Existence of insensitizing controls for a semilinear heat equation with a superlinear nonlinearity

EXISTENCE OF INSENSITIZING CONTROLS FOR A SEMILINEAR HEAT EQUATION WITH A SUPERLINEAR NONLINEARITY O. Bodart∗ , M. Gonz´alez-Burgos†and R. P´erez-Garc´ıa† ——— This work has been partially financed by D.G.E.S. (Spain), Grant PB98–1134. Abstract In this paper we consider a semilinear heat equation (in a bounded domain Ω of IRN) with a nonlinearity that has a superlinear growth at infinity. We prove the existence of a control, with support in an open set ω⊂Ω, that insensitizes the L2−norm of the observation of the solution in another open subset O ⊂ Ω when ω∩ O 6=∅, under suitable assumptions on the nonlinear term f(y) and the right hand side term ξof the equation. The proof, involving global Carleman estimates and regularizing properties of the heat equation, relies on the sharp study of a similar linearized problem and an appropriate fixed-point argument. For certain superlinear nonlinearities, we also prove an insensitivity result of a negative nature. The crucial point in this paper is the technique of construction of Lr–controls (rlarge enough) starting from insensitizing controls in L2. ∗Laboratoire de Math´ematiques Appliqu´ees, Universit´e Blaise Pascal (Clermont-Ferrand 2), 63177 Aubi`ere Cedex, France, E-mails: Olivier.Bo[email protected]clermont.fr †Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain, E-mails: [email protected], [email protected] 1 Introduction and main results Problem formulation Let Ω ⊂IRN,N≥1, be a bounded connected open set with boundary ∂Ω∈C2. Let fbe a C1function defined on IR. Let ωand Obe two open subsets of Ω (thought to be small, in practice). For T > 0, we denote Q= Ω×(0, T) and Σ = ∂Ω×(0, T). We consider a semilinear heat equation with partially known initial condition (∂ty−∆y+f(y) = ξ+v1ωin Q, y= 0 on Σ, y(x, 0) = y0(x) + τˆy0(x) in Ω,(1) where ξ∈L2(Q) is a given heat source, y0∈L2(Ω) is a given initial data (although, by the reasons which will be seen later, we will address in this paper the case y0= 0), ˆy0∈L2(Ω) is unknown with kˆy0kL2(Ω) = 1, τis an unknown small real number and v∈L2(Q) is a control function to be determined. Here 1ωis the characteristic function of the control set ω. Let us define φ(y) = 1 2ZZO×(0,T) |y(x, t;τ, v)|2dx dt, (2) y=y(·,·;τ, v) being a solution to (1) associated to τand v. A control function vis said to insensitize φif ∂φ(y(·,·;τ, v)) ∂τ ¯¯¯¯τ=0 = 0,∀ˆy0∈L2(Ω) with kˆy0kL2(Ω) = 1.(3) This insensitivity condition means that we seek a control function v, acting on ω×(0, T), such that φis locally insensitive to small perturbations in the initial condition. In [3] and [15], the existence of a control vsatisfying (3) is proved to be equivalent to the existence of a control vsolving the following problem: (∂ty−∆y+f(y) = ξ+v1ωin Q, y= 0 on Σ, y(x, 0) = y0(x) in Ω,(4) (−∂tq−∆q+f′(y)q=y1Oin Q, q= 0 on Σ, q(x, T) = 0 in Ω,(5) q(x, 0) = 0 in Ω.(6) 2 Thus the problem of seeking a control that insensitizes φboils down to a non-classical null controllability problem. First, it is a null controllability problem of backwardforward nature for a cascade system of heat equations, the first one of semilinear type. In addition, the control enters on the second equation only indirectly through the first one, while qis the function we want to lead to zero after a time interval of length T. Preliminaries and existing results This problem, addressed by J.-L. Lions in [15], has been studied for globally Lipschitzcontinuous nonlinearities and ω∩ O 6=∅(this last hypothesis is absolutely essential and to our knowledge nothing is known when the intersection is empty). First, in [3] the authors relaxed the notion of insensitizing controls, introducing the so–called ε–insensitizing controls: Given ε > 0, a control vis said to ε–insensitize φif ¯¯¯¯ ∂φ(y(·,·;τ, v)) ∂τ ¯¯¯¯τ=0¯¯¯¯ ≤ε, ∀ˆy0∈L2(Ω) with kˆy0kL2(Ω) = 1. In the above-mentioned paper, the existence of ε–insensitizing controls for partially known data, both in the initial and boundary conditions, was proved. This problem is equivalent to an approximate controllability problem for a