On the controllability of the heat equation with nonlinear boundary Fourier conditions

On the controllability of the heat equation with nonlinear boundary Fourier conditions A. Doubova, E. Fern´andez-Cara, M. Gonz´alez-Burgos a,1 aDpto. E.D.A.N., University of Sevilla, Aptdo. 1160, 41080 Sevilla, Spain. Abstract In this paper we analyze the approximate and null controllability of the classical heat equation with nonlinear boundary conditions of the form ∂y ∂n +f(y) = 0 and distributed controls, with support in a small set. We show that, when the function fis globally Lipschitz-continuous, the system is approximately controllable. We also show that the system is locally null controllable and null controllable for large time when fis regular enough and f(0) = 0. For the proofs of these assertions, we use controllability results for similar linear problems and appropriate fixed point arguments. In the case of the local and large time null controllability results, the arguments are rather technical, since they need (among other things) H¨older estimates for the control and the state. Key words: Controllability, heat equation, nonlinear boundary conditions 1 Introduction Let Ω ⊂RNbe a bounded connected open set whose boundary ∂Ω is regular enough (N≥1). Let O ⊂ 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. In the sequel, γ0will stand for the usual trace operator γ0:H1(Ω) 7→ H1/2(∂Ω). On the other hand, we will denote by C,C1,C2, . . . generic positive constants (usually depending on Ω, O,Tand possibly other data). Email address: [email protected], [email protected], [email protected] (A. Doubova, E. Fern´andez-Cara, M. Gonz´alez-Burgos). 1This work has been partially supported by D.G.E.S. (Spain), Grants PB98–1134 and BFM2000–1317. Preprint submitted to Elsevier Science 29 August 2003 We will consider the heat equation with nonlinear Fourier (or Robin) conditions                    ∂y ∂t −∆y=v1Oin Q, ∂y ∂n +f(y) = 0 on Σ, y(x, 0) = y0(x) in Ω. (1) Here, we assume that v∈L2(O × (0, T)) (at least), 1Ois the characteristic function of O,y0∈L2(Ω) and f:R7→ Ris a given function. In (1), y=y(x, t) is the state and v=v(x, t) is the control; it is assumed that we can act on the system only through O × (0, T). For the existence, uniqueness, regularity and general properties of the solutions to problems like (1), see for instance [1], [2] and [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 body (with respect to the exterior surrounding air). The parabolic equation in (1) means that a heat source v1Oacts 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 fis nondecreasing and f(0) = 0. Of course, the simplified linear model corresponds to the case −∂y ∂n =ay, where ais a constant. For the reasons above, it is natural to assume that a > 0. The main goal of this paper is to analyze the controllability properties of (1). System (1) is said to be approximately controllable in L2(Ω) at time Tif, for any y0, y1∈L2(Ω) and ε > 0, there exist a control v∈L2(O × (0, T)) and an associated solution y∈C0([0, T]; L2(Ω)) satisfying ky(·, T)−y1kL2≤ε. (2) On the other hand, it will be said that system (1) is null controllable at time Tif, for each y0∈L2(Ω), there exist v∈L2(O × (0, T)) and an associated 2 solution y∈C0([0, T]; L2(Ω)) such that y(x, T) = 0 in Ω.(3) The controllability properties of linear and semilinear time dependent systems have been studied intensively these last years, see for instance [8], [10], [15], [17], [22] and [24]. In this paper, we will be concerned with (1), where the nonlinearity is in the boundary condition. This is more difficult to analyze than the cases considered in [5], [8] and [10], where the boundary condition is linear and the equations are of the form ∂y ∂t −∆y+F(y) = v1O or ∂y ∂t −∆y+F(y, ∇y) = v1O. In order to justify this assertion, let us consider the following relatively simple system, one-dimensional in space:                      ∂y ∂t −∂2y ∂x2=v1(α,β)in (0,1) ×(0, T), −∂y ∂x +a0y!(0, t) = ∂y ∂x +a1y!