Exact controllability to the trajectories of the heat equation with Fourier boundary conditions: the semilinear case

ESAIM: COCV ESAIM: Control, Optimisation and Calculus of Variations July 2006, Vol. 12, 466–483 www.edpsciences.org/cocv DOI: 10.1051/cocv:2006011 EXACT CONTROLLABILITY TO THE TRAJECTORIES OF THE HEAT EQUATION WITH FOURIER BOUNDARY CONDITIONS: THE SEMILINEAR CASE ∗ Enrique Fern´ andez-Cara1, Manuel Gonz´ alez-Burgos1, Sergio Guerrero2 and Jean-Pierre Puel3 Abstract. This paper is concerned with the global exact controllability of the semilinear heat equation (with nonlinear terms involving the state and the gradient) completed with boundary conditions of the form ∂y ∂n +f(y) = 0. We consider distributed controls, with support in a small set. The null controllability of similar linear systems has been analyzed in a previous first part of this work. In this second part we show that, when the nonlinear terms are locally Lipschitz-continuous and slightly superlinear, one has exact controllability to the trajectories. Mathematics Subject Classification. 35K20, 93B05. Received February 17, 2005. Revised May 30, 2005 and June 13, 2005. 1. Introduction Let Ω ⊂RN(N≥1) be a bounded connected open set whose boundary ∂Ω is regular enough (for instance ∂Ω∈C2). Let ω⊂Ω be a (small) nonempty open subset and let T>0. We will use the notation Q=Ω×(0,T) and Σ = ∂Ω×(0,T) and we will denote by n(x) the outward unit normal to Ω at the point x∈∂Ω. We will consider the semilinear heat equation with nonlinear Fourier (or Robin) boundary conditions ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ yt−∆y+F(y,∇y)=v1ωin Q, ∂y ∂n +f(y)=0 onΣ, y(x, 0) = y0(x)inΩ. (1) Here, we assume that v∈L2(ω×(0,T)) (at least), 1ωis the characteristic function of ω,y0∈L∞(Ω) and F:R×RN→ Rand f:R→ Rare given functions. In (1), y=y(x, t) is the state and v=v(x, t)isthe control; it is assumed that we can act on the system only through ω×(0,T). Keywords and phrases. Controllability, heat equation, Fourier boundary conditions, semilinear. ∗This work has been partially supported by D.G.E.S. (Spain), Grants BFM2000–1317 and BFM2003–06446. 1Dpto. E.D.A.N., University of Sevilla, Aptdo. 1160, 41080 Sevilla, Spain; [email protected];[email protected]; [email protected] 2Laboratoire Jacques-Louis Lions, Universit´e Pierre et Marie Curie, boˆıte courrier 187, 75035 Cedex 05, Paris, France; [email protected] 3Laboratoire de Math´ematiques Appliqu´ees, Universit´e de Versailles – St. Quentin, 45 avenue des ´ Etats-Unis, 78035 Versailles, France; [email protected]e.fr c EDP Sciences, SMAI 2006 Article published by EDP Sciences and available at http://www.edpsciences.org/cocvor http://dx.doi.org/10.1051/cocv:2006011 E. FERN´ ANDEZ-CARA ET AL. 467 For the existence, uniqueness, regularity and general properties of the solutions to problems like (1), see for instance [1, 2, 7]. An illustrative interpretation of the data and variables in (1) is the following. The function y=y(x, t) can be viewed as the relative temperature of a medium (with respect to the exterior surrounding air) subject to transport and chemical reactions. The parabolic equation in (1) means, among other