Partial null controllability of parabolic linear systems

0.41 MB·

by  Farid Ammar Khodja

Partial null controllability of parabolic linear systems
Farid AMMAR KHODJA∗ Franz CHOULY∗ Michel DUPREZ∗†
7 January 20, 2017
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0
2
Abstract

n
a Thispaperisdevotedtothepartialnullcontrollability issueofparaboliclinearsystemswith
J n equations. Given a bounded domain Ω in RN (N ∈ N∗), we study the effect of m localized

9 controls in a nonempty open subset ω only controlling p components of thesolution (p,m6n).
1 Thefirst main result of thispaperis anecessary and sufficient condition when thecouplingand
control matrices are constant. The second result provides, in a first step, a sufficient condition
]
of partial nullcontrollability when thematrices only dependon time. In a second step, through
P
an example of partially controlled 2×2 parabolic system, we will provide positive and negative
A
results on partial null controllability when thecoefficients are space dependent.
.
h
t
a 1 Introduction and main results
m
[ Let Ω be a bounded domain in RN (N ∈ N∗) with a C2-class boundary ∂Ω, ω be a nonempty

1 open subset of Ω and T > 0. Let p, m, n ∈ N∗ such that p,m 6 n. We consider in this paper the
v following system of n parabolic linear equations
3
8 ∂ y =∆y+Ay+B1 u in Q :=Ω×(0,T),
4 t ω T
y =0 on Σ :=∂Ω×(0,T), (1.1)
5  T
0  y(0)=y0 in Ω,
.
1
where y ∈L2(Ω)n is the initial data, u∈L2(Q )m is the control and
0 0 T
7
1 A∈L∞(QT;L(Rn)) and B ∈L∞(QT;L(Rm,Rn)).
:
v
Inmanyfieldssuchaschemistry,physicsorbiologyitappearedrelevanttostudythecontrollability
i
X of such a system (see [4]). For example, in [11], the authors study a system of three semilinear heat
r equations which is a model coming from a mathematical description of the growth of brain tumors.
a
The unknowns are the drug concentration, the density of tumors cells and the density of wealthy
cells and the aim is to control only two of them with one control. This practical issue motivates the
introduction of the partial null controllability.
For an initial condition y(0) = y ∈ L2(Ω)n and a control u ∈ L2(Q )m, it is well-known that
0 T
System (1.1) admits a unique solution in W(0,T)n, where
W(0,T):={y ∈L2(0,T;H1(Ω)),∂ y ∈L2(0,T;H−1(Ω))},
0 t
with H−1(Ω):=H1(Ω)′ and the following estimate holds (see [22])
0
kyk +kyk 6C(ky k +kuk ), (1.2)
L2(0,T;H01(Ω)n) C0([0,T];L2(Ω)n) 0 L2(Ω)n L2(QT)m
∗Laboratoire de Mathématiques de Besançon UMR CNRS 6623, Université de Franche-Comté, 16 route de Gray,
25030BesançonCedex,France
†Corresponding author: Email: [email protected]
1
where C does not depend on time. We denote by y(·;y ,u) the solution to System (1.1) determined
0
by the couple (y ,u).
0
Let us consider Π the projection matrix of L(Rn) given by Π :=(I 0 ) (I is the identity
p p p p,n−p p
matrix of L(Rp) and 0 the null matrix of L(Rn−p,Rp)), that is,
p,n−p
Π : Rn → Rp,
p
(y ,...,y ) 7→ (y ,...,y ).
1 n 1 p
System (1.1) is said to be
• Π -approximately controllable on the time interval (0,T), if for all real number ε>0 and
p
y , y ∈L2(Ω)n there exists a control u∈L2(Q )m such that
0 T T
kΠ y(T;y ,u)−Π y k 6ε.
p 0 p T L2(Ω)p
• Π -null controllable onthe time interval(0,T), if for allinitial conditiony ∈L2(Ω)n, there
p 0
exists a control u∈L2(Q )m such that
T
Π y(T;y ,u)≡0 in Ω.
p 0
Beforestating our mainresults, letus recallthe few knownresults aboutthe (full) null controlla-
bility of System (1.1). The first of them is about cascade systems (see [20]). The authors prove the
null controllability of System (1.1) with the control matrix B :=e (the first vector of the canonical
1
basis of Rn) and a coupling matrix A of the form
α α α ··· α
1,1 1,2 1,3 1,n
α α α ··· α
2,1 2,2 2,3 2,n
 
0 α α ··· α
A:= 3,2 3,3 3,n , (1.3)
 ... ... ... ... ... 
 
