Degenerate parabolic operators of Kolmogorov type with a geometric control condition.

Transcription

1 Degenerate parabolic operators of Kolmogorov type with a geometric control condition. Karine Beauchard, Bernard Helffer, aphael Henry, Luc obbiano To cite this version: Karine Beauchard, Bernard Helffer, aphael Henry, Luc obbiano. Degenerate parabolic operators of Kolmogorov type with a geometric control condition.. submitted, 203, pp.submitted. <hal > HAL Id: hal Submitted on 24 Oct 203 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Degenerate parabolic operators of Kolmogorov type with a geometric control condition Karine Beauchard, Bernard Helffer, aphael Henry, Luc obbiano Abstract We consider Kolmogorov-type equations on a rectangle domain x, v Ω = T,, that combine diusion in variable v and transport in variable x at speed v γ, γ N, with Dirichlet boundary conditions in v. We study the null controllability of this equation with a distributed control as source term, localized on a subset ω of Ω. In dimension one, when the control acts on a horizontal strip ω = T a, b with 0 < a < b <, then the system is null controllable in any time T > 0 when γ =, and only in large time T > T min > 0 when γ = 2 see [0]. In this article, we prove that, when γ > 3, the system is not null controllable whatever T is in this conguration. This is due to the diusion weakening produced by the rst order term. When the control acts on a vertical strip ω = ω, with ω T, we investigate the null controllability on a toy model, where x, x T is replaced by /2, x Ω, and Ω is an open subset of N. As the original system, this toy model satises the controllability properties listed above. We prove that, for γ =, 2 and for appropriate domains Ω, ω, then null controllability does not hold whatever T > 0 is, when the control acts on a vertical strip ω = ω, with ω Ω. Thus, a geometric control condition is required for the null controllability of this toy model. This indicates that a geometric control condition may be necessary for the original model too. Introduction. Origin of the problem The goal of this article is to study the null controllability of Kolmogorov-type equations t ft, x, v v γ x ft, x, v vft, 2 x, v = ut, x, v ω x, v, t, x, v 0, T Ω, ft, x, ± = 0, t, x 0, T T, f0, x, v = f 0 x, v, x, v Ω,. where Ω = T,, γ N, T > 0, and the control is a source term ut, x, v localized on a nonempty open subset ω of Ω. This equation, with γ =, is close to linearizations of Prandt or Crocco-type equations for uids [54, 7, 6]; this motivates the study of the controllability of.. For other values of γ N, there are less physical motivations, but Centre de Mathématiques Laurent Schwartz, Ecole Polytechnique, 928 Palaiseau Cedex, France. Karine.Beauchard@math.polytechnique.fr Département de Mathématiques, Batiment 425, Université Paris Sud, 9405 Orsay Cedex, France. Bernard.Heler@math.u-psud.fr Département de Mathématiques, Batiment 425, Université Paris Sud, 9405 Orsay Cedex, France. aphael.henry@math.u-psud.fr Laboratoire de Mathématiques de Versailles LM-Versailles, Université de Versailles Saint-Quentinen-Yvelines, CNS UM 800, 45 Avenue des Etats-Unis, Versailles, France. luc.robbiano@uvsq.fr The authors were partially supported by the Agence Nationale de la echerche AN, Projet Blanc EMAQS number AN-20-BS and Projet NOSEVOL number AN-20-BS

3 the behavior of the system with respect to null controllability is extremely interesting, from a theoretical point of view nite speed of propagation, geometric control condition, the rst order term weakens diusion in variable v. Denition. Null controllability. Let T > 0 and γ N. System. is null controllable in time T if, for any f 0 L 2 Ω, there exists u L 2 0, T Ω such that the solution of. satises ft,, = 0. By duality, null controllability is equivalent to observability for the adjoint system t gt, x, v + v γ x gt, x, v vgt, 2 x, v = 0, t, x, v 0, + Ω, gt, x, ± = 0, t, x 0, T T, g0, x, v = g 0 x, v, x, v Ω..2 Denition.2 Observability. Let T > 0, γ N and ω be a non empty open subset of Ω. System.2 is observable in ω in time T if there exists C > 0 such that, for any g 0 L 2 Ω, the solution of the Cauchy problem.2 satises T gt, x, v 2 dxdv C gt, x, v 2 dxdvdt. Ω Equation.2 combines diusion in variable v and transport in variable x at speed v γ. Thanks to the interplay between these two phenomena, the equation diuses both in variables v and x see Proposition 6.2, contrarily to equation t 2 vgt, x, v = 0. But, the global diusion is weaker than for the 2D heat equation t 2 x 2 vgt, x, v = 0. Thus, natural questions are the following ones. Question : Is the diusion in variable v strong enough for observability to hold when the control acts on a horizontal strip ω = T a, b with 0 < a < b <, whatever γ N is? i.e. as for equation t 2 vgt, x, v = 0, t, x, v 0, T T, Question 2: Is the diusion in variable x sucient for null controllability to hold when the control acts on a vertical strip ω = ω, where ω T? i.e. as for the 2D heat equation t 2 x 2 vgt, x, v = 0, t, x, v 0, T T, The goal of this article is to answer the rst question and to study the second one for a toy-model. Null controllability of Equation.2 is studied in [0] when the control is localized in a horizontal strip ω = T a, b with < a < b <. Precisely, the following theorem is proved by the rst author in [0]. Theorem If γ = and ω = T a, b with < a < b <, then System.2 is observable in ω in any time T > If γ = 2 and ω = T a, b with 0 < a < b < then there exists T a 2 /2 such that System.2 is observable in ω in any time T > T ; System.2 is not observable in ω in time T < T. 3. If γ = 2 and ω = T a, b with < a < 0 < b < then System.2 is observable in any time T > 0. When γ =, Statement above illustrates that there is an innite speed of propagation in the direction v. When γ = 2, Statements 2 and 3 above illustrate a dierent situation: a nite speed of propagation in variable v occurs and the information needs time to reach the degeneracy {v = 0} from the observation location ω when ω {v = 0} =. ω 2

