Stabilization and approximate null-controllability for a large class of diffusive equations from thick control supports

HAL Id: hal-03104300

https://hal.archives-ouvertes.fr/hal-03104300v2

Preprint submitted on 2 May 2021

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished 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.

large class of diffusive equations from thick control supports

Paul Alphonse, Jérémy Martin

To cite this version:

Paul Alphonse, Jérémy Martin. Stabilization and approximate null-controllability for a large class of diffusive equations from thick control supports. 2021. �hal-03104300v2�

FOR A LARGE CLASS OF DIFFUSIVE EQUATIONS FROM THICK CONTROL SUPPORTS

PAUL ALPHONSE AND JÉRÉMY MARTIN

Abstract. We prove that the thickness property is a necessary and sufficient geometric condition that ensures the (rapid) stabilization or the approximate null-controllability with uniform cost of a large class of evolution equations posed on the whole space Rⁿ. These equations are associated with operators of the formF(|Dx|), the function F : [0,+∞) →Rbeing continuous and bounded from below. We also provide explicit feedbacks and constants associated with these stabilization properties. The notion of thickness is known to be a necessary and sufficient condition for the null-controllability of the fractional heat equations associated with the functions F(t) = t^2s in the case s > 1/2. Our results apply in particular for this class of equations, but also for the half heat equation associated with the function F(t) = t, which is the most diffusive fractional heat equation for which null-controllability is known to fail from general thick control supports.

1. Introduction

This paper is devoted to investigate the stabilization and approximate null-controllability for control systems of the following form:

(E_F)

(∂_tf(t, x) +F(|D_x|)f(t, x) =h(t, x)¹_ω(x), t >0, x∈Rⁿ,

f(0,·) =f₀∈L²(Rⁿ),

where the operatorF(|D_x|)is the Fourier multiplier associated with the symbolF(|ξ|), with

| · | the canonical Euclidean norm inRⁿ, the function F : [0,+∞) →R being continuous and bounded from below, andω ⊂Rⁿ is a Borel set with positive Lebesgue measure.

The study of the (rapid) stabilization and the (approximate) null-controllability of evolu- tion equations of the form (E_F) has been much addressed recently [2,3,9,12,16,17,19,21].

The Schrödinger counterparts of these equations and the same equations posed on bounded domains have also been studied, respectively in [18] and [15,23]. In this work, we consider control supportsω⊂Rⁿ which are thick:

Definition 1.1. Givenγ ∈(0,1) andL >0, the set ω⊂Rⁿ is said to beγ-thick at scale Lwhen it is measurable and satisfies

∀x∈Rⁿ, Leb(ω∩(x+ [0, L]ⁿ))≥γLⁿ, where Lebdenotes the Lebesgue measure inRⁿ.

This notion of thickness has appeared to play a key role in the null-controllability theory since the works [7, 22], where the authors established that it is a necessary and sufficient geometric condition that ensures the null-controllability of the heat equation posed onRⁿ, which is the equation (E_F) associated with the functionF(t) =t². The same phenomenon holds true more generally for the evolution equations associated with fractional Laplacians (−∆)^s (case where F(t) = t^2s) in the same setting and when s > 1/2, as proven in [3], and also quite surprisingly for the Schrödinger counterparts of these equations in the one dimensional setting and when s ≥ 1/2, see [18] (Corollary 2.8). It is also known from

2020 Mathematics Subject Classification. 93D15, 93B05, 35R11, 26E10.

Key words and phrases. Stabilization, Approximate null-controllability, Thick sets, Quasi-analytic se- quences, Diffusive equations.

1

the works [11, 12] that in the case 0 < s ≤ 1/2, the fractional heat equation (E_t^2s) is not anymore null-controllable from thick control supports in general. Other classes of degenerate parabolic equations of hypoelliptic type, as evolution equations associated with accretive quadratic operators or (non-autonomous) Ornstein-Uhlenbeck operators, were also proven to be null-controllable from thick control supports, see e.g. [2,4, 5]. In this work, we prove that a very general class of equations of the form (E_F) is approximately null-controllable with uniform cost from the control support ω ⊂ Rⁿ if and only if ω is a thick set. Our results hold in particular for the half heat equation (E_t) which is not null-controllable from general thick control supports (at least whenn= 1, the case n≥2 remaining open for the moment), see e.g. [12] (Theorem 2.3) or [16] (Theorem 1.1).

