Partial null controllability of parabolic linear systems

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Partial null controllability of parabolic linear systems

Farid Ammar-Khodja, Franz Chouly, Michel Duprez

To cite this version:

Farid Ammar-Khodja, Franz Chouly, Michel Duprez. Partial null controllability of parabolic lin- ear systems. Mathematical Control and Related Fields, AIMS, 2017, 6 (2), pp.Pages: 185 - 216.

�10.3934/xx.xx.xx.xx�. �hal-01115812v2�

AIMS’ Journals

VolumeX, Number0X, XX200X pp.X–XX

PARTIAL NULL CONTROLLABILITY OF PARABOLIC LINEAR SYSTEMS

Farid Ammar Khodja, Franz Chouly and Michel Duprez^∗

Laboratoire de Math´ematiques de Besan¸con Universit´e de Franche-Comt´e

16, Route de Gray 25030 Besan¸ccon Cedex, France

(Communicated by the associate editor name)

Abstract. This paper is devoted to the partial null controllability issue of parabolic linear systems withnequations. Given a bounded domain Ω inR^N (N∈N^∗), we study the effect ofmlocalized controls in a nonempty open subset ωonly controlling pcomponents of the solution (p, m 6n). The first main result of this paper is a necessary and sufficient condition when the coupling and control matrices are constant. The second result provides, in a first step, a sufficient condition of partial null controllability when the matrices only depend on time. In a second step, through an example of partially controlled 2×2 parabolic system, we will provide positive and negative results on partial null controllability when the coefficients are space dependent.

1. Introduction and main results. Let Ω be a bounded domain inR^N (N∈N^∗) with aC²-class boundary∂Ω, ω be a nonempty open subset of Ω andT >0. Let p,m,n∈N^∗ such thatp, m6n. We consider in this paper the following system of nparabolic linear equations







∂ty= ∆y+Ay+B1^ωu inQT := Ω×(0, T),

y= 0 on ΣT :=∂Ω×(0, T),

y(0) =y0 in Ω,

(1) wherey0∈L²(Ω)ⁿ is the initial data,u∈L²(QT)^mis the control and

A∈L^∞(QT;L(Rⁿ)) andB∈L^∞(QT;L(R^m,Rⁿ)).

In many fields such as chemistry, physics or biology it appeared relevant to study the controllability of such a system (see [4]). For example, in [10], the authors study a system of three semilinear heat equations which is a model coming from a mathematical description of the growth of brain tumors. 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.

2010Mathematics Subject Classification. Primary: 93B05, 93B07; Secondary: 93C20, 93C05, 35K40.

Key words and phrases. Controllability, Observability, Kalman condition, Moment Method, Parabolic Systems.

This work was partially supported by R´egion de Franche-Comt´e (France).

∗Corresponding author: mduprez@math.cnrs.fr.

1

For an initial condition y(0) = y0 ∈ L²(Ω)ⁿ and a controlu ∈ L²(QT)^m, it is well-known that System (1) admits a unique solution inW(0, T)ⁿ, where

W(0, T) :={y∈L²(0, T;H₀¹(Ω)), ∂_ty∈L²(0, T;H⁻¹(Ω))}, withH⁻¹(Ω) :=H₀¹(Ω)⁰ and the following estimate holds (see [21])

kyk_L2(0,T;H₀¹(Ω)ⁿ)+kyk_C0([0,T];L²(Ω)ⁿ)6C(ky0k_L2(Ω)ⁿ+kuk_L2(QT)^m), (2) whereC does not depend on time. We denote byy(·;y0, u) the solution to System (1) determined by the couple (y0, u).

Let us consider Πp theprojection matrix ofL(Rⁿ) given by Πp := (Ip 0p,n−p) (I_p is the identity matrix ofL(R^p) and 0_p,n−pthe null matrix ofL(R^n−p,R^p)), that is,

Πp: Rⁿ → R^p,

(y₁, ..., y_n) 7→ (y₁, ..., y_p).

System (1) is said to be

• Πp-approximately controllable on the time interval (0, T), if for all real number ε >0 andy₀, y_T ∈L²(Ω)ⁿ there exists a controlu∈L²(Q_T)^msuch that

kΠ_py(T;y₀, u)−Π_py_Tk_L2(Ω)^p6ε.

• Πp-null controllable on the time interval (0, T), if for all initial condition y0∈L²(Ω)ⁿ, there exists a controlu∈L²(QT)^msuch that

Πpy(T;y0, u)≡0 in Ω.

Before stating our main results, let us recall the few known results about the (full) null controllability of System (1). The first of them is about cascade systems (see [19]). The authors prove the null controllability of System (1) with the control matrixB:=e₁(the first vector of the canonical basis ofRⁿ) and a coupling matrix Aof the form

A:=















α_1,1 α_1,2 α_1,3 · · · α_1,n α_2,1 α_2,2 α_2,3 · · · α_2,n 0 α_3,2 α_3,3 · · · α_3,n ... ... . .. . .. ... 0 0 · · · α_n,n−1 αn,n















, (3)

where the coefficientsα_i,jare elements ofL^∞(Q_T) for alli, j∈ {1, ..., n}and satisfy for a positive constantC and a nonempty open setω₀ofω

αi+1,i>Cin ω0 or −αi+1,i>Cin ω0 for alli∈ {1, ..., n−1}.

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 de- pendent coefficients and non cascade structure is studied in [7] and [22] (see also [19]).