4 .2 Main results The rst goal of this article is to prove that observability does not hold, when γ 3 and the control acts on a horizontal strip: the presence of the rst order term v γ x f in the equation reduces diusion in the variable v so strongly that observability becomes false. Thus, Theorems.3 and.4 below answer Question. Theorem.4. If γ 3 and ω = T a, b with < a < b <, then System.2 is not observable in ω whatever T > 0 is. The second goal of this article is to investigate null controllability of Equation.2 for γ {, 2} when the control acts on a vertical strip ω = ω, where ω T. Unfortunately, we are not able to work directly on Equation.2. Thus, we consider the following toy model. where t gt, x, v + iv γ D x β gt, x, v vgt, 2 x, v = 0, t, x, v 0, T Ω, gt, x, ± = 0, t, x 0, T Ω, g0, x, v = g 0 x, v, x, v Ω,, Ω := Ω,, Ω is a bounded open subset of N and N N, D x is the Dirichlet-Laplace operator on Ω γ N, β 0,. D D x = H 2 H 0 Ω, D x g = g, Of course, the case β = /2 is of particular interest for System.2. We use the same denition for the observability of Systems.2 and.3..3 We are able to deny observability with explicit counterexamples, under an appropriate assumption Ps on the open sets Ω, ω. In order to express this assumption, we introduce the non decreasing sequence λ n n N of the eigenvalues of D x on Ω and a corresponding orthonormal sequence of associated eigenfunctions, ϕ n x = λ n ϕ n x, x Ω, ϕ n x = 0, x Ω, ϕ n L 2 Ω =..4 Denition.5 Property Ps. Let s 0, /2 and ω be an open subset of Ω. The pair Ω, ω satises the property Ps if [ ] lim ln ϕ n x 2 dx = +. n + ω λ s n This assumption is related to the classical problem of high-frequency localization of the eigenfunctions of the Laplacian. Note that /2 is the optimal upperbound for possible values of s see [47, Theorem 5.4 and Proposition 5.5]. Particular examples of pairs Ω, ω satisfying Property Ps for any s 0, /2 are discussed in Section 4. For instance, if Ω is a conical open subset of d d 2 generated by an open subset U of S d, Ω = {x = rx ; 0 < r <, x U}, and ω is an open subset of Ω that does not intersect its boundary Ω, then the pair Ω, ω satises Property Ps for every s 0, /2. One can indeed construct a subsequence of eigenfunctions ϕ k localized near the boundary Ω, called whispering gallery eigenmodes. Our rst nonobservability result concerns System.3 for γ =. 3

5 Theorem.6. We assume γ =.. If β > 0 and ω = Ω a, b where 0 < a < b < then System.3 is observable in ω in any time T > If β 0, 3/4 and Ω, ω satises Property P 2β 3, then System.3 is not observable in ω = ω, whatever T > 0 is. In particular, when β = /2, the diusion in the variable v is strong enough for System.3 to be observable in a horizontal strip ω = Ω a, b in any positive time T. On the contrary, the diusion in the variable x is too weak for System.3 to be observable in a vertical strip ω = ω, in nite time T, at least for appropriate pairs Ω, ω that satisfy Property P/3 which happens, for instance, when Ω is a bounded conical open subset of d and ω Ω. Thus a Geometric Control Condition GCC on Ω, ω is required for.3 to be observable in ω. Theorem.6 indicates that System.2, with γ =, may require a GCC for being observable. This is a conjecture for the answer of Question 2. Our second noncontrollability result concerns System.3 for γ = 2. Theorem.7. We assume γ = 2.. If β > 0 and ω = Ω a, b where 0 < a < b < then there exists T a 2 /2 such that System.3 is observable in ω in any time T > T, System.3 is not observable in ω in time T < T. 2. If β 0, and Ω, ω satises Property P in ω = ω, whatever T > 0 is. β 2, then System.3 is not observable In particular, when β = /2, the diusion in the variable v is strong enough for System.3 to be observable in a horizontal strip ω = Ω a, b, but there is a nite speed of propagation of the information from the observation location ω to the degeneracy set {v = 0}. On the contrary, the diusion in the variable x is too weak for.3 to be observable in a vertical strip ω = ω, in nite time T, at least for appropriate pairs Ω, ω. Thus a GCC on Ω, ω is required for.3 to be observable in ω. Theorem.7 encourages to conjecture that a GCC condition should be required for System.2, with γ = 2, to be observable..3 Bibliographical comments.3. Null controllability of the heat equation The null and approximate controllabilities of the heat equation are essentially well understood subjects for both linear and semilinear equations, for bounded or unbounded domains [3, 27, 30, 32, 33, 34, 37, 44, 46, 48, 5, 52, 62, 63] and also with discontinuous [28, 2, 3, 57] or singular [6, 29] coecients. In particular, the heat equation on a smooth bounded domain Ω of d d N, with a source term located on an open subset ω of Ω, is null controllable in arbitrarily small time T and with an arbitrarily small control support ω. This result is related to the innite speed of propagation of information in heat equation. It is proved, for the case d = by H. Fattorini and D. ussell [3, Theorem 3.3], and, for d 2 by O. Imanuvilov [42, 43] see also the book [36] by A. Fursikov and O.Imanuvilov and G. Lebeau and L. obbiano [46]. It is then natural to wonder whether the same result holds for degenerate parabolic equations. 4

6 .3.2 Boundary-degenerate parabolic equations The null controllability of parabolic equations degenerating on the boundary of the domain in one space dimension is well understood, but much less is known in higher dimension. Given 0 < a < b < and γ > 0, let us consider the D equation t wt, x + x x 2γ x wt, x = ut, x a,b x, t, x 0, + 0,, with suitable boundary conditions. Then, null controllability holds if and only if γ 0, [22, 23], while, for γ, the best result one can obtain is the so called regional null controllability[2], which consists in controlling the solution within the domain of inuence of the control. Several extensions of the above results are available in one space dimension, see [2, 49] for equations in divergence form, [20, 9] for operators in nondivergence form, and [8, 35] for cascade systems. Fewer results are available for multidimensional problems, and they are mainly obtained in the case of two dimensional parabolic operators which simply degenerate in the normal direction to the boundary of the space domain, see [24]..3.3 Parabolic equations degenerating inside the domain In [50], P. Martinez, J. Vancostenoble and J.-P. aymond study linearized Crocco type equations { t ft, x, v + x ft, x, v vv ft, x, v = ut, x, v ω x, v, t, x, v 0, T T 0,, ft, x, 0 = ft, x, = 0, t, x 0, T T. For a given strict open subset ω of T 0,, they prove that null controllability does not hold: the optimal result is regional null controllability. Note that, for Kolmogorovtype equations.2, the coupling between diusion in v and transport in x at speed v γ generates diusion both in variables x and v see Proposition 6.2. Thus, the controllability results are dierent. In [], K. Beauchard, P. Cannarsa and. Guglielmi study Grushin-type equations { t ft, x, y xft, 2 x, y x 2γ yft, 2 x, y = ut, x, y ω x, y, t, x, y 0, T Ω, ft, x, y = 0, t, x, y 0, T Ω,.5 where Ω :=, 0,, ω 0, 0,, and γ > 0. Here, the parabolic operator degenerates along the line {0} 0,. They prove that null controllability holds in any time T > 0 when γ 0, ; null controllability does not hold whatever T > 0 when γ > ; when γ = and ω = a, b 0, with 0 < a < b <, there exists T min a 2 /2 such that null controllability holds when T > T min and does not hold when T < T min. Note that, contrary to Grushin-type equations.5, in Kolmogorov-type equations.2, the parabolic operator degenerates everywhere on the domain..3.4 Unique continuation for Kolmogorov-type equations In this section, we focus on unique continuation for Kolmogorov-type equations.2, i.e. whether the property gt, x, v 0 on 0, T ω does imply g 0 on 0, T Ω, for a given open subset ω of Ω. When ω = T a, b is an horizontal strip, then the unique continuation of equation.2 holds for every γ N, as a consequence of Holmgren theorem the coecients of the operator are analytic and the hypersurface T {a, b} is noncharacteristic. In particular, 5

