Null controllability results for parabolic equations in unbounded domains

XX Congreso de Ecuaciones Diferenciales y Aplicaciones X Congreso de Matem´ atica Aplicada Sevilla, 24-28 septiembre 2007 (pp. 1–7) Null controllability results for parabolic equations in unbounded domains Manuel Gonz´ alez-Burgos1, Luz de Teresa2 1Dpto. E.D.A.N., Universidad de Sevilla, Aptdo. 1160, E-41080 Sevilla. E-mails: [email protected]. 2Instituto de Matem´aticas, Universidad Nacional Aut´onoma de M´exico, Circuito Exterior, C.U., 04510 D.F., M´exico. E-mail: [email protected]. Palabras clave: Controllability, parabolic PDE, unbounded domains. Resumen In this talk we present some results concerning the null controllability for a heat equation in unbounded domains. We characterize the conditions that must satisfy the weight function, introduced by Fursikov and Imanuvilov, in order to prove a global Carleman inequality for the adjoint problem and then to get a null controllability result. We give some examples of unbounded domains (Ω, ω) that satisfy these sufficient conditions. Finally, when Ω \ωis bounded, we prove the null controllability of the semi-linear heat equation when the nonlinearity is slightly superlinear. 1. Introduction Let Ω ⊂RNbe an unbounded connected open set with boundary ∂Ω of class C0,1 uniformly. Let ω⊂Ω be a nonempty open subset and assume T > 0. We will consider the parabolic linear problem      yt−∆y+B· ∇y+ay =v1ωin Q= Ω ×(0, T), y= 0 on Σ = ∂Ω×(0, T ), y(x, 0) = y0(x) in Ω, (1) where B∈L∞(Q)N,a∈L∞(Q), y0∈L2(Ω), and v∈L2(Q). In (1), 1ωdenotes the characteristic function of the subset ω,y=y(x, t) is the state and v=v(x, t) is the control function. 1 M. Gonz´alez-Burgos, L. de Teresa We will also consider the nonlinear problem      yt−∆y+f(y, ∇y) = v1ωin Q, y= 0 on Σ, y(x, 0) = y0(x) in Ω, (2) where f:R×RN→Ris a locally Lipschitz-continuous function, y0is given in an appropriate Banach space and vis a control which also acts on the system through the open subset ω. The main goal of this paper is to analyze the null controllability properties of (1) and (2). We recall that (1) (resp., (2)) is null controllable at time Tif for every y0∈L2(Ω) (resp., y0∈W1,∞(Ω) ∩H1 0(Ω)) there exists a control v∈L2(Q) such that the solution y of (1) satisfies y(·, T) = 0 in Ω,(3) (resp., (2) possesses a solution y∈C0([0, T]; L2(Ω)) satisfying (3)). There are few results on the null controllability of (1) and (2) when Ω is an unbounded set. In fact, the first result was negative: in [10] the authors considered the case Ω = (0,∞)⊂R,a≡0 and B≡0, and proved that there are no initial data in any negative Sobolev space that may be driven to zero at a finite time by a control function acting on the boundary {x= 0}(see also [11]). The first positive result on null controllability of (1) and (2) was obtained in [2]. There, the authors considered the semilinear heat equation (2) with a globally-Lipschitz continuous function fand proved the null controllability of the problem when the distributed control acts on a set ωsatisfying that Ω \ωis bounded. In [3] the authors studied the null controllability of system (1) when a≡0, B≡0, Ω = (0,∞) and ωis an unbounded open set of the form ω=Sn≥0ωn.Under some technical assumptions on ωn, they proved two null controllability results: one result with control functions