things, that a heat source v1ωis applied on a part of the body. On the boundary, −∂y ∂n can be viewed as the normal heat flux, inwards directed, up to a positive coefficient. Thus, the equality −∂y ∂n =f(y) means that this flux is a (nonlinear) function of the temperature. Accordingly, it is reasonable to assume that f is nondecreasing and f(0) = 0. A simplified linear model which was considered in a previous paper [10] is the following: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ yt−∆y+a(x, t)y+B(x, t)·∇y=v1ωin Q, ∂y ∂n +β(x, t)y=0 onΣ, y(x, 0) = y0(x)inΩ. (2) Here, it is assumed that the coefficients a,Band βsatisfy a∈L∞(Q),B∈L∞(Q)N,β∈L∞(Σ) (3) and, for the reasons above, it is also natural to assume that β≥0 (although this assumption was not used in [10]). The main goal of this paper is to analyze the controllability properties of the nonlinear system (1). More precisely, we will try to reach exactly uncontrolled solutions of (1), i.e. functions y=y(x, t) satisfying ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ yt−∆y+F(y,∇y)=0 inQ, ∂y ∂n +f(y)=0 onΣ, y(x, 0) = y0(x)inΩ. (4) It will be said that (1) is (globally) exactly controllable to the trajectories at time Tif, for any solution of (4) with “suitable” regularity and any y0∈L∞(Ω), there exist controls v∈L2(ω×(0,T)) and associated solutions y∈C0([0,T]; L2(Ω)) such that y(x, T )=y(x, T )inΩ.(5) Here, by suitable regularity we mean the following: y∈L2(0,T;H1(Ω)) ∩C0([0,T]; L2(Ω)) ∩L∞(Q), y0∈L∞(Ω).(6) The controllability properties of semilinear time-dependent systems have been studied intensively these last years. See for instance [8,11–13,15,16], where nonlinearities of the form f(y) are considered. See also the general treatise [14]. In particular, for parabolic systems completed with Dirichlet boundary conditions, nonlinear terms f(y,∇y) depending on both the state and the gradient have been taken into account in [6, 9]. For the similar linear system (2), the null controllability was analyzed more in detail in [10]. In the case of (1), some partial results have been given in [5]. 468 CONTROLLABILITY OF SEMILINEAR HEAT EQUATION Our main result concerns the global exact controllability to the trajectories of (1). It is the following: Theorem 1. Let us assume that Fand fare locally Lipschitz-continuous and satisfy lim |s|→∞ |F(s, p)−F(r, p)| |s−r|log3/2(1 + |s−r|)=0,(7) uniformly in (r, p)∈[−K, K]×RN∀K>0, ⎧ ⎪ ⎨ ⎪ ⎩ ∀L>0,∃M>0such that |F(s, p)−F(r, p)|≤M|s−r|,|F(s, p)−F(s, q)|≤M|p−q| ∀(s, r, p, q)∈[−L, L]2×RN×RN (8) and lim |s|→∞ |f(s)−f(r)| |s−r|log1/2(1 + |s−r|)=0 (9) uniformly in r∈[−K, K]∀K>0. Then, for each T>0, the nonlinear system (1) is exactly controllable to the trajectories at time Twith L∞controls. Remark 1. Conditions (7)–(9) are satisfied if Fand fare globally Lipschitz continuous. Notice that (7) means that the function Fcan only be slightly superlinear in s, uniformly in p. In the similar case of Dirichlet boundary conditions, it is known that conditions like these are sharp. Indeed, for instance, when Fdoes not depend on pand |F(s)−F(r)|∼|s−r|logβ(1 + |s−r|),β>2, due to blow-up phenomena, the system fails to be controllable whenever ω= Ω (see [11]). On the other hand, (9) is also a slightly superlinear growth assumption for f. It would be interesting to know whether a more superlinear fleading to blow up in the absence of control can also be an obstruction for the null controllability of (1). But this question does not seem obvious and remains open. Remark 2. A result proved in [5] says that when F≡0, fis smooth near zero and f(s)s≥0∀s∈R,(10) the nonlinear system (1) is null controllable for large T. That is to say, under these assumptions, for each y0∈L2(Ω) there exist T(y0)>0andcontrolsvin L∞(ω×(0,T) such that the associated states ysatisfy y(x, T (y0)) = 0 in Ω. By inspection of the proof of theorem 1, we see that the same result holds for (1) with F≡0 whenever fis locally Lipschitz-continuous and satisfies the good sign condition (10). For the proof of Theorem 1, we will first establish a null controllability result for (2) (see Prop. 1 below). This will be used, together with an appropriate fixed point argument, to deduce the desired result. This strategy was introduced in [15] in the framework of the exact controllability of the semilinear wave equation. See also [8,12] for similar results concerning the approximate and null controllability of the semilinear heat equation with Dirichlet or Neumann boundary conditions. Our null controllability result for (2) is the following: Proposition 1. For every T>0,system(2) is null controllable at time T, with controls in L∞(ω×(0,T)). More precisely, for each y0∈L2(Ω),thereexistsv∈L∞(ω×(0,T)) such that the associated solution to (2) satisfies (5). Furthermore, the control vcan be found satisfying vL∞(ω×(0,T)) ≤eC(Ω,ω)K(T,a∞,B∞,β∞)y0L2(Ω) ,(11) E. FERN´ ANDEZ-CARA ET AL. 469 where K=1+1/T +a2/3 ∞+B2 ∞+β2 ∞+T(1 + a∞+B2 ∞+β2 ∞).(12) For the proof of proposition 1, we first introduce a control L2(ω×(0,T)) which leads the solution of (2) to zero at time T. In a second step, arguing as in Section 2 in [4], a regularizing argument will lead to the desired L∞control. The rest of this paper is organized as follows. In Section 2, we prove Proposition 1. Section 3 is devoted to the proof of Theorem 1. For completeness, we have also included an Appendix where the proof of a rather technical local regularity result is given in detail. In the sequel, Cdenotes a generic positive constant only depending on Ω and ω. 2. A null controllability result for the linear system In this section we present the proof of Proposition 1. Let y0∈L2(Ω) be given and let us introduce two open sets ωand ω,withω ⊂⊂ ω⊂⊂ ω. Then, we can use the main result in [10] (Th. 2) with control region ω ×(0,T) to deduce the existence of a control v∈L2(ω ×(0,T)) such that the associated solution to (2) verifies (5) and also the estimate vL2(ω ×(0,T)) ≤eC(Ω,ω)K(T,a∞,B∞,β∞)y0L2(Ω),(13) where Kis of the form (12). Let us denote by ythe state associated to v. We now introduce a cut-off function η=η(t) satisfying η∈C∞([0,T]),η(t)=1in(0,T/4),η(t)=0in(3T/4,T) and 0≤η(t)≤1,|η(t)|≤C tin (0,T) and we denote by χthe solution to the system ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ χt−∆χ+a(x, t)χ+B(x, t)·∇χ=0 inQ, ∂χ ∂n +β(x, t)χ=0 onΣ, χ(x, 0) = y0(x)inΩ. Then, the function