Finite-time boundary stabilization of general linear hyperbolic balance laws via Fredholm backstepping transformation $\star$

0.16 MB·

by  Jean-Michel Coron

Finite-timeboundarystabilizationofgenerallinear
hyperbolicbalancelawsviaFredholmbackstepping
⋆
transformation
7
Jean-MichelCorona,LongHub,GuillaumeOlivec
1
0
2 aSorbonne Universit´es, UPMC UnivParis 06, UMR7598, Laboratoire Jacques-Louis Lions,4 placeJussieu, F-75005 Paris,

n France.
a
J bSchool ofMathematics, Shandong University, Jinan, Shandong 250100, China.

8 cUniversit´e deBordeaux, UMR5251, Institut deMath´ematiques de Bordeaux, 351cours delaLib´eration, F-33405 Talence,
1
France.

]
C
O
Abstract
.
h
t Thispaperisdevotedtoasimpleandnewproofontheoptimalfinitecontroltimeforgenerallinearcoupledhyperbolicsystem
a byusingboundary feedback on oneside. The feedback control law isdesigned byfirst usinga Volterra transformation of the
m
secondkindandthenusinganinvertibleFredholmtransformation.Bothexistenceandinvertibilityofthetransformationsare
[ easily obtained.

1
v Keywords: Boundarystabilization; Coupled hyperbolicsystems;Optimal finitetime; Fredholm transformation.
7
6
0
5 1 Introduction where
0
1. Inthispaper,weinvestigatethe stabilizationofthe fol- Λ− =diag(λ1,··· ,λm), Λ+ =diag(λm+1,··· ,λn),
0 lowingn×n linearcoupledhyperbolic system:
7 arediagonalsubmatricesand
1
: ut(t,x)+Λ(x)ux(t,x)=Σ(x)u(t,x), λ (x)<···<λ (x)<0<λ (x)<···<λ (x),
v 1 m m+1 n
rXi tu−∈((t0,1,+)=∞F),(xu∈(t)()0,,1u)+,(t,0)=Qu−(t,0), (1) fmor∈al{l1x,.∈..,[0n,−1]1.}N.Fotineatlhlya,tthweemaastsruimxΣe t∈haCt0n([0≥,12])na×ndn
a
where u = (utr,utr)tr is the state and F is the feed- caonudptlehsetchoenesqtaunattiomnastroifxthQe ∈sysRte(mn−min)s×idme cthouepdleosmtahine
− +
back. We assume that the matrix Λ ∈ C1([0,1])n×n is equationsofthe systemonthe boundary.
diagonal:Λ=diag(λ ,··· ,λ )andsuchthatλ (x)6=0
1 n i
and λ (x) 6= λ (x) for every x ∈ [0,1], for every i ∈ Note that the Riesz representationtheorem shows that
i j
{1,...,n} andfor everyj ∈{1,...,n}\{i}.Therefore, everyboundedlinearfeedbackF ∈L(L2(0,1)n,Rm)has
withoutlossofgenerality,we assumethat necessarilythe form
Λ− 0 n 1
Λ= , Fu= f (x)u (x)dx ,
ij j
0 Λ+! j=1Z0 
X 1≤i≤m
⋆  u=(u ,...,u )tr ∈L2(0,L)n, (2)
1 n
Emailaddresses: [email protected] (Jean-Michel
Coron), [email protected] (LongHu), for some fij ∈ L2(0,1), i ∈ {1,...,m}, j ∈ {1,...,n}.
[email protected] (Guillaume Olive). We can prove that, with this type of boundary condi-
Preprint 19 January 2017
tions,theclosed-loopsystem(1)iswell-posed:forevery authorsprovideacounterexamplethatshowsthatthere
F ∈L(L2(0,1)n,Rm)andu0 ∈L2(0,1)n,there exists a exist linear hyperbolic balance laws, which are control-
unique (weak)solutionu∈C0([0,+∞);L2(0,1)n)to lablebyopen-loopboundarycontrols,butareimpossible
tobe stabilizedunder this kindofboundaryfeedback.
u (t,x)+Λ(x)u (t,x)=Σ(x)u(t,x),
t x
This limitation can be overcome by using the so-called
u−(t,1)=F (u(t)), u+(t,0)=Qu−(t,0), backstepping method, which connects the original sys-
u(0,x)=u0(x), (3) tteiems (teo.ga. teaxrpgoentesnytsiatelmstawbiitlhityd)esviiraabaleVsotlatebrirliatytrparnospfoerr--
t∈(0,+∞), x∈(0,1), mation the second kind. This method was introduced
 apnadrtidceuvlaelro,pthedesbeymMin.alKarrsttiiccleasn[d3,1h8is,2c3o]-awnodrktehrest(usteoer,iianl
Thepurposeofthispaperistofindafull-statefeedback book[15]).In[10],theauthorsdesignedafull-statefeed-
control law F such that the corresponding closed-loop backcontrollaw,withactuationononlyonesideofthe
system (1) vanishes after some time, that is such that boundary, in order to achieve H2 exponential stability
thereexistsT >0suchthat,foreveryu0 ∈L2(0,1)nfor oftheclosed-loop2×2quasilinearhyperbolicsystemby
the solutionu∈C0([0,+∞);L2(0,1)n)to (3),wehave usingVolterra-typebacksteppingtransformation.More-
over,withthismethodwecanevensteerthecorrespond-