in L2(Q) for initial data in some weighted L2(Ω)-spaces and another one for arbitrary initial data in L2(Ω) by using control functions in a weighted L2(Q)-space. Finally, in [12] the author also considered the heat equation (a≡0 and B≡0) with a Dirichlet boundary condition in an unbounded domain Ω and proved several results about the null controllability of this system with distributed and boundary controls. To be precise, the author shows the lack of null controllability of the heat equation under a geometric condition on Ω and ωand also proves a positive null controllability result for unbounded product domains. Our first result in this talk is related to the results in the previous papers but our approach is different from the approaches of the previous authors. In fact we give sufficient conditions on the weight function (and in this sense on Ω and ω) that allow to construct a global Carleman inequality for the so-called adjoint problem. As a consequence, we will be able to recover the results of [2] and prove the results of [12] about the null controllability in L2(Ω) of system (1) when a∈L∞(Q) and B∈L∞(Q)N. Our second main result concerns the controllability properties of system (2) when Ω \ωis bounded and the function fhas a superlinear growth at infinity. In this case the techniques used in [4] do not apply since they use the fact that Lq(Ω) ⊂Lp(Ω) for 1 ≤p < q ≤ ∞ when Ω is of finite measure. 2 Controllability results for the heat equation in unbounded domains 2. A global Carleman inequality All along this paper we will assume that Ω ⊂RNis an unbounded connected open set with boundary at least of class C0,1uniformly, such that DΩ(−∆) = H2(Ω) ∩H1 0(Ω).(4) This “non classical” assumption is needed to give examples in unbounded set that are the cartesian product of regular sets. Observe that even in the simple case in which Ω is the cartesian product of two (bounded or unbounded) intervals, Ω is only a C0,1set, however it is a “good one” in the sense that (4) holds. We also assume that ω⊂Ω is a nonempty open subset such that there exist an open set ω0and a positive function η0∈C2(RN) satisfying              ω0⊂ω⊂Ω with d0= dist (ω0,Ω\ω)>0, |∇η0| ≥ C0>0 in Ω \ω0,∂η0 ∂n ≤0 on ∂Ω, |η0|+|∇η0|+X i,j ¯¯¯¯ ∂2η0 ∂xi∂xj¯¯¯¯≤C1in Ω, (5) for two positive constants C0and C1. In the next section we will give some examples of unbounded domains and control regions that satisfy the previous hypotheses. As said above, we first prove a global Carleman inequality for the linear problem            −ϕt−∆ϕ=F0+ N X i=1 ∂Fi ∂xi in Q, ϕ= 0 on Σ, ϕ(x, T) = ϕ0(x) in Ω, (6) where F0,Fi∈L2(Q) (1 ≤i≤N) and ϕ0∈L2(Ω). To this end, let us introduce for (x, t)∈Q, α(x, t) = e2λm||η0||∞−eλ(m||η0||∞+η0(x)) t(T−t), ξ(x, t) = eλ(m||η0||∞+η0(x)) t(T−t),(7) with sand λpositive real numbers and m > 1 a fixed real number. One has: Theorem 2.1. Under the previous assumptions, there exist three positive constants σ1, λ1and C1only depending on C0,C1and d0such that, for every ϕ0∈L2(Ω) and F0, Fi∈ L2(Q)(1≤i≤N), the corresponding solution ϕto (6) satisfies            sλ2ZZQ e−2sαξ|∇ϕ|2+s3λ4ZZQ e−2sαξ3|ϕ|2≤C1Ãs3λ4ZZω×(0,T) e−2sαξ3|ϕ|2 +ZZQ e−2sα|F0|2+sλ2 N X i=1 ZZQ e−2sαξ2|Fi|2!, (8) for all s≥s1=σ1(T+T2)and λ≥λ1. The proof of this inequality follows the method introduced by Imanuvilov and Yamamoto in [9] (see also [6] and [5]) and will be given in [8]. 