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boundary and in-domain couplings
Mathias Dus, Francesco Ferrante, Christophe Prieur
To cite this version:
Mathias Dus, Francesco Ferrante, Christophe Prieur. Spectral stabilization of linear transport equa-
tions with boundary and in-domain couplings. Comptes Rendus. Mathématique, Académie des sci-
ences (Paris), In press. �hal-03208688v2�
Draft
Spectral stabilization of linear transport equations with boundary and in-domain couplings
Mathias Dusa, Francesco Ferranteband Christophe Prieurc
aIMT Toulouse, 118 Route de Narbonne, 31400 Toulouse
bDepartment of Engineering, University of Perugia, Via G. Duranti, 67, 06125 Perugia cUniv. Grenoble Alpes, CNRS, Grenoble-INP, GIPSA-lab, F-38000, Grenoble, France E-mails:[email protected] (M. Dus), [email protected] (F. Ferrante), [email protected] (C. Prieur)
Abstract.In this work, the problem of stabilization of general systems of linear transport equations with in- domain and boundary couplings is investigated. It is proved that the unstable part of the spectrum is of finite cardinal. Then, using the pole placement theorem, a linear full state feedback controller is synthesized to stabilize the unstable finite-dimensional part of the system. Finally, by a careful study of semigroups, we prove the exponential stability of the closed-loop system. As a by product, the linear control constructed before is saturated and a fine estimate of the basin of attraction is given.
Keywords.Stabilization, Spectral analysis, Saturation, Pole Placement.
2020 Mathematics Subject Classification.93D05, 93D15, 93D20.
Funding.This work has been partially supported by MIAI@Grenoble Alpes (ANR- 19-P3IA-0003) . This article is a draft (not yet accepted!)
1. Introduction 1.1. Literature review
In the work, we investigate boundary stabilization of a class of linear first-order hyperbolic systems of Partial Differential Equations (PDEs) on a finite space domainx∈[0, 1]. Such systems are predominant in modeling of traffic flow [1], heat exchangers [32], open channel flow [3, Chapter 1.4] or multiphase flow [11, 13, 15]. The couplings between states traveling in opposite directions, both in-domain and at the boundaries, may induce instability leading to undesirable behaviors. For example, oscillatory two-phase flow regimes occurring on oil and gas production systems directly result, in some cases, from these mechanisms [13]. The dynamics of most of these industrial systems are described by nonlinear transport equations. If we linearize systems presented before, one obtains a system of equations of the form:
Rt+ΛRx =M R R1(t, 0) =u(t) R2(t, 1) =H R1(t, 1)
(1) whereR=(R1,R2)∈Rd1×Rd2and:
Λ= µΛ1 0
0 −Λ2
¶ ,M=
µM11M12 M21M22
¶ . Velocity matrices have dimensionsΛ1∈D+d
1(R),Λ2∈D+d
2(R) where for allm∈N,D+m(R) stands for the set of definite positive diagonal matrices. For couplings, dimensions areM12∈Md1×d2(R), M21∈Md2×d1(R),M11∈Md1×d1(R) andM22∈Md2×d2(R). It should be noticed that in most of the cases presented in the first paragraph, the linearized system is not homogeneous in the sense that matrices depend on the space variable.
To exponentially stabilize system (1), feedback controlsu(t) depending on the boundary val- uesR(t, 0),R(t, 1) were designed in the literature. Lyapunov techniques allows to establish expo- nential stabilization in Sobolev orCpspaces when termMis supposed to be small. Applications to linearized Saint Venant systems are given in [4, 10, 14, 19, 20].
However when the in-domain coupling term M is too large, simple quadratic Lyapunov function may not be found [2, 19]. Moreover, spectral analysis shows that when the entries of Mexceed a certain amplitude, the system is unstable for any control of the formu(t)=F R(t, 1) (F ∈Md1×d2(R)) [3, Proposition 5.2]. Note that in [3, Proposition 5.2], this was proven only for d1=d2=1 andM=
µ0c c 0
¶ .
To overcome this problem, one can relax the assumption of a control depending only on the value of the state at the boundary. Doing so, it is possible to construct a full-state feedback depending on the value of R on all the domain [0, 1]. In this work, we consider an integral feedback of the formu(t)=R1
0k(x)R(x)d xwherekis a kernel to be defined. As a consequence, to use the proposed method, one needs to measure the stateR on all the domain, which is sometimes impossible in industrial applications (see [11] for example). Some works [5, 12, 16, 18, 31] solve this difficulty designing a boundary observer of the stateR. Here for simplicity, it is assumed that the observation of the state is complete in order to focus only on the effect of the control.
