Rough controls for Schrödinger operators on 2-tori

N IC O L A S B U R Q M A C IE J Z W O R S K I

R O U G H C O N T R O L S F O R

S C H R Ö DI N G E R O P E R AT O R S O N 2 - T O R I

C O N T R Ô L E S P E U R É G U L IE R S P O U R

L ’ E Q U AT IO N D E S C H R Ö DI N G E R S U R L E S T O R E S D E DI M E N SIO N 2

Dedicated to the memory of Jean Bourgain 1954–2018

Abstract. — The purpose of this note is to use the results and methods of [BBZ13]

and [BZ12] to obtain control and observability by rough functions and sets on 2-tori,T² = R²/Z⊕γZ. We show that for a non-trivialW ∈L^∞(T²), solutions to the Schrödinger equation, (i∂t+ ∆)u= 0, satisfyku|t=0k_L2(T²)6KTkW uk_L2([0,T]×T²). In particular, any set of positive Lebesgue measure can be used for observability. This leads to controllability with localization functions inL²(T²) and controls inL⁴([0, T]×T²). For continuous W this follows from the results of Haraux [Har89] and Jaffard [Jaf90], while forT² =R²/(2πZ)² (the rational torus) andT > πthis can be deduced from the results of Jakobson [Jak97].

Résumé. — On se propose dans cette note d’utiliser les résultats et les méthodes de [BBZ13] et [BZ12] pour obtenir des résultats de contrôle et de stabilisation par des fonctions de localisation peu régulières sur les tores de dimension 2, T² = R²/Z⊕γZ. On démontre que pour toute fonctionW ∈L^∞(T²) non triviale, les solutions de l’équation de Schrödinger, (i∂_t+ ∆)u= 0, vérifientku|t=0kL²(T²)6K_TkW ukL²([0,T]×T²). En particulier tout ensemble de

Keywords:Schrödinger equation, control/observability for PDE, semiclassical analysis.

2010Mathematics Subject Classification:35Q41, 35Q93, 49J20, 81Q93.

DOI:https://doi.org/10.5802/ahl.19

mesure de Lebesgue strictement positive suffit pour l’observabilité. On en déduit des résultats de contrôlabilité exacte par des fonctions de localisation dans L²(T²) et des contrôles dans L⁴([0, T]×T²). Pour desW continus, ce résultat est conséquence des travaux de Haraux [Har89]

and Jaffard [Jaf90], tandis que pour T² = R²/(2πZ)² (le tore rationnel) et T > π on peut obtenir ce résultat comme conséquence des travaux de Jakobson [Jak97].

1. Introduction

The purpose of this paper is to investigate the general question of control theory with localized control functions. When the localization is performed by a continuous function, the question is completely settled for wave equations [BLR92, BG96] and well understood for Schrödinger equations on tori [Har89, Jaf90, Kom92, BZ12, AM14].

In this paper we localize only to sets of positive measure or more generally use control functions in L². The understanding is then much poorer and only partial results are available even for the simpler case of wave equations [BG18, Bur]. Using the work with Bourgain [BBZ13] and [BZ12] we completely settle the question for Schrödinger equation on the two dimensional torus taking advantage, as in previous papers, of the particular simplicity of the dynamical structure.

To state the control result consider

(1.1) T² :=R²/Z×γZ, γ ∈R\ {0}, a∈L²(T²), (i∂_t+ ∆)u(t, z) =a(z)1_(0,T)f , u(0, z) =u₀(z),

where a is a localisation function and f a control. From [BBZ13, Proposition 2.2]

(see Theorem 1.4 below) we know that for f ∈ L⁴(T²;L²(0, T)) (so that af ∈ L^4/3(T²;L²(0, T))), and any u₀ ∈L²(T²), there exists a unique solution

u∈C⁰([0, T];L²(T²))∩L⁴(T²;L²(0, T)).

A classical question of control is to fix a and ask for which u₀ ∈ L² does there exist a control f such that the solution of (1.1) satisfies u|_t>T = 0? We show that onT² it is always the case as soon asa ∈L² is non-trivial:

Theorem 1.1. — Leta∈L²(T²),kak_L² >0andT >0. Then for anyu0 ∈L²(T²) there exists f ∈L⁴(T²;L²(0, T))such that the solutionu of (1.1) satisfiesu|_t=T = 0.

