Local controllability of reaction-diffusion systems around nonnegative stationary states

We consider a $n \times n$ nonlinear reaction-diffusion system posed on a smooth bounded domain $\Omega$ of $\mathbb{R}^N$. This system models reversible chemical reactions. We act on the system through $m$ controls ($1 \leq m<n$), localized in some arbitrary nonempty open subset $\omega$ of the domain $\Omega$. We prove the local exact controllability to nonnegative (constant) stationary states in any time $T>0$. A specificity of this control system is the existence of some invariant quantities in the nonlinear dynamics that prevents controllability from happening in the whole space $L^\infty(\Omega)^n$. The proof relies on several ingredients. First, an adequate affine change of variables transforms the system into a cascade system with second order coupling terms. Secondly, we establish a new null-controllability result for the linearized system thanks to a spectral inequality for finite sums of eigenfunctions of the Neumann Laplacian operator, due to David Jerison, Gilles Lebeau and Luc Robbiano and precise observability inequalities for a family of finite dimensional systems. Thirdly, the source term method, introduced by Yuning Liu, Tak\'eo Takahashi and Marius Tucsnak, is revisited in a $L^{\infty}$-context. Finally, an appropriate inverse mapping theorem enables to go back to the nonlinear reaction-diffusion system.

(1) α 1 A 1 + · · · + α n A n ⇋ β 1 A 1 + · · · + β n A n , where n ∈ N * , A 1 , . . . , A n denote n chemical species and (α 1 , . . . , α n ), (β 1 , . . . , β n ) belongs to (N) n and are such that for every 1 ≤ i ≤ n, α i = β i . For 1 ≤ i ≤ n, let u i (t, .) : Ω → R be the concentration of the chemical component A i at time t. By using the law of mass action and Fick's law, (u i ) 1≤i≤n satisfies the following reaction-diffusion system: where T ∈ (0, +∞), Ω is a bounded, connected, open subset of R N (with N ≥ 1) of class C 2 , ν is the outer unit normal vector to ∂Ω and for every 1 ≤ i ≤ n, d i ∈ (0, +∞) is the diffusion coefficient of the chemical species A i . In general, global existence of classical solutions (in the sense of [35, Definition (1.5)]) or weak solutions (in the sense of [35,Definition (5.12)]) for (2) is an open problem.
Let us also mention that global existence of renormalized solutions holds in all cases for (2) (see [19]).