The full-state feedback strategy has already succeeded in stabilizing hyperbolic systems like (1). One can cite [9,21] where authors use the backstepping method to locally stabilize quasilinear hyperbolic systems inH2. Additionally, backstepping can be used to stabilize system (1) in finite time [8]. For an introduction to this method, the book [23] gives a wide overview of the topic.
In this work, the method used differs from backstepping and is based on spectral theory applied to a well-behaved open-loop operator for which the spectrum is reduced to its point spectrum. Our analysis is greatly inspired from works [24, 25, 28] where the authors study a vast class of linear hyperbolic system with in domain and boundary couplings. In order to explain the great lines of the proof of exponential stabilization, it is needed to define the problem in a semigroup form. This is the object of the next two sections.
1.2. Preliminaries
LetH be an Hilbert space. The scalar product onH is denoted〈·,·〉H and the associated norm is given byk · kH.
The following formalism is taken from the book [30, Chapter 2]. Here, concepts are introduced without proof to ease the presentation. For more details, we refer to [30, Chapter 2].
1.2.1. Generator of a strongly continuous semigroup
The notion of semigroup is fundamental in this article. Its definition is given here:
Definition 1. A family(Tt)t≥0of operators inL(H)is a strongly continuous semigroup onH if:
• T0=I .
• ∀t,τ≥0,Tt+τ=TtTτ(the semigroup property)
• ∀z∈H, limt→0Ttz=z (the strong continuity property) The generator of the semigroup (Tt)t≥0is defined as follows:
Definition 2. Let D be the subset ofH such that:
D:=
½
R∈H |lim
t→0+
TtR−R
t exists inH
¾ . Then, the operatorA:D→H is defined such that:
AR= lim
t→0+
TtR−R
t ,∀R∈D.
This operator is called the generator of(Tt)t≥0and for the rest of the paper, we use the very classic notation Tt←eAt,D←D(A). It is also said thatAgenerates the semigroup eAt.
The Lumer-Phillips Theorem states necessary and sufficient conditions on an unbounded operator (A,D(A)) to generate a strongly continuous semigroup.
Theorem 3 (Lumer-Phillips). LetA be an unbounded operator defined on D(A)⊂H. The operatorAgenerates a strongly continuous semigroup eAtif and only if:
• D(A)dense inH
• A is closed
• ∀R∈D(A),〈R,AR〉H ≤ζkRk2H whereζ∈R. This property is called theζdissipativity of A.
• The resolvent set ofA ρ(A) :=©
λ∈C|λI−A is invertible and(λI−A)−1∈L(H)ª is not empty.
Remark 4. The operator (λI−A)−1appearing in the definition ofρ(A) is denoted byR(λ,A) in this paper. Moreover, one can easily generalize the notion of spectrum to unbounded operators;
the spectrum ofAis given by:
σ(A) :=C\ρ(A).
The point spectrum ofAis included inσ(A) and is given as:
σp(A) :={λ∈σ(A)|λI−A is not injective}.
IfR0∈D(A), it is not difficult to prove thatT(t)R0∈D(A) for all timet≥0 and:
d
d teAtR0=AeAtR0,∀t≥0.
In other words, if we noteR(t) :=eAtR0: d
d tR(t)=AR(t),∀t≥0. (2)
Hence, the semigroup representation can be very useful to treat PDEs when the initial data is in the domain of the operator considered.
1.2.2. The adjoint semigroup
WhenR0is not inD(A), things become more difficult since we are not allowed to differentiate eAtR0. In order to consider less regular solutions, we need to introduce duality. The adjoint semigroup is a key tool to understand this aspect.
Let us define the space:
D(A?) := (
ϕ∈H | sup
R∈D(A),R6=0
〈AR,ϕ〉H kRkH < ∞
)
and the adjoint operatorA?is defined by the Riesz representation theorem as:
〈AR,ϕ〉H= 〈R,A?ϕ〉H,∀R∈D(A),ϕ∈D(A?).
Hence whenRis supposed to be inH only, we can still defineARweakly, writing〈AR,ϕ〉 =
〈R,A?ϕ〉whenϕ∈D(A?) so that the problem (2) is written in the duality form:
〈 d
d tR(t),ϕ〉H= 〈R(t),A?ϕ〉H,∀ϕ∈D(A?).
Hereϕcorresponds to the test functions of our problem. Moreover and by density, one can prove thateAtR0is a solution to previous equation and we just found a solution to problem (2) when R0is inH only.
1.2.3. The embeddingH1d⊂H ⊂H−1
In this paper, the control operator is singular in the sense that it does not belong toL(U,H) (Uis the control space). Hence, we cannot use the bracket〈·,·〉H when the control operator is involved. Fortunately, one can generalize the notion of duality to a larger space thanH.