The next result shows that adding an L² damping term results in exponential decay:

Theorem 1.2. — For a ∈ L²(T²), a > 0, kak_L² > 0, there exist C, c > 0 such that for any u₀ ∈L²(T²), the equation

(1.2) (i∂t+ ∆ +ia)u= 0, u|t=0 =u0,

has a unique global solution u∈L^∞(R;L²(T²))∩L⁴(T²;L²_loc(R)) and (1.3) kuk_L²_(T²₎(t)6Ce^−ctku₀k_L²_(T²₎.

As shown in Section 4 both results follow from an the observability estimate. We should think ofain Theorem 1.1 asW² where W appears in the following statement:

Theorem 1.3. — Suppose that W ∈ L⁴(T²), kWk_L⁴ > 0. Then for any T > 0 there exists K such that for u∈L²(T²),

(1.4) kuk_L²_(T²₎6KkW e^it∆uk_L²_((0,T)t×T²).

To keep the paper easily accessible we present proofs in the case when γ ∈ Q in (1.1). Irrational tori require a more complicated reduction to rectangular coor- dinates (see [BZ12, Lemma 2.7 and Figure 1] but the modification can be done as in that paper). The crucial [BBZ13, Proposition 2.2] is valid for all tori. An- other approach to treating (higher dimensional) irrational tori can be found in the work of Anantharaman–Fermanian-Kammerer–Macià, see [AFKM15, Corollary 1.19, Theorem 1.20].

Since, as is already clear, [BBZ13, Proposition 2.2] plays a central role in many proofs we recall it in a version used here:

Theorem 1.4. — LetT > 0. There existsC =C_T such that for u₀ ∈L²(T²), f ∈L⁴³(T²;L²(0, T)),

the solution to (i∂_t+ ∆)u=f, u|_t=0 =u₀, satisfies kuk_L^∞_((0,T);L²_(T²)∩L⁴(T²;L²((0,T)))6C

ku₀k_L²_(T²₎+kfk

L¹((0,T);L²(T²))+L⁴³(T²;L²(0,T))

. Remarks

(1) Theorem 1.3 is equivalent to the same statement with W ∈ L^∞(T²) (by replacing W ∈ L⁴ by 1^|W|6NW ∈ L^∞ with N sufficiently large noting that kW vk_L²_(T²₎ > kW1^|W|6NW vk_L²_(T²₎ and that kWk_L⁴ > 0 implies kW1|W|6Nk_L^∞ > 0 if N is large enough. Our formulation is more consis- tent with the statements of Theorems 1.1 and 1.2.

(2) For rational tori and for T > π, Theorem 1.3, and by Proposition 4.1 below, Theorems 1.1 and 1.2, follow from the results of Jakobson [Jak97]. That is done by using the complete description of microlocal defect measures for eigenfunctions of R²/2πZ². We explain this in detail in the appendix.

(3) The starting point of [Jak97] and [BBZ13] was the classical inequality of Zygmund:

(1.5) ∀λ∈N,

X

|n|²=λ

c_ne^in·z

2

L⁴(T²z)

6

√5 2π

X

|n|²=λ

|c_n|²,

z ∈T² =R²/2πZ², n∈Z². In particular for T² = R²/2πZ², we easily see how the homogeneous part (f = 0) in Theorem 1.4 follows from (1.5). For that put u = ^P_λu_λ, u_λ =

P

|n|2=λc_ne^in·x. Then, using (1.5) in the third line, ke^it∆uk⁴_L4(T²,L²((0,2π))) =

Z

T²



 Z 2π

0

X

λ

e^itλuλ(z)

2

dt





2

dz

= (2π)²

Z

T²

X

λ

|uλ(z)|²

!2

dz

= (2π)²

Z

T²

X

λ,µ

|u_λ(z)|²|u_µ(z)|²dz 6(2π)^2X

λ,µ

ku_λk²_L4ku_µk²_L4

65^X

λ,µ

ku_λk²_L2ku_µk²_L2 = 5 ^X

λ

ku_λk²_L2

!2

= 5kuk⁴_L2. Generalizations for the time dependent Schrödinger equation in higher dimen- sions were obtained by Aïssiou–Jakobson–Macià [AJM12].

