Control of the Grushin equation: non-rectangular control region and minimal time

This paper is devoted to the study of the internal null-controllability of the Grushin equation. We determine the minimal time of controllability for a large class of non-rectangular control region. We establish the positive result thanks to the fictitious control method and the negative one by interpreting the associated observability inequality as an $L^2$ estimate on complex polynomials.


Bibliographical comments
The null-controllability of the heat equation is well known in dimension one since 1971 [17] and in higher dimension since 1995 [19,22]: the heat equation on a C 2 bounded domain is null-controllable in any non-empty open subset and in arbitrarily small time. But for degenerate parabolic equations, i.e. equations where the laplacian is replaced by an elliptic operator that is not uniformly elliptic, we only have results for some particular equations. For instance, control properties of one and two dimensional parabolic equations where the degeneracy is at the boundary are now understood [11]. Other examples of control properties of degenerate parabolic equations that have been investigated include some Kolmogorov-type equation [8], quadratic differential equations [10] and the heat equation on the Heisenberg group [5] (see also references therein).
About the Grushin equation, Beauchard, Cannarsa and Guglielmi [9] proved in 2014 that if ω is a vertical strip that does not touch the degeneracy line {x = 0}, there exists T ⋆ > 0 such that the Grushin equation is not null-controllable if T < T ⋆ and that it is null-controllable if T > T ⋆ ; moreover, they proved that if a is the distance between the control domain ω and the degeneracy line {x = 0}, then a 2 /2 ≤ T ⋆ . Then, Beauchard, Miller and Morancey [9] proved that if ω is a vertical strip that touches the degeneracy line, this minimal time T ⋆ was 0, and that if ω is two symmetric vertical strips, the minimal time is exactly a 2 /2, where a is the distance between ω and {x = 0}. The minimal time if we control from the left (or right) part of the boundary was computed by Beauchard, Dardé and Ervedoza [7], and our positive result is based on that. On the other hand, the second author proved that if ω does not intersect a horizontal strip, then the Grushin equation is never null-controllable [21], and our negative result is proved with the methods of this reference.
So, the Grushin equation needs a minimal time for the null-controllability to hold, as do the Kolmogorov equation and the heat equation on the Heisenberg group (see above references); a feature more expected for hyperbolic equations than parabolic ones. Note, however, that degenerate parabolic equations are not the only parabolic equations that exhibit a minimal time of null-controllability. In dimension one, Dolecki has proved there exists a minimal time for the punctual controllability of the heat equation to hold [14], and parabolic systems may also present a minimal time of null-controllability (see e.g. [2][3][4]15]).

Main results
Our first result is about the null-controllability in large time if the control domain contains an ε-neighborhood of a path that goes from the bottom boundary to the top boundary: (see Fig. 1 We prove this Theorem in Section 5. Since the Grushin equation (1.1) is not null-controllable when ω is the complement of a horizontal strip [21], Assumption (3.1) is quasi optimal. We now state our second main result: y 0 x y ω Figure 3. In green, the domain ω in Theorem 3.3. Even if when |x| tends to ∞, the complement of ω narrows, the Grushin equation is never null-controllable.
We prove this Theorem in Section 6.1. We also mention that with the same method, we can also prove the following Theorem 3.3, which answers a question that was asked to the second author by Yves Colin de Verdière. Theorem 3.3 (Negative result on the whole real line). Let y 0 ∈ (0, π), f : R → R ⋆ + a continuous function that is never zero and ω = {(x, y) ∈ R × (0, π), |y − y 0 | > f (x)} (see Fig. 3). Then Grushin equation on Ω = R × (0, π) is never null-controllable on ω.
We prove this Theorem in Section 6.3. Note that both of these negative results are valid if we take y ∈ R/2πZ instead of y ∈ (0, π). Theorems 3.1 and 3.2 allow us to deduce a condition on the control region ω for which we are able to give an explicit minimal time of null-controllability: Corollary 3.4. Assume that there exists ω 0 satisfying (3.1) and y 0 ∈ (0, π) such that {(x, y 0 ), −a < x < a} is disjoint from ω, where a is given in (3.2

Comments and open problems
Note that most of the existing results results for the controllability of degenerate parabolic equations (see section 2) were only concerned with rectangular control domains. The reason is that these results were based on Fourier series techniques, which can only treat tensorised domains. Our results are built on the previous ones, but by adding some arguments (the fictitious control method for the positive result, and fully treating x as a parameter in the negative one) we can accurately treat a large class of non-rectangular domains. But even then, there remains some open problems.