IfA∈ L(Rⁿ) andB ∈ L(R^m,Rⁿ) (the constant case), it has been proved in [3]

that System (1) is null controllable on the time interval (0, T) if and only if the following condition holds

rank [A|B] =n, (4)

where [A|B], the so-calledKalman matrix, is defined as

[A|B] := (B|AB|...|Aⁿ⁻¹B). (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(R^m;Rⁿ)). In this case, the associated Kalman matrix is defined as follows. Let us define

B0(t) :=B(t),

Bi(t) :=A(t)Bi−1(t)−∂tBi−1(t) for alli∈ {1, ..., n−1}

and denote by [A|B](·)∈ C¹([0, T];L(R^nm;Rⁿ)) the matrix function given by [A|B](·) := (B0(·)|B1(·)|...|Bn−1(·)). (6) In [2] the authors prove first that, if there existst0∈[0, T] such that

rank [A|B](t0) =n, (7)

then System (1) is null controllable on the time interval (0, T). Secondly that System (1) is null controllable on every interval (T0, T1) with 06T0 < T1 6T if and only if there exists a dense subsetE of (0, T) such that

rank [A|B](t) =nfor every t∈E. (8) In the present paper, the controls are acting on several equations but on one subsetω of Ω. Concerning the case where the control domains are not identical, we refer to [24].

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)andB∈ L(R^m,Rⁿ). The condition

rank Πp[A|B] =p (9)

is equivalent to the Πp-null/approximate controllability on the time interval(0, T) of System (1).

The Condition (9) for Πp-null controllability reduces to Condition (4) whenever p=n. A second result concerns the non-autonomous case:

Theorem 1.2. Assume that

A∈ Cⁿ⁻¹([0, T];L(Rⁿ))andB∈ Cⁿ([0, T];L(R^m;Rⁿ)).

If

rank Πp[A|B](T) =p, (10)

then System (1) is Πp-null/approximately controllable on the time interval (0, T).

In Theorems1.1and1.2, we control the pfirst components of the solutiony. If we want to control 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 (7) is necessary for the null controllability of System (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 existst₀∈[0, T] such that : rank Π_p[A|B](t0) =p,

is necessary to the Π_p-null controllability of System (1).

2. As told before, under Condition (7), System (1) is null controllable. But unlike the case where all the components are controlled, the Πp-null controllability at a timet0smaller thanT does not imply this property on the time interval (0, T). This roughly explains Condition (10). Furthermore this condition can not be necessary under the assumptions of Theorem1.2(for a counterexample we refer to [2]).

Remark 2. In the proofs of Theorems 1.1 and 1.2, we will use a result of null controllability for cascade systems (see Section2) proved in [2,19] where the authors consider a time-dependent second order elliptic operatorL(t) given by

L(t)y(x, t) =−

N

X

i,j=1

∂

∂xi

αi,j(x, t)∂y

∂xj

(x, t)

+

N

X

i=1

bi(x, t)∂y

∂xi

(x, t) +c(x, t)y(x, t), (11) with coefficientsαi,j,bi,csatisfying

αi,j∈W_∞¹(QT), bi, c∈L^∞(QT) 16i, j6N, αi,j(x, t) =αj,i(x, t)∀(x, t)∈QT, 16i, j6N and the uniform elliptic condition: there existsa0>0 such that

N

X

i,j=1

α_i,j(x, t)ξ_iξ_j>a₀|ξ|², ∀(x, t)∈Q_T.

Theorems1.1and1.2remain true if we replace−∆ by an operatorL(t) in System (1).

Now the following question arises: what happens in the case of space and time dependent coefficients ? As it will be shown in the following example, the answer seems to be much more tricky. Let us now consider the following parabolic system of two equations











∂ty= ∆y+αz+1^ωu inQT,

∂tz= ∆z inQT,

y=z= 0 on ΣT,

y(0) =y0, z(0) =z0 in Ω,

(12)

for given initial datay₀, z₀∈L²(Ω), a controlu∈L²(Q_T) and where the coefficient α∈L^∞(Ω).

Theorem 1.3. (1) Assume thatα∈C¹([0, T]). Then System (12) isΠ1-null con- trollable for any open setω⊂Ω⊂R^N (N ∈N^∗), that is for all initial conditions y₀, z₀ ∈L²(Ω), there exists a control u∈L²(Q_T)such that the solution (y, z) to System (12) satisfiesy(T)≡0in Ω.

(2) Let Ω := (a, b) ⊂ R (a, b ∈ R), α ∈ L^∞(Ω), (w_k)_k>1 be the L²-normalized eigenfunctions of−∆inΩwith Dirichlet boundary conditions and for allk, l∈ N^∗,

α_kl:=

Z

Ω

α(x)w_k(x)w_l(x)dx.

If the function αsatisfies

|αkl|6C1e^−C2|k−l|for all k, l∈N^∗, (13) for two positive constants C1>0 andC2> b−a, then System (12) is Π1-null controllable for any open set ω⊂Ω.

(3) Assume that Ω := (0,2π) and ω ⊂ (π,2π). Let us consider α ∈ L^∞(0,2π) defined by

α(x) :=

∞

X

j=1

1

j²cos(15jx) for allx∈(0,2π).

Then System (12) is notΠ1-null controllable. More precisely, there existsk1∈ {1, ...,7} such that for the initial condition (y0, z0) = (0,sin(k1x)) and any control u ∈ L²(Q_T) the solution y to System (12) is not identically equal to zero at time T.

We will not prove item (1) in Theorem1.3, because it is a direct consequence of Theorem1.2.

Remark 3. Suppose that Ω := (0, π). Considerα∈L^∞(0, π) and the real sequence (αp)p∈N such that for allx∈(0, π)

α(x) :=

∞

X

p=0

α_pcos(px).