system of coupled heat equations and it was solved by using the techniques in [9]. The first results on the existence and non-existence of insensitizing controls were proved in [16]. To be precise, the author showed that when f≡0 and Ω\ω6=∅, there exists y0∈L2(Ω) such that, for every v∈L2(Q), the corresponding solution (y, q) to (4)–(5) satisfies q(0) 6= 0, that is to say, the functional φcannot be insensitized (see Theorem 2 in [16]). On the other hand, when ω∩ O 6=∅,y0= 0 and f is a C1globally Lipschitz-continuous function such that f(0) = 0, in [16] it is proved: If ξ∈L2(Q; exp(M/2t)) with M>0large enough, there exists v∈L2(Q) such that the solution (y, q)to (4)–(5) satisfies (6) (see Theorem 1 in [16]). Here L2(Q; exp(M/2t)) stands for the weighted Hilbert space L2(Q; exp(M/2t)) = {ξ∈L2(Q) : kexp(M/2t)ξkL2(Q)<∞}. The proof combines a similar null controllability result for linear coupled parabolic systems and an appropriate fixed-point argument. More precisely, the author first linearizes the system and shows its null controllability with controls uniformly bounded in L2(Q) when the potentials of the linearized system lie in a bounded set of L∞(Q). Since fis a globally Lipschitz-continuous function, this fact suffices 3 for proving that the fixed-point mapping maps L2(Q) into a convex compact set of L2(Q). The main goal of the present paper is to analyze the existence of controls that insensitize φ, that is to say, to study the null controllability properties of the coupled system (4)–(5) when fhas a superlinear growth at infinity. In accordance with Theorems 1 and 2 in [16], we will assume from now on that y0= 0 and ω∩ O 6=∅. In the study of the null controllability of semilinear parabolic systems with superlinear nonlinearities, additional technical difficulties arise. Let us recall what happens in the simpler case of one semilinear heat equation. During the last years, the controllability properties of (∂ty−∆y+f(y) = v1ωin Q, y= 0 on Σ, y(x, 0) = y0(x) in Ω,(7) with a superlinear nonlinearity f(y) has been thoroughly studied by several authors. As in the sublinear case, the technique to deal with this problem combines a fixed-point reformulation together with the study of the null controllability of linear problems of the form (∂ty−∆y+ay =v1ωin Q, y= 0 on Σ, y(x, 0) = y0(x) in Ω, where a∈L∞(Q). Due to the superlinear growth of the nonlinearity it is necessary, to perform a fixed point argument, that the solution of the controlled linear problem belongs to L∞(Q). To this aim controls in Lr(Q), with r > (N+ 2)/2, must be built. Moreover, in the linear case it is necessary to analyze how the Lr–norm of the controls depends on kak∞. Let us mention some papers on this issue: 1. In [2], V. Barbu obtained null Lr(N)–controls vsuch that kvkLr(N)(Q)≤C(kak∞)ky0kL2(Ω), with r(N)∈·2,2(N+ 2) N−2¸if N≥3, r(N)∈[2,∞) if N= 2, and r(N)∈ [2,∞] if N= 1. The proof of this estimate is based on a global Carleman inequality for the adjoint problem (−∂tϕ−∆ϕ+aϕ = 0 in Q, ϕ= 0 on Σ, ϕ(x, T) = ϕ0(x) in Ω,(8) 4 due to A.V. Fursikov and O.Yu. Imanuvilov (see [11]). In fact, the author proved a sharp estimate of the constant appearing on the right-hand side of the Carleman inequality with respect to kak∞. Nevertheless, this technique can only be applied to the study of the null controllability of the superlinear heat equation (7) when N < 6. For other controllability results proved in a similar way, see [1]. 2. A second approach was developed by E. Fern´andez-Cara and E. Zuazua in [10]. They proved the “refined” observability inequality kϕ(0)k2 L2(Ω) ≤C(T, kak∞)µZZω×(0,T ) |ϕ|dx dt¶2 ,(9) for the solutions to (8), which implies the existence of a null control v∈L∞(Q) satisfying kvkL∞(Q)≤C(T, kak∞)ky0kL2(Ω). In this case, the observability inequality (9) was deduced combining a global Carleman estimate for the adjoint problem and the regularizing effect of the heat equation. The authors give the explicit dependence on Tand kak∞of the constant C(T, kak∞) in (9), which is essential to prove their nonlinear null controllability result. This technique was later used in [8] for a nonlinear heat equation with a