Insensitizing controls for a heat equation with a nonlinear term involving the state and the gradient

Insensitizing controls for a heat equation with a nonlinear term involving the state and the gradient ? O. Bodart aM. Gonz´alez-Burgos bR. P´erez-Garc´ıa b,∗ aUniversit´e Blaise-Pascal, Laboratoire de Math´ematiques Appliqu´ees, UMR CNRS 6620, Clermont-Ferrand 2, 63177 Aubi`ere, France bUniversidad de Sevilla, Dpto. Ecuaciones Diferenciales y An´alisis Num´erico, Aptdo. 1160, 41080 Sevilla, Spain Abstract In this paper we present two results on the existence of insensitizing controls for a heat equation in a bounded domain of IRN. We first consider a semilinear heat equation involving gradient terms with homogeneous Dirichlet boundary conditions. Then a heat equation with a nonlinear term F(y) and linear boundary conditions of Fourier type is considered. The nonlinearities are assumed to be globally Lipschitz-continuous. In both cases, we prove the existence of controls insensitizing the L2−norm of the observation of the solution in an open subset Oof the domain, under suitable assumptions on the data. Each problem boils down to a special type of null controllability problem. General observability inequalities are proved for linear systems similar to the linearized problem. The proofs of the main results in this paper involve such inequalities and rely on the study of these linear problems and appropriate fixed point arguments. Key words: controllability, nonlinear PDE of parabolic type, nonlinear gradient terms 1991 MSC: 93B05, 35K55, 35K05 ?This work has been partially financed by D.G.E.S. (Spain), Grant PB98–1134. ∗Corresponding author. Email addresses: [email protected]lermont.fr (O. Bodart), [email protected] (M. Gonz´alez-Burgos), [email protected] (R. P´erez-Garc´ıa). Preprint submitted to Nonlinear Analysis 20 February 2004 1 Setting the problems and main results Let Ω ⊂IRN,N≥1, be a bounded connected open set with boundary ∂Ω∈ C2. For T > 0, we denote Q= Ω ×(0, T ) and Σ = ∂Ω×(0, T ). Let ωand O be nonempty open subsets of Ω. We first consider the nonlinear heat equation:      ∂ty−∆y+f(y, ∇y) = ξ+v1ωin Q, y= 0 on Σ, y(x, 0) = y0(x) + τˆy0(x) in Ω, (1) where fis a C1globally Lipschitz-continuous function defined on IR ×IRN, ξ∈L2(Q) and y0∈L2(Ω) are given, ˆy0∈L2(Ω) is unknown with |ˆy0|L2(Ω) = 1, τis a small unknown real number, and v∈L2(Q) is a control function to be determined. Here, ∂tdenotes the time derivative and 1ωis the characteristic function of the set ω. Let us define Φ(y(·,·;τ, v)) = 1 2ZZO×(0,T )|y(x, t;τ, v)|2dx dt, (2) where y(·,·;τ, v) is the solution of (1) associated to τand v. A control function vis said to insensitize the functional Φ if ∂Φ(y(·,·;τ, v)) ∂τ τ=0 = 0,∀ˆy0∈L2(Ω) with |ˆy0|L2(Ω) = 1.(3) This problem, originally addressed by J.-L. Lions in [1], has been studied in the semilinear case for globally Lipschitz-continuous nonlinearities f=f(y). In [2], the authors weakened the underlying problem, defining approximately insensitizing controls. They proved the existence of such controls for unknown data in both the initial and boundary conditions. In [3] two mains results are given. On one hand, the author proves that one cannot expect the existence of insensitizing controls for every y0∈L2(Ω) when Ω \ω6=∅, even if f≡0. On the other hand, for y0= 0 and suitable assumptions on ξ, L. de Teresa proves the existence of controls such that (3) holds (see Theorem 1 in [3]). This result is generalized in [4] and [5] to nonlinearities with certain superlinear growth at infinity. One of the purposes of this paper is to extend Theorem 1 in [3] to the case of a semilinear heat equation where the nonlinearity is allowed to depend on both the state yand its gradient. Then, an insensitivity result for a semilinear heat equation with a nonlinear