(4) Other than tori, the only other manifolds for which (1.4) is known for any non-trivial continuousW are compact hyperbolic surfaces. That was proved by Jin [Jin17] using results of Bourgain–Dyatlov [BD18] and Dyatlov–Jin [DJ18].

Acknowledgements. This research was partially supported by Agence Nationale de la Recherche through project ANAÉ ANR-13-BS01-0010-03 (NB), by the NSF grant DMS-1500852 and by a Simons Fellowship (MZ). MZ gratefully acknowledges the hospitality of Université Paris-Sud during the writing of this paper. We would also like to thank Alexis Drouot, Semyon Dyatlov and Fabricio Macià for helpful comments on the first version. We are also grateful to the referee for the careful reading of the first version of the paper and for the many helpful comments.

2. Semiclassical observability

We follow the strategy of [BZ12] and [BBZ13] and first prove a semiclassical observability result. For that we define

(2.1) Π_h,ρ(u₀) :=χ −h²∆−1 ρ

!

u₀, ρ >0,

where χ∈ C_c^∞((−1,1)) is equal to 1 near 0. With this notation the main result of this section is

Proposition 2.1. — Suppose thata ∈L²(T²), a>0, kak_L² >0. For any T >0 there existK,ρ₀ >0 and h₀ >0such that for any u₀ ∈L²(T²),

(2.2) kΠ_h,ρu₀k²_L2 6K

Z T 0

Z

T²

a(z)|e^it∆Π_h,ρu₀(z)|²dzdt, for0< ρ < ρ₀ and 0< h < h₀.

The proof of the Proposition proceeds by contradiction: if (2.2) does not hold then there exists T >0 such that for any n∈N there exist 0< hn<1/n, 0< ρn <1/n and un∈L² for which

(2.3) 1 = ku_nk²_L2 > n

Z T 0

Z

T²

a(z)|e^it∆u_n(z)|²dzdt, u_n= Π_hn,ρnu_n. We will use semiclassical limit measures associated to subsequences of u_n’s.

2.1. Semiclassical limit measures

Each sequenceu_n(t) :=e^it∆u_n, is bounded inL²_loc(R×T²). After possibly choosing a subsequence,u_n’s define a semiclassical defect measureµon Rt×T^∗(T²z) such that for any functionϕ∈ C_c⁰(Rt) and any A ∈ C_c^∞(T^∗T²z), we have

(2.4) hµ, ϕ(t)A(z, ζ)i= lim

n→∞

Z

Rt

ϕ(t)hA(z, h_nD_z)u_n(t), u_n(t)i_L²_(T²₎dt . The measureµenjoys the following properties:

µ((t₀, t₁)×T^∗T²z) =t₁−t₀, suppµ⊂Σ :={(t, z, ζ)∈Rt×T²z×R²ζ| |ζ|= 1},

∂_s

Z

R

Z

T^∗T²

ϕ(t)A(z+sζ, ζ)dµ= 0, ϕ∈ C_c⁰(Rt), A∈ C_c^∞(T^∗T²z), (2.5)

see [Mac09] for the derivation and references.

We have an additional property which follows from an easy part of Theorem 1.4 (in the rational case related to the Zygmund inequality (1.5); see also Aïssiou–Jakobson–

Macià [AJM12, Theorem 1.2]): for anyτ >0 there existsm_τ ∈L²(T²) such that for allf ∈ C(T²)

(2.6)

Z τ 0

Z

T^∗T²

f(z)dµ(t, z, ζ) =

Z

T²

m_τ(z)f(z)dz.

In fact, Theorem 1.4 shows that

(2.7) U_n^τ(z) :=

Z τ 0

|u_n(t, z)|²dt satisfies

(2.8) kU_n^τk_L²_(T²_z) =ku_n(t, z)k²_L4(T²z,L²((0,τ)) 6Cku_nk²_L2(T²) =C.