superlinear term f(y, ∇y). The study of the null controllability properties of the superlinear coupled system (4)–(5) is more intricate. In this case, as in [1], [2] and [10], null Lr–controls (with r > (N+2)/2) for the corresponding coupled linear system must also be built. Again, Lr–estimates of the controls are needed. The technique introduced by V. Barbu could be applied in this case if the condition N < 6 is imposed (which does not seem to be a natural restriction on N). On the other hand, the approach proposed by E. Fern´andez-Cara and E. Zuazua cannot be applied since the regularizing effect of the associated linear adjoint problem involves two functions ϕand ψ(see (21)– (22)) while the corresponding “refined” observability inequality should only involve ψ(recall that the control vonly appears in (4)). In [4], the authors introduced a new technique of construction of null Lr–controls for coupled linear parabolic systems. This strategy made it possible to generalize Theorem 1 in [16] to more general nonlinearities. The proof of this insensitivity result as well as the above-mentioned technique, sketched in [4], are developed in the present paper. To our knowledge, this is the first insensitivity result in the literature for semilinear heat equations with a superlinear nonlinearity. 5 Main results The first relevant result in this paper is the following one: Theorem 1.1 Assume that ω∩ O 6=∅and y0= 0. Let fbe a C1function defined on IR verifying f′′ ∈L∞ loc(IR),f(0) = 0 and lim |s|→∞ f′(s) log(1 + |s|)= 0.(10) Let r∈µN 2+ 1,∞¶be given. Then, for any ξ∈Lr(Q)such that ZZQ exp µ1 t3¶|ξ|2dx dt < ∞,(11) there exists a control function v∈Lr(Q)insensitizing the functional φgiven by (2). ¤ Condition (11) means that the given source term ξis asked to decay rapidly to zero close to the initial time t= 0. As seen before, a similar assumption, if a weaker one, is required in the case when a globally Lipschitz-continuous nonlinearity is considered. Observe that hypothesis f(0) = 0 is in accordance with assumptions on ξ(see [16] for both considerations). As usual in the study of controllability problems for nonlinear equations, a controllability result for a linearized version of (4)–(6) will be first proved. We will then apply a fixed-point argument to deal with the general case. The structure of the proof is quite general (for other controllability results proved in a similar way, see [10], [17], ...). Remark 1Hypothesis (10) is fulfilled by certain superlinear nonlinearities fsuch as |f(s)|=|p1(s)|logα(1 + |p2(s)|) for all |s| ≥ s0>0, with α∈[0,1), where p1and p2are real affine functions. For nonlinearities f∈C1(IR) satisfying hypothesis (10), system (1) admits a global solution when the data ξ,v,y0and ˆy0are regular enough. For instance, by linearization and the later application of a fixed-point argument, one can prove that for given ξand vin Lr(Q) and y0,ˆy0∈W2−2/r,r(Ω) ∩W1,r 0(Ω), with r > N/2 + 1, system (1) possesses a unique solution in Lr(0, T;W2,r(Ω)). Let us remark that throughout the paper, this regularity will be assumed on the data. ¤ 6 Our second main result is of a negative nature. Theorem 1.2 There exist C1functions fverifying f′′ ∈L∞ loc(IR),f(0) = 0 and |f(s)| ∼ |s|logα(1 + |s|)as |s| → ∞,(12) with α > 2, and there exist source terms ξ∈Lr(Q)satisfying (11), for which it is not possible to find control functions insensitizing the functional φgiven by (2).¤ For the proof of this result, we choose f(s) = Z|s| 0 logα(1 + |σ|)dσ for all s∈IR and we prove a localized estimate in Ω\ωof the corresponding solution yof (4) that shows that for certain source terms ξ, the control vcannot compensate the blow up phenomena occurring in Ω \ω. In view of Theorems 1.1 and 1.2, it would be interesting to analyze what happens when fsatisfies (12), with 1 ≤α≤2 (see Subsection 6.4 for further comments). The rest of this paper will be organized as follows. In the following Section, we present some technical results which will be proved in an appendix. Section 3 provides an exhaustive study of the linear case. In Section 4, we analyze the nonlinear case and prove Theorem 1.1. The fifth Section is devoted to the proof of Theorem 1.2. In Section 6 we give other insensitivity results and discuss some open problems. 