e ins alled, hese
a e used in ela i ely simple eedback loops (PI con olle s)
o s abilize he low, wi h success [8], [9]. Thus, he case o
in e es o he design o mo e ad anced con ol law, is he
one whe e he senso s a e loca ed a he ou le , i.e. a he
igh bounda y o he domain.
Un o una ely, we ha e no been able ye o design a
colloca ed con olle o his 3 ×3 sys em because he
app oach we ha e ollowed in his pape , elying on he
2×2 case, does no seem o ex end o he obse e design.
Howe e , acco ding o [10], a su icien condi ion o exac
obse abili y o he quasilinea sys em is ha he senso is
2as long as he anspo speeds a e C1and he coupling e ms a e C0
unc ions.

loca ed whe e he mos quan i ies “exi ” he domain. In ou
case, i is he igh bounda y, and he condi ion is ul illed.
This gi es us con idence ha i may possible o design an
obse e o he 3 ×3 linea case wi h a ying coe icien .
Re e ences
[1] G. Bas in and J.-M. Co on. Fu he esul s on bounda y eedback
s abiliza ion o 2x2 hype bolic sys ems o e a bounded in e al.
P oceedings o IFAC Nolcos 2010, Bologna, I aly, 2010.
[2] S.P.C. Bel oid, R.J. P an De Linden, G.J.N. Albe s, and
R. Aasheim R., Schulkes. Se e e slugging in oil-well simula ions
using a d i lux code. pages 357–371, 2010. ci ed By (since 1996)
0.
[3] C. E. B ennen. Fundamen als o mul iphase low. Camb idge Uni
P ess, 2005.
[4] J.-M. Co on. Con ol and Nonlinea i y. Ame ican Ma hema ical
Socie y, 2007.
[5] J.M. Co on, G. Bas in, and B. DAnd ´
ea-No el. Dissipa i e bounda y
condi ions o one-dimensional nonlinea hype bolic sys ems. SIAM
Jou nal on Con ol and Op imiza ion, 47(3):1460–1498, 2008.
[6] R. Cou an and D. Hilbe . Me hods o Ma hema ical Physics, Vol. II.
Wiley-In e science, New Yo k, 1962.
[7] F. Di Meglio, G.-O. Kaasa, N. Pe i , and V. Als ad. Slugging in
mul iphase low as a mixed ini ial-bounda y alue p oblem o a
quasilinea hype bolic sys em. 2011 Ame ican Con ol Con e ence
( o appea ), 2011.
[8] J.-M. Godha n, M. P. Fa d, and P. H. Fuchs. New slug con ol
s a egies, uning ules and expe imen al esul s. Jou nal o P ocess
Con ol, 15:547–557, 2005.
[9] K. Ha e, K.O. S o nes, and H. S ay. Taming slug low in pipelines.
ABB e iew, 4:55–63, 2000.
[10] T. T. Li. Con ollabili y and Obse abili y o Quasilinea Hype bolic
Sys ems, olume 3. Highe Educa ion P ess, Beijing, 2009.
[11] X. Li ico and V. F omion. Bounda y con ol o hype bolic conse -
a ion laws using a equency domain app oach. In Decision and
Con ol, 2006 45 h IEEE Con e ence on, pages 5341 –5346, dec. 2006.
[12] W. Liu. Bounda y eedback s abiliza ion o an uns able hea equa ion.
SIAM Jou nal on Con ol and Op imiza ion, Vol. 42, No. 3:1033–1043,
2003.
[13] R. Vazquez and M. K s ic. Con ol o Tu bulen and Magne ohyd o-
dynamic Channel Flows. Sp inge , 2008.
[14] R. Vazquez, M. K s ic, and J.-M. Co on. Backs epping bounda y
s abiliza ion and s a e es ima ion o a 2x2 linea hype bolic sys em. in
P oceedings o he 50 h Con e ence on Decision and Con ol, O lando
( o appea ), 2011.
[15] R. Vazquez, M. K s ic, J.-M. Co on, and G. Bas in. Local Exponen-
ial H2S abiliza ion o a 2x2 Quasilinea Hype bolic Sys em using
Backs epping. in P oceedings o he 50 h Con e ence on Decision
and Con ol, O lando ( o appea ), 2011.
[16] C.-Z. Xu and G. Salle . Exponen ial s abili y and ans e unc ions
o p ocesses go e ned by symme ic hype bolic sys ems. ESAIM:
Con ol, Op imisa ion and Calculus o Va ia ions, 7:421–442, 2002.
Appendix
A. Linea iza ion o he d i - lux model o gas-liquid low
Conside a small a ia ion δζ( ,x) a ound an equilib ium
p o ile ¯
ζ(x). Neglec ing second-o de e ms in δζ, Sys em (1)
becomes
∂δζ
∂ +A(¯
ζ(x))∂δζ
∂x+˜
S(x)δζ =0 (48)
wi h
˜
S(x)= ∂A
∂ζ1
(¯
ζ)¯
ζ0(x)∂A
∂ζ2
(¯
ζ)¯
ζ0(x)∂A
∂ζ3
(¯
ζ)¯
ζ0(x)!
In (48), A(¯
ζ(x)) is diagonalizable, i.e.
L(x)A(¯
ζ(x)) = Λ(¯
ζ(x))L(x)
whe e L(x) is a ma ix o le eigen ec o s and Λ(x)=










