Controllability of a 2x2 parabolic system by one force with a space-dependent coupling term of order one

0.35 MB·

by  Michel Duprez

ESAIM: Control, Optimisation and Calculus of Variations Willbesetbythepublisher
URL:http://www.emath.fr/cocv/
7
1
0 CONTROLLABILITY OF A 2×2 PARABOLIC SYSTEM BY ONE FORCE WITH
2 ∗
SPACE-DEPENDENT COUPLING TERM OF ORDER ONE.

n
a
J Michel Duprez1

9
1
Abstract. This paper is devoted to the controllability of linear systems of two coupled parabolic
]
equations when the coupling involves a space dependent first order term. This system is set on an
P
bounded interval I ⊂⊂ R, and the first equation is controlled by a force supported in a subinterval
A
of I or on the boundary. In the case where the intersection of the coupling and control domains is
.
h nonempty, we prove null controllability at any time. Otherwise, we provide a minimal time for null
t controllability. Finally wegiveanecessary andsufficientcondition fortheapproximatecontrollability.
a
The main technical tool for obtaining these results is themoment method.
m
[ 1991 Mathematics Subject Classification. 93B05, 93B07.

1
January 6, 2016.
v
5
1
3
5 1. Introduction and main results
0
1. Let T > 0, ω := (a,b) ⊆ (0,π) and QT := (0,π)×(0,T). We consider in the present paper the following
0 distributed control system
7
1 ∂ y −∂ y =1 v in Q ,
t 1 xx 1 ω T
:
v ∂ty2−∂xxy2+p(x)∂xy1+q(x)y1 =0 in QT,
i  y (0,·)=y (π,·)=y (0,·)=y (π,·)=0 on (0,T), (1.1)
X  y1(·,0)=y10, y (·,0)2=y0 2 in (0,π)
r 1 1 2 2
a

and boundary control system
∂ z −∂ z =0 in Q ,
t 1 xx 1 T
∂ z −∂ z +p(x)∂ z +q(x)z =0 in Q ,
t 2 xx 2 x 1 1 T
 (1.2)
z (0,·)=u, z (π,·)=z (0,·)=z (π,·)=0 on (0,T),
 1 1 2 2
z (·,0)=z0, z (·,0)=z0 in (0,π),
1 1 2 2

where y0 := (y0,y0) ∈ L2(0,π)2 and z0 := (z0,z0) ∈ H−1(0,π)2 are the initial conditions, v ∈ L2(Q ) and
1 2 1 2 T
u∈L2(0,T) are the controls, p∈W1 (0,π), q ∈L∞(0,π).
∞
Keywords and phrases: Controllability,Observability,MomentMethod,ParabolicSystems.
∗ This work was partially supported by R´egion de Franche-Comt´e (France).
1LaboratoiredeMath´ematiquesdeBesan¸conUMRCNRS6623,Universit´edeFranche-Comt´e,16routedeGray,25030Besan¸con
Cedex,France,E-mail: [email protected]
(cid:13)c EDPSciences,SMAI1999
2 TITLEWILLBESETBYTHEPUBLISHER
It is known (see [20, p. 102] (resp. [16, Prop. 2.2])) that for given initial data y0 ∈ L2(0,π)2 (resp. z0 ∈
H−1(0,π)2) and a control v ∈ L2(Q ) (resp. u ∈ L2(0,T)) System (1.1) (resp. (1.2)) has a unique solution
T
y =(y ,y ) (resp. z =(z ,z )) in
1 2 1 2
L2(0,T;H1(0,π)2)∩C([0,T];L2(0,π)2)
0
(resp. L2(Q )2∩C([0,T];H−1(0,π)2) ),
T
which depends continuously on the initial data and the control, that is
kyk +kyk 6C (ky0k +kvk )
L2(0,T;H01(0,π)2) C([0,T];L2(0,π)2) T L2(0,π)2 L2(QT)
(resp. kzkL2(QT)2 +kzkC([0,T];H−1(0,π)2) 6CT(kz0kH−1(0,π)2 +kukL2(0,T)) ),
where C does not depend on y0, v, z0 and u.
T
Let us introduce the notion of null and approximate controllability for this kind of systems.