 0 0 ··· α α 
 n,n−1 n,n 
 
where the coefficients α are elements of L∞(Q ) for all i,j ∈ {1,...,n} and satisfy for a positive
i,j T
constant C and a nonempty open set ω of ω
0
α >C in ω or −α >C in ω for all i∈{1,...,n−1}.
i+1,i 0 i+1,i 0
A similar result on parabolic systems with cascade coupling matrices can be found in [1].
The null controllability of parabolic 3×3 linear systems with space/time dependent coefficients
and non cascade structure is studied in [8] and [23] (see also [20]).
If A∈L(Rn) and B ∈L(Rm,Rn) (the constantcase), it has been provedin [3] that System (1.1)
is null controllable on the time interval (0,T) if and only if the following condition holds
rank [A|B]=n, (1.4)
where [A|B], the so-called Kalman matrix, is defined as
[A|B]:=(B|AB|...|An−1B). (1.5)
For time dependent coupling and control matrices, we need some additional regularity. More
precisely, we need to suppose that A ∈ Cn−1([0,T];L(Rn)) and B ∈ Cn([0,T];L(Rm;Rn)). In this
case, the associated Kalman matrix is defined as follows. Let us define
B (t):=B(t),
0
B (t):=A(t)B (t)−∂ B (t) for all i∈{1,...,n−1}
i i−1 t i−1
(cid:26)
2
and denote by [A|B](·)∈C1([0,T];L(Rnm;Rn)) the matrix function given by
[A|B](·):=(B (·)|B (·)|...|B (·)). (1.6)
0 1 n−1
In [2] the authors prove first that, if there exists t ∈[0,T] such that
0
rank [A|B](t )=n, (1.7)
0
then System (1.1) is null controllable on the time interval (0,T). Secondly that System (1.1) is null
controllableoneveryinterval(T ,T ) with 06T <T 6T if andonly if there exists a dense subset
0 1 0 1
E of (0,T) such that
rank [A|B](t)=n for every t∈E. (1.8)
In the present paper, the controls are acting on several equations but on one subset ω of Ω.
Concerning the case where the control domains are not identical, we refer to [25].
Our first result is the following:
Theorem 1.1. Assume that the coupling and control matrices are constant in space and time, i.
e., A∈L(Rn) and B ∈L(Rm,Rn). The condition
rank Π [A|B]=p (1.9)
p
is equivalent to the Π -null/approximate controllability on the time interval (0,T) of System (1.1).
p
The Condition (1.9) for Π -null controllability reduces to Condition (1.4) whenever p = n. A
p
second result concerns the non-autonomous case:
Theorem 1.2. Assume that A∈Cn−1([0,T];L(Rn)) and B ∈Cn([0,T];L(Rm;Rn)). If
rank Π [A|B](T)=p, (1.10)
p
then System (1.1) is Π -null/approximately controllable on the time interval (0,T).
p
InTheorems1.1and1.2,wecontrolthepfirstcomponentsofthesolutiony. Ifwewanttocontrol
some other components a permutation of lines leads to the same situation.
Remark 1. 1. When the components of the matrices A and B are analytic functions on the time
interval [0,T], Condition (1.7) is necessary for the null controllability of System (1.1) (see Th.
1.3 in [2]). Under the same assumption, the proof of this result can be adapted to show that
the following condition
there exists t ∈[0,T] such that:
0
rank Π [A|B](t )=p,
p 0
(cid:26)
is necessary to the Π -null controllability of System (1.1).
p
2. As told before, under Condition (1.7), System (1.1) is null controllable. But unlike the case
where all the components are controlled, the Π -null controllability at a time t smaller than
p 0
T does not imply this property on the time interval (0,T). This roughly explains Condition
(1.10). Furthermorethis conditioncannotbe necessaryunder the assumptionsofTheorem1.2