7 Theorem.4 emphasizes that, when γ 3, then observability does not hold even if unique continuation holds. To our best knowledge, when ω is a general open subset of Ω, then unique continuation for Kolmogorov-type equations.2 is an open problem. J.-M. Bony proved in [4] that Hörmander's operators of the form P = j X2 j i.e. such that the Lie algebra generated by the X j has maximal rank at any point with analytic coecients, satisfy the unique continuation, in the following sense: if, for some f with non zero gradient, f a is a strongly noncharacteristic surface and u is a distribution such that P u = 0 and u = 0 on f [, a], then u 0 on a neighborhood of f a. The validity of the same result for Hörmander's operators of the form P = X 0 + j X2 j generalizing our Kolmogorov operator K = t + v γ x v 2 is an open problem. When coecients are not analytic, but only C, unique continuation may not hold. For instance, S. Alinhac and C. Zuily built in [4] a zero order C -perturbation of the Kolmogorov operator K = t + v γ x v 2 for which unique continuation does not hold. There exist C -functions ut, x, v and at, x, v on a neighborhood V of 0 in 3 such that Ku + au = 0, ut, x, v = at, x, v = 0 when v < 0, and 0 Suppu. And the same result holds with any surface {v = constant}. The result of S. Alinhac and C. Zuily leaves open the question of the unique continuation for System.2. Indeed, their counterexample does not satisfy the boundary conditions of.2 and it cannot be built with a = 0. However, it suggests that unique continuation for System.2 is a subtle issue..4 Structure of the paper The article is organized as follows. Section 2 is devoted to the proof of Theorem.4. In Section 3, we prove the negative statements of Theorems.6 and.7. These results rely on a ne semi classical analysis of the complex Airy and Davies operators. In Section 4, we propose examples of pairs Ω, ω satisfying Property Ps for any s 0, /2. The proof of the positive results of Theorems.6 and.7 relies on the decomposition of the solution of.3 on a Hilbert basis of L 2 Ω, called 'Fourier decomposition' with a slight abuse of vocabulary. Thus, the validity of this decomposition and associated well-posedness results are treated in Section 5. In Section 6, we prove the positive results of Theorems.6 and.7. The strategy is the same as in [0], but intermediate results have been improved. Hence we rewrite the proof completely. First, we state a Carleman estimate for the D-heat equation satised by the Fourier components. Then, we quantify the dissipation of Fourier modes; this result is stronger than in [0]. Then, we combine these two tools to prove the rst statements of Theorems.6 and.7. 2 Nonobservability when γ 3 The goal of this section is the proof of Theorem.4. The strategy is the same as in [0, Section 5.3] but intermediate results are dierent. Let γ N, a, b, T be xed, in the whole section, such that γ 3, T > 0 and 0 < a < b <. Step : Approximate solution. Let ɛ > 0 be such that b < ɛ and θ ± C be such that Suppθ ɛ, + ɛ, 6

8 Suppθ + ɛ, + ɛ and θ ± ± =. Let µ C be some eigenvalue, with smallest real part, of the operator 2 y + iy γ, with domain D γ := {u H 2 s. t. y γ u L 2 }. Note that this operator has compact resolvent see [39]; moreover, µ is a simple eigenvalue and a real number if γ = 3. Let ξ be an associated eigenfunction { ξ y + iy γ ξy = µ ξy, y, ξ L2 =. We recall that see [58, Chapter 0, Sections 59 and 60] for some constants C, c > 0. For n N, we dene g n t, v := n 22+γ ξ 2+γ c y ξy Ce 2, y 2. n 2+γ v σ {,+} ξ σn 2+γ θ σ v e µn 2 2+γ t. We have { t g n t, v + in v γ g n t, v v 2 g n t, v = E n t, v, t, v 0, T,, g n t, ± = 0, t 0, T, where E n t, v = n 22+γ σ {,+} µ n 2 2+γ in v γ θ σ v + θ σv ξ σ n 2+γ e µ n 2+γ 2 t. 2.2 Let g n be the solution of t g n t, v + in v γ g n t, v vg 2 n t, v = 0, t, v 0, T,, g n t, ± = 0, t 0, T, g n 0, v = g n 0, v, v,. We have d 2 dt g n g n t 2 L 2, = v g n g n t 2 L 2, +e E n t, v g n g n t, vdv. By Poincaré and Cauchy-Schwarz Inequalities, we deduce that, for every t [0, T ], d dt g n g n t 2 L 2, π2 4 g n g n t 2 L 2, + 4 π 2 E nt 2 L 2,. From this inequality and 2.2, we deduce that, for every t [0, T ] g n g n t 2 L 2, 4 t π 2 0 E nτ 2 L 2, e π 2 C n 2+ 2+γ C n 2 2+γ σ {,} σ {,} ξ ξ 4 t τ dτ σ n 2+γ 2 t σ n 2+γ 2 0 e 2e µ n 2+γ 2 + π2 4 τ dτ 7