LetH1dbe a normed dense subset ofH. For allR∈H, theH−1norm ofRwrites:
kRkH−1:= sup
ϕ∈H1d,kϕkHd 1≤1
〈R,ϕ〉H. The spaceH−1is then defined as follows:
Definition 5. The spaceH−1is the completion ofH with respect to the normk · kH−1 and hence H1d⊂H ⊂H−1. Moreover, for all R∈H−1and all sequence(Rn)n of elements ofH such that limn→∞kR−RnkH−1=0, we define the duality bracket as:
∀ϕ∈Hd1,〈R,ϕ〉H−1,Hd
1 := lim
n→∞〈Rn,ϕ〉H.
The spaceH which defines the scalar product is called the pivot space. The spaceH−1is named as the dual ofH1dwith respect to the pivot spaceH.
One can extend the operatorAand its associated semigroup using the following theorem.
Theorem 6. Letλ∈ρ(A)and define:
H1d:=(D(A?),k( ¯λI−A?)· k)
andH−1be the completion ofH with respect to the normk(λI−A)−1·k. SpacesH1dandH−1are independent on the choice ofλ.
Additionally, the spaceH−1is the dual ofH1d with respect to the pivot spaceH. Moreover, one can extend the operatorAso that (keeping the same notation for the extension)A∈L(H,H−1) and the associated semigroup(eAt)can be extended inL(H−1).
Remark 7. In the proof of last theorem, the extension ofA is built fromA??.
As a consequence, if the operator of controlBis inL(U,H−1), it is easy to define a solution to:
d
d tR=AR+Bu
withu∈L2([0,T],U) (T>0). Using the duality bracket, one can write the last equation as:
〈d
d tR,ϕ〉H−1,Hd
1 = 〈R,A?ϕ〉H−1,Hd
1 + 〈Bu,ϕ〉H−1,Hd
1,∀ϕ∈D(A?) and the closed-loop problem is well-defined.
1.3. The abstract problem
Now that the semigroup framework has been recalled, we can apply it to our problem. Let H=L2([0, 1];Cd) whered:=d1+d2, be the base space embedded with the usual scalar product:
〈f,g〉H:=
d
X
i=1
Z 1
0
fi(x)gi(x)d x.
The corresponding norm onH is denotedk · kH. The open-loop operatorA is given by:
½D(A)=©
R∈H |R0∈H,R1(0)=0,R2(1)=H R1(1)ª AR= −ΛR0+M R.
It is easy to check (left for the reader) thatA is a closed densely defined operator that satisfies the hypothesis of Lumer-Phillips Theorem. Hence, it generates a strongly continuous semigroup denoted (eAt)t≥0onH. For its adjointA?, it can be shown that:
½D(A?)=©
R∈H |R0∈H,R2(0)=0,Λ1R1(1)=HTΛ2R2(1)ª A?R=ΛR0+MTR.
The control spaceL2(R+,U) :=L2(R+,Rd1) (U:=Rd1) is embedded with the canonical norm ofL2(R+,Rd1). To define the control operator, we introduce the spaceH−1which is the dual of D(A?) when we takeH as pivot space, namely:
H−1:=(D(A),kR(λ,A)· kH).
whereλ∈ρ(A) is taken arbitrarily in the resolvant set ofA. Moreover, the primal ofH−1is denoted:
H1d:=(D(A?),k(λI−A?)· kH).
The control operatorB∈L(U,H−1) writes:
Bu:=(p
Λ1u, 0d2)δ(x)∈H−1
whereδis the usual Dirac delta distribution. Its dualB?∈L(Hd1,U) then writes:
∀ϕ∈Hd1,B?ϕ=p
Λ1ϕ1(x=0) (3) whereϕ1is the function gathering the firstd1components ofϕ.
Note that in this paper, we will not make the difference betweenA,A?and their canonical extension (A)??, (A?)??inL(H,H−1).
With this notation, the abstract evolution problem onH−1is defined here (see Definition 9 for a rigorous definition) :
d R
d t =AR+Bu(t) R(0)=R0.
(4) The notation being introduced, one can present the main theorem of this work and the rest of the paper is dedicated to its proof.
Theorem 8. There exists a linear feedback control of the form u:=KR withK ∈L(H−1,U)for which there exists C,δ>0such that for all initial condition R0∈H−1, the system(4)is well-posed and the corresponding unique solution R∈C(R+,H−1)to(4)verifies:
kR(t)kH−1≤C e−δtkR0kH−1,∀t≥0.
A sketch of the proof of this exponential stabilizability result is given below.
• First, we prove thatA has only a discrete spectrum with a finite number of unstable eigenvalues.