2 Some technical results In this Section we state some technical results which will be used later. They are known results on the local regularity for the solutions to the linear heat equation. Nevertheless, we include the proof of these results in an appendix so as to obtain the explicit dependence of the constants on the potentials, which will be essential in our analysis. First, let us present the following notation, which is used all along this paper. For r∈[1,∞] and a given Banach space X,k · kLr(X)will denote the norm in Lr(0, T;X). For simplicity, the norm in Lr(Q) will be represented by k · kLr, for r∈[1,∞), and k · k∞will denote the norm in L∞(Q). For r∈[1,∞) and any open set V ⊂ IRN, we will consider the Banach space Xr(0, T;V) = ©u∈Lr(0, T;W2,r(V)) : ∂tu∈Lr(0, T;Lr(V))ª, 7 and its norm, defined by kukXr(0,T;V)=kukLr(W2,r(V)) +k∂tukLr(Lr(V)). In particular, we will consider the space Xr=Xr(0, T; Ω) and its norm, denoted by k · kXr. The norm in the space L2(0, T;H2(Ω)) ∩C([0, T]; H1(Ω)) will be denoted by k · kL2(H2)∩C(H1). On the other hand, for α∈(0,1) and u∈C0(Q), we define the quantity [u]α,α 2= sup Q |u(x, t)−u(x′, t)| |x−x′|α+ sup Q |u(x, t)−u(x, t′)| |t−t′|α 2 . We will consider the space Cα, α 2(Q) = ©u∈C0(Q) : [u]α,α 2<∞ª,which is a Banach space with its natural norm |u|α, α 2;Q=kuk∞+ [u]α,α 2.We will also consider the Banach space defined by C1+α, 1+α 2(Q) = (u∈C0(Q) : ∂u ∂xi ∈Cα,α 2(Q)∀i, sup Q |u(x, t)−u(x, t′)| |t−t′|1+α 2 <∞). The following holds: Proposition 2.1 Let a∈L∞(Q)and F∈L2(Q)be given. Let us consider a solution y∈L2(0, T;H2(Ω)) ∩C([0, T]; H1(Ω)) to (∂ty−∆y+ay =Fin Q, y= 0 on Σ, y(x, 0) = 0 in Ω.(13) a) Let V ⊂ Ω(resp. B ⊂⊂ Ω) be an open set. Let us suppose that F∈ Lr(0, T;Lr(V)) (resp. F∈Lr(0, T;Lr(Ω \B)), with r∈(2,∞). Then, for any open set V′⊂⊂ V (resp. B ⊂⊂ B′⊂⊂ Ω) one has y∈Xr(0, T;V′)(resp. y∈Xr(0, T; Ω \B′)). Moreover, there exist positive constants C=C(Ω, T, N, r, V,V′)(resp. C= C(Ω, T, N, r, B,B′)) and K=K(N)such that kykXr(0,T;V′)≤C(1 + kak∞)K£kFkLr(Lr(V)) +kykL2(H2)∩C(H1)¤.(14) (resp. kykXr(0,T ;Ω\B′)≤C(1 + kak∞)KhkFkLr(Lr(Ω\B)) +kykL2(H2)∩C(H1)i). 8 b) Assume, in addition, that F∈Lr(0, T;W1,r(V)), with ras above, and ∇a∈ Lγ(Q)N, with γ=       max µr, N 2+ 1¶if r6=N 2+ 1, N 2+ 1 + εif r=N 2+ 1, (15) and εbeing an arbitrarily small positive number. Then, for any open set V′⊂⊂ V, one has y∈Lr(0, T;W3,r(V′)), ∂ty∈Lr(0, T;W1,r(V′)) and, for a new positive constant C=C(Ω, T, N, r, V,V′), the following estimate holds kykLr(W3,r(V′)) +k∂tykLr(W1,r(V′)) ≤CH£kFkLr(W1,r(V)) +kykL2(H2)∩C(H1)¤, where H=H(N, kak∞,k∇akLγ) = (1 + kak∞)K+1(1 + k∇akLγ), K=K(N)being as in (14).¤ We will also use the following result, which is immediately obtained rewriting Lemma 3.3, p. 80, in [14] with our notation. Lemma 2.2 Let Ω⊂IRN,N≥1, be an open set with ∂Ω∈C2. The following continuous embeddings hold: i. If r < N 2+ 1, then Xr֒→Lp(Q), where 1 p=1 r−2 N+ 2. ii. If r=N 2+ 1, then Xr֒→Lq(Q)for all q < ∞. iii. If N 2+ 1 < r < N + 2, then Xr֒→Cβ, β 2(Q), with β= 2 −N+ 2 r. iv. If r=N+ 2, then Xr֒→Cl, l 2(Q)for all l∈(0,1). v. If r > N + 2, then Xr֒→C1+α, 1+α 2(Q), where α= 1 −N+ 2 r.¤ 9 Let us remark that in the context of the heat equation, the previous technique provides a new method of construction of regular controls starting from controls in L2(Q). This will make it possible to give a new proof of known null controllability results for nonlinear heat equations. A local result on the null controllability for the classical heat equation and a local result on insensitizing controls for a semilinear heat equation, both with nonlinear Fourier boundary conditions, can also be obtained by using a similar construction (see [7] and [6]). 4 The nonlinear case: proof of Theorem 1.1 In this Section, we will apply an appropriate fixed-point argument to treat the nonlinear case. In a first step, fwill be assumed to be a C2function. The general case will be studied in Subsection 4.2. 