3 M. Gonz´alez-Burgos, L. de Teresa 3. Null controllability of the linear problem. Examples In this Section, we will assume that y0∈L2(Ω) and that (5) holds. One has, Theorem 3.1. There exists a control bv∈L2(Q)such that the corresponding solution by to system (1) satisfies (3) and ||bv||2 L2(Q)≤exp(M(T))||y0||2 L2(Ω) .(9) with M(T) = ³C³1 + T+1 T+T||a||∞+||a||2/3 ∞+ (1 + T)||B||2 ∞´´. In addition, let us assume that either ∂ω ∩∂Ωis the empty set or ∂ω ∩∂Ωis of class C2uniformly. Then, given y0∈L2(Ω) there exists a new control v∈L2(Q)∩L∞(Q) satisfying ||v||2 L2(Q)+||v||2 ∞≤exp(M(T))||y0||2 L2(Ω) (10) which also gives the null controllability of system (1). In (9),Cis a positive constant only depending on C0,C1and d0; in (10),Calso depends on Ωand ω. Sketch of the proof. The proof of this theorem is done in two steps: First step. Since ωsatisfies the assumptions in Theorem 2.1, there exists another subset ω1such that ω0⊂ω1⊂ω, dist (ω0,Ω\ω1) = d0/2, and dist (ω1,Ω\ω)>0. By using the Carleman inequality (8) (for the set ω1) it is by now classical (e.g. [6]) to obtain an observability inequality for the adjoint equation of (1) that allows the construction of a control bv∈L2(Q) satisfying (9), with supp bv⊂ω1×[0, T] and such that the corresponding solution byto (1) satisfies by(·, T ) = 0 in Ω. Second step. We construct a new null control v∈L∞(Q)∩L2(Q) for system (1). This can be carried out by adapting the technique introduced in [1]. We proceed as follows. Let η∈C∞([0, T]) be such that η≡1 in [0, T/3], η≡0 in [2T/3, T], and 0 ≤η≤1, |η0(t)| ≤ C/T in [0, T ]. Due to the properties of ω1, it is possible to construct a function θ∈C∞(RN) such that (supp θ∩(Ω \ω) = ∅, θ(x) = 1 ∀x∈ω1, 0≤θ(x)≤1,|Dαθ(x)| ≤ C(α)d−|α| 0∀x∈RN,∀α∈NN. Let Y∈L2(0, T;H1 0(Ω)) be the weak solution of system (1) corresponding to v≡0. Let us consider y(x, t) = [1 −θ(x)]by(x, t) + θ(x)η(t)Y(x, t)∀(x, t)∈Q. (11) It is not difficult to see that yis the solution of (1) corresponding to the control v= 2∇θ· ∇[by−η(t)Y]+∆θ[by−η(t)Y]−(B· ∇θ)[by−η(t)Y] + θη0(t)Y . This solution satisfies (3). In addition, by construction of θ, supp v⊂ω×[0, T] and the local parabolic regularity allows to see that v∈L∞(Q) (see [8]). Now, it is interesting to exhibit some examples of unbounded open domains and control regions where the assumptions above hold (for the details, see [8]: 1. When Ω ⊂RNis an unbounded connected open set with boundary ∂Ω of class C2 uniformly and ω⊂Ω satisfies that Ω \ωis bounded. 4 Controllability results for the heat equation in unbounded domains 2. When Ω = (0,∞)⊂R(resp., Ω = R) and one considers ω=[ n≥1 (an, bn),(resp., ω=[ n≥1 ((an, bn)∪(a−n, b−n))), where, for every n≥1, 0 < an< bn< an+1, lim an= lim bn=∞(resp., 0 > a−n> b−n> a−n−1, lim a−n= lim b−n=−∞) and (an+1 −bn≤M < ∞ bn−an≥m > 0(resp., (b−n−a−n−1≤M < ∞ a−n−b−n≥m > 0). 