term F(y) and linear boundary conditions of Fourier type is given. The first insensitivity result we present in this paper is the following one: Theorem 1.1 Assume that ω∩O 6=∅and y0= 0. Let f: IR ×IRN→IR be aC1globally Lipschitz–continuous function such that f(0,0) = 0. Then, there 2 exists a positive constant Mdepending on Ω,ω,O,T, and fsuch that for any ξ∈L2(Q)verifying ZZQexp M t|ξ|2dx dt < ∞,(4) one can find a control function v∈L2(Q)insensitizing the functional Φgiven by (2). Adapting the computations in [1] and [2] to the present case, one gets that the existence of a control vsuch that (3) holds is equivalent to the existence of a control vsuch that the solution (y, q) of      ∂ty−∆y+f(y, ∇y) = ξ+v1ωin Q, y= 0 on Σ, y(x, 0) = y0(x) in Ω, (5)     −∂tq−∆q+∂sf(y, ∇y)q−∇·(∂pf(y, ∇y)q) = y1Oin Q, q= 0 on Σ, q(x, T) = 0 in Ω, (6) verifies q(x, 0) = 0 in Ω.(7) Here we noted (s, p)7→ f(s, p), s∈IR, p∈IRN,∂sfthe derivative of f with respect to s, and ∂pfthe gradient of fwith respect to p. Thus, so as to prove Theorem 1.1, we will restrict our attention to solve the nonstandard null controllability problem (5)–(7) for y0= 0. Let us now consider a semilinear heat equation with linear boundary conditions of Fourier type and partially known initial data:      ∂ty−∆y+F(y) = ξ+v1ωin Q, ∂ny+hy = 0 on Σ, y(x, 0) = y0(x) + τˆy0(x) in Ω, (8) where F: IR →IR is a C1globally Lipschitz-continuous function, h∈L∞(Σ) (at least), ξ,y0,τ, and ˆy0are as in (1), and v∈L2(Q) is again a control function to be determined. Here, ∂ndenotes the derivation with respect to the unit outward normal to ∂Ω and the norm in L∞(Σ) will be denoted by k·k∞;Σ. The next aim in this paper is to prove the existence of controls insensitizing the L2–norm of the observation of the solution of (8) in the open set O. Theorem 1.2 Assume that ω∩O 6=∅and y0= 0. Let F∈C1(IR) be a globally Lipschitz–continuous function (with Lipschitz constant L>0) satisfying F(0) = 0 and let h∈L∞(Σ) be such that ∂th∈L∞(Σ). Then, there exists a positive constant N(depending on Ω,ω,O,T,L,khk∞;Σ, and k∂thk∞;Σ) 3 such that, for any ξ∈L2(Q)verifying ZZQexp N t|ξ|2dx dt < ∞,(9) one can find a control function v∈L2(Q)insensitizing the functional defined in (2),y(·,·;τ, v)being the solution of (8) associated to τand v. In this case, there exists a control function vsuch that (3) holds if and only if there exists a control vsuch that the solution (y, q) of      ∂ty−∆y+F(y) = ξ+v1ωin Q, ∂ny+hy = 0 on Σ, y(x, 0) = y0(x) in Ω, (10)     −∂tq−∆q+F0(y)q=y1Oin Q, ∂nq+hq = 0 on Σ, q(x, T) = 0 in Ω, (11) verifies (7). To prove Theorem 1.2, it will then suffice to find an L2–control solving this new null controllability problem for y0= 0. As in [2] and [5], one can expect to choose a control function vsuch that the associated solution (y, q) of (5), (6) (with y0= 0), in addition to insensitize the functional Φ, it also verifies y(x, T ) = 0 in Ω. This can be done with an extra assumption on ξ: Theorem 1.3 Assume that ω∩O 6=∅and y0= 0. Let fbe as in Theorem 1.1. Then, there exists M>0(depending on Ω,ω,O,T, and f) such that for any ξ∈L2(Q)verifying ZZQexp M t(T−t)!|ξ|2dx dt < ∞, one can find a control function v∈L2(Q)insensitizing the functional Φgiven by (2) and such that the solution y(·,·;τ, v)|τ=0 of (1) (with y0= 0) satisfies y(x, T;τ, v)|τ=0 = 0 in Ω. We will not give the proof of this result, since it is similar to the one of Theorem 1.1. The rest of this paper is organized as follows: in section 2, we first prove an observability inequality that generalizes the one in [3]. This result is indeed one of the main results in this work and we will use it in other forthcoming papers (cf. [4], [5]). We also give an observability inequality for the case of linear Fourier boundary conditions, which will also be used in [7]. In section 3, we prove Theorems 1.1 and 1.2. We end with comments and conclusions. 