Backstepping stabilization of an underactuated 3 X 3 linear hyperbolic system of fluid flow transport equations

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).