(1, t) = 0 for t∈(0, T), y(x, 0) = y0(x) in (0,1). (4) Here, we assume that 0 < α < β < 1 and a0and a1are given in C0([0, T]) (for instance). Let us introduce the function ˜a, with ˜a(x, t) = −a0(t)x+ (a0(t) + a1(t))x2 2 and the new variable z, with z=e˜a(x,t)y. Then ysolves (4) for some v∈L2((α, β)×(0, T)) and y0∈L2(0,1) if and only if zsatisfies                    ∂z ∂t −Lz −∂˜a ∂t z=e˜a(x,t)v1(α,β)in (0,1) ×(0, T), ∂z ∂x(0, t) = ∂z ∂x(1, t) = 0 for t∈(0, T), z(x, 0) = e˜a(x,0)y0(x) in (0,1), (5) 3 where we have set Lz =∂2z ∂x2−2∂˜a ∂x ∂z ∂x −∂2˜a ∂x2z+ ∂˜a ∂x!2 z. Therefore, the approximate (resp. null) controllability of (4) is equivalent to the approximate (resp. null) controllability of a linear heat equation with a possibly singular coefficient ∂˜a ∂t in the zero order term, completed with homogeneous Neumann conditions. This indicates that the case under study in this paper is indeed more intrincate. Remark 1 Recall that the linear heat equation completed with terms of the form B· ∇yand Dirichlet boundary conditions has been considered in [13]. There, null controllability is established under the assumption B∈L∞(Q)N. The proof relies on an appropriate Carleman estimate for the solutions of the adjoint equation −∂ϕ ∂t −∆ϕ− ∇ · (ϕB) = 0. Trying to apply the same techniques to (5), we readily see that what is needed is a Carleman estimate for the solutions to the equation −∂ ∂t ((1 + ˜a)ϕ)−L∗ϕ= 0 in (0,1) ×(0, T), where L∗is the adjoint of L. But this seems much more complicate. The first main result in this paper concerns the approximate controllability of (1). It is the following: Theorem 2 Assume that f:R7→ Ris globally Lipschitz-continuous and T > 0. Then (1) is approximately controllable in L2(Ω) at time T. Notice that, under these assumptions, using standard arguments, it can be shown that for each y0∈L2(Ω) and each v∈L2(O × (0, T)) the nonlinear system (1) possesses exactly one solution ythat satisfies: y∈L2(0, T;H1(Ω)) ∩C0([0, T]; L2(Ω)),∂y ∂t ∈L2(0, T ;H−1(Ω)).(6) Remark 3 The global null controllability of (1) for a globally Lipschitz-continuous function fwithout any assumption on the size and regularity of y0is an open problem. In fact, at present, this is an unsolved question even for similar linear systems, when the nonlinear boundary Fourier condition in (1) is replaced by ∂y ∂n +a(x, t)y= 0 on Σ.(7) 4 Indeed, if the coefficient ais only assumed to be in L∞(Σ) (and this seems to be the natural assumption), the null controllability of the system is unknown (see [11] and remark 15 in Section 3). In order to state our second main result, it will be convenient to introduce some notation. For α, β ∈[0,1), Cα,β(Q) will stand for the space formed by all functions u∈C0(Q) such that [u]α,β = sup Q |u(x, t)−u(x′, t)| |x−x′|α+ sup Q |u(x, t)−u(x, t′)| |t−t′|β<+∞. The natural norm in Cα,β(Q) is kukα,β =kukL∞(Q)+ [u]α,β . With this norm, Cα,β(Q) is a Banach space. The second main result in this paper concerns the local null controllability of (1). It is the following: Theorem 4 Assume that f∈C3(R)and f(0) = 0. Then we can find a positive η=η(Ω,O, α, T)with the following property: If we have y0∈C2+α(Ω) for some α∈(0,1), the compatibility condition ∂y0 ∂n +f(y0) = 0 on ∂Ω (8) is fulfilled and ky0kC2+α(Ω) ≤η, there exists a control v∈Cα,α/2(Q)such that the associated solution yof (1) satisfies (3). This theorem indicates that the nonlinear system (1) is locally null controllable when fis regular enough and vanishes at 0. It will be clear from the proof that the same local property holds when fis C3just in a neighbourhood of 0. Our third main result deals with the case in which fis nondecreasing. It is a consequence of theorem 4 and reads as follows: Theorem 5 Assume that f∈C4(R),f(0) = 0 and f′(s)≥0for all s∈ R. Then (1) is null controllable in large time intervals. In other words, for every y0∈L2(Ω) there exist T=T(y0)and v∈L2(O × (0, T(y0))) such the associated solution to (1) satisfies (3). Again, it will be noticed in the proof of this result that fhas only to be C4 in a neighborhood of 0. The rest of this paper is organized as follows. In Section 2, we prove theorem 2. It will be seen that the proof relies on an approximate controllability result 5 for a linear system similar to (1) where the boundary condition is again of the kind (7) and an appropriate fixed point argument. In Section 3, we give the proof of theorem 4. In this case, we have to introduce and estimate controls in a much more regular space (in fact, this is the reason the argument works only when y0is sufficiently close to zero). Section 4 deals with the proof of theorem 5. This is achieved in several steps: we start from y0at t= 0 and we first choose a control such that the associated state becomes small in the C2+α-norm at t=T∗for T∗large enough; then we apply theorem 4 and we find a control that leads the state to zero at a time T(y0)> T ∗. Finally, in Section 5 we make some comments. 