But then, after passing to a subsequence,U_n^τ converges weakly tomτ ∈L²(T²). Since

Z τ 0

Z

T^∗T²

f(z)dµ(t, z, ζ) = lim

n→∞

Z τ 0

Z

T²

f(z)U_n^τ(z)dz, this proves (2.6).

Using (2.3) and then the fact that U_n^T given in (2.7) converge weakly to m_T in L² we obtain

(2.9)

Z

T²

a(z)m_T(z)dz = 0.

The next lemma shows that our measure has most of its mass on the set of rational directions:

Lemma 2.2. — Suppose that µ is defined by u_n satisfying (2.3). For m ∈ N define,

W^m:=

(

(z, ζ)∈T^∗T²

ζ = (p, q)

√p²+q², max(|p|,|q|)6m,(p, q)∈Z², gcd(p, q) = 1

)

, its complement,W_m:={W^m, and a measure µ_eT on T^∗T²:

(2.10)

Z

T^∗T²

A(z, ζ)dµ_eT :=

Z T 0

Z

T^∗T²

A(z, ζ)dµ(t, z, ζ), A∈ C_c^∞(T^∗T²).

Then,

(2.11) ∀ >0 ∃m such that µe_T(W_m)< . Proof. — We choose

(2.12) a_j ∈ C^∞(T²), a_j >0, lim

j→∞ka−a_jk_L² = 0.

(2.13) T hen

Z

T^∗T²

a_j(z)dµ_eT(z, ζ) =

Z

T²

(a_j(z)−a(z))m_T(z)dz =O(ka−a_jk_L²).

With the notation hbi_S(z, ζ) := _S^{1 R}₀^Sb(z +sζ)ds, b ∈ C^∞(T²), the last property in (2.5) and the fact that W_∞ is invariant under the flow shows that for anyS > 0,

Z

W∞

a_j(z, ζ)dµe_T(z, ζ) =

Z

W∞

ha_ji_S(z, ζ)dµe_T(z, ζ).

We note that

(2.14) W_m+1 ⊂W_m, W∞:=

∞

\

m=1

W_m ={(z, ζ)| |ζ|= 1, ζ ∈R²\Q²}.

For (z, ζ) ∈ W∞, unique ergodicity of the flow z 7→ z +sζ shows that hajiS → haji:=^R

T²aj(z)dz/(2π)². Fatou’s Lemma then shows that

Z

W∞

a_j(z)dµ_eT(z, ζ) = lim inf

S→∞

Z

W∞

ha_ji_S(z, ζ)dµ_eT(z, ζ)

>

Z

W∞

lim inf

S→∞ hajiS(z, ζ)dµeT(z, ζ) =µeT(W∞)haji.

Combining this with (2.13) shows that

µe_T(W∞)6 Cka−a_jk_L²

ha_ji →0, j→ ∞,

(since kak_L² > 0 and a > 0, ha_ji → hai > 0) which gives µe_T(W∞) = 0. But then (2.14) implies that limm→∞µeT(Wm) = µeT(W∞) = 0, concluding the proof.

2.2. Reduction to one dimension

We start with the following

Lemma 2.3. — Suppose that in (2.11) m₀ is large enough so that µe_T(W_m0)< T and that (z, ζ₀)∈supp(µe_T|_{Wm

0). Then there exists F ∈L²(T²) such that (2.15) µ_eT|_T²×{ζ₀} =F ⊗δ_ζ=ζ0, kFk_L²_(T²₎ 6= 0, F >0.

Proof. — Let π : T^∗T² → T² be the natural projection map, π(x, ξ) = x. Then, using (2.6) and (2.10), for any Lebesgue measurable set E ⊂T²,

(2.16) π∗(µ_eT|_T²_×{ζ0})(E)6π∗(µ_eT)(E) =

Z

E

m_T(z)dz, m_T ∈L².

The Radon–Nikodym theorem then shows that π∗(µe_T|_T²×{ζ₀}) = gm_T where g is measurable, mT-a.e. finite. The inequality (2.16) gives F := gmT 6 mT almost

everywhere which shows thatF ∈L².

From now on we fixF and ζ₀ such that (2.15) holds.