1.2.
Control system and open question. We assume that one can act on the system through controls localized on a nonempty open subset ω of Ω. From a chemical viewpoint, it means that one can add or remove chemical species at a specific location of the domain Ω. More precisely, let (3) J ⊂ {1, . . . , n} and := #J < n be the number of controls. 2 We consider the control system: in Ω, where 1 i∈J := 1 if i ∈ J and 0 if i / ∈ J. Here, (u i (t, .)) 1≤i≤n : Ω → R n is the state to be controlled, (h i (t, .)) i∈J : Ω → R m is the control input supported in ω.
(4) ∀1 ≤ i ≤ n, u * i ∈ [0, +∞) and Note that the nonnegative stationary solutions of (2) do not depend on the space variable (see Proposition A.1 in Appendix A.1). Thus, it is not restrictive to assume that (u * i ) 1≤i≤n ∈ [0, +∞) n . The question we ask is the following one: For a given initial condition (u i,0 ) 1≤i≤n , does there exist (h i ) i∈J such that the solution (u i ) 1≤i≤n of (NL-U) satisfies (5) ∀i ∈ {1, . . . , n}, u i (T, .) = u * i ? Under appropriate assumptions (see Assumption 2.4 and Assumption 3.2 below), we prove the controllability of (NL-U), in an appropriate subspace of L ∞ (Ω) n , locally around (u * i ) 1≤i≤n , with controls in L ∞ ((0, T ) × Ω) m (see Theorem 4.2 below). By an adequate affine transformation, the proof relies on the study of the nullcontrollability of an equivalent cascade system with second order coupling terms (see Section 3.1 below).
1.3. Bibliographical comments. In this section, we recall some known results about the null-controllability of linear and semilinear parabolic systems with Neumann boundary conditions. We investigate the case of one control, i.e., m = 1 in this section to simplify. We introduce the notation Q T := (0, T ) × Ω.
1.3.1. Linear results. The null-controllability of the heat equation was proved independently by Gilles Lebeau, Luc Robbiano in 1995 (see [30], [24] and the survey [29]) and by Andrei Fursikov, Oleg Imanuvilov in 1996 (see [20] and [17] In the work [16], Enrique Fernández-Cara, Manuel González-Burgos, Sergio Guerrero and Jean-Pierre Puel prove the same null-controllability result for more general parabolic operators, i.e., ∂ t z − ∆z + B(t, x).∇z + a(t, x)z with a ∈ L ∞ (Q T ), B ∈ L ∞ (Q T ) n and linear Robin conditions, i.e., ∂z Then, the null-controllability of coupled linear parabolic systems has been a challenging issue. Let us now focus on a cascade system with coupling terms of zero order. The following result comes from an easy adaptation of Manuel González-Burgos and Luz de Teresa's proof in the case of Dirichlet boundary conditions to Neumann boundary conditions. Theorem 1.2. [22, Theorem 1.1] Let (d i ) 1≤i≤n ∈ (0, +∞) n , (a i,j ) 1≤i,j≤n ∈ R n×n and assume that a i+1,i = 0 for every Roughly speaking, the component z 1 is controlled by the control input h, the component z 2 is controlled by z 1 thanks to the coupling term a 2,1 z 1 , . . . , the component z n is controlled by z n−1 thanks to the coupling term a n,n−1 z n−1 .
The following result for a 2 × 2 linear parabolic system with a "cross-diffusion" term is due to Sergio Guerrero. We introduce the function space in Ω, The main difference with Theorem 1.2 is that now, the coupling term is of second order ∆z 1 . The condition Ω z 2,0 = 0 is necessary because by integrating with respect to the space variable the second equation of the system, we get d dt Ω z 2 (t, .) = 0.
In particular if z 2 (T, .) = 0, then we need Ω z 2,0 = 0. In view of Theorem 1.2 and Theorem 1.3, a natural question is: are cascade crossdiffusion systems (of arbitrary size n ≥ 2) null-controllable? A byproduct of this article is a positive answer to this question.
In this case, the Kalman condition [2,Theorem 5.3] (that can be easily extended to Neumann boundary conditions instead of Dirichlet boundary conditions) yields null-controllability for initial data in L 2 σ (Ω) n which is smaller that L 2 (Ω) × L 2 σ (Ω) n−1 . For a recent survey on the null-controllability of linear parabolic systems, see [3] and references therein.
in Ω, the usual strategy consists in deducing a local null-controllability result for (NL) from a (global) null-controllability result for the linearized system around ( in Ω.
In this article, we use the powerful source term method, introduced by Yuning Liu, Takéo Takahashi and Marius Tucsnak in [33]. This method enables to prove null- where S i (1 ≤ i ≤ n) has a prescribed decay rate at t = T , depending on the cost of null-controllability for (L). Then, for (z i,0 ) 1≤i≤n sufficiently small (in an appropriate norm), a fixed-point strategy in suitable spaces is applied to the map: where (z i ) 1≤i≤n is the solution associated to the optimal control (in an appropriate norm) of (L+S). Consequently, the local null-controllability for (NL) comes from the null-controllability of only one linear system (L).
In this article, we adapt the source term method in a L ∞ -context in the following way.
• The source term method in L 2 enables to prove a strong observability inequality (see Corollary 6.4). This estimate looks like a global Carleman estimate (see for example [17,Lemma 1.3]), whereas the method to get it is very different.
• By using the Penalized Hilbert Uniqueness Method, introduced by Viorel Barbu in [4], we construct L ∞ -controls (see Theorem 7.1). • We use another time the source term method in L ∞ (see Proposition 7.5).
• We conclude by an appropriate inverse mapping theorem (see Section 8). For other results using the source term method, see for instance [6], [18] and [34].
Another strategy to get local controllability result for (NL), called the Small L ∞perturbations method is used in [1], [4], [28], [31] and [38]. This method requires the null-controllability of a family of linear parabolic systems. Thus, this type of result is proved by using global Carleman estimates that enable to treat parabolic operators as ∂ t z − ∆z + a(t, x)z with a ∈ L ∞ (Q T ).
Nevertheless, for the linearized system around ((z i ) 1≤i≤n , h) = (0, 0) of Section 3.2 (see below), a technical difficulty appears when we want to prove an observability inequality for the adjoint system by using global Carleman estimates when n > 4 (see Remark 1.5).
Let us also mention the new method of [27] to prove the global null-controllability of reaction-diffusion systems of two species with only one control force by constructing controls of the heat equation behaving as odd regular functions.

