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à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Sylvain Ervedoza
email: sylvain.ervedoza@math.u-bordeaux.fr

Jérôme Lemoine
email: jerome.lemoine@uca.fr.

Arnaud Münch
email: arnaud.munch@uca.fr.

Exact controllability of semilinear heat equations through a constructive approach

Keywords: AMS Classifications: 35K58, 93B05 Semilinear heat equation, Null controllability, Carleman estimates, Fixed point

The exact distributed controllability of the semilinear heat equation ∂ty -∆y + f (y) = v 1ω posed over multi-dimensional and bounded domains, assuming that f is locally Lipschitz continuous and satisfies the growth condition lim sup |r|→∞ |f (r)|/(|r| ln 3/2 |r|) β for some β small enough has been obtained by Fernández-Cara and Zuazua in 2000. The proof based on a non constructive fixed point arguments makes use of precise estimates of the observability constant for a linearized heat equation. Under the same assumption, by introducing a different fixed point application, we present a simpler proof of the exact controllability, which is not based on the cost of observability of the heat equation with respect to potentials. Then, assuming that f is locally Lipschitz continuous and satisfies the growth condition lim sup |r|→∞ |f (r)|/ ln 3/2 |r| β for some β small enough, we show that the above fixed point application is contracting yielding a constructive method to compute the controls for the semilinear equation. Numerical experiments illustrate the results.

Introduction and main results

Let Ω be a bounded connected open set of R d (d ∈ N ) with C 2 boundary, ω ⊂⊂ Ω be any non-empty open set and let T > 0. We set Q T = Ω × (0, T ), q T = ω × (0, T ) and Σ T = ∂Ω × (0, T ). We are concerned with the null controllability problem for the following semilinear heat equation

∂ t y -∆y + f (y) = v1 ω in Q T , y = 0 on Σ T , y(•, 0) = u 0 in Ω, (1)
where u 0 ∈ L 2 (Ω) is the initial state of y, v ∈ L 2 (q T ) is a control function and f : R → R is a nonlinear function. Recall that if f is locally Lipschitz-continuous and satisfies the condition |f (r)| C(1 + |r| 4+d ) for all r ∈ R, then (1) possesses exactly one local in time solution Moreover, in accordance with the results in [8, Section 5], under the growth condition |f (r)| C(1 + |r| ln(1 + |r|)) for all r ∈ R and some C > 0, the solutions to [START_REF] Badra | Local controllability to trajectories for nonhomogeneous incompressible Navier-Stokes equations[END_REF] are globally defined in [0, T ] and one has y ∈ C 0 ([0, T ]; L 2 (Ω)) ∩ L 2 (0, T ; H 1 0 (Ω)).

(

) 2
Without the above growth condition, the solutions to (1) can blow up before t = T ; in general, the blow-up time depends on f and the size of u 0 L 2 (Ω) . We refer to [START_REF] Lacey | Global blow-up of a nonlinear heat equation[END_REF] and to [20, Section 2 and Section 5] for a survey on this issue.

The system (1) is said to be exactly controllable to trajectories at time T if, for any globally defined trajectory y ∈ C 0 ([0, T ]; L 2 (Ω)) (corresponding to data u 0 ∈ L 2 (Ω) and v ∈ L 2 (q T )), for any u 0 ∈ L 2 (Ω), there exist controls v such that v1 ω ∈ L 2 (q T ) and associated states y satisfying (2) and y(x, T ) = y (x, T ), x ∈ Ω.

(

) 3
When such property is true for a specific trajectory y and for any choice of initial datum u 0 , we say that system (1) is globally exactly controllable to y at time T . The uniform controllability strongly depends on the nonlinearity f . Assuming a growth condition on f at infinity, Fernández-Cara and Zuazua in [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF] showed the following result.

Theorem 1. [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF] Let T > 0 be given. Assume that (1) admits at least one solution y , globally defined in [0, T ] and bounded in Q T associated with v ∈ L ∞ (q T ). Assume that f : R → R is locally Lipschitz continuous and satisfies

(H 0 ) |f (r)| C(1 + |r| 4+d ) a.

e. in R.

There exists a β = β(y ) > 0 such that if

(H 1 ) lim sup |r|→∞ |f (r)| |r| ln 3/2
+ |r| β then system (1) is globally exactly controllable to y at time T with controls in L ∞ (q T ).

Here and in the sequel, we note

ln + |r| = 0 if |r| 1 ln |r| else.
Therefore, if |f (r)| does not grow at infinity faster than |r| ln p + (|r|) for any p < 3/2, then (1) is exactly controllable to trajectories. We also mention [START_REF] Barbu | Exact controllability of the superlinear heat equation[END_REF] which gives a similar result assuming the additional sign condition f (r)r -C(1 + r 2 ) for all r ∈ R and some C > 0. On the contrary, if f is too "superlinear" at infinity, precisely, if p > 2, then for some initial data, the control cannot compensate the blow-up phenomenon occurring in Ω\ω (see [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF]Theorem 1.1]). The problem remains open when f behaves at infinity like |r| ln p |r| with 3/2 p 2. We mention however the recent work of Le Balc'h [START_REF] Le Balc'h | Global null-controllability and nonnegative-controllability of slightly superlinear heat equations[END_REF] where uniform controllability results in large time are obtained for p 2 assuming additional sign conditions on f , notably that f (r) > 0 for r > 0 or f (r) < 0 for r < 0, a condition not satisfied for f (r) = -r ln p + |r|. Eventually, we also mention [START_REF] Coron | Global steady-state controllability of one-dimensional semilinear heat equations[END_REF] where a positive boundary controllability result is proved for a specific class of initial and final data and T large enough.

Theorem 1 is deduced in [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF] from a null controllability result corresponding to the null trajectory, i.e. y ≡ 0 corresponding to v ≡ 0, u 0 ≡ 0 and assuming f (0) = 0. The proof of the null controllability given in [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF] is based on a fixed point method, initially introduced in [START_REF] Zuazua | Exact controllability for semilinear wave equations in one space dimension[END_REF] for a one dimensional semilinear wave equation. Precisely, it is shown that the operator Λ : L ∞ (Q T ) → L ∞ (Q T ), where y := Λ(z) is a null controlled solution corresponding to the control of minimal L ∞ norm of the linear boundary value problem

∂ t y -∆y + y f (z) = v1 ω in Q T y = 0 on Σ T , y(•, 0) = u 0 in Ω , f (r) := f (r)/r r = 0 f (0) r = 0 (4)
maps a closed ball B(0, M ) ⊂ L ∞ (Q T ) into itself, for some M > 0. The Kakutani's theorem then provides the existence of a fixed point for the operator Λ, which is also a controlled solution for [START_REF] Badra | Local controllability to trajectories for nonhomogeneous incompressible Navier-Stokes equations[END_REF].

This allows to obtain controlled solutions in L ∞ (Q T ) but requires refined L 1 Carleman type estimates (see [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF]Proposition 3.2]).

The general goal considered in this work is the approximation of the controllability problem associated with [START_REF] Badra | Local controllability to trajectories for nonhomogeneous incompressible Navier-Stokes equations[END_REF], that is to construct an explicit sequence (v k ) k∈N converging strongly toward a control for [START_REF] Badra | Local controllability to trajectories for nonhomogeneous incompressible Navier-Stokes equations[END_REF]. The controllability of nonlinear partial differential equations has attracted a large number of works in the last decades (see the monography [START_REF] Coron | Control and nonlinearity[END_REF] and references therein). However, as far as we know, few are concerned with the approximation of exact controls for nonlinear partial differential equations, and the construction of convergent control approximations for nonlinear equations remains a challenge. A natural strategy is to take advantage of the method used in [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF][START_REF] Le Balc'h | Global null-controllability and nonnegative-controllability of slightly superlinear heat equations[END_REF] and consider, for any element y 0 ∈ L ∞ (Q T ), the Picard iterations defined by y k+1 = Λ(y k ), k 0 associated with the operator Λ. Numerical experiments reported in [START_REF] Fernández | Numerical null controllability of semi-linear 1-D heat equations: fixed point, least squares and Newton methods[END_REF] exhibit the non convergence of the sequences (y k ) k∈N and (v k ) k∈N for some initial conditions large enough. This phenomenon is related to the fact that the operator Λ is in general not contracting, including the cases for which f is globally Lipschitz. We also refer to [START_REF] Boyer | On the penalised HUM approach and its applications to the numerical approximation of null-controls for parabolic problems[END_REF][START_REF] Boyer | Carleman estimates for semi-discrete parabolic operators and application to the controllability of semi-linear semi-discrete parabolic equations[END_REF] where this strategy is implemented.

In the one-dimensional case, a least-squares type approach, based on the minimization over

Z := L 2 ((T -t) -1 , Q T ) of the functional R : Z → R + defined by R(z) := z -Λ(z) 2
Z has been introduced and analyzed in [START_REF] Fernández | Numerical null controllability of semi-linear 1-D heat equations: fixed point, least squares and Newton methods[END_REF].

Assuming u 0 ∈ L ∞ (Ω), f ∈ C 1 (R) and ( f ) ∈ L ∞ (R), it is proved that R ∈ C 1 (Z; R + ) and that, for some constant C > 0 (1 -C ( f ) L ∞ (R) u 0 ∞ ) 2R(z) R (z) L 2 (Q T ) ∀z ∈ L 2 (Q T ) implying that if ( f ) ∞ u 0 ∞ is
small enough, then any critical point for R is a fixed point for Λ (see [START_REF] Fernández | Numerical null controllability of semi-linear 1-D heat equations: fixed point, least squares and Newton methods[END_REF]Proposition 3.2]). Under such smallness assumption on the data, numerical experiments reported in [START_REF] Fernández | Numerical null controllability of semi-linear 1-D heat equations: fixed point, least squares and Newton methods[END_REF] display the convergence of gradient based minimizing sequences for R and a better behavior than the ones associated with the Picard iterates for Λ. The analysis of convergence is however not performed.

More recently, a constructive method has been developed for the one-dimensional case in [START_REF] Lemoine | Constructive exact control of semilinear 1d heat equation[END_REF] by introducing the following (non convex) least-squares problem inf

(y,v)∈A E s (y, v), E s (y, v) := ∂ t y -∆y + f (y) -v 1 ω 2 L 2 (ρ0(s),Q T )
where A is a convex space which incorporates the initial and controllability requirement and where ρ 0 denotes a Carleman type weight parametrized by s and blowing up as t → T -. Assuming slightly stronger assumption on f than in Theorem 1, a strong convergent approximation of a controlled pair is obtained: Theorem 2. [START_REF] Lemoine | Constructive exact control of semilinear 1d heat equation[END_REF] Let T > 0 be given. Let d = 1. Assume that (1) admits at least one solution y , globally defined in [0, T ] and bounded in

Q T associated with v ∈ L 2 (q T ) and u 0 ∈ L 2 (Ω). Assume that f ∈ C 1 (R) satisfies the growth condition (H 1 ) ∃α > 0, s.t. |f (r)| (α + β ln + |r|) 3/2 , ∀r ∈ R
for some β = β (y ) > 0 small enough and

(H p ) ∃p ∈ [0, 1] such that sup a,b∈R a =b |f (a) -f (b)| |a -b| p < +∞.
Then, for any u 0 ∈ L 2 (Ω), one can construct a sequence (y k , v k ) k∈N converging strongly to a controlled pair for (1) satisfying (3). Moreover, after a finite number of iterations, the convergence is of order at least 1 + p.

The hypothesis on f are stronger here than in Theorem 1: it should be noted however that the function f (r) = a + br + βr ln(1 + |r|) 3/2 , a, b ∈ R which is somehow the limit case in (H 1 ) satisfies (H 1 ) and (H 1 ). On the other hand, Theorem 2 devoted to the one dimensional case is constructive, contrary to Theorem 1. A similar construction is performed in a multi-dimensional case with d 3 in [START_REF] Lemoine | Approximation of null controls for semilinear heat equations using a least-squares approach[END_REF] assuming that f is globally Lipschitz. The extension of Theorem 2 to the case d 3 can be obtained as well. The minimizing sequence for E s constructed in [START_REF] Lemoine | Approximation of null controls for semilinear heat equations using a least-squares approach[END_REF][START_REF] Lemoine | Constructive exact control of semilinear 1d heat equation[END_REF] are related to the operator Λ N : A → A defined by y = Λ N (z) controlled solution of

∂ t y -∆y + f (z)y = v1 ω + f (z)z -f (z) in Q T y = 0 on Σ T , y(•, 0) = u 0 in Ω , (5)
through the control v of minimal L 2 (ρ 0 (s), q T ) norm. The analysis in [START_REF] Lemoine | Constructive exact control of semilinear 1d heat equation[END_REF] makes use of global L 2 Carleman estimates as initially introduced in this context in [START_REF] Fursikov | Controllability of evolution equations[END_REF].

