Null-controllability properties of the generalized two-dimensional Baouendi-Grushin equation with non-rectangular control sets

We consider the null-controllability problem for the generalized Baouendi-Grushin equation $(\partial_t - \partial_x^2 - q(x)^2\partial_y^2)f = 1_\omega u$ on a rectangular domain. Sharp controllability results already exist when the control domain $\omega$ is a vertical strip, or when $q(x) = x$. In this article, we provide upper and lower bounds for the minimal time of null-controllability for general $q$ and non-rectangular control region $\omega$. In some geometries for $\omega$, the upper bound and the lower bound are equal, in which case, we know the exact value of the minimal time of null-controllability. Our proof relies on several tools: known results when $\omega$ is a vertical strip and cutoff arguments for the upper bound of the minimal time of null-controllability; spectral analysis of the Schr\"odinger operator $-\partial_x^2 + \nu^2 q(x)^2$ when $\Re(\nu)>0$, pseudo-differential-type operators on polynomials and Runge's theorem for the lower bound.


Introduction and statements of the main results
1.1.The Baouendi-Grushin equation.In this article, we study some controllability properties of the two-dimensional generalized Baouendi-Grushin equation.
Note that, because q(0) = 0, the equation degenerates on the vertical axis {0} × T. Nevertheless, the equation is well posed.Precisely, the Friedrichs extension (see [24,Section 4.3]) of the operator generates an analytic semigroup, which allows to define a solution of the generalized Baouendi-Grushin equation (1.1) in the sense of semigroups [12].In our case, this solution is smooth in the following sense: Proposition 1.1.For any source term F ∈ L 2 ((0, T ); L 2 (I × T)) and any initial condition f 0 ∈ L 2 (I × T), there exists a unique f ∈ C 0 ([0, T ]; L 2 (I × T)) ∩ L 2 ((0, T ); V ) solution of the generalized Baouendi-Grushin equation (1.1), with This result is proved in [7] in the case q(x) = |x| γ with γ > 0. The proof is easily generalized to our case of interest.
It is known that, contrary to usual non-degenerate parabolic equations like the heat equation, due to the degeneracy of q on {0} × T, the null-controllability properties of (1.2) strongly depend on the control set ω and the time horizon T .More precisely, for certain control sets ω, there is no time T > 0 such that eq.(1.2) is null-controllable, whereas for other control sets ω a minimal time of null-controllability appears.We refer to the bibliographical comments, section 1.4 below, for a detailed description of the known results on the subject.
In the present paper, we aim to give precise null-controllability results for equation (1.2), for a large class of control sets ω and a large class of functions q.
1.3.Main results.We are interested in the case where the equation is degenerate on {x = 0}.Thus, we assume that q(x) = 0 only when x = 0, and we assume without loss of generality that q (0) > 0.
Before presenting the main results of our study, we introduce the so-called Agmon distance of a point x ∈ I to the origin, defined by:1 This quantity appears naturally in the computation of the minimal time of null-controllability for the generalized Baouendi-Grushin equation.
1.3.1.Lack of null-controllability in small time for a class of control sets ω.Our main result is a negative result of null-controllability for small times.We show that if the control set ω stays at positive distance from a horizontal segment of the form (a, b) × {y 0 }, with −L − a < 0 < b L + and y 0 ∈ T, then equation (1.2) is not null-controllable on ω for time T smaller than a precisely given critical time.
This theorem is a generalization of [20, theorem 3.3], where the result is proved for I symmetric with respect to the origin and, more restrictively, for q(x) = x.A key step in our proof of theorem 1.3 is the study of spectral properties of the family of operators defined on L 2 (I) by (1.4) P ν : − ∂ 2 x + ν 2 q(x) 2 , Dom(P ν ) = H 2 (I) ∩ H 1 0 (I). .In green, an example of a domain ω with, in thick black, an example of a horizontal segment that stays at positive distance of ω.In this example, theorem 1.3 implies that the generalised Baouendi-Grushin equation is not null-controllable in time T < d Agm (a).
Because of technical reasons, we have to consider P ν for every Re(ν) > 0, which makes P ν non self-adjoint.
In the previous article [20], the corresponding results were obtained using explicit solutions of particular ordinary differential equations.These explicit formulae are not available in our general setting.
Note that [25] contains closely related spectral asymptotics.However, we cannot apply them directly since our domain has a boundary and we need uniform estimates with respect to the parameter ν.We first mention a natural adaptation of [20, theorem 3.1].It was actually claimed in [20, Remark 3.2], but the statement was imprecise if q is not odd.We take the opportunity to correct the statement: Theorem 1.4.Assume that q ∈ C 3 (I) is such that q(0) = 0 and min I q > 0. Let ω be an open subset of I × T. Assume that there exists a closed path γ = (γ x , γ y ) ∈ C 0 (T; ω) such that {−L − } × T and {L + } × T are included in different connected components of (I × T) \ γ(T) (see fig. 2).
2 For ν ∈ R * + , we localize the smallest eigenvalue of Pν , and for ν complex, we localize its analytic continuation. ω x y Figure 3.In green, an example of a domain ω that satisfies the hypotheses of theorem 1.6.
The generalized Baouendi-Grushin equation (1.2) is null-controllable on ω in time T such that Remark 1.5.In this theorem, we can replace the hypothesis "{−L − } × T and {L + } × T are included in different connected components of (I × T) \ γ(T)" by "γ is not homotopic to a constant path".These two conditions are essentially equivalent.We discuss this in propositions B.2 and B.3 and remark B.4.
Then the generalized Baouendi-Grushin equation (1.2) is null-controllable on ω in any time T > T * , but it is not null-controllable on ω in time T < T * .
This theorem is proved in section 5.
1.3.3.Comments.Before proceeding further, we make some additional comments on our results.
• The assumptions regarding the function q in theorem 1.3 are slightly more general than in theorem 1.4.They seem also more natural in the context of our study.Therefore, we conjecture that theorem 1.4 holds for functions q ∈ C 3 (I) satisfying q(0) = 0, q (0) > 0, q(x) = 0 for all x ∈ I.