If the control region ω if not connected, the positive result of Theorem 3.1 might not apply. For example in Figure 4a, the negative result of Theorem 3.2 says only that the minimal time for the null-controllability of the Grushin equation is greater than a 2 /2, but we don't know if the Grushin equation is null-controllable in any time greater than a 2 /2 or not.
If the domain is "pinched", as in Figure 4b, the open segment {(x, π/2) : −1 < x < 1} is disjoint from ω but not from ω: we are in a limit case of Theorem 3.2. Again, we only know that the minimal time is greater than a 2 /2, but we don't know if the Grushin equation is ever null-controllable at all.
For ω := {(x, y) : γ 1 (y) < x < γ 2 (y)} with γ 1 , γ 2 ∈ C([0, π], (−1, 1)) such that γ 1 < γ 2 , Corollary 3.4 determines the minimal time (see Remark 3.5) which is not necessary the case if the control region does not have this form. For instance, if there is a "cave" in the control domain, as in Figure 4c, the results of the present paper only ensure that the minimal time is greater or equal than a 2 /2 and smaller or equal than b 2 /2.
The null-controllability in the critical time T = a 2 /2 in Corollary 3.4 also remains an open problem. Since for ω := [(−b, −a) ∪ (a, b)] × (0, 1) with 0 < a < b ≤ 1 the minimal time is equal to a 2 /2 and the Grushin equation is not null-controllable at this time [9], we can conjecture that it is also the case in Corollary 3.4.

Proof of the positive result
This section is devoted to the proof of Theorem 3.1. To this end, we will use the fictitious control method consisting in building a solution of the control problem thanks to algebraic combinations of solution to controlled problems. Concerning the controllability of partial differential equations, the fictitious control method has already been used for instance in [20], [13] and [16].
The observability estimates obtained in [7, Th. 1.4] can be interpreted in terms of boundary controllability of System (1.1) in Ω a := (−a, 0) × (0, π) and acting on the subset of the boundary {−a} × (0, π) ⊂ ∂Ω a , which is equivalent to the internal controllability of System (1.1) by acting on (−1, −a) × (0, π) with a ∈ (0, 1): with a ∈ (0, 1). The following holds: The equivalence between the boundary controllability and the internal controllability is standard for the heat equation (see for instance [1,Theorem 2.2]). However, the equivalence proved in the reference is for boundary controls, but not in L 2 . Moreover, the Grushin equation is degenerate, so we can't directly apply the theorem of the reference. So we will prove Theorem 5.1 in Appendix A.
We will now give a proof of Theorem 3.1 thanks to the fictitious control method: Proof of Theorem 3.1.
Consider ω 0 and a given in (3.1) and (3.2). Let T > a 2 /2. Define . Such construction is illustrated in Figure 5. Consider the following control problems Since T > a 2 /2, using Item (i) of Theorem 5.1, null-controllabity problems (5.1) and (5.2) admit solutions. We now glue this two solutions f left and f right with an appropriate cutoff, given by Lemma 5.2: By looking at Fig. 5, one should be convinced such a cutoff do exists; nevertheless, we provide a rigorous proof in Appendix B. We define x y Figure 6. In green, the domain ω. If we have a horizontal segment that does not touch ω (except maybe the extremities), we can find a horizontal rectangle about the horizontal segment (in blue), with its length as close as we want to the length of the horizontal segment. In the rest of the proof, we will assume that ω is the complement of this blue rectangle.
6. Proof of the negative results

Proof of Theorem 3.2.
We prove in this section the first non-null-controllability result (Theorem 3.2). We do this by pushing the method used by the second author to disprove the null-controllability when ω is the complement of a horizontal strip [21]. First, we note that according to the hypothesis that {(x, y 0 ), −a < x < a} does not intersect ω, for every a ′ < a, there exists a rectangle of the form {−a ′ < x < a ′ , |y − y 0 | < δ} that does not intersect ω (see Fig. 6). Thus, we assume in the rest of this proof that ω is the complement of this rectangle, i.e.
To disprove the null-controllability, we disprove the observability inequality, which is equivalent to the nullcontrollability (see Coron's book [12, Theorem 2.44] for a proof of this equivalence): there exists C > 0 such that for all f 0 in L 2 (Ω), the solution f to For n > 0, let us note v n the first eigenfunction of −∂ 2 x + (nx) 2 with Dirichlet boundary condition on (−1, 1), and with associated eigenvalue λ n . Then the function v n (x) sin(ny) is an eigenfunction of −∂ 2 x − x 2 ∂ 2 y with Dirichlet boundary condition on ∂Ω, and with eigenvalue λ n . We will disprove the observability inequality (6.1) with solutions f of the form f (t, x, y) = a n v n (x)e −λnt sin(ny).