(for a counterexample we refer to [2]).
Remark 2. IntheproofsofTheorems1.1and1.2,wewillusearesultofnullcontrollabilityforcascade
systems (see Section 2) proved in [2, 20] where the authors consider a time-dependent second order
elliptic operator L(t) given by
N N
∂ ∂y ∂y
L(t)y(x,t)=− α (x,t) (x,t) + b (x,t) (x,t)+c(x,t)y(x,t), (1.11)
i,j i
∂x ∂x ∂x
i,j=1 i (cid:18) j (cid:19) i=1 i
X X
3
with coefficients α , b , c satisfying
i,j i
α ∈W1 (Q ), b ,c∈L∞(Q ) 16i,j 6N,
i,j ∞ T i T
α (x,t)=α (x,t) ∀(x,t)∈Q , 16i,j 6N
i,j j,i T
(cid:26)
and the uniform elliptic condition: there exists a >0 such that
0
N
α (x,t)ξ ξ >a |ξ|2, ∀(x,t)∈Q .
i,j i j 0 T
i,j=1
X
Theorems 1.1 and 1.2 remain true if we replace −∆ by an operator L(t) in System (1.1).
Now the following question arises: what happens in the case of space and time dependent coeffi-
cients ? As it will be shownin the following example, the answer seems to be much more tricky. Let
us now consider the following parabolic system of two equations
∂ y =∆y+αz+1 u in Q ,
t ω T
∂ z =∆z in Q ,
 t T (1.12)
y =z =0 on Σ ,
 y(0)=y , z(0)=z in Ω,T
0 0
for given initial data y , z ∈L2(Ω), a control u∈L2(Q ) and where the coefficient α∈L∞(Ω).
0 0 T
Theorem 1.3. (1) Assume that α ∈ C1([0,T]). Then System (1.12) is Π -null controllable for
1
any open set ω ⊂Ω⊂RN (N ∈N∗), that is for all initial conditions y ,z ∈L2(Ω), there exists
0 0
a control u∈L2(Q ) such that the solution (y,z) to System (1.12) satisfies y(T)≡0 in Ω.
T
(2) Let Ω :=(a,b)⊂R (a,b∈ R), α∈L∞(Ω), (w ) be the L2-normalized eigenfunctions of −∆
k k>1
in Ω with Dirichlet boundary conditions and for all k,l ∈N∗,
α := α(x)w (x)w (x)dx.
kl k l
ZΩ
If the function α satisfies
|α |6C e−C2|k−l| for all k,l ∈N∗, (1.13)
kl 1
for two positive constants C >0 and C >b−a, then System (1.12) is Π -null controllable for
1 2 1
any open set ω ⊂Ω.
(3) Assume that Ω:=(0,2π) and ω ⊂(π,2π). Let us consider α∈L∞(0,2π) defined by
∞
1
α(x):= cos(15jx) for all x∈(0,2π).
j2
j=1
X
Then System (1.12) is not Π -null controllable. More precisely, there exists k ∈ {1,...,7} such
1 1
that for the initial condition (y ,z ) = (0,sin(k x)) and any control u ∈ L2(Q ) the solution y
0 0 1 T
to System (1.12) is not identically equal to zero at time T.
We will not prove item (1) in Theorem 1.3, because it is a direct consequence of Theorem 1.2.
Remark 3. Suppose that Ω:=(0,π). Considerα∈L∞(0,π) andthe realsequence (αp)p∈N suchthat
for all x∈(0,π)
∞
α(x):= α cos(px).
p
p=0
X
4
Concerning item (2), we remarkthat Condition (1.13) is equivalent to the existence of two constants
C >0, C >π such that, for all p∈N,
1 2
|α |6C e−C2p.
p 1
As it will be shown, the proof of item (3) in Theorem 1.3 can be adapted in order to get the same
conclusion for any α∈Hk(0,2π) (k ∈N∗) defined by
∞
1
α(x):= cos((2k+13)jx) for all x∈(0,2π). (1.14)
jk+1
j=1
X
These givenfunctions α belong to Hk(0,π) but not to D((−∆)k/2). Indeed, in the proofof the third
item in Theorem 1.3, we use the fact that the matrix (αkl)k,l∈N∗ is sparse (see (5.28)), what seems
true only for coupling terms α of the form (1.14). Thus α is not zero on the boundary.
Remark 4. From Theorem 1.3, one can deduce some new results concerning the null controllability
of the heat equation with a right-hand side. Consider the system
∂ y =∆y+f +1 u in (0,π)×(0,T),
t ω
y(0)=y(π)=0 on (0,T), (1.15)