The study of the (rapid) stabilization of the control system (EF), as for it, has been addressed very recently in the works [9, 17, 21]. It has been proven in [9] (Theorem 1.1) that for alls >0, the fractional heat equation (E_t^2s) is exponentially stabilizable from the supportω if and only if ω is thick. It is also known from [17] (Example 1) that the very same equation (E_t^2s) is rapidly stabilizable from complements of Euclidean balls in Rⁿ when0< s <1. In this paper, we establish that the control system (E_F) is exponentially stabilizable from ω if and only if ω is a thick set when infF ≤ 0 (in the case infF > 0, the control system (EF) is stable) and lim inf+∞F > −infF. Moreover, we provide explicit formulas for the feedbacksK and the constants associated with this stabilization, which allows us to prove that when lim_+∞F = +∞, the control system (E_F) is rapidly stabilizable from ω if and only if ω is thick. In particular, we recover [9] (Theorem 1.1) (with new explicit feedbacks) and we generalize [17] (Example 1). We also prove that when ωis not dense in Rⁿ, the equation (E_F) is never rapidly stabilizable in the particular case where F admits a finite limit at+∞.

In a nutshell, our results highlight the importance of the notion of thickness not only in the null-controllability theory, but also for properties of stabilization and approximate null- controllability with uniform cost, as it turns out to be a necessary and sufficient geometric condition ensuring these two properties for a large class of diffusive equations (E_F).

Outline of the work. In Section 2, we present in details the main results contained in this work. Section3is devoted to the proofs of the results concerning the stabilization and the rapid stabilization of the control system (E_F). Basic properties of quasi-analytic sequences are presented in Section4, which allow to establish the results concerning the approximate null-controllability of the evolution equation (E_F) in Section 5. Finally, Section 6 is an Appendix concerning the proof of an observability result used in Section5.

Notations. The following notations and conventions will be used all over this work:

1. The canonical Euclidean scalar product of Rⁿ is denoted by · and | · | stands for the associated canonical Euclidean norm.

2. For all measurable subsetsω⊂Rⁿ, the inner product ofL²(ω) is defined by hu, viL²(ω)=

Z

ω

u(x)v(x) dx, u, v∈L²(ω),

whilek · kL²(ω) stands for the associated norm. Moreover, L(L²(ω)) stands for the set of bounded operators onL²(ω).

3. For all functionu∈S₍Rⁿ), the Fourier transform ofu is denotedbuorFu, and defined by

b

u(ξ) = (F_{u)(ξ) =} Z

Rⁿ

e^−ix·ξu(x) dx, ξ∈Rⁿ. With this convention, Plancherel’s theorem states that

∀u∈L²(Rⁿ), kbukL²(Rⁿ)= (2π)^n/2kukL²(Rⁿ).

4. We denote by ∇x the gradient and we set D_x = −i∇x. Moreover, F(|D_x|) stands for the Fourier multiplier associated with the symbolF(|ξ|) for all continuous function F : [0,+∞)→R.

5. For all measurable subsetsω⊂Rⁿ,¹_ω stands for the characteristic function ofω.

2. Statement of the main results

This section is devoted to present in details the main results contained in this work. Let us begin by defining precisely the different concepts related to the control system (E_F) we are interested in:

(i) The control system (E_F) is said to benull-controllable from the control support ω in time T > 0 when for all f₀ ∈ L²(Rⁿ), there exists a control h ∈ L²((0, T)×ω) such that the mild solution of (EF) satisfies f(T,·) = 0.

(ii) The control system (E_F) is said to beapproximately null-controllable from the con- trol supportω in time T >0if for all ε >0and f0 ∈L²(Rⁿ), there exists a control h∈L²((0, T)×ω)such that the mild solution of (E_F) satisfies kf(T,·)kL²(Rⁿ)≤ε.