w=y−ηχ satisfies ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩wt−∆w+a(x, t)w+B(x, t)·∇w=−η(t)χ+v1ω in Q, ∂w ∂n +β(x, t)w=0 onΣ, w(x, 0) = 0,w(x, T )=0 inΩ. Our aim is to construct a control v∈L∞(ω×(0,T)) which drives the solution of (2) to zero at time t=T.To this end, we will need a local regularity result for the solutions to linear heat equations with L∞coefficients a and B. This will be used below for the functions χand wand reads as follows: Lemma 1. Let us denote by Ythe space L2(0,T;H1(Ω)) ∩C0([0,T]; L2(Ω)).Lety∈Ybe a solution to the equation yt−∆y+a(x, t)y+B(x, t)·∇y=f, (14) where a∈L∞(Q),B∈L∞(Q)Nand f∈L2(Q).LetO⊂Ωbe a nonempty open set and assume that fis L∞ in the cylinder O×(0,T).Then y∈L∞(δ, T ;W1,∞(O)) 470 CONTROLLABILITY OF SEMILINEAR HEAT EQUATION for any δ∈(0,T)and any nonempty open set O⊂⊂ O. Furthermore, there exists a positive constant C(O) such that the following estimate holds: yL∞(δ,T ;W1,∞(O)) ≤C(O)(T1/2+TN/2)1+δ−1+a∞+B∞N+1 yY+fL∞(O×(0,T )).(15) The previous regularity also holds with δ=0if, besides (14), we have y(x, 0) = 0 in Ω. In that case, one has an estimate similar to (15) without the term in δ. This lemma is implied by well known parabolic regularity theory. For completeness, its proof is given in an Appendix, at the end of this paper. Let us now consider an open set ω0with ω⊂⊂ ω0⊂⊂ ωand a cut-off function ξ,with ξ∈C2 0(ω0),ξ≡1inω and let us set w=(1−ξ)w.Thenwehave: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ wt−∆w+a(x, t)w+B(x, t)·∇w=−η(t)χ+v1ωin Q, ∂w ∂n +β(x, t)w=0 onΣ, w(x, 0) = 0,w(x, T )=0 inΩ, with v=ηξχ+2∇ξ·∇w+∆ξw−B·∇ξw. (16) Let us remark that supp v⊂ω×[0,T]. Therefore, if we prove that v∈L∞(ω×(0,T)), we will have that the function y=w+ηχ solves (together with v) the null controllability problem for (2). Thus, let us check that v∈L∞(ω×(0,T)) and let us estimate its norm in this space: •The regularity of the first term in the right hand side of (16) is implied by the interior regularity of χ not only in space but in time as well. From Lemma 1 with O=ω, we deduce that χ∈L∞(ω0×(δ, T )) with supp ξ⊂ω0⊂⊂ ω(we even have χ∈L∞(δ, T ;W1,∞ loc (ω))) and χL∞(ω0×(δ,T )) ≤C(T1/2+TN/2)1+δ−1+a∞+B∞N+1 χY; recall that Y=L2(0,T;H1(Ω)) ∩C0([0,T]; L2(Ω)). Consequently taking for instance δ=T/8, since η≡0in(0,T/4), we get ηξχL∞(ω×(0,T )) ≤CT−1(T1/2+TN/2)1+T−1+a∞+B∞N+1 χY. •The regularity of the other three terms in the right hand side of (16) is related to the interior space regularity of w. Thus, let us introduce ω1with ω0⊂⊂ ω1⊂⊂ ωand let us apply Lemma 1 with O=ω1\ω. This gives w∈L∞(0,T;W1,∞(ω0\ω)) and the estimate wL∞(0,T;W1,∞(ω0\ω)) ≤C(T1/2+TN/2)(1+a∞+B∞)N+1 (wY+ηχL∞(ω1×(0,T))), whence 2∇ξ·∇w+∆ξw−B·∇ξwL∞(ω×(0,T )) ≤C(T1/2+TN/2) ×(1 + a∞+B∞)N+2 (wY+ηχL∞(ω1×(0,T))). E. FERN´ ANDEZ-CARA ET AL. 471 Putting the previous estimates together, we find that v∈L∞(ω×(0,T)) and vL∞(ω×(0,T)) ≤C(1 + TN−1)1+T−1+a∞+B∞2N+3 (wY+χY).