4.1 The case when fis a C2function Let f∈C2(IR) be a function verifying f(0) = 0 and (10). Let ξ∈Lr(Q) satisfy hypothesis (11), with r > N/2 + 1. Let us define g(s) =    f(s) sif s6= 0, f′(0) if s= 0. Then g,f′∈C0(IR) and f(s) = g(s)sfor all s∈IR. Since f(0) = 0, hypothesis (10) on f′implies a similar one on g, that is, lim |s|→∞ g(s) log(1 + |s|)= 0. Thus, for each ε > 0, there exists a positive constant Cε(which only depends on ε and on the function f) such that |g(s)|+|f′(s)| ≤ Cε+εlog(1 + |s|) for all s∈IR.(37) Let us recall that, for r > N 2+ 1, we defined Zr=C0(Q)∩Lr(0, T;W1,r(Ω)). 16 For any z∈B(0; R)⊂Zr,R > 0 to be determined later, we consider the linear controllability problem (∂ty−∆y+g(z)y=ξ+v1ωin Q, y= 0 on Σ, y(x, 0) = 0 in Ω,(38) (−∂tq−∆q+f′(z)q=y1Oin Q, q= 0 on Σ, q(x, T) = 0 in Ω,(39) q(x, 0) = 0 in Ω. (40) Let us observe that (38)–(39) are of the form (16)–(17) with potentials (a=az=g(z)∈L∞(Q), b=bz=f′(z)∈L∞(Q)∩Lr(0, T;W1,r(Ω)) (γ=rin this case). In view of Corollary 3.4, for any z∈B(0; R)⊂Zrthere exists a control vz∈ Lr(Q) such that the corresponding solution (yz, qz) to (38)–(39) lies in Zr×Zrand satisfies (40). Moreover, estimates kyzkZr≤C1(Ω, ω, O, T, z)µkξkLr+ exp µC 2Hz¶° ° ° ° exp µCMz 2t¶ξ° ° ° °L2¶(41) and kvzkLr≤C2(Ω, ω, O, T, z)µkξkLr+ exp µC 2Hz¶° ° ° ° exp µCMz 2t¶ξ° ° ° °L2¶(42) hold, where C1(Ω, ω, O, T, z) = exp [C(1 + kg(z)k∞+kf′(z)k∞)] , C2(Ω, ω, O, T, z) = exp [C(1 + kg(z)k∞+kf′(z)k∞)] (1 + kf′′(z)∇zkLr), Mz= 1 + kg(z)k2/3 ∞+kf′(z)k2/3 ∞+kg(z)−f′(z)k1/2 ∞, Hz= 1 + kg(z)k∞+kf′(z)k∞+kg(z)k2/3 ∞+kf′(z)k2/3 ∞+kg(z)−f′(z)k1/2 ∞, and C=C(Ω, ω, O, T)>0. Due to hypothesis (11) on ξ, one has ZZQ exp µCMz t¶|ξ|2dx dt =ZZQ exp µCMz t−1 t3¶exp µ1 t3¶|ξ|2dx dt ≤exp ¡C M3/2 z¢ZZQ exp µ1 t3¶|ξ|2dx dt, (43) 17 C=C(Ω, ω, O, T) being a new positive constant. Then, from inequalities (41), (42) and (43), and using the convexity of the real function s7→ s3/2, it can be estimated ||yz||Zr≤C1(Ω, ω, O, T, z)µkξkLr+° ° ° ° exp µ1 2t3¶ξ° ° ° °L2¶,(44) and ||vz||Lr≤C2(Ω, ω, O, T, z)µkξkLr+° ° ° ° exp µ1 2t3¶ξ° ° ° °L2¶ ≤˜ C(Ω, ω, O, T, R)µkξkLr+° ° ° ° exp µ1 2t3¶ξ° ° ° °L2¶, (45) where ˜ C(Ω, ω, O, T, R) is a positive constant independent of z. Let us define A:z∈B(0; R)⊂Zr7−→ A(z)⊂Lr(Q), with A(z) = {v∈Lr(Q) : (y, q) satisfies (38)–(40), v verifying (45)}, and let Λ be the set-valued mapping defined on Zras follows: Λ : z∈B(0; R)⊂Zr7−→ Λ(z)⊂Zr, with Λ(z) = {y∈Zr: (y, q) solves (38)–(39) with v∈ A(z), y satisfying (44)}. Let us prove that Λ fulfills the assumptions of Kakutani’s fixed-point Theorem. In the first place, one can check that Λ(z) is a non-empty closed convex subset of Zrfor fixed z∈Zr, due to the linearity of systems (38) and (39). In fact, in view of Theorem 9.1 in [14] and (45), Λ(z) is uniformly bounded in Xr, the space introduced in Section 2. Recall that for r > N/2 + 1, Xris continuously embedded in the H¨older space Cβ, β 2(Q), with β= 2 −(N+ 2)/r (see Lemma 2.2). Then, there exists a compact set K⊂Zr,Konly depending on R, such that Λ(z)⊂K∀z∈B(0; R).(46) Let us now prove that Λ is an upper hemicontinuous multivalued mapping, that is to say, for any bounded linear form µ∈Z′ r, the real-valued function z∈B(0; R)⊂Zr7−→ sup y∈Λ(z) hµ, yi 18 is upper semicontinuous. Correspondingly, let us see that Bλ,µ =(z∈B(0; R) : sup y∈Λ(z) hµ, yi ≥ λ) is a closed subset of Zrfor any λ∈IR, µ∈Z′ r(see [8] for a similar proof). To this end, we consider a sequence {zn}n≥1⊂Bλ,µ such that zn→zin Zr. Our aim is to prove that z∈Bλ,µ. Since all the Λ(zn) are compact sets, by the definition of Bλ,µ, for any n≥1 there exists yn∈Λ(zn) such that hµ, yni= sup y∈Λ(zn) hµ, yi ≥ λ. (47) Recalling now the definition of A(zn) and Λ(zn), let vn∈ A(zn), qn∈Zrbe such that (yn, qn) solves (38)–(40) with control vnand potentials g(zn), f′(zn). From (44) and (45), ynand vnsatisfy ||yn||Zr≤C1(Ω, ω, O, T, zn)µkξkLr+° ° ° ° exp µ1 2t3¶ξ° ° ° °L2¶, and ||vn||Lr≤C2(Ω, ω, O, T, zn)µkξkLr+° ° ° ° exp µ1 2t3¶ξ° ° ° °L2¶. Thus, {yn}(resp. {vn}) is uniformly bounded in Zr(resp. in Lr(Q)). In particular, (46) gives us {yn} ⊂ K. Hence there exist subsequences, still denoted by {yn}and {vn}, such that yn→ystrongly in Zr, and vn→vweakly in Lr(Q). Since gand f′′ are continuous functions, one also has g(zn)→g(z) in C0(Q), f′(zn)→f′(z) in C0(Q), and f′′(zn)∇zn→f′′(z)∇zin Lr(Q)N. 19 Passing to the limit, one deduces that yand the associated function qsolve (38)–(40) with control function v(and potentials g(z), f′(z)). Moreover, ¯yand ¯vsatisfy (44) and (45), that is, v∈ A(z) and y∈Λ(z). Then, taking limits in (47), it holds that sup y∈Λ(z) hµ, yi ≥ hµ, yi ≥ λ, whence it is deduced that z∈Bλ,µ and hence, Λ is upper hemicontinuous. Finally, let us see that there exists R > 0 such that Λ¡B(0; R)¢⊂B(0; R).