4 2 The Observability Inequalities In this section we first prove an observability inequality that is a generalization of the one given in [3] to the case of linear systems with first order terms. This inequality will be the main tool in the proof of Theorem 1.1. We also give an observability inequality for linear systems with linear boundary conditions of Fourier type, which will be essential to prove Theorem 1.2. Let us consider ϕand ψsolving the following systems:      ∂tϕ−∆ϕ+cϕ +D·∇ϕ= 0 in Q, ϕ= 0 on Σ, ϕ(x, 0) = ϕ0(x) in Ω, (12)     −∂tψ−∆ψ+aψ −∇·(Bψ) = ϕ1Oin Q, ψ= 0 on Σ, ψ(x, T ) = 0 in Ω, (13) with a, c ∈L∞(Q), B, D ∈L∞(Q)N, and ϕ0∈L2(Ω). In the sequel, k · k∞ will denote the norm in both L∞(Q) and L∞(Q)N. It is known (cf. [8], p. 356) that ϕ, ψ ∈L2(0, T ;H1 0(Ω)) ∩C([0, T]; L2(Ω)), ∂tϕ, ∂tψ∈L2(0, T;H−1(Ω)). The main result in this section is the following one: Theorem 2.1 Assume that ω∩ O 6=∅. Then, there exist positive constants Mand Hsuch that, for every ϕ0∈L2(Ω), the corresponding solution (ϕ, ψ) of (12) and (13) satisfies ZZQexp −M t|ψ|2dx dt ≤HZZω×(0,T )|ψ|2dx dt. More precisely, M=C1 + TM0and H= exp"C M0+1 T+T1 + kak∞+kck∞+kBk2 ∞+kDk2 ∞!#, where C=C(Ω,ω,O)and M0is given by M0= 1 + kak2/3 ∞+kck2/3 ∞+ka−ck1/2 ∞+kBk∞+kB−Dk∞+kBk2 ∞+kDk2 ∞. The basic tool to prove this theorem is a global Carleman inequality for linear 5 systems of the form      ∂tz−∆z=Fin Q, z= 0 on Σ, z(x, 0) = z0(x) in Ω, (14) with z0∈L2(Ω) and Fin L2(Q) or in L2(0, T ;H−1(Ω)). For this we need to introduce an auxiliary function whose existence is guaranteed by the following result (see Lemma 1.1. in [9]): Lemma 2.2 Let B ⊂⊂ Ωbe a nonempty open subset. Then there exists a function η0∈C2(Ω) such that η0>0in Ω,η0= 0 on ∂Ωand |∇η0|>0in Ω\B. For a fixed nonempty open subset B ⊂⊂ Ω, let us set α0(x) = e2C∗kη0k∞−eC∗η0(x), x ∈Ω (15) and e α0(x) = e2C∗kη0k∞−e−C∗η0(x), x ∈Ω,(16) C∗being an appropriate positive constant depending on Ω and B. Using results in [9] and [10], one can prove the following Lemma 2.3 Let zbe the solution of (14) associated to z0∈L2(Ω). Let Bbe an open subset of Ω. There exist positive constants C0,σ0, and σ0(depending only on Ωand B) such that: (1) If F∈L2(Q), for every s≥s0=σ0(Ω,B) (T+T2)one has 1 sZZQe−2sαt(T−t)|∂tz|2+|∆z|2+sZZQe−2sαt−1(T−t)−1|∇z|2 +s3ZZQe−2sαt−3(T−t)−3|z|2≤C0 s3ZZB×(0,T )e−2sαt−3(T−t)−3|z|2 +ZZQe−2sα|F|2 , with αdefined by α(x, t) = α0(x) t(T−t), x ∈Ω, t ∈(0, T ), and α0given by (15). 6 (2) If F=f0+ N X i=1 ∂fi ∂xi ,with fi∈L2(Q),i= 0,1,...,N, then sZZQe−2sαt−1(T−t)−1|∇z|2+s3ZZQe−2sαt−3(T−t)−3|z|2 ≤C0 s3ZZB×(0,T )e−2sαt−3(T−t)−3|z|2+ZZQe−2sα|f0|2 +s2 N X i=1 ZZQe−2sαt−2(T−t)−2|fi|2 , for s≥s0=σ0(Ω,B) (T+T2),αbeing as above. The explicit dependence of s0on Thas been analyzed in [6]. Arguing in a similar way, we can obtain the precise way s0depends on T(also see [11]). We will also need the following technical lemma, which proof will be given further for the sake of clarity. Lemma 2.4 Let α0and αbe given as in Lemma 2.3, m0= minΩα0, and M0= maxΩα0. (1) One has s4e−2sαt−7(T−t)−7≤22e−77 m04 T−6, for every s≥7T2 23m0 and (x, t)∈Q. (2) For s≥3T2 2M0 , one has e−2sαt−3(T−t)−3≥Asexp (−Ms/t), for (x, t)∈ Ω×(0, T/2), with As= 26T−6exp −4M0s/T2, Ms= 2M0s/T. (17) (3) For every s≥0, one has e2sαt3(T−t)3≤2−6T6exp 25M0s 3T2!,(x, t)∈Ω×(T/4,3T/4) . Proof of Theorem 2.1: The structure of the proof is similar to that of Proposition 2 in [3]. In the first place, using appropriate Carleman inequalities, we prove an inequality involving the functions ϕand ψwhich solve (12) and (13). This inequality allows us to bound the function ϕin terms of ψ(see (32)). Combining it with energy estimates yields the result. Here we adapt the method exhibited in [3] to the lack of regularity in the term ∇·(Bψ) in equation (13). Moreover, the constants in the inequalities are explicit. Let us consider two open sets B1and B2such that B1⊂⊂ B2⊂ω∩ O. Applying Lemma 2.3 to the solution ϕof (12) with F=−cϕ −D·∇ϕand B=B1, there exist positive constants C1=C1(Ω, B1) and σ1=σ1(Ω, B1) 7 such that sZZQe−2sαt−1(T−t)−1|∇ϕ|2+s3ZZQe−2sαt−3(T−t)−3|ϕ|2 ≤C1s3ZZB1×(0,T )e−2sαt−3(T−t)−3|ϕ|2, (18) for every s≥s1, with s1=σ1(Ω, B1)T+T2+T2kck2/3 ∞+T2kDk2 ∞.