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

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

Constructive exact control of semilinear 1D heat equations

Keywords: AMS Classifications: 35K58, 93B05, 93E24, 49M15 Semilinear heat equation, Exact controllability, Least-squares approach

Constructive exact control of semilinear 1D heat equations

Introduction

Let Ω = (0, 1), 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 -∂ xx y + g(y) = f 1 ω in Q T , y = 0 on Σ T , y(•, 0) = u 0 in Ω, (1)
where u 0 ∈ H 1 0 (Ω) is the initial state of y and f ∈ L 2 (q T ) is a control function. We assume moreover that the nonlinear function g : R → R is, at least, locally Lipschitz-continuous and, following [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF], that g satisfies |g (r)| ≤ C(1

+ |r| 5 ) ∀r ∈ R. (2)
Under this condition, (1) possesses exactly one local in time solution. Moreover, we recall in accordance with the results in [7, Section 5] that under the growth condition

|g(r)| ≤ C(1 + |r| ln(1 + |r|)) ∀r ∈ R, (3)
the solutions to (1) are globally defined in [0, T ] and one has

y ∈ C 0 ([0, T ]; H 1 0 (Ω)) ∩ L 2 (0, T ; H 2 (Ω)). ( 4
)
Without a growth condition of the kind (3), the solutions to (1) can blow up before t = T ; in general, the blow-up time depends on g and the size of u 0 L 2 (Ω) (see [START_REF] Lacey | Global blow-up of a nonlinear heat equation[END_REF]).

System ( 1) is said to be exactly controllable to trajectories at time T if, for any u 0 ∈ L 2 (Ω) and any globally defined bounded trajectory y ∈ C 0 ([0, T ]; L 2 (Ω)) (corresponding to data u 0 ∈ L 2 (Ω) and f ∈ L 2 (q T )), there exist controls f ∈ L 2 (q T ) and associated states y that are again globally defined in [0, T ] and satisfy (4) and y(x, T ) = y (x, T ), x ∈ Ω.

(5)

The uniform controllability strongly depends on the nonlinearity g. Assuming a growth condition on the nonlinearity g at infinity, this problem has been solved by Fernández-Cara and Zuazua in [17, Theorem 1.2] (which also covers the multi-dimensional case for which Ω ⊂ R d is a bounded connected open set with Lipschitz boundary).

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 . Assume that g : R → R is C 1 and satisfies [START_REF] Barbu | Exact controllability of the superlinear heat equation[END_REF] and Therefore, if |g(r)| does not grow at infinity faster than |r| ln p (1 + |r|) for any p < 3/2, then (1) is controllable. We also mention [START_REF] Barbu | Exact controllability of the superlinear heat equation[END_REF] which gives the same result assuming additional sign condition on g, namely g(r)r ≥ -C(1 + r 2 ) for all r ∈ R and some C > 0. On the contrary, if g is too "super-linear" 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 g behaves at infinity like |r| ln p (1 + |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 g, notably that g(r) > 0 for r > 0 or g(r) < 0 for r < 0, a condition not satisfied for g(r) = -r ln p (1+|r|). Eventually, in the case p > 2 for which the blow-up phenomenon can not be compensated by means of control, we also mention [START_REF] Coron | Global steady-state controllability of one-dimensional semilinear heat equations[END_REF] in the one dimensional case where a positive boundary controllability result is proved for steady-state initial and final data and for T large enough.

(
In the sequel, for simplicity, we shall assume that g(0) = 0 and that f ≡ 0, u 0 ≡ 0 so that y is the null trajectory. The proof 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 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 of the linear boundary value problem

∂ t y -∂ xx y + y g(z) = f 1 ω in Q T y = 0 on Σ T , y(•, 0) = u 0 in Ω , g (r)
:= g(r)/r r = 0 g (0) r = 0 [START_REF] Bristeau | On the numerical solution of nonlinear problems in fluid dynamics by least squares and finite element methods. I. Least square formulations and conjugate gradient[END_REF] 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 at least one 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]. The control of minimal L ∞ (q T ) norm is considered in [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF]. This allows, including in the multi-dimensional case, to obtain controlled solutions in L ∞ (Q T ).

The main goal of this work is to determine an 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 (f k ) k∈N converging strongly toward a null control for [START_REF] Badra | Local controllability to trajectories for nonhomogeneous incompressible Navier-Stokes equations[END_REF]. 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. The resulting sequence of controls (f k ) k∈N is so that f k+1 ∈ L 2 (q T ) is a null control for y k+1 solution of ∂ t y k+1 -∂ xx y k+1 + y k+1 g(y k ) = f k+1 1 ω in Q T , y k+1 = 0 on Σ T , y k+1 (•, 0) = u 0 in Ω. [START_REF] Cazenave | An introduction to semilinear evolution equations[END_REF] 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 (f 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 g is globally Lipschitz. We also refer to [START_REF] Boyer | On the penalised HUM approach and its applications to the numerical approximation of nullcontrols 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 that g ∈ C 1 (R) and g ∈ L ∞ (R), it is proved that R ∈ C 1 (Z; R + ) and satisfies, for some constant C > 0, the following inequality

(1 -C g L ∞ (R) u 0 L ∞ (Ω) ) 2R(z) ≤ R (z) L 2 (Q T ) ∀z ∈ L 2 (Q T ).
This implies that if g L ∞ (R) u 0 L ∞ (Ω) 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, 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 Picard iterates. The analysis of convergence is however not performed. As is usual for nonlinear problems and also considered in [START_REF] Fernández | Numerical null controllability of semi-linear 1-D heat equations: fixed point, least squares and Newton methods[END_REF], we may employ a Newton type method to find a zero of the mapping F : Y → W defined by

F (y, f ) = (∂ t y -∂ xx y + g(y) -f 1 ω , y(• , 0) -u 0 ) ∀(y, f ) ∈ Y ( 8
)
where the Hilbert space Y and W are defined as follows

Y := (y, f ) : ρy ∈ L 2 (Q T ), ρ 0 (∂ t y -∂ xx y) ∈ L 2 (Q T ), y = 0 on Σ T , ρ 0 f ∈ L 2 (q T )
and

W := L 2 (ρ 0 ; Q T ) × L 2 (Ω)
for some appropriates weights ρ and ρ 0 (defined in the next section). Here

L 2 (ρ 0 ; Q T ) stands for {z : ρ 0 z ∈ L 2 (Q T )}. It is shown (see [13, Section 3.3]) that, if g ∈ C 1 (R) and g ∈ L ∞ (R), then F ∈ C 1 (Y ; W ) allowing to derive the Newton iterative sequence: given (y 0 , f 0 ) in Y , define (y k , f k ) k∈N in Y N as follows (y k+1 , f k+1 ) = (y k , f k ) -(Y k , F k ) where F k is a control for Y k solution of ∂ t Y k -∂ xx Y k + g (y k ) Y k = F k 1 ω + ∂ t y k -∂ xx y k + g(y k ) -f k 1 ω in Q T , Y k = 0 on Σ T , Y k (•, 0) = u 0 -y k (•, 0) in Ω (9) such that Y k (•, T ) = -y k (•, T ) in Ω.
Numerical experiments 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 lack of convergence of the Newton method for large enough initial condition, for which the solution y is not close enough to the zero trajectory. 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.

In this article, given any initial data u 0 ∈ H 1 0 (0, 1), we design an algorithm providing a sequence (y k ) k∈N converging to a controlled solution for (1), under assumptions on g that are slightly stronger than (H 1 ). Our main result can be formulated as follows.

Theorem 2. Let T > 0 be given. Assume that g ∈ C 1 (R) satisfies g(0) = 0 and the growth condition

(H 1 ) ∃α > 0, s.t. |g (r)| ≤ α + β ln 3/2 (1 + |r|), ∀r ∈ R for some β = β (Ω, T ) > 0 small enough and (H p ) ∃p ∈ [0, 1] such that sup a,b∈R a =b |g (a)-g (b)| |a-b| p < +∞.
Then, for any u 0 ∈ H 1 0 (Ω), one can construct a sequence (y k , f k ) k∈N converging strongly to a controlled pair for [START_REF] Badra | Local controllability to trajectories for nonhomogeneous incompressible Navier-Stokes equations[END_REF]. Moreover, after a finite number of iterations, the convergence is of order at least 1 + p. This result is achieved (in the spirit of [START_REF] Münch | Numerical null controllability of the heat equation through a least squares and variational approach[END_REF] devoted to a linear case) by minimizing the non convex functional

E s (y, f ) := 1 2 ρ 0 (s)(∂ t y -∂ xx y + g(y) -f 1 ω ) 2 L 2 (Q T ) (10)
over a convex space A(s) which incorporates the initial and controllability requirement and where ρ 0 denotes a Carleman type weight parametrized by s ≥ 1 and which blows as t → T -. The least-squares functional E s measures how much a pair (y, f ) in A(s) is close to a controlled solution for [START_REF] Badra | Local controllability to trajectories for nonhomogeneous incompressible Navier-Stokes equations[END_REF]. The controllability for (1) is reflected by the fact that the global minimum of the nonnegative functional E s is zero, over all pairs (y, f ) solutions of (1) and satisfying the controllability requirement at the final time.

The paper is organized as follows. In Section 2, we derive a controllability result for a linearized heat equation with potential in L ∞ (Q T ) and source term in L 2 (Q T ). Then, in Section 3, we define the least-squares functional E s and the corresponding non convex optimization problem [START_REF] Saramito | A damped Newton algorithm for computing viscoplastic fluid flows[END_REF] over the Hilbert space A(s). We show that E s is Gateaux-differentiable over A(s) and that any critical point (y, f ) for E s for which g (y) belongs to L ∞ (Q T ) is also a zero of E s (see Proposition 2). This is done by introducing a pair (Y 1 , F 1 ) for E s (y, f ) for which E s (y, f ) • (Y 1 , F 1 ) is proportional to E s (y, f ). Then, in Section 4, assuming that the nonlinear function g is such that g is uniformly Hölder continuous with exponent p, for some p ∈ [0, 1], we determine a minimizing sequence based on (Y 1 , F 1 ) which converges strongly for s large enough to a controlled pair for the semilinear heat equation (1) (see Theorem 4). Section 5 gathers several remarks on the approach: we notably emphasize that this least-squares approach coincides with the damped Newton method one may use to find a zero of the mapping F mentioned above; this explains the super-linear convergence stated in Theorem 2. We also discuss some other linearizations of the system (1) and show that the analysis extends to the exact controllability to trajectories. We conclude in Section 6 with some perspectives, notably the extension of our analysis to the multi-dimensional case.

As far as we know, the method introduced and analyzed in this work is the first one providing an explicit, algorithmic construction of exact controls for semilinear heat equations with non Lipschitz nonlinearity. It extends the study [START_REF] Lemoine | Approximation of null controls for semilinear heat equations using a least-squares approach[END_REF] where it is assumed that g ∈ L ∞ (R) (i.e. β = 0 in (H 1 )) which allows to obtain directly a uniform bound of the observability constant. The stronger case β > 0 small considered here requires a refined analysis similar to the one recently developed by the authors in [START_REF] Bottois | Constructive exact control of semilinear multidimensional wave equations[END_REF][START_REF] Münch | Constructive exact control of semilinear 1d wave equations by a least-squares approach[END_REF] for the wave equation. The parabolic case is however much more intricate (than the hyperbolic one) as it makes appear Carleman type weights depending on the controlled solution. These works devoted to controllability problems take their roots in the works [START_REF] Lemoine | A fully space-time least-squares method for the unsteady Navier-Stokes system[END_REF][START_REF] Lemoine | Resolution of the implicit Euler scheme for the Navier-Stokes equation through a least-squares method[END_REF] concerned with the approximation of solution of Navier-Stokes type problem, through least-squares methods: they refine the analysis performed in [START_REF] Lemoine | Analysis of continuous H -1 -least-squares approaches for the steady Navier-Stokes system[END_REF][START_REF] Münch | A least-squares formulation for the approximation of controls for the Stokes system[END_REF] inspired from the seminal contribution [START_REF] Bristeau | On the numerical solution of nonlinear problems in fluid dynamics by least squares and finite element methods. I. Least square formulations and conjugate gradient[END_REF].

Notations. Throughout, we denote by • ∞ the usual norm in L ∞ (R), by (•, •) X the scalar product of X (if X is a Hilbert space) and by •, • X,Y the duality product between X and Y .

Given any p ∈ [0, 1], we introduce for any g ∈ C 1 (R) the following hypothesis :

(H p ) [g ] p := sup a,b∈R a =b |g (a)-g (b)| |a-b| p < +∞
meaning, for p ∈ (0, 1], that g is uniformly Hölder continuous with exponent p. For p = 0, by extension, we set [g ] 0 := 2 g ∞ . In particular, g satisfies (H 0 ) if and only if g ∈ C 1 (R) and g ∈ L ∞ (R), and g satisfies (H 1 ) if and only if g is Lipschitz continuous (in this case, g is almost everywhere differentiable and g ∈ L ∞ (R)), and we have [g ] 1 ≤ g ∞ . We also denote by C a positive constant depending only on Ω, ω and T that may vary from lines to lines.