Using [BZ12, Lemma 2.7] (see also [BZ12, Figure 1]) we can assume (by changing the torus but not ∆_z) that ζ₀ = (0,1), z = (x, y), x∈R/A₁Z,y∈R/B₁Z, A₁/B₁ ∈ Q. Abusing the notation we will keep the notation u_n and µ for the transformed functions. Lemma 2.3 and the invariance property in (2.5) (which implies that F(x, y) = g(x) is independent ofy) show now that

(2.17)

(µeT|_T²×{(0,1)}) = g(x)dxdy⊗δ0(ξ)⊗δ1(η), g∈L²(T¹), g >0

Z

T²

g(x)a(x, y)dxdy= 0, kgk_L²_(T²₎ 6= 0.

Form which will be chosen later (independently of the m₀ used in Lemma 2.3) we chooseχ=χ_m ∈ C_c^∞(R²; [0,1]) supported near |ζ|= 1 and such that

(2.18) (T²×suppχ)∩{W_m ={(0,1)}, χ(0,1) = 1.

That is always possible as the set {ζ|(z, ζ)∈{W_m} is discrete (see the definitions in Lemma 2.2).

We then definev_n:= χ(hD_z)u_n andν :=|χ(ζ)|²µ6= 0. Definition (2.4) shows that ν is the semiclassical defect measure associated to v_n(t) := e^it∆v_n = χ(hD_z)e^it∆u_n which in particular shows that

(2.19) lim

n→∞kv_nk²_L2(T²) =T⁻¹ν([0, T]×T^∗T²) =T⁻¹

Z

T²

g(x)dxdy=:β >0.

The reduction to a one dimensional problem is based, as in [BZ12], on a Fourier expansion in y (assuming B₁ = 2π for notational simplicity):

(2.20) v_n(t)(x, y) := [e^it∆v_n](x, y) = ^X

k∈Z

[e^it∂x2v_n,k](x)e^−itk2+iky

We will now use a one dimensional analogue of Theorem 1.3 withW ∈L²(T¹). It will be proved in Section 2.3 below.

Lemma 2.4. — Suppose thatb ∈L¹(T¹), b>0,kbk_L¹ >0and that T > 0. Then there exists C such that for w∈L²(T¹),

(2.21) kwk²_L2(T¹)6C

Z T 0

Z

T¹

b(x)[e^it∂x2w](x)|²dxdt.

Let a_j be again given by (2.12). We apply (2.21) to (2.20) with b = hai_y :=

1 2π

R

T¹a(x, y)dy. This and (2.19) give, for n large enough (and using Theorem 1.4 to

pass froma to a_j in the third line), 0< ¹₂β <kv_nk²_L2(T²) = 2π^X

k∈Z

kv_n,kk²_L2(T¹)6C⁰

Z T 0

Z

T¹

hai_y(x)^X

k∈Z

|[e^it∂2xv_n,k](x)|²dxdt

=C

Z T 0

Z

T²

hai_y(x)|[e^it∆v_n](x, y)|²dxdydt

=C

Z T 0

Z

T²

ha_ji_y(x)|[e^it∆v_n](x, y)|²dxdydt+O(ka−a_jk_L²_(T²₎)

−→C

Z

T^∗T²

hajiy(x)dνeT(x, y, ξ, η) +O(ka−ajk_L²_(T²₎), n −→ ∞, where ν_eT =|χ(ξ, η)|²µeT (see (2.10)). In particular for everyj,

(2.22) 0< α6

Z

T^∗T²

ha_ji_y(x)dν_eT(x, y, ξ, η) +O(ka−a_jk_L²_(T²₎), α:= β C. We now decompose the integral in (2.22) as I₁+I₂ and use (2.17):

I₁ :=

Z

T²×{(0,1)}ha_ji_y(x)dν_eT(x, y, ξ, η) =

Z

T²

g(x)a_j(x, y)dxdy

=

Z

T²

g(x)(a_j(x, y)−a(x, y))dxdy 6√

2πkgk_L²_(T¹₎ka_j−ak_L²_(T²₎. (2.23)

We then use (2.18) to estimate the remainder:

I₂ :=

Z

(ξ,η)6=(0,1)ha_ji_y(x)dν_eT(x, y, ξ, η)6

Z

Wm

ha_ji_y(x)dν_eT(x, y, ξ, η) 6ka_jk_L^∞ν_eT(W_m).