9 where the constant C may change from line to line. By 2., we deduce that g n g n t L 2, Cn 3+2γ 22+γ e c n, t [0, T ]. 2.3 Step 2: Conclusion. Working by contradiction, we assume that System.2 is observable in ω in time T. The observability inequality applied to the solution gt, x, v := g n t, ve inx of.2 gives T b g n T, v 2 dv C g n t, v 2 dvdt, n N. 0 a We deduce from the triangular inequality, the previous relation and 2.3 that g n T L2, C T 0 C T + C T 0 However, there exists C > 0 such that b /2 a g nt, v 2 dvdt + gn g n T L2, 0 b a g n g n t, v 2 dvdt /2 b a g nt, v 2 dvdt /2 + + T CC n 3+2γ 22+γ e c n. 2 e µ n 2+γ T g n T L 2 Ce and = = T b /2 a g nt, v dvdt 2 0 T 0 b n 2+γ b a n 2+γ ξ n 2+γ 2 /2 v e 2e µn 2+γ 2 t dvdt because b < ɛ a n C n 2+γ /2 T ξy 2 dy 2+γ 0 b n 2+γ 2+γ a n 2+γ y 2 e 2c dy C n 22+γ e c a 2+γ 2 n. 2 /2 µ n 2+γ e 2e t dt /2 by 2. This gives a contradiction, when n +, because 2 2+γ < 2 when γ > 2. 3 Nonobservability on a vertical strip The goal of this section is the proof of the nonobservability results of Theorems.6 and Accurate spectral analysis In this section, we are interested in the spectrum of the operators A, := d2 dy 2 + iy and H, := d2 dy 2 + iy2 dened on the segment,, > 0, with Dirichlet boundary conditions at y = ±, with domains DA, = DH, = H 2 H 0,, C. More precisely, we study the asymptotic behavior, as +, of the bottom of the spectrum of A, and H, and we prove the following two theorems, in Subsections 3.3 and 3.4 respectively. 8

10 Theorem 3.. Let µ < 0 be the rst zero of the Airy function. Then, lim inf e σa, = µ 2, 3. where σa, denotes the spectrum of A,. Moreover, for every ε > 0, there exists ε > 0 and M ε > 0 such that, for every ε, A, γ + iν LL2, M ε. 3.2 sup γ µ /2 ε, ν Now, let us consider the case of the Davies operator. Theorem 3.2. We have 2 lim inf e σh, = Moreover, for every ε > 0, there exists ε > 0 and M ε > 0 such that, for every ε, H, γ + iν LL2, M ε. 3.4 sup γ 2/2 ε, ν Analogous questions have been considered in [5, 8, 7, 9] and [6] in relation with problems occuring in superconductivity. We study these two operators using the techniques developed in these references. The study of more general cases dimension 2 complementary to those studied in [5] and [6] will be done in [4]. 3.2 Proof of the negative statements of Theorems.6 and.7 The goal of this subsection is the proof of the second statements of Theorems.6 and.7, by application of the results of the previous subsection. Thus, in the whole subsection, γ, β, Ω and ω are xed such that either γ =, β 0, 3/4 and Ω, ω satises Property P2β/3, or γ = 2, β 0, and Ω, ω satises Property Pβ/2. For n N, we introduce the operator A n,γ dened by DA n,γ := H 2 H 0,, C, A n,γ ψ := d2 ψ dv 2 + iλβ nv γ ψ. β 2+γ By rescaling y = λn v and using Theorems 3. and 3.2, there exist C, C 2 > 0 and n N such that, for every n n, A n,γ has an eigenvalue µ n satisfying C λ 2β 2+γ n 2β 2+γ e µ n C 2 λn. 3.5 We introduce a normalized eigenfunction ψ n of A n,γ associated with the eigenvalue µ n, ψ nv + iλ β nv γ ψ n v = µ n ψ n v, v,, ψ n ± = 0, ψ n L2, =. Then the function g n t, x, v := ϕ n xψ n ve µnt is a solution of.3. The second statement of Theorems.6 and.7 is a consequence of the following proposition. 9

11 Proposition 3.3. For every T > 0, we have T 0 ω lim g nt, x, v 2 dxdvdt n + Ω g = 0. nt, x, v 2 dxdv Proof of Proposition 3.3: We have because ψ n and ϕ n are normalized in L 2. By Fubini's Theorem, we get T 0 ω g nt, x, v 2 T dxdvdt = = Ω g n T, x, v 2 dv = e 2 e µnt, 0 e 2 e µn t dt e 2 e µn T 2 e µ n Thus, T 0 ω g nt, x, v 2 dxdvdt Ω g nt, x, v 2 dxdv Let C be a positive constant such that ψ nv dv 2 ω ϕ nx 2 dx ω ϕ n x 2 dx. = e2 e µn T 2 e µ n ω ϕ n x 2 dx. C > 2 C 2 T, 3.6 where C 2 is as in 3.5. Let s := 2β 2+γ. By Property Ps, there exists a subsequence n k k N such that ln ϕ nk x 2 dx C, k N, ω or, equivalently λ s n k ϕ nk x 2 dx e C λ n k, k N. ω Then, T 0 ω g n k t, x, v 2 dxdvdt Ω g n k T, x, v 2 dxdv by 3.6, which gives the conclusion. s s C λn e2c2t k 2C λ s n k 0, k Semi classical analysis of the complex Airy operator γ = The goal of this subsection is the proof of Theorem 3.. We introduce two model-operators, that have well known spectral and pseudospectral behavior. Let A,+ and A, be the Dirichlet realizations of the operator d2 dy +iy 2 on the intervals, + and, respectively. We are going to approximate the resolvent of A, by the one of A,+ or A, depending on where we are, respectively close to or close to +. Let us remark that, if T : ux ux + and U : ux u x 3.7 then T A,+ λt = A 0,+ λ + i, 3.8 U A, λu = A 0,+ λ i, 3.9 0

12 thus inf e σ A, because inf e σ A 0,+ = µ /2, see [5]. = inf e σ A, = µ 2, 3.0 Step : We prove lim inf e σ A, µ and 3.2. Let ε > 0. We search ε > 0 such that ε, σ A, ], µ /2 ε] + i =. 3.2 We recall that, by [38], there exists C ε > 0 such that A 0,+ γ + iν LL20,+ C ε, 3.3 sup γ µ /2 ε, ν sup γ µ /2 ε, ν A 0,+ γ + iν LL C ε ,+ Let λ = γ + iν ], µ /2 ε] + i and h +, h C ; [0, ] be such that For > 0, we dene and Supp h, /2, h on, /2], Supp h + /2, +, h + on [/2, +, h 2 + h 2 + on, +. λ = η x η ± x = h ±, x 3.5 η η A,+ λ + η+ + A, λ. 3.6 λ will be used as an approximation of the resolvent of A,. We have η A, λ λ = I + [A,, η ] A,+ λ η + [A,, η + A ] +, λ 3.7 as an equality between operators on L 2,. We estimate the second term on the right hand side. In what follows, the estimates are uniform with respect to ν = Im λ. We have η [A,, η A ],+ λ = η 2η d η A,+ λ dy, 3.8 Using η L, = O and η L, = O 2, we get, by 3.8 and 3.3, η η A,+ λ = O LL 2, Moreover, for every v L 2, +, v d dy A,+ λ L 2,+ /2 A,+ λ + γ A,+ λ v L2,