In this paper, we prove the following result.

Theorem 3. Let T > 0 be given. Let d 5. Assume that (1) admits at least one solution y , globally defined in [0, T ] and bounded in Q T associated with a control function v ∈ L 2 (q T ) and u 0 ∈ L ∞ (Ω).

Assume that f : R → R is locally Lipschitz continuous.

• Assume that there exists β > 0 small enough such that f satisfies (H 1 ). Then system (1) is globally exactly controllable from any initial datum in L ∞ (Ω) to y at time T with controls in {v * } + L p d (q T ), where p d is defined in [START_REF] Fursikov | Controllability of evolution equations[END_REF].

• If f ∈ C 1 (R) satisfies (H 1 ) for β * > 0 small enough, then for any u 0 ∈ L ∞ (Ω), one may construct a sequence (y k , v k ) k∈N converging strongly in L 2 (Q T ) × L 2 (q T ) to a controlled pair (y ∞ , v ∞ ) for (1). Besides, the convergence of (y k -y ∞ , v k -v ∞ ) k∈N holds at least with a linear rate for the norm L 2 (ρ 0 (s), Q T ) × L 2 (ρ 0 (s), q T )
, where ρ 0 is defined in [START_REF] Fernández | Strong convergent approximations of null controls for the 1D heat equation[END_REF] and s is chosen suitably large depending on u 0 L ∞ (Ω) .

The first part of Theorem 3 differs from Theorem 1 on the functional spaces as it is obtained from a null controllability result based on a different fixed point application leading to a simpler proof. In particular, it is not based on the analysis of the cost of observability of the heat equation with potential. We believe that this different approach could possibly help analyzing non-linearities f which behave like |r| log + (|r|) p for p ∈ [3/2, 2].

On the other hand, the second part relaxed the Hölder type assumption (H p ) on f but still leads in multi-dimensional cases to a constructive method. As we shall see, this is related to an appropriate choice of the parameter s related to the norm of the initial condition.

The null controllability result we shall prove and leading to Theorem 3 (by simply considering y -y * instead of y and v -v * instead of v, see [18, p.603]; the proof of Theorem 3 is then left to the reader) reads as follows.

Theorem 4. Let T > 0 be given. Let d 5 and s > 0 large enough.

• There exists β > 0 such that if f ∈ C 0 (R) satisfies (H 2 ) ∃α > 0, s.t. |f (r)| |r| α + β ln + |r| 3/2 , ∀r ∈ R then system (1) is globally null-controllable at time T for initial data in L ∞ (Ω) with controls in L 2 (q T ) ∩ L p d (q T )
, where p d is defined in [START_REF] Fursikov | Controllability of evolution equations[END_REF] and corresponding controlled solution in L ∞ (Q T ).

• If f ∈ C 1 (R) satisfies (H 1 ) for β > 0 small enough and f (0) = 0 then for any u 0 ∈ L ∞ (Ω), one can construct a sequence (y k , v k ) k∈N converging strongly in L 2 (Q T ) × L 2 (Q T )
to a controlled pair for [START_REF] Badra | Local controllability to trajectories for nonhomogeneous incompressible Navier-Stokes equations[END_REF]. Besides, the convergence of (y k , v k ) k∈N holds at least with a linear rate for the norm L 2 (ρ 0 (s), Q T ) × L 2 (ρ 0 (s), q T ), where ρ 0 is defined in [START_REF] Fernández | Strong convergent approximations of null controls for the 1D heat equation[END_REF] and s is chosen suitably large depending on u 0 L ∞ (Ω) .

Remark that the first part of Theorem 4 relaxes the regularity assumption to f ∈ C 0 (R) instead of f locally Lipschitz continuous. Therefore, with such non-linearities, we do not know the local existence in time for the uncontrolled system. It should be noted that (H 2 ) implies, in addition to f (0) = 0 that |f (r)|/|r| is bounded in any neighborhood of 0.

Theorem 4 is obtained by introducing the following linearized controllability problem: for y in a suitable class C(s) depending on a free parameter s 1, find the control v such that the solution y of

∂ t y -∆y = v1 ω -f ( y) in Q T y = 0 on Σ T , y(•, 0) = u 0 in Ω , (6)
satisfies y(•, T ) = 0 in Ω, and (y, v) corresponds to the minimizer of a functional J s depending on s and involving Carleman weight functions (see Remark 1), which can also be computed as the solution of an affine problem involving Carleman weight functions (see [START_REF] Victor | The problem of blow-up in nonlinear parabolic equations[END_REF]). This will define an operator Λ s : y → y from some suitable class C(s) into itself, on which we can use fixed point theorems for s sufficiently large depending on u 0 L ∞ (Ω) , namely Schauder fixed point theorem for the first item of Theorem 4, and Banach-Picard fixed point theorem for the second item, allowing to exhibit a simple sequence of convergent approximations of the control and controlled trajectory.

The main trick here is thus to keep the parameter s free in the whole construction of the fixed point operator and to get suitable estimates on the control and controlled trajectories.

In order to do that, we will use the Carleman estimate introduced in [START_REF] Badra | Local controllability to trajectories for nonhomogeneous incompressible Navier-Stokes equations[END_REF], which presents the advantage of not degenerating as t → 0, allowing to handle initial conditions in a somewhat more natural way than with the classical parabolic Carleman estimate of [START_REF] Fursikov | Controllability of evolution equations[END_REF].

Let us also point out that this idea of introducing Carleman estimates within the control process and choose the Carleman parameter large to limit the influence of lower order terms is also very natural in the context of inverse problems, and in some extent, it should be compared to the techniques developed in [START_REF] Baudouin | Global Carleman estimates for waves and applications[END_REF] or in [START_REF] Michael | Globally strictly convex cost functional for a 1-D inverse medium scattering problem with experimental data[END_REF] for instance, where Carleman estimates have been used to design numerical methods to recover the unknown coefficient.

The paper is organized as follows. In Section 2, we derive a controllability result for the linear heat equation with precise estimates. Then, in Section 3, we prove, for any time T > 0 and control domain ω the uniform null controllability of (1) assuming that f is continuous and satisfies the condition (H 2 ). Then in Section 4, assuming the hypothesis (H 1 ), we show that the operator Λ s is contracting, yielding the convergence of the Picard iterates y k+1 = Λ s (y k ). Section 5 illustrates the result with some numerical experiments in the one dimensional case while Section 6 concludes with some perspectives.

Notations. In this article, C denotes generic constants depending on Ω, ω, and T , which may change from line to line, but is independent of the Carleman parameter s.

Controllability result for the linear heat equation

This section is devoted to a controllability result for a linear heat equation with a right hand side B = B(x, t) in some class defined in the sequel. More precisely we are interested by the existence of a control function v such that the solution z of

∂ t z -∆z = v1 ω + B in Q T , z = 0 on Σ T , z(•, 0) = u 0 in Ω, ( 7
) satisfies z(•, T ) = 0 in Ω. ( 8
)
This control problem has already been analyzed many times in the literature since the pioneering work [START_REF] Fursikov | Controllability of evolution equations[END_REF], and it is known that null-controllability can be achieved only if the source term B vanishes exponentially fast close to T .

In this section, we follow the usual strategy of [START_REF] Fursikov | Controllability of evolution equations[END_REF] to construct a solution to the null-controllability problem ( 7)-( 8), using Carleman estimates as a fundamental tool, and getting suitable estimates on this linear control problem.

In order to do that, instead of using the classical Carleman estimates of [START_REF] Fursikov | Controllability of evolution equations[END_REF], we use the one in [START_REF] Badra | Local controllability to trajectories for nonhomogeneous incompressible Navier-Stokes equations[END_REF], for which it is easier to deal with initial data as the weight function does not blow up as t → 0.

Carleman estimates

Before introducing the Carleman estimate, we define the several weight functions which will be involved in it.

We start by choosing a function ψ ∈ C 2 (Ω; [0, 1]) satisfying ψ = 0 on ∂Ω and inf Ω\ω |∇ ψ| > 0. We then set ψ = ψ + 6.

We then introduce two free parameters s 1 and λ 1.

Setting µ = sλ 2 e 2λ , we then choose a function θ ∈ C 2 ([0, T )), depending on the free parameters s and λ through the choice of µ, such that

θ(t) =                  1 + 1 - 4t T µ ∀t ∈ [0, T /4], 1 ∀t ∈ [T /4, T -T /2], θ is increasing on [T -T /2, T -3T /4], 1 + 1 T -t ∀t ∈ [T -3T /4, T ). (9)
(In [1, Theorem 2.5], the function θ was chosen equal to 1/(T -t) on [T -3T /4, T ], which is possible only if T 4, but one can easily adapt the proof of [1, Theorem 2.5] to handle a function θ as in [START_REF] Ciarlet | The finite element method for elliptic problems[END_REF], which also allows to consider any time horizon T .)

We then set ϕ, ξ and ρ as follows

ϕ(x, t) = θ(t) λ exp(12λ) -exp(λ ψ(x)) , (x, t) ∈ Ω × [0, T ), (10)
ξ(x, t) = θ(t) exp(λ ψ(x)), (x, t) ∈ Ω × [0, T ), (11)
ρ(x, t) = exp sϕ(x, t) ,

(x, t) ∈ Ω × [0, T ). ( 12
)
Finally, we introduce the weight functions ρ0 , ρ1 , and ρ2 as follows:

ρ0 = ξ -3/2 ρ, ρ1 = ξ -1 ρ, ρ2 = ξ -1/2 ρ. (13)
We emphasize that these weights ρ, ρ0 , ρ1 , and ρ2 all blow up at t → T -and not at t = 0. Let us also point out that, although it does not explicitly appear in the above notations (as it is usually done), all the above weight functions depend on the parameters s 1 and λ 1. In fact, to make the dependence in the parameter s explicit, we will write the weights ρ, ρ0 , ρ1 , and ρ2 as functions of (x, t) ∈ Ω × [0, T ) and s 1.

The controllability property for the linear system ( 7) is based on the following Carleman estimate.

Lemma 1. [1, Theorem 2.5] There exists λ 0 1 and s 0 1 such for all λ λ 0 and for all s s 0 , for all p ∈ L 2 (0, T ;

H 2 ∩ H 1 0 (Ω)) ∩ H 1 (0, T ; L 2 (Ω)), Ω ρ -2 (•, 0, s)|∇p(0)| 2 + s 2 λ 3 e 14λ Ω ρ -2 (•, 0, s)|p(0)| 2 + sλ 2 Q T ρ-2 2 (s)|∇p| 2 + s 3 λ 4 Q T ρ-2 0 (s)|p| 2 C Q T ρ -2 (s)| -∂ t p -∆p| 2 + s 3 λ 4 q T ρ-2 0 (s)|p| 2 . ( 14
)
In the following, we fixed λ = λ 0 and we only keep the parameter s free. This allows to replace in [START_REF] Fernández | Numerical null controllability of semi-linear 1-D heat equations: fixed point, least squares and Newton methods[END_REF] all the weights ρ0 , ρ1 , and ρ2 in (13) by the following ones:

ρ 0 = θ -3/2 ρ, ρ 1 = θ -1 ρ, ρ 2 = θ -1/2 ρ. (15)
Modifying the constant if needed, it is easy to check that Lemma 1 implies that there exists a constant C > 0 such that for all s s 0 , for all p ∈ L 2 (0, T ;

H 2 ∩ H 1 0 (Ω)) ∩ H 1 (0, T ; L 2 (Ω)), Ω ρ -2 (•, 0, s)|∇p(0)| 2 + s 2 Ω ρ -2 (•, 0, s)|p(0)| 2 + s Q T ρ -2 2 (s)|∇p| 2 + s 3 Q T ρ -2 0 (s)|p| 2 C Q T ρ -2 (s)| -∂ t p -∆p| 2 + s 3 q T ρ -2 0 (s)|p| 2 . ( 16
)
For further purposes, we will also need some additional L 2 (0, T ; H 2 (Ω))∩H 1 (0, T ; L 2 (Ω)) estimates on p from the right hand side. This is the goal of the main Lemma.