We now combine these two estimates with (2.22) to obtain:

0< α6Kkaj−ak_L²_(T²₎+kajkL^∞νeT(Wm), where the constant K does not depends on χ and m.

Hence, we first choosej large enough so that Kka_j−ak_L²_(T²₎ < α/2 and then m large enough andχ satisfying (2.18) so that (2.11) gives

ka_jk_L^∞ν_eT(W_m)< ka_jk_L^∞ < α/2.

This provides a contradiction and proves Proposition 2.1.

2.3. One dimensional estimate

We now prove Lemma 2.4. The semiclassical part proceeds along the lines of the proof of Proposition 2.1. The derivation of (2.21) from the semiclassical estimate follows the same arguments needed in Section 3 and we will refer to that section for details.

Proof of Lemma 2.4. — We start with a semiclassical statement: for everyT there exist K, ρ0 and h0 such that for 0 < h < h0 and 0< ρ < ρ0 we have the analogue of (2.2):

(2.24) kπ_h,ρu₀k²_L2(T¹) 6K

Z T 0

Z

T¹

b(z)|e^it∆π_h,ρu₀(z)|²dzdt,

π_h,ρ(u₀) :=χ h²D_x²−1 ρ

!

u₀. We proceed by contradiction which leads to an analogue of (2.3) and then to a measureω_T analogous toµ_eT (see (2.10)) onT^∗T¹and satisfying: suppω_T ⊂ {ξ =±1},

∂_xω_T = 0, where the derivative is taken in the distributional sense.

From [BBZ13, Proposition 2.1]⁽¹⁾ and the argument in Lemma 2.3 (with weak convergence inL² replaced by the weak∗ convergence inL^∞ = (L¹)^∗) we obtain

dωT =^X

±

f±(x)dx⊗δ±1(ξ)dξ, f± ∈L^∞(T¹), f_±>0.

But the fact that∂_xω_T = 0 and the analogue of (2.9) show that f±(x) =c±>0, c₊+c−>0, (c₊+c−)

Z

T¹

b(x)dx= 0, which is a contradiction proving (2.24).

From the semiclassical estimate we obtain ku₀k_L²_(T¹₎ 6C

Z T 0

Z

T¹

b(z)|e^it∆u₀(z)|²dzdt+Cku₀k_H⁻²_(T¹₎.

That is done by the same argument recalled in Section 3.1 below. Finally the error term ku0k_H⁻¹_(T¹₎ is removed (see Section 3.2 for review of the procedure for doing

(applying [BBZ13, Proposition 2.1] again)).

3. Observability estimate

To prove Theorem 1.3 we first prove a weaker statement involving an error term:

Proposition 3.1. — Suppose that W ∈ L⁴(T²), a > 0 and kWk_L⁴ 6= 0. Then for any T >0, there exists K such that foru∈L²,

(3.1) kuk²_L2(T) 6C

Z T 0

Z

T²

|W(z)e^it∆u(z)|²dzdt+Ckuk²_H−2(T²).

(1)See https://math.berkeley.edu/~zworski/corr_bbz.pdffor a corrected version. That cor- rection is relevant only when treating the irrational case: see [BZ12] and [BBZ13]. That proposition is an easy one dimensional analogue of Theorem 1.4.

3.1. Dyadic decomposition

The proof of (3.1) uses a dyadic decomposition as in [BBZ13, Section 5.1] and [BZ12, Section 4] and we recall the argument adapted to the setting of this paper. For that let 1 =ϕ₀(r)²+^P∞_k=1ϕ_k(r)², where for k >1

ϕ_k(r) :=ϕ(R^−k|r|), R >1,

ϕ∈ C_c^∞((R⁻¹, R); [0,1]), (R⁻¹, R)⊂ {r|χ(r/ρ)> ¹₂},

withχandρsame as in (2.1) and (2.2). Then, we decomposeu₀ dyadically:ku₀k²_L2 =

P∞

k=0kϕ_k(−∆)u₀k²_L2, which will allow an application of Proposition 2.1.