• System(1.1)(resp. System(1.2))isnullcontrollable attimeT ifforeveryinitialconditiony0 ∈L2(0,π)2
(resp. z0 ∈ H−1(0,π)2) there exists a control v ∈ L2(Q ) (resp. u ∈ L2(0,T)) such that the solution
T
to System (1.1) (resp. System (1.2)) satisfies
y(T)≡0 (resp. z(T)≡0) in (0,π).
• System (1.1) (resp. System (1.2)) is approximately controllable at time T if for all ε > 0 and all
y0, y1 ∈ L2(0,π)2 (resp. z0, z1 ∈ H−1(0,π)2) there exists a control v ∈ L2(Q ) (resp. u ∈ L2(0,T))
T
such that the solution to System (1.1) (resp. System (1.2)) satisfies
ky(T)−y1kL2(0,π)2 6ε (resp. kz(T)−z1kH−1(0,π)2 6ε).
We recall that null-controllability at some time T implies approximate controllability at the same time T for
linear parabolic systems. This follows from the backward uniqueness result of [17, Th. 1.1] for first order
perturbations and Propositions 2.5 and 2.6. Moreover the approximate controllability does not depend on the
time of control T since we consider autonomous systems. It is a consequence of the analyticity in time of the
adjoint semigroup.
Themaingoalofthisarticleistoprovideacompleteanswertothenullandapproximatecontrollabilityissues
for System (1.1) and (1.2). For a survey and some applications in physics, chemistry or biology concerning the
controllability of this kind of systems, we refer to [6]. In the last decade, many papers studied this problem,
howevermostofthemarerelatedtosomeparabolicsystemswithzeroordercouplingterms. Withoutfirstorder
couplingterms,someKalmancouplingconditionsaremadeexplicitin[3,Th. 1.4],[4,Th. 1.1]and[16,Th. 1.1]
for distributed null controllability of systems of more than two equations with constant matrices and in higher
space dimension and in the case of time dependent matrices, some Silverman-Meadowscoupling conditions are
given in [3, Th. 1.2].
Concerning the null and approximate controllability of Systems (1.1) and (1.2) in the case p ≡ 0 and q 6≡ 0
in (0,π), a partialansweris givenin [1,2,13,23] under the sign conditionq 60 or q >0 in (0,π). These results
are obtained as a consequence of controllability results of a hyperbolic system using the transmutation method
π
(see [21]). One can find a necessary and sufficient condition in [7] when q(x)dx 6= 0. Finally, in [11], the
0
authors gives a complete characterization of the approximate controllability and, in the recent work [8,9], we
R
can find a complete study of the null controllability.
When p 6= 0, the approximate controllability of systems (1.1) and (1.2) in any dimension is studied in [22].
The author gives a sufficient condition for the approximate controllability on the boundary and, in the case
of analytic coupling coefficients p and q, a necessary and sufficient condition for the internal approximate
controllability.
TITLEWILLBESETBYTHEPUBLISHER 3
Let us now remind known results concerning null controllability for systems of the following more general
form. Let Ω be a bounded domain in RN (N ∈N∗) of class C2 and ω an arbitrary nonempty subset of Ω. We
0
denote by ∂Ω the boundary of Ω. Consider the system of two coupled linear parabolic equations
∂ y =∆y +g ·∇y +g ·∇y +a y +a y +1 v in Ω×(0,T),
t 1 1 11 1 12 2 11 1 12 2 ω0
∂ y =∆y +g ·∇y +g ·∇y +a y +a y in Ω×(0,T),
 t 2 2 21 1 22 2 21 1 22 2 (1.3)
 y =0 on ∂Ω×(0,T),
y(·,0)=y0 in Ω,
where y0 ∈L2(Ω)2, g ∈L∞(Ω×(0,T))N and a ∈L∞(Ω×(0,T)) for all i,j ∈{1,2}.
ij ij
As a particular case of the result in section 4 of [18] (see also [5]), System (1.3) is null controllablewhenever
g ≡0 in ω and (a >C in ω or a <−C in ω ), (1.4)
21 1 21 1 21 1
for a positive constant C and ω a non-empty open subset of ω .