A controllability result for a linearized heat equation with potential and right hand side

This section is devoted to a controllability result for a linear heat equation with potential in L ∞ (Q T ) and right hand side in L 2 (ρ 0 (s); Q T ) for a precise weight ρ 0 (s) parametrized by s ∈ R + defined in the sequel. More precisely we are interested in the existence of a control v such that the solution z of

∂ t z -∂ xx z + Az = v1 ω + B in Q T , z = 0 on Σ T , z(•, 0) = z 0 in Ω (11) satisfies z(•, T ) = 0 in Ω. ( 12
)
We follow the usual strategy of [START_REF] Fursikov | Controllability of evolution equations[END_REF] to construct a solution of the null controllability problem, using Carleman type estimates. Instead of using the classical 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 non zero initial data as the weight function does not blow up as t → 0.

Carleman estimate

For any s ≥ 1, we consider the weight functions ρ(s) = ρ(x, t, s), ρ 0 (s) = ρ 0 (x, t, s) and ρ 1 (s) = ρ 1 (x, t, s) which are continuous, strictly positives and in L ∞ (Q T -δ ) for any δ > 0. Precisely, we use the weights introduced in [1]: ρ 0 (s) = ξ -3/2 ρ(s), ρ 1 (s) = ξ -1 ρ(s), ρ 2 (s) = ξ -1/2 ρ(s) where ρ(s) and ξ are defined, for all s ≥ 1 and λ ≥ 1, as follows

ρ(x, t, s) = exp(sϕ(x, t)), ξ(x, t) = θ(t) exp(λ ψ(x)) (13)
where θ ∈ C 2 ([0, T )) is defined such that, noting µ = sλ 2 e 2λ and 0 < T 1 < min( 1 4 , 3T 8 ),

θ(t) =                  1 + 1 - 4t T µ ∀t ∈ [0, T /4] 1 ∀t ∈ [T /4, T -2T 1 ] θ is increasing on [T -2T 1 , T -T 1 ], 1 T -t ∀t ∈ [T -T 1 , T ) (14)
and ϕ ∈ C 1 ([0, T )) is defined by

ϕ(x, t) = θ(t) λ exp(12λ) -exp(λ ψ(x)) (15)
with ψ = ψ + 6, where ψ ∈ C 1 (Ω) satisfies ψ ∈ (0, 1) in Ω, ψ = 0 on ∂Ω and |∂ x ψ(x)| > 0 in Ω\ω. We emphasize that the weights blow up at t → T -but not as t → 0 + .

Remark 1. We shall use in the sequel that

1 < ρ 0 ≤ ρ 1 ≤ ρ in Q T . Indeed, since ξ ≥ 1, ρ 0 ≤ ρ 1 ≤ ρ.
Moreover, for all (x, t) ∈ Q T and λ ≥ 1, we check that ϕ(x, t) ≥ 3 2 ξ(x, t) and thus, since s ≥ 1 and ξ(x, t) ≥ 1, we get

ρ 0 (x, t, s) = ξ -3/2 (x, t)ρ(x, t, s) ≥ ξ -3/2 (x, t) exp 3 2 sξ(x, t) ≥ e 3/2s ∀(x, t) ∈ Q T .
The controllability property for the linear system [START_REF] Doubova | On the controllability of parabolic systems with a nonlinear term involving the state and the gradient[END_REF] is based on the following Carleman estimate.

Lemma 1. Let P 0 := {q ∈ C 2 (Q T ) : q = 0 on Σ T }. There exist λ 0 ≥ 1 and s 0 ≥ 1 such for all λ ≥ λ 0 and for all s ≥ max(

A 2/3 L ∞ (Q T ) , s 0 ), the following Carleman estimate holds Ω ρ -2 (0, s)|∂ x 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)|∂ x p| 2 + s 3 λ 4 Q T ρ -2 0 (s)|p| 2 ≤ C Q T ρ -2 (s)| -∂ t p -∂ xx p + Ap| 2 + Cs 3 λ 4 q T ρ -2 0 (s)|p| 2 , ∀p ∈ P 0 . (16)
Proof. This estimate is deduced from the one obtained in [1, Theorem 2.5] devoted to the case A ≡ 0: there exist λ 1 ≥ 1 and s 0 ≥ 1 such that for every smooth function z on Q T satisfying z = 0 on Σ T and for all s ≥ s 0 and λ ≥ λ

1 Ω ρ -2 (0, s)|∂ x 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)|∂ x p| 2 + s 3 λ 4 Q T ρ -2 0 (s)|p| 2 ≤ C Q T ρ -2 (s)|∂ t p + ∂ xx p| 2 + Cs 3 λ 4 q T ρ -2 0 (s)|p| 2 .
Writing

Q T ρ -2 (s)|∂ t p + ∂ xx p| 2 ≤ 2 Q T ρ -2 (s)| -∂ t p -∂ xx p + Ap| 2 + 2 Q T ρ -2 (s)|Ap| 2 ≤ 2 Q T ρ -2 (s)| -∂ t p -∂ xx p + Ap| 2 + 2 A 2 L ∞ (Q T ) Q T ρ -2 (s)|p| 2 we infer, since ρ 0 ≤ ρ, that Ω ρ -2 (0, s)|∂ x 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)|∂ x p| 2 + s 3 λ 4 Q T ρ -2 0 (s)|p| 2 ≤ C Q T ρ -2 (s)| -∂ t p -∂ xx p + Ap| 2 + C A 2 L ∞ (Q T ) Q T ρ -2 0 (s)|p| 2 + Cs 3 λ 4 q T ρ -2 0 (s)|p| 2 .
Taking λ ≥ λ 0 = max(λ 1 , (2C) 1/4 ) and s ≥ max(

A 2/3
L ∞ (Q T ) , s 0 ) leads to [START_REF] Fernández | The cost of approximate controllability for heat equations: the linear case[END_REF].

In the sequel we assume that λ = λ 0 and denote by C any constant depending only on Ω, ω, λ 0 and T .

We then define and check that the bilinear form (p, q) P :=

Q T ρ -2 (s)L A p L A q + s 3 λ 4 0 q T ρ -2 0 (s)p q
where L A q := -∂ t q -∂ xx q + Aq for all q ∈ P 0 is a scalar product on P 0 (see [START_REF] Fernández | Strong convergent approximations of null controls for the 1D heat equation[END_REF]). The completion P of P 0 for the norm • P associated with this scalar product is a Hilbert space. By density arguments, [START_REF] Fernández | The cost of approximate controllability for heat equations: the linear case[END_REF] remains true for all p ∈ P , that is, for

λ = λ 0 , Ω ρ -2 (0, s)|∂ x p(0)| 2 +s 2 λ 3 0 e 14λ0 Ω ρ -2 (0, s)|p(0)| 2 +sλ 2 0 Q T ρ -2 2 (s)|∂ x p| 2 +s 3 λ 4 0 Q T ρ -2 0 (s)|p| 2 ≤ C p 2 P (17) for all s ≥ max( A 2/3 L ∞ (Q T ) , s 0 ).
Remark 2. We denote by P (instead of P A ) the completion of P 0 for the norm • P since P does not depend on A (see [START_REF] Fernández | Numerical null controllability of semi-linear 1-D heat equations: fixed point, least squares and Newton methods[END_REF]Lemma 3.1]).

Lemma 2. Let s ≥ max( A 2/3 L ∞ (Q T ) , s 0 ).
There exists a unique solution p ∈ P of (p, q) P = Ω z 0 q(0)

+ Q T Bq, ∀q ∈ P. ( 18
)
This solution satisfies the following estimate (with c := ϕ(•, 0) L ∞ (Ω) )

p P ≤ Cs -3/2 ρ 0 (s) B L 2 (Q T ) + e cs z 0 L 2 (Ω) . (19)
Proof. The linear map L 1 :

P → R, q → Q T Bq is continuous. Indeed, for all q ∈ P 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 the Carleman estimate [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF] we have

Q T |ρ -1 0 (s)q| 2 1/2 ≤ Cs -3/2 q P , therefore |L 1 (q)| = Q T Bq ≤ Cs -3/2 ρ 0 (s)B L 2 (Q T ) q P .
Thus L 1 is continuous.

From [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF] we deduce that the linear map L 2 : P → R, q → Ω z 0 q(0) is continuous. Indeed, noting c := ϕ(•, 0) ∞ and using s ≥ 1, we obtain for all q ∈ P that:

|L 2 (q)| = s -3/2 e cs z 0 L 2 (Ω) s 3/2 e -cs q(0) L 2 (Ω) ≤ s -3/2 e cs z 0 L 2 (Ω) s 3/2 q(0)e -sϕ(x,0) L 2 (Ω) = s -3/2 e cs z 0 L 2 (Ω) s 3/2 ρ -1 (0, s)q(0) L 2 (Ω) ≤ Cs -3/2 e cs z 0 L 2 (Ω) q P .
Using Riesz's theorem, we conclude that there exists exactly one solution p ∈ P of ( 18) and this solution satisfies [START_REF] Glowinski | Exact and approximate controllability for distributed parameter systems[END_REF].

Application to controllability

We now show how Lemma 1 and Lemma 2 imply a controllability result for the linear system 11. This part is mainly classical and follows closely [START_REF] Fursikov | Controllability of evolution equations[END_REF] but we present it with some details as it is an essential part of our arguments developed in Section 3. The main result giving a precise bound of a control pair in term of the potential and right hand side is as follows.

Theorem 3. Assume A ∈ L ∞ (Q T ), s ≥ max( A 2/3 L ∞ (Q T ) , s 0 ), B ∈ L 2 (ρ 0 (s), Q T ) and z 0 ∈ L 2 (Ω).
Then there exists a control v ∈ L 2 (ρ 0 (s); q T ) such that the weak solution z of ( 11) satisfies [START_REF] Duyckaerts | On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials[END_REF].

Moreover, the unique control v which minimizes together with the corresponding solution z the functional J :

L 2 (ρ(s); Q T )×L 2 (ρ 0 (s); q T ) → R + defined by J(z, v) := 1 2 ρ(s) z 2 L 2 (Q T ) + s -3 λ -4 0 2 ρ 0 (s) v 2 L 2 (q T )
satisfies the following estimates

ρ(s) z L 2 (Q T ) + s -3/2 λ -2 0 ρ 0 (s) v L 2 (q T ) ≤ Cs -3/2 ρ 0 (s)B L 2 (Q T ) + e cs z 0 L 2 (Ω) ( 20
)
with c := ϕ(•, 0) L ∞ (Ω) and ρ 1 (s)z L ∞ (0,T ;L 2 (Ω)) + ρ 1 (s)∂ x z L 2 (Q T ) ≤ C 1 (s, A) ρ 0 (s)B L 2 (Q T ) + e cs z 0 L 2 (Ω) (21)
where

C 1 (s, A) := Cs -1/2 (1 + A 1/2 L ∞ (Q T ) ). Moreover, if z 0 ∈ H 1 0 (Ω) then z ∈ L ∞ (Q T ) and z L ∞ (Q T ) ≤ Ce -3 2 s (1 + A L ∞ (Q T ) ) ρ 0 (s)B L 2 (Q T ) + e cs z 0 H 1 0 (Ω) . (22)
We refer to [START_REF] Fernández | The cost of approximate controllability for heat equations: the linear case[END_REF][START_REF] Fursikov | Controllability of evolution equations[END_REF] for estimates of the null control of minimal L 2 (q T )-norm (corresponding to ρ 0 ≡ 1 and ρ = 0) in the case B ≡ 0, refined later on in [START_REF] Duyckaerts | On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials[END_REF][START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF]. For simplicity, we divide the proof of Theorem 3 in several technical lemma. We introduce the convex set 11)-( 12) in the transposition sense .

C(z 0 , T ) := (z, v) : ρ(s)z ∈ L 2 (Q T ), ρ 0 (s)v ∈ L 2 (q T ), (z, v) solves (
Let us remark that (z, v) solves ( 11)-( 12) in the transposition sense if (z, v) is solution of

Q T zL A q = q T vq + Ω z 0 q(0) + Q T Bq, ∀q ∈ P. ( 23
) Therefore if (z, v) ∈ C(z 0 , T ), then since v ∈ L 2 (q T ) and B ∈ L 2 (Q T )
, z coincides with the unique weak solution of ( 11) associated with v. We can now claim that C(z 0 , T ) is non empty.

Lemma 3. Let s ≥ max( A 2/3 L ∞ (Q T )
, s 0 ), p ∈ P the unique solution of ( 18) and (z, v) be defined by

z = ρ -2 (s)L A p and v = -s 3 λ 4 0 ρ -2 0 (s)p| q T . (24)
Then (z, v) ∈ C(z 0 , T ) and satisfies the estimate [START_REF] Lacey | Global blow-up of a nonlinear heat equation[END_REF].

Proof. From the definition of P , ρ(s

)z ∈ L 2 (Q T ) and ρ 0 (s)v ∈ L 2 (q T ) and therefore, since ρ ≥ ρ 0 > 1, z ∈ L 2 (Q T ) and v ∈ L 2 (q T ).
In view of ( 18), (z, v) is solution of ( 23) and satisfies [START_REF] Lacey | Global blow-up of a nonlinear heat equation[END_REF] that is, z is the solution of ( 11)-( 12) associated with v in the transposition sense. Thus (z, v) ∈ C(z 0 , T ).