Concerning item (2), we remark that Condition (13) is equivalent to the existence of two constantsC1>0, C2> π such that, for allp∈N,

|α_p|6C₁e^−C2p.

As it will be shown, the proof of item (3) in Theorem1.3 can be adapted in order to get the same conclusion for anyα∈H^k(0,2π) (k∈N^∗) defined by

α(x) :=

∞

X

j=1

1

j^k+1cos((2k+ 13)jx) for allx∈(0,2π). (14) These given functionsαbelong toH^k(0, π) but not toD((−∆)^k/2). Indeed, in the proof of the third item in Theorem1.3, we use the fact that the matrix (α_kl)_k,l∈N^∗ is sparse (see (103)), what seems true only for coupling termsαof the form (14).

Thusαis 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







∂_ty= ∆y+f+1ωu in (0, π)×(0, T), y(0) =y(π) = 0 on (0, T), y(0) =y₀ in (0, π),

(15)

where y0 ∈L²(0, π) is the initial data and f, u ∈ L²(QT) are the right-hand side and the control, respectively. Using the Carleman inequality (see [16]), one can prove that System (15) is null controllable when f satisfies

e^T−tC f ∈L²(QT), (16)

for a positive constant C. For more general right-hand sides it was rather open.

The second and third points of Theorem 1.3 provide some positive and negative null controllability results for System (15) with right-hand side f which does not fulfil Condition (16).

Remark 5. Consider the same system as System (12) except that the control is now on the boundary, that is











∂ty= ∆y+αz in (0, π)×(0, T),

∂_tz= ∆z in (0, π)×(0, T),

y(0, t) =v(t), y(π, t) =z(0, t) =z(π, t) = 0 on (0, T), y(x,0) =y₀(x), z(x,0) =z₀(x) in (0, π),

(17)

wherey0, z0∈H⁻¹(0, π). In Theorem5.4, we provide an explicit coupling function α for which the Π1-null controllability of System (17) does not hold. Moreover one can adapt the proof of the second point in Theorem 1.3 to prove the Π1-null controllability of System (17) under Condition (13).

If the coupling matrix depends on space, the notions of Π1-null and approximate controllability are not necessarily equivalent. Indeed, according to the choice of the coupling functionα∈L^∞(Ω), System (12) can be Π₁-null controllable or not. But this system is Π₁-approximately controllable for allα∈L^∞(Ω):

Theorem 1.4. Let α∈L^∞(QT). Then System (12) is Π1-approximately control- lable for any open setω⊂Ω⊂R^N (N ∈N^∗), that is for all y₀, y_T, z₀∈L²(Ω)and all ε >0, there exists a control u∈L²(Q_T)such that the solution (y, z)to System (12) satisfies

ky(T)−y_TkL²(Ω)6ε.

This result is a direct consequence of the unique continuation property and exis- tence/unicity of solutions for a single heat equation. Indeed System (12) is Π₁- approximately controllable (see Proposition1) if and only if for allφ₀∈L²(Ω) the solution to the adjoint system











−∂tφ= ∆φ in QT,

−∂tψ= ∆ψ+αφ in QT,

φ=ψ= 0 on ΣT,

φ(T) =φ0, ψ(T) = 0 in Ω

(18)

satisfies

φ≡0 inω×(0, T) ⇒(φ, ψ)≡0 inQT.

If we assume that, for an initial data φ0 ∈ L²(Ω), the solution to System (18) satisfiesφ≡0 inω×(0, T), then using Mizohata uniqueness Theorem in [23],φ≡0 in QT and consequently ψ ≡0 in QT. For another example of parabolic systems for which these notions are not equivalent we refer for instance to [5].

Remark 6. The quantity αkl, which appears in the second item of Theorem 1.3, has already been considered in some controllability studies for parabolic systems.

Let us define for allk∈N^∗







I1,k(α) :=

Z a 0

α(x)wk(x)²dx, Ik(α) :=αkk.

In [6], the authors have proved that the system











∂ty = ∆y+αz in (0, π)×(0, T),

∂tz= ∆z+1^ωu in (0, π)×(0, T), y(0, t) =y(π, t) =z(0, t) =z(π, t) = 0 on (0, T), y(x,0) =y0(x), z(x,0) =z0(x) in (0, π),

(19)

is approximately controllable if and only if

|Ik(α)|+|I1,k(α)| 6= 0 for allk∈N^∗.

A similar result has been obtained for the boundary approximate controllability in [9]. Consider now

T₀(α) := lim sup

k→∞

−log(min{|I_k|,|I_1,k|})

k² .

It is also proved in [6] that: If T > T0(α), then System (19) is null controllable at time T and ifT < T0(α), then System (19) is not null controllable at time T. As in the present paper, we observe 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 pre- liminary results on the null controllability for the cascade systems and on the dual concept associated to the Πp-null controllability. Theorem 1.1 is proved in a first step with one force i.e. B∈Rⁿin Section3.1and in a second step withmforces in Section3.2. Section4is devoted to proving Theorem1.2. We consider the situations of the second and third items of Theorem1.3 in Section 5.1 and 5.2 respectively.

This paper ends with some numerical illustrations of Π₁-null controllability and non Π₁-null controllability of System (12) in Section5.3.

2. Preliminaries. In this section, we recall a known result about cascade systems and provide a characterization of the Πp-controllability through the corresponding adjoint system.