1 0
In [19, Th. 4], the authorsupposes that a , g , a , g areconstantand the first ordercoupling operator
11 11 22 22
g ·∇+a can be written as
21 21
g ·∇+a =P ◦θ in Ω×(0,T), (1.5)
21 21 1
where θ ∈ C2(Ω) satisfies |θ| > C in ω ⊆ ω for a positive constant C and P is given by P := m ·∇+m ,
1 0 1 1 0 1
for some m ,m ∈R. Moreover the operator P has to satisfy
0 1 1
kuk 6CkP∗uk ∀u∈H1(Ω).
H1(Ω) 1 L2(Ω) 0
Under these assumptions, the author proves the null controllability of System (1.3) at any time.
In [10, Th. 2.1], the authors prove that the same property holds true for System (1.3) if we assume that
a ∈ C4(Ω×(0,T)), g ∈ C1(Ω×(0,T))N for all i,j ∈ {1,2}, g ∈ C3(Ω×(0,T)) and the geometrical
ij ij 21
condition
∂ω∩∂Ω contains an open subset γ for which the interior˚γ is non-empty,
(1.6)
∃x ∈γ s.t. g (t,x )·ν(x )6=0 for all t∈[0,T],
(cid:26) 0 21 0 0
where ν represents the exterior normal unit vector to the boundary ∂Ω.
Lastly,forconstantcoefficients,itisprovedin[14,Th. 1]thatSystem(1.3)isnull/approximatelycontrollable
at any time T if and only if
g 6=0 or a 6=0.
21 21
In [14,Th. 2], the authors give alsoa conditionof null/approximatecontrollabilityin dimension one which can
be written for system (1.1) as: p∈C2(ω ), q ∈C3(ω ) and
0 0
−4∂ (q)∂ (p)p+∂ (q)p2+2q∂ (q)p−3pq∂ p+6q(∂ p)2−2q2∂ p
x x xx x xx x x
−∂ (p)p2+5∂ (p)∂ (p)p−4(∂ p)3 6=0 in ω
xxx x xx x 0
for a subinterval ω of ω.
0
Now let us go back to Systems (1.1) and (1.2) for which we will provide a complete description of the null
and approximate controllability. Our first and main result is the following
Theorem 1.1. Let us suppose that p∈W1 (0,π)∩W2 (ω), q ∈L∞(0,π)∩W1 (ω) and
∞ ∞ ∞
(Supp p∪Supp q)∩ω 6=∅. (1.7)
Then System (1.1) is null controllable at any time T.
Let us compare this result with the previously described results to highlight our main contribution:
4 TITLEWILLBESETBYTHEPUBLISHER
(1) Even though System (1.1) is considered in one space dimension, we remark first that our coupling
operator has a more general form than the one in (1.5) assumed in [19]. Moreover unlike [14], its
coefficients are non-constant with respect to the space variable.
(2) We do not have the geometricalrestriction (1.6) assumed in [10]. More precisely we do not require the
control support to be a neighbourhood of a part of the boundary.
(3) As said before, in [22, Theorem 4.1], the author gives a necessary and sufficient condition for the
approximate controllability of System (1.1) when p and q are analytic. We deduce that the condition
of [22] is satisfied in dimension one as soon as p or q is not equal to zero.
For all k ∈ N∗, we denote by ϕ : x 7→ 2 sin(kx) the normalized eigenvector of the Laplacian operator,
k π
with Dirichlet boundary condition, and consqider the two following quantities
a π
I (p,q):= q− 1∂ p ϕ2 and I (p,q):= q− 1∂ p ϕ2, (1.8)
a,k 2 x k k 2 x k
Z0 Z0
(cid:0) (cid:1) (cid:0) (cid:1)
for all k ∈ N∗. Combined with the criterion of Fattorini (see [15, Cor. 3.3] or Th. 5.1 in the present paper),
Theorem 1.1 leads to the following characterization:
Theorem 1.2. Let us suppose that p ∈ W1 (0,π) ∩W2 (ω) and q ∈ L∞(0,π) ∩ W1(ω). System (1.1) is
∞ ∞ ∞
approximately controllable at any time T >0 if and only if
(Supp p∪Supp q)∩ω 6=∅ (1.9)
or
|I (p,q)|+|I (p,q)|6=0 for all k ∈N∗. (1.10)
k a,k
This last result recovers the case p ≡ 0 studied in [11] for Supp q∩ω = ∅, where the authors also use the
criterion of Fattorini.