3. The example we are going to present is, as far as we know, authentically new. We give an example of unbounded sets Ω and ωsuch that Ω \ωis an unbounded set with infinite measure and where the observability inequality holds. In fact, let Ω be an unbounded open set uniformly of class C2. The set ωis going to be constructed from Ω by eliminating an “infinite strip” around the boundary. That is, ω={x∈Ω : dist (x, ∂Ω) > δ0} for some δ0>0 that is going to be fixed later. For the choice of δ0we will need to recall some geometric results and properties which are inherent to uniformly C2 unbounded sets. To this aim, we will use results appearing in the paper of Fornaro et al. [7]. We recall that a set Ω satisfies a uniform interior sphere condition if for each point y0∈∂Ω there exists a ball By0, depending on y0, contained in Ω and such that By0∩∂Ω = {y0}, moreover the radii of these balls are bounded from below by a positive constant. The following results are extracted from [7]: Proposition 3.2. If ∂Ωis uniformly of class C2, then it satisfies a uniform interior sphere condition. Proposition 3.3. Assume that ∂Ωis uniformly of class C2and let b δbe a positive constant such that at each point of ∂Ωthere exists a ball which satisfies the interior sphere condition at y0with radius greater or equal to b δ. Then: a) For every x∈Ωb δ={y∈Ω : dist (y, ∂Ω) <b δ}there exists a unique ξ=ξ(x)∈ ∂Ωsuch that |x−ξ|=dist (x, ∂Ω). b) Let us define the function d(x) = dist (x, ∂Ω), then d∈C2 b(Ωb δ). c)∇d(x) = −n(ξ(x)) for every x∈Ωb δand where n(ξ)denotes the unit outward normal vector to ∂Ωat point ξ. With the previous results in mind, we are now able to define δ0and prove that the sets Ω and ωsatisfy (5) (obviously, (4) holds). In fact, let us take δ0=b δ−ε, where b δ > 0 is as in Proposition 3.3 and ε∈(0,b δ), and ω0={x∈Ω : dist (x, Ω) >b δ−ε/2}. 5 M. Gonz´alez-Burgos, L. de Teresa It is easy to see that the first statement of (5) holds with d0≡ε/2. On the other hand, we define ˜η0(x) = (d(x)≡dist (x, ∂Ω) if x∈Ω\ω0, b δ−ε/2 if x∈ω0. Let us remark that Ω \ω0⊂Ωb δand dist (Ω \ω0,Ω\Ωb δ) = ε/2>0. We can then consider ρ∈C2(Ω) with 0 ≤ρ(x)≤1 in Ω, ρ(x)≡1 in Ω \ω0, supp ρ⊂ Ωb δand |Dαρ(x)| ≤ C/ε|α|for every x∈Ω and α∈NNwith |α| ≤ 2. Finally, if we define η0(x) = ˜η0(x)ρ(x) and taking into account Propostion 3.3 and the previous properties we readily obtain that η0also satisfies the two last statement of condition (5). 4. Let us suppose that Ω ⊂RNis an open set which satisfies (4), ωis an open subset of Ω and (Ω, ω) satisfies (5) with constants C0,C1and d0. Then, if O⊂RM(M≥1) is another open set with boundary of class C0,1uniformly, and such that the open set e Ω = Ω ×O⊂RN+Msatisfies (4), then e Ω with eω=ω×Oalso satisfy assumption (5) with the same constants (and therefore, with constants which are independent of O). 5. The previous example can be readily generalized as follows: Let us assume that Ω1⊂RN, Ω2⊂RM(N, M ≥1) are two open sets satisfying condition (4) and such that for Ω = Ω1×Ω2condition (4) holds. Suppose that ω1⊂Ω1,ω2⊂Ω2are two open subsets such that (Ω1, ω1) and (Ω2, ω2) satisfy (5). Then, condition (5) comes true for (Ω, ω) with ω=ω1×ω2. Remark 1.The previous examples generalize some known results on null controllability of system (1) in unbounded domains that have been studied before. Firstly, we again obtain the null controllability result of the linear system (1) established in [2] when (Ω, ω) satisfies the conditions in example 1. Secondly, by means of the examples 4 and 5, we generalize the positive results on null controllability of the heat equation obtained in [12]. Observe that our approach allows us to consider general coefficients a∈L∞(Q) and B∈L∞(Q)N in system (1). 