HAL Id: hal-01162105
https://hal.archives-ouvertes.fr/hal-01162105
Submitted on 9 Jun 2015
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Indirect controllability of some linear parabolic systems of m equations with m – 1 controls involving coupling
terms of zero or first order
Michel Duprez, Pierre Lissy
To cite this version:
Michel Duprez, Pierre Lissy. Indirect controllability of some linear parabolic systems of m equations with m – 1 controls involving coupling terms of zero or first order. Journal de Mathématiques Pures et Appliquées, Elsevier, 2016, 106 (5), pp.905-934. �10.1016/j.matpur.2016.03.016�. �hal-01162105�
equations with m −1 controls involving coupling terms of zero or first order.
Michel Duprez∗,Pierre Lissy†
June 9, 2015
Abstract
This paper is devoted to the study of the null and approximate controllability for some classes of linear coupled parabolic systems with less controls than equations. More precisely, for a given bounded domain Ω in RN (N ∈ N∗), we consider a system of m linear parabolic equations (m> 2) with coupling terms of first and zero order, and m−1 controls localized in some arbitrary nonempty open subset ωofΩ. In the case of constant coupling coefficients, we provide a necessary and sufficient condition to obtain the null or approximate controllability in arbitrary small time. In the case m= 2and N = 1, we also give a generic sufficient condition to obtain the null or approximate controllability in arbitrary small time for general coefficients depending on the space and times variables, provided that the supports of the coupling terms intersect the control domainω. The results are obtained thanks to the fictitious control method together with an algebraic method and some appropriate Carleman estimates.
1 Introduction
1.1 Presentation of the problem and main results
LetT > 0, let Ωbe a bounded domain in RN (N ∈ N∗) of class C2 and let ω be an arbitrary nonempty open subset ofΩ. LetQT := (0, T)×Ω, qT := (0, T)×ω, ΣT := (0, T)×∂Ωandm>2.
We consider the following system ofm parabolic linear equations, where the m−1 first equations are controlled:
∂ty1= div(d1∇y1) +Pm
i=1g1i· ∇yi+Pm
i=1a1iyi+1ωu1 in QT,
∂ty2= div(d2∇y2) +Pm
i=1g2i· ∇yi+Pm
i=1a2iyi+1ωu2 in QT, ...
∂tym−1= div(dm−1∇ym−1) +Pm
i=1g1(m−1)i· ∇yi+Pm
i=1a(m−1)iyi+1ωum−1 in QT,
∂tym= div(dm∇ym) +Pm
i=1gmi· ∇yi+Pm
i=1amiyi in QT,
y1=. . .=ym= 0 on ΣT,
y1(0,·) =y01, . . . , ym(0,·) =y0m in Ω,
(1.1)
wherey0:= (y01, . . . , ym0)∈L2(Ω)mis the initial condition and u:= (u1, . . . , um−1)∈L2(QT)m−1 is the control. The zero and first order coupling terms(aij)16i,j6m and (gij)16i,j6m are assumed to
∗Laboratoire de Mathématiques de Besançon UMR CNRS 6623, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France,michel.duprez@univ-fcomte.fr,
†CEREMADE, UMR 7534, Université Paris-Dauphine & CNRS, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France,lissy@ceremade.dauphine.fr.
1.1 Presentation of the problem and main results June 9, 2015 M. Duprez, P. Lissy
be respectively inL∞(QT) and inL∞(0, T;W∞1(Ω)N). Given some l∈ {1, ..., m}, the second order elliptic self-adjoint operatordiv(dl∇)is given by
div(dl∇) =
m
X
i,j=1
∂i(dijl ∂j),
with
dijl ∈W∞1(QT), dijl =djil inQT, where the coefficientsdijl satisfy the uniform ellipticity condition
N
X
i,j=1
dijl ξiξj>d0|ξ|2in QT, ∀ξ∈RN,
for a constantd0>0.
In order to simplify the notation, from now on, we will denote by D:= diag(d1, . . . , dm), G:= (gij)16i,j6m∈ Mm(RN), A:= (aij)16i,j6m∈ Mm(R), B:=
diag(1, . . . ,1) 0
∈ Mm,m−1(R),
so that we can write System (1.1) as
∂ty= div(D∇y) +G· ∇y+Ay+1ωBu inQT,
y= 0 onΣT,
y(0,·) =y0 in Ω.