Proof of Proposition 3.1. — Let ψ ∈ C_c^∞((0, T); [0,1]) satisfy ψ(t) > 1/2, on T /3< t <2T /3. Proposition 2.1 applied witha=W²,u₀ =e^{i(T /3)∆}Π_h,ρu₀ and with T replaced by T /3, shows that

(3.2) kΠ_h,ρu₀k²_L2 6K

Z

R

ψ(t)²kWe^it∆Π_h,ρu₀k²_L2(T²)dt, 0< h < h₀. TakingK large enough so that R^−K 6h₀ we apply (3.2) to the dyadic pieces:

ku₀k²_L2 = ^X

k∈Z

kϕ_k(−∆)u₀k²_L2

6

K

X

k=0

kϕ_k(−∆)u₀k²_L2 +C

∞

X

k=K+1

Z T 0

ψ(t)²kW ϕ_k(−∆) e^it∆u₀k²_L2(T²)dt

=

K

X

k=0

kϕk(−∆)u₀k²_L2 +C

∞

X

k=K+1

Z

R

kψ(t)W ϕk(Dt) e^it∆u₀k²_L2(T²)dt.

In the last equality we used the equation and replacedϕ(−∆) by ϕ(D_t).

We need to consider the commutator of ψ ∈ C_c^∞((0, T)) and ϕ_k(D_t) =ϕ(R^−kD_t).

Ifψ^e ∈ C_c^∞((0, T)) is equal to 1 on suppψ then the semiclassical pseudo-differential calculus withh=R^−k (see for instance [Zwo12, Chapter 4]) gives

(3.3) ψ(t)ϕ_k(D_t) =ψ(t)ϕ_k(D_t)ψ(t) +^e E_k(t, D_t), ∂^αE_k=O(hti^−Nhτi^−NR^{−N k}), for all N and uniformly ink.

The errors obtained from E_k can be absorbed into the ku₀k_H⁻²_(T²₎ term on the right-hand side. Hence we obtain,

ku₀k²_L2 6Cku₀k²_H−2(T²)+C

∞

X

k=0

Z T 0

kψ(t)ϕ_k(D_t)ψ(t)W^e e^it∆u₀k²_L2(T²)dt 6Cku^e ₀k²_H−2(T²)+K

∞

X

k=0

hϕ_k(D_t)²ψ(t)^e We^it∆u₀, ψ(t)²ψ(t)^e We^it∆u₀,i_L²_(Rt×T²₎ 6Cku^e ₀k²_H⁻²_(T2)+K

Z

R

kψ(t)^e We^it∆u₀k²_L2(T²)dt 6Cku^e ₀k²_H⁻²_(T2)+K

Z _T

0

kWe^it∆u₀k²_L2(T²)dt,

where the last inequality is (3.1) in the statement of the proposition.

3.2. Elimination of the error term

We now eliminate the error term on the right hand side of (3.1). For that we adapt the now standard method of Bardos–Lebeau–Rauch [BLR92] just as we did at the end of [BZ12, Section 4]. The argument recalled there shows that if

(3.4) N :={u∈L²(T²)|W e^it∆u≡0 on (0, T)×T²}

is non-trivial then sinceiW e^it∆∆u=∂tW e^it∆u≡0 on (0, T)×T², thenN is invariant by the action of ∆, and hence it contains a nontrivialw∈L²(T²) such that for some λ,

(−∆−λ)w= 0, W w≡0.

But then w is a trigonometric polynomial vanishing on a set of positive measure which implies that w≡0. Hence

(3.5) N ={0}.

Proof of Theorem 1.3. — Suppose the conclusion (1.4) were not to valid. Then there exists a sequenceu_n ∈L²(T²) such that

(3.6) ku_nk_L²_(T²₎ = 1, kW e^it∆u_nk_L²_((0,T)×T²) →0, n → ∞.

By passing to a subsequence we can then assume thatu_nconverging weakly inL²(T²) and strongly inH⁻²(T²) to some u∈L². From Proposition 3.1 we would also have

1 = kunk²_L2(T²) 6C

Z T 0

kW e^it∆unk²_L2(T²)dt+Ckunk²_H⁻²_(T2). Hence,

(3.7) 16C lim

n→∞ku_nk_H⁻²_(T²₎ =Ckuk_H⁻²_(T²₎ =⇒ u6≡0.