2.1. Cascade systems. Some theorems of this paper use the following result of null controllability for the following cascade system ofn equations controlled byr distributed functions







∂tw= ∆w+Cw+D1^ωu in QT,

w= 0 on ΣT,

w(0) =w0 in Ω,

(20)

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(R^r,Rⁿ) are given by

C(t) :=











C₁₁(t) C₁₂(t) · · · C_1r(t) 0 C₂₂(t) · · · C_2r(t)

... ... . .. ... 0 0 · · · Crr(t)











(21)

with

Cii(t) :=















αⁱ₁₁(t) αⁱ₁₂(t) αⁱ₁₃(t) · · · αⁱ_1,si(t) 1 αⁱ₂₂(t) αⁱ₂₃(t) · · · αⁱ_2,s

i(t) 0 1 αⁱ₃₃(t) · · · αⁱ_3,si(t) ... ... . .. . .. ... 0 0 · · · 1 αⁱ_si,si(t)













 ,

si ∈ N, Pr

i=1si = n and D := (eS1|...|eSr) with S1 = 1 and Si = 1 +Pi−1 j=1sj, i∈ {2, ..., r} (ej is thej-th element of the canonical basis ofRⁿ).

Theorem 2.1. System (20) is null controllable on the time interval(0, T), i.e. for all w₀ ∈ L²(Ω)ⁿ there exists u∈L²(Ω)^r such that the solution w in W(0, T)ⁿ to System (20) satisfiesw(T)≡0in Ω.

The proof of this result uses a Carleman estimate (see [16]) and can be found in [2] or [19].

2.2. Partial null controllability of a parabolic linear system by m forces and adjoint system. It is nowadays well-known that the controllability has a dual concept calledobservability (see for instance [4]). We detail below the observability for the Πp-controllability.

Proposition 1. 1. System (1) isΠ_p-null controllable on the time interval(0, T) if and only if there exists a constant C_obs > 0 such that for all initial data ϕ₀= (ϕ⁰₁, ..., ϕ⁰_p)∈L²(Ω)^p the solutionϕ∈W(0, T)ⁿ to the adjoint system







−∂tϕ= ∆ϕ+A^∗ϕ in QT,

ϕ= 0 on ΣT,

ϕ(·, T) = Π^∗_pϕ0= (ϕ⁰₁, ..., ϕ⁰_p,0, ...,0) in Ω

(22) satisfies the observability inequality

kϕ(0)k²_L2(Ω)ⁿ 6C_obs Z T

0

kB^∗ϕk²_L2(ω)^mdt. (23) 2. System (1) isΠp-approximately controllable on the time interval(0, T)if and

only if for allϕ0∈L²(Ω)^p the solutionϕto System (22) satisfies B^∗ϕ≡0 in (0, T)×ω ⇒ϕ≡0 in Q_T.

Proof. For ally0∈L²(Ω)ⁿ, andu∈L²(QT)^m, we denote byy(t;y0, u) the solution to System (1) at timet∈[0, T]. For allt∈[0, T], let us consider the operatorsSt

andLtdefined as follows S_t: L²(Ω)ⁿ → L²(Ω)ⁿ

y0 7→ y(t;y0,0) and L_t: L²(Q_T)^m → L²(Ω)ⁿ

u 7→ y(t; 0, u). (24) 1. System (1) is Πp-null controllable on the time interval (0, T) if and only if

∀y0∈L²(Ω)ⁿ, ∃u∈L²(QT)^msuch that

ΠpLTu=−ΠpSTy0. (25)

Problem (25) admits a solution if and only if

Im Π_pS_T ⊂Im Π_pL_T. (26)

The inclusion (26) is equivalent to (see [11], Lemma 2.48 p. 58)

∃C >0 such that∀ϕ0∈L²(Ω)^p,

kS_T^∗Π^∗_pϕ0k²_L2(Ω)ⁿ6CkL^∗_TΠ^∗_pϕ0k²_L2(QT)^m. (27) We note that

S_T^∗Π^∗_p: L²(Ω)^p → L²(Ω)ⁿ

ϕ₀ 7→ ϕ(0) and L^∗_TΠ^∗_p: L²(Ω)^p → L²(QT)^m ϕ₀ 7→ 1ωB^∗ϕ, where ϕ ∈ W(0, T)ⁿ is the solution to System (22). Indeed, for all y0 ∈ L²(Ω)ⁿ, u∈L²(Q_T)^mand ϕ₀∈L²(Ω)^p

hΠ_pS_Ty₀, ϕ₀i_L2(Ω)^p = hy(T;y₀,0), ϕ(T)i_L2(Ω)ⁿ

= Z T

0

h∂ty(s;y0,0), ϕ(s)iL²(Ω)ⁿds +

Z T 0

hy(s;y₀,0), ∂_tϕ(s)i_L2(Ω)ⁿds+hy₀, ϕ(0)i_L2(Ω)ⁿ

= hy0, ϕ(0)iL²(Ω)ⁿ

(28) and

hΠpLTu, ϕ0iL²(Ω)^p = hy(T; 0, u), ϕ(T)iL²(Ω)ⁿ

= Z T

0

h∂ty(s; 0, u), ϕ(s)i_L2(Ω)ⁿds +

Z T 0

hy(s; 0, u), ∂tϕ(s)iL²(Ω)ⁿds

= h1^ωBu, ϕi_L2(QT)ⁿ=hu,1^ωB^∗ϕi_L2(QT)^m.

(29)

The inequality (27) combined with (28)-(29) lead to the conclusion.

2. In view of the definition in (24) ofST andLT, System (1) is Πp-approximately controllable on the time interval (0, T) if and only if

∀(y0, yT)∈L²(Ω)ⁿ×L²(Ω)^p, ∀ε >0, ∃u∈L²(QT)^m such that kΠpLTu+ ΠpSTy0−yTkL²(Ω)^p6ε.