Definition and general properties of the trajectories
In this section, we introduce the concept of trajectory of (NL-U) which requires a well-posedness result (see Definition-Proposition 2.2).
2.1. Usual notations. Let k, l ∈ N * . We denote by M k (R) (respectively M k,l (R)) the algebra of matrices with k lines and k columns (respectively the algebra of matrices with k lines and l columns with entries in R. The matrix A tr ∈ M l,k (R) denotes the transpose of the matrix A ∈ M k,l (R). For τ > 0, we introduce Q τ := (0, τ ) × Ω.
We must be careful with the dependence on the constants appearing in the estimates with respect to T (when T is small). That is why, from now and until the end of the article, we assume that (10) T ∈ (0, 1).
Unless otherwise specified, we denote by C various positive constants varying from line to line.
The following result introduces the notion of solution for linear parabolic systems and provides estimates in terms of the initial data and the source term.
This means that U is the unique function in W k T that satisfies the variational formulation Moreover, there exists C > 0 independent of U 0 and S such that Finally, if U 0 ∈ L ∞ (Ω) k and S ∈ L ∞ (Q T ) k , then U ∈ L ∞ (Q T ) k and there exists C > 0 independent of U 0 and S such that The proof of Definition-Proposition 2.1 can be found in [28,Proposition 2.3].
The following result introduces the notion of trajectory associated to the nonlinear system (NL-U) (see Section 1.2). (8) and (9)) is a trajectory of (NL-U) if (7) and Moreover, (U, H J ) is a trajectory of (NL-U) reaching U * (see (8)) in time T if The proof of the uniqueness of Definition-Proposition 2.2 comes from the fact that F is locally Lipschitz on R n (see the proof of [27, Definition-Proposition 2.4]).

Invariant quantities of the nonlinear dynamics.
In this section, we show that in the system (NL-U) (see Section 1.2), some quantities are invariant. They impose some restrictions on the initial condition, for the controllability results.
In particular, for every k = l ∈ {m + 1, . . . , n}, The proof of Proposition 2.3 is done in Appendix A.2. We prove (20) by integrating with respect to the space variable an appropriate linear combination of equations of (NL-U) and by using the Neumann boundary conditions. We prove (21) by the backward uniqueness of the heat equation applied to an appropriate linear combination of equations of (NL-U).
The equation (21) implies that we can reduce the number of components of (u i ) 1≤i≤n of (NL-U) when some diffusion coefficients d i are equal for m + 1 ≤ i ≤ n. This simplify the study, thus in order to treat the more difficult case, we make the following hypothesis. Remark 2.5. It will be interesting to note that the mass condition (22) is equivalent to (24) ∀k ≥ m + 2, 3. An adequate change of variables and linearization 3.1. Change of variables -Cross diffusion system. The goal of this section is to transform the controlled system (NL-U) (see Section 1.2) satisfied by U into another system of cascade type for which we better understand the controllability properties. Roughly speaking, for 1 ≤ i ≤ m, the component u i is easy to control thanks to the localized control term h i 1 ω . Thus, the challenge is to understand how the reaction term f i (U ) (see (6)) acts on the component We multiply the (m+1)-th equation of (NL-U) by 1/((β m+1 −α m+1 )(d m+1 −d m+2 )) and the (m + 2)-th equation of (NL-U) by 1/((β m+2 − α m+2 )(d m+2 − d m+1 )) and we sum: Roughly speaking, this linear combination enables to "kill" the reaction-term and to create a coupling term of second order.
By iterating this strategy, we construct a linear transformation V = P U such that u m+1 acts on v m+2 , v m+2 acts on v m+3 , ..., v n−1 acts on v n through cross diffusion terms. Moreover, we transform the problem of controllability for U to U * into a null-controllability problem for (25) Z where P is the invertible triangular matrix defined by: with the convention ∅ = 1.