13 Indeed, let w := A,+ λ v, i.e. { w y + iywy λwy = vy, y, +, w = w+ = 0. We have w 2 L 2,+ + = e wyw ydy + = e w[iyw + λw + v] + = γ w e wv γ w 2 L 2,+ + w L 2,+ v L 2,+. By taking the square root of this inequality, we get w L 2,+ γ w L 2,+ + w /2 L 2,+ v /2 L 2,+, which proves By applying 3.20 to v = η u, u L2, we get η d A,+ λ dy which gives, with 3.8 and 3.9, [A,, η ] A,+ λ In the same way, we verify that [A,, η + ] A, λ Equality 3.7 can be written η η η + = O LL2, = O LL 2, = O LL 2, A, λ λ = I + E λ,, with E λ LL 2, = O, uniformly with respect to λ ], µ /2 ε] + i. We deduce the existence of ε > 0 such that, for every ε, A, λ is invertible, with inverse A, λ = λ I + E λ. We have proved 3.2. Moreover, according to the denition 3.6 of λ, 3.8, 3.9, 3.3 and 3.4 yield the estimate 3.2. Step 2: We prove that lim inf e σ A, µ First, we reduce the study to the complex Airy operator A 0, on the interval 0,. Indeed, applying the translation T : ux ux +, we get T A, λt = A 0,2 λ + i, 2

14 thus e σa, = e σa 0,2. Therefore, in order to prove 3.24, we are going to prove that µ lim inf e σ A0, Let θ, θ 2 C ; [0, ] be such that For j =, 2 and > 0, we dene We want to prove that Supp θ, 2/3, θ on, /2, Supp θ 2 /2, +, θ 2 on 2/3, +, θ 2 + θ2 2 on. χ j x = θ j x 0, x , 0, A 0, + A 0,+ + in LL Let us remark that σ 0, 0, A 0, + = σ A 0, + with non vanishing eigenvalues that have the same multiplicity for both operators. Step 2.a: We prove that 0, A 0, + 0, χ A 0,+ + χ + 0 in LL2 +. For this, we use the following approximations of the resolvent of A 0, +, Then, we have = χ χ χ A 0,+ + + χ 2 2 A 0, +. A 0, + χ = I + [A 0, +, χ ] A 0,+ + χ + [A 0, +, χ 2 2 ] A 0,2 +, thus, by composing on the left by 0, A 0, + 0,, we get 0, χ 0, A 0, + χ χ A 0,+ + = χ 2 2 A 0, + 0, χ 0, A 0, + [A 0, +, χ ] A 0,+ + 0, χ 0, A 0, + [A 0, +, χ 2 2 ] A 0, Now, we control the dierent terms on the right hand side. The terms involving commutators can be estimated as in Step, thanks to 3.2, and we get 0, χ 0, A 0, + [A 0, +, χ ] A 0,+ + LL2+ = O,

15 0, χ 0, A 0, + [A 0, +, χ 2 2 LL ] A 0, = O Moreover, for u L 2 0,, C, we have Im A 0, + u, u = yu, u 3.3 where.,. denotes the L 2 0,, C-hermitian product. χ This relation, applied to u = χ 2 2 A 0, + f, f L 2 0, +, which is supported in /2,, gives Im A 0, + u, u 2 u 2. Moreover, χ A 0, + u = χ 2 2 f + [A 0, +, χ 2 2 ] A 0, + f. Thus, estimating the commutator as in Step, we get Im A0, + u, u C + f u. Therefore, We have proved that 2 u 2 C + f u. χ χ 2 2 A 0, + LL20,+ = O By 3.28, 3.29, 3.30 and 3.32, we have 0, χ 0, A 0, + χ A 0,+ + LL20,+ = O 3.33 which ends Step 2.a. Step 2.b: We verify that χ χ A 0,+ + A 0,+ + in LL 2 0, +, which ends the proof of To simplify notation, let us introduce First, we write A + = A 0,+ +. χ A + χ A + = χ 2 χ A + [A +, χ ], then, composing on the right by A + and using that χ 2 = χ 2 2, A + χ A + χ = χ 2 2 A + + χ A + [A +, χ ]A The term involving a commutator can be estimated as in Step, χ A + [A +, χ ]A + = O LL2 + 4

16 For f L 2 0, +, we have 2 χ2 2 A + f 2 y /2 χ 2 2 A + f 2 because Supp χ 2 /2, = Im A + χ 2 2 A + f, χ 2 2 A + f A + χ 2 2 A + f χ 2 2 A + f χ 2 2 f + [A +, χ 2 2 ]A + f χ 2 2 A + f, where.,. denotes the L 2 0, +, C-hermitian product and. is the associated norm. Estimating the term with a commutator as in Step, we get χ 2 2 A + f L 2 0,+ C + f L 2 0,+. Thus χ 2 2 A + = O LL2 0,+ Finally, 3.35, 3.36 and 3.37 imply Step 2.c: Conclusion. Step 2.a and Step 2.b prove The eigenvalues of A + are isolated, thus we can apply [45, Section IV, Ÿ3.5]. For any subsequence j + and any eigenvalue λ σa + \ {0}, there exists a sequence λ j such that, for every j large enough 0,j λ j σ 0,j A 0,j + \ {0} = σ A 0,j + \ {0} and λ j λ when j +. In particular, with λ = / λ +, where λ = e iπ/3 µ σa 0,+ is the eigenvalue of A 0,+ with smallest real part see [5], we get a sequence λ j = /λ j σa 0,j such that e λ j e λ = µ /2, from which we deduce Semi classical analysis of the Davies operator γ = 2 The goal of this section is the proof of Theorem 3.2, which is similar to the one of Theorem 3.. Step : Let ε > 0. We search ε > 0 such that ε, σ H,, 2/2 ε + i = 3.38 and we prove 3.4. Let α 0, /3 and ζ, ζ2, ζ3 C ; [0, ] be such that Supp ζ, + α, ζ on, + α /2, Supp ζ 2 + α /2, α /2, ζ 2 on + α, α, Supp ζ 3 α, +, ζ 3 on α /2, +, ζ 2 + ζ ζ 3 2 on, ζ j L = Close to y =, we have O + α, ζ j L = O + 2α, 3.39 y 2 = 2y o y +. 5

