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 \ ω is bounded, we prove the null controllability of the semi-linear heat equation when the nonlinearity is slightly superlinear

González Burgos, Manuel

Teresa de Oteyza, María de la Luz de

idUS. Depósito de Investigación Universidad de Sevilla

XX Congreso de Ecuaciones Diferenciales y AplicacionesX Congreso de Matema´tica AplicadaSevilla, 24-28 septiembre 2007(pp. 1–7)Null controllability results for parabolic equations inunbounded domainsManuel Gonza´lez-Burgos1, Luz de Teresa21 Dpto. E.D.A.N., Universidad de Sevilla, Aptdo. 1160, E-41080 Sevilla. E-mails: manoloburgos@us.es.2 Instituto de Matema´ticas, Universidad Nacional Auto´noma de Me´xico, Circuito Exterior, C.U., 04510D.F., Me´xico. E-mail: deteresa@matem.unam.mx.Palabras clave: Controllability, parabolic PDE, unbounded domains.ResumenIn this talk we present some results concerning the null controllability for a heatequation in unbounded domains. We characterize the conditions that must satisfy theweight function, introduced by Fursikov and Imanuvilov, in order to prove a globalCarleman inequality for the adjoint problem and then to get a null controllability re-sult. We give some examples of unbounded domains (Ω, ω) that satisfy these sufficientconditions. Finally, when Ω \ ω is bounded, we prove the null controllability of thesemi-linear heat equation when the nonlinearity is slightly superlinear.1. IntroductionLet Ω ⊂ RN be an unbounded connected open set with boundary ∂Ω of class C0,1uniformly. Let ω ⊂ Ω be a nonempty open subset and assume T > 0. We will consider theparabolic linear problemyt −∆y +B · ∇y + ay = v1ω in 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), 1ω denotes thecharacteristic function of the subset ω, y = y(x, t) is the state and v = v(x, t) is thecontrol function.1M. Gonza´lez-Burgos, L. de TeresaWe will also consider the nonlinear problemyt −∆y + f(y,∇y) = v1ω in Q,y = 0 on Σ,y(x, 0) = y0(x) in Ω,(2)where f : R × RN → R is a locally Lipschitz-continuous function, y0 is given in anappropriate Banach space and v is a control which also acts on the system through theopen 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 T if for every y0 ∈ L2(Ω)(resp., y0 ∈ W 1,∞(Ω) ∩H10 (Ω)) there exists a control v ∈ L2(Q) such that the solution yof (1) satisfiesy(·, 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 unboundedset. 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 negativeSobolev space that may be driven to zero at a finite time by a control function acting onthe 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 f and proved the null controllability of theproblem when the distributed control acts on a set ω satisfying that Ω \ ω is bounded.In [3] the authors studied the null controllability of system (1) when a ≡ 0, B ≡ 0,Ω = (0,∞) and ω is an unbounded open set of the form ω = ⋃n≥0 ωn. Under sometechnical assumptions on ωn, they proved two null controllability results: one result withcontrol functions in L2(Q) for initial data in some weighted L2(Ω)-spaces and another onefor 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) witha Dirichlet boundary condition in an unbounded domain Ω and proved several resultsabout the null controllability of this system with distributed and boundary controls. Tobe precise, the author shows the lack of null controllability of the heat equation under ageometric condition on Ω and ω and also proves a positive null controllability result forunbounded product domains.Our first result in this talk is related to the results in the previous papers but ourapproach is different from the approaches