Let us now consider the following extremal problem, introduced by Fursikov and Imanuvilov in [START_REF] Fursikov | Controllability of evolution equations[END_REF]

     Minimize J s (z, v) = 1 2 Q T ρ 2 (s)|z| 2 + s -3 λ -4 0 2 q T ρ 2 0 (s)|v| 2 Subject to (z, v) ∈ C(z 0 , T ). (25)
Then (z, v) → J s (z, v) is strictly convex and continuous on L 2 (ρ(s); Q T ) × L 2 (ρ 0 (s); q T ). Therefore [START_REF] Lemoine | Analysis of continuous H -1 -least-squares approaches for the steady Navier-Stokes system[END_REF] possesses at most a solution in C(z 0 , T ). More precisely we have :

Lemma 4. Let s ≥ max( A 2/3 L ∞ (Q T ) , s 0 ). Then (z, v) ∈ C(z 0 , T ) defined in Lemma 3 is the unique solution of (25). Proof. Let (y, w) ∈ C(z 0 , T ). Since J s is convex and differentiable on L 2 (ρ(s); Q T ) × L 2 (ρ 0 (s); q T ) we have: J s (y, w) ≥ J s (z, v) + Q T ρ 2 (s)z(y -z) + s -3 λ -4 0 q T ρ 2 0 (s)v(w -v) = J s (z, v) + Q T L A p(y -z) - q T p(w -v) = J s (z, v)
y being the solution of ( 11) associated with w in the transposition sense.

Proof of Theorem 3. Lemma 4 gives the existence of a control v ∈ L 2 (ρ 0 (s); q T ) such that the corresponding solution z of ( 11) satisfies [START_REF] Duyckaerts | On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials[END_REF]. Moreover, this control is the unique control which minimizes together with the corresponding solution z the functional J s and satisfies [START_REF] Lacey | Global blow-up of a nonlinear heat equation[END_REF]. To finish the proof of Theorem 3, it suffices to prove that (z, v) satisfies the estimate [START_REF] Le Balc'h | Global null-controllability and nonnegative-controllability of slightly superlinear heat equations[END_REF] and if z 0 ∈ H 1 0 (Ω), then z ∈ L ∞ (Q T ) and satisfies the estimate [START_REF] Lemoine | Approximation of null controls for semilinear heat equations using a least-squares approach[END_REF].

Multiplying (11) by ρ 2 1 (s)z and integrating by part we obtain

1 2 Ω (∂ t |z| 2 ) ρ 2 1 (s) + Ω ρ 2 1 (s)|∂ x z| 2 + 2 Ω ρ 1 (s)z∂ x ρ 1 (s) • ∂ x z + Ω ρ 2 1 (s)Azz = ω vρ 2 1 (s)z + Ω Bρ 2 1 (s)z. ( 26
) But Ω (∂ t |z| 2 )ρ 2 1 (s) = ∂ t Ω |z| 2 ρ 2 1 (s) -2 Ω |z| 2 ρ 1 (s)∂ t ρ 1 (s) and ∂ t ρ 1 (s) = -∂tθ θ ρ 1 (s) + s ∂tθ θ ϕρ 1 (s).
Moreover, from the definition of θ and ϕ we have:

∂ t θ θ (t) ≤      Cs ∀t ∈ [0, T /4] 0 ∀t ∈ [T /4, T -2T 1 ] θ ∀t ∈ [T -T 1 , T ) and ∂tθ θ (t) ≤ C ∀t ∈ [T -2T 1 , T -T 1 ] since ∂ t θ θ (T -2T 1 ) = 0, ∂ t θ θ (T -T 1 ) = 1 T1 and θ is C 2 . Since θ ≤ ξ and s ≥ 1, on [0, T ): ∂ t θ θ ≤ Csξ.
From the definition of ϕ, ϕ ≤ Cθ ≤ Cξ and thus on [0, T ) :

∂ t θ θ ϕ ≤ Csξ 2 .
Thus, since s ≥ 1, ξ ≥ 1 and ρ(s) = ξρ 1 (s), on [0, T ) :

-

Ω |z| 2 ρ 1 (s)∂ t ρ 1 (s) = Ω ∂ t θ θ |ρ 1 (s)z| 2 -s Ω ∂ t θ θ ϕ|ρ 1 (s)z| 2 ≤ Cs 2 Ω ρ 2 (s)|z| 2 .
On the other hand

∂ x ρ 1 (s) = ∂ x (ξ -1 ρ(s)) = ∂ x (ξ -1 )ρ(s) + ξ -1 ∂ x ρ(s) = -∂ x ψ λ 0 ξ -1 ρ(s) + sλ 0 ρ(s) = -∂ x ψλ 0 ρ(s) ξ -1 + s
and thus, since ξ ≥ 1 and s ≥ 1, we write (ξ -1 + s) ≤ 2s and

Ω ρ 1 (s)z∂ x ρ 1 (s) • ∂ x z ≤ 2 ∂ x ψ L ∞ (Ω) λ 0 Ω s|ρ(s)z| |ρ 1 (s)∂ x z| ≤ Cs 2 Ω |ρ(s)z| 2 + 1 2 Ω |ρ 1 (s)∂ x z| 2 .
We also have, since ρ(s) = ξρ 1 (s) and ξ ≥ 1 the estimate,

Ω ρ 2 1 (s)Azz ≤ C A L ∞ (Q T ) ρ(s)z 2 L 2 (Ω) . Finally, since ρ 2 1 (s) = ξ -1/2 ρ 0 (s)ρ(s) and ξ -1/2 ≤ 1, we infer that ω vρ 2 1 (s)z ≤ ω ρ 0 (s)vξ -1/2 ρ(s)z ≤ ω |ρ 0 (s)v| 2 1/2 ρ(s)z L 2 (Ω)
and

| Ω Bρ 2 1 (s)z| ≤ ρ 0 (s)B L 2 (Ω) ρ(s)z L 2 (Ω)
. Thus [START_REF] Münch | A least-squares formulation for the approximation of controls for the Stokes system[END_REF] implies that

∂ t Ω ρ 2 1 (s)|z| 2 + Ω ρ 2 1 (s)|∂ x z| 2 ≤C s 2 + A L ∞ (Q T ) ρ(s)z 2 L 2 (Ω) + ω |ρ 0 (s)v| 2 1/2 + ρ 0 (s)B L 2 (Ω) ρ(s)z L 2 (Ω)
and therefore for all t ∈ [0, T ), since ρ 1 (s, 0)z 0 2

L 2 (Ω) ≤ e cs z 0 2 L 2 (Ω) , we get Ω ρ 2 1 (s)|z| 2 (t) + Qt ρ 2 1 (s)|∂ x z| 2 ≤ C s 2 + A L ∞ (Q T ) ρ(s)z 2 L 2 (Q T ) + ρ 0 (s)v L 2 (q T ) + ρ 0 (s)B L 2 (Q T ) ρ(s)z L 2 (Q T ) + e cs z 0 2 L 2 (Ω) .
Using [START_REF] Lacey | Global blow-up of a nonlinear heat equation[END_REF], we obtain since s ≥ 1 that for all t ∈ [0, T ) :

Ω ρ 2 1 (s)|z| 2 (t) + Qt ρ 2 1 (s)|∂ x z| 2 ≤ Cs -1 1 + A L ∞ (Q T ) ρ 0 (s)B 2 L 2 (Q T ) + e cs z 0 2 L 2 (Ω) (27)
which gives [START_REF] Le Balc'h | Global null-controllability and nonnegative-controllability of slightly superlinear heat equations[END_REF].

If now z 0 ∈ H 1 0 (Ω), then since z is a weak solution of (11) associated with v, maximal regularity in L 2 and Sobolev embeddings give that z ∈ L 2 (0, T ;

H 2 (Ω)), ∂ t z ∈ L 2 (Q T ) and therefore z ∈ L ∞ (Q T ).
Moreover, multiplying [START_REF] Doubova | On the controllability of parabolic systems with a nonlinear term involving the state and the gradient[END_REF] by ∂ xx z and integrating by part we obtain

1 2 ∂ t Ω |∂ x z| 2 + Ω |∂ xx z| 2 ≤ -Az + v1 ω + B L 2 (Ω) ∂ xx z L 2 (Ω)
and thus, since ρ(s) ≥ ρ 0 (s) ≥ e

∂ t Ω |∂ x z| 2 ≤ Ω | -Az + v1 ω + B| 2 ≤ 3 Ω |Az| 2 + Ω |v1 ω | 2 + Ω |B| 2 ≤ 3e -3s A 2 L ∞ (Q T ) ρ(s)z 2 L 2 (Ω) + ρ 0 (s)v1 ω 2 L 2 (Ω) + ρ 0 (s)B 2 L 2 (Ω)
which gives, a.e in t ∈ (0, T )

Ω |∂ x z| 2 (t) ≤ 3e -3s A 2 L ∞ (Q T ) ρ(s)z 2 L 2 (Q T ) + ρ 0 (s)v 2 L 2 (q T ) + ρ 0 (s)B 2 L 2 (Q T ) + z 0 2 H 1 0 (Ω) . Since z(•, t) ∈ H 1 0 (Ω), a.e in t ∈ (0, T ), z(x, t) = x 0 ∂ x z(r, t)dr ≤ ∂ x z 2 (t)
and thus, using [START_REF] Lacey | Global blow-up of a nonlinear heat equation[END_REF], since s ≥ 1 :

z L ∞ (Q T ) ≤ ∂ x z L ∞ (0,T ;L 2 (Ω)) ≤ √ 3e -3 2 s A L ∞ (Q T ) ρ(s)z L 2 (Q T ) + ρ 0 (s)v L 2 (q T ) + ρ 0 (s)B L 2 (Q T ) + z 0 H 1 0 (Ω) ≤ Ce -3 2 s (1 + A L ∞ (Q T ) ) ρ 0 (s)B L 2 (Q T ) + e cs z 0 H 1 0 (Ω)
that is [START_REF] Lemoine | Approximation of null controls for semilinear heat equations using a least-squares approach[END_REF].

Remark 3. Remark that c = ϕ(•, 0) L ∞ (Ω) > 3/2 so that the bound (22) of z L ∞ (Q T )
is not uniform with respect to the parameter s ≥ 1.

The least-squares method

In this section, we assume that the nonlinear function g satisfies the hypothesis (H p ) for some p ∈ [0, 1] and that (H 2 ) There exists α ≥ 0 and β > 0 such that |g (r)| ≤ α + β ln 3/2 (1 + |r|) for every r in R.

We introduce the notation ψ(r)

:= α + β ln 3/2 (1 + |r|), ∀r ∈ R. ( 28
)
We also assume that g(0) = 0 leading in particular to the estimate |g(r)| ≤ |r|(α + β ln 3/2 (1 + |r|)) for every r ∈ R. The case p = 0 in (H p ) corresponds to β = 0 and α = g ∞ and thus ψ(r) ≤ g ∞ for every r ∈ R. Remark that (H 2 ) implies ( 2) and (H 1 ).

The least-squares method

We introduce, for all s ≥ s 0 , the vector space A 0 (s)

A 0 (s) := (y, f ) : ρ(s) y ∈ L 2 (Q T ), ρ 0 (s)f ∈ L 2 (q T ), ρ 0 (s)(∂ t y -∂ xx y) ∈ L 2 (Q T ), y(•, 0) = 0 in Ω, y = 0 on Σ T
where ρ(s), ρ 1 (s) and ρ 0 (s) are defined in [START_REF] Fernández | Numerical null controllability of semi-linear 1-D heat equations: fixed point, least squares and Newton methods[END_REF]. Note that A 0 (s) endowed with the scalar product

(y, f ), (y, f ) A0(s) := ρ(s)y, ρ(s)y L 2 (Q T ) + ρ 0 (s)f, ρ 0 (s)f L 2 (q T ) + ρ 0 (s)(∂ t y -∂ xx y), ρ 0 (s)(∂ t y -∂ xx y) L 2 (Q T )
is a Hilbert space. The corresponding norm is (y, f ) A0(s) := ((y, f ), (y, f )) A0(s) . We also consider the affine space

A(s) := (y, f ) : ρ(s) y ∈ L 2 (Q T ), ρ 0 (s)f ∈ L 2 (q T ), ρ 0 (s)(∂ t y -∂ xx y) ∈ L 2 (Q T ), y(•, 0) = u 0 in Ω, y = 0 on Σ T (29)
so that we can write A(s) = (y, f ) + A 0 (s) for any element (y, f ) ∈ A(s). We endow A(s) with the metric d A(s) induced by the norm of A 0 (s), that is

d A(s) (y 1 , f 1 ), (y 2 , f 2 ) = (y 1 -y 2 , f 1 -f 2 ) A0(s) , for all (y 1 , f 1 ) ∈ A(s) and (y 2 , f 2 ) ∈ A(s). Clearly, if (y, f ) ∈ A(s), then y ∈ C([0, T ]; L 2 (Ω)) and since ρ(s) y ∈ L 2 (Q T ), then y(•, T ) = 0.
The null controllability requirement is therefore incorporated in the spaces A 0 (s) and A(s).