17 Thus, we are going to approximate H,, close to y =, by the complex Airy type operator on, + A := d2 dy 2 2iy + + i2. In the same way, we will approximate H, close to y = + by the complex Airy type operator on, + A + := d2 dy 2 2i y + i2. Then, we remark that, if T and U are dened by 3.7, then we have A = T à 2T + i2 and A + = U à 2U + i2, where à is the Dirichlet realization of the complex Airy operator d2 dy 2 + iy on 0, +. Following [38], we deduce that inf e σ A + = inf e σ A = 2 2/3 µ 2, 3.40 and, for every ε > 0, there exists C ε > 0 such that sup A ± γ + iν γ [0, 2/3 µ /2 ε], ν C ε /3 We call H 0 the complex harmonic oscillator d2 dy 2 + iy 2 on, that will serve to approximate H, on the support of ζ 2. We recall that inf e σh 0 = cos π/4 = 2/2 see [25] and sup γ 2/2 ε, ν for some C ε > 0, see for instance [56]. Now, we take λ = γ + iν 0, 2/2 ε + i and we set Q λ = ζ Then, we have H 0 γ + iν C ε, 3.42 ζ A λ ζ + ζ 2 2 ζ H 0 λ + ζ 3 A + λ ζ H, λq λ = I + [H,, ζ] A λ ζ +[H,, ζ] 2 2 ζ H 0 λ + [H,, ζ] 3 A + λ 3 ζ +ζh, A A λ ζ + ζh 3, A + A + λ 3, as equality between operators on L 2,. The terms involving commutators can be estimated as in Step of the previous section, by using 3.39, 3.4, 3.42 and we get ζ [H,, ζ] A λ LL2 ζ + [H,, ζ] 2 2 H 0 λ LL2,, ζ + [H,, ζ] 3 A + λ 3 LL = O α. 2, Moreover, we have, by denition of A, H, A uy = iy + 2 uy, 6

18 and on the support of ζ, we have y + α. Therefore, by 3.4 ζ H, A A λ ζ LL2, In the same way, we verify 2α A λ LL2,+ C ε 2α /3. ζ 3 H, A + A + λ ζ 3 LL2, C ε 2α /3. Thus, we have proved that H, λq λ = I + Ẽλ, with Ẽλ 0 as +, uniformly with respect to λ in the interval 0, 2/2 ε+i. Thus, there exists ε > 0 such that, for every ε, H, λ is invertible, with H, λ Ẽλ. = Q λ I This proves the existence of ɛ > 0 such that 3.38 holds. The resolvent estimate 3.4 follows from 3.4, 3.42 and Step 2: We prove Let ϕ, ϕ2 C, [0, ] be such that lim inf e σ H, Supp ϕ, /2 /2, +, ϕ on, 2/3 2/3, +, Supp ϕ 2 2/3, 2/3, ϕ 2 on /2, /2, ϕ 2 + ϕ 2 2 on, ϕ j L = O, ϕ j L = O 2. We recall that H 0 denotes the operator d2 dx + ix 2 dened on, and we set 2 Q = ϕ 2 ϕ 2 ϕ H ϕ H, +. Thus, we have H, + Q = I + P, where ϕ P = [H,, ϕ 2 2 ϕ ] H [H,, ϕ ] H, +, and By composing on the left with H, +, we get P LL2, = O ϕ 2 ϕ 2 ϕ P H, + H 0 + = ϕ H, + H, By going back over the proof of 3.32 and replacing 3.3 by Im H, u, u = x 2 u, u,

19 we get ϕ ϕ H, + LL2, = O. By 3.47, the previous relation, together with 3.46 and 3.4 imply H, + ϕ 2 ϕ 2 H 0 + LL2, = O Then, we prove that the operator ϕ 2 H 0 + ϕ 2 converges to H 0 + in LL 2, when +, with the same arguments as in Step 2.b of the previous section. Thus, 3.45 is proved, with the same arguments as in Step 2.c of the previous section, and this ends the proof of Theorem Examples of Ω, ω satisfying Property Ps The goal of this section is to give examples of pairs Ω, ω that satisfy Property Ps for any s 0, /2. Precisely, we prove that it is the case if Ω is a conical bounded subset of d and ω is any open subset of Ω that does not intersect the boundary Ω. Note that the result covers the situation where Ω is a disk or a circular sector in 2D, a ball in any space dimension. Proposition 4.. Let d N, d 2 and U be an open subset of S d. Let Ω be the conical open subset of d dened by Ω := {x = rx ; 0 < r <, x U}. Let ω be an open subset compactly embedded in Ω. There exist constants C, K > 0, a sequence λ k k N of eigenvalues of the operator D Ω with domain H 2 H0 Ω and associated normalized eigenvectors ϕ k k N such that λk ϕ k x 2 dx Ke C, k N. ω In particular Ω, ω satises Property Ps for any s 0, /2. We refer to [53] for other similar results. Our proof of Proposition 4. relies on properties of Bessel functions, recalled in the next statement. Proposition 4.2. The Bessel functions of the rst kind J ν satisfy 0 < J ν νx e νgx, ν 0, +, x 0,, 4. where J ννx < + x2 /4 e νgx x 2πν J ν ν ν +, ν 0, +, x 0,, 4.2 a, 4.3 ν/3 gx := lnx + x 2 ln[ + x 2 ] and a := 2 /3 3 2/3 Γ2/3 > 0. Inequalities 4. and 4.2 are proved in [59]; inequality 4.3 is in [, Formula 9.3.3, Page 368]. Note that g is negative and increasing on 0, and that g = 0. 8