Remark 1.3. We will see in the proof of Theorems 1.1 and 1.2 that only the following regularity are needed
for p and q
p∈W1 (0,π)∩W2(ω),
∞ ∞
q ∈L∞(0,π)∩W1 (ω),
(cid:26) ∞
for an open subinterval ω of ω. These hypotheses are used in Defienition (1.8) of I (p,q) and I (p,q) and the
k a,k
changeofunknowndescribedinSection3.2. For moregeneralcoeuplingterms, these controlproblemsareopen.
e
When the supports of the control and the coupling terms are disjoint in System (1.1), following the ideas
in [9, Th. 1.3] where the authors studied the case p≡0, we obtain a minimal time of null controllability:
Theorem 1.4. Let p∈W1 (0,π), q ∈L∞(0,π). Suppose that Condition (1.10) holds and
∞
(Supp p∪Supp q)∩ω =∅. (1.11)
Let T (p,q) be given by
0
min(−log|I (p,q)|,−log|I (p,q)|)
k a,k
T (p,q):=limsup . (1.12)
0 k2
k→∞
One has
(1) If T >T (p,q), then System (1.1) is null controllable at time T.
0
(2) If T <T (p,q), then System (1.1) is not null controllable at time T.
0
TITLEWILLBESETBYTHEPUBLISHER 5
Concerning the boundary controllability, in [22, Th. 3.3], using the criterion of Fattorini, the author proves
that System (1.2) is approximately controllable at time T if and only if
I (p,q)6=0 for all k∈N∗. (1.13)
k
About null controllability of System (1.2), we can again generalize the results given in [9, Th. 1.1] to obtain a
minimal time:
Theorem 1.5. Let p∈W1 (0,π), q ∈L∞(0,π) and suppose that Condition (1.13) is satisfied. Let us define
∞
−log|I (p,q)|
k
T (p,q):=limsup . (1.14)
1 k2
k→∞
One has
(1) If T >T (p,q), then System (1.2) is null controllable at time T.
1
(2) If T <T (p,q), then System (1.2) is not null controllable at time T.
1
Remark 1.6. Using Riemann-Lebesgue Lemma, sequences (Ik(p,q))k∈N∗ and (Ia,k(p,q))k∈N∗ are convergent,
more precisely
1 π 1 a
lim I (p,q)=I(p,q):= (q− 1∂ p) and lim I (p,q)=I (p,q):= (q− 1∂ p).
k→∞ k π Z0 2 x k→∞ a,k a π Z0 2 x
Thus, if one of the two limits I(p,q) and I (p,q) (resp. the first limit) is not equal to zero, then the minimal
a
time T (p,q) (resp. T (p,q)) is equal to zero.
0 1
This article is organized as follows. In the second section, we present some preliminary results useful to
reducethenullcontrollabilityissuestothemomentproblem. Inthethirdandfourthsections,westudythenull
controllability issue of System (1.1) in the two cases when the intersectionofthe coupling andcontrolsupports
is empty or not. Then we give the proof of Theorems 1.2 and 1.5 in Section 5 and 6, respectively. We finish
with some comments and open problems in Section 7.
2. Preliminary results
Consider the differential operator
L: D(L)⊂L2(0,π)2 → L2(0,π)2
f 7→ −∂ f +A (p∂ f +qf),
xx 0 x
where the matrix A is given by
0
0 0
A := ,
0 1 0
(cid:18) (cid:19)
the domain of L and its adjoint L∗ is given by D(L) =D(L∗)= H2(0,π)2∩H1(0,π)2. In section 2.1, we will
0
first establish some properties of the operatorL that will be useful for the moment method and, in section 2.2,
we will recall some characterizations of the approximate and null controllability of system (1.1).
2.1. Biorthogonal basis
Let us first analyze the spectrum of the operators L and L∗.
Proposition 2.1. For all k ∈N∗ consider the two vectors
ψ∗ ϕ
Φ∗ := k ,Φ∗ := k ,
1,k ϕ 2,k 0
k
(cid:18) (cid:19) (cid:18) (cid:19)
6 TITLEWILLBESETBYTHEPUBLISHER
where ψ∗ is defined for all x∈(0,π) by
k
1 x
ψ∗(x)=α∗ϕ (x)− sin(k(x−ξ))[I (p,q)ϕ (ξ)+∂ (p(ξ)ϕ (ξ))−q(ξ)ϕ (ξ)]dξ,
k k k k k k x k k
 1 π x Z0
 α∗k = k sin(k(x−ξ))[Ik(p,q)ϕk(ξ)+∂x(p(ξ)ϕk(ξ))−q(ξ)ϕk(ξ)]ϕk(x)dξdx.