Remark 4. For any (y, f ) ∈ A(s), since ρ 0 (s) ≥ 1 (see Remark 1), we get that

∂ t y -∂ xx y ∈ L 2 (Q T ); since u 0 ∈ H 1 0 (Ω), standard arguments imply that y ∈ L ∞ (Q T ) with y L ∞ (Q T ) ≤ C u 0 H 1 0 (Ω) + ∂ t y -∂ xx y L 2 (Q T ) .
In particular, for any (y, f

) ∈ A 0 (s), y L ∞ (Q T ) ≤ Ce -3 2 s (y, f ) A0(s) for some C > 0 independent of s.
For any fixed (y 0 , f 0 ) ∈ A(s) and s ≥ 0, we can now consider the following non convex extremal problem:

inf (y,f )∈A0(s) E s (y 0 + y, f 0 + f ) ( 30
)
where the least-squares functional E s : A(s) → R is defined as follows

E s (y, f ) := 1 2 ρ 0 (s) ∂ t y -∂ xx y + g(y) -f 1 ω 2 L 2 (Q T ) . (31)
We check that ρ 0 (s)g(y) ∈ L 2 (Q T ) for any (y, f ) ∈ A(s) so that E s is well-defined. Precisely, using that |g(r)| ≤ |r| α + β ln 3/2 (1 + |r|) = |r|ψ(r) for every r and that ρ 0 ≤ ρ, we write

ρ 0 (s)g(y) L 2 (Q T ) ≤ ρ 0 (s)|y|ψ(y) L 2 (Q T ) ≤ ψ y L ∞ (Q T ) ρ(s)y L 2 (Q T ) ≤ ψ y L ∞ (Q T ) (y, f ) A(s) . (32)
Any pair (y, f ) ∈ A for which E s (y, f ) vanishes is a controlled pair of (1), and conversely. In this sense, the functional E s is a so-called error functional which measures the deviation of (y, f ) from being a solution of the underlying nonlinear equation. Moreover, although the hypothesis (H 2 ) is stronger than (H 1 ), Theorem 1 proved in [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF] does not imply the existence of zero of E s in A(s), since controls of minimal L ∞ (q T ) norm are considered in [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF]. Nevertheless, our constructive approach will show that, for s large enough, the extremal problem (30) admits solutions (y, f ) ∈ A(s) for which E s vanishes.

We also emphasize that the L 2 (Q T ) norm in E s indicates that we are looking for regular weak solutions of the parabolic equation [START_REF] Badra | Local controllability to trajectories for nonhomogeneous incompressible Navier-Stokes equations[END_REF]. We refer to [START_REF] Lemoine | Approximation of null controls for semilinear heat equations using a least-squares approach[END_REF] devoted to the case g ∈ L ∞ (R) and the multidimensional case where the L 2 (0, T ; H -1 (Ω)) norm is considered leading to weaker solutions.

A practical way of taking a functional to its minimum is through some use of its derivative. In doing so, the presence of local minima is always something that may dramatically spoil the whole scheme. The unique structural property that discards this possibility is the convexity of the functional E s . However, for nonlinear equation like [START_REF] Badra | Local controllability to trajectories for nonhomogeneous incompressible Navier-Stokes equations[END_REF], one cannot expect this property to hold for the functional E s . Nevertheless, we are going to construct a minimizing sequence which always converge to a zero of E s . To do so, we introduce the following definition. Definition 1. For any s large enough and (y, f ) ∈ A(s), we define the unique pair

(Y 1 , F 1 ) ∈ A 0 (s) solution of ∂ t Y 1 -∂ xx Y 1 + g (y)Y 1 = F 1 1 ω + ∂ t y -∂ xx y + g(y) -f 1 ω in Q T , Y 1 = 0 on Σ T , Y 1 (•, 0) = 0 in Ω ( 33
)
and which minimizes the functional J s defined in Theorem 3. In the sequel, it is called the minimal controlled pair.

The next proposition shows that there do exists some (Y 1 , F 1 ) in A 0 (s). We emphasize that F 1 is a null control for the solution Y 1 . Preliminary, we prove the following result.

Lemma 5. There exists (y,

f ) ∈ L 2 (Q T ) × L 2 (q T ) such that (y, f ) ∈ A(s) for all s ≥ 0.
Proof. Let y be the solution of

∂ t y -∂ xx y = 0 in Q T , y = 0 on Σ T , y (•, 0) = u 0 ∈ H 1 0 (Ω) in Ω, so that y ∈ L 2 (0, T ; H 2 (Ω)) and ∂ t y ∈ L 2 (0, T ; L 2 (Ω)). Let now any function φ ∈ C ∞ ([0, T ]), 0 ≤ φ ≤ 1 such that φ(0) = 1 and φ ≡ 0 in [T /2, T ].
Then, we easily check that the pair (y, 0) with y := φ y belongs to A(s) for any s ≥ 0.

Proposition 1. Let (y, f ) ∈ A(s) with s ≥ max g (y) 2/3 L ∞ (Q T ) , s 0 .
There exists a minimal controlled pair (Y 1 , F 1 ) ∈ A 0 (s) solution of (33). It satisfies the estimate:

(Y 1 , F 1 ) A0(s) ≤ C E s (y, f ) (34)
for some C > 0.

Proof. For all (y, f

) ∈ A(s), ρ 0 (s)(∂ t y -∂ xx y + g(y) -f 1 ω ) ∈ L 2 (Q T ).
The existence of a null control F 1 is therefore given by Proposition 3. Choosing the control F 1 which minimizes together with the corresponding solution Y 1 the functional J s defined in Theorem 3, we get from ( 20)-( 21) the following estimate (recall that Y 1 (•, 0) = 0) :

ρ(s) Y 1 L 2 (Q T ) + s -3/2 λ -2 0 ρ 0 (s)F 1 L 2 (q T ) ≤ Cs -3/2 ρ 0 (s)(∂ t y -∂ xx y + g(y) -f 1 ω ) L 2 (Q T ) ≤ Cs -3/2 E s (y, f ). ( 35
)
Eventually, from the equation solved by Y 1 ,

ρ 0 (s)(∂ t Y 1 -∂ xx Y 1 ) L 2 (Q T ) ≤ ρ 0 (s)F 1 L 2 (q T ) + ρ 0 (s)g (y)Y 1 L 2 (Q T ) + ρ 0 (s)(∂ t y -∂ xx y + g(y) -f 1 ω ) L 2 (Q T ) ≤ ρ 0 (s)F 1 L 2 (q T ) + ρ 0 (s)g (y)Y 1 L 2 (Q T ) + 2E s (y, f ).
But, since ρ 0 (s) ≤ ρ(s), using (35), we have

ρ 0 (s)g (y)Y 1 2 L 2 (Q T ) ≤ g (y) 2 L ∞ (Q T ) ρ(s)Y 1 2 L 2 (Q T ) ≤ Cs -3 g (y) 2 L ∞ (Q T ) E s (y, f ) (36) thus ρ 0 (s)(∂ t Y 1 -∂ xx Y 1 ) L 2 (Q T ) ≤ C 1 + s -3/2 g (y) L ∞ (Q T ) E s (y, f ) (37)
which proves that (Y 1 , F 1 ) belongs to A 0 (s). Eventually,

(Y 1 , F 1 ) 2 A0(s) = ρ(s)Y 1 2 L 2 (Q T ) + ρ 0 (s)F 1 1 ω 2 L 2 (Q T ) + ρ 0 (s)(∂ t Y 1 -∂ xx Y 1 ) 2 L 2 (Q T ) ≤ ρ(s)Y 1 2 L 2 (Q T ) + 4 ρ 0 (s)F 1 1 ω 2 L 2 (Q T ) + 3 ρ 0 (s)g (y)Y 1 2 L 2 (Q T ) + 3( 2E s (y, f )) 2 ≤ C s -3 E s (y, f ) + s -3 g (y) 2 L ∞ (Q T ) E s (y, f ) + E s (y, f ) ≤ CE s (y, f )(1 + s -3 + s -3 g (y) 2 L ∞ (Q T ) ).
Using that s ≥ max( g (y)

2/3

L ∞ (Q T ) , s 0 ) ≥ 1, we get s -3 ≤ 1 and s -3 g (y) 2 L ∞ (Q T ) ≤ 1 leading to the result. Remark 5. From (33), we observe that z := y -Y 1 ∈ L 2 (ρ(s); Q T ) is a null controlled solution satisfying ∂ t z -∂ xx z + g (y)z = (f -F 1 )1 ω + g (y)y -g(y) in Q T , z = 0 on Σ T , z(•, 0) = u 0 in Ω ( 38
)
by the control (f -F 1 ) ∈ L 2 (ρ 0 (s); q T ).

Remark 6. We emphasize that the presence of a right hand side term in (33), namely ∂ t y -∂ xx y + g(y) -f 1 ω , forces us to introduce the non trivial weights ρ 0 (s), ρ 1 (s) and ρ(s) in the space A(s). This can be seen in the equality (18): since ρ -1 0 (s)q belongs to L 2 (Q T ) for all q ∈ P , we need to impose that ρ 0 (s)B ∈ L 2 (Q T ) with here B = ∂ t y -∂ xx y + g(y) -f 1 ω . Working with the linearized equation (6) (introduced in [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF]) which does not make appear any right hand side, we may avoid the introduction of Carleman type weights. Actually, [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF] considers controls of minimal L ∞ (q T ) norm. Introduction of weights allows however the characterization [START_REF] Fursikov | Controllability of evolution equations[END_REF], which is very convenient at the practical level. We refer to [START_REF] Fernández | Strong convergent approximations of null controls for the 1D heat equation[END_REF] where this is discussed at length.

We also emphasize that we have considered bounded weights at the initial time t = 0 because of the constraints "ρ(s)y ∈ L 2 (Q T )" and "y(•, 0) = u 0 in Ω" appearing in the set A(s).

Main properties of the functional E s

The interest of the minimal controlled pair (Y 1 , F 1 ) ∈ A 0 (s) lies in the following result.

Proposition 2. For any (y, f ) ∈ A(s) and s ≥ max g (y)

2/3 L ∞ (Q T ) , s 0 , let (Y 1 , F 1 ) ∈ A 0 (s) defined in Definition 1. Then the derivative of E s at the point (y, f ) ∈ A(s) along the direction (Y 1 , F 1 ) given by E s (y, f ) • (Y 1 , F 1 ) := lim η→0,η =0 Es((y,f )+η(Y 1 ,F 1 ))-Es(y,f ) η satisfies E s (y, f ) • (Y 1 , F 1 ) = 2E s (y, f ). ( 39
)
Proof. We preliminary check that E s is for all (Y, F ) ∈ A 0 (s), E s is Gâteaux-differentiable at the point (y, f ) ∈ A(s) along the direction (Y, F ) ∈ A 0 (s). For all λ ∈ R, simple computations lead to the equality

E s (y + λY, f + λF ) = E s (y, f ) + λE s (y, f ) • (Y, F ) + h s, (y, f ), λ(Y, F ) with E s (y, f ) • (Y, F ) = ρ 0 (s)(∂ t y -∂ xx y + g(y) -f 1 ω ), ρ 0 (s)(∂ t Y -∂ xx Y + g (y)Y -F 1 ω ) L 2 (Q T ) (40)
and

h(s, (y, f ), λ(Y, F )) :=λ ρ 0 (s)(∂ t Y -∂ xx Y + g (y)Y -F 1 ω ), ρ 0 (s)l(y, λY ) L 2 (Q T ) + λ 2 2 ρ 0 (s)(∂ t Y -∂ xx Y + g (y)Y -F 1 ω ) 2 L 2 (Q T ) + ρ 0 (s)(∂ t y -∂ xx y + g(y) -f 1 ω ), ρ 0 (s)l(y, λY ) L 2 (Q T ) + 1 2 ρ 0 (s)l(y, λY ) 2 L 2 (Q T )
where l is defined by

l(y, Y ) := g(y + Y ) -g(y) -g (y)Y. ( 41
)
The application (Y,

F ) → E s (y, f ) • (Y, F
) is linear and continuous from A 0 (s) to R as it satisfies

|E s (y, f ) • (Y, F )| ≤ ρ 0 (s)(∂ t y -∂ xx y + g(y) -f 1 ω ) L 2 (Q T ) ρ 0 (s)(∂ t Y -∂ xx Y + g (y)Y -F 1 ω ) L 2 (Q T ) ≤ 2E s (y, f ) ρ 0 (s)(∂ t Y -∂ xx Y ) L 2 (Q T ) + ρ 0 (s)F L 2 (q T ) + ρ 0 (s)g (y)Y L 2 (Q T ) ≤ 2E s (y, f ) ρ 0 (s)(∂ t Y -∂ xx Y ) L 2 (Q T ) + ρ 0 (s)F L 2 (q T ) + g (y) L ∞ (Q T ) ρ(s)Y L 2 (Q T ) ≤ C 1 + g (y) L ∞ (Q T ) E s (y, f ) (Y, F ) A0(s) . (42) Similarly, for all λ ∈ R 1 λ h s, (y, f ), λ(Y, F ) ≤ |λ| ρ 0 (s)(∂ t Y -∂ xx Y + g (y)Y -F 1 ω ) L 2 (Q T ) + 2E s (y, f ) + 1 2 ρ 0 (s)l(y, λY ) L 2 (Q T ) 1 |λ| ρ 0 (s)l(y, λY ) L 2 (Q T ) + |λ| 2 ρ 0 (s)(∂ t Y -∂ xx Y + g (y)Y -F 1 ω ) 2 L 2 (Q T ) .
Since g ∈ C(R) we have, a.e in Q T : 1 λ l(y, λY ) = g(y+λY )-g(y) λ -g (y)Y → 0 as λ → 0 and, since

Y ∈ L ∞ (Q T ) and y ∈ L ∞ (Q T ), a.e in Q T 1 λ l(y, λY ) = g(y + λY ) -g(y) λ -g (y)Y ≤ sup θ∈[0,1] g (y + θY ) L ∞ (Q T ) + g (y) L ∞ (Q T ) |Y |
and therefore (recalling that ρ 0 ≤ ρ)

1 |λ| ρ 0 (s)l(y, λY ) L 2 (Q T ) ≤ sup θ∈[0,1] g (y + θY ) L ∞ (Q T ) + g (y) L ∞ (Q T ) ρ 0 Y L 2 (Q T ) ≤ sup θ∈[0,1] g (y + θY ) L ∞ (Q T ) + g (y) L ∞ (Q T ) ρY L 2 (Q T ) .
It then follows from the Lebesgue dominated convergence theorem that 1 λ ρ 0 (s)l(y, λY ) L 2 (Q T ) → 0 as λ → 0 and therefore that h(s, (y, f ), λ(Y, F )) = o(λ). Thus the functional E s is Gâteaux-differentiable at the point (y, f ) ∈ A(s) along the direction (Y, F ) ∈ A 0 (s). Eventually, the equality (39) follows from the relation (33) satisfied by the pair (Y 1 , F 1 ).