• Then, the focus is on the unstable finite-dimensional part M of the system using a projection on the unstable eigenspace. Proving a controlability result for the operator (A|M,B|M), it is possible to use the pole placement theorem to find a state feedback control stabilizing the unstable part.
• Finally, we prove that the whole closed-loop system is well-posed and exponentially stable. This is not immediate since the control synthesized from the finite-dimensional unstable part can destabilize the remaining one.
The article is organized as follows:
Outline: In Section 2, we give the definition of solution, state and prove an admissibility condition for well-posedness. Section 3 is dedicated to the spectral study of the open-loop operatorA. Then, Section 4 gives a rigorous proof of Theorem 8. Next, the control constructed in Theorem 8 is saturated in a certain sense and we give a local stability result in Section 5. In the same part, numerical illustrations of the results previously presented, are exposed. Finally, as a conclusion, perspectives and open problems are stated.
2. Solution definition and admissibility condition
Letu∈L2(0,T,U) be given. The definition of a solution to (4) is proposed below:
Definition 9. The function R∈C1([0,T],H−1)∩C([0,T];H)is a solution to(4)if for allϕ∈H1d
〈R(t),ϕ〉H−1,Hd
1 = 〈R0,eA?tϕ〉H−1,Hd
1 +
Z t
0 〈u(s),B?eA?(t−s)ϕ〉U,Ud s,∀0≤t≤T. (5) whereUis identified with its dual.
The following lemma is an admissibility result allowing to prove that the solution has regularity inH.
Lemma 10. The admissibility property holds for all time T≥0:
Z T
0 kB?eA?(T−t)ϕk2Ud t≤C2kϕk2H,∀ϕ∈H. (6) where C is a constant depending on the parameters of the problem.
Proof. Letϕ∈H1d; the caseϕ∈H is easily deduced by a density argument. Letz(t) :=etA?ϕ the solution to:
zt−Λzx =MTz
z2(0)=0, Λ1z1(1)−HTΛ2z2(1)=0 z(t=0) =ϕ.
We define the functionalV by:
V(z) := Z 1
0
zT1z1e−γx+z2Tz2e−γ(1−x)d x= Z 1
0
zTΓzd x
whereΓ(x) :=diag(e−γxId1,e−γ(1−x)Id2) andγ>0 will be chosen later. Using integration by parts, we get:
dV(z) d t =
Z 1 0
zTtΓz+zTΓztd x
= Z 1
0
(Λzx+MTz)TΓz+zTΓ(Λzx+MTz)d x
=2 Z1
0
zTΛΓzxd x+ Z 1
0
zT(ΓMT+MΓ)zd x
=[zTΛΓz]10− Z 1
0
zTΛΓxzd x+ Z1
0
zT(ΓMT+MΓ)zd x.
Using the fact thatz∈H1dfor all time, the boundary terms are estimated below:
[zTΛΓz]10= −z1(0)TΛ1z1(0)+z2(0)TΛ2z2(0)e−γ +z1(1)TΛ1z1(1)e−γ−z2(1)TΛ2z2(1)
= −z1(0)TΛ1z1(0)
+z2(1)TΛ2HTΛ−11HTΛ2z2(1)e−γ−z2(1)TΛ2z2(1).
Thus, forγ>0 large enough:
[zTΛΓz]10≤ −1
2z1(0)TΛ1z1(0)= −1
2kB?etA?ϕk2U where the last equality comes from the definition (3) ofB?. Hence:
dV d t ≤ −1
2kB?etA?ϕk2U− Z1
0
zTΛΓxzd x+ Z 1
0
zT(ΓMT+MΓ)zd x.
Using the fact thatΓxΛ= −γΓ|Λ|, the following estimate onV holds:
dV d t ≤ −1
2kB?eA?tϕk2U+Cm,γV whereCm,γdepends onM,γ. Integrating, one obtains:
V(z(t))−V(z(0))eCm,γt≤ −1 2
Z t
0 kB?eA?sϕk2UeCm,γ(t−s)d s
which immediately gives the existence of a constantCdepending ontand the parameters of the problem such that:
C2V(z(0)=ϕ)≥ Z T
0 kB?eA?(T−t)ϕk2Ud t
which is the required result sinceV is equivalent to the square norm onH. The proof of the
general case ofϕ∈H follows by density ofH1dinH.
By [7, Theorem 2.37], Lemma 10 gives the following results.
Lemma 11. If u∈L2(0,T,U)then there exists a unique solution to(4)in the sense of Definition 9.