(1.2)
It is well-known (see for instance [16, Th. 3 & 4, p. 356-358]) that for any initial datay0∈L2(Ω)m andu∈L2(0, T;H−1(Ω))m−1, System (1.2) admits a unique solution y inW(0, T)m, where
W(0, T) :=L2(0, T;H01(Ω))∩H1(0, T;H−1(Ω)),→ C0([0, T];L2(Ω)). (1.3) Moreover, one can prove (see for instance [16, Th. 5, p. 360]) that if y0 ∈ H01(Ω)m and u ∈ L2(QT)m−1, then the solution yis in W22,1(QT)m, where
W22,1(QT) :=L2(0, T;H2(Ω)∩H01(Ω))∩H1(0, T;L2(Ω)),→ C0([0, T];H01(Ω)). (1.4) The main goal of this article is to analyse the null controllability and approximate controllability of System (1.1). Let us recall the definition of these notions. It will be said that
• System (1.1) is null controllable at time T if for every initial condition y0 ∈ L2(Ω)m, there exists a controlu∈L2(QT)m−1such that the solution yin W(0, T)mto System (1.1) satisfies
y(T)≡0 in Ω.
• System (1.1) isapproximately controllable at timeT if for every ε >0, every initial condition y0 ∈ L2(Ω)m and every yT ∈ L2(Ω)m, there exists a control u ∈ L2(QT)m−1 such that the solutiony inW(0, T)mto System (1.1) satisfies
ky(T)−yTk2L2(Ω)m 6ε.
Let us remark that if System (1.1) is null controllable on the time interval (0, T), then it is also approximately controllable on the time interval(0, T)(this is an easy consequence of usual results of backward uniqueness concerning parabolic equations as given for example in [10]).
Our first result gives a necessary and sufficient condition for null (or approximate) controllability of System (1.1) in the case of constant coefficients. This condition is the natural one that can be expected, since it only means that the last equation is coupled with at least one of the others.
Theorem1. Let us assume thatD,GandAareconstant in space and time. Then System(1.1) is null (resp. approximately) controllable at timeT >0if and only if there exists i0∈ {1, ..., m−1}
such that
gmi0 6= 0 or ami06= 0. (1.5) Our second result concerns the case of general coefficients depending on space and time variables, in the particular case of two equations and one space dimension (i.e. m= 2andN = 1), and gives a controllability result under some technical conditions on the coefficients (see (1.7) and (1.8)) coming from the algebraic solvability (see Section3.1). To understand why this kind of condition appears here, we refer to the simple example given in [26, Ex. 1, Sec. 1.3]. Let us emphasize that Condition (1.8), which is clearly technical since it does not even cover the case of constant coefficients, isgeneric as soon as we restrict to the coupling coefficients verifying g21 6= 0 on(0, T)×ω, in the following sense: it only requires some regularity on the coefficients and a given determinant, involving some coefficients and their derivatives, to be nonzero. Since this condition may seem a little bit intricate, we will give in Remark 1 some particular examples that clarify the scope of Theorem2.
Theorem2. Let us assume thatm= 2,N= 1,Ω = (0, L) withL >0, and consider the following system:
∂ty1=∂x(d1∂xy1) +g11∂xy1+g12∂xy2+a11y1+a12y2+1ωu in QT,
∂ty2=∂x(d2∂xy2) +g21∂xy1+g22∂xy2+a21y1+a22y2 in QT, y1(·,0) =y2(·,0) =y1(·, L) =y2(·, L) = 0 on (0, T), y1(0,·) =y01, y2(0,·) =y20, in (0, L),
(1.6)
wherey0= (y01, y02)∈L2(0, L)2 is the initial condition.