of the previous authors. In fact we give sufficientconditions on the weight function (and in this sense on Ω and ω) that allow to construct aglobal Carleman inequality for the so-called adjoint problem. As a consequence, we will beable to recover the results of [2] and prove the results of [12] about the null controllability inL2(Ω) of system (1) when a ∈ L∞(Q) and B ∈ L∞(Q)N . Our second main result concernsthe controllability properties of system (2) when Ω \ ω is bounded and the function f hasa superlinear growth at infinity. In this case the techniques used in [4] do not apply sincethey use the fact that Lq(Ω) ⊂ Lp(Ω) for 1 ≤ p < q ≤ ∞ when Ω is of finite measure.2Controllability results for the heat equation in unbounded domains2. A global Carleman inequalityAll along this paper we will assume that Ω ⊂ RN is an unbounded connected open setwith boundary at least of class C0,1 uniformly, such thatDΩ(−∆) = H2(Ω) ∩H10 (Ω). (4)This “non classical” assumption is needed to give examples in unbounded set that are thecartesian product of regular sets. Observe that even in the simple case in which Ω is thecartesian product of two (bounded or unbounded) intervals, Ω is only a C0,1 set, howeverit 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 openset ω0 and 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|+∑i,j∣∣∣∣ ∂2η0∂xi∂xj∣∣∣∣ ≤ C1 in Ω,(5)for two positive constants C0 and C1. In the next section we will give some examples ofunbounded 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∑i=1∂Fi∂xiin 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 s and λ positive real numbers and m > 1 a fixed real number. One has:Theorem 2.1. Under the previous assumptions, there exist three positive constants σ1,λ1 and C1 only depending on C0, C1 and d0 such that, for every ϕ0 ∈ L2(Ω) and F0, Fi ∈L2(Q) (1 ≤ i ≤ N), the corresponding solution ϕ to (6) satisfiessλ2∫∫Qe−2sαξ|∇ϕ|2 + s3λ4∫∫Qe−2sαξ3|ϕ|2 ≤ C1(s3λ4∫∫ω×(0,T )e−2sαξ3|ϕ|2+∫∫Qe−2sα|F0|2 + sλ2N∑i=1∫∫Qe−2sαξ2|Fi|2),(8)for all s ≥ s1 = σ1(T + T 2) and λ ≥ λ1.The proof of this inequality follows the method introduced by Imanuvilov and Ya-mamoto in [9] (see also [6] and [5]) and will be given in [8].3M. Gonza´lez-Burgos, L. de Teresa3. Null controllability of the linear problem. ExamplesIn this Section, we will assume that y0 ∈ L2(Ω) and that (5) holds. One has,Theorem 3.1. There exists a control v̂ ∈ L2(Q) such that the corresponding solution ŷto system (1) satisfies (3) and||v̂||2L2(Q) ≤ exp(M(T ))||y0||2L2(Ω) . (9)with M(T ) =(C(1 + T + 1T + T ||a||∞ + ||a||2/3∞ + (1 + T )||B||2∞)).In addition, let us assume that either ∂ω ∩ ∂Ω is the empty set or ∂ω ∩ ∂Ω is of classC2 uniformly. Then, given y0 ∈ L2(Ω) there exists a new control v ∈ L2(Q) ∩ L∞(Q)satisfying||v||2L2(Q) + ||v||2∞ ≤ exp(M(T ))||y0||2L2(Ω) (10)which also gives the null controllability of system (1). In (9), C is a positive constant onlydepending on C0, C1 and d0; in (10), C also depends on Ω and ω.Sketch of the proof. The proof of this theorem is done in two steps:First step. Since ω satisfies the assumptions in Theorem 2.1, there exists another subsetω1 such that ω0 ⊂ ω1 ⊂ ω, dist (ω0,Ω \ ω1) = d0/2, and dist (ω1,Ω \ ω) > 0. By usingthe Carleman inequality (8) (for the set ω1) it is by now classical (e.g. [6]) to obtain anobservability inequality for the adjoint equation of (1) that allows the construction of acontrol v̂ ∈ L2(Q) satisfying (9), with supp v̂ ⊂ ω1× [0, T ] and such that the correspondingsolution ŷ to (1) satisfies ŷ(·, T ) = 0 in Ω.Second step. We construct a new null control v ∈ L∞(Q) ∩ L2(Q) for system (1). Thiscan 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,|η′(t)| ≤ C/T in [0, T ]. Due to the properties of ω1, it is possible to construct a functionθ ∈ 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 ;H10 (Ω)) be the weak solution of system (1) corresponding to v ≡ 0. Letus considery(x, t) = [1− θ(x)]ŷ(x, t) + θ(x)η(t)Y (x, t) ∀(x, t) ∈ Q. (11)It is not difficult to see that y is the solution of (1) corresponding to the controlv = 2∇θ · ∇[ŷ − η(t)Y ] + ∆θ [ŷ − η(t)Y ]− (B · ∇θ)[ŷ − η(t)Y ] + θη′(t)Y .This solution satisfies (3). In addition, by construction of θ, supp v ⊂ ω × [0, T ] and thelocal parabolic regularity allows to see that v ∈ L∞(Q) (see [8]).Now, it is interesting to exhibit some examples of unbounded open domains and controlregions where the assumptions above hold (for the details, see [8]:1. When Ω ⊂ RN is an unbounded connected open set with boundary ∂Ω of class C2uniformly and ω ⊂ Ω satisfies that Ω \ ω is bounded.4Controllability results for the heat equation in unbounded domains2. When Ω = (0,∞) ⊂ R (resp., Ω = R) and one considersω =⋃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. Wegive an example of unbounded sets Ω and ω such that Ω \ ω is an unbounded setwith infinite measure and where the observability inequality holds. In fact, let Ω bean unbounded open set uniformly of class C2. The set ω is going to be constructedfrom 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 δ0 we will needto recall some geometric results and properties which are inherent to uniformly C2unbounded sets. To this aim, we will use results appearing in the paper of Fornaroet al. [7].We recall that a set Ω satisfies a uniform interior sphere condition if for each pointy0 ∈ ∂Ω there exists a ball By0 , depending on y0, contained in Ω and such thatBy0 ∩ ∂Ω = {y0}, moreover the radii of these balls are bounded from below by apositive constant. The following results are extracted from [7]:Proposition 3.2. If ∂Ω is uniformly of class C2, then it satisfies a uniform interiorsphere condition.Proposition 3.3. Assume that ∂Ω is uniformly of class C2 and let δ̂ be a positiveconstant such that at each point of ∂Ω there exists a ball which satisfies the interiorsphere condition at y0 with radius greater or equal to δ̂. Then:a) For every x ∈ Ωδ̂= {y ∈ Ω : dist (y, ∂Ω) < δ̂} there exists a unique ξ = ξ(x) ∈∂Ω such that |x− ξ| = dist (x, ∂Ω).b) Let us define the function d(x) = dist (x, ∂Ω), then d ∈ C2b (Ωδ̂).c) ∇d(x) = −n(ξ(x)) for every x ∈ Ωδ̂and where n(ξ) denotes the unit outwardnormal vector to ∂Ω at point ξ.With the previous results in mind, we are now able to define δ0 and prove that thesets Ω and ω satisfy (5) (obviously, (4) holds). In fact, let us take δ0 = δ̂ − ε, whereδ̂ > 0 is as in Proposition 3.3 and ε ∈ (0, δ̂), andω0 = {x ∈ Ω : dist (x,Ω) > δ̂ − ε/2}.5M. Gonza´lez-Burgos, L. de TeresaIt is easy to see that the first statement of (5) holds with d0 ≡ ε/2. On the otherhand, we defineη˜0(x) ={d(x) ≡ dist (x, ∂Ω) if x ∈ Ω \ ω0 ,δ̂ − ε/2 if x ∈ ω0 .Let us remark that Ω \ ω0 ⊂ Ωδ̂ and dist (Ω \ ω0,Ω \ Ωδ̂) = ε/2 > 0. We canthen consider ρ ∈ C2(Ω) with 0 ≤ ρ(x) ≤ 1 in Ω, ρ(x) ≡ 1 in Ω \ ω0, supp ρ ⊂Ωδ̂and |Dαρ(x)| ≤ C/ε|α| for every x ∈ Ω and α ∈ NN with |α| ≤ 2. Finally,if we define η0(x) = η˜0(x)ρ(x) and taking into account Propostion 3.3 and theprevious properties we readily obtain that η0 also satisfies the two last statement ofcondition (5).4. Let us suppose that Ω ⊂ RN is an open set which satisfies (4), ω is an open subset ofΩ and (Ω, ω) satisfies (5) with constants C0, C1 and d0. Then, if O ⊂ RM (M ≥ 1) isanother open set with boundary of class C0,1 uniformly, and such that the open setΩ˜ = Ω×O ⊂ RN+M satisfies (4), then Ω˜ with ω˜ = ω×O also 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 thatΩ1 ⊂ RN , Ω2 ⊂ RM (N,M ≥ 1) are two open sets satisfying condition (4) and suchthat for Ω = Ω1 × Ω2 condition (4) holds. Suppose that ω1 ⊂ Ω1, ω2 ⊂ Ω2 are twoopen subsets such that (Ω1, ω1) and (Ω2, ω2) satisfy (5). Then, condition (5) comestrue for (Ω, ω) with ω = ω1 × ω2.Remark 1. The previous examples generalize some known results on null controllability ofsystem (1) in unbounded domains that have been studied before. Firstly, we again obtainthe null controllability result of the linear system (1) established in [2] when (Ω, ω) satisfiesthe conditions in example 1. Secondly, by means of the examples 4 and 5, we generalizethe positive results on null controllability of the heat equation obtained in [12]. Observethat our approach allows us to consider general coefficients a ∈ L∞(Q) and B ∈ L∞(Q)Nin system (1).4. Null controllability of the nonlinear problemFinally, when Ω \ ω is bounded, we have the following result concerning the nonlinearsystem (3). From now on, we consider a locally Lipschitz-continuous function f : R×RN →R such that f(0, 0) = 0. Under these assumptions we can decompose f as f(s, p) =g(s, p)s+G(s, p) · p with g,G ∈ L∞loc . One has:Theorem 4.1. Let Ω be an unbounded connected open set of class C2 uniformly andω ⊂ Ω an open subset such that Ω \ ω is bounded. Suppose that f as above satisfiesl´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 ∈ W 1,∞(Ω) ∩ H10 (Ω) there exists a control v ∈ L2(Ω) ∩ L∞(Ω) suchthat (2) admits a solution y that satisfies (3).6Controllability results for the heat equation in unbounded domainsIdea of the proof. The proof is obtained by a fixed-point argument (see [4] for the bound-ed case) and is a consequence of Theorem 3.1. In order to avoid the lack of compactnessof the Sobolev embeddings when Ω is unbounded, we use the construction of y as 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.AcknowledgementsThe 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. Gonza´lez-Burgos, R. Pe´rez-Garc´ıa, Existence of insensitizing controls for a semilinearheat 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 thesemilinear 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 unboundeddomains by a finite measure control region, ESAIM Control Optim. Calc. Var. 10 (2004), no. 3,381–408.[4] A. Doubova, E. Ferna´ndez-Cara, M. Gonza´lez-Burgos, E. Zuazua, On the controllability of parabolicsystems with a nonlinear term involving the state and the gradient, SIAM J. Control Optim. 41(2002), no. 3, 798–819.[5] E. Ferna´ndez-Cara, S. Guerrero, Global Carleman inequalities for parabolic systems and applicationsto controllability, SIAM J. Control Optim. 45 (2006), no. 4, 1399–1446.[6] E. Ferna´ndez-Cara, E. Zuazua, The cost of approximate controllability for heat equations: The linearcase, Adv. Differential Equations 5 (2000), no. 4–6, 465–514.[7] S. Fornaro, G. Metafune, E. Priola, Gradient estimates for Dirichlet parabolic problems in unboundeddomains. J. Differential Equations 205 (2004), no. 2, 329–353.[8] M. Gonza´lez-Burgos, L. de Teresa, Some results on controllability for linear and nonlinear heatequations in unbounded domains, submited.[9] O. Yu. Imanuvilov and M. Yamamoto, Carleman inequalities for parabolic equations in Sobolevspaces 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

Null controllability results for parabolic equations in unbounded domains

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Null controllability results for parabolic equations in unbounded domains

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