3.2.
Linearization. We will work under the following hypothesis which will guarantee the null-controllability of this linearized system.
The linearized system of (NL-Z) satisfied by Z around (0, 0) is Up to a renumbering of the first m equations, we can assume that j = m. Then, by Assumption 3.2, we have (37) a m+1,m = 0.
Roughly speaking, we summarize the controllability properties established in the following diagram:

Main results
The goal of this section is to state the main results of the paper. First, we prove a local null-controllability result for the system (NL-Z) (see Section 3.1). Then, we deduce a local controllability result around U * for (NL-U) (see Section 1.2).
We have seen in Proposition 2.3 that a trajectory (U, H J ) reaching U * has to verify the condition (20). Thus, it prevents local-controllability from happening for arbitrary initial data. This is why we introduce a notion of local controllability adapted to (22).
Let p ∈ [1, +∞]. We introduce the following subspace of L p (Ω) n : We deduce from Theorem 4.1 the following local controllability result.
In this case, we check that Assumption 3.2 is

Linear null-controllability under constraints in L 2
The main result of this section, stated in the following theorem, is the nullcontrollability in L 2 inv for the linear system (L-Z) (see Section 3.2).
inv . More precisely, there exists C > 0 such that for every T > 0 and Z 0 ∈ L 2 inv , there exists a control The goal of the next two subsections is to prove Theorem 5.1. The proof is based on the Lebeau-Robbiano's method, introduced for the first time to prove the nullcontrollability of the heat equation (see [30]). First, it consists in establishing a null-controllability result in finite dimensional subspaces of L 2 inv with a precise estimate of the cost of the control (see Proposition 5.2). This first step is based on two main results: the spectral inequality for eigenfunctions of the Neumann-Laplace operator (see Lemma 5.4) and precise observability estimates of linear finite dimensional systems associated to the adjoint system of (L-Z) (see Lemma 5.5). Secondly, we conclude by a time-splitting procedure: the control H J is built as a sequence of active controls and passive controls. The passive mode allows to take advantage of the natural parabolic exponential decay of the L 2 norm of the solution. This decay enables to compensate the cost of the control which steers the low frequencies to 0 (see Section 5.2).

5.1.
A null-controllability result for the low frequencies. We define H 2 N e (Ω) := y ∈ H 2 (Ω) ; ∂y ∂ν = 0 . The unbounded operator on L 2 (Ω): (−∆, H 2 N e (Ω)) is selfadjoint and has compact resolvent. Thus, we introduce the orthonormal basis (e k ) k≥0 of L 2 (Ω) of eigenfunctions associated to the increasing sequence of eigenvalues (λ k ) k≥0 of the Laplacian operator, i.e., we have −∆e k = λ k e k and (e k , e l ) L 2 (Ω) = δ k,l . For The goal of this section is to prove the following null-controllability result in a finite dimensional subspace of L 2 inv . Proposition 5.2. There exist C > 0, p 1 ∈ N such that for every τ ∈ (0, T ), λ > 0, such that the solution Z of From Proposition 5.2, for every τ, λ > 0 and Z 0 ∈ E λ ∩ L 2 inv we introduce the notation: such that the solution Z of (43) satisfies Z(τ, .) = 0 and H J is the minimal-norm element of L 2 (Q τ ) m satisfying the estimate (42). In other words, H J is the projection of 0 in the nonempty closed convex set of controls satisfying (42) and driving the solution Z of (43) in time τ to 0.
By the Hilbert Uniqueness Method (see [9, Theorem 2.44]), in order to prove Proposition 5.2, we need to prove an observability inequality for the solution of the adjoint system of (43).
Proof. The proof is inspired by [32,Section 3]. Let τ > 0, λ > 0 and ϕ τ ∈ E λ ∩ L 2 inv . We have: Then, the solution ϕ of (45) is where ϕ k is the unique solution of the ordinary differential system We recall the spectral inequality for eigenfunctions of the Neumann-Laplace operator. There exists C > 0 such that for every sequence (a k ) k≥0 ⊂ C N and for every λ > 0, we have: By using (50) for a k = ϕ k,i (t) with 1 ≤ i ≤ m and by summing on 1 ≤ i ≤ m, we obtain that there exists C > 0 such that By integrating with respect to the time variable between 0 and τ the inequality (51), we obtain Now, our goal is to establish the following lemma.