y(0)=y in (0,π),
 0
where y ∈ L2(0,π) is the initial data and f,u ∈ L2(Q ) are the right-hand side and the control,
0 T
respectively. Using the Carleman inequality (see [17]), one can prove that System (1.15) is null
controllable when f satisfies
eTC−tf ∈L2(QT), (1.16)
for a positive constant C. For more general right-hand sides it was rather open. The second and
thirdpointsofTheorem1.3providesomepositiveandnegativenullcontrollabilityresultsforSystem
(1.15) with right-hand side f which does not fulfil Condition (1.16).
Remark 5. ConsiderthesamesystemasSystem(1.12)exceptthatthecontrolisnowontheboundary,
that is
∂ y =∆y+αz in (0,π)×(0,T),
t
∂ z =∆z in (0,π)×(0,T),
 t (1.17)
y(0,t)=v(t), y(π,t)=z(0,t)=z(π,t)=0 on (0,T),
 y(x,0)=y (x), z(x,0)=z (x) in (0,π),
0 0
where y , z ∈H−1(0,π). In Theorem 5.1, we provide an explicit coupling function α for which
0 0
the Π -null controllability of System (1.17) does not hold. Moreover one can adapt the proof of the
1
second point in Theorem 1.3 to prove the Π -null controllability of System (1.17) under Condition
1
(1.13).
If the coupling matrix depends on space, the notions of Π -null and approximate controllability
1
are not necessarily equivalent. Indeed, according to the choice of the coupling function α ∈ L∞(Ω),
System (1.12) can be Π -null controllable or not. But this system is Π -approximately controllable
1 1
for all α∈L∞(Ω):
Theorem 1.4. Let α ∈ L∞(Q ). Then System (1.12) is Π -approximately controllable for any
T 1
open set ω ⊂Ω⊂RN (N ∈N∗), that is for all y ,y ,z ∈L2(Ω) and all ε>0, there exists a control
0 T 0
u∈L2(Q ) such that the solution (y,z) to System (1.12) satisfies
T
ky(T)−y k 6ε.
T L2(Ω)
5
This result is a direct consequence of the unique continuation property and existence/unicity of
solutions for a single heat equation. Indeed System (1.12) is Π -approximately controllable (see
1
Proposition 2.1) if and only if for all φ ∈L2(Ω) the solution to the adjoint system
0
−∂ φ=∆φ in Q ,
t T
−∂ ψ =∆ψ+αφ in Q ,
 t T (1.18)
φ=ψ =0 on Σ ,
 φ(T)=φ , ψ(T)=0 in ΩT
0
satisfies 
φ≡0 in ω×(0,T) ⇒(φ,ψ)≡0 in Q .
T
If we assume that, for an initial data φ ∈ L2(Ω), the solution to System (1.18) satisfies φ ≡ 0 in
0
ω×(0,T), then using Mizohata uniqueness Theorem in [24], φ ≡ 0 in Q and consequently ψ ≡ 0
T
in Q . For another example of parabolic systems for which these notions are not equivalent we refer
T
for instance to [5].
Remark 6. The quantity α , which appears in the second item of Theorem 1.3, has already been
kl
considered in some controllability studies for parabolic systems. Let us define for all k ∈N∗
a
I (α):= α(x)w (x)2dx,
1,k k
 Z0
 I (α):=α .
k kk
In [6], the authors have proved that the system
∂ y =∆y+αz in (0,π)×(0,T),
t
∂ z =∆z+1 u in (0,π)×(0,T),
 t ω (1.19)
y(0,t)=y(π,t)=z(0,t)=z(π,t)=0 on (0,T),
 y(x,0)=y (x), z(x,0)=z (x) in (0,π),
0 0
is approximately controllable if and only if
|I (α)|+|I (α)|6=0 for all k ∈N∗.
k 1,k
A similar result has been obtained for the boundary approximate controllability in [10]. Consider
now
−log(min{|I |,|I |})
k 1,k
T (α):=limsup .
0 k2
k→∞
It is also proved in [6] that: If T > T (α), then System (1.19) is null controllable at time T and if
0
T <T (α),thenSystem(1.19)isnotnullcontrollableattimeT. Asinthepresentpaper,weobserve
0
a difference between the approximate and null controllability, in contrast with the scalar case (see
[4]).
In this paper, the sections are organized as follows. We start with some preliminary results on
the null controllability for the cascade systems and on the dual concept associated to the Π -null
p
controllability. Theorem 1.1 is provedin a first step with one force i.e. B ∈Rn in Section 3.1 and in
asecondstepwithmforcesinSection3.2. Section4isdevotedtoprovingTheorem1.2. Weconsider
the situations of the second and third items of Theorem 1.3 in Section 5.1 and 5.2 respectively. This
paperendswithsomenumericalillustrationsofΠ -nullcontrollabilityandnonΠ -nullcontrollability
1 1
of System (1.12) in Section 5.3.
2 Preliminaries
In this section, we recall a known result about cascade systems and provide a characterizationof
the Π -controllability through the corresponding adjoint system.
p
6
2.1 Cascade systems
2.1 Cascade systems
Sometheoremsofthispaperusethefollowingresultofnullcontrollabilityforthefollowingcascade
system of n equations controlled by r distributed functions
∂ w=∆w+Cw+D1 u in Q ,
t ω T
w =0 on Σ , (2.1)
 T
w(0)=w in Ω,
 0
where w0 ∈ L2(Ω)n, u = (u1,...,ur) ∈ L2(QT)r, with r ∈ {1,...,n}, and the coupling and control
matrices C ∈C0([0,T];L(Rn)) and D ∈L(Rr,Rn) are given by
C (t) C (t) ··· C (t)
11 12 1r
0 C (t) ··· C (t)
22 2r
C(t):= ... ... ... ...  (2.2)
 