20 Proof of Proposition 4.: We recall that, in coordinates r, x, the Dirichlet-Laplacian writes D Ω ϕ = 2 ϕ r 2 d ϕ r r + r 2 D U ϕ. Let λ k k N be the increasing sequence of eigenvalues of D U and X k k N be associated eigenfunctions D U X kx = λ k X kx, x U, X k x = 0, x U, X k L 2 U =. For k N, we dene ν k := λ k + d 2 2 and j k the rst positive zero of the Bessel function of rst kind J νk. Note that ν k < j k < ν k + δν /3 k, k N, 4.4 for some constant δ > 0 see [, Formula 9.5.4, Page 37]. Let Then, for every k N, the function /2 C k := r d 2 + J νk j k r 2 r dr d, k N. 0 ϕ k rx := C k r d 2 + J νk j k rx k x, r 0,, x U, is a normalized eigenfunction of D Ω associated to the eigenvalue λ k := j 2 k. 4.5 Step : We prove the existence of C > 0 such that, for k large enough C k C. 4.6 ν 3/4 k Let ɛ 0, 5/6. Performing changes of variables, we get, for k large enough C k = /2 0 J ν k j k r 2 rdr = j k jk 0 J ν k ρ 2 ρdρ /2 νk j k J 0 νk ρ 2 ρdρ /2 0 J ν k ν k r 2 rdr ν k jk C ν 5 6 ɛ k /2 J νk ν k r 2 dr by 4.4 /2 by For r ν 5 6 ɛ, and ν large enough, we have J ν νr J ν ν ν r sup{ J ννσ ; σ r, } a ν 5 2ν /3 6 ɛ C ν by 4.2 and 4.3 a ν /3 2 C ν ɛ a 4ν /

21 We deduce from 4.7 and 4.8 that 4.6 holds for some constant C > 0. Step 2: Conclusion. Let ω be an open subset of d such that ω Ω. There exists a 0, such that ω {x = rx ; 0 < r < a, x U}. Thus, for every k N, ω ϕ k x 2 dx a 0 a2 2C 2 k 2 r d 2 + J νk j k r C k r d dr sup {J νk j k r; 0 < r < a}. Let b a,. By 4.4, we have j ka ν k < b < for k large enough. Then, by 4. for every r 0, a, j k r 0 < J νk j k r = J νk ν k e ν jk r kg ν k. ν k Explicit computations show that g x > 0, for every x 0,, thus jk r g < g b < 0, r 0, a. ν k Therefore, ϕ k x ω 2 dx a2 2Ck 2 e gb ν k. By 4.6, 4.4 and 4.5, we get the conclusion. Finally, let us quote, without proof, other examples of pairs Ω, ω satisfying Property Ps for appropriate values of s. If Ω is a lled ellipse and ω is an open subset of Ω that does not intersect Ω, then the pair Ω, ω satises property Ps for any s 0, /2. This can be proved by working in separate variables as in [53] and constructing "whispery galleries" solutions. The same result holds if ω intersects Ω but does not intersect the small axis of Ω see [53, Theorem 3., page 786]. This time this corresponds to "focusing solutions". All these results can be proved with semi-classical analysis see, for instance [60] and [26]. 5 Well posedness and Fourier decomposition In this section γ N and β 0, are xed. For f C c and f V := v fx, v 2 dxdv Ω V := Adh. V [C c Ω, C]. Ω, C, we dene Observe that H 0 Ω V L 2 Ω, thus V is dense in L 2 Ω. We dene the operator A γ,β by DA γ,β := {f V ; 2 vf + iv γ x β f L 2 Ω}, /2 A γ,β f := 2 vf + iv γ x β f. 20

22 Then DA γ,β is dense in L 2 Ω, A γ,β, DA γ,β is a closed operator and both A γ,β and A γ,β are dissipative, thus A γ,β, DA γ,β generates an strongly continuous semigroup of contractions of L 2 Ω see the Lumer-Phillips Theorem [55, Corollary 4.4, Chapter, page 5], or the Hille Yosida Theorem [5, Theorem VII.4, page 05]. We consider a solution g C 0 [0, T ], L 2 Ω of.3. Then, the function x gt, x, v belongs to L 2 Ω for almost every t, v [0, +,, thus, it can be developed on the Hilbert basis ϕ n n N see.4 as follows gt, x, v = g n t, vϕ n x where g n t, v := gt, x, vϕ n xdx, n N. n N T 5. In what follows, with a slight abuse of vocabulary, this decomposition is called 'Fourier decomposition' and the functions g n t, v are called 'Fourier components'. Proposition 5.. For every n N, g n is the unique solution of t g n t, v + iλ β nv γ g n t, v vg 2 n t, v = 0, t, v 0, +,, g n t, ± = 0, t 0, +, g n 0, v = g 0,n v, v,, where g 0,n L 2, is given by g 0,n v := g 0 x, vϕ n xdx, v,. Ω This result can be proved by following the same steps as in [, Section 2.2]. 6 Observability on a horizontal strip 5.2 The goal of this section is the proof of the statements of Theorems.6 and.7. Note that the negative part of the rst statement of Theorem.7 i.e. no null controllability, when γ = 2 and T < T can be done exactly as in [0]. 6. Global Carleman estimate The goal of this subsection is the statement of a global Carleman estimate, proved in [0, Appendix] and useful for the proof of the statements of Theorems.6 and.7. For λ and γ {, 2}, we introduce the operator P λ,γ g := t g + iλv γ g 2 vg. Proposition 6.. Let a, b be such that < a < b <. There exist a weight function β C [, ], +, positive constants C, C 2 such that, for every λ, γ {, 2}, T > 0 and g C 0 [0, T ], L 2, L 2 0, T ; H0, the following inequality holds T C 0 T 0 M g tt t tt t dvdt v t, v 2 + M 3 gt, tt t v 2 e Mβv 3 P λ,γgt, v 2 e Mβv tt t dvdt + T 0 where M := C 2 max{t + T 2 ; λ T 2 }. b a 6. M 3 tt t gt, v 2 e Mβv 3 tt t dvdt, In this proposition, the weight β is the usual one for Carleman estimates for D heat equations; since its explicit expression will not be used in this article, we do not specify its properties. Note that we have sharp dependency of M on λ and T. In particular, if we treat the term iλv γ g as a lower-order term, to apply the Carleman estimate for the operator t 2 v, then, we can obtain a less sharp dependency M = Oλ 2/3, which is not sucient in this article. The proof of this Carleman estimate is done in [0, Appendix], by revisiting the usual proof. 2