2 Proof of the approximate controllability result This Section is devoted to prove theorem 2. As usual, the proof relies on an approximate controllability result for similar linear problems and a fixed point argument. This strategy was introduced in [22], in the framework of the controllability of the semilinear wave equation. See also [8] and [10] for similar results concerning the semilinear heat equation with Dirichlet boundary conditions. 2.1 The approximate controllability of similar linear problems We consider the following linear system:                    ∂y ∂t −∆y=v1Oin Q, ∂y ∂n +a(x, t)y= 0 on Σ, y(x, 0) = 0 in Ω, (9) where the coefficient a∈L∞(Σ). For each v∈L2(O × (0, T)), (9) possesses exactly one solution ysatisfying (6). We have the following result: Lemma 6 Assume that T > 0and a∈L∞(Σ). Then (9) is approximately controllable in L2(Ω) at time T. In other words, for each z1∈L2(Ω) and each ε > 0, there exists a control v∈L2(O × (0, T )) such that the corresponding 6 solution of (9) satisfies ky(·, T)−z1kL2(Ω) ≤ε. (10) Furthermore, the control vcan be found such that kvkL2(O×(0,T)) ≤C1(Ω,O, T, ε, kakL∞(Σ),kz1kL2),(11) where C1(Ω,O, T, R, kz1kL2)is nondecreasing in R. Sketch of the proof: For the proof, we will adapt the arguments in [8] (more details are given in [4]). Let T > 0 and a∈L∞(Σ) be given. We will use the well known fact that the approximate controllability of the linear problem (9) is equivalent to the unique continuation property for the solutions to the following adjoint system (where ϕ0∈L2(Ω)):                    −∂ϕ ∂t −∆ϕ= 0 in Q, ∂ϕ ∂n +a(x, t)ϕ= 0 on Σ, ϕ(x, T) = ϕ0(x) in Ω. (12) That is to say, (9) is approximately controllable in L2(Ω) at time Tif and only if the following holds: If ϕ0∈L2(Ω),ϕis the associated solution to (12) and we have ϕ= 0 in O × (0, T ), then ϕ≡0. It is clear that this property holds. Actually, we have a much stronger result in which the boundary conditions play no role: If ϕ∈L2 loc(Q)(for instance), ∂ϕ ∂t −∆ϕ= 0 in Qand we have ϕ= 0 in O × (0, T ), then ϕ≡0. In fact, this is also true for much more general parabolic equations, see for instance [21]. Thus, if z1is given in L2(Ω) and ε > 0 is fixed, there exist controls v∈L2(O×(0, T )) such that the corresponding solution of (9) verifies (10). 7 It is also clear that vcan be chosen of minimal L2-norm. Let us introduce the functional Jε(·;a, z1), with Jε(ϕ0;a, z1) = 1 2ZZO×(0,T)|ϕ|2dx dt +εkϕ0kL2−(z1, ϕ0)L2(13) for all ϕ0∈L2(Ω), where ϕis the associated solution of (12). This is a continuous and strictly convex functional on L2(Ω). Furthermore, using the previous unique continuation property, it can be proved that Jε(·;a, z1) is coercive on L2(Ω). Assume the minimum is attained at ˆϕ0. We can then take ˆv= ˆϕ|O×(0,T ), where ˆϕis the solution to (12) for ϕ0= ˆϕ0. This control ˆvis such that (10) holds. Moreover, ˆvis the unique control with the following property: If vis another control such that the solution of (9) verifies (10), then kˆvkL2(O×(0,T)) ≤ kvkL2(O×(0,T)) . We can now argue as in [8] to deduce that ˆvsatisfies (11) for some C1= C1(Ω,O, T, R, kz1kL2) that is nondecreasing in R. In fact, we have the following stronger result, whose proof is essentially based on the arguments of [8]: Lemma 7 Let Φ : L∞(Σ) ×L2(Ω) 7→ L2(Ω) be given by Φ(a, y1) = ˆϕ0, where ˆϕ0is the unique minimizer of Jε(·;a, y1)in L2(Ω). If Bis a bounded subset of L∞(Σ) and Kis a compact subset of L2(Ω), then Φ(B×K)is a bounded subset of L2(Ω). Moreover, if aµ→aweakly-∗in L∞(Σ) and y1 µ→y1strongly in L2(Ω), then ˆϕ0 µ→ˆϕ0weakly in L2(Ω). This ends the proof of lemma 6. 2.2 Proof of theorem 2. The fixed point argument We will first consider the case in which fis C1in (−1,1). Let us take y0, y1∈ L2(Ω) and ε > 0. We denote by gthe following function: g(s) =        f(s)−f(0) sif s6= 0, f′(0) if s= 0. (14) 8 Then gis continuous and uniformly bounded (because fis globally Lipschitzcontinuous) and we have |g(s)| ≤ L∀s∈R.