The observability inequality (6.1) applied to these functions reads: 2) Note that we have not supposed the v n to be normalized in L 2 . Indeed, we will find more convenient to normalize them by the condition v n (0) = 1. Let us remind that thanks to Sturm-Liouville theory, v n is even, and v n (0) = 0.
The main idea is to relate this inequality to an estimate on polynomials. Let ε ∈ (0, 1/2) be a small real number to be chosen later (it will depend only on T and a ′ ). Then we have Lemma 6.1: Figure 7. The domain U . Fig. 7). The inequality (6.2) implies that there exists N > 0 and C > 0 such that for all polynomials of the form p(z) = n>N a n z n , Proof. About the left-hand side of the observability inequality (6.2), we first note that by writing the integral on a disk D = D(0, r) of z nzm in polar coordinates, we find that the functions z → z n are orthogonal on D(0, r). So, we have for all polynomials 3 n>0 a n z n−1 : and computing D(0,e −T ) |z| 2n−2 dλ(z) in polar coordinates: Moreover, we know from basic spectral analysis that writing λ n = n + ρ n , (ρ n ) n≥0 is bounded (see [6,Section 3.3]) and that |v n | 2 L 2 (−1,1) ≥ c|n| −1/2 (see for instance [21,Lemma 21]), so that = C|f (T, ·, ·)| 2 L 2 (Ω) . Thus, the observability inequality (6.2) implies for another constant C (that depends on time but it does not matter): To bound the right-hand side, we begin by writing sin(ny) = (e iny − e −iny )/2i, so that the right-hand side satisfies 3 We denote λ the Lebesgue measure on C ≃ R 2 . I.e. for a function f : Figure 8. Above: The domainω is the union of ω and of its symmetric with respect to the axis x = 0. Below: the domain Dx is defined to be the set of complex numbers z of modulus between e −T −(1−ε)x 2 /2 and e −(1−ε)x 2 /2 , and with argument such that (x, arg(z)) ∈ω. It is a partial ring if |x| < a ′ and a whole ring if |x| > a ′ . Indeed, if we take a slice ofω by fixing x, when |x| < a ′ , we don't have the whole interval (−π, π), but when |x| > a ′ , the slice is the whole interval (−π, π).
Then, we again write λ n = n + ρ n and v n (x) = e −(1−ε)nx 2 /2 w n (x), so that with z x (t, y) = e −t+iy−(1−ε)x 2 /2 , the previous implies: [0,T ]×ω n>0 a n w n (x)e −ρnt z x (t, y) n 2 dt dx dy. (6.5) Now for each x ∈ (−1, 1), we make the change of variables z x = e −t+iy−(1−ε)x 2 /2 , for which dt dy = |z x | −2 dλ(z). For each x, let us note D x the image of the set we integrate on for this change of variables (see Fig. 8), that is, We get [0,T ]×ω n>0 a n w n (x)e −ρnt z x (t, y) n 2 dt dx dy = 1 −1 Dx n>0 a n w n (x)e −ρnt z n−1 2 dλ(z) dx, where we kept the notation e −ρnt for simplicity instead of expressing it as a function of z and x (we have e −ρnt = |e (1−ε)x 2 /2 z| ρn ). With this change of variables, the inequality (6.5) becomes Figure 9. The domain K, in green, is (the closure of) the union of the domains Dx described in Fig. 8. For the domain K the radius of the inner part-of-circle is the largest radius of the Dx that is a full ring, i.e. e −(1−ε)a ′2 /2 . We also show U for comparison. Notice that U has been defined to be a neighborhood of K that is star-shaped with respect to 0.
We want to bound the right-hand side by | n>0 a n z n−1 | 2 L ∞ (U) . To do this, we use the following Lemma 6.2, that we prove in Section 6.2. This Lemma is a rigorous statement of the fact that ρ n and w n are small. 4 Lemma 6.2. Let K be a compact subset of C and V a bounded neighborhood of K that is star-shaped with respect to 0. Then, there exists C > 0 and N > 0 such that for every x ∈ (−1, 1), for every τ ∈ [0, T ], and for every polynomial n>N a n z n−1 : n>N a n w n (x)e −ρnτ z n−1 .
From now on, we assume that a n = 0. In the Lemma, we chose K = 1 x=−1 D x and V = U , where U was defined in the statement of Lemma 6.1 (see Fig. 9). Notice that by definition of D x , K is the union of the ring {e −T −(1−ε)/2 ≤ |z| ≤ e −(1−ε)a ′2 /2 } and of the partial ring {e −T −(1−ε)a ′2 /2 ≤ |z| ≤ 1, || arg(z)| − y 0 | ≥ δ}, and so U is a neighborhood of K that is star-shaped with respect to 0, and the hypotheses of Lemma 6.2 are satisfied. We get, by also taking τ = t (let us remind that t is a function of z and x): n>N a n w n (x)e −ρnt z n−1 . (6.7) Then, we plug this estimate (6.7) into the estimate (6.6) to get for some C .