(19) Then applying Lemma 2.3 to the solution ψof (13) with B=B1⊂B2and F=−aψ +∇ · (Bψ) + ϕ1O, there exist positive constants C2=C2(Ω, B1) and s2=σ2(Ω, B1) (T+T2+T2kak2/3 ∞+T2kBk2 ∞) such that, for s≥s2, one has sZZQe−2sαt−1(T−t)−1|∇ψ|2+s3ZZQe−2sαt−3(T−t)−3|ψ|2 ≤C2 s3ZZB2×(0,T )e−2sαt−3(T−t)−3|ψ|2+ZZO×(0,T )e−2sα|ϕ|2!. (20) In a first step, we prove an inequality which bounds ϕwith respect to ψ. Consider a function ξ1∈C∞ 0(Ω) such that 0≤ξ1≤1 in Ω, ξ1= 1 in B1,supp ξ1⊂B2⊂ω∩O,(21) ∆ξ1/ξ1/2 1∈L∞(Ω),and ∇ξ1/ξ1/2 1∈L∞(Ω)N.(22) This is achieved by setting ξ1=ζ4, with ζ∈C∞ 0(Ω) verifying (21). To simplify notations, we set u=e−2sαs3t−3(T−t)−3.(23) Let s≥s1,s1given by (19). Multiplying (13) by ϕξ1u, integrating over Q, and taking into account that u(0) vanishes in Ω, we have ZZO×(0,T )e−2sαs3t−3(T−t)−3|ϕ|2ξ1=ZZQ(a−c)ϕψξ1u +ZZQ(B−D)·∇ϕ ψξ1u+ϕψξ1∂tu−∆(ξ1u) + B·∇(ξ1u) −2ZZQ∇(ξ1u)·∇ϕ ψ := I1+I2+I3+I4+I5+I6. (24) Let us estimate each Ii,1≤i≤6. In the sequel, Cwill denote a positive constant depending only on Ω and B1(thus on B2) which may change from one line to another. In the first place, using H¨older and Young inequalities, we have I1=ZZQ(a−c)ϕψξ1u≤δ1ZZQξ1u|ϕ|2+1 4δ1ka−ck2 ∞ZZQξ1u|ψ|2,(25) 8 for any δ1>0. Then, I2=ZZQ(B−D)·∇ϕ ψξ1u≤γ1ZZQe−2sαst−1(T−t)−1|∇ϕ|2ξ1 +1 4γ1kB−Dk2 ∞ZZQe−2sαs5t−5(T−t)−5|ψ|2ξ1, (26) for any γ1>0. Let us now observe that |∂tu| ≤ T s3e−2sαt−5(T−t)−5Cs + 3T2/4≤CTs4e−2sαt−5(T−t)−5, since s≥σ1(Ω, B1)T2. Thus, we can estimate I3≤ZZQ|ϕ||ψ|ξ1|∂tu| ≤ ZZQCTe−2sαs4t−5(T−t)−5|ϕ||ψ|ξ1 ≤δ2ZZQe−2sαs3t−3(T−t)−3|ϕ|2ξ1+CT2 δ2ZZQe−2sαs5t−7(T−t)−7|ψ|2ξ1 ≤δ2ZZQξ1u|ϕ|2+C δ2ZZQe−2sαs7t−7(T−t)−7|ψ|2ξ1, (27) for δ2>0, since s≥σ1(Ω, B1)T. In order to estimate I4=−ZZQϕψ∆(ξ1u), let us observe that ∆(ξ1u) = s3t−3(T−t)−3(∆ξ1)e−2sα + 2∇ξ1·∇(e−2sα) + ξ1∆(e−2sα), with |∇(e−2sα)|= 2se−2sαt−1(T−t)−1|∇α0| ≤ Cse−2sαt−1(T−t)−1, |∆(e−2sα)| ≤ 2se−2sαt−2(T−t)−2(2s|∇α0|2+t(T−t)|∆α0|) ≤Cse−2sαt−2(T−t)−2(s+T2)≤Cs2e−2sαt−2(T−t)−2. Taking these considerations and (22) into account, we have I4≤CZZQe−2sαs3t−3(T−t)−3|ϕ||ψ|ξ1/2 1 +ZZQe−2sαs4t−4(T−t)−4|ϕ||ψ|ξ1/2 1+ZZQe−2sαs5t−5(T−t)−5|ϕ||ψ|ξ1. We now use H¨older and Young inequalities and (21) to get I4≤δ3ZZQξ1u|ϕ|2+C δ3ZZQe−2sαs3t−3(T−t)−3|ψ|21B2 +C δ3ZZQe−2sαs5t−5(T−t)−5|ψ|21B2+C δ3ZZQe−2sαs7t−7(T−t)−7|ψ|21B2, 9 for δ > 0. We can also obtain the corresponding estimates for Ii+e Ii,i= 3,4,6, similar to (27), (29) and (31), respectively, and valid for any s≥ max {s1, C(T+T2)},with C > 0 depending only on Ω and B1. Taking such estimates to (46) and using (44) (and (21)), we can estimate s3ZZQρt−3(T−t)−3|ϕ|2≤Cka−ck2 ∞ZZB2×(0,T )ρs3t−3(T−t)−3|ψ|2 +CZZB2×(0,T )ρs7t−7(T−t)−7|ψ|2, for s≥max {s1, C(T+T2))}. Then, if s≥s3= max ns1, C T+T2+T2ka−ck1/2 ∞o, the following estimate for ϕholds ZZQρt−3(T−t)−3|ϕ|2≤C3ZZB2×(0,T )ρs4t−7(T−t)−7|ψ|2,(47) with C3>0 depending on Ω, B1,T,kkk∞;Σ, and k∂tkk∞;Σ. Now, using (45) and (47), a new estimate for ψanalogous to (34) is obtained. More precisely, there exists C4=C4(Ω, B1,T,khk∞;Σ,k∂thk∞;Σ,kkk∞;Σ,k∂tkk∞;Σ)>0 such that ZZQρt−3(T−t)−3|ψ|2≤C4ZZB2×(0,T )ρs4t−7(T−t)−7|ψ|2,(48) for any s≥s4, with s4= max ns1, s2, C T+T2+T2ka−ck1/2 ∞o.(49) On the other hand, multiplying the equation in (41) by ϕand integrating over Ω, we get 1 2 d dt ZΩ|ϕ(t)|2+ZΩ|∇ϕ(t)|2≤ kkk∞;Σ Z∂Ω|ϕ(t)|2dσ +kck∞ZΩ|ϕ(t)|2,(50) for ta.e. in (0, T ). We claim that d dt|ϕ(t)|2 L2(Ω) ≤K1|ϕ(t)|2 L2(Ω),a.e. in (0, T ),(51) K1being a positive constant depending on kck∞and kkk∞;Σ. Indeed, in view of the chain of embeddings H1(Ω) ⇒Hγ(Ω) ,→L2(Ω), γ < 1, the first one being compact, for any ε > 0 there exists C(ε)>0 such that kuk2 Hγ(Ω) ≤εZΩ|∇u|2dx +C(ε)|u|2 L2(Ω),∀u∈H1(Ω). 