In order to state it precisely, we define two additional weight functions ϕ * = ϕ * (t, s) and ρ 3 = ρ 3 (t, s) depending only on the time variable and on the free parameter s s 0 : We then get the following result.

∀t ∈ [0, T ), ϕ * (t) = sup x∈Ω ϕ(x, t), ∀t ∈ [0, T ), ρ 3 (t, s) = e
Lemma 2. There exists a constant C > 0 such that for all s s 0 , for all p ∈ L 2 (0, T ;

H 2 ∩ H 1 0 (Ω)) ∩ H 1 (0, T ; L 2 (Ω)), ∂ t (ρ -1 3 (s)p) 2 L 2 (Q T ) + ρ -1 3 (s)p 2 L 2 (0,T ;H 2 (Ω)) C Q T ρ -2 (s)|-∂ t p-∆p| 2 +s 3 q T ρ -2 0 (s)|p| 2 . ( 17
)
Proof. Denoting p 1 = ρ -1 3 (s)p we have

∂ t p 1 + ∆p 1 = ρ -1 3 (s)(∂ t p + ∆p) + 3 2 s∂ t ϕ * (t)ρ -1 3 (s)p,
and from maximal regularity results for the heat equation

∂ t p 1 2 L 2 (Q T ) + p 1 2 L 2 (0,T ;H 2 (Ω)) C ρ -1 3 (s)(∂ t p + ∆p) 2 L 2 (Q T ) + s∂ t ϕ * (t)ρ -1 3 (s)p 2 L 2 (Q T ) + ∇p 1 (0) 2 L 2 (Ω) ). (18)
Let us estimate each term of the right hand side of this inequality. Since ρ -1 3 (s) ρ -1 (s), the first term of the right hand-side ( 18) is easily bounded as follows:

ρ -1 3 (s)(∂ t p + ∆p) 2 L 2 (Q T ) ρ -1 (s)(∂ t p + ∆p) 2 L 2 (Q T ) .
For the second term of the right hand-side [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF], from the definition of ϕ * we have ∂ t ϕ * = ∂tθ θ ϕ * . Besides, from the definition of θ we have, for some constant C independent of s:

∂ t θ θ (t)      Cs ∀t ∈ [0, T /4], 0 ∀t ∈ [T /4, T -T /2], Cθ ∀t ∈ [T -T /2, T ).
Since θ 1, we deduce that |∂ t θ/θ| Csθ for some C independent of s, and thus we get that

|sϕ * | Csθϕ * Csθ 2 .
Accordingly, for s s 0 ,

|s∂ t ϕ * (t)ρ -1 3 (s)| Csθ 2 ρ -1 3 (s) Csθ 2 exp(-sϕ * /2)ρ -1 (s) Cs 3/2 θ 3/2 ρ -1 (s) = Cs 3/2 ρ -1 0 (s),
where we used that sθ 2 exp(-sϕ * /2) Cs 3/2 θ 3/2 for s s 0 : this is obvious since ϕ * cθ * for some strictly positive constant c > 0, s 1/2 θ 1/2 exp(-csθ) τ 1/2 exp(-cτ ) L ∞ (R+) and s 1. These estimates entail that

s∂ t ϕ * (t)ρ -1 3 (s)p 2 L 2 (Q T ) Cs 3 ρ -1 0 (s)p 2 L 2 (Q T ) .
Since ρ -1 3 (0, s) ρ -1 (x, 0, s) for all x ∈ Ω, the third term of the right hand-side ( 18) is easily bounded as follows:

∇p 1 (0) 2 L 2 (Ω) = ρ -1 3 (0, s)∇p(0) 2 L 2 (Ω) ρ -1 (0, s)∇p(0) 2 L 2 (Ω) .
The inequality [START_REF] Fernández-Cara | On the numerical controllability of the two-dimensional heat, Stokes and Navier-Stokes equations[END_REF] then follows from the definition of p 1 , the estimate ( 18) and the Carleman inequality [START_REF] Fernández | Numerical exact controllability of the 1D heat equation: duality and Carleman weights[END_REF].

Application to controllability

In the next section, we see how Lemma 1 and 2 imply control results for the controllability problem ( 7)-( 8).

This part is merely classical, and the results presented below differs only slightly from the ones in [1, Theorem 2.6] and [START_REF] Ervedoza | Local exact controllability for the two-and threedimensional compressible Navier-Stokes equations[END_REF]Theorem 3.3]. Still, we present it below with some details, as it is an essential part of our argument and requires to be adequately adapted from [1, Theorem 2.6] and [START_REF] Ervedoza | Local exact controllability for the two-and threedimensional compressible Navier-Stokes equations[END_REF]Theorem 3.3] In particular, we will give a construction which, in dimension d, provides a control v ∈ L p d (q T ), where p d is given by

p d =          + ∞ if d = 1, any number ∈ [2, ∞) if d = 2, 2(d + 2) d -2 if d 3. ( 19
)
Before going further, for s s 0 , let us introduce the bilinear form (p, q) P,s :=

Q T ρ -2 (s)L p L q + s 3 q T ρ -2 0 (s)p q
where L q := -∂ t q -∆q defined for p, q ∈ L 2 (0, T ;

H 2 ∩ H 1 0 (Ω)) ∩ H 1 (0, T ; L 2 (Ω)). It is easily seen that (•, •) P,s is a scalar product on P = L 2 (0, T ; H 2 ∩ H 1 0 (Ω)) ∩ H 1 (0, T ; L 2 (Ω)). We thus introduce P s as the completion of L 2 (0, T ; H 2 ∩ H 1 0 (Ω)) ∩ H 1 (0, T ; L 2 (Ω))
for the norm • P,s associated with this scalar product, which therefore endows P s with an Hilbert structure.

Besides, by density arguments, ( 16) remains true for all s s 0 and for all p ∈ P s .

We can now state the main result of this section.

Theorem 5. For s s 0 , B ∈ L 2 (ρ 0 (s), Q T ) and u 0 ∈ L 2 (Ω), there exists a unique function p s ∈ P s , depending linearly on (B, u 0 ) such that

(p s , q) P,s = Ω u 0 q(0) + Q T Bq, ∀q ∈ P s . (20)
Then v s = -s 3 ρ -2 0 (s)p s 1 ω is a control function for [START_REF] Carthel | On exact and approximate boundary controllabilities for the heat equation: a numerical approach[END_REF], the corresponding controlled trajectory is given by z s = ρ -2 (s)L p s , and the operator Λ 0 s : (B, u 0 ) → z s is linear and continuous from

L 2 (ρ 0 (s), Q T ) × L 2 (Ω) to L 2 (ρ(s), Q T ).
Furthermore, we have the following estimates, for some constant C 1 = C 1 (Ω, ω, T ) > 0 which does not depend on s:

ρ(s) z s L 2 (Q T ) + s -1 ρ 1 (s)∇z s L 2 (Q T ) d + s -1 ρ 1 (s)z s L ∞ (0,T ;L 2 (Ω)) + s -3/2 ρ 0 (s) v s L 2 (q T ) C 1 s -3/2 ρ 0 (s)B L 2 (Q T ) + s -1 e c2s u 0 L 2 (Ω) ( 21
)
with c 2 := ϕ(•, 0) ∞ . Eventually, v s ∈ L p d (q T )
, and for some constant

v s L p d (q T ) C ρ 0 (s)B L 2 (Q T ) + e c2s u 0 L 2 (Ω) . (22)
Proof. For simplicity, we divide the proof of Theorem 5 in several lemmas.

We first analyze the solvability of the equation [START_REF] Victor | The problem of blow-up in nonlinear parabolic equations[END_REF].

Lemma 3. Under the assumptions of Theorem 5, there exists a unique p s ∈ P s solution of (20). This function p s satisfies the following estimate (with

c 2 = ϕ(•, 0) ∞ ) p s Ps C s -3/2 ρ 0 (s) B L 2 (Q T ) + s -1 e c2s u 0 L 2 (Ω) . (23)
for some constant C = C(Ω, ω, T ) > 0. Finally, p s depends linearly on the couple (B, u 0 ).

Proof of Lemma 3. Since (•, •) Ps is a scalar product on P s , Lemma 3 is a consequence of the fact that the right hand-side of (20) corresponds to a linear continuous form on P s . The linear map L 1 :

P s → R, q → Q T Bq is continuous since ρ(s)B ∈ L 2 (Q T ). Indeed, for all q ∈ P s Q T Bq Q T |ρ 0 (s)B| 2 1/2 Q T |ρ -1 0 (s)q| 2 1/2
and since from ( 16)

Q T |ρ -1 0 (s)q| 2 1/2
Cs -3/2 q Ps , it follows that

|L 1 (q)| = Q T Bq Cs -3/2 ρ 0 (s)B L 2 (Q T ) q Ps .
Thus L 1 is continuous on P s . From ( 16) we deduce that the linear map L 2 : P s → R, q → Ω u 0 q(0) is continuous. Indeed, using s 1, we obtain for all q ∈ P s that:

|L 2 (q)| = s -1 e c2s u 0 L 2 (Ω) se -c2s q(0) L 2 (Ω) s -1 e c2s u 0 L 2 (Ω) s q(0)e -sϕ(x,0) L 2 (Ω) = s -1 e c2s u 0 L 2 (Ω) s ρ -1 (s, 0)q(0) L 2 (Ω) Cs -1 e c2s u 0 L 2 (Ω) q Ps .
Accordingly, the right hand-side of (20) corresponds to a linear continuous form on P s and Riesz representation theorem gives the existence of a unique p s ∈ P s solving [START_REF] Victor | The problem of blow-up in nonlinear parabolic equations[END_REF], which additionally satisfies [START_REF] Michael | Globally strictly convex cost functional for a 1-D inverse medium scattering problem with experimental data[END_REF].

We then establish the link between the equation [START_REF] Victor | The problem of blow-up in nonlinear parabolic equations[END_REF] and the control problem ( 7)- [START_REF] Cazenave | An introduction to semilinear evolution equations[END_REF].

Lemma 4. Let s s 0 and p s ∈ P s be the unique solution of [START_REF] Victor | The problem of blow-up in nonlinear parabolic equations[END_REF] given in Lemma 3 and define (z s , v s ) by

z s = ρ -2 (s)L p s and v s = -s 3 ρ -2 0 (s)p s 1 ω . ( 24
)
Then z s solves ( 7)-( 8)

with v = v s , ρ(s)z s ∈ L 2 (Q T ), ρ 0 (s)v s ∈ L 2 (q T )
, and satisfies the following estimate

ρ(s) z s L 2 (Q T ) + s -3/2 ρ 0 (s) v s L 2 (q T ) C 1 s -3/2 ρ 0 (s)B L 2 (Q T ) + s -1 e c2s u 0 L 2 (Ω) . (25)
Proof of Lemma 4. From the definition of P s , ρ(s)z s ∈ L 2 (Q T ) and ρ 0 (s)v s ∈ L 2 (q T ), and [START_REF] Lacey | Global blow-up of a nonlinear heat equation[END_REF] immediately follows from [START_REF] Michael | Globally strictly convex cost functional for a 1-D inverse medium scattering problem with experimental data[END_REF].

In view of ( 20), (z s , v s ) satisfies, for all q ∈ L 2 (0, T ;

H 2 ∩ H 1 0 (Ω)) ∩ H 1 (0, T ; L 2 (Ω)), Q T z s L q = Q T v s 1 ω q + Ω u 0 q(0) + Q T Bq. (26)
Therefore, z s is a solution of ( 7)-( 8) corresponding to the choice v = v s in the sense of transposition.