Z0 Z0
One has 
(1) The spectrum of L∗ is given by σ(L∗)={k2 :k ∈N∗}.
(2) For k >1, the eigenvalue k2 of L∗ is simple (algebraic multiplicity 1) if and only if I (p,q)6=0. In this
k
case, Φ∗ and Φ∗ are respectively an eigenfunction and a generalized eigenfunction of the operator L∗
2,k 1,k
associated with the eigenvalue k2, more precisely
(L∗−k2Id)Φ∗ =I Φ∗ ,
1,k k 2,k (2.1)
( (L∗−k2Id)Φ∗2,k =0.
(3) For k >1, the eigenvalue k2 of L∗ is double (algebraic multiplicity 2) if and only if I (p,q)=0. In this
k
case, Φ∗ and Φ∗ are two eigenfunctions of the operator L∗ associated with the eigenvalue k2, that is
1,k 2,k
for i=1,2
(L∗−k2Id)Φ∗ =0.
i,k
Proof. The adjointoperatorL∗ ofL isgivenby D(L∗)=D(L)andL∗f =−∂ f+A∗(−∂ (pf)+qf). We can
xx 0 x
remark first that the resolvent of L∗ is compact. Thus the spectrum of L∗ reduces to its point spectrum. The
eigenvalue problem associated with the operator L∗ is
−∂ ψ−∂ (p(x)ϕ)+q(x)ϕ=λψ in (0,π),
xx x
−∂ ϕ=λϕ in (0,π), (2.2)
 xx
 ϕ(0)=ψ(0)=ϕ(π)=ψ(π)=0,
where(ψ,ϕ)∈D(L∗)andλ∈C. Forϕ≡0in(0,π)andψ =ϕ in(0,π),λ=k2 isaneigenvalueofL∗ andthe
k
vector Φ∗ := (ϕ ,0) is an associated eigenfunction. If now ϕ 6≡ 0 in (0,π), then λ = k2 is an eigenvalue and
2,k k
ϕ = κϕ with κ ∈ R∗. We remark that System (2.2) has a solution if and only if I (p,q) = 0. If I (p,q) = 0,
k k k
Φ∗ := (ψ∗,ϕ ) is a second eigenfunction of L∗ linearly independent of Φ∗ , where, applying the Fredholm
1,k k k 2,k
alternative, ψ∗ is the unique solution to the non-homogeneous Sturm-Liouville problem
k
−∂ ψ−k2ψ =f in (0,π),
xx
(2.3)
π
ψ(0)=ψ(π)=0, ψ(x)ϕ (x)dx=0.
(cid:26) 0 k
π R
with f := ∂ (p(x)ϕ )−q(x)ϕ . We recall that fϕ = 0, since I = 0. Solving System (2.3) leads to the
x k k 0 k k
expression of ψ∗ given in Proposition 2.1. The expression of α is given by the equality πψ∗(x)ϕ (x)dx = 0
k R k 0 k k
and the identity πfϕ = 0 leads to ψ (π) = 0. Thus, in the case I (p,q) = 0, λ = k2 is a double eigenvalue
0 k k k R
of L∗. Items 1 and 3 are now proved.