16 Taking also into account the continuous embedding of Hγ(Ω) into L2(∂Ω), for γ > 1/2, there exists C(kkk∞;Σ)>0 such that kkk∞;Σ Z∂Ω|ϕ(t)|2dσ ≤1 2ZΩ|∇ϕ(t)|2+C(kkk∞;Σ)|ϕ(t)|2 L2(Ω), for 1/2< γ < 1. Combining this estimate with (50), yields (51), with K1given by K1= 2(C(kkk∞;Σ) + kck∞). Then |ϕ(t+T/4)|2 L2(Ω) ≤exp (K1T/4) |ϕ(t)|2 L2(Ω),∀t∈(T/4,3T/4) , and hence ZZΩ×(T/2,T )|ϕ|2≤exp (K1T/4) ZZΩ×(T/4,3T/4) |ϕ|2.(52) Now, multiply the equation in (42) by ψand integrate over Ω. Using again a compactness–uniqueness argument, we obtain −d dt|ψ(t)|2 L2(Ω) ≤K2|ψ(t)|2 L2(Ω) +|ϕ(t)|2 L2(O),a.e. in (0, T ), with K2=K2(kak∞,khk∞;Σ)>0. Then |ψ(t)|2 L2(Ω) ≤ZT texp (K2(s−t)) |ϕ(s)|2 L2(O)ds, ∀t∈(0, T), whence ZZΩ×(T/2,T )|ψ|2≤exp (K2T)ZZO×(T/2,T )|ϕ|2.(53) The form of the weight function ρdefined in Lemma 2.5 allows one to prove estimates similar to those in Lemma 2.4, with e−2sα replaced by ρ, valid for s≥CT2. Let us fix s= max {s4, CT 2}, with s4given by (49). We can thus bound both sides of (48) and deduce ZZΩ×(0,T/2) exp −N t|ψ|2≤C5ZZB2×(0,T )|ψ|2,(54) with N > 0 and C5>0 depending on Ω, B1,T,kak∞,kck∞,khk∞;Σ,kkk∞;Σ, k∂thk∞;Σ, and k∂tkk∞;Σ. In addition, due to (53), (52), and (47), we can also estimate (see a similar proof in page 12) ZZΩ×(T/2,T )exp −N t|ψ|2≤exp K1 T 4+K2T+C6ZZB2×(0,T )|ψ|2,(55) with C6>0 depending on Ω, T,kak∞,kck∞,khk∞;Σ,kkk∞;Σ,k∂thk∞;Σ, k∂tkk∞;Σ, and B1, thus on ωand O. Finally, gathering (54) and (55) yields the desired observability inequality, since B2⊂ω, with Nas in (54) and K=C5+ exp (K1T/4 + K2T+C6).  17 3 Proof of Theorems 1.1 and 1.2 We devote this section to prove Theorems 1.1 and 1.2. Both proofs, which are inspired in those of other known controllability results for nonlinear systems (see [12], [13], [3], [14],...), rely on controllability results for linear problems similar to the linearized system and appropriate fixed point arguments. The proof of Theorem 1.2 is similar to the one of Theorem 1.1 and it will be omitted here. Proof of Theorem 1.1: We start with the existence of approximately insensitizing controls for a linearized version of (5), (6) for y0= 0. For given a, c ∈L∞(Q), B, D ∈L∞(Q)Nand ξ∈L2(Q), we consider the linear systems      ∂ty−∆y+ay +B·∇y=ξ+v1ωin Q, y= 0 on Σ, y(x, 0) = 0 in Ω, (56)     −∂tq−∆q+cq −∇·(Dq) = y1Oin Q, q= 0 on Σ, q(x, T ) = 0 in Ω, (57) and the corresponding adjoint systems (12) and (13). The following result holds: Proposition 3.1 Assume that ω∩ O 6=∅. Let Mand Hbe the positive constants provided by Theorem 2.1. For any ε > 0, there exists a control function vε∈L2(ω×(0, T)) such that the associated solution (yε, qε)of (56), (57) satisfies |qε(0)|L2(Ω) ≤ε. (58) In addition, if ξ∈L2(Q)satisfies ZZQexp M t|ξ|2dx dt < ∞,(59) then the controls {vε}ε>0are uniformly bounded in L2(ω×(0, T )). More precisely, kvεkL2(ω×(0,T )) ≤√HZZQexp M t|ξ|2dx dt1/2 ,∀ε > 0.(60) Proof: The structure of the proof being identical to the one in [2] and [3], we will not go into details. For fixed ε > 0, we introduce the functional defined on L2(Ω) J(ϕ0;a, c, B, D) = 1 2ZZω×(0,T )|ψ|2+ε|ϕ0|L2(Ω) +ZZQξψ, (61) 18 where ψsolves (13), ϕbeing the solution of (12) with initial data ϕ0∈L2(Ω). In view of a unique continuation property for the adjoint systems (which follows, for instance, from Theorem 2.1), the continuous and convex functional J(·;a, c, B, D) is strictly convex and satisfies lim inf |ϕ0|L2(Ω)→+∞ J(ϕ0;a, c, B, D) |ϕ0|L2(Ω) ≥ε. (62) Thus, J(·;a, c, B, D) is coercive and therefore it reaches its minimum at a unique ϕ0 ε∈L2(Ω). Set vε=ψε1ω,(63) (ϕε, ψε) solving (12), (13) with initial data ϕ0 ε. Then, the solution (yε, qε) of (56), (57) associated to vεsatisfies (58). Indeed, vεis the unique control of minimal L2–norm solving (56)–(58). Now, assume that ξsatisfies (59). The optimality condition for ϕ0 εand Theorem 2.1 give ZZω×(0,T )|ψε|2+ε|ϕ0 ε|L2(Ω) =−ZZQξ ψε ≤ HZZω×(0,T )|ψε|2!1/2ZZQexp M t|ξ|21/2 , which yields, together with (63), the uniform estimate (60).  