ing linearized hyperbolic system to rest in finite time,
u(t)=0, ∀t≥T, (4)
that is what is called finite time stabilization. The pre-
sented method can also be extended to linear systems
andto obtainthe besttime T suchthat(4)holds.
withonlyonenegativecharacteristicvelocity(see[11]).
In[13],afully generalcaseofcoupledheterodirectional
The boundary stabilization problem of 1-D hyperbolic
hyperbolicPDEs,allowinganarbitrarynumberofPDEs
systems have been widely investigated in the literature
convecting in each direction and the boundary controls
foralmosthalfacentury.Thepioneerworksdatebackto
appliedon one side, is presented.The proposedbound-
[20] and [21] for linear coupled hyperbolic systems and
arycontrolsalsoyieldthefinite-timeconvergencetozero
[22], [12] for the corresponding nonlinear setting, espe-
withthe controltime givenby
cially for the quasilinear wave equation. For such sys-
tems, many articles are based on the boundary condi-
1 1 m 1 1
tionswith the followingspecific form
t = dx+ dx. (6)
F
λ (x) |λ (x)|
Z0 m+1 i=1Z0 i
X
u (t,0) u (t,1)
+ +
=G , (5) However,this time t is largerthan the theoretical op-
u−(t,1)! u−(t,0)! timaloneweexpectFandthatis givenin[17],namely
where G : Rn → Rn is a suitable smooth vector func- 1 1 1 1
tion.Withthisboundarycondition(5),twomethodsare T = dx+ dx. (7)
opt
λ (x) |λ (x)|
distinguished to deal with the stability problem of the Z0 m+1 Z0 m
linear and nonlinear hyperbolic system. The first one
is the so-called characteristic method, which allows us In[1],theauthorsfoundaminimumtimestabilizingcon-
to estimate the related bounds along the characteristic troller which makes the coupled hyperbolic system (1)
curves.Thismethod waspreviouslyinvestigatedin[12] withconstantcoefficientsvanishesafterTopt byslightly
for 2×2 systems and in [19,16,24] for a generalization changing the target system in [13], in which only local
to n×n homogeneous nonlinear hyperbolic systems in cascadecouplingterms areinvolvedinthe PDEs.
the framework of C1 norm. The second one is the con-
trolLyapunovfunctionmethod,whichwasintroducedin Inthispaper,weshowthatthiskindofcontrollercanbe
[5,6,7]toanalyzetheasymptoticbehaviorofthenonlin- establishedinamucheasierway.Inspiredbytheknown
earhyperbolicequationsinthecontextofC1andH2so- resultsof[13]and[14],wewillmaptheinitialcoupledhy-
lutions.Bothofthesetwoapproachesguaranteetheex- perbolicsystem(1)toanewtargetsysteminwhichthe
ponentialstabilityofthenonlinearhomogeneoushyper- cascade coupling terms of the previous works (namely,
bolicsystemsprovidedthattheboundaryconditionsare G(x)β(t,0) in [13] and Ω(x)β(t,x) in [1]) can be com-
dissipative to some extent. Dissipative boundary con- pletely cancelled. Our strategy is to first transform (1)
ditions are standard static boundary output feedback to the target system of [14] by a Volterra transforma-
(thatis,afeedbackofthestatevaluesattheboundaries tionofthesecondkind,whichisalwaysinvertibleifthe
only). However, there is a drawback of these boundary kernelbelongstoL2.Then,regardingthetargetsystem
conditionswheninhomogeneoushyperbolicsystemsare obtainedas the initial hyperbolic systemto be studied,
considered, especially the coupling of which are strong byusingaFredholmtransformationasintroducedin[9],
enough.In Section 5.6 ofthe recent monograph[2], the we then map this intermediate system to a new target
2
system,vanishingafterT ,withoutanycouplingterms whereG ∈L∞(0,1)m×m hasthe cascadestructure
opt 1
in the PDEs other than a simple trace coupling term.
Moreover, the existence and the invertibility of such a
0 ··· ··· 0
transformationwill be easily proved(we point out here
thatthesetransformationsarenotalwaysinvertible,see g ... ... ...
21
[c8a]s,cbaudtesthtriuscwtuilrleinofdtehedekbeerntehleincvaoslevehderientohuarnFkrsedtohotlhme G1 = ... ... ... ..., (10)
 