4. Null controllability of the nonlinear problem Finally, when Ω \ωis bounded, we have the following result concerning the nonlinear system (3). From now on, we consider a locally Lipschitz-continuous function f:R×RN→ Rsuch that f(0,0) = 0. Under these assumptions we can decompose fas f(s, p) = g(s, p)s+G(s, p)·pwith g, G ∈L∞ loc . One has: Theorem 4.1. Let Ωbe an unbounded connected open set of class C2uniformly and ω⊂Ωan open subset such that Ω\ωis bounded. Suppose that fas above satisfies l´ım |(s,p)|→∞ |g(s, p)| log3/2(1 + |s|+|p|)= 0,l´ım |(s,p)|→∞ |G(s, p)| log1/2(1 + |s|+|p|)= 0.(12) Then, for every y0∈W1,∞(Ω) ∩H1 0(Ω) there exists a control v∈L2(Ω) ∩L∞(Ω) such that (2) admits a solution ythat satisfies (3). 6 Controllability results for the heat equation in unbounded domains Idea of the proof. The proof is obtained by a fixed-point argument (see [4] for the bounded case) and is a consequence of Theorem 3.1. In order to avoid the lack of compactness of the Sobolev embeddings when Ω is unbounded, we use the construction of yas in (11), and take advantage of the fact that supp (y−η(t)θ(x)Y) is contained in a bounded set (key point in [2]). See [8] for a complete proof. Acknowledgements The first author was partially supported by D.G.E.S. (Spain), grant BFM2003–06446. The second authors was partially supported by C.O.N.A.C.yT. (Mexico). References [1] O. Bodart, M. Gonz´alez-Burgos, R. P´erez-Garc´ıa, Existence of insensitizing controls for a semilinear heat equation with a superlinear nonlinearity, Comm. PDE 29 (2004), no. 7-8, 1017–1050. [2] V.R. Cabanillas, S.B. de Menezes, E. Zuazua, Null controllability in unbounded domains for the semilinear heat equation with nonlinearities involving gradient terms, J. Optim. Theory Appl. 110 (2001), no. 2, 245–264. [3] P. Cannarsa, P. Martinez, J. Vancostenoble, Null controllability of the heat equation in unbounded domains by a finite measure control region, ESAIM Control Optim. Calc. Var. 10 (2004), no. 3, 381–408. [4] A. Doubova, E. Fern´andez-Cara, M. Gonz´alez-Burgos, E. Zuazua, On the controllability of parabolic systems with a nonlinear term involving the state and the gradient, SIAM J. Control Optim. 41 (2002), no. 3, 798–819. [5] E. Fern´andez-Cara, S. Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability, SIAM J. Control Optim. 45 (2006), no. 4, 1399–1446. [6] E. Fern´andez-Cara, E. Zuazua, The cost of approximate controllability for heat equations: The linear case, Adv. Differential Equations 5 (2000), no. 4–6, 465–514. [7] S. Fornaro, G. Metafune, E. Priola, Gradient estimates for Dirichlet parabolic problems in unbounded domains. J. Differential Equations 205 (2004), no. 2, 329–353. [8] M. Gonz´alez-Burgos, L. de Teresa, Some results on controllability for linear and nonlinear heat equations in unbounded domains, submited. [9] O. Yu. Imanuvilov and M. Yamamoto, Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations, Publ. Res. Inst. Math. Sci. 39 (2003), no. 2, 227–274. [10] S. Micu, E. Zuazua, On the lack of null-controllability of the heat equation on the half-line, Trans. Amer. Math. Soc. 353 (2001), no. 4, 1635–1659. [11] S. Micu, E. Zuazua, On the lack of null-controllability of the heat equation on the half space, Port. Math. (N.S.) 58 (2001), no. 1, 1–24. [12] L. Miller, On the null-controllability of the heat equation in unbounded domains, Bull. Sci. Math. 129 (2005), no. 2, 175–185. 7