This is equivalent to

∀ε >0, ∀zT ∈L²(Ω)^p,∃u∈L²(QT)^msuch that kΠpLTu−zTk_L2(Ω)^p6ε.

That means

ΠpLT(L²(QT)^m) =L²(Ω)^p. In other words

kerL^∗_TΠ^∗_p={0}.

Thus System (1) is Πp-approximately controllable on the time interval (0, T) if and only if for allϕ0∈L²(Ω)^p

L^∗_TΠ^∗_pϕ0=1^ωB^∗ϕ≡0 inQT ⇒ϕ≡0 inQT.

Corollary 1. Let us suppose that for allϕ0∈L²(Ω)^p, the solutionϕto the adjoint System (22) satisfies the observability inequality (23). Then for all initial condition

y0 ∈ L²(Ω)ⁿ, there exists a control u∈L²(qT)^m (qT :=ω×(0, T)) such that the solution y to System (1) satisfiesΠpy(T)≡0 in Ωand

kuk_L2(q_T)^m 6p

C_obsky₀k_L2(Ω)ⁿ. (30) The proof is classical and will be omitted (estimate (30) can be obtained directly following the method developed in [15]).

3. Partial null controllability with constant coupling matrices. Let us con- sider the system







∂_ty= ∆y+Ay+B1ωu inQ_T,

y= 0 on Σ_T,

y(0) =y₀ in Ω,

(31) wherey₀∈L²(Ω)ⁿ,u∈L²(Q_T)^m,A∈ L(Rⁿ) andB∈ L(R^m;Rⁿ). Let the natural numbersbe defined by

s:= rank [A|B] (32)

andX ⊂Rⁿ be the linear space spanned by the columns of [A|B].

In this section, we prove Theorem1.1 in two steps. In subsection3.1, we begin by studying the case whereB ∈Rⁿand 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 y₀ ∈L²(Ω)ⁿ, u∈ L²(Q_T)^m and P ∈ C¹([0, T],L(Rⁿ)) such that P(t) is invertible for allt ∈[0, T]. Then the change of variable w=P⁻¹(t)y transforms System (31) into the equivalent system







∂_tw= ∆w+C(t)w+D(t)1ωu in Q_T,

w= 0 on Σ_T,

w(0) =w₀ in Ω,

(33)

withw0:=P⁻¹(0)y0,C(t) :=−P⁻¹(t)∂tP(t)+P⁻¹(t)AP(t)andD(t) :=P⁻¹(t)B.

Moreover

Πpy(T)≡0 in Ω ⇔ ΠpP(T)w(T)≡0 in Ω.

If P is constant, we have

[C|D] =P⁻¹[A|B].

3.1. One control force. In this subsection, we suppose that A∈ L(Rⁿ),B ∈Rⁿ and denote by [A|B] =: (kij)_16i,j6n and s := rank [A|B]. We begin with the following observation.

Lemma 3.2. {B, ..., A^s−1B} is a basis of X.

Proof. Ifs= 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, ..., A^s−1B} is not a basis of X, that is for somei∈ {0, ..., s−2}the family{B, ..., AⁱB}is linearly independent andAⁱ⁺¹B∈span(B, ..., AⁱB), that isAⁱ⁺¹B =Pi

k=0αkA^kB withα0, ..., αi∈R. Multiplying byAthis expression, we deduce thatAⁱ⁺²B∈span(AB, ..., Aⁱ⁺¹B) = span(B, ..., AⁱB). Thus, by induction, A^lB ∈ span(B, ..., AⁱB) for all l ∈ {i+ 1, ..., n−1}. Then rank (B|AB|...|Aⁿ⁻¹B) = rank (B|AB|...|AⁱB) = i+ 1 < s, contradicting with (32).

Proof of Theorem 1.1. Let us remark that

rank Π_p[A|B] = dim Πp[A|B](Rⁿ)6rank [A|B] =s. (34) Lemma3.2yields

rank (B|AB|...|A^s−1B) = rank [A|B] =s. (35) Thus, for alll∈ {s, s+ 1, ..., n} andi∈ {0, ..., s−1}, there existα_l,i such that

A^lB=

s−1

X

i=0

αl,iAⁱB. (36)

Since, for alll∈ {s, ..., n}, ΠpA^lB=Ps−1

i=0αl,iΠpAⁱB, then

rank Πp(B|AB|...|A^s−1B) = rank Πp[A|B]. (37) We first prove in (a) that condition (9) is sufficient, and then in (b) that this condition is necessary.

(a) Sufficiency part: Let us assume first that condition (9) holds. Then, using (37), we have

rank Πp(B|AB|...|A^s−1B) =p. (38) Let be y0 ∈ L²(Ω)ⁿ. We will study the Πp-null controllability of System (31) according to the values ofpands.

Case 1 : p = s. The idea is to find an appropriate change of variable P to the solutiony to System (31). More precisely, we would like the new variable w:=P⁻¹yto 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|...|A^s−1B|Ps+1(t)|...|Pn(t)), (39) where, for all l ∈ {s+ 1, ..., n}, Pl(t) is the solution in C¹([0, T])ⁿ to the system of ordinary differential equations

∂tPl(t) =APl(t) in [0, T],

Pl(T) =el. (40)

Using (39) and (40), we can write P(T) =

P₁₁ 0 P21 I_n−s

, (41)

whereP11:= Πp(B|AB|...|A^s−1B)∈ L(R^s),P21∈ L(R^s,R^n−s) andI_n−sis the identity matrix of sizen−s. Using (38),P11is invertible and thusP(T) also. Furthermore, sinceP(t)∈ C¹([0, T],L(Rⁿ)) continuous in time on the time interval [0, T], there existsT^∗ ∈[0, T) such that P(t) is invertible for allt∈[T^∗, T].