But up to our knowledge, this is still an open question.
• There are still numerous geometrical configurations not included in theorem 1.6.Nevertheless, in many situations, theorem 1.4 and theorem 1.3 give information about null-controllability properties.As an example, in the geometrical configuration described in fig.4, combining theorem 1.4 and theorem 1.3, we obtain the existence of a critical time such that the Baouendi-Grushin equation is null-controllable on ω in time T > T * , and is not null-controllable on ω in time T < T * .In the two geometrical configurations presented in fig.5, theorem 1.3 implies that the Baouendi-Grushin equation is not null-controllable on • These results are stated for the generalized Baouendi-Grushin equation posed on I × T. They can be adapted to the equation posed on I × (0, π) with Dirichlet boundary conditions, with very similar proofs.We refer to appendix A for details on the statements and the corresponding proofs.
1.4.1.On the Baouendi-Grushin equation.The study of controllability properties of system (1.2) began with the pioneering work [7], where the authors study the null-controllability of the equation They prove that in the case γ ∈ (0, 1) (weak degeneracy), the Baouendi-Grushin equation (1.2) is null-controllable for any control set ω and any time T > 0, whereas in the case γ > 1 (strong degeneracy), it is not null-controllable for any control set ω and any time T > 0, except if ω contains {0} × T in which case it is null-controllable in any positive time T .More surprisingly, in the case γ = 1, which corresponds to the Baouendi-Grushin equation (1.2) with q(x) = x, and for ω = (a, b) × T, with 0 < a < b, there exists a critical time T * a 2 2 such that the Baouendi-Grushin equation (1.2) is null controllable on ω in time T , for every T > T * , and is not null controllable on ω in time T , for every T < T * .It is also proved that if γ = 1 and ω contains the vertical line {0} × T, equation (1.5) is null controllable in any time T > 0. Such a minimal time of null-controllability would not be surprising for equations with finite speed of propagation, such as the wave equation [3], but the Baouendi-Grushin equation has a infinite speed of propagation.
Many works followed, trying to characterize precisely the critical time T * , and to generalize the result to different geometrical settings and different functions q.The first exact characterization of T * is given in [10] in the case q(x) = x and with two symmetric vertical strips as control set, that is ω = (−1, −a) × (a, 1), a ∈ (0, 1).Using the transmutation method and sideways energy estimates, the authors prove that eq.(1.2) is null-controllable in ω in any time T > a 2 2 , and is not null-controllable in ω in any time T < a 2 2 .When ω is a vertical strip of the form (a, b) × T, with a > 0, as in [7], the precise value of the critical time T * was obtained independently in the works [1,8,30].More precisely, in [1], using new estimates for biorthogonal sequences to real exponentials and the moments method, the authors prove that in the case q(x) = x, the critical time is a 2 2 .In [8], with a function q satisfying the assumptions of theorem 1.6, the authors use a Carleman strategy to obtain that eq.(1.2) is null controllable on ω in any time T > T * , and not null-controllable on ω in any time T < T * , with Very recently, this result was obtained in [30] using the moments method, with a stronger smoothness assumption on q (see [30, Remark 1.12 and Proposition 1.13]).
All the strategies developed in [7, 10, 1, 8], although very different, rely on a Fourier expansion of system (1.2) with respect to the y-variable and the study of the obtained family of one dimensional parabolic equations in the variables t, x.As a consequence, the control set ω has to contain a vertical strip, which seems to be an important restriction of the proposed methods.Nevertheless, in [20], the authors generalize the positive null-controllability results obtained in [8] to a large class of control sets: in the setting of theorem 1.6 and with the additional assumptions that I is symmetric and q is odd, system (1.2) is null controllable in any time T > T * , with In the specific case q(x) = x, they also prove that if there exist a, b ∈ I, a < 0 < b, and y 0 ∈ T such that distance((a, b) × {y 0 } ∩ ω) > 0, then (1.2) is not null controllable in time T < min(a 2 , b 2 )/2, whereas if there is y 0 ∈ T such that distance(I × {y 0 } ∩ ω) > 0, then (1.2) is not null controllable on ω in any positive time T .Theorem 1.6 is the generalization of this result to a wider class of functions q.
To end this overview on controllability issues for the parabolic Baouendi-Grushin equation, we point out that partial controllability results are known in some multidimensional configurations [8] while precise results are known for cascade systems of two-dimensional Baouendi-Grushin equations with one control, in the case q(x) = x [1].1.4.2.Some related problems.Let us briefly mention the literature on related problems, in several directions: other degenerate parabolic equations, minimal time of null controllability for parabolic systems, and other type of degenerate equations.
Since the pioneering works [21,22] on the null-controllability of the one-dimensional heat equation, the null-controllability of non-degenerate parabolic equations has been extensively studied.The nullcontrollability of degenerate parabolic equations is a more recent subject of study.The case of a degeneracy at the boundary of the domain is now well-understood [14] (see also the references therein).
When the degeneracy occurs in the domain, we lack for the moment a general theory, and equations are studied case by case.The two-dimensional Baouendi-Grushin equations is arguably the simplest and best understood equation of that type.Very similar results, including a minimal time of nullcontrollability for quadratic degeneracy, have been observed for the heat equation on the Heisenberg group [5,8], and the Kolmogorov equation [4,6,9,16,28].
The related problem of approximate controllability for degenerate parabolic equations has been studied in a somewhat general framework [31].
A minimal time of null-controllability might also appear for the heat equation with punctual control [19] and for systems of parabolic equations, degenerate or not [2,11].
Finally, let us mention than the subelliptic wave equation is not controllable [32], and that the Grushin-Schrödinger equation has a minimal time of controllability [13,33].