(iii) The control system (E_F) is said to beapproximately null-controllable with uniform cost from the control support ω in time T > 0 if for all ε > 0, there exists a positive constant C_ε,T > 0 such that for all f₀ ∈ L²(Rⁿ), there exists a control h ∈L²((0, T)×ω) such that the mild solution of (E_F) satisfieskf(T,·)kL²(Rⁿ)≤ε with moreover Z T

0 kh(t,·)k²_L²_(ω) dt≤C_ε,Tkf₀k²_L²₍Rⁿ).

(iv) The control system (E_F) is said to be exponentially stabilizable from the control support ω at rate α > 0 if there exist a positive constant M_α ≥1 and a feedback K_α ∈ L(L²(Rⁿ)) such that for allt≥0,

(2.1) e^{−t(F(|Dx|)+1ωKα)}

L(L²(Rⁿ))≤M_αe^−αt.

When the feedback K_α can be chosen equal to zero, the control system (E_F) is said to be stable. The existence of the semigroup generated by the operatorF(|D_x|) +

1ωK_α is ensured by the theory of bounded perturbation of semigroups, see e.g. [8]

(Theorem III.1.3).

(v) The control system (E_F) is said to be rapidly stabilizable from the control support ω if it is exponentially stabilizable from ω at any rateα >0.

2.1. Stabilization. First of all, we are interested in tackling stabilization issues for the evolution system (EF). Let us begin by noticing that when infF > 0, we get from Plancherel’s theorem that

∀t≥0, e^−tF(|Dx|)

L(L²(Rⁿ))≤e^−(infF)t,

so the control system (EF) is stable. The interesting case is therefore when infF ≤ 0.

In this case, we prove that the thickness of the supportω ⊂Rⁿ is a necessary geometric condition that ensures the stabilization of the equation (EF), and a sufficient one when assuming in addition that lim inf_+∞F > |infF|. We also provide explicit feedbacks and quantitative estimates associated with this stabilization.

Theorem 2.1. Let F : [0,+∞) → R be a continuous function bounded from below and ω⊂Rⁿ be a measurable set.

(i) If infF ≤0and the evolution system (E_F) is exponentially stabilizable fromω, then the set ω is thick.

(ii) When lim inf+∞F >infF and ω is a thick set, then there exist some positive con- stants C=C(ω)≥1 and R₀ =R₀(F)>0 such that for all R≥R₀ and t≥0, (2.2) e^{−t(F(|Dx|)+CeCR(αR−infF)1ωKR)}

L(L²(Rⁿ))≤√

2Ce^CRe^{−(αR+infF)t/2},

where we set α_R = inf_r≥RF(r) and where K_R stands for the following orthogonal projection

KR:L²(Rⁿ)→

f ∈L²(Rⁿ) : Suppfb⊂B(0, R) ,

withB(0, R) the closed Euclidean ball centered in 0 with radius R >0.

(iii) When lim inf_+∞F >|infF|andω is a thick set, then the evolution system (E_F) is exponentially stabilizable from ω.

Let us check that the assertion (iii) in Theorem2.1is a straightforward consequence of the assertion (ii) in the same result. Obviously, the assumption lim inf_+∞F >|infF|is read aslim inf_+∞F >infF andlim inf_+∞F >−infF. On the one hand, the assumption lim inf+∞F >infF implies that the estimates (2.2) hold according to the assertion (ii).

On the other hand, assuming thatlim inf_+∞F >−infF, we get thatα_R+ infF >0when R≫ 1 is large enough. The estimates (2.2) then imply that the evolution equation (E_F) is exponentially stabilizable, according to the definition (2.1) presented in the beginning of this section.

By gathering the results stated in the assertions(i)and(iii)in Theorem2.1, we directly obtain the following corollary:

Corollary 2.2. LetF : [0,+∞)→Rbe a continuous function bounded from below satisfy- inginfF ≤0 and lim inf_+∞F >−infF, and ω ⊂Rⁿ be a measurable set. The evolution system (EF) is exponentially stabilizable fromω if and only ifω is thick.

It is a very interesting issue to know whether the control system (E_F) is exponentially stabilizable wheninfF ≤0andlim inf+∞F ≤ −infF. We shall not tackle such a question in this work.