Let us suppose first that T^∗ = 0. Since P(t) ∈ C¹([0, T],L(Rⁿ)) and invertible, in view of Lemma 3.1: for a fixed controlu∈L²(Q_T),y is the solution to System (31) if and only ifw:=P(t)⁻¹yis the solution to System (33) whereC,D are given by

C(t) :=−P⁻¹(t)∂tP(t) +P⁻¹(t)AP(t) and D(t) :=P⁻¹(t)B, for allt∈[0, T]. Using (36) and (40), we obtain







−∂_tP(t) +AP(t) = (AB|...|A^sB|0|...|0) =P(t)

C₁₁ 0

0 0

in [0, T],

P(t)e1=B in [0, T],

(42)

where

C₁₁:=















0 0 0 . . . α_s,0 1 0 0 . . . α_s,1 0 1 0 . . . αs,2

... ... . .. . .. ... 0 0 . . . 1 α_s,s−1















∈ L(R^s). (43)

Then

C(t) =

C11 0

0 0

andD(t) =e1. (44)

Using Theorem 2.1, there exists u ∈ L²(QT) such that the solution to System (33) satisfies w1(T)≡...≡ws(T)≡0 in Ω. Moreover, using (41), we have

Πsy(T) = (y1(T), ..., ys(T)) =P11(w1(T), ..., ws(T))≡0 in Ω.

If nowT^∗6= 0, lety be the solution inW(0, T^∗)ⁿto System (31) with the initial condition y(0) =y₀ in Ω and the control u≡0 in Ω×(0, T^∗).

We use the same argument as above to prove that System (31) is Π_s-null controllable on the time interval [T^∗, T]. Let v be a control in L²(Ω× (T^∗, T)) such that the solution z in W(T^∗, T)ⁿ to System (31) with the initial conditionz(T^∗) =y(T^∗) in Ω and the controlv satisfies Πsz(T)≡0 in Ω. Thus if we definey anduas follows

(y, u) :=

(y,0) if t∈[0, T^∗], (z, v) ift∈[T^∗, T],

then, for this controlu,yis the solution inW(0, T)ⁿ to System (31). More- overy satisfies

Πsy(T)≡0 in Ω.

Case 2 : p < s. In order to use Case 1, we would like to apply an appropriate change of variable Q to the solution y to System (31). If we denote by [A|B] =: (kij)ij, equalities (35) and (38) can be rewritten

rank







k11 · · · k1s

... ... k_n1 · · · k_ns





=s and rank







k11 · · · k1s

... ... k_p1 · · · k_ps





=p.

Then there exist distinct natural numbers λp+1, ..., λs such that we have {λp+1, ..., λs} ⊂ {p+ 1, ..., n} and

rank





















k11 · · · k1s

... ...

kp1 · · · kps

k_λp+11 · · · k_λp+1s

... ...

k_λs1 · · · k_λss





















=s. (45)

LetQbe the matrix defined by

Q:= (e₁|...|e_p|e_λp+1|...|e_λn)^t,

where {λs+1, ..., λn} := {p+ 1, ..., n}\{λp+1, ..., λs}. Q is invertible, so taking w := P⁻¹y with P := Q⁻¹, for a fixed control u in L²(Q_T), y is

solution to System (31) if and only ifw is solution to System (33) where w0 :=Qy0, C :=QAQ⁻¹ ∈ L(Rⁿ) and D := QB ∈ L(R;Rⁿ). Moreover there holds

[C|D] =Q[A|B].

Thus, equation (45) yields

rank Πs[C|D] = rank ΠsQ[A|B] = rank





















k11 · · · k1n

... ...

kp1 · · · kpn

kλ_p+11 · · · kλ_p+1n

... ...

kλs1 · · · kλsn





















=s.

Since rank [C|D] = rank [A|B] = s, we proceed as in Case 1 forward deduce that System (33) is Π_s-null controllable, that is there exists a control u∈L²(Q_T) such that the solutionwto System (33) satisfies

Πsw(T)≡0 in Ω.

Moreover the matrixQcan be rewritten Q=

Ip 0 0 Q22

, where Q₂₂∈ L(R^n−p). Thus

Πpy(T) = ΠpQy(T) = Πpw(T)≡0 in Ω.

(b) Necessary part: Let us denote by [A|B] =: (kij)ij. We suppose now that (9) is not satisfied: there exist p∈ {1, ..., p} and βi for alli ∈ {1, ..., p}\{p} such thatkpj=

p

P

i=1,i6=p

βikij for allj ∈ {1, ..., s}. The idea is to find a change of variable w:=Qythat allows to handle more easily our system. We will achieve this in three steps starting from the simplest situation.

Step 1. Let us suppose first that

k11=...=k1s= 0 and rank







k21 · · · k2s

... ... ks+1,1 · · · ks+1,s





=s. (46) We want to prove that, for some initial condition y0 ∈ L²(Ω)ⁿ, a control u∈L²(Q_T) cannot be found such that the solution to System (31) satisfies y₁(T)≡0 in Ω. Let us consider the matrixP∈ L(Rⁿ) defined by

P := (B|...|A^s−1B|e₁|e_s+2|...|e_n). (47) Using the assumption (46), P is invertible. Thus, in view of Lemma 3.1, for a fixed control u∈L²(QT), y is a solution to System (31) if and only if w:=P⁻¹y is a solution to System (33) whereC, D are given by C :=

P⁻¹AP andD:=P⁻¹B. Using (36) we remark that A(B|AB|...|BA^s−1) = (B|AB|...|BA^s−1)

C11

0

,

withC11 defined in (43). ThenC can be rewritten as C=

C11 C12

0 C22

, (48)

whereC12∈ L(R^n−s,R^s) andC22∈ L(R^n−s). Furthermore D=P⁻¹B=P⁻¹P e1=e1.

and with the Definition (47) ofP we get y₁(T) =w_s+1(T) in Ω.