The aim of this paper is to stabilize (4) showing the existence of an admissible operator (the notion of admissible operator will be defined latter)K :H−1→Usuch that:
½d R
d t =(A+BK)R R(0)=R0
(7) is well-posed and its solution verifies the bound:
kR(t,·)kH−1≤C e−δtkR0kH−1,∀t≥0 whereδ,C>0 do not depend onR0.
3. Spectral analysis of the open-loop problem
Before going into the proof of Theorem 8, we need some information the open-loop operatorA. More precisely, it is proved thatA has a spectrum reduced to its point spectrum with a finite number of unstable eigenvalues.
Proposition 12. The spectrum of the open-loop operator verifies:
• σ(A)=σp(A).
• There exists r>0such that:
σp(A)⊂{z∈C| ℜz<r}.
• The unstable part of the spectrum;σp(A)∩{λ∈C| ℜλ≥0}has a finite cardinal.
The proof will be the object of this section.
3.1. Structure of the spectrum
The first lemma states that the spectrum of A is in fact its point spectrum. Its proof is an adaptation of the proof of [24, Lemma 2.2].
Lemma 13. The spectra ofA consists of isolated eigenvalues of finite geometric multiplicity ie σ(A)=σp(A).
Proof. Letλ∈Cand consider the unique solution to:
−ΛR0+M R=λR R1(0) =0 R2(1) =H R1(1).
(8) In particular, by defining:
T(x,y,λ)=e−Λ−1(λId−M)(x−y). The solution to (8) is given by:
R(x)=T(x, 0,λ)(0,Id2)Tv(0)
wherev(0)∈Cd2. In order to satisfy the right border boundary condition, we need to impose:
(H,−Id2)T(1, 0,λ)(0,Id2)Tv(0)=0d1. Hence, denoting:
U(λ) :=(H,−Id2)T(1, 0,λ)(0,Id2)T,
the point spectrum of the system is given by the zeros of the following characteristic equation:
det(U(λ))=0. (9)
Moreover, for allλ∈σp(A), the corresponding eigenspace is given by:
Ei g(A,λ)=©
T(x, 0,λ)(0,Id2)Tv(0)|v(0)∈ker(U(λ))ª .
The geometric multiplicity is less thand2(the dimension ofv(0)) and forλ∈C\σp(A):
R(λ,A)G=T(x, 0,λ)(0,Id2)v(0)− Z x
0
T(x,y,λ)Λ−1Gd y,∀G∈H (10) with:
v(0)=U(λ)−1(H,−Id2) Z 1
0
T(x,y,λ)Λ−1Gd y.
With (10), it is easy to see that R(λ,A) is bounded inL(H) whenλ∉σp(A) and hence
σ(A)=σp(A).
Remark 14. In the proof of the previous lemma, we have shown that:
R(λ,A)=U(λ)−1E(λ),∀λ∈ρ(A) (11) whereE(λ) is aH valued entire function ofλ.
The expression (11) gives immediately that:
Lemma 15. The algebraic multiplicity ofλ∈σ(A)is given by the multiplicity of the zeros of κ(λ) :=det(U(λ)).
3.2. Analysis of the unstable part of the spectrum
In this section, we end the proof of Proposition 12. The result given next ensures that the characteristic equation of the spectrum (9) can be approximated by the same equation removing the effect of non-diagonal 0th order term coming from the matrixMfor which the spectrum is known. To be clear, the velocity matrix can always be decomposed as follows:
Λ=blockdiag( ¯λiIδi)1≤i≤n1+n2 whereδi>0,δi ∈Nsuch thatPn1
1 δi=1=d1,Pn2
i=n1+1δi =d2. Moreover, for all 1≤i ≤n1, ¯λi>0 and forn1+1≤i ≤n2, ¯λi <0. Concerning the matrixM, one can also use the same type of decompositionM=( ¯Mi j)1≤i,j≤n1+n2where ¯Mi j∈Mδi×δj(R). The matrixM0is then defined by:
M0:=blockdiag( ¯Mi i)1≤i≤n1+n2. andA0the operator inH:
½D(A0)=©
R∈H |R0∈H,R1(0)=0,R2(1)=H R1(1)ª A0R= −ΛR0+M0R.
Obviously, we can prove Lemma 13 for operatorA0and exhibit a characteristic equation for A0:
κ0(λ) :=det(U0(λ))=0. (12)
where:
U0(λ) :=(H,−Id2)e−Λ−1(λId−M0)(0,Id2)T. (13)
By simple computations:
det(U0(λ))=det(−e−Λ−21(λ−M0,22))
where M0,22 :=(M0,i j)d1<i,j≤d1+d2 ∈ Md2×d2(R) and this functional does not have zeros. This means that the operatorA0generates a semigroup andρ(A0)=C. We present the following result taken from [24] which states that spectrum ofA0andAare closed for large imaginary part:
Lemma 16. Let r>0be such thatCr:={z∈C| |ℜz| ≤r}. We have the following:
|ℑλ|→∞lim |U(λ)−U0(λ)| =0 (14) and the convergence is uniform onCr for all r>0.