(15) Let us introduce the mapping Γ : L2(Σ) 7→ L2(Σ) as follows: For each z∈ L2(Σ), we put Γ(yz) = γ0yz, where yz=uz+wz,uzis the solution of                    ∂uz ∂t −∆uz= 0 in Q, ∂uz ∂n +g(z)uz=−f(0) on Σ, uz(x, 0) = y0(x) in Ω (16) and wzis (together with vz) the solution to the approximate controllability problem                              ∂wz ∂t −∆wz=vz1Oin Q, ∂wz ∂n +g(z)wz= 0 on Σ, wz(x, 0) = 0 in Ω, kwz(·, T)−(y1−uz(·, T)) kL2≤ε (17) furnished by lemma 6 (thus, vzis the unique minimal L2-norm control for which the inequality kwz(·, T)−(y1−uz(·, T)) kL2≤εis satisfied). We then have                    ∂yz ∂t −∆yz=vz1Oin Q, ∂yz ∂n +g(z)yz=−f(0) on Σ, yz(x, 0) = y0(x) in Ω, kyz(·, T)−y1kL2≤ε and kvzkL2(O×(0,T)) ≤C1(Ω,O, T, ε, L, ky1−uz(·, T)kL2). We will see that Schauder’s theorem can be applied to Γ. This will serve to deduce that Γ possesses a fixed point and will suffice to prove theorem 2 in this case. 9 support in O × [0, T] such that w(x, T) = 0 in Ω.(27) In a first step, we will construct a control ˜vin L2(O×(0, T)) with this property. Then, using the regularizing property of the heat equation, we will be able to find a more regular control vsuch that (27) also holds. First of all, let us recall from [11] a global Carleman inequality for the adjoint system (12). To this end, let us introduce a nonempty open set O0satisfying O0⊂⊂ O and a function α0=α0(x) satisfying α0∈C4(Ω) and α0>0 in Ω, α0= 0 on ∂Ω and ∇α06= 0 in Ω\ O0. The existence of such a function α0is justified in [11]. One has the following: Lemma 10 Assume that a∈L∞(Σ) and at∈L∞(Σ). There exists a positive number λ1depending on Ω,O,T,kakL∞(Σ) and katkL∞(Σ), with the following property: For each λ≥λ1, there exist positive constants Cand s1, again depending on Ω,O,T,kakL∞(Σ) and katkL∞(Σ) , such that ZZQ(e−2sα +e−2s˜α)t−3(T−t)−3|ϕ|2dx dt ≤CZZO0×(0,T )(e−2sα +e−2s˜α)t−3(T−t)−3|ϕ|2dx dt for all s≥s1. Here, ϕis the solution of (12) associated to ϕ0∈L2(Ω) and the functions α=α(x, t)and ˜α= ˜α(x, t)are given by α(x, t) = e2λkα0k∞−eλα0 t(T−t),˜α(x, t) = e2λkα0k∞−e−λα0 t(T−t). For the proof of this result, see [11]. We can now deduce an observability estimate for the solutions to (12) whose proof is postponed to the end of this paragraph: Lemma 11 There exist positive constants C4and Mdepending on Ω,O,T, kakL∞(Σ) and katkL∞(Σ) such that ZZQe−M T−t|ϕ|2dx dt ≤C4ZZO0×(0,T )|ϕ|2dx dt (28) for any ϕ0∈L2(Ω). Arguing as in [9], we can deduce from (28) that (26) is null controllable with L2-controls supported in O0×[0, T]. More precisely, let y0∈L2(Ω) be given 16 and let us introduce the functional Kε(·;a), with          Kε(ϕ0;a) = 1 2ZZO0×(0,T )|ϕ|2dx dt +εkϕ0kL2−ZZQθ′(t)qϕ dx dt ∀ϕ0∈L2(Ω) (recall that qis the solution to (25)). Then Kε(·;a) is continuous, strictly convex and coercive in L2(Ω). This is due to the unique continuation property of the solutions to the adjoint system (12). Let ϕ0 εbe the unique minimizer of Kε(·;a) and let ϕεbe the associated solution to (12). Then the control vε=ϕε|O0×(0,T )is such that the corresponding solution wεto (26) (with Oreplaced by O0) satisfies kwε(·, T)kL2≤ε. On the other hand, thanks to the fact that θ′= 0 near t=T, we have ZZQeM T−t|θ′(t)q|2dx dt1/2 ≤Cky0kL2(29) for some Cdepending only on Ω, O,T,kakL∞(Σ) and katkL∞(Σ) . Then the optimality conditions satisfied by ϕ0 εgive ZZO0×(0,T )|ϕε|2dx dt +εkϕ0 εkL2=ZZQθ′(t)qϕεdx dt ≤ZZQeM T−t|θ′(t)q|2dx dt1/2ZZQe−M T−t|ϕε|2dx dt1/2 . Therefore, from the estimates (28) and (29), we easily find that kvεkL2(O0×(0,T)) = ZZO0×(0,T )|ϕε|2dx dt!1/2 ≤Cky0kL2, for a new constant Conly depending on Ω, O,T,kakL∞(Σ) and katkL∞(Σ) . Thus, at least for a subsequence, we have vε→˜vweakly in L2(O0×(0, T )). In this way, we have found a control ˜vthat vanishes outside O0×(0, T), satisfies k˜vkL2(O0×(0,T)) ≤C(Ω,O, T, kakL∞(Σ) ,katkL∞(Σ))ky0kL2(30) and is such that the solution to (26) associated to ˜vsatisfies (27). Obviously, this proves that (23) is null controllable with controls in L2(O0×(0, T)). Let us finally indicate the way we can obtain from ˜va second (regular) control vwith similar properties. 17 Let us introduce a C∞function ξ=ξ(x) such that ξ= 1 in a neighborhood of O0and ξ∈ D(O). Let us set w= (1 −ξ) ˜w, where ˜wis the solution to (26) associated to ˜v. Then wis the solution of                    ∂w ∂t −∆w=−θ′(t)q(x, t) + v1Oin Q, ∂w ∂n +a(x, t)w= 0 on Σ, w(x, 0) = 0, w(x, T) = 0 in Ω, where v=ξ(x)θ′(t)q+ 2∇ξ· ∇ ˜w+ (∆ξ) ˜w. We have therefore built a new control vwhich provides the null controllability of (23). In view of the interior regularity properties for the solution of (25), we have q∈C∞(Ω′×(ε, T)) and kqkCℓ(Ω′×(ε,T)) ≤C(Ω,Ω′, ε, T, ℓ, kakL∞(Σ))ky0kL2(31) for any integer ℓ≥0, any ε > 0 and any open set Ω′⊂⊂ Ω. Using this fact, the interior