transformation).Finally,thetargetsystemandtheorig-  
inal system share the same stability properties due to gm1 ··· gmm−1 0
 
the invertibilityofthe transformation.
for some g ∈ L∞(0,1), i ∈ {2,...,m}, j ∈
ij
The mainresultofthis paperis the following: {1,...,i−1}, and G2 ∈ L∞(0,1)(n−m)×m. We
recall that, for every H ∈ L(L2(0,1)n,Rm) and
Theorem1 There exists F ∈ L(L2(0,1)n,Rm) γ0 ∈ L2(0,1)n, there exists a unique (weak) solution
such that, for every u0 ∈ L2(0,1)n, the solution γ ∈C0([0,+∞);L2(0,1)n)to(8)satisfyingγ(0,·)=γ0.
u∈C0([0,+∞);L2(0,1)n)to (3)satisfies
Taking into accountthe form of the feedbacks (see (2))
u(t)=0, ∀t≥T , we can use the standard backstepping method and es-
opt
tablishthe following result,in the exactsame wayas it
whereT is given by (7). wasdone in[14]forthe caseH =0:
opt
Remark1 We recall that this result has already been Lemma1 There exist G ∈ L∞(0,1)n×n with the
obtained in [1] in the case of constant matrices Λ and structure (9)-(10) and an invertible bounded lin-
Σ.Therefore,Theorem1generalizesthisresult.Wealso ear map V : L2(0,1)n −→ L2(0,1)n such that,
believe that, even in the case of constant matrices, the for every H ∈ L(L2(0,1)n,Rm), there exists F ∈
approach we shall present below, based on an invertible L(L2(0,1)n,Rm) such that, for every u0 ∈ L2(0,1)n, if
Fredholm transformation and a simple target system, is γ ∈ C0([0,+∞),L2(0,1)n) denotes the solution to (8)
easier than the one presented in [1], where a Volterra satisfying theinitial data γ(0,·)=V−1u0,then
transformationandadifferenttargetsystemareused.In
particular, we do not need repeatedly use the successive u(t)=Vγ(t),
approximation approach tofindthekernels in thetrans-
is thesolution totheCauchy problem (3).
formation.
Forthe restofthe paper, Gis fixedasinLemma1.
Therestofthe paperisorganizedasfollows.InSection
2, we first recall the results of [14] and then we present
In[14],theauthorschosethesimplestpossibilityH =0
a new target system which vanishes after the optimal
sothat,duetothe cascadestructure(9)-(10),anysolu-
time T . Then, in Section 3, we prove the existence
opt tion to the resulting system (8) defined at time 0 van-
ofaninvertibleFredholmtransformationthatmapsthe
ishesafterthetimet givenby(6)(see[14,Proposition
target system introduced in [14] into the new designed F
2.1] for more details). However, this appears to be not