Moreover, we have the following property which will be exploited for the complex functional κ0defined in (12):
Lemma 17. [24] Let f be an exponential polynomial of the form f(λ)=Pr
i=1ajebkλ(λ,aj ∈ C,bj∈R). Let Z:=©
λ∈C|f(λ)=0ª
denotes the zero set of f. For allδ>0,α,β∈Rwithα<βthere exists a constant m(δ,α,β)>0such that for allλ∈Csatisfying dist(λ,Z)>δ,α< ℜλ<β, we have
|f(λ)| >m(δ,α,β).
A direct consequence of previous lemma is that on all strips of the form α < ℜλ < β, inf|κ0(λ)| >0 (κ0does not admit zeros and is an exponential polynomial).
Corollary 18. For allα<β, we have
λ∈C,α<ℜλ<βinf |κ0(λ)| >0.
To conclude, Rouché’s Theorem is recalled here:
Theorem 19 (Rouché). Let U⊂Cbe an open connected set and f,g two meromorphic functions on U with finite number of zeros and poles. Letγbe a closed smooth curve in U that does not intersect the set of zeros of f or g and that forms the border∂K of a compact set K . If
|f(z)−g(z)| < |g(z)|,∀z∈γ, then:
Zf−Pf =Zg−Pg
where Zf,Zg designate the number of zeros of f and g in K and Pf,Pg designate the number of poles of f and g in K .
By applying Rouché’s Theorem tof =κandg=κ0and using Lemma 16 and Corollary 18, one has thatκ(λ) has zeros located near the real axis. Owing this and the fact thatκ(λ) is an entire function, it has a finite number of zeros in the right-half plane. Combining this with Lemma 13, we easily conclude on the proof of Proposition 12.
4. Proof of Theorem 8
Let us denote M the finite-dimensional unstable generalized eigenspace (Jordan blocks) of A andM0 its topological complement. We denote by α:=dim(M). Let P :H →M be the projection ontoM defined as [22, Theorem 6.17]:
P= − 1 2iπ
I
ΓR(λ,A)dλ
whereΓis any contour enclosing the unstable eigenvalues ofA (this is possible because of the separation of unstable eigenvalues). The following technical lemma allows to extendP as an element ofL(H−1,H):
Lemma 20. The projector P∈L(H)can be extended as an elementP of˜ L(H−1,H). Moreover,
RanP˜=RanP=M. (15)
Proof. First, we prove thatPhas an extension inH−1. We define ˜Pby duality, as follows:
∀R∈H−1,ϕ∈Hd1,〈P R,˜ ϕ〉H−1,Hd1:= 〈R,P?ϕ〉H−1,Hd1.
The operator ˜Pis obviously an extension ofPinH−1. Now we will prove that ˜P∈L(H−1,H) and to do so let us takeR∈H−1,ϕ∈Hd1andλ∈ρ(A).
〈P R,ϕ〉H−1,H1
d= 〈R,P?ϕ〉H−1,H1
d
≤ kRkH−1kP?ϕkH1
d
≤ kRkH−1k( ¯λI−A?)P?ϕkH. The operator ( ¯λI−A?)P?writes:
( ¯λI−A?)P?=( ¯λI−A?) 1 2πi
Z
Γ( ¯ξI−A?)−1dξ
= 1 2πi
Z
Γ( ¯ξ−λ¯)( ¯ξI−A?)−1dξ∈L(H) where we used the resolvant formula from [6, Proposition 3.18].
Hence,
〈P R,ϕ〉H−1,H1
d≤ kRkH−1k( ¯λI−A?)P?kL(H)kϕkH andP∈L(H−1,H) withkPkL(H−1,H)≤ k( ¯λI−A?)P?kL(H).
Finally, we have to prove (15). Let R∈H−1and let (Rn)n be a sequence of elements ofH converging toRinH−1. AsM is of finite dimensionα, there exists a family (ei)i which is an orthogonal basis ofM.
Letλ∈ρ(A). In the finite-dimensional spaceM,λI−A as an operator fromM toM is an automorphism (it can be identified to a matrix with Jordan blocks). We define:
½fi:=(λI−A)ei∈M
f˜i:=R( ¯λ,A?)R(λ,A)fi∈Hd1.
and (fi)i, ( ˜fi)iare still a basis ofMsince they are the image of a basis by an automorphism.