Then System (1.1) is null (resp. approximately) controllable at time T if there exists an open subset(a, b)× O ⊆qT where one of the following conditions is verified:
(i) Coefficients of System (1.6)satisfydi, gij, aij∈ C1((a, b),C2(O))fori, j= 1,2 and g21= 0 anda216= 0 in (a, b)× O,
1/a21∈L∞(O) in (a, b). (1.7)
(ii) Coefficients of System (1.6)satisfydi, gij, aij∈ C3((a, b),C7(O))fori= 1,2and
|det(H(t, x))|> C for every(t, x)∈(a, b)× O, (1.8) where
H:=
−a21+∂xg21 g21 0 0 0 0
−∂xa21+∂xxg21 −a21+ 2∂xg21 0 g21 0 0
−∂ta21+∂txg21 ∂tg21 −a21+∂xg21 0 g21 0
−∂xxa21+∂xxxg21 −2∂xa21+ 3∂xxg21 0 −a21+ 3∂xg21 0 g21
−a22+∂xg22 g22−∂xd2 −1 −d2 0 0
−∂xa22+∂xxg22 −a22+ 2∂xg22−∂xxd2 0 g22−2∂xd2 −1 −d2
.
(1.9)
Remark 1. (a) We will see during the proof that, in Item (ii) of Theorem 2, taking into account the derivatives of the appearing in (1.9), (3.5) and (3.9), and the regularity needed for the control in Proposition 3.4, we need only the following regularity for the coefficients:
di∈ C1((a, b),C2(O)), gij ∈ C0((a, b),C2(O)), aij ∈ C0((a, b),C1(O)) (1.10)
1.1 Presentation of the problem and main results June 9, 2015 M. Duprez, P. Lissy
for alli, j∈ {1,2}in the first case (i) and
d1∈ C1((a, b),C4(O))∩ C2((a, b),C1(O)), g11, g12∈ C0((a, b),C4(O))∩ C1((a, b),C1(O)), a11, a12∈ C0((a, b),C3(O))∩ C1((a, b),C0(O)), g21∈ C0((a, b),C7(O))∩ C3((a, b),C1(O)), d2, g22∈ C0((a, b),C6(O))∩ C2((a, b),C2(O)), a22∈ C0((a, b),C5(O))∩ C2((a, b),C1(O)), a21∈ C0((a, b),C6(O))∩ C3((a, b),C0(O)),
(1.11)
in the second case(ii).
(b) One can easily compute explicitly the determinant of matrixH appearing in (1.8):
det(H) = 2∂a∂x21∂d∂x2g221−4∂a∂x21d2∂g∂x21g21+∂2a21
∂x2 d2g212 + 2a21∂a∂x21d2g21−∂a∂x21g221g22+∂a∂t21g221
−4a21∂d∂x2∂g∂x21g21+a21∂2d2
∂x2g212 +a221∂d∂x2g21−3a21d2∂2g21
∂x2 g21+ 6a21d2(∂g∂x21)2−2a221d2∂g∂x21 +a21∂g∂x21g21g22
−a21∂g21
∂t g21−a21g221∂g∂x22+∂a∂x22g213 −∂2d2
∂x2
∂g21
∂x g212 −2∂d∂x2∂2g21
∂x2 g221+ 3∂d∂x2(∂g∂x21)2g21−d2∂3g21
∂x3 g212 +5d2∂g∂x21∂2g21
∂x2 g21−4d2(∂g∂x21)3+∂g∂x21g221∂g∂x22+∂g21
∂x2g221g22−∂g∂x212g21g22−∂g∂x∂t21g221+∂g∂x21∂g∂t21g21
−g213 ∂g∂x222.
(c) We remark that Condition (1.8) implies in particular that
g216= 0 in (a, b)× O. (1.12) Our conjecture is that, as in the case of constant coefficients, either the first line of (1.7) or (1.12) is sufficient as soon as we restrict to the class of coupling terms that intersect the control region, since it is the minimal conditions one can expect (as in the case of constant coefficients, this only means that the last equation is coupled with one of the others).
(d) Even though Condition (1.8) seems complicated, it can be simplified in some cases. Indeed, for example, System (1.6) is null controllable at timeT if there exists an open subset(a, b)×O ⊆qT such that
g21≡κ∈R∗ in (a, b)× O, a21≡0 in (a, b)× O,
∂xa226=∂xxg22 in (a, b)× O.