Null-controllability in large time
In this section, we prove theorem 1.4.The idea of the proof is to use known controllability results for equation (1.2) when the control set is a vertical strip combined with a cutoff argument.More precisely, we recall the following result [8, theorem 1.4]. 4   Proposition 2.1.Assume that q satisfies the assumptions of theorem 1.4.Let ω = (a, b) × T, with −L − a < b L + .Then . Definition of ω − (red) and ω + (blue).

Lack of null-controllability
In this section, we prove the following case of theorem 1.3.
To completly prove theorem 1.3, there are two more cases: The proofs of these cases are minor modifications of the one of theorem 3.1.We mention in footnotes the most important modifications and leave the details to the reader.
Under the hypotheses of this theorem, there exists a closed interval W 0 that is a neighborhood of y 0 and such that ω ∩ [a, L + ) × W 0 = ∅ (see fig. 7).To prove theorem 3.1, we assume without loss of generality that ω is the complement of the rectangle [a, L + ) × W 0 : In green, the domain ω.If a horizontal segment stays at positive distance from ω, it can be thickened into a rectangle that is disjoint from ω.
3.1.Observability inequality.Using standard duality arguments (see [15, theorem 2.44]), the null-controllability of the generalized Baouendi-Grushin equation (1.2) is equivalent to the following observability inequality: there exists C > 0 such that for every C g 2 L 2 ((0,T )×ω) .To prove theorem 3.1, we proceed in two steps: we prove that the observability inequality (3.3) implies an inequality on polynomials, and then we disprove this new inequality. 5.2.Model case.We start with a model equation, that we study to showcase the main ideas of the proof of theorem 3.1 without some of the more technical aspects.Consider the Baouendi-Grushin equation on R × T: We say that the Baouendi-Grushin equation (3.4) is observable on ω in time T > 0 if there exists C > 0 such that for all g solution of (3.4), the following observability inequality holds: C g 2 L 2 ((0,T )×ω) .We prove the following theorem.
Theorem 3.3.let a > 0, W 0 ⊂ T a closed interval with non-empty interior and The Baouendi-Grushin equation (3.4) is not observable on ω in time T .
Before going into the proof, let us examine some solutions of the Baouendi-Grushin equation that are concentrated around x = 0. Taking the n-th Fourier coefficient in y of g, which we will denote by ĝ(t, x, n), we get The spectral properties of the harmonic oscillator are well-known (see, e.g., [24, §1.3] or appendix C), and in particular the first eigenvalue is |n| with associated eigenfunction ϕ n (x) = (|n|/π) 1/4 e −|n|x 2 /2 .Thus, if (a n ) n>0 is a complex-valued sequence with only a finite number of nonzero terms, the function g defined by is a solution of the Baouendi-Grushin equation (3.4).We will look for a counterexample of the observability inequality (3.5) in this class of functions.This solution can be written as g(t, x, y) = g pol (e iy−t−x 2 /2 ) with (3.9) We will use the fact that g is a polynomial in z = e iy−t−x 2 /2 to rewrite the observability inequality we want to disprove as an inequality on polynomials.More precisely, we have the following estimate.
Lemma 3.4.Assume that the observability inequality for the Baouendi-Grushin equation (3.5) holds.Let U ⊂ C be defined by (see fig. 8) Then, there exists C > 0 such that for every polynomial p ∈ C[X], Proof.
Step 1: Observability inequality.Let p(z) = n 0 a n z n a polynomial and set g pol (z) = zp(z) = n>0 a n−1 z n .The discussion above shows that g defined by Step 2: Left-hand side of the observability inequality (3.10).Since the functions ψ n : (x, y) → e iny−nx 2 /2 are orthogonal in L 2 (R × T), the left-hand side can we rewritten as Elementary computations in polar coordinates prove that the functions z → z n are orthogonal in L 2 (D(0, R), m), where m is the Lebesgue measure on C R 2 , and that for R > 0 Thus, Step 3: Right-hand side of the observability inequality (3.10).We write the right-hand side of the observability inequality by making the change of variables (x, z) = (x, e −t+iy−x 2 /2 ).We have dx dm(z) = |z| 2 dt dx dy.Thus, if we denote by Ω ⊂ R × C the image of (0, T ) × ω by this change of variables, we have When the disk D(0, e −T ) (in red) is not included in U , we can find holomorphic functions that are small in U but arbitrarily large in D(0, e −T ).For instance, we can construct with Runge's theorem a sequence of polynomials that converges to z → (z − z 0 ) −1 away from the blue line.
We have proved that (p k ) is a counterexample to the inequality of lemma 3.4, which concludes the proof of theorem 3.3.