targetsystem.
the bestchoicesince itdoes notgivethe expectedopti-
mal time T . In the present paper, we will show how
opt
2 Newtarget system to properly choose H in order to reduce the vanishing
timetoT .Forthispurpose,theideaistoapplyasec-
opt
In[14,Section2.1]theauthorsintroducedthefollowing ondtimethebacksteppingmethodandfindaFredholm
targetsysteminthe particularcaseH =0: mappingthattransformstheprevioustargetsystem(8)
intothe followingnew targetsystem:
γ (t,x)+Λ(x)γ (t,x)=G(x)γ(t,0),
t x
z (t,x)+Λ(x)z (t,x)=G(x)z(t,0),
γ (t,1)=H(γ(t)), γ (t,0)=Qγ (t,0), (8) t x
− + −
t∈(0,+∞), x∈(0,1), tz−∈(t(,01,)+=∞0),, xz∈+((0t,,01)),=eQz−(t,0), (11)
wThheerme aγtr=ix(Gγ−tr∈,γL+tr∞)tr(0i,s1t)hne×nstiasteaalonwdeHr trisiaangfeueladrbamcak-. wherez = (ztr,ztr)tr is the state and G ∈ L∞(0,1)n×n
− +
trixwith the followingstructure isthe followingmatrix
e
G 0 0 0
1
G= , (9) G(x)= , (12)
G2 0! G2(x) 0!
e
3
where G is defined in (9). We recall that, for every L2(0,1)n:
2
z0 ∈ L2(0,1)n, there exists a unique (weak) solution
z ∈ C0([0,+∞);L2(0,1)n) to (11) satisfying z(0,·) =
z0.Moreoverone hasthe followingproposition:
1
Proposition1 For every z0 ∈ L2(0,1)n, the solution Fz(x)=z(x)− K(x,y)z(y)dy,
z ∈C0([0,+∞);L2(0,1)n)to(11)satisfyingz(0,·)=z0 Z0
x∈(0,1), z ∈L2(0,1)n, (14)
verifies z(t)=0for every t≥T .
opt
Proof. Indeed, using the method of characteristics
and the cascade structure (12) of G, one first gets withakernelK ∈L2((0,1)×(0,1))n×nwiththefollow-
1
that z−(t) = 0 for t ≥ 0 1/|λm(x)|dx and then that ingstructure:
z (t)=0fort≥T . ✷ e
+ opt
R
We willprovethe followingresult:
K 0
1
Proposition2 There exist an invertible bounded K = , (15)
linear map F : L2(0,1)n −→ L2(0,1)n and H ∈ 0 0!
L(L2(0,1)n,Rm) such that, for every γ0 ∈ L2(0,1)n, if
z ∈ C0([0,+∞),L2(0,1)n) denotes the solution to (11)
satisfying theinitial data z(0,·)=F−1γ0, then
inwhichK ∈L2((0,1)×(0,1))m×misalowertriangular
1
matrixwith0diagonalentries,thatishasthefollowing
γ(t)=Fz(t),
cascadestructure
is thesolution to (8)satisfying γ(0,·)=γ0.
Remark2 In Lemma 1 it is showed that we can reach
0 ··· ··· 0
system(1)fromsystem(8)whateverthefeedbackH is,F
beingfixedconsequently.Notethat thereis nosuchfree- k ... ... ...
21
dom inProposition 2asweneedtheboundarycondition K1 = .. .. .. .., (16)
z (t,1)=0inacrucialwayfortheproof,see(18)below.  . . . .
−  
 