Thus we need only to prove that there exists w0 ∈ L²(Ω)ⁿ such that we cannot find a control u ∈ L²(QT) with the corresponding solution w to System (33) satisfyingws+1(T)≡0 in Ω. Therefore we apply Proposition 1and prove that the observability inequality (23) can not be satisfied. More precisely, for allw₀∈L²(Ω)ⁿ, there exists a controlu∈L²(Q_T) such that the solution to System (33) satisfiesw_s+1(T)≡0 in Ω, if and only if there exists C_obs>0 such that for all ϕ⁰_s+1 ∈L²(Ω) the solution to the adjoint system











−∂tϕ = ∆ϕ+

C₁₁^∗ 0 C₁₂^∗ C₂₂^∗

ϕ inQT,

ϕ = 0 on ΣT,

ϕ(T) = (0, ...,0, ϕ⁰_s+1,0, ...,0)^t=es+1ϕ⁰_s+1 in Ω

(49)

satisfies the observability inequality Z

Ω

ϕ(0)²dx6C_obs Z

ω×(0,T)

ϕ²₁dx dt. (50)

But for allϕ⁰_s+1 6≡0 in Ω, the inequality (50) is not satisfied. Indeed, we remark first that, sinceϕ1(T) = ...=ϕs(T) = 0 in Ω, we have ϕ1=...= ϕs= 0 inQT, so thatR

ω×(0,T)ϕ²₁dx= 0,while, if we chooseϕ⁰_s+16≡0 in Ω, using the results on backward uniqueness for this type of parabolic system (see [17]), we have clearly (ϕ_s+1(0), ..., ϕ_n(0))6≡0 in Ω.

Step 2. Let us suppose only thatk₁₁=...=k_1s= 0. Since rank (B|...|A^s−1B) =s, there exists distinctλ₁, ..., λ_s∈ {2, ..., n}such that

rank







k_λ1,1 · · · k_λ1,s ... ... k_λs,1 · · · k_λs,s





=s.

Let us consider the following matrix

Q:= (e₁|e_λ1|...|e_λn−1)^t,

where{λ_s+1, ..., λ_n−1}={2, ..., n}\{λ₁, ..., λ_s}. Thus, forP :=Q⁻¹, again, for a fixed control u∈L²(Q_T), y is a solution to System (31) if and only if w:=P⁻¹y is a solution to System (33) whereC, D are given by C :=

QAQ⁻¹ andD:=QB. Moreover, we have [C|D] =Q[A|B].

If we note (˜kij)ij := [C|D], this implies ˜k11=...= ˜k1s= 0 and

rank







˜k₂₁ · · · ˜k_2s ... ...

˜k_s+1,1 · · · k˜_s+1,s





= rank







k_λ11 · · · k_λ1s ... ... kλ_s,1 · · · kλ_s,s





=s.

Proceeding as in Step 1 for w, there exists an initial condition w₀ such that for all control u in L²(Q_T) the solution w to System (33) satisfies w₁(T)6≡0 in Ω. Thus, for the initial conditiony₀:=Q⁻¹w₀, for all control uinL²(Q_T), the solutiony to System (31) satisfies

y1(T) =w1(T)6≡0 in Ω.

Step 3. Without loss of generality, we can suppose that there exists βi for all i ∈ {2, ..., p} such that k1j =

p

P

i=2

βikij for all j ∈ {1, ..., s} (otherwise a permutation of lines leads to this case). Let us define the following matrix

Q:= (e₁−

p

X

i=2

β_ie_i)|e₂|...|e_n

!t

.

Thus, for P :=Q⁻¹, again, for a fixed initial condition y0 ∈ L²(Ω)ⁿ and a control u ∈ L²(QT), consider System (33) with w := P⁻¹y, y being a solution to System (31). We remark that if we denote by (˜kij) := [C|D], we have ˜k₁₁=... = ˜k_1s = 0. Applying step 2 tow, there exists an initial conditionw₀such that for all controluinL²(Q_T) the solutionwto System (33) satisfies

w1(T)6≡0 in Ω. (51)

Thus, with the definition ofQ, for all controluinL²(Q_T) the solutiony to System (31) satisfies

w1(T) =y1(T)−

p

X

i=2

βiyi(T) in Ω.

Suppose Π_py(T)≡0 in Ω, thenw₁(T)≡0 in Ω and this contradicts (51).

As a consequence of Proposition 1, the Πp-null controllability implies the Πp- approximate controllability of System (33). If now Condition (9) is not satisfied, as for the Π_p-null controllability, we can find a solution to System (49) such that φ₁≡0 inω×(0, T) andφ6≡0 inQ_T and we conclude again with Proposition1.

3.2. m-control forces. In this subsection, we will suppose that A ∈ L(Rⁿ) and B∈ L(R^m,Rⁿ). We denote byB =: (b¹|...|b^m). To prove Theorem1.1, we will use the following lemma which can be found in [2].