3.3.
From the model case to the generalized Baouendi-Grushin equation.Now, our goal is to adapt the strategy used in the model case to the generalized Baouendi-Grushin equation (1.2).In the generalized Baouendi-Grushin equation, if we take the n-th Fourier coefficient in y of g, we get Recall that for n 0, P n is the unbounded operator −∂ 2 x + n 2 q 2 on L 2 (I) with Dirichlet boundary conditions.We will denote by λ n the first eigenvalue of P n and by ϕ n a corresponding eigenfunction.Notice that ϕ n is not required to be normalized in L 2 (I).Then, we will look for a counterexample of the observability inequality (3.3) with solutions of the generalized Baouendi-Grushin equation (3.2) of the form Heuristically, this should work because we expect the eigenfunction ϕ n to be localized around x = 0 as n → +∞, in which case the operator −∂ 2 x + n 2 q 2 looks like −∂ 2 x + n 2 q (0) 2 x 2 , and the eigenvalue and eigenfunction look like λ n ≈ nq (0) and ϕ n (x) ≈ n 1/4 e −nq (0)x 2 /2 .So the solutions g defined above look like the solutions used to treat the model case (eq.(3.8)), up to a factor q (0).
In fact, a better approximation of ϕ n would be the so-called WKB approximation6 (3.16) Thus, we have a n e n(iy−q (0)t−dAgm(x)) , i.e., g can almost be written as g(t, x, y) ≈ c 0 (x)g pol (e iny−tq (0)−dAgm(x) ), where g pol is the polynomial Let us write this in an exact way.Consider φn (x) := n 1/4 e −nq (0)x 2 /2 and let Π n be the spectral projection associated to the first eigenvalue λ n of P n .We define 7(3.18) ϕ n := Π n φn .
We will see later that ϕ n = 0, at least if n is large enough.Let ε ∈ (0, 1), that we need for technical reason, and that we will later choose close to 0. We define γ t,x (n) by The shift of n in the definition is linked to the fact that we will consider p(z) = g pol (z)/z, as we did in the model case.Then, the solution g defined in eq.(3.15) can be written as In some sense, this formula tells us that g can be written as "pseudo-differential-type" operator applied to the "model solution" g pol (e iy−q (0)t−(1−ε) dAgm(x) ).To successfully adapt the strategy used for the model Baouendi-Grushin equation, we need some continuity estimates for these "pseudo-differential-type" operators.We claim that the following estimate holds.
Figure 10.The domains U and V .
Lemma 3.5.Let T > 0 and ε > 0. Define γ t,x as in eq.(3.19).Let γ t,x (z∂ z ) be the operator on polynomials defined by Let X be a compact subset of C. Let V be an open neighborhood of X that is star-shaped with respect to 0. There exist C > 0 and N ∈ N such that for every polynomial p ∈ C[X] with a zero of order N at 0 and for every 0 < t < T and x ∈ I, As γ t,x (n) is related to the eigenvalues and eigenfunctions of P n , proving this lemma requires a spectral analysis of this operator.What is more surprising is that we actually need a spectral analysis of P ν when ν is not necessarily real, meaning we have to do some nonselfadjoint spectral analysis.We will prove lemma 3.5 in section 4.3 with the spectral analysis done in the rest of section 4 and a general estimate on operators on polynomials [27, theorem 18].