Since (z s , v s ) ∈ L 2 (Q T ) × L 2 (q T )
, by uniqueness of solutions of ( 7)-( 8) in the sense of transpositions, z s also solves ( 7)-( 8) in the weak sense.

Remark 1. Following [START_REF] Fursikov | Controllability of evolution equations[END_REF], it is not difficult to check that the functions z s and v s provided by Lemma 4 can also be characterized as the unique minimizer of the functional J s defined as

J s (z, v) = s 3 2 Q T ρ 2 (s)|z| 2 + 1 2 q T ρ 2 0 (s)|v| 2 (27)
over the set

(z, v) : ρ(s)z ∈ L 2 (Q T ), ρ 0 (s)v ∈ L 2 (q T
), (z, v1 ω ) solves ( 7)- [START_REF] Cazenave | An introduction to semilinear evolution equations[END_REF] in the transposition sense .

The next lemma gives additional estimates on the trajectory z s given by ( 4), based on the fact that it solves a parabolic equation with a source term in L 2 (Q T ) and an initial datum in L 2 (Ω), thus belonging to the space L 2 (0, T ; H 1 0 (Ω)) ∩ H 1 (0, T ; H -1 (Ω)), and strongly inspired by [1, Theorem 2.6] in which the initial datum was zero (see also [START_REF] Ervedoza | Local exact controllability for the two-and threedimensional compressible Navier-Stokes equations[END_REF]Theorem 3.3], where similar results are obtained): Lemma 5. With the notations and assumptions of Lemma 4, we further have that ρ

1 (s)z s ∈ L ∞ (0, T ; L 2 (Ω)), ∇z s ∈ L 2 (ρ 1 (s), Q T ) d ,
and there exists a constant C independent of s s 0 such that

ρ 1 (s)z s L ∞ (0,T ;L 2 (Ω)) + ρ 1 (s)∇z s L 2 (Q T ) d C s -1/2 ρ 0 (s)B L 2 (Q T ) + e c2s u 0 L 2 (Ω) . ( 28
) Proof of Lemma 5. Since z s ∈ L 2 (0, T ; H 1 0 (Ω)), ∂ t z s ∈ L 2 (0, T ; H -1 (Ω)) and ρ 1 (s) ∈ C ∞ ([0, T [; C ∞ (Ω))
, multiplying (7) by ρ 2 1 (s)z s and integrating by part we obtain in D (0, T )

1 2 ∂ t Ω |z s | 2 ρ 2 1 (s) + Ω ρ 2 1 (s)|∇z s | 2 = ω v s ρ 2 1 (s)z s + Ω Bρ 2 1 (s)z s + Ω |z s | 2 ρ 1 (s)∂ t ρ 1 (s) -2 Ω ρ 1 (s)z s ∇ρ 1 (s) • ∇z s . ( 29
)
Let us estimate each term of the right side of this inequality. We have ∂ t ρ 1 (s) = -∂tθ θ ρ 1 (s) + s ∂tθ θ ϕρ 1 (s) and thus, since from the definition of ϕ, ϕ Cθ and ∂tθ θ ϕ Csθ 2 and since s 1, θ 1 and ρ(s) = θρ 1 (s) we deduce that, on [0, T ) :

Ω |z s | 2 ρ 1 (s)∂ t ρ 1 (s) Ω ∂ t θ θ |ρ 1 (s)z s | 2 + s Ω ∂ t θ θ ϕ |ρ 1 (s)z s | 2 Cs 2 Ω ρ 2 (s)|z s | 2 .
On the other hand ∇ρ 1 (s) = θ -1 ∇ρ(s) Csρ(s), and thus,

Ω ρ 1 (s)z s ∇ρ 1 (s) • ∇z s C Ω s|ρ(s)z s | |ρ 1 (s)∇z s | Cs 2 Ω |ρ(s)z s | 2 + 1 2 Ω |ρ 1 (s)∇z s | 2 .
Finally, since ρ 2 1 (s) = θ -1/2 ρ 0 (s)ρ(s) and θ -1/2 1, we infer that

ω v s ρ 2 1 (s)z s ω ρ 0 (s)v s θ -1/2 ρ(s)z s ρ 0 (s)v s L 2 (ω) ρ(s)z s L 2 (Ω)
and

| Ω Bρ 2 1 (s)z s | ρ 0 (s)B L 2 (Ω) ρ(s)z s L 2 (Ω)
. Thus [START_REF] Münch | A mixed formulation for the direct approximation of L 2 -weighted controls for the linear heat equation[END_REF] implies that

∂ t Ω ρ 2 1 (s)|z s | 2 + Ω ρ 2 1 (s)|∇z s | 2 Cs 2 ρ(s)z s 2 L 2 (Ω) + ρ 0 (s)v s L 2 (ω) + ρ 0 (s)B L 2 (Ω) ρ(s)z s L 2 (Ω)
and therefore, since

ρ 1 (0, s)u 0 2 L 2 (Ω) e 2c2s u 0 2 L 2 (Ω) , we get, for all t ∈ [0, T ), Ω ρ 2 1 (s)|z s | 2 (t) + Qt ρ 2 1 (s)|∇z s | 2 Cs 2 ρ(s)z s 2 L 2 (Q T ) + ρ 0 (s)v s L 2 (q T ) + ρ 0 (s)B L 2 (Q T ) ρ(s)z s L 2 (Q T ) + e 2c2s u 0 2 L 2 (Ω) .
Since s 1, using [START_REF] Lacey | Global blow-up of a nonlinear heat equation[END_REF] we obtain, for all t ∈ [0, T ),

Ω ρ 2 1 (s)|z s | 2 (t) + Qt ρ 2 1 (s)|∇z s | 2 C s -1 ρ 0 (s)B 2 L 2 (Q T ) + e 2c2s u 0 2 L 2 (Ω)
which gives [START_REF] Lemoine | Constructive exact control of semilinear 1d heat equation[END_REF].

The next result is slightly less classical and proves that the control v s produced by Lemma 4 enjoys some nice integrability property. Lemma 6. Let s s 0 . Let v s be given by Lemma 4. Then v s ∈ L p d (q T ) (recall the definition of p d in [START_REF] Fursikov | Controllability of evolution equations[END_REF]) and satisfies, for some C > 0,

v s L p d (q T ) C( ρ 0 (s)B L 2 (Q T ) + e c2s u 0 L 2 (Ω) ).
Proof of Lemma 6. From estimates ( 16)-( 17), for all p ∈ L 2 (0, T ;

H 2 ∩ H 1 0 (Ω)) ∩ H 1 (0, T ; L 2 (Ω)), ρ -1 3 (s)p H 1 (0,T ;L 2 (Ω)) + ρ -1 3 (s)p L 2 (0,T ;H 2 (Ω)) C p P,s .
Therefore, by density, this estimate is still true for all p ∈ P s . Thus, by interpolation, for all p ∈ P s ,

ρ -1 3 p ∈ H τ (0, T ; H 2(1-τ ) (Ω)) for all τ ∈ [0, 1].
In particular, with the choice τ = 2/(d+2), in dimension d 3, using Sobolev's embedding theorem, we get that H τ (0, T ) and H 2(1-τ ) (Ω) respectively embeds in L p d (0, T ) and L p d (Ω). Accordingly, we get, for all p ∈ P s ,

ρ -1 3 p L p d (Q T ) C p Ps .
In dimension 1 and 2, this estimate also holds with p d as in ( 19): indeed, one can simply use that L 2 (0, T ;

H 2 ∩ H 1 0 (Ω)) ∩ H 1 (0, T ; L 2 (Ω)) embeds into L ∞ (0, T ; H 1 0 (Ω))
and that H 1 0 (Ω) embeds into L ∞ (Ω) in dimension 1, and in any L p (Ω) with p < ∞ when d = 2.

Let p s ∈ P s be the unique solution of [START_REF] Victor | The problem of blow-up in nonlinear parabolic equations[END_REF] given in Lemma 3, z s = ρ -2 (s)L p s and v s = -s 3 ρ -2 0 (s)p s 1 ω . Then ρ -1 3 (s)p s 1 ω = -s -3 θ -3 ρ 2 (s)ρ -1 3 (s)v s . We then have, using s 1 and ( 23):

s -3 θ -3 ρ 2 ρ -1 3 v s L p d (q T ) C p s P C( ρ 0 (s)B L 2 (Q T ) + e c2s u 0 L 2 (Ω) ). (30)
Now, to conclude the proof of Lemma 6, we simply check that there exists C 3 > 0 such that s -3 θ -3 ρ 2 ρ -1 3 C 3 . From the definition of ρ and ρ 3 we have

ρ 2 (s)ρ -1 3 (s) = e 2sϕ e -3 2 sϕ * e sθ( 1 2 λe 12λ -2e λ ψ ) e csθ
for some positive constant c > 0, thus making obvious the existence of a positive C 3 such that for all s 1, s -3 θ -3 ρ 2 ρ -1

3 C 3 .
We can now conclude easily Theorem 5 by putting together Lemma 3, Lemma 4, Lemma 5 and Lemma 6.

Additional properties of the controlled trajectories given by Theorem 5

In this section, we provide two properties of the controlled trajectory given by Theorem 5, which will be useful in our fixed point argument.

The first one concerns the integrability of the controlled trajectory. To better understand the origin of this property, we first state two classical results on the heat equation:

Lemma 7 (Maximal regularity in L q class). [31, Theorem 9.1 p.341] Let q ∈ (1, ∞). For all F ∈ L q (Q T ), the unique weak solution z F of ∂ t z F -∆z F = F in Q T , z F = 0 on Σ T , z F (•, 0) = 0 in Ω ( 31
)
satisfies z F ∈ L q (0, T ; W 2,q (Ω)) ∩ W 1,q (0, T ; L q (Ω)) and z F L q (0,T ;W 2,q (Ω))∩W 1,q (0,T ;L q (Ω)

) C F L q (Q T ) . (32)
As an immediate corollary of Lemma 7, from [31, Lemma 3.4 p.83], z ∈ C 0 ([0, T ]; W 2-2 q ,q (Ω)) and therefore from the Sobolev-Gagliardo-Nirenberg Theorem (see [12, Theorems 6.7 and 8.2] and [6, Corollary 9.14 p. 284]) and since, for q > d+2 2 , L q (0, T ; W 2,q (Ω)) ∩ W 1,q (0, T ; L q (Ω)) → C 0 (Q T ) is compact (see [START_REF] Simon | Compact sets in the space L p (0, T ; B)[END_REF]Corollary 8 p. 90 and Lemma 12 p. 91], we easily get the following result: Corollary 1. With the notations of Lemma 7, if q > q d , where q d is defined by

q d = d + 2 2 , ( 33
)
there exists a constant C > 0 such that for all F ∈ L q (Q T ), the solution z F of (31) satisfies

z F L ∞ (Q T ) C F L q (Q T ) . ( 34
) Furthermore, if (F n ) n∈N is a bounded sequence of L q (Q T ), (z Fn ) n∈N is compact in C 0 (Q T ).
Accordingly, using [START_REF] Hecht | New development in Freefem++[END_REF], an interesting L ∞ (Q T ) estimate on the controlled trajectory given by Theorem 5 can be proved when u 0 ∈ L ∞ (Ω) and B ∈ L q (Q T ) for q > q d when p d > q d , i.e. d 5: Theorem 6. Within the setting of Theorem 5, when d 5, if we further assume that B ∈ L q (Q T ) for some q > q d , and u 0 ∈ L ∞ (Ω), we have the following additional estimate: there exists a constant C q > 0 such that the controlled trajectory provided by Theorem 5 belongs to L ∞ (Q T ) and satisfies:

z s L ∞ (Q T ) C q ( B L q (Q T ) + ρ 0 (s)B L 2 (Q T ) + e c2s u 0 L ∞ (Ω) ). ( 35
)
Proof. According to Theorem 5, the control v s provided by Theorem 5 belongs to L p d (Q T ) and its

L p d (Q T ) norm is bounded by C ρ 0 (s)B L 2 (Q T ) + e c2s u 0 L 2 (Ω)
. Since d 5, p d > q d , and we can then use Corollary 1 to estimate the solution z F of (31) corresponding to the choice

F = B + v s 1 ω : z F L ∞ (Q T ) C( B L q (Q T ) + ρ 0 (s)B L 2 (Q T ) + e c2s u 0 L 2 (Ω) ). ( 36
)
We then write that the controlled trajectory z s given by Theorem 5 as z F + z u0 , where z u0 is the solution of

∂ t z u0 -∆z u0 = 0 in Q T , z u0 = 0 on Σ T , z u0 (•, 0) = u 0 in Ω, (37)
for which the maximum principle immediately yields z u0 ∈ L ∞ (Q T ) and

z u0 L ∞ (Q T ) u 0 L ∞ (Ω) . (38)
Combining the estimates (36) and (38) easily gives the estimate (35).