23 6.2 Dissipation of Fourier components The Dirichlet realization of the operator v 2 + iλ β nv γ on, is not a normal operator. Thus it is not obvious that the exponential decay of the solutions of 5.2 is given by the smallest real part of the eigenvalues of this operator. This question is answered in the following statement. Proposition 6.2. Let γ {, 2} and d := 2γβ 2 + γ. There exist K, δ > 0 such that, for every n N and g 0,n L 2,, the solution of 5.2 satises g n t L 2, Ke δλd n t g 0,n L 2,, t > Moreover, for every ɛ > 0, there exists n > 0 such that, for every n > n, 6.2 holds with K = K ɛ and { µ /2 ε if γ =, δ = 6.3 2/2 ε if γ = 2, where µ is the rst zero from the right of the Airy function. Finally, the exponent d of λ n in 6.2 is optimal, and the critical value of δ in 6.3 is also optimal. This result is stronger than [0, Propositions 0 and 7] because in 6.2, we have L 2 - norms on both sides, whereas in [0] there was an H -norm on the right hand side. We study this problem in semi-classical formulation take h n = λ β/2 n and s = h n t. Let h 0 > 0. For h 0, h 0 and ψ 0,h L 2,, we consider the equation h t ψ h t, v h 2 vψ 2 h t, v + iv γ ψ h t, v = 0, t, v 0, +,, ψ h t, ± = 0, t 0, +, ψ h 0, v = ψ 0,h v, v,. 6.4 Proposition 6.3. Let e = 2γ/γ + 2. There exist K, δ > 0 such that, for every h 0, h 0 and ψ 0,h L 2,, the unique solution of 6.4 satises ψ h t L 2, Ke δhe t ψ 0,h L 2,, t > Moreover, for every ε > 0, there exists h 0, h 0 such that, for every h 0, h, 6.5 holds with K = K ε and 6.3 where µ is the rst zero from the right of the Airy function. Finally, the exponent d of h in 6.5 is optimal, and the critical value of δ in 6.3 is also optimal. Proof of Proposition 6.3: Let A h be the operator dened by A h = h 2 d2 dv 2 + ivγ, DA h = H 2, H 0,. By rescaling = h = h e/γ and y = v and using Theorems 3. and 3.2, we have { µ /2 if γ =, lim h 0 h e inf e σa h = 6.6 2/2 if γ = 2. Thus, we can consider δ := min h 0,h h e inf e σa h >

24 Let δ 0, δ. By Theorems 3. and 3.2, there exists C δ such that sup A h δh e C δ iν h e. Thus, ν sup ν Ah h δhe iν C δh e. 6.7 Moreover, the operator h A h is maximally accretive, thus it generates a semigroup of contractions: ψ h t L2, ψ 0,h L2,, t > We can apply [40, Theorem.5], with ω = δh e < 0, rω C δ h e, mt and a = ã = t/2. Note that Thus, we obtain ψ h t, L 2, 2 L 2 0,t/2;e ωt dt = eωt/2 ω Let c 0 > 0 and t h = 2c 0 h e /δ. Then, by 6.9,. δc δ e t e δhe t/2 e δh ψ 0,h L 2,, t > ψ h t, L 2, K e δhe t ψ 0,h L 2,, t t h with Moreover, by 6.8, K = δc δ e c0. ψ h t L 2, K 2 e δhe t ψ 0,h L 2,, t t h with K 2 = e 2c0. Thus, ψ h t L2, Ke δhe t ψ 0,h L2,, t > with K = maxk, K 2. Finally, if ε > 0 is xed, by 6.6 there exists h 0, h 0 such that all the previous estimates hold for h 0, h and δ as in 6.3. Indeed, we have δ < δ := min h 0,h h e inf e σa h. To prove the optimality of exponent e of h in 6.5, we just consider ψ 0,h kera h λ 0,h h e, where λ 0,h satises h e λ 0,h σa h and h e e λ 0,h = inf e σa h. Then, we have ψ h t, v = e λ 0,hh e t ψ 0,h v. Thus, by 6.6, for every t > 0 and ε > 0, there exists h > 0 such that, for every h 0, h, ψ h t, L2, = e λ 0,hh e t g 0,n L2, with ν = µ /2 if γ = and ν = 2/2 if γ = 2. e ν+εhe t ψ 0,h L2,, 23

25 6.3 Proof of the positive statements of Theorems.6 and.7 The positive statements in Theorems.6 and.7 are consequences of the following proposition and of the Bessel-Parseval equality. Proposition 6.4. Let β 0, and 0 < a < b <. If γ =, then, for every T > 0, there exists C > 0 such that for every n N and g 0,n L 2,, the solution of 5.2 satises T b g n T, v 2 dv C g n t, v 2 dvdt a If γ = 2, then, there exists T > 0 such that, for every T > T, there exists C > 0 such that for every n N and g 0,n L 2,, the solution of 5.2 satises 6.. Proof of Proposition 6.4: We deduce from Proposition 6. that C 3 λ 3β/2 n e c λ β/2 n 2T/3 T/3 g n t, v 2 dvdt C 4 T 0 b a g n t, v 2 dvdt 6.2 for n large enough, where C 3 := C 2 max{4c ; 4C 3 }, c := 9 2 C 2 max{βv; v [, ]}, C 4 := max{x 3 e β x ; x 0} and β := min{βv; v a, b}. Moreover, thanks to Proposition 6.2, we have where C 5 := K 2 C 4 /C 3. g nt, v 2 dv 3K2 T e 2δλd n T/3 2T/3 T/3 C5 e c λ β/2 n λ 3β/2 n 2δλd n T/3 T 0 g nt, v 2 dvdt b a g nt, v 2 dvdt 6.3 Case : γ =. Then d = 2β 3 > β 2, thus the observability constant above converges to zero as n +. This proves the existence of a uniform observability constant for high frequencies: there exists C H > 0 and n 0 N such that g n T, v 2 dv C H T 0 b a g n t, v 2 dvdt, g 0 n L 2,, n > n 0. Moreover, for every n {,..., n 0 }, there exists a constant C n > 0 such that g n T, v 2 dv C n T 0 b a g n t, v 2 dvdt, g 0 n L 2, usual observability inequality for D heat equations. Thus, the uniform observability constant C := max{c H, C n ; n n 0 } gives the conclusion. Case 2: γ = 2. Then d = β 2, thus, when T > T := 3c 2δ, the observability constant in 6.3 converges to zero as n + and the proof can be ended as in the previous case. eferences [] M. Abramowitz and I.A. Stegun. Handbook of mathematical functions with formulas graphs and mathematical tables. Milton Ed., New York: Dover, 972. [2] F. Alabau-Boussouira, P. Cannarsa, and G. Fragnelli. Carleman estimates for degenerate parabolic operators with applications to null controllability. J. Evol. Equ., 62:6204,

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