On can writeP Rnin the basis (fi)i:
P Rn= Xα
i=1
βinfi where theβinare the coefficient ofP Rnin this basis. Forj≤α:
〈P Rn, ˜fj〉H−1,Hd
1 =Pα
i=1βin〈fi, ˜fj〉H−1,Hd
1
=Pα
i=1βin〈fi,R( ¯λ,A?)R(λ,A)fj〉H−1,Hd
1
=Pα
i=1βin〈R(λ,A)fi,R(λ,A)fj〉H−1,Hd
1
=Pα
i=1βin〈ei,ej〉H−1,Hd
1
=Pα
i=1βin〈ei,ej〉H
=βnj.
Aside from that, 〈P Rn, ˜fj〉H−1,Hd
1 = 〈Rn,P?f˜j〉H−1,Hd
1 →n→+∞ 〈R,P?f˜j〉H−1,Hd
1 since lim+∞Rn =R inH−1. Consequently, for all j ≤α the sequence (βj)n is convergent and its limit is〈R,P?f˜j〉H−1,Hd
1. To conclude, we have that:
P R˜ = Xα
i=1
〈R,P?f˜j〉H−1,Hd
1
fi which is an element ofM. This concludes the proof of the lemma.
From now on, we will not make a difference between P and its extension ˜P keeping the notation without tilda.
Corollary 21. The operator P is an element ofL(H−1).
Proof. The proof is immediate owing Lemma 20 and the fact thatH is continuously embedded
inH−1.
AsP andA commute, it is possible to decomposeA on the topological sumML
M0=H and the abstract stabilization problem becomes:
½ PR˙=PAP R+PBu(t)
(I−P) ˙R=(I−P)A(I−P)R+(I−P)Bu(t). (16)
4.1. Stabilization of the finite-dimensional part
First, we stabilize the finite-dimensional part without considering the infinite-dimensional part takingu(t) of the formu(t)=K P R(t) whereK is a matrix of dimensiond1×α. Hence, we have to solve a finite-dimensional stabilization problem where the open-loop matrix is the restriction denotedAMofAonMand the control matrix isBM=PB∈Mα×d1(R).
Proposition 22. The system(AM,BM)is controllable. Hence, there exists K∈Md1×α(R)such that AM+BMK is Hurwitz.
Proof. We show the result by using the Fattorini-Hautus test (also known as the Popov–Belevitch–Hautus test). It is necessary to prove that:
ker(λI−AM?)∩kerBM?={0},∀λ∈C (17) which reduces the analysis to eigenspaces only (and not generalized eigenspaces). In order to prove (17), the eigenvalues ofA?are calculated. This is equivalent to solve:
ΛR0+MTR=λR
R1(1)=Λ−11 HTΛ2R2(1) R2(0)=0.
(18) To do so, we introduce the operator:
T˜(x,y,λ) :=eΛ−1(λId−MT)(x−y). The solution to (18) is given by:
R(x)=T˜(x, 0,λ)(Id1, 0)Tv?(0)
wherev?(0)∈Cd1. In order to satisfy the right-border boundary condition, it is needed to impose:
(−Id1,Λ−11 HTΛ2) ˜T(1, 0,λ)(Id1, 0)Tv?(0)=0d1. Hence the spectrum ofA?is given by the equation:
det( ˜U(λ)) :=det((−Id1,Λ−11 HTΛ2) ˜T(1, 0,λ)(0,Id2)T)=0, (19) where:
U˜(λ)=(−Id1,Λ−11HTΛ2) ˜T(1, 0,λ)(0,Id2)T.
Similarly toA, the spectrum ofA?corresponds to its point spectrum and for allλ∈σp(A?):
Ei g(A?,λ)=©T˜(x, 0,λ)(Id1, 0)Tv(0)|v(0)∈ker( ˜U(λ))ª . To conclude, it suffices to remark that for allλ∈σ(A?) and allv?(0)∈ker( ˜U(λ)):
B?T˜(x, 0,λ)(Id1, 0)Tv?(0)=0 ⇐⇒v?(0)=0.
This proves Fattorini’s condition:
ker(λI−A?)∩kerB?={0},∀λ∈C
which immediately implies (17). The finite-dimensional system (AM,BM) is controllable and we can apply a pole placement theorem to find a matrix gainKsuch thatAM+BMKis Hurwitz.