3 A controllability result for (1) with f ∈ C 0 (R): proof of the first item of Theorem 4

The goal of this section is to prove the first item of Theorem 4, in a slightly more precise form, namely:

Theorem 7. Let T > 0 and Ω ⊂ R d , 1 d 5.
There exists β > 0 (given in (50) afterwards) such that if the function f ∈ C 0 (R) satisfies (H 2 ), for all u 0 ∈ L ∞ (Ω), there exists a control v ∈ L p d (q T ) and a solution y ∈ L 2 (0, T ;

H 1 0 (Ω)) ∩ L ∞ (Q T ) of (1) associated with v1 ω such that y(•, T ) = 0. Moreover, there exists s 1 such that (y, v) ∈ L 2 (ρ(s), Q T ) × L 2 (ρ 0 (s), q T )
, where ρ and ρ 0 are respectively defined in [START_REF] Di Nezza | Hitchhiker's guide to the fractional Sobolev spaces[END_REF] and (15). Remark 2. We emphasize that, contrary to Theorem 1, we do not assume that f is locally Lipschitz continuous. In particular, we do not assume a priori the existence of local solutions in time for (1). Remark 3. We also emphasize that the controllability time is arbitrary. However, the specific structure of the control v s together with lower bound on s forces the control to act from the beginning so as to prevent the solution to blow up.

For s s 0 a parameter to be fixed later, for all y ∈ L 2 (ρ 0 (s), Q T )∩L ∞ (Q T ), we solve the linearized null controllability problem

     ∂ t y -∆y = v1 ω -f ( y) in Q T , y = 0 on Σ T , y(., 0) = u 0 in Ω, y(•, T ) = 0 in Ω (39)
and we will prove the existence of a fixed point of Λ s : y → y, (equivalently, with the notation of Theorem 5, Λ s ( y

) = Λ 0 s (-f ( y), u 0 )) ( 40
)
where y is the solution of the null controllability problem (39) associated with v1 ω given in Theorem 5 for B = -f ( y).

In order to that, we will employ the Schauder fixed point theorem.

To be more precise, for s s 0 , we first introduce the Banach space L ∞ (Q T ) ∩ L 2 (ρ 0 (s), Q T ), endowed with the norm

|||y||| s := y L ∞ (Q T ) + y L 2 (ρ0(s),Q T ) .
For R > 0 and s s 0 , we then introduce the class C R (s), defined as the non empty closed convex set of L 2 (ρ 0 (s),

Q T ) ∩ L ∞ (Q T ) given by C R (s) = y ∈ L ∞ (Q T ), y L ∞ (Q T ) R, ρ 0 (s)y L 2 (Q T ) R 1/2 . ( 41
)
Our goal will then be to prove that:

• the map Λ s is well-defined on L 2 (ρ 0 (s), Q T ) ∩ L ∞ (Q T ); see Section 3.1;

• If β > 0 is small enough in (H 2 ), there exists s and R large enough so that Λ s (C R (s)) ⊂ C R (s); see Section 3.2; Accordingly, by Schauder fixed point theorem, Λ s will have a fixed point on C R (s), and this fixed point will provide the controlled trajectory y of Theorem 7, see Section 3.5.

• Λ s (C R (s)) is compact in C R (s)

The map Λ

s is well-defined on L 2 (ρ 0 (s), Q T ) ∩ L ∞ (Q T )
To properly define the map Λ s , we need to check the following lemma: Lemma 8. Let d 5 and 2 q +∞. Under the assumptions of Theorem 7. For all s s 0 ,

y ∈ L 2 (ρ 0 (s), Q T ) ∩ L ∞ (Q T ), f ( y) ∈ L 2 (ρ 0 (s), Q T ) ∩ L q (Q T ) and ρ 0 (s)f ( y) L 2 (Q T ) α + β ln + y L ∞ (Q T ) 3/2 ρ 0 (s) y L 2 (Q T ) , (42)
f ( y) L q (Q T ) α + β ln + y L ∞ (Q T ) 3/2 ρ 0 (s) y 2/q L 2 (Q T ) y 1-2/q L ∞ (Q T ) . (43)
Proof. Assumption (H 2 ) immediately implies (42).

To estimate f ( y) in L q (Q T ), we write that f ( y

) L q (Q T ) f ( y) θ L 2 (Q T ) f ( y) 1-θ L ∞ (Q T ) where 1 q = θ 2 + 1 -θ ∞ , that is θ = 2 q
. Since ρ 0 1, we obviously have from (42) that

f ( y) L 2 (Q T ) ρ 0 f ( y) L 2 (Q T ) α + β ln + y L ∞ (Q T ) 3/2 ρ 0 (s) y L 2 (Q T ) ,
while we easily have

f ( y) L ∞ (Q T ) α + β ln + y L ∞ (Q T ) 3/2 y L ∞ (Q T ) .
These last estimates easily give (43).

An immediate consequence of Lemma 8, Theorem 5 and Theorem 6 is the following one:

Proposition 1. Under the assumptions of Theorem 7, the map Λ s defined in (40) as the solution of the null-controllability problem (39) using Theorem 5 is well-defined for s s 0 on L 2 (ρ 0 (s), Q T )∩L ∞ (Q T ), and for y ∈ L 2 (ρ 0 (s), Q T ) ∩ L ∞ (Q T ), y = Λ s ( y) satisfies the following estimates:

ρ(s) y L 2 (Q T ) + s -1 ρ 1 (s)∇y L 2 (Q T ) d + s -1 ρ 1 (s)y L ∞ (0,T ;L 2 (Ω)) C 1 s -3/2 α + β ln + y L ∞ (Q T ) 3/2 ρ 0 (s) y L 2 (Q T ) + C 1 s -1 e c2s u 0 L 2 (Ω) , (44)
and for q > max{q d , 2}, we have

y L ∞ (Q T ) C q α + β ln + y L ∞ (Q T ) 3/2 ρ 0 (s) y 2/q L 2 (Q T ) y 1-2/q L ∞ (Q T ) + C q α + β ln + y L ∞ (Q T ) 3/2 ρ 0 (s) y L 2 (Q T ) + C q e c2s u 0 L ∞ (Ω) . ( 45
)
In particular,

Λ s maps L 2 (ρ 0 (s), Q T ) ∩ L ∞ (Q T ) into itself.
Furthermore, the control v provided by Theorem 5 satisfies

v L p d (q T ) C α + β ln + y L ∞ (Q T ) 3/2 ρ 0 (s) y L 2 (Q T ) + e c2s u 0 L 2 (Ω) . (46)

A stable class for suitable choices of parameters

Our next goal is to show the following result:

Lemma 9. Under the assumptions of Theorem 7, if β in (H 1 ) is small enough, there exist s and

R > 0 such that Λ s (C R (s)) ⊂ C R (s), (47)
where the class C R (s) is given in (41).

Proof. Of course, the proof relies on the estimates given by Proposition 1.

We fix q > max{q d , 2} with q finite. Then, for s s 0 and y ∈ C R (s), we have from (44) (recalling s s 0 1 and the inequality ρ 0 ρ) that y = Λ s ( y) satisfies

ρ 0 (s) y L 2 (Q T ) ρ(s) y L 2 (Q T ) C 1 s -3/2 α + β ln + R 3/2 R 1/2 + C 1 e c2s u 0 L 2 (Ω) .
On the other hand, for s s 0 and y ∈ C R (s), from (45), y = Λ s ( y) satisfies

y L ∞ (Q T ) C q α + β ln + R 3/2 (R 1-1/q + R 1/2 ) + C q e c2s u 0 L ∞ (Ω) .
In view of these two estimates, it is natural to impose the relation

s = 1 4c 2 ln + R , (48)
taking R large enough so that s s 0 . Indeed, with this relation, the two above estimates yield that for y ∈ C R (s), y = Λ s (y) satisfies:

ρ 0 (s) y L 2 (Q T ) C 1 (4c 2 ) 3/2 α ln + (R) + β 3/2 R 1/2 + C 1 R 1/4 u 0 L 2 (Ω) , y L ∞ (Q T ) C q α + β ln + R 3/2 (R 1-1/q + R 1/2 ) + C q R 1/4 u 0 L ∞ (Ω) .
Therefore, if (2C 1 ) 2/3 4c 2 β < 1, that is if β is small enough, for R large enough, we have

                       C 1 (4c 2 ) 3/2 α ln + (R) + β 3/2 1 2 , C 1 R 1/4 u 0 L 2 (Ω) 1 2 R 1/2 . C q α + β ln + R 3/2 (R 1-1/q + R 1/2 ) 1 2 R, C q R 1/4 u 0 L ∞ (Ω) 1 2 R. (49)
According to the previous estimates, we have thus proved that if (2C 1 ) 2/3 4c 2 β < 1, imposing the relation (48), and taking R large enough so that s s 0 and R satisfies (49), for all y ∈ C R (s), y = Λ s ( y) belongs to C R (s).

Remark 4. The smallness condition on β is explicit:

β < 1 4c 2 (2C 1 ) 2/3 , ( 50
)
where the constants C 1 and c 2 are the constants appearing in Theorem 5.

Remark 5. In fact, the above proof shows that, provided we impose the relation (48), C R (s) is stable for Λ s for any R R 0 (equivalently s s 0 ) for a suitably large R 0 (equivalently s 0 ). Furthermore, with the above choices, in view of (49), the lower bound R 0 depends on u 0 L ∞ (Ω) as a power of u 0 L ∞ (Ω) , so that the lower bound s 0 can be chosen as depending logarithmically on u 0 L ∞ (Ω) . Proof. Let (y n ) n∈N be a sequence of Λ s (C R ) bounded for the norm |||•||| s . We have to prove that there exists a subsequence (y n k ) k∈N of (y n ) n∈N and y ∈ C R such that y n k → y in L 2 (ρ 0 (s), Q T ) and

Λ

s (C R (s)) is a relatively compact subset of C R for the norm |||•||| s .
y n k → y in L ∞ (Q T ). First, since Λ s (C R (s)) ⊂ C R (s), (y n ) n∈N
is a bounded sequence of the Hilbert space L 2 (ρ 0 (s), Q T ), there exists a subsequence (y n k ) k∈N and y ∈ L 2 (ρ 0 (s), Q T ) such that (y n k ) k∈N weakly converge to y in L 2 (ρ 0 (s), Q T ), and therefore y L 2 (ρ0(s),Q T ) R 1/2 . Furthermore, since (y n k ) k∈N is bounded in L ∞ (Q T ), extracting another subsequence if necessary, which we still denote the same for simplicity, the sequence (y n k ) k∈N -weakly converges to y in L ∞ (Q T ) and we also have

y L ∞ (Q T ) R.
On the other hand, there exists (

y n k ) k∈N a sequence of C R (s) such that y n k = Λ s ( y n k ) for all k ∈ N, meaning that there exists v n k ∈ L p d (q T ) (recall (46)) such that ∂ t y n k -∆y n k = v n k 1 ω -f ( y n k ) in Q T , y n k = 0 on Σ T , y n k (., 0) = u 0 in Ω.
In particular, each y n k can be decomposed as y n k = z Fn k + z u0 where z Fn k solves [START_REF] Ladyzenskaja | Linear and Quasi-linear Equations of Parabolic Type[END_REF] with

F n k = v n k 1 ω -f ( y n k ) and z u0 as in (37).
Since ( y n k ) k∈N belongs to C R (s), from (43), (f (

y n k )) k∈N is bounded in L ∞ (Q T ), and from (46), (v n k 1 ω ) k∈N is bounded in L p d (Q T ). Since d 5, p d > q d ,
and thus, according to Corollary 1, there exists a subsequence, also denoted by (z Fn k ) k∈N and z

∈ C 0 (Q T ) such that (z Fn k ) k∈N converges to z in C 0 (Q T ) as k → ∞.
Since z u0 is independent of n and belongs to L ∞ (Q T ) by the maximum principle, (y n k ) k∈N strongly converges to y = z + z u0 in L ∞ (Q T ). By uniqueness of the limit in D (Q T ), y = y.