R
Let us now suppose that I (p,q) 6= 0. The eigenvalue λ = k2 is simple, Φ∗ := (ϕ ,0) is an eigenfunction
k 2,k k
and a solution Φ∗ :=(ψ,ϕ) to (L∗−k2Id)Φ∗ =I (p,q)Φ∗ , that is
1,k 1,k k 2,k
−∂ ψ−∂ (p(x)ϕ)+q(x)ϕ=k2ψ+I (p,q)ϕ in (0,π),
xx x k k
−∂ ϕ=k2ϕ in (0,π), (2.4)
 xx
 ϕ(0)=ψ(0)=ϕ(π)=ψ(π)=0,

TITLEWILLBESETBYTHEPUBLISHER 7
is a generalized eigenfunction of L∗. We deduce that ϕ=ϕ in (0,π) and ψ is solution to the Sturm-Liouville
k
problem (2.3) with f =I (p,q)ϕ +∂ (p(x)ϕ )−q(x)ϕ . We obtain the expressionof ψ∗ given in Proposition
k k x k k k
2.1. (cid:3)
The function ψ∗ given in Proposition 2.1 will play an important role in this paper. Since ϕ , ϕ′ and I are
k k k k
bounded, we have the following lemma
Lemma 2.2. There exists a positive constant C such that
C C
|α∗k|6 k, kψk∗kL∞(0,π) 6 k, k∂xψk∗kL∞(0,π) 6C, ∀k ∈N∗. (2.5)
Since the eigenvalues of the operator L∗ are real, we deduce that L and L∗ have the same spectrum and the
associated eigenspaces have the same dimension. The eigenfunctions and the generalized eigenfunctions of L
can be found as previously.
Proposition 2.3. For all k ∈N∗ consider the two vectors
0 ϕ
Φ := ,Φ := k ,
1,k ϕ 2,k ψ
k k
(cid:18) (cid:19) (cid:18) (cid:19)
where ψ is defined for all x∈(0,π) by
k
1 x
ψ (x):=α ϕ (x)− sin(k(x−ξ))[I (p,q)ϕ (ξ)−p(ξ)∂ (ϕ (ξ))−q(ξ)ϕ (ξ)]dξ,
k k k k k x k k
k
 1 π x Z0
 αk := sin(k(x−ξ))[Ik(p,q)ϕk(ξ)−p(ξ)∂x(ϕk(ξ))−q(ξ)ϕk(ξ)]ϕk(x)dξdx,
k
Z0 Z0
One has 
(1) The spectrum of L is given by σ(L)=σ(L∗)={k2 :k ∈N∗}.
(2) For k >1, the eigenvalue k2 of L is simple (algebraic multiplicity 1) if and only if I (p,q)6=0. In this
k
case, Φ and Φ are an eigenfunction and a generalized eigenfunction of the operator L associated
1,k 2,k
with the eigenvalue k2, more precisely
(L−k2Id)Φ =0,
1,k
(2.6)
(L−k2Id)Φ =I Φ .
(cid:26) 2,k k 1,k
(3) For k >1, the eigenvalue k2 of L is double (algebraic multiplicity 2) if and only if I (p,q)=0. In this
k
case, Φ and Φ are two eigenfunctions of the operator L associated with the eigenvalue k2, that is
1,k 2,k
for i=1,2
(L−k2Id)Φ =0.
i,k
Lemma 2.3 and Corollary 2.6 in [9] can be adapted easily to prove the following proposition.
Proposition 2.4. Consider the families
B :={Φ ,Φ :k ∈N∗} and B∗ := Φ∗ ,Φ∗ :k∈N∗ .
1,k 2,k 1,k 2,k
n o
Then
(1) The sequences B and B∗ are biorthogonal Riesz bases of L2(0,π)2.
(2) The sequence B∗ is a Schauder basis of H1(0,π)2 and B is its biorthogonal basis in H−1(0,π).
0
We recall that B and B∗ are biorthogonal in L2(0,π)2 if hΦ ,Φ∗ i = δ δ for all k,l ∈ N∗ and
i,k j,l L2(0,π)2 i,j k,l
i,j ∈{1,2}.
8 TITLEWILLBESETBYTHEPUBLISHER
2.2. Duality
As it is well known, the controllability has a dual concept called observability (see for instance [6, Th.