(17) At this point, notice that for any f∈L2(Q)andanyy0∈L2(Ω) the solution yto the linear system ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ yt−∆y+a(x, t)y+B(x, t)·∇y=fin Q, ∂y ∂n +β(x, t)y=0 onΣ, y(x, 0) = y0(x)inΩ (18) satisfies yY≤eCT(1+a∞+B2 ∞+β2 ∞)(fL2(Q)+y0L2(Ω)). For a detailed proof, see for example proposition 1 in [10]. ThiscanbeusedtoestimatewYand χYin terms of vL2(ω×(0,T)) and y0L2(Ω). In view of (17), we see that vL∞(ω×(0,T)) ≤L(vL2(ω ×(0,T )) +y0L2(Ω)),(19) where L=CT−1(1 + TN−1)1+T−1+a∞+B∞2N+3 exp{CT(1 + a∞+B2 ∞+β2 ∞)}. Combining this estimate and (13), we finally obtain that vL∞(ω×(0,T)) ≤eCK(T,a∞,B∞,β∞)y0L2(Ω),(20) where Kis given by (12). This ends the proof of Proposition 1. 3. Controllability of the nonlinear system In this section we will prove Theorem 1. The following auxiliary result will be needed: Proposition 2. Let us assume that, in (18), we have f∈L∞(Q)and y0∈L∞(Ω). Let us also assume that the coefficients a,Band βsatisfy (3).Theny∈L∞(Q)and y∞≤eCT(1+a∞+B2 ∞+β2 ∞)y0∞+f∞.(21) for some C=C(Ω). Proof. We will consider two different situations: Case 1. We will first assume that a≥1andβ≥0 and we will establish (21) in this case. In fact, we will show that, under these assumptions, y∞≤y0∞+f∞.(22) To this end, let us introduce the system ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ zt−∆z+a(x, t)z+B(x, t)·∇z=hin Q, ∂z ∂n +β(x, t)z=kon Σ, z(x, 0) = z0(x)inΩ, 472 CONTROLLABILITY OF SEMILINEAR HEAT EQUATION where h∈L∞(Q), k∈L∞(Σ) and z0∈L∞(Ω) and let us show that, if h,z0and kare nonnegative, then this is also the case for z. Indeed, by multiplying the equation satisfied by zby z−(·,t) (the negative part of z(·,t)) for each t∈(0,T) and integrating in Ω, after several simplifications, we find: 1 2 d dtΩ |z−(x, t)|2dx+Ω |∇z−(x, t)|2dx +∂Ω β(x, t)|z−(x, t)|2dσ+∂Ω k(x, t)z−(x, t)dσ+Ω a(x, t)|z−(x, t)|2dx= −Ω h(x, t)z−(x, t)dx−Ω B(x, t)·∇z−(x, t)z−(x, t)dx. From this identity, in view of the positiveness of a,h,βand k, we easily deduce that d dtΩ |z−(x, t)|2dx≤B2 ∞Ω |z−(x, t)|2dx, whence z≥0inQ. Now, let M>0 be a large constant (to be chosen below). The function z=M−ysatisfies ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ zt−∆z+a(x, t)z+B(x, t)·∇z=a(x, t)M−fin Q, ∂z ∂n +β(x, t)z=β(x, t)Mon Σ, z(x, 0) = M−y0(x)inΩ. Therefore, if we take M≥max{fL∞(Q),y0L∞(Ω)}, we can apply the previous argument and deduce that y≤M. In a similar way, one can deduce that y≥−M and, consequently, |y|≤M. This proves that whenever a≥1andβ≥0, the estimate (22) holds. Case 2. We will now prove (21) for general L∞coefficients aand β. Let γ∈C2(Ω) be a function satisfying γ≥0inΩ,∂γ ∂n ≤−β∞on ∂Ω,γ∞≤1, ∇γ∞≤Cβ∞,D2γ∞≤Cβ2 ∞. (23) We give here a sketch of the proof of the existence of such a function γ. To this end, let δ>0beaparameter (depending on Ω) such that x∈Ωδ→ dist(x, ∂Ω) is C2,withΩ δ={x∈Ω:dist(x, ∂Ω) <δ}. We distinguish two cases. Letusfirstassumethatβ∞≥1/δ.Thenwetakeγ(x)≡1inΩ\Ωδ,γ(x)=β∞dist(x, ∂Ω) in Ωεwith ε=1/(2β∞) and a regularization of γin Ωδ\Ωε. This gives the desired properties for γ. On the other hand, if β∞<1/δ,wetakeγ(x)=δβ∞in Ω \Ωδ,γ(x)=β∞dist(x, ∂Ω) in Ωδ/2and a regularization in Ωδ\Ωδ/2. This also provides a desired function in this case. E. FERN´ ANDEZ-CARA ET AL. 473 Let us now set y=e γ(x)y.Thenysatisfies ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩yt−∆y+a(x, t)y+ B(x, t)·∇y=e γ(x)fin Q, ∂y ∂n + β(x, t)y=0 onΣ, y(x, 0) = eγ(x)y0(x)inΩ, (24) where a=a+∆γ−|∇γ|2−B·∇γ,  