Remark 1 In view of (60), for any ξ∈L2(Q)verifying (59), one can prove the existence of a control v∈L2(ω×(0, T )) such that the associated solution (y, q)of (56),(57) satisfies (7). Moreover, this control vsatisfies the estimate kvkL2(ω×(0,T )) ≤√HZZQexp M t|ξ|2dx dt1/2 , with Mand Has above. That is to say, an insensitivity result in the linear case can also be proved. We now apply a fixed point argument to prove an approximate insensitivity result in the nonlinear case. Proposition 3.2 For fixed ε > 0, under the assumptions in Theorem 1.1, there exist a positive constant M(depending on Ω,ω,O,T, and f) such that for any ξ∈L2(Q)satisfying (4), one can find a control vε∈L2(ω×(0, T)) so that the associated solution (yε, qε)of (5),(6) satisfies (58). Furthermore, kvεkL2(ω×(0,T )) ≤ HZZQexp M t|ξ|2dx dt1/2 ,∀ε > 0,(64) Hbeing a new positive constant depending on Ω,ω,O,T, and f. 19 Proof: For a given function fas in Theorem 1.1, we can write f(s, p) = g(s, p)s+G(s, p)·pfor all (s, p)∈IR ×IRN, where g: IR ×IRN→IR and G: IR ×IRN→IRNare the bounded continuous functions defined by g(s, p) = Z1 0∂sf(σs, σp)dσ, G(s, p) = Z1 0∂pf(σs, σp)dσ. (65) Since it is a fixed parameter, the dependence on εwill be omitted in this proof. For any z∈L2(0, T;H1 0(Ω)), we consider the linear systems (56) and (57), with a=az=g(z, ∇z), c =cz=∂sf(z, ∇z)∈L∞(Q) and B=Bz= G(z, ∇z), D =Dz=∂pf(z, ∇z)∈L∞(Q)N. Indeed, the hypothesis on fgives kazk∞,kczk∞,kBzk∞,kDzk∞≤L, ∀z∈L2(0, T;H1 0(Ω)),(66) where L > 0 is a bound of ∂sfand ∂pfin IR×IRN. In view of Proposition 3.1, there exists a control vz∈L2(ω×(0, T)) such that the corresponding solution (yz, qz) of these systems satisfies |qz(0)|L2(Ω) ≤ε. (67) Let Mzand Hzbe the positive constants provided by Theorem 2.1 for a=az, c=cz,B=Bz, and D=Dz. Recalling the expressions of Mzand Hz, and using (66), there exist positive constants Mand Hof the form      M=C(Ω,ω,O) (1 + T(1 + L2)) , H= exp C(Ω,ω,O)1 + 1 T+T+ (1 + T)L2,(68) such that, for all z∈L2(0, T ;H1 0(Ω)), Mz≤ M and √Hz≤ H. Then, if ξ satisfies (4), using (60) we have the following estimate (uniform with respect to zand ε) kvzkL2(ω×(0,T )) ≤ HZZQexp M t|ξ|21/2 ,∀z∈L2(0, T;H1 0(Ω)).(69) We now consider the mapping Λε:L2(0, T;H1 0(Ω)) →L2(0, T;H1 0(Ω)) defined by Λε(z) = yz,yzbeing the solution of (56) associated to the potentials a=az and B=Bzand the control vzprovided by Proposition 3.1. We will apply the Schauder fixed point theorem to prove that Λεpossesses at least one fixed point. First, by classical regularity results on the heat equation, yzlies in the space Y={u:u∈L2(0, T;H2(Ω) ∩H1 0(Ω)), ∂tu∈L2(Q)}, with kyzkY≤exp hC1 + T+T1/2kazk∞+TkBzk2 ∞ikξ+vz1ωkL2(Q) 20 (here kyzkY=kyzkL2(H2∩H1 0)+k∂tyzkL2(Q)and k·kL2(H2∩H1 0)denotes the norm in L2(0, T;H2(Ω) ∩H1 0(Ω))). Taking into account (66) and (69), one deduces that Λεmaps L2(0, T;H1 0(Ω)) into a bounded set of Y. This space being compactly embedded in L2(0, T;H1 0(Ω)), there exists a fixed compact set Kin L2(0, T;H1 0(Ω)) such that Λε(L2(0, T;H1 0(Ω))) ⊂K. (70) Thus, Λεis a compact mapping. Now, let {zj} ⊂ L2(0, T ;H1 0(Ω)) be such that zj→zin L2(0, T;H1 0(Ω)). From (66) and the regularity assumptions on f, one has azj=g(zj,∇zj)* az, czj=∂sf(zj,∇zj)* czweak-?in L∞(Q), Bzj=G(zj,∇zj)* Bz, Dzj=∂pf(zj,∇zj)* Dzweak-?in L∞(Q)N. (71) Let ˆϕ0(resp. ˆϕ0 j,j≥1) be the unique minimizer in L2(Ω) of the functional J defined by (61) with a=az,c=cz,B=Bzand D=Dz(resp. with a=azj, c=czj,B=Bzjand D=Dzj). Reasoning as in [12] and [15], the coercivity property (62) is proved to be hold uniformly on potentials a,c,Band D uniformly bounded. Then, one can see that the sequence {ˆϕ0 j}is bounded in L2(Ω) and, finally, one proves that ˆϕ0 j→ˆϕ0in L2(Ω).