Remark that from the equality (40), the derivative E s (y, f ) is independent of (Y, F ). We can then define the norm E s (y, f ) A 0 (s) := sup (Y,F )∈A0(s),(Y,F ) =(0,0)

E s (y,f )•(Y,F )
(Y,F ) A 0 (s) associated to A 0 (s), the set of the linear and continuous applications from A 0 (s) to R.

Combining (34) and (39), we deduce the following estimates of E s (y, f ) in term of the norm of E s (y, f ). Proposition 3. For any (y, f ) ∈ A(s) and s ≥ max g (y)

2/3 L ∞ (Q T ) , s 0 , the inequalities hold true

1 C 1 + g (y) L ∞ (Q T ) E s (y, f ) A 0 (s) ≤ E s (y, f ) ≤ C E s (y, f ) A 0 (s)
for some C > 0.

Proof. (39

) rewrites E s (y, f ) = 1 2 E s (y, f ) • (Y 1 , F 1 )
where (Y 1 , F 1 ) ∈ A 0 (s) is solution of (33) and therefore, with (34)

E s (y, f ) ≤ 1 2 E s (y, f ) A 0 (s) (Y 1 , F 1 ) A0(s) ≤ C E s (y, f ) A 0 (s) E s (y, f ).
On the other hand, using (42), for all (Y, F ) ∈ A 0 (s):

|E s (y, f ) • (Y, F )| ≤ C 1 + g (y) L ∞ (Q T ) E s (y, f ) (Y, F ) A0(s)
leading to the left inequality.

In particular, any critical point (y, f ) ∈ A(s) for E s (i.e. for which E s (y, f ) vanishes) is a zero for E s , a pair solution of the controllability problem. In other words, any sequence (y k , f k ) k∈N of A(s) satisfying E s (y k , f k ) A 0 (s) → 0 as k → ∞ and for which ( g (y k ) ∞ ) k∈N is bounded is such that E s (y k , f k ) → 0 as k → ∞. We insist that this property does not imply the convexity of the functional E(s) (nor a fortiori the strict convexity of E s , which actually does not hold here in view of the multiple zeros for E s ) but show that a minimizing sequence for E s can not be stuck in a local minimum. Our least-squares algorithm, designed in the next section, is based on that property.

Eventually, the left inequality indicates that the functional E s is flat around its zero set. As a consequence, gradient based minimizing sequences for E s are inefficient as they usually achieve a low rate of convergence (we refer to [START_REF] Münch | Numerical null controllability of the heat equation through a least squares and variational approach[END_REF] and also [START_REF] Lemoine | Analysis of continuous H -1 -least-squares approaches for the steady Navier-Stokes system[END_REF] devoted to the Navier-Stokes equation where this phenomenon is observed).

We end this section with the following crucial estimate.

Lemma 6. Assume that g satisfies (H p ) for some p ∈ [0, 1]. For any (y, f ) ∈ A(s) with s ≥ max g (y)

2/3 L ∞ (Q T ) , s 0 , let (Y 1 , F 1 ) ∈ A 0 (s) be given in Definition 1 associated with (y, f ). For any λ ∈ R + the following estimate holds

E s (y, f ) -λ(Y 1 , F 1 ) ≤ E s (y, f ) |1 -λ| + λ p+1 c 1 (s) E s (y, f ) p (43) with c 1 (s) := C 1+p 1 + p s -3/2 e -3p 2 s [g ] p . (44)
Proof. The expansion of E s (y, f ) -λ(Y 1 , F 1 ) together with Definition 1 of (Y 1 , F 1 ) leads to

2E s (y, f ) -λ(Y 1 , F 1 ) = ρ 0 (s) ∂ t y -∂ xx y + g(y) -f 1 ω -λρ 0 (s) ∂ t Y 1 -∂ xx Y 1 + g (y)Y 1 -F 1 ω + ρ 0 (s)l(y, -λY 1 ) 2 L 2 (Q T ) = ρ 0 (s)(1 -λ) ∂ t y -∂ xx y + g(y) -f 1 ω + ρ 0 (s)l(y, -λY 1 ) 2 L 2 (Q T ) ≤ ρ 0 (s)(1 -λ) ∂ t y -∂ xx y + g(y) -f 1 ω L 2 (Q T ) + ρ 0 (s)l(y, -λY 1 ) L 2 (Q T ) 2 ≤ 2 |1 -λ| E s (y, f ) + ρ 0 (s)l(y, -λY 1 ) L 2 (Q T ) 2 ( 45
)
where l is defined in (41). For any (x, y) ∈ R 2 , y = 0, and λ ∈ R, we write g(x + λy) -g(x) = λ 0 yg (x + ξy)dξ leading to

|g(x + λy) -g(x) -λg (x)y| ≤ λ 0 |y||g (x + ξy) -g (x)|dξ ≤ λ 0 |y| 1+p |ξ| p |g (x + ξy) -g (x)| |ξy| p dξ ≤ [g ] p |y| 1+p |λ| 1+p 1 + p . It follows that |l(y, λY 1 )| = g(y + λY 1 ) -g(y) -λg (y)Y 1 ≤ [g ] p |Y 1 | 1+p |λ| p+1 1 + p and thus, since Y 1 ∈ L ∞ (Q T ) (see Remark 4
) and ρ 0 (s) ≤ ρ(s) :

ρ 0 (s)l(y, -λY 1 ) L 2 (Q T ) ≤ [g ] p |λ| p+1 1 + p ρ 0 (s)|Y 1 | 1+p L 2 (Q T ) ≤ [g ] p |λ| p+1 1 + p Y 1 p L ∞ (Q T ) ρ(s)Y 1 L 2 (Q T ) . (46)
Eventually, Remark 4 and estimates (34) and (35) in term of E s (y, f ) imply

Y 1 p L ∞ (Q T ) ρ(s)Y 1 L 2 (Q T ) ≤ Cs -3/2 e -3p 2 s C p E s (y, f ) p+1 2 .
Combining this estimate with (45) and ( 46), we get (43).

Convergence of the least-squares method

We now examine, for s large enough, the convergence of a minimizing sequence (y k , f k ) ∈ A(s) for E s . In this respect, we observe from the equality (39) that, for any (y, f ) ∈ A(s), -(Y 1 , F 1 ) given in Definition 1, is a descent direction for the functional E s at the point (y, f ), as soon as s ≥ max( g (y)

2/3 L ∞ (Q T ) , s 0 ). Therefore, we can define at least formally, for any fixed m ≥ 1, a minimizing sequence (y k , f k ) k∈N ∈ A(s) as follows:

     (y 0 , f 0 ) ∈ A(s), (y k+1 , f k+1 ) = (y k , f k ) -λ k (Y 1 k , F 1 k ), k ≥ 0, λ k = argmin λ∈[0,m] E s (y k , f k ) -λ(Y 1 k , F 1 k ) (47)
where

(Y 1 k , F 1 k ) ∈ A 0 (s) is the minimal controlled pair solution of ∂ t Y 1 k -∂ xx Y 1 k + g (y k )Y 1 k = F 1 k 1 ω + ∂ t y k -∂ xx y k + g(y k ) -f k 1 ω in Q T , Y 1 k = 0 on Σ T , Y 1 k (•, 0) = 0 in Ω (48)
associated with (y k , f k ) ∈ A(s). In particular, the pair (Y 1 k , F 1 k ) vanishes when E s (y k , f k ) vanishes. The real number m ≥ 1 is arbitrarily fixed and is introduced in order to keep the sequence (λ k ) k∈N bounded.

We highlight that, in order to give a meaning to (47), we need to prove that we can choose the parameter s independent of k, that is s ≥ max g (y k ) 2/3 L ∞ (Q T ) , s 0 for all k ∈ N. In this respect, it suffices to prove that there exists M > 0 such that y k L ∞ (Q T ) ≤ M for every k ∈ N. Under (H 2 ), this implies that g (y k ) L ∞ (Q T ) ≤ ψ(M ) for every k ∈ N, where ψ is defined in [START_REF] Münch | A mixed formulation for the direct approximation of L 2 -weighted controls for the linear heat equation[END_REF].

In the next section, assuming the parameter s large enough, we prove the existence of such M by induction making use of the properties of the functional E s proven in Section 3.2.

Boundedness of the sequence (y

k ) k∈N in L ∞ (Q T )
The boundedness is obtained after several technical results. We first prove an estimate of y n+1 L ∞ (Q T ) assuming a bound of y k L ∞ (Q T ) for 0 ≤ k ≤ n and s large enough. Proposition 4. Assume that g satisfies (H 2 ) and (H p ) for some p ∈ [0, 1]. Let M > 0 large enough and s ≥ max C(p)ψ(M ) 2/3 , s 0 with C(p) = 1 if p ∈ (0, 1], and C(0) = (2C) 3/2 . Let (y 0 , f 0 ) ∈ A(s) such that M ≥ y 0 L ∞ (Q T ) . Assume that, for some n ≥ 0, (y k , f k ) 0≤k≤n defined from (47) satisfies

y k L ∞ (Q T ) ≤ M for every 0 ≤ k ≤ n. Then y n+1 L ∞ (Q T ) ≤ y 0 L ∞ (Q T ) + Cm max p + 1 p E s (y 0 , f 0 ), (1 + p) 1 p +1 p c 1/p 1 (s)E s (y 0 , f 0 ) ( 49
)
if p ∈ (0, 1] and

y n+1 L ∞ (Q T ) ≤ y 0 L ∞ (Q T ) + Cm E s (y 0 , f 0 ) 1 -c 1 (s) ( 50
)
if p = 0. Recall that c 1 (s) is defined in (44).

We point out that the existence of (y 0 , f 0 ) ∈ A(s) follows from Lemma 5.

Proof. The inequality

y n L ∞ (Q T ) ≤ M implies that g (y n ) 2/3 L ∞ (Q T ) ≤ ψ(M ) 2/3 and then s ≥ max ψ(M ) 2/3 , s 0 ≥ max g (y n ) 2/3 L ∞ (Q T ) , s 0 . Proposition 1 allows to construct the pair (Y 1 n , F 1 n ) ∈ A 0 (
s) solution of (33). Then, (47) allows to define (y n+1 , f n+1 ). Estimate (43) implies that

E s (y k , f k ) -λ(Y 1 k , F 1 k ) ≤ E s (y k , f k ) |1 -λ| + λ p+1 c 1 (s) E s (y k , f k ) p and then E s (y k+1 , f k+1 ) ≤ E s (y k , f k ) min λ∈[0,m] p k (s, λ) (51) with p k (s, λ) := |1 -λ| + λ p+1 c 1 (s)E s (y k , f k ) p/2 , ∀λ ∈ R, ∀s > 0. ( 52
)
Since E s (y k , f k ) 0≤k≤n decreases, p k (s, λ) 0≤k≤n decreases for all λ (p k do not depend on k if p = 0) and thus, defining p k (s, λ k ) := min λ∈[0,m] p k (s, λ), p k (s, λ k ) 0≤k≤n decreases as well. (51) then implies, for all 0 ≤ k ≤ n -1, that

E s (y k+1 , f k+1 ) ≤ E s (y k , f k )p k (s, λ k ) ≤ E s (y k , f k )p 0 (s, λ 0 ). ( 53
)
First case : p ∈ (0, 1]. We prove that

n k=0 E s (y k , f k ) ≤ max p + 1 p E s (y 0 , f 0 ), (1 + p) 1 p +1 p c 2 (s)E s (y 0 , f 0 ) ( 54
)
where c 2 (s) := c 1/p 1 (s) . Since ∂ λ p 0 (s, 0) = -1, p 0 (s, λ 0 ) < p 0 (s, 0) = 1 we deduce from (53) that :

n k=0 E s (y k , f k ) ≤ E s (y 0 , f 0 ) 1 -p 0 (s, λ 0 ) n+1 1 -p 0 (s, λ 0 ) ≤ E s (y 0 , f 0 ) 1 -p 0 (s, λ 0 ) . ( 55
) If c 2 (s) E s (y 0 , f 0 ) < 1 (p+1) 1/p , we check that p 0 (s, λ 0 ) ≤ p 0 (s, 1) = c 1 (s) E s (y 0 , f 0 ) p ≤ 1 p+1 and thus E s (y 0 , f 0 ) 1 -p 0 (s, λ 0 ) ≤ p + 1 p E s (y 0 , f 0 ). If c 2 (s) E s (y 0 , f 0 ) ≥ 1 (p+1) 1/p , then for all λ ∈ [0, 1], ∂ λ p 0 (s, λ) = -1 + (p + 1)λ p c 1 (s)E s (y 0 , f 0 ) p/2 and thus ∂ λ p 0 (s, λ) = 0 if and only if λ = 1 (p+1) 1/p c2(s) √ Es(y0,f0) leading to p 0 (s, λ 0 ) = 1 - p (1 + p) 1 p +1 1 c 2 (s) E s (y 0 , f 0 ) and E s (y 0 , f 0 ) 1 -p 0 (s, λ 0 ) ≤ (1 + p) 1 p +1 p c 2 (s)E s (y 0 , f 0 ).
(55) then leads to (54). Then (47) implies that y n+1 = y 0 -n k=0 λ k Y 1 k and thus, using (34)

y n+1 L ∞ (Q T ) ≤ y 0 L ∞ (Q T ) + m n k=0 Y 1 k L ∞ (Q T ) ≤ y 0 L ∞ (Q T ) + Cm n k=0 (Y 1 k , F 1 k ) A0(s) ≤ y 0 L ∞ (Q T ) + Cm n k=0 E s (y k , f k )
which gives (49), using (54).