As a consequence of the quantitative stabilization estimates (2.2), we directly obtain the following result concerning the rapid stabilization of the evolution system (E_F) under the assumption lim_+∞F = +∞, by applying Theorem 2.1to the function F−infF.

Corollary 2.3. Let F : [0,+∞) → R be a continuous function bounded from below sat- isfying lim_+∞F = +∞, and ω ⊂Rⁿ be a measurable set. The evolution system (E_F) is rapidly stabilizable fromω if and only ifω is thick.

Example 2.4. For all positive real numbers s > 0, let us consider the function F_s : [0,+∞) → [0,+∞) defined for all t ≥ 0 by Fs(t) = t^2s. We also consider ω ⊂ Rⁿ a measurable set with positive Lebesgue measure. It follows from Corollaries2.2and2.3that the associated control system (E_Fs) is exponentially stabilizable from the control support ω if and only if it is rapidly stabilizable from ω if and only if ω is a thick set. Moreover, we deduce from Theorem 2.1that when ω is thick, there exist a positive constant C ≥1 andR0>0such that for all R≥R0 and t≥0,

e^{−t((−∆)s+CeCRR2s1ωKR)}

L(L²(Rⁿ))≤√

2Ce^CRe^−R2st/2,

where K_R stands for the following orthogonal projection K_R:L²(Rⁿ)→

f ∈L²(Rⁿ) : Suppfb⊂B(0, R) .

These explicit stabilization estimates allow to recover [9] (Theorem 1.1) and also to gen- eralize [17] (Example 1).

In the case where lim+∞F <+∞, we only provide a necessary condition for the control system (E_F) to be rapidly stabilizable. The following result implies in particular that when the functionF has a finite limit in+∞ and when the supportω ⊂Rⁿ is not dense inRⁿ, then the equation (E_F) is not rapidly stabilizable fromω.

Proposition 2.5. Let F : [0,+∞)→R be a continuous function bounded from below and ω⊂Rⁿ be a measurable set which is not dense inRⁿ. We assume thatlim_+∞F exists and is a non-negative real number (the function F is therefore bounded). Then, if α >supF, the equation (EF) is not exponentially stabilizable from ω at rate α.

2.2. Cost-uniform approximate null-controllability. In the second part of this work, we study the cost-uniform approximate null-controllability of the equations (E_F). We will not address this question for general continuous functions F bounded from below, but only for the ones generating aquasi-analytic sequence. Let us precisely define this class of functions. Associated with the functionF is the following log-convex sequenceM^F whose elementsM_k^F, assumed to be positive real numbers, are defined by

(2.3) 0< M_k^F = sup

r≥0

r^ke^−F(r)<+∞, k≥0.

We say that the sequence M^F is quasi-analytic when, for all real numbers a < b, the associated Denjoy-Carleman class

CM^F([a, b]) =

f ∈ C^∞([a, b],C) :∀k≥0,∀x∈[a, b], |f^(k)(x)| ≤M_k^F , is quasi-analytic, meaning that any functionf in this class satisfying

∃x₀ ∈[a, b],∀k≥0, f^(k)(x₀) = 0,

is identically equal to zero. We refer to Section 4 where the notion of quasi-analytic sequence is discussed, and where Denjoy-Carleman’s theorem is presented, giving a useful characterization of such sequences.

When the functionF generates a quasi-analytic sequenceM^F of positive real numbers, the solutions of the homogeneous counterpart of the equation (EF) belong to quasi-analytic classes of functions (see Subsection 5.1 for more details). By taking advantage of this quasi-analytic regularity, we prove that the notion of thickness is a necessary and sufficient geometric condition that ensures the cost-uniform approximate null-controllability of the evolution equations (E_F) in any positive time.

Theorem 2.6. Let F : [0,+∞) → R be a continuous function bounded from below and ω⊂Rⁿbe a measurable set. We assume that the sequence M^F associated with the function F defined in (2.3) is a quasi-analytic sequence of positive real numbers. Then, for all positive time T > 0, the diffusive equation (E_F) is cost-uniformly approximately null- controllable from the control supportω in time T if and only if ω is thick.