Lemma 3.3. There exist r ∈ {1, ..., s} and sequences {lj}16j6r ⊂ {1, ..., m} and {sj}16j6r⊂ {1, ..., n}with Pr

j=1sj=s, such that B:=

r

[

j=1

{b^lj, Ab^lj, ..., A^sj−1b^lj}

is a basis of X. Moreover, for every 16j 6r, there exist αⁱ_k,s

j ∈Rfor16i6j and16k6sj such that

A^sjb^lj =

j

X

i=1

αⁱ_1,sjb^li+αⁱ_2,sjAb^li+...+αⁱ_si,sjA^si−1b^li

. (52)

Proof of Theorem 1.1. Consider the basisB ofX given by Lemma3.3. Note that rank Πp[A|B] = dim Πp[A|B](Rⁿ)6rank [A|B] =s.

IfM is the matrix whose columns are the elements of B, i.e.

M = (mij)ij := b^l1|Ab^l1|...|A^s1−1b^l1|...|b^lr|Ab^lr|...|A^sr−1b^lr , we can remark that

rank ΠpM = rank Πp[A|B]. (53)

Indeed, relationship (52) yields Π_pA^sjb^lj =

j

X

i=1

αⁱ_1,s

jΠ_pb^li+αⁱ_2,s

jΠ_pAb^li+...+αⁱ_s

i,s_jΠ_pA^si−1b^li . We first prove in (a) that condition (9) is sufficient, and then in (b) that this condition is necessary.

(a) Sufficiency part: Let us suppose first that (9) is satisfied. Let be y₀ ∈ L²(Ω)ⁿ. We will prove that we need only r forces to control System (31). More precisely, we will study the Π_p-null controllability of the system







∂ty= ∆y+Ay+ ˜B1ωv inQT,

y= 0 on ΣT,

y(0) =y0 in Ω,

(54) where ˜B= (b^l1|b^l2| · · · |b^lr)∈ L(R^r,Rⁿ). Using (9) and (53), we have

rank Πp(b^l1|Ab^l1|...|A^s1−1b^l1|...|b^lr|Ab^lr|...|A^sr−1b^lr) =p. (55) Case 1 : p=s. As in the case of one control force, we want to apply a change of variable P to the solutiony to System (54). Let us define for allt∈[0, T] the following matrix

P(t) := (b^l1|Ab^l1|...|A^s1−1b^l1|...|b^lr|Ab^lr|...|A^sr−1b^lr|P_s+1(t)|...|P_n(t))∈ L(Rⁿ), (56) where for alll∈ {s+ 1, ..., n},Plis solution inC¹([0, T])ⁿ to the system of ordinary differential equations

∂_tP_l(t) =AP_l(t) in [0, T],

P_l(T) =e_l. (57)

Using (56) and (57) we have P(T) =

P11 0 P21 I_n−s

, (58)

where P11 := Πs(b^l1|Ab^l1|...|A^s1−1b^l1|...|b^lr|Ab^lr|...|A^sr−1b^lr) ∈ L(R^s) and P21∈ L(R^n−s,R^s). From (55), P11 and thusP(T) are invertible. Further- more, since P is continuous on [0, T], there exists aT^∗ ∈ [0, T) such that P(t) is invertible for allt∈[T^∗, T].

We suppose first that T^∗ = 0. Since P is invertible and continuous on [0, T], for a fixed control v∈L²(Q_T)^r, y is the solution to System (54)

if and only if w:=P(t)⁻¹y is the solution to System (33) where C,D are given by

C(t) :=−P⁻¹(t)∂_tP(t) +P⁻¹(t)AP(t) and D(t) :=P⁻¹(t) ˜B, for allt∈[0, T]. Using (52) and (57), we obtain











−∂tP(t) +AP(t) = (Ab^l1|A²b^l1|...|A^s1b^l1|...|Ab^lr|A²b^lr|...|A^srb^lr|0|...|0),

= P(t)

C˜11 0

0 0

in [0, T], P(t)eS_i = b^li in [0, T],

(59) where S_i= 1 +Pi−1

j=1s_j fori∈ {1, ..., r},

C˜₁₁:=











C₁₁ C₁₂ · · · C_1r 0 C₂₂ · · · C_2r ... ... . .. ... 0 0 · · · Crr











∈ L(R^s) (60)

and for 16i6j6rthe matricesCij ∈ L(R^sj,R^si) are given by

Cii:=















0 0 0 . . . αⁱ_1,s

i

1 0 0 . . . αⁱ_2,si 0 1 0 . . . αⁱ_3,si ... ... . .. . .. ... 0 0 . . . 1 αⁱ_s

i,s_i















and

Cij:=















0 0 0 . . . αⁱ_1,s

j

0 0 0 . . . αⁱ_2,sj 0 0 0 . . . αⁱ_3,sj ... ... . .. . .. ... 0 0 . . . 0 αⁱ_si,sj















forj > i.

Then

C(t) =

C˜11 0

0 0

andD(t) = (eS₁|...|eS_r). (61) Using Theorem 2.1, there exists v ∈ L²(Q_T)^r such that the solution to System (33) satisfies w₁(T) =...=w_s(T)≡0 in Ω. Moreover, using (58), we have

Πsy(T) = (y1(T), ..., ys(T)) =P11(w1(T), ..., ws(T))≡0 in Ω.

If now T^∗ 6= 0, we conclude as in the proof of Theorem 1.1 with one force (see§3.1).

Case 2 : p < s. The proof is a direct adaptation of the proof of Theorem1.1with one force, it is possible to find a change of variable in order to get back to the situation of Case 1 (see §3.1).

(b) Necessary part: If (9) is not satisfied, there existp∈ {1, ..., p} and, for all i∈ {1, ..., p}\{p}, scalarsβ_i such that m_pj =

p

P

i=1,i6=p

β_im_ij for allj ∈ {1, ..., s}. As