Second case : p = 0. Recall that for p = 0, ψ(r) = g ∞ for every r ∈ R. Then simply p k (s, λ) = |1 -λ| + λc 1 (s) for all k and

p 0 (s, λ 0 ) = min λ∈[0,m] p 0 (s, λ) = min λ∈[0,1] p 0 (s, λ) = p 0 (s, 1) = c 1 (s) with c 1 (s) = Cs -3/2 [g ] 0 = 2Cs -3/2 g ∞ = 2Cs -3/2 ψ(M ).
Taking s large enough, precisely s > max((2C) 1/3 ψ(M ) 2/3 , s 0 ), we obtain that c 1 (s) < 1. We then have for all 0 ≤ k ≤ n -1 that

E s (y k+1 , f k+1 ) ≤ E s (y k , f k ) c 1 (s)
and thus

n k=0 E s (y k , f k ) ≤ E s (y 0 , f 0 ) 1 -c 1 (s) n+1 1 -c 1 (s) ≤ E s (y 0 , f 0 ) 1 -c 1 (s) . ( 56
)
Proceeding as before, we get (50).

We now intend to choose s such that y n+1 L ∞ (Q T ) ≤ M . In view of estimates ( 49) and (50), we need to this end an estimate of E s (y 0 , f

0 ) = 1 2 ρ 0 (s)(∂ t y 0 -∂ xx y 0 + g(y 0 ) -f 0 1 ω ) 2 L 2 (Q T ) in terms of s.
Since the weight ρ(s) does not belong to L 2 (Q T ), such estimate is not straightforward for any (y 0 , f 0 ) ∈ A(s). We select the pair (y 0 , f 0 ) ∈ A(s) solution of the linear problem, i.e. g ≡ 0 in (1). Lemma 7. Assume that g satisfies (H 2 ). For any s ≥ s 0 , let (y 0 , f 0 ) ∈ A(s) be the solution of the extremal problem (25) in the linear case for which g ≡ 0. Then,

E s (y 0 , f 0 ) ≤ α + β c 3/2 + ln 3/2 (1 + C u 0 H 1 0 (Ω) ) e cs u 0 H 1 0 (Ω) . ( 57
)
with c = ϕ(•, 0) L ∞ (Ω) .
Proof. Estimate (20) of Proposition 3 with A = 0, B = 0 and z 0 = u 0 leads to

ρ(s)y 0 L 2 (Q T ) + s -3/2 λ -2 0 ρ 0 (s)f 0 L 2 (q T ) ≤ Cs -3/2 e cs u 0 L 2 (Ω) (58)
while ( 22) leads, since s ≥ 1 and ρ 0 ≥ 1, to

y 0 L ∞ (Q T ) ≤ Ce -3 2 s e cs u 0 H 1 0 (Ω) ≤ Ce cs u 0 H 1 0 (Ω) . ( 59
)
It follows that, using (32), since ρ 0 ≤ ρ and s ≥ 1

E s (y 0 , f 0 ) = 1 √ 2 ρ 0 (s)g(y 0 ) L 2 (Q T ) ≤ 1 √ 2 ψ( y 0 L ∞ (Q T ) ) ρ(s)y 0 L 2 (Q T ) ≤ Cψ(Ce cs u 0 H 1 0 (Ω) )s -3/2 e cs u 0 H 1 0 (Ω) ≤ C α + β ln 3/2 (1 + Ce sc u 0 H 1 0 (Ω) ) s -3/2 e cs u 0 H 1 0 (Ω) ≤ C α + β ln 3/2 (e sc (1 + C u 0 H 1 0 (Ω) )) s -3/2 e cs u 0 H 1 0 (Ω) ≤ C α + β (sc) 3/2 + ln 3/2 (1 + C u 0 H 1 0 (Ω) ) s -3/2 e cs u 0 H 1 0 (Ω) ≤ C α + β c 3/2 + ln 3/2 (1 + C u 0 H 1 0 (Ω) ) e cs u 0 H 1 0 (Ω) .
We are now in position to prove to following result.

Proposition 5. Assume that g satisfies (H 2 ) and (H p ) for some p ∈ [0, 1]. Assume moreover that 2cC(p)β 2/3 < 1 and let (y 0 , f 0 ) be the controlled pair given by Lemma 7. There exists M 0 > 0 such that, if we have constructed from (47) the pairs

(y k , f k ) 0≤k≤n ∈ A(s) with s = max(C(p)ψ(M 0 ) 2/3 , s 0 ) satisfying y k L ∞ (Q T ) ≤ M 0 for all 0 ≤ k ≤ n,
then the pair (y n+1 , f n+1 ) constructed from (47) also belongs to A(s) and satisfies

y n+1 L ∞ (Q T ) ≤ M 0 .
Proof. The inequality (a + b) 2/3 ≤ a 2/3 + b 2/3 for all a, b ≥ 0 allows to write, for all M > 0 :

ψ(M ) 2/3 = (α + β ln 3/2 (1 + M )) 2/3 ≤ α 2/3 + β 2/3 ln(1 + M ).
Assume that for some M large enough,

y k L ∞ (Q T ) ≤ M . Estimate (59) with s = max(C(p)ψ(M ) 2/3 , s 0 ) then leads to y 0 L ∞ (Q T ) ≤ C u 0 H 1 0 (Ω) e cs ≤ C u 0 H 1 0 (Ω) e c(s0+C(p)ψ(M ) 2/3 ) ≤ Ce cs0 u 0 H 1 0 (Ω) e cC(p)ψ(M ) 2/3 ≤ Ce cs0 u 0 H 1 0 (Ω) e cC(p) α 2/3 +β 2/3 ln(1+M ) ≤ Ce cs0 u 0 H 1 0 (Ω) e C(p)α 2/3 e cC(p)β 2/3 ln(1+M ) ≤ Ce cs0 u 0 H 1 0 (Ω) e C(p)α 2/3 (1 + M ) cC(p)β 2/3 ≤ c(α, u 0 )(1 + M ) cC(p)β 2/3 . (60)
Similarly, this estimate of e cs and (57) lead to

E s (y 0 , f 0 ) ≤ α + β c 3/2 + ln 3/2 (1 + C u 0 H 1 0 (Ω) ) e cs0 e C(p)α 2/3 (1 + M ) cC(p)β 2/3 u 0 H 1 0 (Ω) ≤ c(α, β, u 0 )(1 + M ) cC(p)β 2/3 . ( 61
)
First case : p ∈ (0, 1]. Since s ≥ 1, the constant c 1 (s) defined in (44) satisfies c 1 (s) ≤ C 1+p 1+p [g ] p .
Therefore, by combining (49), (60) and (61), we get

y n+1 L ∞ (Q T ) ≤c(α, u 0 )(1 + M ) cC(p)β 2/3 + Cm max 1 p c(α, β, u 0 )(1 + M ) cC(p)β 2/3 , (1 + p) p [g ] 1/p p C (1+p)/p c 2 (α, β, u 0 )(1 + M ) 2cC(p)β 2/3 ≤C(p, α, u 0 , [g ] p , β)(1 + M ) 2cC(p)β 2/3 .
finite subset of N. Indeed, for all k ∈ I and for all λ ∈ [0, 1] :

∂ λ p k (s, λ) = -1+(p+1)λ p c 1 (s)E s (y k , f k ) p/2 and thus ∂ λ p k (s, λ) = 0 if and only if λ = 1 (p+1) 1/p c2(s) √ Es(y k ,f k ) , which gives p k (s, λ k ) = min λ∈[0,m] p k (λ) = min λ∈[0,1] p k (λ) = p k 1 (1 + p) 1/p c 2 (s) E s (y k , f k ) = 1 - p (1 + p) 1 p +1 1 c 2 (s) E s (y k , f k ) and thus c 2 (s) E s (y k+1 , f k+1 ) ≤ 1 - p (1 + p) 1 p +1 1 c 2 (s) E s (y k , f k ) c 2 (s) E s (y k , f k ) = c 2 (s) E s (y k , f k ) - p (1 + p) 1 p +1 . (65)
This inequality implies that the sequence c 2 (s) E s (y k , f k ) k∈I strictly decreases and at at least linearly writing that

c 2 (s) E s (y k+1 , f k+1 ) c 2 (s) E s (y k , f k ) ≤ 1 - p (1 + p) 1 p +1 c 2 (s) E s (y k , f k ) ≤ 1 - p (1 + p) 1 p +1 c 2 (s) E s (y 0 , f 0 ) < 1 ∀k ∈ I. (66)
It follows that there exists

k 0 ∈ N such that for all k > k 0 , c 2 (s) E s (y k , f k ) < 1 (p+1) 1/p , that is I is a finite subset of N. Moreover, from k = k 0 + 1,
arguing as in the previous case, the convergence to zero of c 2 (s) E s (y k , f k ) is as least of order 1 + p. The index k 0 is defined in Remark 8 below.

Second case : p = 0. Then for all k ∈ N, since c 1 (s) < 1, p k (s, λ k ) = c 1 (s) (since λ k = 1) and therefore E s (y k+1 , f k+1 ) ≤ c 1 (s) E s (y k , f k ) ≤ c 1 (s) k+1 E s (y 0 , f 0 ). ( 67
) Thus E s (y k , f k ) → 0 as k → ∞.
We can now prove the convergence of the sequence (y k , f k ) k∈N announced in Theorem 2. Precisely, we have the following main result.

Theorem 4. Assume that g satisfies (H 2 ) and (H p ) for some p ∈ [0, 1]. Assume moreover that β is small enough so that 2cC(p)β 2/3 < 1

with c = ϕ(•, 0) L ∞ (Ω)
. Let M 0 be given by (62) for p ∈ (0, 1], by (63) for p = 0 and let s = max(C(p)ψ(M 0 ) 2/3 , s 0 ). Let (y 0 , f 0 ) ∈ A(s) be the solution of the extremal problem (25) in the linear situation for which g ≡ 0 and let (y k , f k ) k∈N be the sequence defined by (47). Then,

(y k , f k ) k∈N → (y, f ) in A(s)
where f is a null control for y solution of (1). The convergence is at least linear, and is at least of order 1 + p after a finite number of iterations.

Proof. For all k ∈ N, let 34)), we write, using (54) and (55) :

F k = - k n=0 λ n F 1 n and Y k = k n=0 λ n Y 1 n . Let us prove that (Y k , F k ) k∈N converges in A 0 (s), i.e. that the series λ n (F 1 n , Y 1 n ) converges in A 0 (s). Using that (Y 1 k , F 1 k ) A0(s) ≤ C(M 0 ) E s (y k , f k ) for all k ∈ N (see (
k n=0 λ n (Y 1 n , F 1 n ) A0(s) ≤ m k n=0 (Y 1 n , F 1 n ) A0(s) ≤ C(M 0 ) k n=0 E s (y n , f n ) ≤ E s (y 0 , f 0 ) C(M 0 ) 1 -p 0 ( λ 0 )
.

We deduce that the series n λ n (Y 1 n , F 1 n ) is normally convergent and so convergent. Consequently, there exists (Y,

F ) ∈ A 0 (s) such that (Y k , F k ) k∈N converges to (Y, F ) in A 0 (s). Denoting y = y 0 + Y and f = f 0 + F , we then have that (y k , f k ) k∈N = (y 0 + Y k , f 0 + F k ) k∈N converges to (y, f ) in A(s).
It suffices now to verify that the limit (y, f ) satisfies E s (y, f ) = 0. Using that (Y 1 k , F 1 k ) goes to zero in A 0 (s) as k → ∞, we pass to the limit in (48) and get that (y, f ) ∈ A(s) solves [START_REF] Badra | Local controllability to trajectories for nonhomogeneous incompressible Navier-Stokes equations[END_REF], that is E s (y, f ) = 0. Moreover, we have

(y, f ) -(y k , f k ) A0(s) ≤ C(M 0 ) E s (y k , f k ), ∀k > 0 (68)
which implies, using Proposition 7, the announced order of convergence. Precisely,

(y, f ) -(y k , f k ) A0(s) = ∞ p=k+1 λ p (Y 1 p , F 1 p ) A0(s) ≤ m ∞ p=k+1 (Y 1 p , F 1 p ) A0(s) ≤ m C(M 0 ) ∞ p=k+1 E s (y p , f p ) ≤ m C(M 0 ) ∞ p=k+1 p 0 ( λ 0 ) p-k E s (y k , f k ) ≤ m C(M 0 ) p 0 ( λ 0 ) 1 -p 0 ( λ 0 ) E s (y k , f k ).
We emphasize, in view of the non uniqueness of the zeros of E s , that an estimate (similar to (68)) of the form (y, f ) -(y, f ) A0(s) ≤ C(M 0 ) E s (y, f ) does not hold for all (y, f ) ∈ A(s). We also mention the fact that the sequence (y k , f k ) k∈N and its limit (y, f ) are uniquely determined from the initial guess (y 0 , f 0 ) and from our criterion of selection of the pair (Y 1 k , F 1 k ) for every k. In other words, the solution (y, f ) is unique up to the element (y 0 , f 0 ) and the functional J s .