Another simple situation is the case where the coefficients depend only on the time variable. In this case, it is easy to check that Condition (1.8) becomes simply: there exists an open interval (a, b)of(0, T)such that
g21(t)∂ta21(t)6=a21(t)∂tg21(t) in (a, b).
(e) In fact, it is likely that one could obtain a far more general result than the one obtained in Theorem 2. By using the same reasoning, one would be able to obtain a result of controllability for arbitrarymandN, however the generic Condition (1.8) would be far more complicated and in general impossible to write down explicitly. That is the reason why we chose to treat only the case m= 2andN = 1.
This paper is organized as follows. In Section 1.2 we recall some previous results and explain precisely the scope of the present contribution. In Section1.3we present the main method used here, that is to say the fictitious control method together with some algebraic method. Section 2 is devoted to the proof of Theorem 1. We finish with the proof of Theorem 2 in Section3.
1.2 State of the art
The study of what is called the indirect controllability for linear or nonlinear parabolic coupled systems has been an intensive subject of interest these last years. The main issue is to try to control many equations with less controls than equations (and ideally only one control if possible), with the hope that one can actindirectly on the equations that are not directly controlled thanks to the coupling terms. For a recent survey concerning this kind of control problems, we refer to [7]. Here, we will mainly present the results related to this work, that is to say the case of the null or approximate controllability of linear parabolic systems with distributed controls.
First of all, in the case of zero order coupling terms, a necessary and sufficient algebraic condition is proved in [5] for the controllability of parabolic systems, for constant coefficients and diffusion coefficientsdithat are equal to the identity matrices. This condition is similar to the usual algebraic Kalman rank condition for finite-dimensional systems. These results were then extended in [6], where a necessary and sufficient condition is given for constant coefficients but different diffusion coefficientsdi(the Laplace operator∆can also be replaced by some general time-independent elliptic operator). Moreover, in [5], some results are obtained in the case of time-dependent coefficients under a sharp sufficient condition which is similar to the sharpSilverman-Meadows Theorem in the finite- dimensional case.
Concerning the case of space-varying coefficients, there is currently no general theory. In the case where the support of the coupling terms intersect the control domain, the most general result is proved in [19] for parabolic systems in cascade form with one control force (and possibly one order coupling terms). We also mention [4], where a result of null controllability is proved in the case of a system of two equations with one control force, with an application to the controllability of a nonlinear system of transport-diffusion equations. In the case where the coupling regions do not intersect the control domain, only few results are known and in general there are some technical and geometrical restrictions (see for example [1], [3] or [27]). These restrictions come from the use of the transmutation method that requires a controllability result on some related hyperbolic system. Let us also mention [8], where the authors consider a system of two equations in one space dimension and obtain a minimal time for null controllability, when the supports of the coupling terms do not intersect the control domain.
Concerning the case of first order coupling terms, there are also only few results. The first one is [22], where the author studies notably the case ofmcoupled heat equations withm−1control forces and obtains the null controllability of System (1.1) at any time when the following estimate holds:
kukH1(Ω)6Ckg21· ∇u+a21ukL2(Ω), (1.13) for all u ∈ H01(Ω). Let us emphasize that inequality (1.13) is very restrictive, because it notably implies that g21 has to be nonzero on each of its components (due to the H1−norm appearing in the left-hand side). Another case is the null controllability at any time T >0 of m equations with one control force, which is studied in [19], under many assumptions: the coupling matrixG has to be upper triangular (except on the controlled equation) and many coefficients ofA have to be non identically equal to zero onω, notably, in the2×2 case, we should have
g21≡0 inqT and
(a21> a0 inqT or a21<−a0in qT),
(1.14)
for a constanta0>0. The last result concerning first order coupling terms is the recent work [11], where the case of 2×2 and 3×3 systems with one control force is studied under some technical assumptions. Notably, in the2×2case, the authors assume that
there exists an nonempty open subsetγ of ∂ω∩∂Ω,
∃x0∈γ s.t. g21(t, x0)·ν(x0)6= 0 for allt∈[0, T], (1.15)
1.3 Strategy June 9, 2015 M. Duprez, P. Lissy
where ν represents the exterior normal unit vector to the boundary ∂Ω. Under these technical restrictions on the control domain and the coupling terms, System (1.1) is null controllable at any timeT >0.