Combiningalltheaforementionedresults,itisnoweasy km1 ··· kmm−1 0
 
toobtainTheorem1:
Proof of Theorem 1.LetF andH be the twomappings
for some k ∈ L2((0,1)×(0,1)), i ∈ {2,...,m}, j ∈
provided by Propositon 2 and then let V and F be the ij
{1,...,i−1}, yet to be determined. Note that F is
correspondingmappingsprovidedbyLemma1.Letz ∈
clearly invertible due to this very particular structure
C0([0,+∞),L2(0,1)n)bethesolutionto(11)associated
(seetheAppendixAfordetails).Therefore,weonlyhave
withthe initialdata z(0,·)=(V ◦F)−1u0.Then, tocheckthatγ definedby
u(t)=V ◦Fz(t), (13)
is the solution to the Cauchy problem (3). By Proposi- 1
γ(t,x)=z(t,x)− K(x,y)z(t,y)dy, (17)
tion 1, we know that z(t) = 0 for every t ≥ T and it
opt
readilyfollowsfrom(13)thatu(t)=0foreveryt≥T Z0
opt
aswell. ✷
Therefore, it only remains to establish Proposition 2. is solution to (8) for some H ∈ L(L2(0,1)n,Rm) to be
This isachievedinthe nextsection. determinedaswell.
3 Existenceofan invertible Fredholmtransfor-
mation
Letusfirstperformsomeformalcomputationstoderive
the equations that the k have to satisfy. Taking the
ij
In this section we prove Proposition 2. To this end, we derivative with respect to time in (17), using the equa-
look for a Fredholm transformation F : L2(0,1)n −→ tion satisfied by z (see the first line of (11)) and inte-
4
gratingby partsyield withthe condition
1 K(x,0)=(G(x)−G(x))Λ−1(0).
γ (t,x)=z (t,x)− K(x,y)z (t,y)dy
t t t
Z0
=−Λ(x)z (t,x)+G(x)z(t,0) In order to guarantee tehe well-posedness of the system
x
satisfiedbyK,weimposethefollowingextracondition:
1
+ K(x,y)Λ(y)ezy(t,y)dy
Z0 K−(0,y)=0, (19)
1
− K(x,y)G(y)z(t,0)dy
(whereK denotesthesubmatrixcontainingthefirstm
Z0 −
rowsofK),whichturnsouttoalsoimplythefollowing,
=−Λ(x)zx(t,x)e+G(x)z(t,0)+K(x,1)Λ(1)z(t,1) becauseofthestructuresofG(see(9))andK(see(15)),
1
−K(x,0)Λ(0)z(t,e0)− Ky(x,y)Λ(y)z(t,y)dy
Z0 G(x)K(0,y)=0,
1
− K(x,y)Λ (y)z(t,y)dy
y andtherefore makesthe kernelsystemmuchsimpler to
Z0 solve.To summarize,K willsatisfy the system
1
− K(x,y)G(y)z(t,0)dy.
Z0 Λ(x)K (x,y)+K (x,y)Λ(y)+K(x,y)Λ (y)=0,
x y y
e
K (0,y)=0,
−
NstoruwctoubrseesrvoefKtha(ts,eesin(1c5e)z)−a(ntd,1G)=(se0ea(n1d2)b),ecwaeuhseavoeftthhee K(x,0)=(G(x)−G(x))Λ−1(0),
followingtwoconditions: x,y ∈(0,1).
K(x,1)Λ(1)z(te,1)=0, (18)  e
Note that the structure (15) of K, (17) and (19) imply
that
K(x,y)G(y)=0.
γ(t,0)=z(t,0).
Therefore,
Therefore, the boundary condition at x = 0 for γ is
e
automaticallyguaranteed:
γ (t,x)=−Λ(x)z (t,x)+ G(x)−K(x,0)Λ(0) z(t,0)
t x
1 (cid:16) (cid:17) γ+(t,0)=z+(t,0)=Qz−(t,0)=Qγ−(t,0).
− K (x,y)Λ(y)+K(ex,y)Λ (y) z(t,y)dy.