2.1], [12, Th. 2.44, p. 5657]). Consider the dual system associated with System (1.1)
−∂ θ−∂ θ+A∗(−∂ (p(x)θ)+q(x)θ)=0 in Q ,
t xx 0 x T
θ(0,·)=θ(π,·)=0 on (0,T), (2.7)

 θ(·,T)=θ0 in (0,π),
where θ0 ∈ L2(0,π)2. LetB the matrix given by B = (1 0)∗. The approximate controllability is equivalent to
a unique continuation property:
Proposition 2.5. (1) System (1.1) is approximately controllable at time T if and only if for all initial
condition θ0 ∈L2(0,π)2 the solution to System (2.7) satisfies the unique continuation property
1 B∗θ ≡0 in Q ⇒ θ ≡0 in Q . (2.8)
ω T T
(2) System (1.2)is approximately controllable attimeT ifandonlyiffor allinitialcondition θ0 ∈H1(0,π)2
0
the solution to System (2.7) satisfies the unique continuation property
B∗∂ θ(0,t)≡0 in (0,T) ⇒ θ ≡0 in Q . (2.9)
x T
The null controllability is characterized by an observability inequality:
Proposition 2.6. (1) System (1.1) is null controllable at time T if and only if there exists a constant C
obs
such that for all initial condition θ0 ∈ L2(0,π)2 the solution to System (2.7) satisfies the observability
inequality
kθ(0)k2 6C |1 (x)B∗θ(x,t)|2dxdt. (2.10)
L2(0,π)2 obs ω
ZZQT
(2) System (1.1) is null controllable at time T if and only if there exists a constant C such that for all
obs
initial condition θ0 ∈H1(0,π)2 the solution to System (2.7) satisfies the observability inequality
0
T
kθ(0)k2 6C |B∗∂ θ(0,t)|2dt. (2.11)
H1(0,π)2 obs x
0 Z0
3. Resolution of the moment problem
In this section, we first establish the moment problem related to the null controllability for System (1.1)
and then we will solve it in section 3.2 (Theorem 1.1). The strategy involves finding an equivalent system (see
Definition 3.1) to System (1.1), which has an associated quantity I satisfying ”some good properties”.
k
3.1. Reduction to a moment problem
Let y0 :=(y0,y0)∈L2(0,π)2. For i∈{1,2} and k ∈N∗, if we consider θ0 :=Φ∗ in the dual System (2.7),
1 2 i,k
we get after an integration by parts
v(x,t)1 (x)B∗θ(x,t)dxdt =hy(T),Φ∗ i −hy0,θ(0)i .
ω i,k L2(0,π)2 L2(0,π)2
ZZQT
Since B∗ is a Riesz basis of L2(0,π)2, System (1.1) is null controllableif andonly if for all y0 ∈L2(0,π)2, there
exists a control v ∈ L2(Q ) such that for all k ∈ N∗ and i ∈ {1,2} the solution y to System (1.1) satisfies the
T
following equality
v(x,t)1 (x)B∗θ (x,t)dxdt=−hy0,θ (0)i , (3.1)
ω i,k i,k L2(0,π)2
ZZQT
TITLEWILLBESETBYTHEPUBLISHER 9
where θ is the solution to the dual system (2.7) with the initial condition θ0 :=Φ∗ .
i,k i,k
In the moment problem (3.1), we will look for a control v of the form
v(x,t):=f(1)(x)v(1)(T −t)+f(2)(x)v(2)(T −t) for all (x,t)∈Q , (3.2)
T
with v(1),v(2) ∈L2(0,T) and f(1),f(2) ∈L2(0,π) satisfying Supp f(1) ⊆ω and Supp f(2) ⊆ω.