Combining this with the estimate (6.4) on the left-hand side, we get , which is, up to a change of summation index n ′ = n − 1, the estimate (6.3) we wanted.
Let us assume T < (1 − 2ε)a ′2 /2. We want to disprove the inequality (6.3) of Lemma 6.1. We will use Runge's theorem to construct a counterexample.  By definition of U , there exists a complex number z 0 ∈ D(0, e −T ) which is non-adherent to U (see Fig. 10). Then, according to Runge's theorem, there exists a sequence of polynomialsp k that converges uniformly on every compact subset of C \ z 0 [1, +∞) to z → (z − z 0 ) −1 , and we define p k (z) = z N +1p k (z). Then, the family p k is a counter example to the inequality on entire polynomials (6.3). Indeed, since z N +1 (z − z 0 ) −1 is bounded on U , p k is uniformly bounded on U , thus, the right-hand side of the inequality (6.3) is bounded. But since z 0 is in D(0, e −T ), z N +1 (z − z 0 ) −1 has infinite L 2 -norm in D(0, e −T ), and thanks to Fatou's Lemma, |p k | L 2 (D(0,e −T )) tends to +∞ as k → +∞.
Thus, the Grushin equation is not null-controllable if T < (1 − 2ε)a ′2 /2. But a ′ can be chosen arbitrarily close to a, and ε arbitrarily small, so the Grushin equation is not null-controllable if T < a 2 /2.
The map γ ∈ S r → H γ satisfies the following continuity-like estimate: for each compact K and each neighborhood V of K that is star-shaped with respect to 0, there exists a constant C > 0 and a finite number of seminorms (p θi,εi ) 1≤i≤n of S r such that for every symbol γ and entire functions f : Theorem 6.6 (Theorem 22 and Proposition 25 of [21]). We remind that λ n is the first eigenvalue of −∂ 2 x +(nx) 2 with Dirichlet boundary conditions on (−1, 1), v n is the associated eigenfunction that satisfies v n (0) = 1 and w n = e −(1−ε)nx 2 /2 v n (x).
There exists a non-decreasing function r : (0, π/2) → R such that for N = ⌊inf θ r(θ)⌋: (i) there exists a symbol γ ∈ S r such that for n > N , λ n = n + γ(n)e −n ; (ii) for each x, there exists a symbol w(x) ∈ S r such that for n > N , w n (x) = w(x)(n), and moreover, the family (w(x)) −1<x<1 is a bounded family of S r .
Proof of Lemma 6.2. We want to bound w n (x)e −ρnτ a n z n by a n z n . To do this, we prove that w n (x)e −ρnτ is a symbol in the sense of Definition 6.4, and then apply the Theorem 6.5.
Moreover, we already know that (w(x)) x∈(−1,1) is a bounded family in S r . Since γ τ,x is the multiplication of w(x) and e −ρτ and since the multiplication is continuous 5 in S r , (γ τ,x ) 0<τ <T,x∈(−1,1) is a bounded family of S r .
So according to the estimate (6.9) of Theorem 6.5, if V is a bounded neighborhood of K that is star-shaped with respect to 0, there exists C > 0 independent of ζ, x, such that: n>N γ τ,x (n)a n z n So, thanks to equation (6.10): 6.3. Sketch of the proof of Theorem 3.3. The proof of Theorem 3.3 goes along the same lines, but is actually simpler: the eigenfunctions are exactly the Gaussian e −nx 2 /2 , with associated eigenvalue exactly n, and we don't need the Lemma 6.2, nor any equivalent.
Following the proof until equation (6.5), with the change of variables z x = e −t+iy−x 2 /2 , we find that the null-controllability on ω (as defined in the statement of the Theorem) would imply that with D .
(This time, we only write |p| L 2 ≤ C|p| L ∞ .) But this does not hold. Indeed, z 0 = e −T /2+iy0 (for instance) is non adherent to U , and we can construct a counter-example to the estimate above with Runge's theorem, as in the end of the proof of theorem 3.3.
Appendix A. Proof of Theorem 5.1 In this section, we will show that the boundary null-controllability of System (1.1) (this notion is recalled in Definition A.3) implies the internal null-controllability of System (1.1). The argument is standard, but for the sake of clarity, we include it in the present paper.