 0 0 ··· C (t) 
 rr 
 
with
αi (t) αi (t) αi (t) ··· αi (t)
11 12 13 1,si
1 αi (t) αi (t) ··· αi (t)
 22 23 2,si 
0 1 αi (t) ··· αi (t)
Cii(t):= 33 3,si ,
 ... ... ... ... ... 
 
 0 0 ··· 1 αi (t) 
 si,si 
 
s ∈N, r s =n and D :=(e |...|e ) with S =1 and S =1+ i−1s , i∈{2,...,r} (e is the
i i=1 i S1 Sr 1 i j=1 j j
j-th element of the canonical basis of Rn).
P P
Theorem 2.1. System(2.1) is nullcontrollable on thetimeinterval(0,T),i.e. forall w ∈L2(Ω)n
0
there exists u ∈ L2(Ω)r such that the solution w in W(0,T)n to System (2.1) satisfies w(T) ≡ 0 in
Ω.
The proof of this result uses a Carleman estimate (see [17]) and can be found in [2] or [20].
2.2 Partial null controllability of a parabolic linear system by m forces
and adjoint system
It is nowadayswell-known that the controllability has a dual concept called observability (see for
instance [4]). We detail below the observability for the Π -controllability.
p
Proposition 2.1. 1. System (1.1) is Π -null controllable on the time interval (0,T) if and
p
only if there exists a constant C >0 such that for all ϕ =(ϕ0,...,ϕ0)∈L2(Ω)p the solution
obs 0 1 p
ϕ∈W(0,T)n to the adjoint system
−∂ ϕ=∆ϕ+A∗ϕ in Q ,
t T
ϕ=0 on Σ , (2.3)
T