5.2.
The Lebeau-Robbiano's method. The goal of this section is to prove Theorem 5.1.
Proof. The proof is inspired by [29,Section 6.2]. The constants C, C ′ will increase from line to line.
We split the interval [0, T ] = ∪ k∈N [a k , a k+1 ] with a 0 = 0, a k+1 = a k + 2T k and T k = T /2 k for k ∈ N. We also define µ k = M 2 2k for M > 0 sufficiently large which 13 will be defined later and for k ∈ N. Then, we define the control H J in the following way: Z(a k , .), a k , T k ) (see the notation (44)) and where S(t) denotes the semigroup of the parabolic system: S(t) = e t(D J ∆+A J ) . In particular, by (16) and (12), S(t) L(L 2 (Ω) n ) ≤ C.
By (42), the choice of H J during the interval time [a k , a k + T k ] implies During the passive period of the control, t ∈ [a k +T k , a k+1 ], the solution exponentially decreases: Thus, by using 2 2k T k = 2 k T , (55) and (56), we have and consequently, By taking M such that C √ M − C ′ M T < 0, for instance M ≥ 2(C/C ′ T ) 2 , we conclude by (57) that we have lim k→+∞ Z(a k , .) = 0, i.e., Z(T, .) = 0 because t → Z(t, .) ∈ C([0, T ]; L 2 (Ω) n ) because H J ∈ L 2 (Q T ) m (see Definition-Proposition 2.1 and (12)) as we will show now.
We have H J 2 L 2 (Q T ) m = +∞ k=0 H J 2 L 2 ((a k ,a k +T k )×Ω) m . Then, by using the estimate (42) of the control on each time interval (a 0 , a 0 + T 0 ) and the estimate (57), we get: This concludes the proof of Theorem 5.1. 14

The source term method in L 2
We use the source term method, introduced by Yuning Liu, Takéo Takahashi and Marius Tucsnak in [33, Proposition 2.3] to deduce a local null-controllability result for a nonlinear system from the null-controllability result for only one linear system (and an estimate of the cost of the control) (see also [6]).
By Theorem 5.1, we have an estimate for the control cost in L 2 , then we fix M > 0 such that C T ≤ M e M/T . Let q ∈ (1, √ 2) and p > q 2 /(2 − q 2 ). We define the weights .
, which will be useful for the estimate of the polynomial nonlinearity (see Section 8).
Then, we define associated spaces for the source term, the state and the control Z r := Z ∈ L r ((0, T ); L r inv ) ; inv . Proposition 6.3. For every S ∈ S 2 and Z 0 ∈ L 2 inv , there exists H J ∈ H 2 , such that the solution Z of (L+S-Z) satisfies Z ∈ Z 2 . Furthermore, there exists C > 0, not depending on S and Z 0 , such that where C T = Ce C/T . In particular, since ρ 0 is a continuous function satisfying ρ 0 (T ) = 0, the above relation (65) yields Z(T, .) = 0.
For the sake of completeness, the proof of Proposition 6.3 is in Appendix A.5 (see Proposition A.6 applied with r = 2). Now, we will deduce an observability estimate for the adjoint system: Corollary 6.4. There exists C > 0 such that for every ϕ T ∈ L 2 inv , the solution of (66) satisfies: Remark 6.5. The estimate (67) looks like a global Carleman inequality for (66). But the strategy to get this type of estimate comes from the null-controllability theorem in L 2 inv for (L-Z) with an estimate of the cost and the source term method: Theorem 5.1 and Proposition 6.3. We insist on the fact that we do not know how to prove the null-controllability in L 2 inv of (L-Z) by the usual global Carleman estimates applied to each equation of (66) when m < n − 4 (see Section 9 and in particular Open problem 9.4).
In the next section, we take advantage of the strong observability estimate (67) to get more regularity in L p -sense for the control H J . 7. Construct L ∞ -controls and the source term method in L ∞ 7.1. Construct L ∞ -controls: the Penalized Hilbert Uniqueness Method. The goal of this section is to prove a null-controllability result in L ∞ with an estimate of the cost of the control.
where C T = Ce C/T . and such that the solution Z of (L-Z) (see Section 3.2) satisfies Z(T, .) = 0.
From now and until the end of the section, we will denote by C T various positive constants which can change from line to line and such that C T ≤ Ce C/T .
In the next four parts, we perform the usual Penalized Hilbert Uniqueness Method, introduced for the first time by Viorel Barbu in [4]. The idea is the following one: it is a well-known fact that the optimal control H J ∈ L 2 ((0, T ) × Ω) m , i.e., the minimal-norm element in L 2 , which steers the solution Z of (L-Z) to 0 in time T can be expressed as a function of a solution of the adjoint system (66) (see [9, Section 1.4] for more details in the context of linear finite dimensional controlled systems). By using the strong observability inequality (67), we will use this link by considering a penalized problem in H 2 ⊂ L 2 ((0, T ) × Ω) m : the behavior at time t = T of the weight ρ 0 will be the key point to produce more regular controls in L p -sense.