(48) Let R > 0 be, to be determined. For any z∈B(0; R)⊂Zr, from (44) and (37) it is observed that each y∈Λ(z) satisfies ||y||Zr≤exp [C(1 + Cε+εlog(1 + ||z||∞))] µkξkLr+° ° ° ° exp µ1 2t3¶ξ° ° ° °L2¶ ≤exp [C(1 + Cε)] (1 + R)Cε µkξkLr+° ° ° ° exp µ1 2t3¶ξ° ° ° °L2¶, with C=C(Ω, ω, O, T)>0. Thus, choosing ε= 1/2C, we get ||y||Zr≤C(1 + R)1/2µkξkLr+° ° ° ° exp µ1 2t3¶ξ° ° ° °L2¶, from which we infer the existence of R > 0 large enough such that (48) is satisfied. The Kakutani Fixed-point Theorem thus applies, which ends the proof when f is a C2function. Remark 2Let us notice that two different nonlinearities ffor which the positive constant Cεin (37) coincide, lead to the same R > 0. This fact will be used in the following Subsection when studying the general case. ¤ 4.2 The general case It is now assumed that fis a C1function satisfying f′′ ∈L∞ loc(IR), f(0) = 0 and (10) and let ξbe as above. We consider a function ρ∈ D(IR) such that ρ≥0 in IR,supp ρ⊂[−1,1] and ZIR ρ(s)ds = 1. 20 For any n≥1, let us set ρn(s) = nρ(ns) for all s∈IR, Fn=ρn∗f, fn(·) = Fn(·)−Fn(0) and gn(s) =    fn(s) sif s6= 0, f′ n(0) if s= 0. Due to the properties of ρnand the convolution, as well as the hypothesis on f, one can prove that fnand gnhave the following properties: (i)gnand f′′ nare continuous functions and fn(0) = 0 for all n≥1. (ii)fn→fin C1(K) for all compact set K⊂IR. (iii)gn→guniformly on compact sets of IR. (iv) For any given M > 0 there exists a positive constant C(M) such that sup |s|≤M (|gn(s)|+|f′ n(s)|+|f′′ n(s)|)≤C(M) for all n≥1. (v) It also holds that: lim |s|→∞ |f′ n(s)|+|gn(s)| log(1 + |s|)= 0 uniformly in n, that is, for any ε > 0 there exists Mε>0 such that |gn(s)|+|f′ n(s)| ≤ εlog(1 + |s|) for all |s| ≥ Mεand n≥1. In particular, the last two properties imply that, for any ε > 0, there exists Cε>0, only depending on εand not on the function fn, such that |gn(s)|+|f′ n(s)| ≤ Cε+εlog(1 + |s|) for all s∈IR and n≥1.(49) As it was proved in the previous Subsection, for any n≥1 there exists a control function vn∈Lr(Q), with supp vn⊂ω×[0, T], such that the following cascade of systems (∂tyn−∆yn+fn(yn) = ξ+vn1ωin Q, yn= 0 on Σ, yn(x, 0) = 0 in Ω,(50) 21 (−∂tqn−∆qn+f′ n(yn)qn=yn1Oin Q, qn= 0 on Σ, qn(x, T) = 0 in Ω,(51) admits a solution (yn, qn)∈Zr×Zrsatisfying qn(x, 0) = 0 in Ω.(52) Moreover, estimates ||yn||Zr≤ C1(Ω, ω, O, T, yn)µkξkLr+° ° ° ° exp µ1 2t3¶ξ° ° ° °L2¶,(53) and ||vn||Lr≤ C2(Ω, ω, O, T, yn)µkξkLr+° ° ° ° exp µ1 2t3¶ξ° ° ° °L2¶,(54) hold, where C1(Ω, ω, O, T, yn) = exp [C(1 + kgn(yn)k∞+kf′ n(yn)k∞)] , C2(Ω, ω, O, T, yn) = exp [C(1 + kgn(yn)k∞+kf′ n(yn)k∞)] (1 + kf′′ n(yn)∇ynkLr), and C=C(Ω, ω, O, T)>0. Let us recall that ynis, for any n≥1, a fixed point of a set-valued mapping Λndefined from Zronto itself. In view of estimates (53) and (54), and taking into account (49) and Remark 2, one deduces, arguing as in the previous Subsection, that there exists R > 0 large enough, and independent of n, such that Λn¡B(0; R)¢⊂B(0; R). In addition, {yn}(resp. {vn}) is uniformly bounded in Zr(resp. in Lr(Q)). Indeed, reasoning as in Subsection 4.1, {yn}is uniformly bounded in the space Xrand hence, rbeing grater than N/2 + 1, there exists a compact set Kin Zrsuch that {yn} ⊂ K. Thus, up to a subsequence, one has yn→ystrongly in Zr and vn→vweakly in Lr(Q), with v∈Lr(Q) and y∈K⊂Zr. Taking now into account properties (ii) and (iii), one also has gn(yn)→g(y) and f′ n(yn)→f′(y) in C0(Q). 