regularity properties satisfied by ˜w(the solution to (26) for v= ˜v) and the fact that ξis constant in a neighborhood of O0and outside O, we have that v∈C∞(Q), the estimates (24) hold and, obviously, the associated solution to (26) satisfies (27). This ends the proof of theorem 9. Proof of lemma 11: Let us first apply lemma 10 in the time interval [T/4, T] for fixed and sufficiently large λand s. We obtain: ZZΩ×(T/4,T )(e−2sα +e−2s˜α)t−3(T−t)−3|ϕ|2dx dt ≤CZZO0×(0,T)(e−2sα +e−2s˜α)t−3(T−t)−3|ϕ|2dx dt. (32) In view of the form of the weight functions in (32), we can easily deduce that there exist positive constants K1and Mdepending only on Ω, O,T,kakL∞(Σ) and katkL∞(Σ) such that ZZΩ×(T/4,T )e−M T−t|ϕ|2dx dt ≤K1ZZO0×(0,T )|ϕ|2dx dt. (33) 18 On the other hand, multiplying (12) by ϕand integrating in Ω, we get −1 2 d dt Z Ω |ϕ|2dx +Z Ω |∇ϕ|2dx ≤ kakL∞(Σ) Z ∂Ω |ϕ|2dσ ≤Z Ω |∇ϕ|2dx +C(kakL∞(Σ))Z Ω |ϕ|2dx for every t > 0. From these inequalities, it is immediate that ZZΩ×(0,T/4)|ϕ|2dx dt ≤eC(T,kakL∞(Σ))ZZΩ×(T/4,T/2)|ϕ|2dx dt and we also find that ZZΩ×(0,T/4)|ϕ|2dx dt ≤eC(T,kakL∞(Σ))+2M/T ZZΩ×(T/4,T/2)e−M T−t|ϕ|2dx dt. Using (33), we see that ZZΩ×(0,T/4)|ϕ|2dx dt ≤K2ZZO0×(0,T )|ϕ|2dx dt, (34) where K2=K1exp C(T, kakL∞(Σ)) + 2M/T. Now, from (33) and (34), the desired observability estimate (28) follows with C4=K1+K2. This ends the proof. Remark 12 It is possible to find an estimate of the constant in (30) that is explicit in kakL∞(Σ) and katkL∞(Σ) . This can be made arguing as in [9], using sharp estimates of the constants λ1and s1in the Carleman inequality in lemma 10. All this yields the following estimate of the cost C(y0) of the null controllability of (23) with controls in L2(O × (0, T)): C(y0)≤eC(Ω,O)1+T+1 T+kak2 L∞(Σ)+katkL∞(Σ)+Tkak2 L∞(Σ)ky0kL2. In this estimate, we find kakL∞(Σ) and, unfortunately, also katkL∞(Σ) . This is the main reason we cannot give a positive answer to the global null controllability problem for (1) when fis Lipschitz-continuous (see remark 15 below for additional details). In fact, an estimate of the cost for problem (23) of the form C(y0)≤eC(Ω,O)1+T+1 T+γ(kakL∞(Σ))+Tkak2 L∞(Σ)ky0kL2, where γis a positive increasing function, would lead to the null controllability of (1) even when fis locally Lipschitz-continuous and slightly superlinear at infinity. Results of this kind were deduced in [10] when the nonlinearity is in the partial differential equation and we impose homogeneous Dirichlet conditions. 19 3.2 The local null controllability of the nonlinear problem We will need the (Banach) spaces e C1+α,1(Q) = {u∈C1(Q) : D1 xu∈Cα,α/2(Q)}, e C1+α,1/2+α/2(Q) = {u∈Cα,1/2+α/2(Q) : D1 xu∈Cα,α/2(Q)} and e C2+α,1+α/2(Q) = {u∈C0(Q) : D1 xu∈C1+α,1/2+α/2(Q),∂u ∂t ∈Cα,α/2(Q)}. Here, we have used Dm xuto denote all space derivatives of uof order mput together. We will denote by e Cn+α,r+β(Σ) the Banach space formed by the restrictions to Σ of the functions in e Cn+α,r+β(Q). For linear systems of the form                    ∂z ∂t −∆z=k(x, t) in Q, ∂z ∂n +a(x, t)z= 0 on Σ, z(x, 0) = z0(x) in Ω, (35) one has the following result, whose proof is given in [14], p. 320: Lemma 13 Assume that k∈Cα,α/2(Q),a∈e C1+α,1/2+α/2(Σ),z0∈C2+α(Ω) and the following compatibility condition is satisfied: ∂z0 ∂n +a(x, 0)z0= 0 on ∂Ω. Then (35) possesses exactly one solution z, with z∈e C2+α,1+α/2(Q)and        kzke C2+α,1+α/2(Q) ≤C(Ω, T, kake C1+α,1/2+α/2(Σ))kkkCα,α/2(Q)+kz0kC2+α(Ω). (36) Assume that fis of class C3,f(0) = 0 and y0∈C2+α(Ω) satisfies the compatibility condition (8). Let us introduce the function g, given by (14). Then gis a C2function and g(s) =        f(s) sif s6= 0, f′(0) if s= 0. 20 Let us introduce the Banach space Z=e C1+α,1(Σ) and the closed linear manifold Z0={z∈Z:z(x, 0) = y0(x) on ∂Ω}. For each z∈Z0, we will consider the null controllability problem for the linear system                    ∂y ∂t −∆y=v1Oin Q, ∂y ∂n +g(z(x, t))y= 0 on Σ, y(x, 0) = y0(x) in Ω. (37) This can be solved arguing as in the previous paragraph. Indeed, in view of theorem 9, there exist controls vz∈C∞(Q) satisfying kvzkCα,α/2(Q)≤C5(Ω,O, T, kg(z)kZ)ky0kL2,(38) such that the solution yzto (37) with v=vzsatisfies yz(x, T) = 0 in Ω.