(72) Let now ( ˆϕ, ˆ ψ) (resp. ( ˆϕj,ˆ ψj), j≥1) be the solution of (12), (13) with a=az, c=cz,B=Bz,D=Dz(resp. a=azj,c=czj,B=Bzj,D=Dzj) and the initial condition ˆϕ0(resp. ˆϕ0 j). From (71) and (72), we have ˆϕj→ˆϕ, ˆ ψj→ˆ ψin L2(Q).(73) By definition of Λε, Λε(z) (resp. Λε(zj), j≥1) is the solution of (56) associated to the control ˆv=ˆ ψ1ω(resp. ˆvj=ˆ ψj1ω) with a=az,c=cz,B=Bzand D=Dz(resp. a=azj,c=czj,B=Bzj, and D=Dzj). From (73) one has ˆvj→ˆvin L2(Q),so that from (71) one gets that Λε(zj)→Λε(z) in L2(Q) and also in L2(0, T;H1 0(Ω)), due to (70). This proves the continuity of Λε. All the assumptions of the Schauder theorem being fulfilled, Λεpossesses at least one fixed point yε∈L2(0, T ;H1 0(Ω)). Then, the control vε=vyεis such that yεsolves      ∂tyε−∆yε+g(yε,∇yε)yε+G(yε,∇yε)·∇yε=ξ+vε1ωin Q, yε= 0 on Σ, yε(x, 0) = 0 in Ω, (74) 21 and the solution qεof     −∂tqε−∆qε+∂sf(yε,∇yε)qε−∇·(∂pf(yε,∇yε)qε) = yε1Oin Q, qε= 0 on Σ, qε(x, T ) = 0 in Ω, (75) satisfies (58). In other words, we have found a control function vε∈L2(ω× (0, T)) such that the associated solution of (5), (6) (with y0= 0) verifies (58). Finally, estimate (64) follows readily from (69), which ends the proof of Proposition 3.2.  We will end the proof of Theorem 1.1 by passing to the limit in (74), (75), and (58). Since the controls vεprovided by Proposition 3.2 are uniformly bounded in L2(ω×(0, T)) and (66) holds, due to the regularizing effect of the heat equation, {(yε, qε)}lies in a bounded set of Y×W(0, T ) (Ydefined in page 20 and W(0, T ) := {u:u∈L2(0, T ;H1 0(Ω)), ∂tu∈L2(0, T;H−1(Ω))}) and accordingly, in a compact set of L2(0, T ;H1 0(Ω)) ×L2(Q). Then, up to a subsequence, one has vε* v weakly in L2(ω×(0, T)), (yε, qε)→(y, q) in L2(0, T ;H1 0(Ω)) ×L2(Q), qε(0) →q(0) in L2(Ω), for some v∈L2(ω×(0, T )), y∈Y,q∈W(0, T). Due to the continuity of g and G, one can pass to the limit in (74) and (75), deducing that (y, q) solves (5), (6) with control term vand initial datum y0= 0. Moreover, from (58), the function qsatisfies (7). Thus, the function vis an insensitizing control for the functional Φ given by (2). Finally, (64) and the convergences above allow one to estimate kvkL2(ω×(0,T )) ≤ HZZQexp M t|ξ|2dx dt1/2 ,(76) with Mand Hgiven by (68) and the proof is complete.  Remark 2 The method used in Theorem 1.1 to obtain such a control vprovides an upper bound of the cost of insensitizing the functional Φ. Indeed, in the proof of the theorem it is shown that the control function vcan be chosen satisfying estimate (76), with Mand Hgiven by (68). Inspired in [6], denote by Uad the nonempty set Uad ={v∈L2(ω×(0, T)) : (y, q) satisfies (5)–(7) with y0= 0}. Thus, the quantity Cins = inf{kvkL2(ω×(0,T )) :v∈ Uad},which measures the cost of insensitizing the functional Φ, can be estimated as follows Cins ≤ HZZQexp M t|ξ|2dx dt1/2 . 22 4 Comments and conclusions Boundary Fourier conditions. Proving an insensitivity result for the system:      ∂ty−∆y+f(y, ∇y) = ξ+v1ωin Q, ∂ny+hy = 0 on Σ, y(x, 0) = τˆy0(x) in Ω, with a C1globally Lipschitz-continuous function fis a much more difficult problem. Let us observe that such an insensitivity result is equivalent to the following null controllability problem:      ∂ty−∆y+f(y, ∇y) = ξ+v1ωin Q, ∂ny+hy = 0 on Σ, y(x, 0) = 0 in Ω,     −∂tq−∆q−∇·(∂pf(y, ∇y)q) + ∂sf(y, ∇y)q=y1Oin Q, ∂nq+hq + (∂pf(y, ∇y)·n)q= 0 on Σ, q(x, T ) = 0 in Ω, q(x, 0) = 0 in Ω. This leads us to analyze the null controllability problem for the cascade linear system      ∂ty−∆y+ay +B·∇y=ξ+v1ωin Q, ∂ny+hy = 0 on Σ, y(x, 0) = 0 in Ω, (77)     −∂tq−∆q−∇·(Dq) + cq =y1Oin Q, ∂nq+ (h+D·n)q= 0 on Σ, q(x, T ) = 0 in Ω, (78) under the hypothesis a, c ∈L∞(Q) and B, D ∈L∞(Q)N(which are the natural assumptions on these potentials for the given function f). For this, an observability inequality for the corresponding adjoint problem should be proved. This adjoint problem is:      ∂tϕ−∆ϕ+cϕ +D·∇y= 0 in Q, ∂nϕ+hϕ = 0 on Σ, ϕ(x, 0) = ϕ0in Ω,     −∂tψ−∆ψ−∇·(Bψ) + aψ =ϕ1Oin Q, ∂nψ+ (h+B·n)ψ= 0 on Σ, ψ(x, T) = 0 in Ω. In order to obtain such an observability inequality, we need a Carleman inequality for these adjoint problems. The presence of the term (B·n)ψ in the boundary condition for ψand the unique hypothesis B∈L∞(Q)N makes it quite difficult (even in the case of a null controllability problem for 23 a unique linear heat equation with the same kind of boundary conditions) and this is out of the scope of this paper. Superlinear nonlinearities. The observability results proved in this paper are of wider use than the scope of this article. First, Theorem 2.1 is used in [4] and [5] for a semilinear heat equation with a superlinear nonlinearity f(y). Theorem 2.6 is also used in [7] for the case of nonlinear Fourier boundary conditions. It is of interest to notice that in Theorem 2.6, the dependency of the constants with respect to the boundary data hand k is not explicit. This comes from the proof of Lemma 2.5 (see Lemma 1.2 in [9]). In view of known null controllability results, it is natural to think of extending Theorem 1.1 to C1locally Lipschitz-continuous functions fsuch that f(0,0) = 0 and lim |(s,p)|→∞ |g(s, p)| log3/2(1 + |s|+|p|)= 0,lim |(s,p)|→∞ |G(s, p)| log1/2(1 + |s|+|p|)= 0, with gand Gthe functions given by (65) (see Theorem 1.1 in [11]). Observe that such nonlinearities may lead to blow-up phenomena. However, the idea in [11] of taking short control times to avoid blow-up to occur fails here (even if G≡0), since the initial and final times are fixed in insensitivity problems. In [4] and [5] the authors introduce a new technique and prove an insensitivity result for nonlinearities f=f(y) with certain superlinear growth at infinity, e.g. for fsuch as |f(s)|=|p1(s)|logα(1+|p2(s)|) for all |s| ≥ s0>0, with α∈[0,1), p1and p2being first order real polynomial functions. The crucial point in these works is the construction, in the linear case, of regular controls starting from insensitizing controls in L2. The idea is as follows. Let us consider two open sets B0and Bsuch that B0⊂⊂ B ⊂ ω∩O. Let ˆvbe an L2–control, with supp ˆv⊂ B0×[0, T ], such that the corresponding solution (ˆy, ˆq) of (56), (57) for B=D= 0 satisfies (7). Then, setting q= (1 −θ)ˆq, y = (1 −θ) ˆy+ 2∇θ·∇ˆq+ (∆θ)ˆq, with θ∈ D(B) such that θ≡1 in a neighborhood of B0, it is possible to furnish a regular insensitizing control vsupported on B × [0, T ]. This construction uses local regularization properties of the heat equation. This technique does not apply to the case involving gradient terms because of the lack of regularity introduced by the term −∇ · (Dq) in (57). Indeed, in this case, the expression of the regular control we wish to build contains some terms, which are not regular enough to make the state ylie in a suitable space to apply a fixed point argument. This is why in Theorem 1.1 we cannot consider nonlinearities of higher order. In [7], a local result on the existence of insensitizing controls for a semilinear heat equation with nonlinear boundary conditions of Fourier type is proved. Such boundary conditions lead to seek a fixed point, thus also control functions, in certain H¨older spaces. A construction similar to that used in [4] and [5], allows one to build, in the linear case, controls with h¨olderian 24 regularity starting from L2–controls. Again, this is one of the essential points in the referenced work. 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