The solutions θ and θ to the dual System (2.7) with the initial condition Φ∗ and Φ∗ are givenfor all
1,k 2,k 1,k 2,k
(x,t)∈Q by
T
θ (x,t)=e−k2(T−t) Φ∗ (x)−(T −t)I (p,q)Φ∗ (x) ,
1,k 1,k k 2,k
(3.3)
 θ (x,t)=e−k2(T−t)Φ(cid:16)∗ (x). (cid:17)
 2,k 2,k
Plugging (3.2) and (3.3) in the moment problem (3.1), we get for all k >1

T T
f(1) v(1)(t)e−k2tdt+f(2) v(2)(t)e−k2tdt
k k
 Z0 Z0
T T
 e −Ik(pe,q)fk(1)Z0 v(1=)(t−)tee−−kk22Ttdty−0 I−k(pT,Iq)(fpk(,2q))Zy00 v(,2)(t)te−k2tdt
1,k k 2,k
T T
 fk(1)Z0 v(1)(t)e−k2tdt+fk(2)Z0 v(2)(t)e−k2tdt=(cid:8)−e−k2Ty20,k, (cid:9)
where f(i), f(i) and y0 are given for all i∈{1,2} and k ∈N∗ by
k k i,k
e π π
f(i) := f(i)(x)ϕ (x)dx, f(i) := f(i)(x)ψ∗(x)dx, (3.4)
k k k k
Z0 Z0
e
and
y0 :=hy0,Φ∗ i . (3.5)
i,k i,k L2(0,π)
In[16,Prop. 4.1],theauthorsprovedthatthefamily e :=e−k2t,e :=te−k2t admitsabiorthogonal
1,k 2,k
k≥1
family {q ,q } in the space L2(0,T), i.e. a familny satisfying o
1,k 2,k k≥1
T
e q (t)dt=δ δ , ∀k,l≥1, 1≤i,j ≤2. (3.6)
i,k j,l ij kl
Z0
Moreover for all ε>0 there exists a constant C >0 such that
ε,T
kq k ≤C eεk2, ∀k ≥1, i=1,2. (3.7)
i,k L2(0,T) ε,T
We will look for v(1) and v(2) of the form
v(i)(t)= {v(i)q (t)+v(i)q (t)}, i=1,2 (3.8)
1,k 1,k 2,k 2,k
k>1
X
and prove that the series converges. The moment problem (3.1) can be written as
A V +A V =F , for all k >1, (3.9)
1,k 1,k 2,k 2,k k
10 TITLEWILLBESETBYTHEPUBLISHER
with for all k ∈N∗
f(1) f(2) −I (p,q)f(1) −I (p,q)f(2)
A = k k , A = k k k k , (3.10)
1,k fk(1) fk(2) ! 2,k (cid:18) 0 0 (cid:19)
e e
v(1) v(1)
V := 1,k , V := 2,k (3.11)
1,k v(2) ! 2,k v(2) !
1,k 2,k
and
−e−k2T y0 −TI (p,q)y0
F = 1,k k 2,k . (3.12)
k (cid:16)−e−k2Ty0 (cid:17) !
2,k
3.2. Solving the moment problem
In this section, we will prove the null controllability of System (1.1) at any time T when the supports of
p or q intersects the control domain ω (Theorem 1.1). In [18], the authors obtain the null controllability of
System (1.1) at any time under Condition (1.4), so we will not consider this case and we will always suppose
that Supp p∩ω 6=∅. This implies that there exists x ∈ω such that p(x )6=0. By continuity of p, we deduce
0 0
that |p|>C in ω for a positive constant C and an open subinterval ω of ω.
Definition 3.1. Let p , p ∈W1 (0,π) and q , q ∈L∞(0,π). Consider the systems given for i∈{1,2} by
1 2 ∞ 1 2
e e
For given y0 ∈L2(0,π)2, v ∈L2(Q ),
T
Find y :=(y ,y )∈L2(0,T;H1(0,π)2)∩C([0,T];L2(0,π)2) such that:
 1 2 0
 ∂∂ttyy12−−∂∂xxxxyy12+=p1iω(xv)∂xy1+qi(x)y1 =0 iinn QQTT,, (Si)
y(0,·)=y(π,·)=0 on (0,T),
 y(·,0)=y0 in (0,π).
We say that System (S ) is equivalent to System (S ) if System (S ) is null controllable at time T if and only
1 2 1
if System (S ) is null controllable at time T.
2
Let us present the main technique used all along this section. Suppose that System (1.1) is null controllable
at time T. Let v a control such that the solution y to System (1.1) verifies y(T)=0 in (0,π) and ω :=(α,β)
0
a subinterval of ω =(a,b). Consider a function θ ∈W2 (0,π) satisfying
∞
θ ≡κ in (0,α),
1
θ ≡κ in (β,π), (3.13)
 2
 θ >κ3 in (0,π),
with κ , κ , κ ∈R∗. Thus if we consider the change of unknown
1 2 3 +
y :=(y ,y ) with y :=θ−1y , (3.14)
1 2 1 1
then y is solution in L2(0,T;H1(0,π)2)∩C([0,T];L2(0,π)2) to the system
0 b b b
∂ y −∂ y =1 v in Q ,
b t 1 xx 1 ω T
∂ y −∂ y +p∂ y +qy =0 in Q ,
 t 2 xx 2 x 1 1 T (3.15)
 y(0b,·)=y(bπ,·)=0b on (0,T),
y(·,0)=y0 b b bb in (0,π),
 b b
b b