Furthermore, it is easy to check from the above arguments that a.e in t ∈ [0, T ], y n k (t) → y(t) strongly in L ∞ (Ω).

To finish the proof, it suffices to prove that

y n k → y in L 2 (ρ 0 (s), Q T ). Since (y n k ) k∈N weakly converge to y in L 2 (ρ 0 (s), Q T ) it suffices to prove that y n k L 2 (ρ0(s),Q T ) → y L 2 (ρ0(s),Q T ) .
From the previous step, we have the strong convergence y n k (t) → y(t) a.e in t ∈ [0, T ] in L ∞ (Ω) and thus, since for all t ∈ [0, T ),

ρ 0 (•, t, s) ∈ C 0 (Ω), ρ 0 (•, t, s)y n k (t) → ρ 0 (•, t, s)y(t) in L 2 (Ω) a.e in t ∈ [0, T ).
Moreover we deduce from (44) and the fact that ( y n k ) k∈N is a sequence in C R (s) that there exists a constant C > 0 such that for every

k ∈ N ρ 1 (s)y n k (t) L ∞ (0,T ;L 2 (Ω)) C. Since ρ 0 ρ 1 , this obviously implies that sup k∈N ρ 0 (s)y n k (t) L ∞ (0,T ;L 2 (Ω)) < ∞.
We then deduce from the Lebesgue dominated convergence theorem that ρ 0 (s)

y n k L 2 (Ω) → ρ 0 (s)y L 2 (Ω) in L 2 (0, T ), which gives the strong convergence y n k → y in L 2 (ρ 0 (s), Q T ) as k → ∞. Therefore Λ s (C R (s)) is a relatively compact subset of C R (s) for the norm |||•||| s . 3.4 Λ s : C R (s) → C R (s) is a continuous mapping for the norm |||•||| s .
We have the following result:

Proposition 3. Under the asumption of Lemma 9, Λ s : C R (s) → C R (s) is a continuous mapping for the norm |||•||| s .
Proof. For all n ∈ N , let ( y n ) n∈N be a sequence of C R (s) and y 0 ∈ C R (s) such that y n → y 0 for the norm |||•||| s .

Setting

y n = Λ s ( y n ) for all n ∈ N, let us prove that y n → y 0 for the norm |||•||| s . First, recall that Λ s ( y n ) = Λ 0 s (-f ( y n ), u 0 )
, where Λ 0 s is the operator given in Theorem 5, which is continuous from

L 2 (ρ 0 (s), Q T ) × L 2 (Ω) to L 2 (ρ(s), Q T ).
Accordingly, if we manage to show that the sequence (f ( y n )) n∈N weakly converges to f ( y 0 ) in L 2 (ρ 0 (s), Q T ), the sequence (y n ) would weakly converge to y 0 in L 2 (ρ(s), Q T ) as n → ∞, and thus in L 2 (ρ 0 (s), Q T ) since ρ 0 ρ. Furthermore, from the compactness of Λ s (C R (s)) for the topology induced by |||•||| s (proved in Proposition 2), this would entail the continuity of Λ s on C R (s) for the topology induced by |||•||| s .

Therefore, it is sufficient to prove that the sequence (f (

y n )) n∈N weakly converges to f ( y 0 ) in L 2 (ρ 0 (s), Q T ) as n → ∞. First, y n → y 0 in L ∞ (Q T ) and f being continuous, it is uniformly continuous on [-R, R]. There- fore, the sequence (f ( y n )) n∈N converges to f ( y 0 ) in L ∞ (Q T ). Secondly, since (f ( y n )) n∈N is bounded in L 2 (ρ 0 (s), Q T ) (recall (42) and that ( y n ) n∈N is a sequence of C R (s)), there exists a subsequence (f ( y n k )) k∈N and Y ∈ L 2 (ρ 0 (s), Q T ) such that (f ( y n k )) k∈N weakly converge to Y in L 2 (ρ 0 (s), Q T ).
By uniqueness of the limit in D (Q T ), Y = f ( y 0 ) and all the sequence (f ( y n )) n∈N weakly converges to f ( y 0 ) in L 2 (ρ 0 (s), Q T ). As said above, this suffices to conclude the proof of Proposition 3.

Conclusion: proof of Theorem 7.

Taking β small enough so that Lemma 9 applies, with s and R given by Lemma 9, we can apply Schauder fixed point theorem to Λ s on C R (s). Indeed, we have:

• C R (s) is a closed convex set of the Banach space L 2 (ρ 0 (s), Q T ) ∩ L ∞ (Q T ); • C R (s) is stable by Λ s by Lemma 9;
• Λ s is continuous on C R (s) for the topology induced by |||•||| s from Proposition 3;

• Λ s (C R (s)) is compact for the topology induced by |||•||| s from Proposition 2.
Therefore, we deduce from the Schauder fixed-point theorem that there exists a fixed point y ∈ C R (s) of Λ s . By construction of Λ s , there exists v s ∈ L 2 (ρ 0 (s), q T )∩L p d (q T ) such that y is the solution of the null controllability problem (39) with v = v s and y = y.

4 A contracting property for the fixed point operator Λ s :

Proof of the second item of Theorem 4

In this section, under additional regularity assumptions on the nonlinear function f , we prove that the operator Λ s used in the previous section to prove Theorem 7 using Schauder fixed point theorem is a contracting mapping leading to a constructive method to find its fixed point.

In this section, we assume that the nonlinear function f is locally Lipschitz-continuous and satisfies the hypothesis (H 1 ) with β as in (50).

Since (H 1 ) obviously implies (H 2 ) with the same β (recall f (0) = 0), Theorem 7 obviously applies and we can follow the main steps of its proof. Therefore, we define Λ s as in (40), C R (s) as in (41), and we choose the parameters s and R such that Lemma 9 applies.

Our goal then is to check the following property:

Proposition 4. Assume that f satisfies (H 1 ) with β defined in (50) and s and R chosen as in Lemma 9. Then, for y 1 , y 2 ∈ C R (s), we have

Λ s ( y 2 ) -Λ s ( y 1 ) L 2 (ρ0(s),Q T ) 1 2 y 2 -y 1 L 2 (ρ0(s),Q T ) . (51)
In particular, Λ s is a contraction mapping from C R (s) into itself for the weighted norm L 2 (ρ 0 (s), Q T ).

Proof. Let y 1 , y 2 ∈ C R (s), and recall that for i = 1, 2, Λ s ( y i ) = Λ 0 s (-f ( y i ), u 0 ). Therefore, Λ s ( y 2 ) -Λ s ( y 1 ) = Λ 0 s (f ( y 1 ) -f ( y 2 ), 0). According to Theorem 5, using that ρ 0 ρ, we thus have:

Λ s ( y 2 ) -Λ s ( y 1 ) L 2 (ρ0(s),Q T ) C 1 s -3/2 f ( y 2 ) -f ( y 1 ) L 2 (ρ0(s),Q T ) .
In particular, taking y i ∈ C R (s) for i = 2, we can use (H 1 ) to deduce

f ( y 2 ) -f ( y 1 ) L 2 (ρ0(s),Q T ) (α + β ln + (R)) 3/2 y 2 -y 1 L 2 (ρ0(s),Q T ) . (52)
Now, with s and R as in Lemma 9, i.e. as in (48), we immediately derive

Λ s ( y 2 ) -Λ s ( y 1 ) L 2 (ρ0(s),Q T ) C 1 (4c 2 ) 3/2 α ln + (R) + β 3/2 y 2 -y 1 L 2 (ρ0(s),Q T ) . (53)
In view of the constraint (49) (1) satisfied by s and R in Lemma 9, we immediately deduce (51).

Remark 6. Interestingly, estimate (53) underlines that the Lipschitz constant of Λ s seems to get smaller as R → ∞ (which is allowed, recall Remark 5), that is, in view of the relation (48), when s → ∞, and approach the constant C 1 (4c 2 β * ) 3/2 at the limit. In particular, if one considers a semilinear function f which satisfies

lim |r|→∞ |f (r)| ln 3/2 + |r| = 0,
then, for any ε > 0, we can guarantee that Λ s gets ε-contractive by taking s large enough. This suggests that the speed of convergence of the sequence (y k ) k∈N increases with s, a fact which we will illustrate numerically afterwards.

5.1 Construction of the sequence (y k , v k ) k∈N

Based on Theorem 5, for k ∈ N, the controlled trajectory y k and the corresponding control function v k are computed as follows.

For s large enough according to Remark 5, we first find p k ∈ P s the solution of the variational formulation:

(p k , p) P,s = -

Q T f (y k-1 )p + Ω u 0 p(0) ∀p ∈ P s . (57)
Then, we simply set

y k = ρ -2 (s)L p k , v k = -s 3 ρ -2 0 (s)p k 1 ω .
The numerical approximation of the variational formulation (57) (of second order in time and fourth order in space) has been discussed at length in [START_REF] Fernández | Strong convergent approximations of null controls for the 1D heat equation[END_REF]. In particular, a conformal parametrized approximation, say P h of P s , leads to a strong convergent approximation p k,h of p k as the discretization parameter h goes to 0, i.e. p k,h -p k P,s → 0 as h → 0. From p k,h , an approximation of the controlled state is given by ( [START_REF] Fernández | Strong convergent approximations of null controls for the 1D heat equation[END_REF][START_REF] Fernández | Numerical exact controllability of the 1D heat equation: duality and Carleman weights[END_REF], in order to solve (57), it is very convenient to preliminary perform the change of variable

y k,h , v k,h ) := (ρ -2 (s)L p k,h , -s 3 ρ -2 0 (s)p k,h 1 ω ). As discussed in
m k = ρ -1 0 (s)p k , z k = ρ -1 (s)L p k so that z k = ρ -1 (s)L (ρ 0 (s)m k
) and y k = ρ -1 (s)z k and then replace the formulation (57) by the equivalent and well-posed following mixed formulation:

find (z k , m k , η k ) ∈ L 2 (Q T ) × ρ -1 0 P s × L 2 (Q T ) solution of                  Q T z k z + s 3 q T m k m + Q T (T -t) 1/2 η k z -ρ -1 (s)L (ρ 0 (s)m) = - Q T ρ 0 (s)f (ρ -1 (s)z k-1 )m + Ω ρ 0 (s, 0)u 0 m(0) ∀(m, z) ∈ ρ -1 (s)P s × L 2 (Q T ), Q T (T -t) 1/2 η z k -ρ -1 (s)L (ρ 0 (s)m k ) = 0, ∀η ∈ L 2 (Q T ).
(58) The variable η k stands as a Lagrange multiplier for the constraint z k -ρ -1 (s)L (ρ 0 (s)m k ) = 0 in Q T .

We check the following equality for every m ∈ ρ -1 (s)P s

-ρ -1 (s)L (ρ 0 (s)m) = (g 1 (θ, ϕ) + g 2 (θ, ϕ))m + θ -3/2 (∂ t m + ∆m) + g 3 (θ, ϕ) • ∇m with        g 1 (θ, ϕ) := ρ -1 (s)∂ t ρ 0 (s) = ∂ t (θ -3/2 ) + θ -3/2 s(∂ t ϕ), g 2 (θ, ϕ) := θ -3/2 (s∆ϕ + s 2 (∇ϕ) 2 ), g 3 (θ, ϕ) := ρ -1 (s)∇ρ 0 (s) = θ -3/2 s∇ϕ ( 59
)
where ϕ and θ are defined in [START_REF] Coron | Control and nonlinearity[END_REF] and [START_REF] Ciarlet | The finite element method for elliptic problems[END_REF]. We observe that g 2 is slightly singular like (T -t) -1/2 for t T -T 1 and thus justifies the introduction of the weight function (T -t) 1/2 in (58). Eventually, with f (r) = c f (α + β ln + |r|) 3/2 r, we get

- Q T ρ 0 (s)f (ρ -1 (s)z k-1 )m = -c f Q T θ -3/2 (α + β ln + |ρ -1 (s)z k-1 |) 3/2 z k-1 m, ∀m ∈ ρ -1 (s)P s .
In the experiments, we use the equivalent formulation (58) instead of (57) as it allows, first to eliminate the singularity of the coefficients for t close to T and second to obtain simultaneously the control and the controlled solution.