(39) Furthermore, the constant C5in (38) can be chosen nondecreasing with respect to the last argument kg(z)kZ. From the compatibility condition (8), the fact that z∈Z0and lemma 5, we deduce that yz∈e C2+α,1+α/2(Q) and an estimate like (36) holds. Notice that, here, we are using the fact that gis twice continuously differentiable, which gives g(z)∈e C1+α,1/2+α/2(Σ). This is why we need fof class C3. Let A(z) be the family formed by all the controls in Cα,α/2(Q) such that (38) and (39) hold and let us set Λ(z) = {γ0yz:yzis the solution of (37) associated to v∈A(z)}. Notice that Λ(z)⊂Z0for all z∈Z0. Then, for all q∈Λ(z), we have kqkZ≤C6(Ω,O, α, T, kg(z)kZ)ky0kC2+α(Ω) (40) 21 and kqke C2+α,1+α/2(Σ) ≤C7(Ω,O, α, T, kg(z)kZ)ky0kC2+α(Ω) (41) for some constants C6and C7again nondecreasing in kg(z)kZ. We will consider the set-valued mapping z7→ Λ(z). We will check that, for some η(Ω,O, α, T)>0, the inequality ky0kC2+α(Ω) ≤ηis sufficient to ensure that Λ possesses at least one fixed point in Z. To this end, we will check that, under these conditions, Kakutani’s fixed point theorem can be applied to Λ (for the statement and proof of this result, see for instance [3]). Of course, this will imply the existence of a control v∈Cα,α/2(Q) such that the corresponding solution to (1) satisfies (3). Indeed, it is not difficult to see that Λ(z) is, for each z∈Z0, a nonempty closed convex set in Z0. Furthermore, from (41) and the compactness of the embedding e C2+α,1+α/2(Σ) ֒→Z, we deduce that for each z∈Z0there exists a compact set Kz⊂Z0such that Λ(z)⊂Kz. We also have the following result, whose proof is given below: Lemma 14 Under the assumptions of theorem 4 and with the previous notation, the set-valued mapping z→Λ(z)is upper hemicontinuous. In other words, for each bounded linear form ξ∈Z′, the real-valued function z7→ sup q∈Λ(z) hξ, qi is upper semicontinuous. Now, let R > 0 be given, let us assume that z∈Z0satisfies kzkZ≤R and let us denote by M(R) the following quantity: M(R) = sup kzkZ≤R C6(Ω,O, α, T, kg(z)kZ) Let us set η=R/M(R) and let us assume that the initial state y0satisfies ky0kC2+α(Ω) ≤η(besides (8)). Let us put K(y0) = {z∈Z0:kzkZ≤R}. 22 Then K(y0) is a nonempty closed convex set in Z. In view of (40) and (41), Λ maps K(y0) into a fixed compact set K⊂K(y0). Consequently, all hypotheses of Kakutani’s theorem are certainly satisfied and the existence of a fixed point of Λ in K(y0) is ensured. This ends the proof of theorem 4. Proof of lemma 14: Let us see that the set B(κ, ξ) = {z∈Z0: sup q∈Λ(z) hξ, qi ≥ κ} is closed for every κ∈Rand every ξ∈Z′. Thus, assume that zm∈B(κ, ξ) for all mand zm→zin Z. Our aim is to prove that z∈B(κ, ξ) . In view of the regularity of g, we have g(zm)→g(z) in Z. Since all sets Λ(zm) are compact, for each mwe must have κ≤sup q∈Λ(zm) hξ, qi=hξ, qmi(42) for some qm∈Λ(zm)⊂K. From the definitions of Λ(zm) and A(zm), there must exist controls vm∈Cα,α/2(Q) and associated states ymsatisfying                    ∂ym ∂t −∆ym=vm1Oin Q, ∂ym ∂n +g(zm(x, t))ym= 0 on Σ, ym(x, 0) = y0(x), ym(x, T) = 0 in Ω and qm=γ0ym. We also have kvmkCα,α/2(Q)≤C5(Ω,O, T, kg(zm)kZ)ky0kL2 and kqmke C2+α,1+α/2(Σ) ≤C7(Ω,O, α, T, kg(zm)kZ)ky0kC2+α(Ω) . Hence, qm(resp. vm) is uniformly bounded in e C2+α,1+α/2(Σ) (resp. Cα,α/2(Q)). Therefore, we can write the following at least for a subsequence: qm→ˆqstrongly in Z, vm→ˆvstrongly in C0(Q) 23 and ˆv∈Cα,α/2(Q). Now, it is easy to deduce that ˆv∈A(z) and ˆq=γ0ˆy, with                    ∂ˆy ∂t −∆ˆy= ˆv1Oin Q, ∂ˆy ∂n +g(z(x, t))ˆy= 0 on Σ, ˆy(x, 0) = y0(x),ˆy(x, T) = 0 in Ω. In particular, we have ˆq∈Λ(z). Now, we can take limits in (42) and this gives κ≤ hξ, ˆqi ≤ sup q∈Λ(z) hξ, qi, that is to say, z∈B(κ, ξ). This proves that z7→ Λ(z) is upper hemicontinuous. Remark 15 To prove a (global) null controllability result for (1), a natural strategy is a fixed point approach similar to the argument we have used in Section 2. But the requirement at∈L∞(Σ), which seems to be necessary in the proofs of lemma 10 and theorem 9, is apparently too strong. Indeed, we would need in practice functions zsuch that the trace of the time derivative of g(z) belongs to L∞(Σ). Thus, we are not too far from ∂z ∂t ∈L∞(0, T ;W1,N+κ(Ω)), with κ > 0. But the spaces of this kind seem to be too small to permit compactness and good estimates for the fixed point mapping. Hence, as we already mentioned at the end of Section 1, the global null controllability of (1) is an open question. 