ϕ(·,T)=Π∗ϕ =(ϕ0,...,ϕ0,0,...,0) in Ω
 p 0 1 p
satisfies the observability inequality
T
kϕ(0)k2 6C kB∗ϕk2 dt. (2.4)
L2(Ω)n obs L2(ω)m
Z0
2. System (1.1) is Π -approximately controllable on the time interval (0,T) if and only if for all
p
ϕ ∈L2(Ω)p the solution ϕ to System (2.3) satisfies
0
B∗ϕ≡0 in (0,T)×ω ⇒ϕ≡0 in Q .
T
7
2.2 Adjoint system
Proof. For all y ∈ L2(Ω)n, and u ∈L2(Q )m, we denote by y(t;y ,u) the solution to System (1.1)
0 T 0
at time t∈[0,T]. For all t∈[0,T], let us consider the operators S and L defined as follows
t t
S : L2(Ω)n → L2(Ω)n L : L2(Q )m → L2(Ω)n
t and t T (2.5)
y 7→ y(t;y ,0) u 7→ y(t;0,u).
0 0
1. System (1.1) is Π -null controllable on the time interval (0,T) if and only if
p
∀y ∈L2(Ω)n, ∃u∈L2(Q )m such that
0 T (2.6)
Π L u=−Π S y .
p T p T 0
Problem (2.6) admits a solution if and only if
Im Π S ⊂Im Π L . (2.7)
p T p T
The inclusion (2.7) is equivalent to (see [12], Lemma 2.48 p. 58)
∃C >0 such that ∀ϕ ∈L2(Ω)p,
0
(2.8)
kS∗Π∗ϕ k2 6CkL∗Π∗ϕ k2 .
T p 0 L2(Ω)n T p 0 L2(QT)m
We note that
S∗Π∗ : L2(Ω)p → L2(Ω)n L∗Π∗ : L2(Ω)p → L2(Q )m
T p and T p T
ϕ 7→ ϕ(0) ϕ 7→ 1 B∗ϕ,
0 0 ω
where ϕ∈W(0,T)n is the solutionto System (2.3). Indeed, for all y ∈L2(Ω)n, u∈L2(Q )m
0 T
and ϕ ∈L2(Ω)p
0
hΠ S y ,ϕ i = hy(T;y ,0),ϕ(T)i
p T 0 0 L2(Ω)p 0 L2(Ω)n
T
= h∂ y(s;y ,0),ϕ(s)i ds
t 0 L2(Ω)n
Z0 T (2.9)
+ hy(s;y ,0),∂ ϕ(s)i ds+hy ,ϕ(0)i
0 t L2(Ω)n 0 L2(Ω)n
Z0
= hy ,ϕ(0)i
0 L2(Ω)n
and
hΠ L u,ϕ i = hy(T;0,u),ϕ(T)i
p T 0 L2(Ω)p L2(Ω)n
T T
= h∂ y(s;0,u),ϕ(s)i ds+ hy(s;0,u),∂ ϕ(s)i ds
t L2(Ω)n t L2(Ω)n
= Zh10 Bu,ϕi =hu,1 B∗ϕi Z0 .
ω L2(QT)n ω L2(QT)m
(2.10)
The inequality (2.8) combined with (2.9)-(2.10) lead to the conclusion.
2. In view of the definition in (2.5) of S and L , System (1.1) is Π -approximately controllable
T T p
on the time interval (0,T) if and only if
∀(y ,y )∈L2(Ω)n×L2(Ω)p, ∀ε>0, ∃u∈L2(Q )m such that
0 T T
kΠ L u+Π S y −y k 6ε.
p T p T 0 T L2(Ω)p
This is equivalent to
∀ε>0, ∀z ∈L2(Ω)p,∃u∈L2(Q )m such that
T T
kΠ L u−z k 6ε.
p T T L2(Ω)p
That means
Π L (L2(Q )m)=L2(Ω)p.
p T T
8
In other words
ker L∗Π∗ ={0}.
T p
ThusSystem(1.1)isΠ -approximatelycontrollableonthe timeinterval(0,T)ifandonlyiffor
p
all ϕ ∈L2(Ω)p
0
L∗Π∗ϕ =1 B∗ϕ≡0 in Q ⇒ϕ≡0 in Q .
T p 0 ω T T
Corollary 2.1. Let us suppose that for all ϕ ∈ L2(Ω)p, the solution ϕ to the adjoint System (2.3)
0
satisfies the observability inequality (2.4). Then for all initial condition y ∈ L2(Ω)n, there exists a
0
control u ∈ L2(q )m (q := ω×(0,T)) such that the solution y to System (1.1) satisfies Π y(T) ≡
T T p
0 in Ω and
kuk 6 C ky k . (2.11)
L2(qT)m obs 0 L2(Ω)n
p
The proof is classical and will be omitted (estimate (2.11) can be obtained directly following the
method developed in [16]).
3 Partial null controllability with constant coupling matrices
Let us consider the system
∂ y =∆y+Ay+B1 u in Q ,
t ω T
y =0 on Σ , (3.1)
T