We also have the following convergence of the optimal sequence (λ k ) k∈N .

Lemma 8. Under hypotheses of Theorem (4) with p ∈ (0, 1], the sequence (λ k ) k∈N defined in (47) converges to 1 as k → ∞.

Proof. If p ∈ (0, 1], in view of (45) we have, as long as

E s (y k , f k ) > 0, since λ k ∈ [0, m] (1 -λ k ) 2 = E s (y k+1 , f k+1 ) E s (y k , f k ) -(1 -λ k ) ρ 0 (s) ∂ t y k + ∆y k + g(y k ) -f k 1 ω , ρ 0 (s)l(y k , -λ k Y 1 k ) L 2 (Q T ) E s (y k , f k ) - ρ 0 (s)l(y k , -λ k Y 1 k ) 2 L 2 (Q T ) 2E s (y k , f k ) ≤ E s (y k+1 , f k+1 ) E s (y k , f k ) -(1 -λ k ) ρ 0 (s) ∂ t y k + ∆y k + g(y k ) -f k 1 ω , ρ 0 (s)l(y k , -λ k Y 1 k ) L 2 (Q T ) E s (y k , f k ) ≤ E s (y k+1 , f k+1 ) E s (y k , f k ) + √ 2m E s (y k , f k ) ρ 0 (s)l(y k , -λ k Y 1 k ) L 2 (Q T ) E(s, y k , f k ) ≤ E s (y k+1 , f k+1 ) E s (y k , f k ) + √ 2m ρ 0 (s)l(y k , -λ k Y 1 k ) L 2 (Q T ) E s (y k , f k ) .
But, from (46), (34) and Remark 4, we infer that

ρ 0 (s)l(y k , -λ k Y 1 k ) L 2 (Q T ) ≤ [g ] p λ p+1 k p + 1 Y p L ∞ (Q T ) ρ(s)Y L 2 (Q T ) ≤ λ p+1 k Cs -3/2 e -3p 2 s p + 1 [g ] p E s (y k , f k ) p+1 2 ≤ C(s)m p+1 [g ] p E s (y k , f k ) p+1 2 22
and thus

(1 -λ k ) 2 ≤ E s (y k+1 , f k+1 ) E s (y k , f k ) + m p+2 C(s)[g ] p (E s (y k , f k )) p/2 .
Consequently, since E s (y k , f k ) → 0 and Es(y k+1 ,f k+1 )

Es(y k ,f k )
→ 0, we deduce that (1 -λ k ) 2 → 0 as k → ∞.

If p = 0 and if s is large enough, then c 1 (s) < 1 and λ k = 1 for every k ∈ N leading to the decay of (E s (y k , f k )) k∈N to 0 (see (67)). Moreover, estimate (43) implies that the sequence (λ k ) k∈N with λ k = 1 for every k also leads to the decay (E s (y k , f k )) k∈N with an order at least linear. Whether or not this constant sequence if the optimal one (as defined in (47)) is an open question. Remark 7. In Theorem 4, the sequence (y k , f k ) k∈N is initialized with the solution of minimal norm corresponding to g ≡ 0. This natural choice in practice leads to a precise estimate of E s (y 0 , f 0 ) with respect to the parameter s. Many other pairs are available such as for instance the pair (y 0 , f 0 ) = (y, 0) = (φ y , 0) constructed in Lemma 5 since it leads to the following estimate in term of s:

E s (φy , 0) = 1 √ 2 ρ 0 ∂ t (φy ) -∆(φy ) + g(φy ) L 2 (Q T ) = 1 √ 2 ρ 0 (s) ∂ t φ y + g(φy ) L 2 (Q T ) ≤ T 2 ρ 0 (s) L ∞ (Q T /2 ) ∂ t φ L ∞ (Q T /2 ) y L ∞ (Q T /2 ) + ρ 0 (s)g(φy )) L 2 (Q T ) ≤ C(T ) ρ 0 (s) L ∞ (Q T /2 ) u 0 H 1 (Ω) + ψ( φy L ∞ (Q T /2 ) ) ρ(s)φy L ∞ (Q T /2 ) ≤ C(T ) ρ(s) L ∞ (Q T /2 ) u 0 H 1 (Ω) 1 + ψ( u 0 L ∞ (Ω) ) ≤ C(T ) u 0 H 1 (Ω) 1 + α + β ln 3/2 (1 + u 0 H 1 (Ω) ) e s ϕ L ∞ (Q T /2 ) .
Remark 8. As stated in Theorem 4, the convergence is at least of order 1 + p after a number k 0 of iterations. Using (65), k 0 is given by

k 0 = 1 + p p (1 + p) 1/p c 2 (s) E s (y 0 , f 0 ) -1 + 1, (69)
(where • is the integer part) if (1 + p) 1/p c 2 (s) E s (y 0 , f 0 ) -1 > 0, and k 0 = 1 otherwise.

Comments

Several comments are in order.

Asymptotic condition. The asymptotic condition (H 2 ) on g is slightly stronger than the asymptotic condition (H 1 ) made in [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF]: this is due to our linearization of (1) which involves r → g (r) while the linearization (6) in [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF] involves r → g(r)/r. There exist cases covered by Theorem 1 in which exact controllability for (1) is true but that are not covered by Theorem 4. Note however that the example g(r) = a + br + cr ln 3/2 (1 + |r|), for any a, b ∈ R and for any c > 0 small enough (which is somehow the limit case in Theorem 1) satisfies (H 2 ) as well as (H p ) for any p ∈ [0, 1]. While Theorem 1 was established in [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF] by a nonconstructive fixed point argument, we obtain here, in turn, a new proof of the exact controllability of semilinear one dimensional heat equations, which is moreover constructive, with an algorithm that converges unconditionally, with order at least 1 + p.

Minimization functional. The estimate (34) is a key point in the convergence analysis and is independent of the choice of the functional J s defined in Proposition 3 in order to select a pair (Y 1 , F 1 ) in A 0 (s). Thus, we may consider other weighted functionals, for instance the choice J s (y, f ) = 1 2 ρ 0 (s)f 2

L 2 (q T )
discussed in [START_REF] Münch | A mixed formulation for the direct approximation of L 2 -weighted controls for the linear heat equation[END_REF].

Link with Newton method. If we introduce F :

A(s) → L 2 (Q T ) by F (y, f ) := ρ -1 0 (s)(∂ t y -∂ xx y + g(y)-f 1 ω ), we get that E s (y, f ) = 1 2 F (y, f ) 2 L 2 (Q T )
and check that, for λ k = 1 for every k, the algorithm (47) coincides with the Newton algorithm associated with the mapping F . This explains the super-linear convergence in Theorem 4. The optimization of the parameter λ k is crucial here as it allows to get a global convergence result. It leads to so-called damped Newton method (for F ) (we refer to [START_REF] Deuflhard | Newton methods for nonlinear problems[END_REF]Chapter 8]). As far as we know, the analysis of damped type Newton methods for partial differential equations has deserved very few attention in the literature. We mention [START_REF] Lemoine | A fully space-time least-squares method for the unsteady Navier-Stokes system[END_REF][START_REF] Saramito | A damped Newton algorithm for computing viscoplastic fluid flows[END_REF] in the context of fluid mechanics.

A variant. To simplify, let us take λ k = 1 for every k, as in the standard Newton method. Then, for each k ∈ N, the optimal pair (Y 1 k , F 1 k ) ∈ A 0 is such that the element (y k+1 , f k+1 ) minimizes over A(s) the functional (z, v) → J s (z -y k , v -f k ). Alternatively, we may select the pair (Y 1 k , F 1 k ) so that the element (y k+1 , f k+1 ) minimizes the functional (z, v) → J(z, v). This leads to the sequence (y k , f k ) k∈N defined by

∂ t y k+1 -∂ xx y k+1 + g (y k )y k+1 = f k+1 1 ω + g (y k )y k -g(y k ) in Q T , y k = 0 on Σ T , y k+1 (•, 0) = u 0 in Ω. ( 70
)
In this case, for every k ∈ N, (y k , f k ) is a controlled pair for a linearized heat equation, while, in the case of the algorithm (47), (y k , f k ) is a sum of controlled pairs (Y 1 j , F 1 j ) for 0 ≤ j ≤ k. This analysis of this variant used in [START_REF] Fernández | Numerical null controllability of semi-linear 1-D heat equations: fixed point, least squares and Newton methods[END_REF] is apparently less straightforward.

Local controllability when removing the growth condition (H 2 ). As in [START_REF] Bottois | Constructive exact control of semilinear multidimensional wave equations[END_REF][START_REF] Münch | Constructive exact control of semilinear 1d wave equations by a least-squares approach[END_REF] devoted to the wave equation, we may expect to remove the growth condition (H 2 ) on g if the initial value E s (y 0 , f 0 ) is small enough. For s fixed, in view of Lemma 7, this is notably true if g(0) = 0 and if the norm u 0 H 1 0 (Ω) of the initial data to be controlled is small enough. This would allow to recover the local controllability of the heat equation (usually obtained by an inverse mapping theorem, see [18, chapter 1]) and would be in agreement with the usual convergence of the standard Newton method. In the parabolic case considered here, the proof is however open, since in order to prove the convergence of (E s (y k , y k )) k∈N to zero, for some s large enough independent of k, we need to prove that the sequence ( y k L ∞ (Q T ) ) k∈N is bounded. This is in contrast with [START_REF] Bottois | Constructive exact control of semilinear multidimensional wave equations[END_REF][START_REF] Münch | Constructive exact control of semilinear 1d wave equations by a least-squares approach[END_REF] where the control of minimal L 2 (Q T ) independently of any parameter s.

Weakening of the condition (H p ). Given any p ∈ [0, 1], we introduce for any g ∈ C 1 (R) the following hypothesis :

(H p ) There exist α, β, γ ∈ R + such that |g (a) -g (b)| ≤ |a -b| p α + β(|a| γ + |b| γ ) , ∀a, b ∈ R which coincides with (H p ) if γ = 0 for α + 2β = [g ] p . If γ ∈ (0, 1
) is small enough and related to the constant β appearing in the growth condition (H 2 ), Theorem 4 still holds if (H p ) is replaced by the weaker hypothesis (H p ).

Influence of the parameter s and a simpler linearization. Taking s large enough in the case p = 0 (corresponding to g ∈ L ∞ (R)) allows to ensure that the coefficient c 1 (s) (see (44)) is strictly less than one, and then to prove the strong convergence of the sequence (y k , f k ) k∈N . This highlights the influence of the parameter s appearing in the Carleman weights ρ, ρ 0 and ρ 1 . Actually, in this case, a similar convergence can be obtained by considering a simpler linearization of the system (1). For any s ≥ s 0 and z ∈ L 2 (ρ(s); Q T ), we define the controlled pair (y, f ) ∈ A(s) solution of

∂ t y -∂ xx y = f 1 ω -g(z) in Q T , y = 0 on Σ T , y(•, 0) = u 0 in Ω,
and which minimizes the weighted cost J s . If g(0) = 0 and g is globally Lipschitz, then ρ 0 (s)g(z) ∈ L 2 (Q T ) and Theorem 3 implies ρ(s) y L 2 (Q T ) ≤ Cs -3/2 ρ 0 (s)g(z) L 2 (Q T ) +e cs u 0 L 2 (Ω) . This allows to define the operator K : L 2 (ρ(s); Q T ) → L 2 (ρ(s); Q T ) by y := K(z). From Lemma 3, for any z i ∈ L 2 (ρ(s); Q T ), i = 1, 2, y i := K(z i ) is given by y i = ρ -2 (s)L 0 p i where p i ∈ P 0 solves (p i , q) P = Ω u 0 q(0) -Q T g(z i )q, ∀q ∈ P 0 .

Taking q := p 1 -p 2 , we then get

p 1 -p 2 2 P ≤ Q T |g(z 1 ) -g(z 2 )||p 1 -p 2 | ≤ ρ 0 (s)(g(z 1 ) -g(z 2 )) L 2 (Q T ) ρ -1 0 (s)(p 1 -p 2 ) L 2 (Q T ) ≤ g ∞ ρ 0 (s)(z 1 -z 2 ) L 2 (Q T ) ρ -1 0 (s)(p 1 -p 2 ) L 2 (Q T ) .
Using that ρ 0 ≤ ρ and Lemma 1, we obtain

ρ(s) K(z 1 ) -K(z 2 ) L 2 (Q T ) ≤ g ∞ Cλ 4 0 s -3/2 ρ(s)(z 1 -z 2 ) L 2 (Q T ) , ∀z 1 , z 2 ∈ L 2 (ρ(s); Q T )
and conclude that, if s > max(s 0 , (Cλ 4 0 g ∞ ) 2/3 ), then the operator K is contracting. This allows to infer the convergence of the sequence (y k ) k∈N defined by y k+1 = K(y k ), k ≥ 0 for any y 0 ∈ L 2 (ρ(s); Q T ) to a controlled solution of (1). In order to replace the assumption g ∈ L ∞ (R) by (H 1 ), one needs to show some compactness properties for K, which is an open question.