λ1(¯
ζ(x)) 0 0
0λ2(¯
ζ(x)) 0
0 0 µ(¯
ζ(x))










he ma ix o anspo
speeds. Thus, conside ing he change o a iables
χ=L(x)δζ
and le -mul iplying (48) by L(x) yields
∂χ
∂ + Λ(x)∂χ
∂z=−˜
Σ(x)χ(49)
wi h
˜
Σ(x)=L(x)˜
S(x)L−1(x)+A(¯u(x)(L−1)0(x)
The exp ession o ˜
Σ(x) is oo complica ed o be w i en in
de ails. Rema kably, he hi d line o ˜
Σis only illed wi h 0.
Indeed, he o iginal s a e a iable ζ1is a Riemann in a ian .
This s uc u e is p ese ed by he p eceding ans o ma ion,
and χ1=u1is also a Riemann in a ian o (49)3. Thus, we
deno e
˜
Σ = 









000
˜σ2,1˜σ2,2˜σ2,3
˜σ3,1˜σ3,2˜σ3,3










Finally, ollowing [1], we de ine he ollowing exp essions
ϕ2(x)=exp Zz
0
˜σ2,2(s)
λ2(s)ds!, ϕ3(x)=exp −Zz
0
˜σ3,3(s)
µ(s)ds!,
ϕ(x)=ϕ1(x)
ϕ2(x)
and make he ollowing change o a iables
u1=χ1,u2=ϕ2(x)χ2, =ϕ3(x)χ3
This yields sys em (3)-(4)-(5) wi h










000
σ2,10σ2,3
σ3,1σ3,20










=









0 0 0
ϕ2(x) ˜σ2,1(x) 0 ϕ(x) ˜σ2,3(x)
ϕ3(x) ˜σ3,1(x)ϕ−1(x) ˜σ3,2(x) 0










Deno ing δζ( ,0) =δζ(0), he linea ized bounda y condi ions
ead
∂hl
∂ζ1
(¯
ζ(0))δζ1(0) +∂hl
∂ζ2
(¯
ζ(0))δζ2(0) +∂hl
∂ζ3
(¯
ζ(0))δζ3(0) =0
∂hl
∂ζ1
(¯
ζ(L))δζ1(L)+∂hl
∂ζ2
(¯
ζ(L))δζ2(L)+∂hl
∂ζ3
(¯
ζ(L))δζ3(L)
+∂hl
∂ζ3
(¯
ζ(L))δZ( )=0
Thus,
∂hl
∂ζ (¯
ζ(0))L−1(¯
ζ(0)) 









1 0 0
0ϕ−1
2(0) 0
0 0 ϕ−1
3(0)










w( ,0) =0
∂h
∂ζ (¯
ζ(L))L−1(¯
ζ(L)) 









ϕ−1
1(L)00
0ϕ−1
2(L) 0
0 0 1










w( ,L)=
−∂hl
∂ζ3
(¯
ζ(L))δZ( )
3This can be e i ied by compu ing explici ly Σo , simply, by linea izing
he ollowing equa ion, e i ied by ζ1:∂ζ1
∂ ( ,x)+λ1( ,x)∂ζ1
∂x( ,x)=0.
E en ually, he bounda y condi ions can be exp essed as
ollows















u1(0)
u2(0) != q1
q2! (0)
u2(L)=q q0 u1(L)
u2(L)!+kδZ( )
Se ing U( )=q q0 u1(L)
u2(L)!+kδZ( ) yields equa-
ions (6)-(7).