B=B+2∇γ,  β=β−∂γ ∂n ≥0onΣ. Notice that, from the inequalities (23) satisfied by γ, we know that |a+∆γ−|∇γ|2−B·∇γ|≤C1(a∞+B2 ∞+β2 ∞)inQ for some C1>0. Now, let us set y=e −(C1(a∞+B2 ∞+β2 ∞)+1)ty. Then ysatisfies ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩yt−∆y+a(x, t)y+(B(x, t)+2∇γ(x)) ·∇y= fin Q, ∂y ∂n + β(x, t)y=0 onΣ, y(x, 0) = eγ(x)y0(x)inΩ, (25) where a=a+∆γ−|∇γ|2−B·∇γ+C1(a∞+B2 ∞+β2 ∞)+1,  f=e −(C1(a∞+B2 ∞+β2 ∞)+1)t+γ(x)f and  β= β. Since a≥1and β≥0, we can apply case 1 to y. This provides the estimates y∞≤y∞≤eCT(1+a∞+B2 ∞+β2 ∞)(y0∞+f∞), whence we deduce (21).  Let us now start with the proof of Theorem 1. Let y0∈L∞(Ω) and ybe given and assume that ysatisfies (6) and (4) in the weak sense. Let us consider the nonlinear system ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ wt−∆w+F1(w,∇w;x, t)w+F2(∇w;x, t)·∇w=v1ωin Q, ∂w ∂n +F3(w;x, t)w=0 onΣ, w(x, 0) = y0(x)−y(x, 0) in Ω, (26) wherewehaveusedthenotation F1(s, p;x, t)=F(y(x, t)+s, ∇y(x, t)+p)−F(y(x, t),∇y(x, t)+p) s,(27) F2=(F21,...,F 2N),F 2j(p;x, t)=1 0 ∂F ∂pj (y(x, t),∇y(x, t)+λp)dλ(28) 474 CONTROLLABILITY OF SEMILINEAR HEAT EQUATION and F3(s;x, t)=f(y(x, t)+s)−f(y(x, t)) s(29) for s∈Rand p∈RN. We will prove that there exist a control v∈L∞(ω×(0,T)) and an associated solution to (26) such that w(x, T )=0 in Ω.(30) With this control and the state y=w+y, we will have solved the exact controllability problem for (1) and we will have thus proved Theorem 1. We will first assume that the functions Fand fare continuously differentiable. Then, by a density argument, we will be able to prove the result in the general case. 3.1. The case in which Fand fare C1 The idea of the proof is well known: we introduce an appropriate (set-valued) fixed point mapping and we check that it possesses at least one fixed point; this will be a solution to the null controllability problem associated to (26). Let R>0 be given and let us introduce the following function: MR(s)=⎧ ⎪ ⎨ ⎪ ⎩ −Rif s<−R, sif −R≤s≤R, Rif s>R. Let us denote by Zthe Hilbert space Z=L2(0,T;H1(Ω)) and let us set for each R>0andeachz∈Z aR,z(x, t)=F1(MR(z(x, t)),∇z(x, t); x, t), Bz(x, t)=F2(∇z(x, t); x, t) and βR,z(x, t)=F3(MR(z(x, t)); x, t). Consider the linear null controllability problem ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ wt−∆w+aR,z(x, t)w+Bz(x, t)·∇w=v1ωin Q, ∂w ∂n +βR,z(x, t)w=0 onΣ, w(x, 0) = y0(x)−y(x, 0) in Ω, (31) together with (30). Let us introduce the function w0,withw0(x)=y0(x)−y(x, 0) for all x∈Ω. From (6), (8) and the fact that f∈C1(R), we have aR,z ∈L∞(Q),B z∈L∞(Q)N,β R,z ∈L∞(Σ). Consequently, in view of Proposition 1, (30)–(31) can be solved with controls in L∞(ω×(0,T)). We are now going to select a particular solution to (30)–(31) constructed as in [11]. To do this, we first set TR=min{T,a−1/3 R}>0, where aR=sup |s|≤R, p∈RN ess sup (x,t)∈Q |F1(s, p;x, t)|. E. FERN´ ANDEZ-CARA ET AL. 481 We have f0∈L2(Q). Consequently, ∆y0∈L2(Q), y0∈C0([0,T]; H1 0(Ω)) and appropriate estimates are satisfied. Indeed, by multiplying the equation satisfied by y0by −∆y0and integrating with respect to xin Ω, we find 1 2 d dt∇y0(·,t)2 L2+Ω |∆y0(x, t)|2dx=−Ω f0(x, t)∆y0(x, t)dx. (45) Since f0L2≤CfL2+(1+δ−1+a∞+B∞)yY, we easily obtain from (45) that y0C0([0,T];H1 0(Ω)) +∆y0L2(Q)≤CfL2+1+δ−1+a∞+B∞yY.