We will also use the relatively elementary bounds on λ n and ϕ n L 2 (I) given by the following proposition: Proposition 3.6.In the limit n → +∞, λ n = nq (0) + o(n).Moreover, there exist c > 0 and N 0 such that for every n N , ϕ n L 2 (I) c.This proposition is standard (see, e.g., [18, theorem 4.23 & Eq.(4.20)]), nevertheless, for the reader convenience, we provide a proof in section 4.2.
With these two estimates, we prove the following version of lemma 3.4 adapted for the generalized Baouendi-Grushin equation.Lemma 3.7.Assume that the observability inequality (3.3) for the generalized Baouendi-Grushin equation holds.Let ε > 0 and let U ⊂ C be defined by (see fig. 10) Let V be a neighborhood of U that is star-shaped with respect to 0. Then, there exist C > 0 and N ∈ N such that for every polynomial p ∈ C[X] with a zero of order N at 0, we have Proof.The proof mostly follows the one of lemma 3.4, but with the error term γ t,x which will be handled by lemma 3.5.Let N > 0 as in proposition 3.6 and lemma 3.7.Let p(z) = n N a n z n a polynomial and g pol (z) = zp(z).The discussion above shows that g defined by (3.21) is a solution of the Baouendi-Grushin equation (3.2).
Step 1: Left-hand side of the observability inequality (3.3).Since the functions ψ n : (x, y) → ϕ n (x)e iny are orthogonal, the left-hand side can we rewritten as using the lower bounds on ϕ n L 2 (I) given by proposition 3.6, we get that ψ n L 2 (I×T) c > 0 for n N .Thus, Now, thanks to the asymptotics for λ n given by proposition 3.6, there exists As in the proof of lemma 3.4, we denote by m the Lebesgue measure on C R 2 , the functions z → z n are orthogonal on L 2 (D(0, R), m) and z n 2 L 2 (D(0,R),m) = πR 2n+2 /(n + 1).Thus, Step 2: Right-hand side of the observability inequality (3.3).We make the analogous change of variables as in the model case, but adapted to our case, i.e., (x, z) = (x, e −q (0)t+iy−(1−ε) dAgm(x) ).We have dx dm(z) = q (0)|z| 2 dt dx dy.Thus, if we denote by Ω ⊂ I × C the image of (0, T ) × ω by this change of variables, which is a subset of I × D(0, 1), we have When the disk D(0, e −q (0)T (1+ε) ) (in red) is not included in U , we can find holomorphic functions that are small in U but arbitrarily big in D(0, e −q (0)T (1+ε) ).
For instance, we can construct with Runge's theorem a sequence of polynomials that converges to z → z N +1 (z − z 0 ) −1 away from the blue line.

Spectral Analysis
As explained in section 3.3, we need some spectral properties on the operator P ν = −∂ 2 x + ν 2 q(x) 2 with Dirichlet boundary conditions on I (defined precisely in eq.(1.4)).We start with an asymptotic of the first eigenvalue, and in following subsection, we prove some Agmon-type upper bound for the associated eigenfunctions.