The beginning of the Penalized Hilbert Uniqueness Method. Let us fix
We define P ε : H 2 → R + , by, for every H J ∈ H 2 , where Z is the solution to the Cauchy problem (L-Z) (see Section 3.2) associated to the control H J . The functional P ε is a C 1 , coercive, strictly convex functional on the Hilbert space H 2 , then P ε has a unique minimum H J,ε ∈ H 2 . Let Z ε be the solution to the Cauchy problem (L-Z) with control H J,ε and initial data Z 0 .
The Euler-Lagrange equation gives where Z is the solution to the Cauchy problem (L-Z) associated to the control H J and initial data Z 0 = 0. We introduce ϕ ε the solution to the adjoint problem (66) with final condition ϕ ε (T, .) = − 1 ε Z ε (T, .). A duality argument between Z and ϕ ε gives Then, we deduce from (70) and (71) that Consequently, we have Another duality argument applied between Z ε and ϕ ε together with (72) gives which yields By Young's inequality and the observability estimate (67) applied to ϕ ε , for δ > 0, we have: Then, by using (72), (73), (74) and by taking δ sufficiently small, we get Remark 7.2. The estimate (75) yields Proposition 6.3 for S = 0 by letting ε → 0.
We remark that we have only used the term ϕ(0, .) 2 L 2 (Ω) n in the left hand side of (67). The second term in the left hand side of (67) enables to get more regularity (in L p -sense) for the control H J (see Section 7.1.3 below).

Maximal regularity theorems and Sobolev embeddings.
In this part, we recall a maximal regularity theorem in L p (1 < p < +∞) for parabolic systems and an embedding result for Sobolev spaces.
We have the following maximal regularity theorem. in Ω.

Moreover, there exists C > 0 independent of S such that
In the next two parts, we will use the key identity between the control H J,ε and the solution of the adjoint system ϕ ε , i.e, (72) in order to deduce L p -regularity for H J,ε from L p -regularity for ϕ ε . This kind of regularity will come from the application of successive L p -parabolic regularity theorems stated in Proposition 7.3 to a modification of ϕ ε called ψ ε,r (see a precise definition in (80) below) which is bounded from below by ρ 2 0 ϕ. The beginning of this bootstrap argument is the strong observability inequality (67). Finally, we will pass to the limit ( (75) and We introduce for every r ∈ N, Then, we have from (59), for every r ∈ N, (78) ρ 2 0 ≤ C T ρ S,r . We remark that we have for every r ∈ N, We define for every r ∈ N, From (66), (77) and (80), we have for every r ∈ N * , in Ω.
Therefore, from (89), (H J,ε ) ε is bounded in L ∞ (Q T ) m , then up to a subsequence, we can assume that there exists H J ∈ L ∞ (Q T ) m such that (89), Definition-Proposition 2.1 applied to (L-Z) satisfied by Z ε , we obtain So, from (92), up to a subsequence, we can suppose that there exists Z ∈ W n T such that and from (12), Then, as we have Z ε (0, .) = Z 0 and Z ε (T, .) → 0 from (75), we deduce that in Ω.
This ends the proof of Theorem 7.1 by using (91) and (96).

7.2.
The come back to the source term method in L ∞ . The goal of this section is to apply the source term method in L ∞ thanks to the null-controllability result in L ∞ : Theorem 7.1.
To simplify the notations, we assume that the control cost in L ∞ of Theorem 7.1 satisfies: C T ≤ M e M/T where M is already defined at the beginning of Section 6.
From Proposition A.6 with r = +∞ proved in Appendix A.5, we deduce the following null-controllability result for (L+S-Z) (see Section 6) in L ∞ . Proposition 7.5. For every S ∈ S ∞ and Z 0 ∈ L ∞ inv , there exists H J ∈ H ∞ , such that the solution Z of (L+S-Z) satisfies Z ∈ Z ∞ . Furthermore, there exists C > 0, not depending on S and Z 0 , such that In particular, since ρ 0 is a continuous function satisfying ρ 0 (T ) = 0, the above relation (97) yields Z(T, .) = 0.