The sequence (y k , v k ) k∈N is initialized with y -1 = 0 (so that f (y -1 ) ≡ 0 and the first iteration computes the control pair (y 0 , v 0 ) corresponding to the controlled trajectory of the linear heat equation with initial datum u 0 and zero source term) and is computed until the following criterion is satisfied

ρ 0 (s)(y k+1 -y k ) L 2 (Q T ) ρ 0 (s)y k L 2 (Q T ) 10 -6 . (60)
We shall denote by k the lowest integer k for which (60) holds true.

Concerning the approximation of the formulation (58), we use a conformal space-time finite element method (as described in [START_REF] Fernández | Strong convergent approximations of null controls for the 1D heat equation[END_REF]). We consider a regular family T = {T h ; h > 0} of triangulation of Q T such that Q T = ∪ K∈T h K. The family T is indexed by h = max K∈T h diam(K). The variable z k and η k are approximated with the space

P h = {p h ∈ C 0 (Q T ); p h | K ∈ P 1 (K), ∀K ∈ T h } ⊂ L 2 (Q T )
where P 1 (K) denotes the space of affine functions both in x and t. The variable m k is approximated with the space V h = {v h ∈ C 1 (Q T ); v h | K ∈ P(K), ∀K ∈ T h } ⊂ M where P(K) denotes the composite Hsieh-Clough-Tocher C 1 element (we refer to [9, page 356]). Figure 1 depicts a typical triangulation T h of Q T corresponding to the value h ≈ 0.0208. Contrarily to (57), theoretical convergence results with respect to h for the approximated solution of (58) are not known. We refer to [START_REF] Münch | A mixed formulation for the direct approximation of L 2 -weighted controls for the linear heat equation[END_REF] for some numerical evidences of the robustness of the method with respect to the parameter h. Eventually, it should be noted that the corresponding finite dimensional linear system associated with (58) is independent of the iteration index k allowing a notable gain in term of computational time. An UMFPACK type solver is used.

Experiments

We present some numerical experiments in the one dimensional setting with Ω = (0, 1). We consider simply connected interval ω ⊂ Ω. We take T = 1/2 and consider data for which the uncontrolled solution of (1) blows up before T . Moreover, in order to reduce the decay of the solution of (1) when f ≡ 0, we replace the term -∆y in (1) by -ν∆y with ν = 10 -1 . The Carleman parameter λ is taken equal to one while the function ψ : Ω → [0, 1] is defined by

ψ(x) = x(1 -x)e -(x-c) 2 x (1 -x )e -(x -c) 2 , c := x - 1 -2x 2x (1 -x )
where x is the mid-point of ω (we refer to [15, Figure 1] for a plot of ψ).

We consider the nonlinear even function f as follows

f (r) = c f α + β ln(1 + |r|) 3/2 r
with α = β = 1 and c f < 0. We easily check that f satisfies (H 1 ), (H 1 ) and also (H p ) for every p ∈ [0, 1]. In particular, f ∈ L ∞ (Q T ). As for the initial condition to be controlled, we consider u 0 (x) = c u0 sin(πx) parametrized by c u0 > 0.

We use a mesh composed of 29132 and 14807 triangles corresponding to h ≈ 1.17 × 10 -2 . For ω = (0.2, 0.8), c u0 = 10 and c f = -5, Figure 2-left depicts the evolution of the relative error

ρ0(s)(y k+1 -y k ) L 2 (Q T ) ρ0(s)y k L 2 (Q T )
with respect to the parameter of iteration k for s ∈ {1, 2, 3, 4}. In agreement with the theoretical part, the convergence is observed for s large enough, here s 2. Moreover, the rate increases with s: the convergence is observed after k iterations equal to 48, 17, 13 for s = 2, 3 and 4 respectively. Figure 2-right depicts the ratio

ρ0(s)(Λs(y k )-Λs(y k-1 )) L 2 (Q T ) ρ0(s)(y k -y k-1 ) L 2 (Q T )
(appearing in Proposition 4) highlighting the lack of contracting property of Λ s for s = 1. Figure 3 depicts the evolution of the L 2 (Ω) norm of the control and corresponding controlled solution with respect to the time variable for s = 2, 3, 4. As expected in view of the definition of the weights, large values of s concentrate the action of the control close to the initial time and leads to large L ∞ (Q T ) norm of the control (see Table 1). Figure 4 and Figure 5 depict the control and corresponding controlled solution in Q T for these values of s. We also observed that the initialization (y 0 , v 0 ) = (ρ -1 (s)u 0 , 0) (valid since u 0 ∈ H 2 (Ω)) leads to the same limit of the algorithm. Table 2 provides some norms of the solution for s = 3 with respect to the fineness h of the triangular mesh used and highlights the stability of the approximation. Actually, the high degree (equal to 3) of the approximation induced by the composite finite element HCT makes the convergence of the approximation quite fast with respect to h. We also observe that the number of iterations to reach the convergence of the sequence (y k ) k 0 is independent of h. For larger data, the algorithm still converges if the parameter s is chosen sufficiently large. Table 3 provides for s = 2 and c u0 = 20 some norms associated with the convergent sequence (y k , f k ) k∈N with respect to the amplitude c f of the nonlinear function. The real C 1 (y, v) is defined by

(s)(y k+1 -y k ) L 2 (Q T ) ρ0(s)y k L 2 (Q T ) (Left) and ρ0(s)(y k+1 -y k ) L 2 (Q T ) ρ0(s)(y k -y k-1 ) L 2 (Q T ) (Right) w.r.t. k for s ∈ {0, 1, 2, 3}. s y k L 2 (Q T ) ρ(s)y k L 2 (Q T ) v k L 2 (q T ) ρ 0 (s)v k L 2 (q T ) v k L ∞ (q T ) k 2
h y k L 2 (Q T ) ρ(s)y k L 2 (Q T ) v k L 2 (q T ) ρ 0 (s)v k L 2 (q T ) v k L ∞ (q T ) k 0.
C 1 (y, v) := ρ(s) y L 2 (Q T ) + s -3/2 ρ 0 (s) v L 2 (q T ) s -3/2 ρ 0 (s)f (y) L 2 (Q T ) + s -1 e c2s u 0 L 2 (Ω) , (61)
and is plotted in Table 3. The ratio C 1 (y, v) notably appears in the estimate ( 25) with B = -f (y). The divergence of the sequence is observed for c f -7 suggesting that, for s = 3, the quantity C 1 (y k , v k ) is not uniformly bounded with respect to |c f |. This does not contradict Theorem 8 where an upper bound is assumed on β. On the contrary, Table 4 and 5 suggest that the quantity C 1 (y k , v k ) associated with (c f , s) = (-2, 2) and (c f , s) = (-2, 3) is uniformly bounded with respect to the parameter c u0 : this is in agreement with the uniform controllability of system (1) with respect to the initial data stated in Theorem 7. Table 6 collects some norms of (y k , v k ) w.r.t. s for the smaller support of control ω = (0.2, 0.6), c u0 = 10 and c f = -2 A smaller support yields to a larger constant C 1 = C 1 (Ω, ω, T ) in [START_REF] Glowinski | Exact and approximate controllability for distributed parameter systems[END_REF] and larger norms for the control and corresponding controlled solution. For instance, for s = 3, we obtain v k L ∞ (q T ) ≈ 1301.97 with ω = (0.2, 0.6) and v k L ∞ (q T ) ≈ 271.86 with ω = (0.2, 0.8) (see the first line of Table 5). Accordingly, the number k of iterations to reach convergence increases when the size of ω is reduced.

c f y k L 2 (Q T ) ρ(s)y k L 2 (Q T ) v k L 2 (q T ) ρ 0 (s)v k L 2 (q T ) v k L ∞ (q T ) C 1 (y k , v k ) k 6
We now consider the nonlinear function, for any η ∈ (0, 1) and α 0 With α = 0, the function f satisfies (H 1 ), (H 1 ) but only (H p ) for p ∈ (0, η). f (r) behaves likes ln(1 + |r|) 3/2 at infinity while in the neighborhood of zero, f is bounded but highly oscillates. For c f = -5, c u0 = 10 and η = 1/10, Table 7 provides some norms of (y k , v k ) w.r.t. the parameter s leading to similar results than in the previous case. On the other hand, with α > 0, the function f still satisfies (H 1 ) but not (H 1 ) nor (H p ). In particular, f takes arbitrarily large values in the neighborhood of zero. The values c f = -5, c u0 = 10 do not lead to the convergence of the algorithm including for large values of s. In order to enhance oscillations for large values, we now consider the function f (r) = c f r| cos(r 2 )| ln 3/2 (1 + |r|), c f < 0 (63) which still satisfies (H 1 ) but not (H 1 ) nor (H p ). For small values of |c f |, the method still produces convergent sequences: Table 8 reports some values for c f = -1; observe that the number k increases 7: c u0 = 10 ; c f = -5; Norms of (y k , v k ) w.r.t. s; f given by (62) with α = 0. from the case s = 1 and the case s = 2. For c f = -2, we observe the convergence only for s 6.

f (r) = c f r| cos(r -3/2+η )| α + ln(1 + |r|) 3/2 , c f < 0. ( 62
) c u0 y k L 2 (Q T ) ρ(s)y k L 2 (Q T ) v k L 2 (q T ) ρ 0 (s)v k L 2 (q T ) v k L ∞ (q T ) C 1 (y k , v k ) k 10
c u0 y k L 2 (Q T ) ρ(s)y k L 2 (Q T ) v k L 2 (q T ) ρ 0 (s)v k L 2 (q T ) v k L ∞ (q T ) C 1 (y k , v k ) k 10
s y k L 2 (Q T ) ρ(s)y k L 2 (Q T ) v k L 2 (q T ) ρ 0 (s)v k L 2 (q T ) v k L ∞ (q T ) k 1
s y k L 2 (Q T ) ρ(s)y k L 2 (Q T ) v k L 2 (q T ) ρ 0 (s)v k L 2 (q T ) v k L ∞ (q T ) k 1
s y k L 2 (Q T ) ρ(s)y k L 2 (Q T ) v k L 2 (q T ) ρ 0 (s)v k L 2 (q T ) v k L ∞ (q T ) k 1

Concluding remarks

By introducing a different fixed point application than in the seminal paper of Fernández-Cara and Zuazua, we have derived under the same assumptions a simpler proof of the exact controllability for semilinear heat equations. The fixed point application involves a linearized heat equation with a right hand side but no Within the Carleman functional setting introduced by Fursikov and Imanuvilov, precise estimates of the cost of observability with respect to the right hand side term allow to apply the Schauder fixed point theorem as soon as the Carleman parameter s is large enough. This is to our knowledge the first time that the analysis of the controllability is not based on the analysis of the cost of observability of the heat equation with potential. Moreover, assuming an additional asymptotic condition on the derivative of the nonlinear function, the fixed point application turns out to be contracting yielding to a constructive sequence of linear controls converging strongly and with a linear rate toward a control for the semilinear equation. Numerical experiments illustrate this property and also that the contraction property is amplified as the Carleman parameter s increases. This construction is original in the multi-dimensional setting: it extends and simplify the one proposed in [START_REF] Lemoine | Constructive exact control of semilinear 1d heat equation[END_REF] based on a Newton type linearization and requiring as well the parameter s to be large enough to ensure a super-linear convergence. The numerical implementation is also much simpler. We also conjecture that the linearization introduced by Fernández-Cara and Zuazua also leads, for s large enough, to a contracting application but this remains to be done.