22 Hence, passing to the limit in (50)–(52), one infers that yand the corresponding qsolve (38)–(40), that is, we have found a control vin Lr(Q) insensitizing φ. This ends the proof of Theorem 1.1. ¤ 5 Proof of Theorem 1.2 This Section is devoted to proving the insensitivity result of a negative nature stated in Theorem 1.2. To do so, we will show that for certain functions fas in the statement and certain source terms ξ∈Lr(Q) vanishing for t∈(0, t0), with t0∈ (0, T), whatever the control vis, the corresponding solution yto (4) blows up before the time t=Tand hence, the functional φcannot be insensitized. We will follow the proof of Theorem 1.1 in [10], where the lack of null controllability of a semilinear heat equation is proved. Let us consider the following function f(s) = Z|s| 0 logα(1 + σ)dσ for all s∈IR, with α > 2. It is easy to check that fis a convex function, f(s)s < 0 for all s < 0 and |f(s)|∼|s|logα(1 + |s|) as |s| → ∞. Let ρ∈ D(Ω) be such that ρ≥0 in Ω, ρ ≡0 in ω, and ZΩ ρ(x)dx = 1. For a fixed t0∈(0, T), we set ξ(x, t) = (0 if t∈[0, t0], −(M+k) if t∈(t0, T], where Mis a positive constant which will be chosen later and kis given by k=1 2ZΩ ρf∗µ2|∆ρ| ρ¶dx. Here f∗is the convex conjugate of the convex function f(the function ρcan be taken in such a way that ρf∗(2|∆ρ|/ρ)∈L1(Ω), see [10]). 23 Let ybe a solution to (4), associated to a control vand ξ, defined in the maximal interval [0, T∗). Multiplying the equation in (4) by ρand integrating in Ω, we get d dt ZΩ ρy dx =ZΩ ρ∆y dx −ZΩ ρf(y)dx +ZΩ ρξ dx. We also set z(t) = −ZΩ ρ(x)y(x, t)dx, ∀t∈[t0, T∗). Using the properties of f(see [10] for the details), we obtain      z′(t)≥M+1 2f(z(t)), t ∈[t0, T∗), z(t0) = z0=−ZΩ ρ(x)y(x, t0)dx. Defining G(z0;s) = Zs z0 2 f(σ) + 2Mdσ, ∀s≥z0, we can prove that T∗≤t0+ sup t∈[t0,T∗) G(z0;z(t)) ≤t0+Z∞ z0 2 f(σ) + 2Mdσ. Thus, for M > 0 large enough, the solution yblows up in L1(Ω) before T. This ends the proof. 6 Further comments, results and open problems 6.1 On the construction of regular controls for parabolic null controllability problems The technique of construction of regular controls (starting from L2–controls) introduced in the present paper can be applied to the study of the null controllability of (7) not only when fjust depends on the state ybut also when f=f(y, ∇y). In fact, controls in Cα, α 2(Q), with α∈(0,1], can be obtained for potentials regular enough. Let us describe this strategy in the linear case. We consider the null controllability problem (∂ty−∆y+B· ∇y+ay =v1ωin Q, y= 0 on Σ, y(x, 0) = y0in Ω,(55) 24 y(x, T) = 0 in Ω, where y0∈L2(Ω), a∈L∞(Q) and B∈L∞(Q)N. The previous null controllability problem is equivalent to      ∂tq−∆q+B· ∇q+aq =−η′(t)Y+v1ωin Q, q= 0 on Σ, q(x, 0) = 0 in Ω, q(x, T) = 0 in Ω, (56) where q=y−η(t)Y,η∈C∞([0, T]) satisfies η≡1 in [0, T/3], η ≡0 in [2T/3, T], and Ysolves (55) with v= 0. Suppose that there exists a control ˆv∈L2(Q) solving (56), with supp ˆv⊂B0×[0, T], and let ˆqbe the associated state (here B0⊂⊂ ωis a non-empty open set). Then q= (1 −θ(x))ˆqtogether with v=θ(x)η′(t)Y+ 2∇θ· ∇ˆq+ (∆θ)ˆq−(B· ∇θ)ˆq solve the null controllability problem (56), where θ∈ D(ω) verifies θ≡1 in B0. Following Subsection 3.2, one can prove that v∈L∞(Q) and kvk∞≤C(T, Ω, ω, kak∞,kBk∞)kˆvkL2. The explicit dependence of the constant Con kak∞and kBk∞can also be obtained. Moreover, if B∈Cα, α 2(Q), the control vis proved to lie in Cα, α 2(Q) and vverifies a Cα,α 2–estimate similar to the previous one. This strategy has been used in [6] to construct h¨olderian controls for a null controllability problem for the heat equation with nonlinear boundary Fourier conditions (see also [7]). 6.2 Other insensitivity results The proof of Theorem 1.1 can be adapted to prove other insensitivity results for system (1). 1. Theorem 1.1 is still true under a slightly more general condition on f. To be precise, there exists l1(Ω, ω, O, T)>0 such that, if hypothesis (10) is replaced by lim sup |s|→∞ |f′(s)| log(1 + |s|)≤l1, a control function insensitizing the functional given by (2) can be found for any given source term ξas in Theorem 1.1. 