Here we detail how our results differ from the existing ones:
1. In the case of constant coefficients, we are able to obtain a necessary and sufficient condition in the case of m equations, m−1 controls and coupling terms of order 0 or 1, which is the main new result. Moreover, the diffusion coefficients can be different, we are able to treat the case of as many equations as wanted and we use an inequality similar to Condition (1.13) but with theL2−norm in the left-hand side (see Lemma 2.4). To finish, we do not need the control domain to extend up to the boundary as in Condition (1.15). The main restriction is that all the coefficients of System (1.1) must be constant.
2. In the case N = 1 and m = 2, we are able to obtain the controllability in arbitrary small time under some generic conditions which are purely technical (at least in the second item). In Theorem 2, the geometric condition (1.15) is not necessary, which is very satisfying, moreover, in the case g21 = 0, we do not need a condition of constant sign like in (1.14) on a21, and the conditions ona21 and g21 given in (1.7) have to be verified only locally in space and time (contrary to (1.14) where it is local only in space).
1.3 Strategy
The method described in this section is sometimes calledfictitious control methodand has already been used for instance in [20], [14] and [2]. One important limitation of this method is that it will never be useful to treat the case where the support of the coupling terms do not intersect the control region, because, in what we call the algebraic resolution, we have to work locally on the control region.
Roughly, the method is the following: we first control the equations withmcontrols (one on each equation) and we try to eliminate the control on the last equation thanks to algebraic manipulations.
Let us be more precise and decompose the problem into two different steps:
Analytic problem:
Find a solution(z, v)in an appropriate space to the control problem bymcontrols which are regular enough and are in the range of a differential operator. More precisely, solve
∂tz= div(D∇z) +G· ∇z+Az+N(1ωev) in QT,
z= 0 on ΣT,
z(0,·) =y0, z(T,·) = 0 in Ω,
(1.16) where N is some differential operator to be chosen later andωe is strongly included in ω. Solving Problem (1.16) is easier than solving the null controllability at timeT of System (1.1), because we control System (1.16) with a control on each equation. The important points (and somehow different from the usual method) are that:
1. The control has to be of a special form (it has to be in the range of a differential operatorN), 2. The control has to be regular enough, so that it can be differentiated a certain amount of times with respect to the space and/or time variables (see the next section about the algebraic resolution).
If we look for a controlv in the weighted Sobolev spaceL2(QT, ρ−1/2) for some weightρ, it is well known (see, e.g. [12, Th. 2.44, p. 56–57]) that the null controllability at timeT of System (1.16) is equivalent to the followingobservability inequality:
Z
Ω
|ψ(0, x)|2dx6Cobs
Z Z
qT
ρ|N∗ψ(t, x)|2dxdt, (1.17)
whereψis the solution to the dual system
−∂tψ= div(D∇ψ)−G∗· ∇ψ+A∗ψ inQT,
ψ= 0 on ΣT,
ψ(T,·) =ψ0 in Ω.
Inequalities like (1.17) can be proved thanks to some appropriateCarleman estimates. The weightρ can be chosen to be exponentially decreasing at timest= 0andt=T, which will be useful later. In fact, we will have to adapt the usual HUM duality method to ensure that one can find such controls.
Algebraic problem:
Forf :=1ωv, find a pair(bz,bv)(wherebvnow acts only on the firstm−1equations) in an appropriate space satisfying the following control problem:
∂tzb= div(D∇z) +b G· ∇bz+Azb+Bbv+Nf in QT,
bz= 0 on ΣT,
bz(0,·) =z(T,b ·) = 0 in Ω,
(1.18) and such that the spatial support ofvbis strongly included in ω. We will solve this problem using the notion ofalgebraic resolvability of differential systems, which is based on ideas coming from [21, Section 2.3.8] and was already widely used in [14] and [2]. The idea is to write System (1.18) as an underdetermined system in the variables zband bv and to see Nf as a source term, so that we can write Problem (1.18) under the abstract form
L(bz,bv) =Nf, (1.19)
where
L(z,b bv) := ∂tzb−div(D∇bz)−G· ∇zb−Azb−Bbv. (1.20) The goal will be then to find a partial differential operatorMsatisfying
L ◦ M=N. (1.21)
When (1.21) is satisfied, we say that System (1.18) is algebraically resolvable. This exactly means that one can find a solution(z,b bv) of System (1.18) which can be written as a linear combination of some derivatives of the source term Nf. The main advantage of this method is that one can only worklocally onqT, because the solution depends locally on the source term and then has the same support as the source term (to obtain a solution which is defined everywhere onQT, one just extends it by0). This part will be explained in more details in Sections2.1and3.1.