4.1.
The first eigenvalue and corresponding spectral projection.For β ∈ C with Re(β) > 0, we denote by H β the non-selfadjoint harmonic oscillator −∂ 2 x + β 2 x 2 on R. We refer to appendix C for the precise definition and properties of H β .
In this paragraph we prove that the operator P ν has an eigenvalue close to the eigenvalue q (0)ν of the model operator H q (0)ν , and that the corresponding spectral projection is also a perturbation of the spectral projection of H q (0)ν .See proposition 4.2.
For this we first prove that the resolvent of P ν is a perturbation of the resolvent of H q (0)ν , in the sense that the difference between these two resolvents is smaller than the resolvent of H q (0)ν .
Notice that the resolvents of P ν and H q (0)ν are not defined on the same space.We denote by 1 I the operator that maps a function v ∈ L 2 (I) to its extension by 0 on R. Then 1 * I is the operator which maps u ∈ L 2 (R) to its restriction on I: Then there exists ν 0 1 such that for ν ∈ Σ θ0 with |ν| ν 0 and z ∈ Zν we have z ∈ ρ(P ν ), and Proof.For ν ∈ Σ θ0 and z ∈ Zν we set We consider a cut-off function Step 1: Approximation close to x = 0. We first prove that if |ν| is large enough then R ν (z)χ ν (P ν − z)−χ ν extends to a bounded operator on L 2 (I) for all z ∈ Zν , and Here and everywhere below it is implicitly understood that ν always belongs to Σ θ0 .Let u ∈ Dom(P ν ).We have χ ν u ∈ Dom(P ν ) and, if |ν| is large enough, 1 I χ ν u belongs to Dom(H q (0)ν ).For x ∈ Ī we set r(x) = q(x) 2 − q (0) 2 x 2 , so that, for |ν| large enough, The commutator ν , P ν ] of χ ν and P ν is equal to By the resolvent estimate (C.3), we have Similarly, Considering the last term in eq. ( 4.4), we have for v ∈ L 2 (R) .
We multiply by e iθ and take the real part.This gives, uniformly in ν ∈ Σ θ0 and z ∈ Zν , (4.7) Taking the adjoint in the first inequality gives, for |ν| large enough, and eq.( 4.3) follows.
We will use proposition 4.3 with κ = (1 − ε) d Agm , where d Agm defined in eq.(1.3).Up to this point, we assumed ϕ to be an eigenfunction of P ν , but we did not specified which one, neither how it is normalized.We do this in the following definition, which is the natural extension of the definition of ϕ n when n ∈ N (eq.(3.18)).For Re(ν) > 0 that satisfies the hypotheses of proposition 4.2, let φν ∈ L 2 (I) be defined by φν (x) := ν 1/4 e −q (0)νx 2 /2 , and (4.13) where Π ν is the spectral projection for P ν associated with λ ν , as defined in proposition 4.2.

C|ν|.
Step 2: Second We again use Agmon's equality (4.16) to get The claimed estimate then follows from Sobolev's embedding of H 1 (I) into L ∞ (I).
We also prove the lower bound of proposition 3.6 for ϕ n when n 0 is large enough. 10roof of proposition 3.6.The part about λ n was already proved in proposition 4.2.By definition of ϕ n , we have Moreover, denoting by ϕ H β,1 (x) = (Re(β)/π) 1/4 e −βx 2 /2 the first eigenvector of H β , we have for The integral above is on I, but if we integrate on R instead, we only add a small error term.Thus, Hence, Since φn L 2 (I) is bounded (proposition 4.4) and since 1 * I ϕ H q (0)n,1 L 2 (I) = 1 + o(1) (thanks to similar computations as above), this proves the claimed lower bound.

4.3.
Estimate for some pseudo-differential type operators on polynomials.In this section, we use the spectral analysis of the operator P ν to deduce the operator estimate of lemma 3.5.In order to do that, we need some definitions and theorems about a general class of operators on polynomials.The following comes from [27, definition 9 & theorem 18].Definition 4.6.Let Ω be an open subset of C. Assume that there exists (r θ ) 0 θ<π/2 with r θ 0 such that 0 θ<π/2 Σ θ \ D(0, r θ ) ⊂ Ω (see fig. 12).
For the next theorem, if U is an open subset of C, we denote the set of bounded holomorphic functions on U that have a zero of order n 0 at 0 by O ∞ n0 (U ).We endow O ∞ n0 (U ) with the L ∞ -norm.Theorem 4.7.Let Ω ⊂ C as in definition 4.6 and set n 0 = min{n ∈ N : [n, +∞) ⊂ Ω}.Let γ in S(Ω) and γ(z∂ z ) be the operator on polynomials with a zero of order n 0 at 0, defined by: . An example of a set Σ θ \ D(0, r θ ) ⊂ Ω.The angle θ is allowed to be arbitrarily close to π/2, but then, the radius r θ of the disk we avoid may blow up arbitrarily fast.For the Ω we will consider, the corresponding r θ does blow up when θ → π/2 (at least, we cannot exclude that it blows up).
Let U be a bounded open subset of C. Let V be a neighborhood of U that is star shaped with respect to 0. Then there exists C > 0 such that for all polynomials p with a zero of order n 0 at 0, Moreover, the constant C above can be chosen continuously in γ ∈ S(Ω): . We now have all the pieces needed to prove lemma 3.5.
Step 2: The family (γ t,x ) 0<t<T,x∈I is a bounded family of S(Ω).According to the definition of Ω as a union of domains that look like the one of fig.12, Ω is stable by ν → ν + 1.Then, the map γ ∈ S → γ(• + 1) ∈ S is well-defined and continuous.Thus, according to the first step and the definition of γ (eq.(4.17)), the family (γ t,x ) 0<t<T,x∈I is indeed a bounded family of S(Ω).
Step 3: Conclusion.Let n 0 = min{n ∈ N : [n, +∞) ⊂ Ω}. 12 Let U be a bounded open neighborhood of X such that U ⊂ V .Then, the sets U and V satisfy the hypotheses of theorem 4.7.Hence, according to theorem 4.7, there exists C > 0 such that for every polynomials p with a zero of order n 0 at 0, and for every x ∈ I and 0 < t < T ,