We emphasize that the method is mainly based on global Carleman estimates, which are by now well-known and employed. Therefore, the approach can very likely be extended to many others PDEs such as the wave equation, the burgers equation (addressed from a numerical perspective in [21, Part 1, Section 4]), the Navier-Stokes system (addressed in [17, section 4]), etc, for which such estimates are available, and may lead to constructive algorithms in semi-linear cases for these systems too.

3 2

 3 sϕ * (t) .

Proposition 2 .

 2 Under the assumptions of Lemma 9, Λ s (C R (s)) is a relatively compact subset of C R (s) for the norm |||•||| s .

Figure 1 :

 1 Figure 1: Regular space-time mesh of Q T := (0, 1)×(0, T ) composed of 1744 triangles and 933 vertices; h ≈ 0.0208.

Figure 2 :

 2 Figure 2: Relative error

  ρ0

Figure 3 :

 3 Figure 3: Evolution of v k (•, t) L 2 (Ω) and y k (•, t) L 2 (Ω) w.r.t. t ∈ [0, T ] for c u0 = 10, c f = -5 and s ∈ {1, 2, 3}.

Figure 4 :

 4 Figure 4: The control v k in Q T for c u0 = 10, c f = -5 and s ∈ {1, 2, 3}.

Figure 5 :

 5 Figure 5: The controlled solution y k in Q T for c u0 = 10, c f = -5 and s ∈ {1, 2, 3}.

  for the topology induced by |||•||| s ; see Section 3.3; • Λ s is continuous on C R (s) for the topology induced by |||•||| s ; see Section 3.4.

Table 1 :

 1 c u0 = 10 ; c f = -5; Norms of (y k , v k ) w.r.t. s.

		2.43	80.50	58.24	208.52	297.56	48
	3	1.415	86.53	51.30	463.69	414.93	17
	4	1.108	173.17	52.83	1366.08	605.20	13
	5	0.9307	429.07	57.04	4328.61	889.05	11

Table 2 :

 2 c u0 = 10 ; c f = -5 ; s = 3; Norms of (y k , v k ) w.r.t. h.

	156205	1.47841	90.9285	51.4646	469.008	420.345	18
	0.0760345	1.46148	87.9869	51.2379	465.822	419.42	17
	0.044171	1.45521	87.0578	51.0243	464.527	416.886	17
	0.0208981	1.45056	86.2678	51.0448	463.253	414.223	17
	0.0117201	1.45203	86.5628	51.1068	463.723	415.114	17

Table 3 :

 3 c u0 = 20 ; s = 3; Norms of (y k , v k ) w.r.t. c f .

		1.5676	39.0933	18.4173	144.851	188.71	9.71 × 10 -5 18
	5	1.63982	41.5574	21.5396	170.772	212.189	1.04 × 10 -4 15
	4	1.71818	44.4218	25.3636	202.888	240.201	1.13 × 10 -4 13
	3	1.80253	47.8002	30.0497	242.771	273.779	1.24 × 10 -4 11
	2	1.89243	51.8744	35.7756	292.257	314.074	1.37 × 10 -4	9
	1	1.98736	56.9667	42.7222	353.371	362.474	1.52 × 10 -4	6
	0	2.08767	63.7034	51.053	428.191	420.662	1.73 × 10 -4	1
	-1	2.19716	73.375	60.8936	518.678	491.01	2.01 × 10 -4	7
	-2	2.32931	88.719	72.3206	626.54	576.832	2.42 × 10 -4	9
	-3	2.52065	115.606	85.3784	753.626	682.763	3.09 × 10 -4 12
	-4	2.85296	166.848	100.33	907.951	815.128	4.30 × 10 -4 16
	-5	3.47542	272.771	119.009	1140.26	982.953	6.70 × 10 -4 25
	-6	4.64025	523.143	148.453	1674.92	1198.33	1.22 × 10 -3 61

Table 4 :

 4 c f = -2 ; s = 2; Norms of (y k , v k ) w.r.t. c u0 .

		1.54301	19.4769	30.6487	91.8124	170.047	0.0573655	11
	100	17.4912	334.631	422.723	1323.21	2255.98	0.0857455	18
	500	107.77	3234.76	2631.81	9152.24	13811.4	0.130185	29
	1000	246.968	9077.9	5870.83	22622.8	30051.5	0.159445	38
	1500	405.983	16828	9489.64	39571.9	47314.5	0.180076	48
	2000	580.37	26235.4	13430	59621.2	65262.9	0.196388	55
	3000	965.936	49498.3	22141.2	108020	102663	0.221715	77
	4000	1391.49	78171.6	31830.1	166352	141531	0.241246	106
	5000	1850.25	111855	42375.6	233684	181553	0.257167	141

Table 5 :

 5 c f = -2 ; s = 3; Norms of (y k , v k ) w.r.t. c u0 .

		1.14031	41.0689	34.0273	292.739	271.86	2.25 × 10 -4	8
	100	12.4721	576.084	420.613	3722.92	3381.68	3.07 × 10 -4 12
	500	69.8357	4170.43	2443.36	22324.9	20055.3	4.28 × 10 -4 15
	1000	149.215	10045.6	5213.5	48679.6	43195	5.05 × 10 -4 17
	2000	322.25	24509.6	11144.3	107260	93000.8	6.05 × 10 -4 20
	3000	507.998	41520.3	17405	171395	145648	6.77 × 10 -4 22
	4000	703.063	60489.4	23901.8	239863	200195	7.34 × 10 -4 23
	5000	905.632	81095	30586.9	311987	256227	7.83 × 10 -4 24
	6000	1114.55	103128	37430.5	387325	313419	8.26 × 10 -4 25
	7000	1329.02	126440	44412	465555	371629	8.65 × 10 -4 26
	8000	1548.43	150917	51516.2	546433	430756	9.01 × 10 -4 28
	9000	1772.31	176471	58731.6	629764	490642	9.34 × 10 -4 28
	10000	2000.29	203029	66048.7	715388	551209	9.65 × 10 -4 30
	20000	4455.89	513872	143566	1.67 × 10 6	1.18 × 10 6	1.20 × 10 -3 39

Table 6 :

 6 c u0 = 10 ; c f = -2; Norms of (y k , v k ) w.r.t. s; ω = (0.2, 0.6)

		5.00473	55.4739	117.728	144.256	1301.97	20
	2	3.34631	94.7894	90.8444	358.58	1483.52	14
	3	2.91028	237.695	109.246	1060	2948.23	12
	4	2.87397	651.291	146.28	3690	5576.49	11

Table

		3.47589	35.7363	62.9227	67.9855	243.604	119
	2	1.72679	25.1665	39.8201	120.101	214.439	20
	3	1.22787	47.0941	39.477	339.374	308.99	12
	4	0.981409	112.296	43.1533	1073.88	460.349	10

Table 8 :

 8 c u0 = 10 ; c f = -1; Norms of (y k , v k ) w.r.t. s; f given by (63).

		2.16033	7.85245	15.7601	14.5863	47.9055	25
	2	1.39592	14.5911	20.81	60.618	125.071	55
	3	1.06386	33.145	27.0012	227.116	220.722	11
	4	0.878876	85.2781	33.3402	815.691	354.915	9

Acknowledgment. The first author thanks the funding by the french ANR Project TRECOS grant ANR-20-CE40-0009 (2021-2024). The third author thanks the funding by the french government research program "Investissements d'Avenir" through the IDEX-ISITE initiative 16-IDEX-0001 (CAP 20-25).

As as corollary of the previous result and the classical Banach-Picard's fixed point theorem, the contraction property of the operator Λ s for β small enough given in (50) and s and R given by Lemma 9 allows to define a convergent sequence (y k , v k ) k∈N to a controlled pair for (1) and prove the following precise version of the second item of Theorem 4: Theorem 8. Let d 5 and u 0 ∈ L ∞ (Ω). Assume that f is locally Lipschitz continuous and satisfies (H 1 ) with β given in (50), and let s and R chosen as in Lemma 9. Then, for any y 0 ∈ C R (s), the sequence (y k ) k∈N ∈ C R (s) given by

where Λ s is defined as in (40) (relying on Λ 0 s defined in Theorem 5) together with the corresponding sequence of controls (v k ) k∈N strongly converge in L 2 (ρ 0 (s), Q T ) × L 2 (ρ 0 (s), q T ) to a controlled pair solution for (1). Moreover, the convergence is at least linear.

Proof. The L 2 (ρ 0 (s), Q T ) convergence of the sequence (y k ) k∈N toward y = Λ s (y) ∈ C R (s) at a linear rate follows from the contracting property of Λ s . From the definition of Λ s in (40), let v ∈ L 2 (ρ 0 (s), q T ) associated with y so that y -

Lemma 4 and estimate (52) then give that

and therefore the convergence at a linear rate in L 2 (ρ 0 (s), q T ) of the sequence (v k ) k∈N toward the null control v for system (1).

Numerical illustrations

We illustrate in this section our results of convergence, precisely Theorem 8 by computing a sequence

,

obtained through the solution of the variational formulation [START_REF] Victor | The problem of blow-up in nonlinear parabolic equations[END_REF] with B = -f (y k-1 ). We first provide some practical details of the algorithm then discuss some experiments in the one dimensional case performed with the software Freefem++ (see [START_REF] Hecht | New development in Freefem++[END_REF]). Approximations of null controls for the (linear) heat equation is a delicate issue: we mention the seminal work [START_REF] Carthel | On exact and approximate boundary controllabilities for the heat equation: a numerical approach[END_REF] dealing with the control of minimal L 2 -norm which is very oscillatory near the final time t = T and therefore difficult to construct (see also [START_REF] Labbé | Uniform controllability of semidiscrete approximations of parabolic control systems[END_REF][START_REF] Münch | Numerical approximation of null controls for the heat equation: ill-posedness and remedies[END_REF] where this is discussed at length). On the other hand, the introduction of Carleman weights in the cost functional J corresponding to the minimization of weighted L 2 -norms (as in Remark 1) leads to robust method and strong convergent approximations with respect to the discretization parameter. We refer to [START_REF] Fernández | Strong convergent approximations of null controls for the 1D heat equation[END_REF][START_REF] Fernández | Numerical exact controllability of the 1D heat equation: duality and Carleman weights[END_REF]. For each k ∈ N , the construction of the pair (y k , v k ) is based on a suitable discretization of [START_REF] Victor | The problem of blow-up in nonlinear parabolic equations[END_REF]. Here, we display numerical evidences of the convergence as k → ∞.


Leo Gerlin

Ludovic Cottret

Sophie Cesbron

Géraldine Taghouti

Marie-Agnès Jacques

Stéphane Genin

Caroline Baroukh

Genome-scale investigation of the metabolic determinants generating bacterial fastidious growth

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Genome-scale investigation of the metabolic determinants generating bacterial fastidious growth

Leo Gerlin ⇤ 1 , Ludovic Cottret 1 , Sophie Cesbron 2 , Géraldine Taghouti 2 , Marie-Agnès Jacques 2 , Stéphane Genin 1 , Caroline Baroukh High proliferation rate and robustness are vital characteristics of bacterial pathogens to successfully colonize their hosts. The observation of drastically slow growth in some pathogens is thus paradoxical and remains unexplained. In this study, we sought to understand the strikingly slow, designated as fastidious, growth of the plant pathogen Xylella fastidiosa. Using genomescale metabolic network reconstruction, modeling and experimental validation, we explored its metabolic capabilities. Despite genome reduction and slow growth, the pathogen's metabolic network is complete but strikingly minimalist and lacking robustness. Most alternative reactions were missing, especially those favoring a fast growth, replaced by less e cient paths. We also unraveled that the production of virulence factors is ine cient and imposes a heavy burden on growth. Interestingly, some specific determinants of fastidious growth were also found in other slow-growing pathogens, enriching the view that these metabolic peculiarities are a pathogenicity strategy to remain at low population level.

  1 Institut de Recherche en Horticulture et Semences -Université d'Angers, Institut National de la Recherche Agronomique : UMR1345, Agrocampus Ouest -AGROCAMPUS OUEST, UMR1345 IRHS, F-49045 Angers, France, France

	1 LIPM -Université de Toulouse, INRA, CNRS -Chemin de Borde-Rouge -BP 27 31326 CASTANET
	TOLOSAN CEDEX, France
	2