Conclusion:
If we can solve the analytic and algebraic problems, then it is easy to check that(y, u) := (z−bz,−bv) will be a solution to System (1.1) in an appropriate space and will satisfyy(T)≡0 in Ω (for more explanations, see [14, Prop. 1] or Sections 2.4 and 3.3 of the present paper).
2 Proof of Theorem 1
Let us remind that in this case, D, G and A are constant. In Section 2.1, 2.2 and 2.3, we will always consider somei0∈ {1, ..., m−1}such that
gmi0 6= 0 orami0 6= 0. (2.1) We will follow the strategy described in Section 1.3, and we first begin with finding some appropriate operatorN.
2.1 Algebraic resolution June 9, 2015 M. Duprez, P. Lissy
2.1 Algebraic resolution
We will here explain how to choose the differential operator N used in the next section. We will assume from now on that all differential operators of this section are defined inC∞(QT). The appropriate spaces will be precised in Section 2.4. We consider N as the operator defined for all f := (f1, . . . , fm)by
N(f) :=
N1f N2f . . . Nmf
:=
(−gmi0· ∇ −ami0)f1
(−gmi0· ∇ −ami0)f2
. . .
(−gmi0· ∇ −ami0)fm
. (2.2)
Let us recall that the definition ofLis given in (1.20).
As explained in Section 1.3, we want find a differential operatorMsuch that
L ◦ M=N. (2.3)
We have the following proposition:
Proposition 2.1. Let N be defined as in (2.2). Then there exists a differential operator M of order1 in time and2 in space, with constant coefficients, such that (2.3)is verified.
Proof of Proposition 2.1. We can remark that equality (2.3) is equivalent to
M∗◦ L∗=N∗. (2.4)
The adjointL∗ of the operatorL is given for allϕ∈ C∞(QT)mby
L∗ϕ :=
L∗1ϕ
. . . L∗2m−1ϕ
=
−∂tϕ1−d1∆ϕ1+Pm
j=1{gj1· ∇ϕj−aj1ϕj} . . .
−∂tϕm−dm∆ϕ1+Pm
j=1{gjm· ∇ϕj−ajmϕj} ϕ1
. . . ϕm−1
. (2.5)
Now we applygmi0·∇−ami0 to the(m+i)thline fori∈ {1, ..., m−1}and we add(∂t+di0∆)L∗m+i0ϕ+
Pm−1
j=1 (−gji0·∇+aji0)L∗m+jϕto the(i0)thline. Hence, remarking thatL∗m+iϕ=ϕi(i∈ {1, ..., m−1}), we obtain
(gmi0· ∇ −ami0)L∗m+1ϕ . . .
(gmi0· ∇ −ami0)L∗2m−1ϕ L∗i0ϕ+ (∂t+di0∆)L∗m+i0ϕ+Pm−1
j=1 (−gji0· ∇+aji0)L∗m+jϕ
=N∗ϕ.
Thus if we defineM∗ forψ:= (ψ1, ..., ψ2m−1)∈ C∞(QT)2m−1 by
M∗
ψ1 . . . ψ2m−1
:=
(gmi0∇ −ami0)ψm+1
. . .
(gmi0∇ −ami0)ψ2m−1 ψi0+ (∂t+di0∆)ψm+i0+Pm−1
j=1 (−gji0· ∇+aji0)ψm+j
, (2.6)
then equality (2.4) is satisfied and hence equality (2.3) also. Moreover, the coefficients ofM∗ are constant, hence it is also the case for the coefficients ofM.