Critical time of null-controllability for some domains
In this section, we prove theorem 1.6.
Step 1: Lower bound of the minimal time.For this step, we only have to treat the case T * > 0. In this case, either a − < 0 or a + > 0. If a − < 0, for any a − < a < 0, the segment [a, L + ] × {y − } stays at positive distance of ω, and thanks to theorem 3.1, the generalized Baouendi-Grushin equation (3.2) is not null-controllable on ω in time T < d Agm (a)/q (0).Similarily, if a + > 0, for any 0 < a < a + , the segment [−L − , a] × {y + } stays at positive distance from ω, and the generalized Baouendi-Grushin equation is not null-controllable in time T < d Agm (a)/q (0).This holds for any a < a < 0 and 0 < a < a + , thus the generalized Baouendi-Grushin equation (3.2) is not null-controllable in time T < T * .
Assume that the generalized Baouendi-Grushin equation with Dirichlet boundary conditions (A.1) is null controllable on ω in time T , then for every complex sequence (a n ) with a finite number of nonzero terms, Sketch of the proof.This lemma is proved by testing the associated observability inequality on the function g(t, x, y) = n 0 a n ϕ n (x) sin(ny)e −λnt , and writing sin(ny) = (e iny − e −iny )/(2i).Thus, with g(t, x, y) = n 0 a n ϕ n (x)e iny−λnt , g(t, x, y) = (g(t, x, y) − g(t, x, −y))/(2i), the right-hand side of the observability inequality satisfies The right-hand side of this inequality is the right-hand side of the claimed estimate.
Theorem A.2 is then proved by remarking that we already disproved such an inequality in section 3.3.
Step  Proof.Assume that for some y 0 , y 1 ∈ T, there exists a continuous path c 1 in (I × T) \ γ(T) from (a, y 0 ) to (b, y 1 ).Since I × T is Hausdorff, we may assume that c 1 is simple.We can also assume that c 1 touches the boundaries {a, b} × T only at the start and end.Now, consider the universal cover , that starts at (a, ỹ0 ) and ends at (b, ỹ1 ).Let c 2 the simple closed loop formed by concatenating c 1 , the vertical segment {b} × [ỹ 1 , ỹ1 + 2π], the reverse of the path c1 + (0, 2π), and finally the vertical segment {a} × [ỹ 0 , ỹ0 + 2π] from top to bottom (see fig. 14).
If we see this path c 2 as a path on R 2 , according to Jordan's theorem, R 2 \ c 2 has two path-connected components, one of them bounded.Let us denote by Ω 1 this bounded component, which, according to Jordan-Schoenflies' theorem, is simply connected.One of the lift of γ lies in Ω 1 , let us call it γ.But γ is not homotopic to a constant path, which contradicts the simple connectedness of Ω 1 .tells us that for any ε > 0 there exists a path γ that stays at distance at most 2ε from γ(T) and that is not homotopic to a constant path.
Proof.The proof uses some basic tools of algebraic topology, in particular van Kampen's theorem (see for instance Hatcher's "Algebraic Topology" Step Since A − is the union of two connected subset that have a non-empty intersection (we saw in step 1 that ∂C − ⊂ C − ∩ ω), A − is connected.Finally, A + is connected because it is the union of the connected subset ω and of the connected components of [a, b] × T \ ω other than C − , which all have a non-empty intersection with ω.
Step 3: Conclusion using van Kampen's theorem.If α is a closed path in a topological space X, we will denote its homotopy class by [α] X .We will denote the fundamental group of X by π 1 (X).We will denote by p − (respectively p + ) the canonical injection of π 1 (A − ) (respectively π 1 (A + )) into the free product π 1 (A − ) * π 1 (A + ).
According to van Kampen's theorem [23,Theorem 1.20], the map k : where none of the terms in the right-hand side are the neutral element of π 1 (A ± ).Since the left-hand side is already a reduced word, by definition of the free product of groups, the two words on the left and right-hand side of this equality are the same.Thus there are exactly two factors in the right-hand side of eq.(B.2) and The proof also uses Jordan's theorem, but in a simpler way than proposition B.2, and is left to the reader.