(46) Clearly, the same estimate holds for yL∞(δ/(N+1),T ;H1(O0)) +∆yL2(O0×(δ/(N+1),T )). Let us now try to improve the local space regularity properties of y. To this end, we will use the following lemma: Lemma 2. Let us set p0=2,letpibe defined by 1 pi =1 pi−1 −1 2N for 1≤i≤N−1and let us set pN=+∞. Let us denote by Xithe space Xi=L∞((i+1)δ/(N+1),T;W1,pi(Oi)) for 0≤i≤Nand suppose that y∈Xj−1for some j. Then we also have y∈Xjand yXj≤C(O)(T1/2f∞+D(T,δ,a∞,B∞)yXj−1), where D(T,δ,a∞,B∞)=(T1/4+T1/2)(1 + δ−1+a∞+B∞).(47) Proof of Lemma 2. Let us introduce ξj∈C2 c(Oj−1)andηj∈C1([0,T]), with ξj(x)=1inOj,η j(t)=1in[(j+1)δ/(N+1),T], ηj(t)=0in[0,jδ/(N+1)],|ηj,t(t)|≤C δin (0,T) and let us put yj=ηjξjy.Thenyjsatisfies the following: ⎧ ⎪ ⎨ ⎪ ⎩ yj,t −∆yj=fjin Q, yj=0 onΣ, yj(x, 0) = 0 in Ω (48) with fj=fj,1+fj,2+fj,3, where fj,1=ηjξjf, fj,2=ηj,t ξjy−ηj∆ξjy−aη jξjy−ηj(B·∇ξj)y, fj,3=−2ηj∇ξj·∇y−ηjξjB·∇y. 482 CONTROLLABILITY OF SEMILINEAR HEAT EQUATION From the fact that the system (48) is linear, we see that yjcan be written as the sum of three solutions to similar systems with right hand sides fj,1,fj,2and fj,3. Let us respectively denote them by yj,1,yj,2and yj,3. We are now going to deduce estimates of yj,k in Xjfor 1 ≤k≤3. To this end, we will use the usual representation of yj,k provided by the semigroup S(t)associatedtothe heat equation with homogeneous Dirichlet conditions, say yj,k(·,t)=t 0 S(t−s)fj,k(·,s)ds for all t∈(0,T). Since f∈L∞(Q), we can write yj,1(·,t)W1,pj(Ω) ≤Ct 0 (t−s)−1/2fj,1(·,s)Lpj(Ω) ds. Therefore, from Young’s inequality we find that yj,1∈L∞(0,T;W1,pj(Ω)) and yj,1L∞(0,T ;W1,pj(Ω)) ≤CT1/2fj,1L∞(0,T ;Lpj(Ω)) ≤C(O)T1/2fL∞(O×(0,T)) . Taking into account that fj,2∈L∞(0,T;Lp∗ j−1(Ω)) with p∗ j−1=⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ∞if j>N−1, p(arbitrary in (1,+∞)) if j=N−1, 2N N−j−1if j<N−1, we see that fj,2is not worse than fj,1and, again, yj,2(·,t)W1,pj(Ω) ≤Ct 0 (t−s)−1/2fj,2(·,s)Lpj(Ω) ds for all t. From Young’s inequality and the assumption y∈Xj−1,wealsogetyj,2∈L∞(0,T;W1,pj(Ω)) and yj,2L∞(0,T ;W1,pj(Ω)) ≤CT1/2fj,2L∞(0,T ;Lp∗ j−1(Ω)) ≤C(O)T1/2(1 + δ−1+a∞)yXj−1. In the definition of fj,3, we find ∇y. Consequently, we can only ensure that fj,3∈L∞(0,T;Lpj−1(Ω)). Since −N 21 pj−1 −1 pj−1 2=−3 4, we have yj,3(·,t)W1,pj(Ω) ≤Ct 0 (t−s)−3/4fj,3(·,s)Lpj−1(Ω) ds and now Young’s inequality gives yj,3∈L∞(0,T;W1,pj(Ω)) and yj,3L∞(0,T ;W1,pj(Ω)) ≤CT1/4fj,3L∞(0,T ;Lpj−1(Ω)) ≤C(O)T1/4(1 + B∞)yXj−1. E. FERN´ ANDEZ-CARA ET AL. 483 Putting the estimates of yj,kL∞(0,T;W1,pj(Ω)) together and taking into account the definitions of ηjand ξj, we obtain the desired inequality for yXj. This concludes the proof of Lemma 2.  Since we already had y∈X0, we deduce from Lemma 2 that y∈XNand yXN≤C(O)(T1/2f∞+D(T,δ,a∞,B∞)yXN−1), where, Dis given by (47). We can apply Lemma 2 subsequently for j=N,N −1,...,1. The estimates we find yield yXN≤C(T1/2(1 + DN−1)fL∞(O×(0,T )) +DNyX0). This, together with (46), yields yXN≤C(T1/2+TN/2)D(T,δ,a∞,B∞)N+1(fL∞(O×(0,T )) +yY), which is exactly (15). References [1] H. 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