Let W_j ∈ C^∞(T²) satisfy kW −W_jk_L⁴_(T²₎ → 0. For ϕ ∈ C_c^∞((0, T)×T²), due to distributional convergence, Theorem 1.4 and (3.6),

|hW_je^it∆u, ϕi|= lim

n→∞|he^it∆u_n, W_jϕi|

6 lim

n→∞

|h(Wj −W)e^it∆un, ϕi|+hW e^it∆un, ϕi

6kϕk_L²k(W_j−W)e^it∆u_nk_L²((0,T)×T²) 6Ckϕk_L²kW_j −Wk_L⁴_(T²₎. On the other hand the same argument shows that

|hW e^it∆u, ϕi|6|hW_je^it∆u, ϕi|+Ckϕk_L²kW_j−Wk_L⁴_(T²₎. Combining the two inequalities we see that

|hW e^it∆u, ϕi|6C lim

j→∞kW_j−Wk_L⁴_(T²₎ = 0

which means that W e^it∆u≡0. Thus u∈N given by (3.4) and by (3.5), u= 0. This

contradicts (3.7) completing the proof.

4. The HUM method: proofs of Theorems 1.1 and 1.2

We now show the equivalence of the stabilization, control and observability prop- erties in our context. The proof is a variation on the classical HUM method [Lio88], but since our damping and localization functions arenot in L^∞it requires additional care.

Proposition 4.1. — The following are equivalent (for fixed T >0).

(1) Let a ∈ L²(T²;R),kak_L² > 0. For any u₀ ∈ L²(T²) there exists f ∈ L⁴(T²;L²(0, T)) such that the solution u of (1.1) satisfies u|_t=T = 0

(2) Let a ∈ L⁴(T²;R),kak_L⁴ > 0. For any u₀ ∈ L²(T²) there exists f ∈ L²((0, T)×T²) such that the solution u of (1.1) satisfies u|_t=T = 0

(3) Let a ∈ L⁴(T²;R),kak_L⁴ > 0. Then there exists C > 0 such that for any v₀ ∈L²(T²),

(4.1) kv₀k²_L2(T²) 6Ckae^it∆v₀k_L²_((0,T)×T²₎.

Proof. — We first prove that (2) and (3) are equivalent, and for that we follow the HUM method. Define the map

R :f ∈L²((0, T)×T²)7→Rf =u|_t=0, where uis the solution of the final value problem

(i∂t+ ∆)u(z) = a(x)1_(0,T)f ∈L^4/3(T²;L²(0, T)), u|T=0 = 0.

By Theorem 1.4 R:L²(T²;L²(0, T))→L²(T²) and

(4.2) (2) ⇐⇒R(L²(T²;L²(0, T))) =L²(T²).

Again by Theorem 1.4,e^it∆v₀ ∈L⁴(T²;L²(0, T)) for v₀ ∈L²(T²), we define S :v₀ ∈L²(T²)7→1_(0,T)×ae^it∆v₀ ∈L²((0, T)×T²),

and

(4.3) (3)⇐⇒ ∃K ∀v₀ ∈L²(T²) kv₀k_L²_(T²₎6KkSv₀k_L²_((0,T)×T²). To relate R and S we integrate by parts:

Z T 0

Z

T²

af vdxdt=

Z T 0

Z

T²

(i∂_t+ ∆)uvdxdt =i

Z

T²

uvdx

T 0

+

Z T 0

Z

T²

u(i∂_t+ ∆)vdxdt

=−i

Z

T²

uvdx|_t=0, which is the same as

(4.4) f, Sv0

L²((0,T)×T²) =−iRf, v0

L²(T²).

Let us assume (2). By (4.2) and the closed graph theorem there exists η > 0 such that the image of the unit ball in L²((0, T) ×T²) by R contains the ball {v₀ ∈L²(T²)| kv₀k_L² 6η}. Hence for allv₀ ∈L²(T²) there existsf ∈L²((0, T)×T²), such that

kfk_L²_((0,T)×T²₎6 1

ηkv₀k_L², Rf =v₀.