Appendix C. Non-selfadjoint harmonic oscillators
Let β ∈ C with Re(β) > 0. We discuss in this appendix the basic properties of the non-selfadjoint harmonic oscillator (or Davies operator) defined on L 2 (R) by x u + β 2 x 2 u is understood in the sense of distributions, and we define H β by eq.(C.1) on Dom(H β ).This defines an unbounded operator on L 2 (R).When β = 1 we recover the usual harmonic oscillator.
The spectral properties of the operator has been studied (see, among others, [17, §14.5], [24, §14.4], [29], and the references therein), and the properties stated in this appendix are standard, at least in spirit.Nevertheless, for the reader convenience, we collect and prove the properties needed in our study.Let θ 0 ∈ 0, π 2 .There exists C > 0 such that if β ∈ Σ θ0 (see eq. , so e −iθ H β is sectorial with angle θ.In particular, (e −iθ H β + 1) is injective.Now let This is a Hilbert space for the natural norm Let f ∈ L 2 (R).By the Lax-Milgram Theorem, there exists a unique u ∈ B 1 (R) such that Q β (u, v) = f, v L 2 (R) for all v ∈ B 1 (R).In the sense of distributions we have e −iθ (−u + β 2 x 2 u) + u = f ∈ L 2 (R), so u ∈ Dom(H β ) and (e −iθ H β + 1)u = f .This proves that e iθ belongs to the resolvent set of deduce that Dom(H β ) is compactly embedded in L 2 (R).Since H β has nonempty resolvent set, it has compact resolvent.In particular, its spectrum consists of isolated eigenvalues of finite multiplicities.
Step we set F (ξ) = R e ixξ u(x)e −βx 2 /2 dx.Then F is analytic and F (n) (0) = 0 for all n ∈ N.This implies that u(x) = 0 for almost all x ∈ R, so the family (ϕ H β,k ) k∈N * is complete.For ρ > 0 we consider on L 2 (R) the unitary operator Θ ρ such that for u ∈ L 2 (R) and x ∈ R we have We observe that Θ −1  .

Figure 1
Figure1.In green, an example of a domain ω with, in thick black, an example of a horizontal segment that stays at positive distance of ω.In this example, theorem 1.3 implies that the generalised Baouendi-Grushin equation is not null-controllable in time T < d Agm (a).

1. 3 . 2 .Figure 2 .
Figure 2. In green, an example of a domain ω, with, in blue, a corresponding path γ that satisfies the hypotheses of theorem 1.4.

Figure 4 .Figure 5 .
Figure 4.In this configuration, we obtain lower and upper bounds of the critical time of null-controllability

Figure 13 .
Figure13.In green, the domain ω.At y = y − , the function γ − 2 takes its maximum a − .Then, the interval (a − , L + ) × {y − } is disjoint from ω. So, the Grushin equation is not null-controllable in time T < d Agm (a − )/q (0).Similarly, the interval (−L − , a + ) × {y + } is disjoint from ω. So, the Grushin equation is not null-controllable in time T < d Agm (a + )/q (0).Also, if we take a path γ (here in blue) that is close to the boundary of ω around y = y − and y = y + , then, we can apply theorem 1.4, and the Grushin equation is null-controllable in time T > max(d Agm (a − ), d Agm (a + ))/q (0).

4 :
Value of χ on ω + \ ω.According to the definition of χ and proposition B.2, χ = 0 on {b} × T. The rest of this step is a copy-paste of the previous step.Now, we justify remark 1.5, with the following two propositions: Proposition B.2.Let a < b and let γ ∈ C 0 (T, (a, b) × T) be a closed path that is not homotopic to a constant path.Then {a} × T and {b} × T are included in different connected components of ([a, b] × T) \ γ(T).

Figure 14 .
Figure 14.Illustration of the path c 2 defined in the proof of proposition B.2.

Proposition B. 3 .
Let ω be a connected open subset of [a, b] × T such that {a} × T and {b} × T are included in different connected components of ([a, b] × T) \ ω.Let ω a connected open subset of [a, b] × T such that ω ⊂ ω.There exists a closed path γ ∈ C 0 (T, ω) that is not homotopic in [a, b] × T to a constant path.Remark B.4.Let γ be a closed path in (a, b) × T such that {a} × T and {b} × T are included in different connected components of ([a, b] × T) \ γ(T).It is possible this path γ is homotopic to a constant path, but proposition B.3 applied with ω := {z : distance(z, γ(T)) < ε} and ω := {z : distance(z, γ(T)) < 2ε} e., C. By contradiction, we see that every x ∈ ∂X is in ∂ω ⊂ ω.Step 2: A + and A − are open and connected.We begin with the openness of A − .If x ∈ A − , there are three cases:• If x is in the interior of C − , it is in the interior of A − by definition.• If x is in ω, since ω is open, x isalso in the interior of A − .If x ∈ ∂C − , according to step 1, x ∈ ω which implies that x is in the interior of A − .The subset A + is open because it is the union of the open subsets ([a, b] × T) \ C − and ω.