4 Proof of the large time null controllability result This Section is devoted to prove theorem 5. To this end, we will argue as follows: •Starting from an arbitrary large y0∈L2(Ω), we first use the local feedback law v=−y1O. This provides a first control v1for t∈[0, T1] which leads the system to a state y1=y(·, T1) which is small in the H1-norm. •Then, we simply take v2= 0 for t∈[T1, T2]. This leads to a second intermediate state y2=y(·, T2) which is small in the H2-norm. 24 •Starting from y2at time t=T2and setting again v3= 0 for t∈[T2, T∗], we arrive now at a state y∗=y(·, T∗) such that ky∗kC2+α≤η(Ω,O, α, ε),(43) where ηis the constant arising in theorem 4 and εis arbitrarily small. •Let us introduce T=T∗+ε. In view of (43) and theorem 4, we can find a control v∗defined for t∈[T∗, T ] such that the associated state y∗satisfies y∗(x, T) = 0 in Ω.(44) Obviously, this ends the proof. Let us now give more details. For simplicity, we will assume that N≤4. This assumption is not strictly necessary but will make the argument easier and will clarify the presentation we can give. We will use well known regularity results for linear and semilinear parabolic systems, see for instance [14] and [18]. Thus, let y0∈L2(Ω) be given and let us choose α∈(0,1) and ε > 0. First Step: Consider the closed-loop controlled system                    ∂y ∂t −∆y=−y1Oin Ω ×(0,+∞), ∂y ∂n +f(y) = 0 on ∂Ω×(0,+∞), y(x, 0) = y0(x) in Ω. (45) This semilinear system possesses exactly one solution ˆy, with ˆy∈L2(0,+∞;H1(Ω)) ∩C0([0,+∞); L2(Ω)). Furthermore, using standard techniques, we see at once that                    1 2kˆy(·, t)k2 L2+Zt τk∇ˆy(·, s)k2 L2ds +Zt τZO|ˆy(x, s)|2dx ds +Zt τZ∂Ωf(ˆy)ˆy dΓds =1 2kˆy(·, τ)k2 L2 (46) for all t, τ ∈[0,+∞) with τ < t. Since f(s)s≥0 for all s, we deduce that kˆy(·, t)k2 L2+CZt τkˆy(·, s)k2 H1ds ≤ kˆy(·, τ)k2 L2(47) 25 In order to clarify this point, let us consider the relatively simple case of a radial solution of the system                    ∂y ∂t −∆y= 0 in BR×(0, T), ∂y ∂n −h(y) = 0 on ∂BR×(0, T ), y(x, 0) = y0(x) in BR. (72) where BRis the open ball in RNof radius R. We will assume here that h∈ C1(R) is nondecreasing, h(s)>0 for all s > 0 and Z+∞ 0 1 h(s)ds < +∞. Assume that y0is a regular radial function such that y0 r(x) = ∇y0(x)·x/|x| ≥ 0 and m(0) = ZBR y0(x)dx > 0. Let us denote by ythe associated solution to (72) and let us set m(t) = ZBR y(x, t)dx for all t. Then, following for instance [19], it is not difficult to prove that m′(t) = ZBR yt(x, t)dx =ZBR ∆y(x, t)dx =Z∂BR ∂y ∂n(x, t)dΓ = Z∂BR h(y(x, t)) dΓ ≥Ah(BZBR y(x, t)dx) and thus m′(t)≥Ah(Bm(t)) (73) for some positive constants Aand B. Notice that we have used here the fact that yr(x, t) = ∇y(x, t)·x/|x| ≥ 0 for all t. In particular, we deduce from (73) that m(t)>0 for all positive t. Let us introduce the functions Hand L, with H(s) = Z+∞ s 1 h(σ)dσ and L=H−1. 32 Then, it can be easily deduced from (73) that, for some C > 0, one has H(Bm(0)) −H(Bm(t)) ≥Ct and m(t)≥1 BL(H(Bm(0)) −Ct) for all t. Since L(H(Bm(0)) −Ct)→+∞as t→1 CH(Bm(0)), we have blow-up before t=T∗=1 CH(Bm(0)). Unfortunately, the arguments in [10] cannot be applied to a system of the kind (1), since they rely strongly on the fact that, there, the nonlinear term interacts with the elliptic operator −∆ in Ω \ω. Indeed, to apply the techniques in [10] in the context of (72), we have to introduce a cut-off function ρ=ρ(x) with support in BR\ωand we have to analyze the evolution of ˜m(t) = ZBR ρ(x)y(x, t)dx. It would be satisfactory to have for ˜ma differential inequality of the kind (73). But this time we have ˜m′(t) = ZBR ρ(x)yt(x, t)dx =ZBR ρ(x)∆y(x, t)dx =ZBR ∆ρ(x)y(x, t)dx +Z∂BR ρ(x)∂y ∂n(x, t)−∂ρ ∂n(x)y(x, t)!dΓ. By choosing ρsuch that ∂ρ ∂n(x) = 0 on ∂BR, we find that ˜m′(t) = ZBR ∆ρ(x)y(x, t)dx +Z∂BR ρ(x)h(y(x, t)) dΓ, but it seems complicate to bound from below the sum of these integrals by an expression of the form Ah(B˜m(t)) −C(notice that we do not have now yr≥0). Thus, for systems like (1) a new argument is required and, for the moment, the question is open. For other basic facts on the blow-up due to the presence of nonlinear boundary conditions, see for instance [6], [19], [20] and [16]. 