The inverse mapping theorem in appropriate spaces
The goal of this section is to prove Theorem 4.1. The proof is based on Proposition 7.5 and an inverse mapping theorem in suitable spaces.
Proof. Let us introduce the following space (see the definitions (62), (63) and (64)): We endow E with the following norm: for every (Z, For every Z ∈ Z ∞ , we introduce the following polynomial nonlinearity of degree more than 2: where G is defined in (31). By denoting γ := max term with respect to Z = (z 1 , . . . , z n ) of degree i. By using (61), we deduce that Q(Z) ∈ S ∞ and for every 2 ≤ i ≤ γ, We introduce the following mapping: Z(0, .)).
This concludes the proof of Theorem 4.1.
9. Comments 9.1. More general semilinearities. In this paper, we have only considered particular semilinearities of the form: But the main result of the article, i.e., Theorem 4.2 holds true with more general polynomial semilinearities satisfying where R[X 1 , . . . , X n ] denotes the space of multivariate polynomials with coefficients in R. In this case, (u * i ) 1≤i≤n is a constant nonnegative stationary state if (u * i ) 1≤i≤n ∈ [0, +∞) n and R(u * 1 , . . . , u * n ) = 0. For example, (104) rewrites as follows 9.2. Degenerate cases. In this part, we assume that Assumption 3.2 is not satisfied. Then, the usual strategy is to perform the return method, introduced by Jean-Michel Coron in [8] (see also [9,Chapter 6]). This method consists in finding a reference trajectory (U , H J ) verifying U (0, .) = U (T, .) = U * of (NL-U) (see Section 1.2) such that the linearized system of (NL-U) around (U , H J ) is null-controllable.
to depend on (t, x) and a 61 (t, x) ≥ ε > 0 on (t 1 , t 2 ) × ω 0 ⊂ (0, T ) × ω. This linear system seems to be null-controllable according to the following heuristic diagram: Unfortunately, we do not know how to prove that the linearized system around this trajectory is null-controllable for technical reasons maybe. It comes from the fact that in this case m = 5 < n−4 = 6. Intuitively, with the proof strategy performed in [28], we have to benefit from one coupling term of order 0 (in L ∞ ) and four coupling terms of order 2. This leads to the following open problem.
We introduce the following notation: is the set of functions A defined on Q T of class C ∞ and such that all the derivatives of A are bounded. We choose to state the following open problem for Dirichlet conditions instead of Neumann conditions to avoid the constraints on the initial data.
Remark 9.5. Open problem 9.4 is closely related to the generalization of [15, Theorem 1.1] to linear parabolic systems with diffusion matrices that contain Jordan blocks of dimension more than 5. Indeed, the diffusion matrix of (108) is D J defined in (32). The submatrix D ♯ (see again (32)) looks like a Jordan block of dimension more than 5 if m < n − 4. Consequently, the strategy of Carleman inequalities applied to each equation of the adjoint system of (108) yields some global terms in the right hand side of the inequality that cannot be absorbed by the left hand side (see [15,Section 2]).
Appendix A.
A.1. Stationary states. We only have considered nonnegative stationary constant solutions of (2). It is not restrictive because of the following proposition. (6). Then, for every 1 ≤ i ≤ n, u i is constant.
The proof relies on an entropy inequality: − n i=1 log(u i )f i (U ) ≤ 0. Proof. Let ε > 0 be a small parameter. For every 1 ≤ i ≤ n, we introduce We have Then, from (109) and (110), we have We sum the n equations of (111), we integrate on Ω and we use the increasing of the function log: Moreover, Consequently, from (112), (113), we get that Consequently, for every 1 ≤ i ≤ n, u i is constant.
Our proof of Theorem 4.2 does not treat the case of stationary states which can change of sign, contrary to the proof of [28, Theorem 3.2] (see [28,Section 6.2]). As in the previous part (see Example 9.3), the proof of [28,Theorem 3.2] can be adapted to local controllability around stationary states of (2) which can change of sign if m ≥ n − 4 (for technical reasons maybe, see Open problem 9.4).
A.2. Proof of the existence of invariant quantities in the system. The goal of this section is to prove Proposition 2.3.
Proof. We introduce the notation R := n k=1 u α k k − n k=1 u β k k and we take m + 1 ≤ i ≤ n. By using the fact that u i ∈ W T and from [14,Lemma 3], we obtain that the mapping t → Ω u i (t, x)dx is absolutely continuous and for a.e. 0 ≤ t ≤ T , Then, by using that ((u i ) 1≤i≤n , (h i ) 1≤i≤m ) is a trajectory of (NL-U) and by taking w = 1 in (18), we find that for a.e. 0 ≤ t ≤ T , Then, by using (114) and (115), we get for a.e. 0 ≤ t ≤ T , 24 Now, let m + 1 ≤ k = l ≤ n. By (116) for i = k and (116) for i = l , we deduce that for a.e. 0 ≤ t ≤ T , Therefore, from (117), we have for every t ∈ [0, T ], If we assume that d := d k = d l , then the equation satisfied by v : The backward uniqueness of the heat equation (see for instance [5, Théorème II.1]) applied to (118) leads to