The linearization (6) associated with the weighted cost J s . Similarly, one can wonder if a parameter s large enough may lead to a contracting property for the operator Λ introduced in [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF] and leading to the linearization [START_REF] Bristeau | On the numerical solution of nonlinear problems in fluid dynamics by least squares and finite element methods. I. Least square formulations and conjugate gradient[END_REF]. For any β > 0, we introduce the hypothesis (H 1 ) lim sup |r|→∞ |g(r)| |r| ln 3/2 |r| ≤ β similar to (H 1 ). Then, as in [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF] (where controls of minimal L ∞ (Q T ) norm are considered), the linearization (6) also leads to a compactness property when associated with the cost J s . Proposition 8. Assume that g satisfies (H 1 ) with

cβ 2/3 < 1 with c = ϕ(•, 0) L ∞ (Ω) . Let z ∈ L ∞ (Q T ) and s = max( g(z) 2/3 L ∞ (Q T ) , s 0 ). Let (y, f ) ∈ A(s) the minimizer of the functional J s and solution of ∂ t y -∂ xx y + y g(z) = f 1 ω in Q T , y = 0 on Σ T , y(•, 0) = u 0 in Ω. There exists M > 0 such that if z L ∞ (Q T ) ≤ M and s := max(ψ(M ) 2/3 , s 0 ) with ψ(r) = α + β ln 3/2 (1 + |r|), then y L ∞ (Q T ) ≤ M . We note Λ s : L ∞ (Q T ) → L ∞ (Q T ) such that y = Λ s (z). Proof. (H 1 ) implies that | g(r)| ≤ ψ(r) := α+β ln 3/2 (1+|r|) for all r ∈ R. Consequently, g(z) L ∞ (Q T ) ≤ ψ( z L ∞ (Q T ) ) = α + β ln 3/2 (1 + z L ∞ (Q T ) ) leading to c g(z) 2/3 L ∞ (Q T ) ≤ cα 2/3 + cβ 2/3 ln(1 + z L ∞ (Q T ) ) and then to e cs ≤ e c(s0+α 2/3 ) (1 + z L ∞ (Q T ) ) cβ 2/3 . Estimate (20) then implies ρ(s) y L 2 (Q T ) + s -3/2 λ -2 0 ρ 0 (s) f L 2 (q T ) ≤ e c(s0+α 2/3 ) (1 + z L ∞ (Q T ) ) cβ 2/3 u 0 L 2 (Ω)
and in particular, since ρ(s) ≥ ρ 0 (s) ≥ e

y L 2 (Q T ) + λ -2 0 f L 2 (q T ) ≤ e c(s0+α 2/3 ) (1 + z L ∞ (Q T ) ) cβ 2/3 u 0 L 2 (Ω) . (71)
Moreover, if u 0 ∈ H 1 0 (Ω), standard estimate for the heat equation reads as

y L ∞ (Q T ) ≤ C g(z) L ∞ (Q T ) y L 2 (Q T ) + f L 2 (q T ) + u 0 L 2 (Ω)
which combined with (71) leads to

y L ∞ (Q T ) ≤ C 1 + α + β ln 3/2 (1 + z L ∞ (Q T ) ) e c(s0+α 2/3 ) (1 + z L ∞ (Q T ) ) cβ 2/3 u 0 L 2 (Ω) + u 0 L 2 (Ω) .
It follows that if cβ 2/3 < 1, then there exists an M > 0 depending on u 0 L 2 (Ω) , α, s 0 , λ 0 , Ω, T such that

z L ∞ (Q T ) ≤ M implies y L ∞ (Q T ) ≤ M since C (1 + α + β ln 3/2 (1 + M ))e c(s0+α 2/3 ) (1 + M ) cβ 2/3 u 0 L 2 (Ω) + u 0 L 2 (Ω) M -→ 0 + , as M → ∞.
Let now z i ∈ L 2 (ρ(s); Q T ) and y i := Λ ρ (z i ) for i = 1, 2. Then, using the estimates of Theorem 3 and the characterization of Lemma 2, we can proved, for all s ≥ max(ψ(M ) 2/3 , s 0 ) that

ρ(s) Λ s (z 1 ) -Λ s (z 2 ) L 2 (Q T ) ≤ C g L ∞ (0,M ) u 0 L 2 (Ω) ζρ -1 (s) L ∞ (Q T ) s -3/2 e cs ρ(s)(z 1 -z 2 ) L 2 (Q T )
(72) for some C = C(Ω, T ) and ζ = (T -t) -1/2 . The existence of a parameter s large enough for which the operator Λ s enjoys a contracting property remains however an open issue.

Controllability to trajectories

The least-squares approach developed here extends to the case of the exact controllability to trajectories, defined in the introduction and studied in [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF]. Let y be a bounded trajectory associated with the data u 0 ∈ H 1 0 (Ω) and f ∈ L 2 (q T ) solution of We aim to find a control f ∈ L 2 (q T ) such that y satisfies (1) and [START_REF] Boyer | Carleman estimates for semi-discrete parabolic operators and application to the controllability of semi-linear semi-discrete parabolic equations[END_REF]. We write that z := y -y and v := f -f solve

∂ t z -∂ xx z + G(x, t, z) = v1 ω in Q T , z = 0 on Σ T , z(•, 0) = u 0 -u 0 in Ω, (74)
with for all r ∈ R, G(x, t, r) := g(y (x, t) + r) -g(y (x, t)). The function satisfies G(•, •, 0) = 0, is C 1 with respect to the variable r if g is C 1 and satisfies the growth condition (H 2 ) if g satisfies (H 2 ) as well. Precisely, using that (a + b) ) and β = √ 2β. Consequently, Theorem 4 applied to system (74) allows to define a sequence (z k , v k ) k∈N which converges toward a controlled pair for (74). This implies the convergence of the sequence (z k + y , v k + f ) k∈N toward a solution of (1) and [START_REF] Boyer | Carleman estimates for semi-discrete parabolic operators and application to the controllability of semi-linear semi-discrete parabolic equations[END_REF].

Extension to the multi-dimensional case The extension of our arguments to dimensions greater than one (as considered in [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF]) remains to be done. We have used the fact that the controlled solution of the linearized problem [START_REF] Doubova | On the controllability of parabolic systems with a nonlinear term involving the state and the gradient[END_REF] belongs to L ∞ (Q T ). This allows to give a meaning to g (y) L ∞ (Q T ) and therefore to use the Carleman estimate of Lemma 1 with potential A in L ∞ (Q T ). In the multidimensional setting, due to the right hand side B in L 2 (Q T ), the controlled solution of ( 11) is not in general in L ∞ (Q T ), including for controls in L ∞ (q T ) but only in L ∞ (0, T ; L p (Ω)) for some p < ∞. The extension to the multi-dimensional case therefore requires a refined Carleman estimate dealing with potentials in L ∞ (0, T ; L q (Ω)) for some q < ∞ related to p (so as to give a meaning to g (y) L ∞ (0,T ;L q (Ω)) for y ∈ L ∞ (0, T ; L p (Ω))) and very likely a stronger condition than (H 2 ) of the form (H 2 ) There exists α ≥ 0 and β > 0 such that |g (r)| ≤ α + β ln m (1 + |r|) for every r in R for some m = m(q) < 3/2 so as to ensure the convergence of the minimizing sequence for E s . This is notably the situation for the wave equation where a stronger growth condition is made on g in [START_REF] Bottois | Constructive exact control of semilinear multidimensional wave equations[END_REF] devoted to the multi-dimensional case than in [START_REF] Münch | Constructive exact control of semilinear 1d wave equations by a least-squares approach[END_REF] devoted to the one dimensional case.

Conclusions

Exact controllability of (1) has been established in [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF], under a growth condition on g, by means of a Kakutani fixed point argument which is not constructive. Under a slightly stronger growth condition and under the additional assumption that g is uniformly Hölder continuous with exponent p ∈ [0, 1], we have designed, in the one dimensional setting, an explicit algorithm and proved its convergence to a controlled solution of (1). Moreover, the convergence is super-linear of order greater than or equal to 1 + p after a finite number of iterations. In turn, our approach gives a new and constructive proof of the exact controllability of (1), which is, at least in the one-dimensional setting, simpler that in [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF] where refined L 1 Carleman estimates are employed. In fact, in the one-dimensional setting, we can achieve the power 3/2 appearing in the growth condition since the controlled sequence (y k ) k∈N belongs to the space L ∞ (Q T ). This is in general no longer true in the multidimensional setting in view of the L 2 (Q T ) right hand side term in (48), even with L ∞ (q T ) controls considered in [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF] leading to an interesting open question.

We also emphasize that the method is general and may be applied to any other equations or systems for which a precise observability estimate for the linearized problem is available. Such estimates are usually obtained by the way of Carleman estimates as initially done in the monography of Imanuvilov-Fursikov [START_REF] Fursikov | Controllability of evolution equations[END_REF], extended later to a very large number of systems and situations. For instance, the method can be extended to the case of boundary controls. This remains however to be done. Moreover, the introduction of the Carleman type weights, which blow up at the final time and which depends on several parameters (itself related to the controlled solution), makes the analysis quite intricate. From this point of view, the case of hyperbolic equations (considered in [START_REF] Münch | Constructive exact control of semilinear 1d wave equations by a least-squares approach[END_REF][START_REF] Bottois | Constructive exact control of semilinear multidimensional wave equations[END_REF]) is simpler. Whether or not an appropriate choice of these parameters may lead directly to some contracting properties for some fixed point operator is also an open and interesting issue. Eventually, it would be also interesting to address other types of linearity involving notably the gradient of the solution (see [START_REF] Doubova | On the controllability of parabolic systems with a nonlinear term involving the state and the gradient[END_REF]): we mention notably the Burgers equation and the Navier-Stokes system, formally solved numerically from a controllability viewpoint in [START_REF] Glowinski | Exact and approximate controllability for distributed parameter systems[END_REF]Part 1] and in [START_REF] Fernández-Cara | On the numerical controllability of the two-dimensional heat, Stokes and Navier-Stokes equations[END_REF] respectively.

H 1 )

 1 lim sup |r|→∞ |g(r)| |r| ln 3/2 |r| = 0 then (1) is controllable in time T .

3 2 s ≥ 1 and s - 3 /2 e 3 2

 333 s ≥ 1 that

∂

  t y -∂ xx y + g(y ) = f 1 ω in Q T , y = 0 on Σ T , y (•, 0) = u 0 in Ω.(73)

  3/2 ≤ √ 2(a 3/2 + b 3/2 ) for all a, b ≥ 0, we infer that, for all (x, t) ∈ Q T , |∂ r G (x, t, r)| = |g (y (x, t) + r)| ≤ α + β ln 3/2 (1 + |y (x, t) + r|)

	≤ α + β ln 3/2 (1 + |y |)(1 +	|z| 1 + |y |	)
	≤ α +	√	2β ln 3/2 (1 + |y |) +	√	2β ln 3/2 1 + |r| 1 + |y |

≤ α + β ln 3/2 (1 + |r|), ∀r ∈ R with α = α+ √ 2β ln 3/2 (1+ y L ∞ (Q T )

Funding by the French government research program "Investissements d'Avenir" through the IDEX-ISITE initiative 16-IDEX-0001 (CAP 20-25) (Corresponding author).

Now, if β is small enough so that 2cC(p)β 2/3 < 1, the real M 0 defined as follows

exists and is independent of n. Consequently, taking M = M 0 and s = max(C(p)ψ(M 0 ) 2/3 , s 0 ) lead to

Second case : p = 0. In this case, c 1 (s) = Cs -3/2 [g ] 0 = 2Cs -3/2 g ∞ < 1 for s large enough. By combining (50), ( 60) and (61), we get

and we conclude as in the previous case with

By induction, we can now conclude to the boundedness of the sequence (y k ) k∈N for the norm L ∞ (Q T ).

Proposition 6. Assume that g satisfies (H 2 ) and (H p ) for some p ∈ [0, 1]. Assume moreover that 2cC(p)β 2/3 < 1. Let M 0 be given by (62) for p ∈ (0, 1], by (63) for p = 0 and s = max(C(p)ψ(M 0 ) 2/3 , s 0 ). Let (y 0 , f 0 ) ∈ A(s) be the solution of the extremal problem (25) in the linear situation for which g ≡ 0.

Then the sequence (y k , f k ) k∈N ∈ A(s) defined by (47) satisfies

Proof. From the construction of M 0 , y 0 L ∞ (Q T ) < M 0 . Therefore, the uniform boundedness of the sequence (y k ) k∈N follows by induction from Proposition 5.

Convergence of the sequence (y k , f k ) k∈N and proof of Theorem 2

Once we know that we can choose the parameter s large enough but independent of k, we may prove the decay to zero of the sequence (E s (y k , f k )) k∈N .

Proposition 7. Assume that g satisfies (H 2 ) and (H p ) for some p ∈ [0, 1]. Assume moreover that 2cC(p)β 2/3 < 1. Let M 0 be given by (62) for p ∈ (0, 1], by (63) for p = 0 and s = max(C(p)ψ(M 0 ) 2/3 , s 0 ). Let (y 0 , f 0 ) ∈ A(s) be the solution of the extremal problem (25) in the linear situation for which g ≡ 0.

Let (y k , f k ) k∈N ∈ A(s) be the sequence defined by (47). Then, the sequence

The convergence is at least linear, and is at least of order 1 + p after a finite number of iterations.

Proof. First case : p ∈ (0, 1]. From the definition of p k given in (52

Thus, if c 2 (s) E s (y 0 , f 0 ) < 1 (p+1) 1/p , then c 2 (s) E s (y k , f k ) → 0 as k → ∞ with a rate 1+p. On the other hand, if c 2 (s) E s (y 0 , f 0 ) ≥ 1 (p+1) 1/p , then we check that I := {k ∈ N, c 2 (s) E s (y k , f k ) ≥ 1 (p+1) 1/p } is a