33 5.3 A variant for systems of the Stokes kind Let us now consider the Stokes system with nonlinear slip boundary conditions                  ∂y ∂t −∆y+∇π=v1O,∇ · y= 0 in Q, y·n= 0,(σ(y, π)·n)tg +f(y)tg = 0 on Σ, y(x, 0) = y0(x) in Ω, (74) where v∈L2(O × (0, T ))N,y0∈Hand f:RN7→ RNis globally Lipschitzcontinuous. Here, we have used the following notation: atg =a−(a·n)nis the tangential component of a, σ(y, π) = −πId + (∇y+t∇y) is the usual stress tensor, H={v∈L2(Ω) : ∇ · v= 0 in Ω, v ·n= 0 on ∂Ω}. Arguing as in Section 2, it can be proved that (74) is approximately controllable in Hfor all T > 0. Furthermore, the control vcan be chosen of the form v= (v1, v2,0), with vi∈L2(O × (0, T)) (see [4]). However, the null controllability of (74) is an open problem. It seems reasonable to expect results similar to theorems 4, 5 and 9. But, again, this is unknown at present. Acknowledgements: The authors are indebted to the anonymous referee for his/her comments and suggestions. These have contributed to a substantial improvement of the paper. References [1] H. Amann, Parabolic evolution equations and nonlinear boundary conditions, J. Diff. Equ. 72 (1988), p. 201–269. [2] J. Arrieta, A. Carvalho and A. Rodr´ıguez-Bernal, Parabolic problems with nonlinear boundary conditions and critical nonlinearities, J. Diff. Equ. 156 (1999), p. 376–406. [3] J.P. Aubin, L’analyse non lin´eaire et ses motivations ´economiques, Paris, Masson 1984. 34 [4] A. Doubova, Thesis, University of Sevilla (Spain), september 2000. [5] A. Doubova, E. Fern´andez-Cara, M. Gonz´alez-Burgos and E. Zuazua, On the controllability of parabolic systems with a nonlinear term involving the state and the gradient, SIAM J. Control Optim. Vol. 41, No. 3 (2002), p. 798–819. [6] M. Chipot, M. Fila and P. Quittner, Stationary solution, blow up and convergence to stationary solution for semilinear parabolic equations with nonlinear boundary conditions, Acta Math. Univ. Comenian. 60 (1991), p. 35– 103. [7] L. Evans, Regularity properties of the heat equation subject to nonlinear boundary constraints, Nonlinear Anal. 1(1997), p. 593–602. [8] C. Fabre, J.P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation, Proc. Royal Soc. Edinburgh, 125A (1995), 31–61. [9] E. Fern´andez-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case, Adv. Diff. Eqs. Vol. 5 (4–6), April-June (2000), 465–514. [10] E. Fern´andez-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations, Ann. Inst. H. Poincar´e, Anal. non lin´eaire, 17 (2000), no. 5, 583–616. [11] A. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes #34, Seoul National University, Korea, 1996. [12] O. Yu. Imanuvilov, Controllability of parabolic equations, Mat. Sb. 186, N. 6, (1995), 102–132. [13] O. Yu. Imanuvilov and M. Yamammoto , Carleman estimate for a parabolic equation in a Sobolev space of negative order and its applications, in “Control of nonlinear distributed parameter systems”, p. 113–137, Lecture Notes in Pure and Appl. Math., 218, Dekker, New York 2001. [14] O.A. Ladyzhenskaya, V.A. Solonnikov and N.N. Uraltzeva, Linear and Quasilinear Equations of Parabolic Type, Translation of Mathematical Monographs, Vol. 23, Providence, Rhode Island 1968. [15] I. Lasiecka and R. Triggiani, Exact controllability of semilinear abstract systems with applications to waves and plates boundary control, Appl. Math. & Optim., 23 (1991), 109–154. [16] Z. Lin and M. Wang, The blow-up properties of solutions to semilinear heat equations with nonlinear boundary conditions, Z. Angew. Math. Phys. 50 (1999), no. 3, 361–374. [17] J.L. Lions, Contrˆolabilit´e exacte, Perturbations et Stabilisation de Syst`emes Distribu´es, Vol. 1 et 2, Masson, Collection RMA, Paris 1988. [18] J.L. Lions and E. Magenes, Probl`emes aux Limites Non Homog`enes et Applications, Vol I, II and III, Dunod, Gauthers-Villars, Paris 1969. 35 [19] J. L´opez-G´omez, V. M´arquez and N. Wolanski, Blow-up results and localization of blow-up points for the heat equation with a nonlinear boundary condition, J. Diff. Equ. 92 (2) (1991), p. 384–401. [20] A. Rodr´ıguez-Bernal and A. Tajdine, Nonlinear balance for reaction diffusion equations under nonlinear boundary conditions: Dissipativity and blow-up, J. Diff. Equ., 169 (2001), p. 332–372. [21] J.C. Saut and B. Scheurer, Unique continuation for some evolution equations, J. Diff. Equ., 66 (1), 1987. [22] E. Zuazua, Exact boundary controllability for the semilinear wave equation, in “Nonlinear Partial Differential Equations and their Applications”, Vol. X, H. Brezis and J.L. Lions eds., Pitman, 1991, 357–391. [23] E. Zuazua, Finite-dimensional null controllability for the semilinear heat equation, J. Math. Pures Appl. (9) 76 (1997), no. 3, 237–264. [24] E. Zuazua, Controllability of the linear system of thermoelasticity, J. Math. Pures Appl., 74 (1995), 303–346. 36