y(0)=y in Ω,
 0
where y ∈ L2(Ω)n, u ∈ L2(Q)m, A ∈ L(Rn) and B ∈ L(Rm;Rn). Let the natural number s be
0 T
defined by
s:=rank [A|B] (3.2)
and X ⊂Rn be the linear space spanned by the columns of [A|B].
In this section, we prove Theorem 1.1 in two steps. In subsection 3.1, we begin by studying the
case where B ∈Rn and the general case is considered in subsection 3.2.
All along this section, we will use the lemma below which proof is straightforward.
Lemma 3.1. Let be y ∈L2(Ω)n, u∈L2(Q )m and P ∈C1([0,T],L(Rn)) such that P(t) is invertible
0 T
forallt∈[0,T]. Thenthechangeofvariablew =P−1(t)y transformsSystem(3.1)intotheequivalent
system
∂ w =∆w+C(t)w+D(t)1 u in Q ,
t ω T
w =0 on Σ , (3.3)
T

w(0)=w in Ω,
 0
with w :=P−1(0)y , C(t):=−P−1(t)∂ P(t)+P−1(t)AP(t) and D(t):=P−1(t)B. Moreover
0 0 t
Π y(T)≡0 in Ω ⇔ Π P(T)w(T)≡0 in Ω.
p p
If P is constant, we have
[C|D]=P−1[A|B].
9
3.1 One control force
3.1 One control force
In this subsection, we suppose that A∈L(Rn), B ∈Rn and denote by [A|B]=:(k ) and
ij 16i,j6n
s:=rank [A|B]. We begin with the following observation.
Lemma 3.2. {B,...,As−1B} is a basis of X.
Proof. If s = rank [A|B] = 1, then the conclusion of the lemma is clearly true, since B 6= 0.
Let s > 2. Suppose to the contrary that {B,...,As−1B} is not a basis of X, that is for some
i ∈ {0,...,s − 2} the family {B,...,AiB} is linearly independent and Ai+1B ∈ span(B,...,AiB),
that is Ai+1B = i α AkB with α ,...,α ∈ R. Multiplying by A this expression, we deduce
k=0 k 0 i
that Ai+2B ∈span(AB,...,Ai+1B)=span(B,...,AiB). Thus, by induction, AlB ∈span(B,...,AiB)
P
for all l ∈ {i+ 1,...,n− 1}. Then rank (B|AB|...|An−1B) = rank (B|AB|...|AiB) = i +1 < s,
contradicting with (3.2).
Proof of Theorem 1.1. Let us remark that
rank Π [A|B]=dim Π [A|B](Rn)6rank [A|B]=s. (3.4)
p p
Lemma 3.2 yields
rank (B|AB|...|As−1B)=rank [A|B]=s. (3.5)
Thus, for all l∈{s,s+1,...,n} and i∈{0,...,s−1}, there exist α such that
l,i
s−1
AlB = α AiB. (3.6)
l,i
i=0
X
Since, for all l ∈{s,...,n}, Π AlB = s−1α Π AiB, then
p i=0 l,i p
rank Π (PB|AB|...|As−1B)=rank Π [A|B]. (3.7)
p p
We firstprovein(a)thatcondition(1.9)issufficient,andthenin(b)thatthis conditionisnecessary.
(a) Sufficiency part: Let us assume first that condition (1.9) holds. Then, using (3.7), we have
rank Π (B|AB|...|As−1B)=p. (3.8)
p
Letbe y ∈L2(Ω)n. Wewillstudy the Π -nullcontrollabilityofSystem(3.1)accordingtothe values
0 p
of p and s.
Case 1 : p=s. The idea is to find an appropriate change of variable P to the solution y to System
(3.1). More precisely, we would like the new variable w := P−1y to be the solution to a
cascade system and then, apply Theorem 2.1. So let us define, for all t∈[0,T],
P(t):=(B|AB|...|As−1B|P (t)|...|P (t)), (3.9)
s+1 n
where, for all l ∈ {s+1,...,n}, P (t) is the solution in C1([0,T])n to the system of ordinary
l
differential equations
∂ P (t)=AP (t) in [0,T],
t l l (3.10)
P (T)=e .
l l
(cid:26)
Using (3.9) and (3.10), we can write
P 0
P(T)= 11 , (3.11)
P I
21 n−s
(cid:18) (cid:19)
where P := Π (B|AB|...|As−1B) ∈ L(Rs), P ∈ L(Rs,Rn−s) and I is the identity
11 p 21 n−s
matrix of size n−s. Using (3.8), P is invertible and thus P(T) also. Furthermore, since
11
10