y y
Z0 (cid:16) (cid:17) Now, because of the structures of K, G and G given in
Ontheotherhand,takingthederivativewithrespectto (15), (12) and (9) respectively, the system for K trans-
spacein(17)we have latesinto the followingsystemforK1:e
1
γx(t,x)=zx(t,x)− Kx(x,y)z(t,y)dy. Λ−(x)(K1)x(x,y)+(K1)y(x,y)Λ−(y)
Z0  +K1(x,y)(Λ−)y(y)=0,
As aresult,weobtain KK1((x0,,y0))==0−,G (x)Λ−1(0),
γt(t,x)+Λ(x)γx(t,x)−G(x)γ(t,0) 1 1 −
=(cid:16)G(x1)−K(x,0)Λ(0)−G(x)(cid:17)z(t,0) x,y ∈(0,1).
−e K (x,y)Λ(y)+K(x,y)Λ (y) Regarding y as the time parameter, this is a standard
y y
time-dependentuncoupledhyperbolicsystemwithonly
Z0 (cid:16) positive speeds λ (x)/λ (y) > 0, i,j ∈ {1,...,m},
+Λ(x)K (x,y)−G(x)K(0,y)) z(t,y)dy, i j
x and therefore it admits a unique (weak) solution
(cid:17) K ∈L2((0,1)×(0,1))m×m.Actually,usingthemethod
1
andtheright-handsidehastobezero.Thisyieldstothe ofcharacteristics,we see that the solutionis explicitely
followingkernelsystem givenby
Λ(x)Kx(x,y)+Ky(x,y)Λ(y) g φ−1 φ (x)−φ (y)
+K(x,y)Λy(y)−G(x)K(0,y)=0 kij(x,y)= ij i −iλ (y) j , (20)
(cid:0) (cid:0) j (cid:1)(cid:1)
5
if i ∈ {2,...,m}, j ∈ {1,...,i−1} and φi(x) ≤ φj(y), [4] Jean-Michel Coron. Stabilization of control systems and
andk (x,y)=0 otherwise,where nonlinearities. In Proceedings of the 8th International
ij
CongressonIndustrial andAppliedMathematics,pages 17–
x 1 40. Higher Ed. Press,Beijing,2015.
φi(x)= dξ, i∈{1,...,m}. [5] Jean-MichelCoronandGeorgesBastin.Dissipativeboundary
λ (ξ)
Z0 i conditions for one-dimensional quasi-linear hyperbolic
systems: Lyapunov stability for the C1-norm. SIAM J.
Note that φ is indeed invertible since it is a monoton- Control Optim., 53(3):1464–1483, 2015.
i
ically decreasing continuous function of x. Finally, we [6] Jean-Michel Coron, Georges Bastin, and Brigitte d’Andr´ea
readilyseefrom(20)that Novel. Dissipative boundary conditions for one-dimensional
nonlinear hyperbolic systems. SIAM J. Control Optim.,
47(3):1460–1498, 2008.
K (1,·)∈L2(0,1)m×m,
1 [7] Jean-Michel Coron, Brigitte d’Andr´ea Novel, and Georges
Bastin. A strict Lyapunov function for boundary control
sothatthe mapH :L2(0,1)n −→Rm givenby of hyperbolic systems of conservation laws. IEEE Trans.
Automat. Control, 52(1):2–11, 2007.
1 [8] Jean-Michel Coron, Long Hu, and Guillaume Olive.
Hγ =− K (1,y) F−1γ (y)dy, γ ∈L2(0,1)n, Stabilization and controllability of first-order integro-
1 −
Z0 differential hyperbolic equations. J. Funct. Anal.,
(cid:2) (cid:3) 271(12):3554–3587, 2016.
is well-defined and H ∈ L(L2(0,1)n,Rm). This con- [9] Jean-Michel Coron and Qi Lu¨. Local rapid stabilization for