25 Then, recalling that w0|V0=y|V0, one infers from (69) and (68) that y∈Lr(0, T;W2,p0(V0)) and the following estimate is satisfied kykLr(W2,p0(V0)) ≤C(1 + kak∞)¡kFkLr(Lr(V)) +kykL2(H2)∩C(H1)¢. If r≤2 + 4 N, that is, if p0=r, the first point is already proved. Let us now suppose that r > 2 + 4 N(i.e. 2 < p0=2Nr Nr −4< r). We will now use the following Lemma, which will be proved at the end of this Appendix. Lemma A.1 Let a∈L∞(Q)and F∈L2(Q)∩Lr(0, T;Lr(V)) be, with V ⊂ Ωan arbitrary open set and r∈(2,∞). Let y∈L2(0, T;H2(Ω))∩C([0, T]; H1(Ω)) satisfy (13). Let ω0and ω1be two open subsets of Ωsuch that ω1⊂⊂ ω0⊂ V. Let us assume that y∈Lr(0, T;W2,r0(ω0)), with r0∈[2, r). Then, y∈Lr(0, T;W2,r1(ω1)) and ∂ty∈Lr(0, T;Lr1(ω1)), with r1=   min µr, Nr0 N−r0¶if r0< N, rif r0≥N. (70) Furthermore, the following estimate holds: kykLr(W2,r1(ω1)) +k∂tykLr(Lr1(ω1)) ≤C(1 + kak∞)¡kFkLr(Lr(V)) +kykLr(W2,r0(ω0))¢, where Cis a positive constant depending on Ω,T,N,r,ω0and ω1.¤ We apply this Lemma for i= 1,...,I, replacing ω0,ω1,r0and r1, respectively, by Vi−1,Vi,pi−1and pi, with 1 pi =1 p0 −i Nfor i= 1,...,I−1 and pI=r. This yields y∈Lr(0, T;W2,pi(Vi)), ∂ty∈Lr(0, T;Lpi(Vi)), 1 ≤i≤I, and the corresponding estimates. In order to determine I, observe that we may go on applying Lemma A.1 while pi−1< N and pi< r, that is to say, while i < N(r−2) −4 2r. 32 Thus, in Isteps, with I=·N(r−2) −4 2r¸+ 1 ([σ] being the integer part of the real number σ), we have y∈Xr(0, T;V′) together with estimate (14), with K=N 2+ 2, which is a uniform bound of I+ 1. b) Suppose now that F∈Lr(0, T;W1,r(V)), ras above, and ∇a∈Lγ(Q)N, with γgiven by (15). Let us consider a new open set ˜ Vsuch that V′⊂⊂ ˜ V ⊂⊂ V. In view of point a), y∈Xr(0, T;˜ V) and kykXr(0,T;˜ V)≤C(1 + kak∞)K£kFkLr(Lr(V)) +kykL2(H2)∩C(H1)¤.(71) To end the proof, let us see that, in addition, ∂iy∈Xr(0, T;V′),1≤i≤N, and that the following estimate is satisfied k∂iykXr(0,T;V′)≤C(1 + kak∞)K+1(1 + k∂iakLγ)£kFkLr(W1,r(V)) +kykL2(H2)∩C(H1)¤, where ∂iydenotes the derivative of ywith respect to xi, 1 ≤i≤N. To do so, let us set wi=ζ1∂iyfor a fixed i∈ {1,...,N}, with ζ1∈ D(˜ V) a function such that ζ1≡1 in V′. Then, wisolves (67) with G=Gi given by Gi=ζ1∂iF−ζ1a∂iy−ζ1y∂ia−2∇ζ1· ∇ (∂iy)−(∆ζ1)∂iy. (72) Let us see that Gi∈Lr(Q). We will study in detail the term ζ1y∂ia. Under assumptions on y,aand F, it is direct to see that the other terms in (72) lie in Lr(Q). Let us observe that ζ1y∈Xr. In view of Lemma 2.2, since ∇a∈Lγ(Q)N, with γ given by (15), and recalling that the H¨older space Cl, l 2(Q) is continuously embedded in C0(Q), for all r∈(2,∞) one may infer that the term into consideration, ζ1y∂ia, lies in Lr(Q) and one has kζ1y∂iakLr≤Ckζ1ykXrk∂iakLγ. 33 Hence, coming back to (72), Gi∈Lr(Q) and one can estimate kGikLr≤Chk∂iFkLr(Lr(V)) + (kak∞+k∂iakLγ+ 1) kykXr(0,T ;˜ V)i.(73) The regularizing effect of the heat equation (see [12] and [13]) yields wi∈Xr, with kwikXr≤CkGikLr.(74) Then, from (74), (73) and (71), one deduces kwikXr≤C(1 + kak∞)K+1 (1 + k∂iakLγ)£kFkLr(W1,r(V)) +kykL2(H2)∩C(H1)¤, with C=C(Ω, T, N, r, V,V′) and K=K(N) the same as above. Finally, just taking into account that wi≡∂iyin V′, point b) is proved. ¤ Remark 3Following the previous proof, one easily observes that the same result can be obtained when replacing hypothesis y∈L2(0, T;H2(Ω)) ∩C([0, T]; H1(Ω)) by y∈L2(0, T;H1 loc(Ω)) ∩L∞(0, T;L2 loc(Ω)). This fact just affects the number Iof steps required to prove point a). ¤ We end this Appendix by giving the Proof of Lemma A.1: Let us consider a function ζ∈ D(ω0) such that ζ≡1 in ω1and let us set u=ζy. Then, usolves (67), with G=ζF −[ζay + 2∇ζ· ∇y+ (∆ζ)y]. The regularity of yand usual Sobolev embeddings give G∈Lr(0, T;Lr1(Ω)), where r1is given by (70), and the estimate kGkLr(Lr1(Ω)) ≤C(1 + kak)∞£kFkLr(Lr(V)) +kykLr(W2,r0(ω0))¤(75) (here Cdepends on the open sets ω0and ω1). Then, again due to the regularizing properties of the heat equation, one deduces that u∈Lr(0, T;W2,r1(Ω)), ∂tu∈Lr(0, T;Lr1(Ω)) and kukLr(W2,r1(Ω)) +k∂tukLr(Lr1(Ω)) ≤CkGkLr(Lr1(Ω)). Finally, taking into account that u|ω1=y|ω1and inequality (75), the result follows. ¤ 34 Acknowledgements. The authors thank the referees for their interesting comments and suggestions. References [1] Anita, S., Barbu, V., Null controllability of a nonlinear convective heat equations, ESAIM:COCV 5 (2000), 157–173. 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