This yields (21).
A.3. Proofs concerning the change of variables.
A.3.1. Proof of the equivalence of the two systems. In this section, we prove Proposition 3.1. It is based on the following algebraic lemma.
Lemma A.2. Let s be an integer such that s ≥ 2. Let (a 1 , . . . , a s ) ∈ C s be such that a i = a j for i = j. Then, we have Proof. Let C(X) be the field of fractional functions with coefficients in C and F ∈ C(X) be defined by The partial fractional decomposition of F is the following one: For 1 ≤ i ≤ s − 1, we compute each b i by multiplying (120) by (X − a i ) and by evaluating X = a i : We deduce that F = 0. By remarking that we conclude the proof of (119) The following result is an easy consequence of Lemma A.2.

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Corollary A.3. For every m + 2 ≤ k ≤ n, we have Proof. By (27), we have by taking s = k − m and a i = d i+m for This ends the proof of Corollary A.3. Now, we turn to the proof of Proposition 3.1.
Proof. We introduce the following notation: u β k k . We assume that (U, H J ) is a trajectory of (NL-U). The equations 1 ≤ i ≤ m + 1 of (28) are clearly satisfied. Let m + 2 ≤ i ≤ n. We have: This ends the proof of "⇒".
A.4. Proof of an observability estimate for linear finite dimensional systems. The goal of this section is to give a self-contained proof of Lemma 5.5. By the Hilbert Uniqueness Method (see [9, Theorem 2.44]), it suffices to show the following null-controllability result for finite dimensional systems.
Remark A.5. We do not treat the case λ 0 = 0 with initial data y 0 ∈ R m+1 × {0} n−m−1 because it is a simple adaptation of the following proof.
Step 1: Construction of the control h by a Brunovsky approach. We start by defining y to be the free solution of the system (126) (take h = 0). We have y(t) = e tA y 0 = e t(−λD J +A J ) y 0 . We easily have that for any l ≥ 0, ≤ C(1 + λ l−1/2 ) y 0 R n .
Step 2: Properties of the solution y and estimate of the control h. First, we remark that,  ≤ C 1 + 1 τ 1/2 + λ 1/2 y 0 R n .
A.5. Source term method in L r for r ∈ {2, +∞}. We use the same notations as in the beginning of Section 6. The goal of this section is to prove Proposition 6.3 and Proposition 7.5. We have the following result.
Therefore, for an appropriate choice of constant C > 0, Z and H J satisfy (136). This concludes the proof of Proposition A.6.
A.6. Proof of a strong observability inequality. We take the same notations as in the beginning of Section 6. The goal of this section is to prove Corollary 6.4.
From [9, Lemma 2.48], we have that Range(F 1 ) ⊂ Range(F 2 ) is equivalent to the observability inequality Consequently, by using the null-controllability result for (L+S-Z): Proposition 6.3, we have that (152) holds true. Moreover, the constant C T in (152) can be chosen such that C T ≤ Ce C/T by using the cost estimate (65) (see the proof of [9, Theorem 2.44] for more details between the constant of cost estimate and the constant of observability inequality).