4.2. Well-posedness of the closed-loop system
Now we take a gain matrixKstabilizing the finite-dimensional part of (16), defineK :=K Pand system (7) split as follows:
½ PR˙=(PAP+PBK P)R
(I−P) ˙R=((I−P)A(I−P)+(I−P)BK P)R. (20) Our notion of solution is given in the following definition:
Definition 23. If R0∈His the initial data considered and T>0. The element R∈C1([0,T],H−1)∩ C([0,T];H)is a solution to(7)if for allϕ∈H1d
〈R(t),ϕ〉H−1,Hd
1 = 〈R0,eA?tϕ〉H−1,Hd
1 +
Z t
0 〈K P R,B?eA?(t−s)ϕ〉U,Ud s,∀0≤t≤T. (21) whereUis identified with its dual.
Proposition 24. There exists a unique solution to(7)in the sense of Definition 23.
Proof. LetT>0. We use a Banach-Picard fixed-point theorem proving existence and uniqueness at the same time. Let us defineT :C([0,T],H)→C([0,T],H) the application such that for all ϕ∈H:
〈(TR)(t),ϕ〉H= 〈R0,eA?tϕ〉H+ Z t
0〈K P R(s),B?eA?(t−s)ϕ〉U,Ud s,∀R∈C([0,T],H), 0≤t≤T.
LetR,Q∈C([0,T],H), we have:
〈(T(R−Q))(t),ϕ〉H= Z t
0 〈K P(R(s)−Q(s)),B?eA?(t−s)ϕ〉U,Ud s,∀0≤t≤T. By Cauchy-Schwartz inequality and Lemma 10:
〈(T(R−Q))(t),ϕ〉H ≤ qRt
0kB?e(t−s)A?ϕk2Ud s× qRt
0kK P(R(s)−Q(s))k2Ud s
≤ q
Ckϕk2H×q
tkK Pk2L(H,U)kR−Qk2C([0,T],H)
where we have used the fact thatK Pis inL(H,U). Indeed,Pis a projection, henceP∈L(H) andKis a matrix. As a consequence,
〈(T(R−Q))(t),ϕ〉H ≤Cp
TkK PkL(H,U)kR−QkC([0,T],H)kϕkH,∀0≤t≤T. TakingTsufficiently small (uniformly with respect toR0), it holds:
〈(T(R−Q))(t),ϕ〉H≤kR−QkC([0,T],H)
2 kϕkH,∀0≤t≤T.
As a consequence,
kT(R−Q))(t)kH ≤kR−QkC([0,T],H)
2 ,∀0≤t≤T.
We can apply Banach-Picard theorem to assert the existence of a unique fixed point ofT in C([0,T],H) forT sufficiently small. By a bootstrap argument, we conclude on the existence and uniqueness inC([0,T],H) for allT>0. This unique solution is denoted byRand forϕ∈H1d:
〈R(t),ϕ〉H−1,Hd
1 = 〈R0,eA?tϕ〉H−1,Hd
1 +
Zt
0 〈K P R(s),B?eA?(t−s)ϕ〉U,Ud s,∀0≤t≤T. (22) The equation (22) is equivalent to:
〈R(t),ϕ〉H−1,Hd
1 = 〈R0,eA?tϕ〉H−1,Hd
1 +
Z t
0 〈BK P R(s),eA?(t−s)ϕ〉H−1,Hd
1d s,∀0≤t≤T.
Owing the fact thatR∈C([0,T],H),B∈L(U,H−1) ands7→eA?sϕ∈C1([0,T],H1d), one deduces thatR∈C1([0,T],H−1). Moreover:
〈d R
d t (t),ϕ〉H−1,Hd
1 = 〈R0,A?eA?tϕ〉H+ 〈K P R(t),B?ϕ〉U,U
− Z t
0〈K P R(s),B?eA?(t−s)A?ϕ〉U,Ud s,∀0≤t≤T.
All terms in last equation are convergent because of Lemma 10. Indeed, Rt
0〈K P R(s),B?A?eA?(t−s)ϕ〉U,Ud s ≤C||A?ϕ||H× ||K P||L(H)||R||C([0,T],H)
≤C||ϕ||Hd
1 × ||K P||L(H)||R||C([0,T],H−1).
This concludes the proof of Proposition 24.
When the initial data is only known to be inH−1, we have the following well-posedness result:
Proposition 25. Let T >0. If R0∈H−1, there exists a unique solution in C([0,T],H−1)to(7)in the sense that for allϕ∈Hd1,(21)holds.
Proof. The proof is very similar to the one of Proposition 24 and this is why, we only give a sketch of the proof here.
Let us defineT :C([0,T],H−1)→C([0,T],H−1) the application such that for allϕ∈Hd1, 0≤ t≤T:
〈(TR)(t),ϕ〉H−1,H1
d= 〈R0,eA?tϕ〉H−1,H1
d+ Z t
0 〈K P R(s),B?eA?(t−s)ϕ〉U,Ud s,∀R∈C([0,T],H−1).