cludes the proofofProposition2. ✷ a Korteweg-de Vries equation with a Neumann boundary
controlontheright. J.Math.PuresAppl.(9),102(6):1080–
1120, 2014.
Remark3 Let us conclude this paper by pointing out
[10]Jean-Michel Coron, Rafael Vazquez, Miroslav Krstic, and
that it would be very interesting to know the target sys- GeorgesBastin. LocalexponentialH2stabilizationofa2×2
tems that can be achieved with general linear transfor-
quasilinear hyperbolic system using backstepping. SIAM J.
mations. We recall that it is proved in [4] that, for the Control Optim., 51(3):2005–2035, 2013.
finitedimensional controlsystemy˙ =Ay+Bu,thetar-
[11]Florent Di Meglio, Rafael Vazquez, and Miroslav Krstic.
getsystemy˙ =Ay−λy+Bucanbeachievedbyalinear Stabilization of a system of n + 1 coupled first-order
transformation for every λ ∈ R, if we assume that it is hyperboliclinearPDEswithasingleboundaryinput. IEEE
controllable (which is a necessary condition to the rapid Trans. Automat. Control, 58(12):3097–3111, 2013.
stabilization). [12]JamesM.GreenbergandTatsienLi. Theeffectofboundary
damping for the quasilinear wave equation. J. Differential
Equations, 52(1):66–75, 1984.
Acknowledgements [13]Long Hu, Florent Di Meglio, Rafael Vazquez, and
Miroslav Krstic. Control of homodirectional and general
heterodirectional linear coupled hyperbolic pdes. IEEE
The authors thank Amaury Hayat and Shengquan Xi-
Trans. Automat. Control, 61:3301–3314, 2016.
angforusefulcomments.Thisprojectwassupportedby
[14]Long Hu, Rafael Vazquez, Florent Di Meglio, and
the ERC advancedgrant266907(CPDENL) of the 7th
Miroslav Krstic. Boundary exponential stabilization
Research Framework Programme (FP7), ANR Project of 1-d inhomogeneous quasilinear hyperbolic systems.
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A Invertibility ofthe Fredholmtransformation
For the completeness we prove in this appendix the in-
vertibilityofthe FredholmtransformationF.
Lemma2 ForanygivenK ∈L2((0,1)×(0,1))n×nwith
the cascade structure (15)-(16), the transformation F
defined by (14) is invertible. Moreover, its inverse has
thesameform:
1
F−1γ(x)=γ(x)− Θ(x,y)γ(y)dy,
Z0
x∈(0,1), γ ∈L2(0,1)n,
e
forsomeΘ∈L2((0,1)×(0,1))n×nwiththesamestruc-
tureas K,that is,
e
Θ 0
Θ= ,
0 0!
e
inwhichΘ∈L2((0,1)×(0,1))m×misalowertriangular
matrix with0diagonal entries as K :
1
0 ··· ··· 0
θ ... ... ...
21
Θ= ,
 ... ... ... ...
 
 
θm1 ··· θmm−1 0
 
 
for some θ ∈ L2((0,1)×(0,1)), i ∈ {2,...,m} and
ij
j ∈{1,...,i−1}.
Proof of Lemma 2. Let γ = Fz, where z ∈ L2(0,1)n is
given.Thanksto (15)and(17),we have
z =γ , ∀i∈{m+1,...,n}.
i i
7