The necessary part of Theorem 2.6 is a consequence of the fact that the cost-uniform approximate null-controllability implies rapid stabilization, see Proposition 5.3, and is therefore a consequence of Corollary2.3(notice that the assumption onF in Theorem2.6 implies in particular thatlim_+∞F = +∞.)

Let us now present explicit examples of functionsF generating quasi-analytic sequences M^F and for which Theorem 2.6therefore applies.

Example 2.7. Let us assume that the non-negative continuous function F : [0,+∞) → [0,+∞) satisfies Θ ≤ F, where the weight Θ : [0,+∞) → [0,+∞) verifies the following properties:

(i) Θ(0) = 0 and Θis non-decreasing with lim_+∞Θ = +∞, (ii) Θis lower-semicontinuous and t∈R7→Θ(e^t) is convex, (iii)

Z +∞

0

Θ(t)

1 +t² dt= +∞.

It follows from the work [10] (see also Proposition4.6in the present work) that the sequence M^Θ associated with the weight Θ, defined in (2.3), is quasi-analytic. Moreover, we have M^F ≤ M^Θ, since Θ ≤ F, and Lemma 4.4 implies that the sequence M^F is also quasi- analytic. We deduce that for all Borel setω⊂Rⁿand all positive timeT >0, the equation (E_F) is cost-uniformly approximately null-controllable from the setω in timeT if and only ifωis thick. A relevant particular example is whenF(t) = Θ(t) =t. Indeed, the associated evolution equation is the half heat equation (Et) posed on the whole space associated with the operator√

−∆, known to be not null-controllable from any non dense control support

ω (at least when n = 1), see [12] (Theorem 2.3) or [16] (Theorem 1.1). This evolution equation is then a relevant example where the thick condition fails to be sufficient for the (strong) null-controllability but appears to be necessary and sufficient for the cost-uniform approximate null-controllability.

Actually, we are able to derive cost-uniform approximate null-controllability results for much less diffusive equations than the half heat equation (E_t), as illustrated in the two following examples.

Example 2.8. Let s≥1, 0≤δ ≤1 be non-negative real numbers and F_s,δ : [0,+∞) → [0,+∞)be the non-negative continuous function defined for all t≥0by

F_s,δ(t) = t^s log^δ(e+t).

We check in Corollary 4.8 that the associated sequenceM^Fs,δ defined in (2.3) is a quasi- analytic sequence of positive real numbers. Therefore, for all Borel set ω ⊂ Rⁿ and all positive timeT >0, the equation (E_Fs,δ) is cost-uniformly approximately null-controllable from the setω in time T if and only ifω is thick.

Example 2.9. Let p ≥ 1 be a positive integer and F_p : [0,+∞) → [0,+∞) be the non- negative continuous function defined for allt≥0 by

F_p(t) = t

g(t)(g◦g)(t)...g^◦p(t), where g(t) = log(e+t),

withg^◦p = g◦. . .◦g (p compositions). We check in Proposition 4.7 that the associated sequenceM^Fpdefined in (2.3) is quasi-analytic. As a consequence, for all Borel setω ⊂Rⁿ and all positive time T > 0, the equation (E_Fp) is cost-uniformly approximately null- controllable from the set ω in time T if and only ifω is thick.

Regarding the weaker notion of approximate null-controllability presented at the be- ginning of this section, the geometry of the allowed control support is much simpler. In- deed, the following proposition ensures that the control system (E_F) is approximately null-controllable in any positive time T > 0 and from any measurable set ω ⊂ Rⁿ with positive Lebesgue measure whenF generates a log-convex quasi-analytic sequenceM^F. Proposition 2.10. Let F : [0,+∞) → R be a continuous function bounded from below and ω ⊂ Rⁿ be a measurable set with positive Lebesgue measure. If the sequence M^F associated with the function F, defined in (2.3), is a quasi-analytic sequence of positive real numbers, then for all positive timeT >0, the diffusive equation (E_F) is approximately null-controllable from the support control ω in timeT.

In particular, diffusive equations discussed in Examples 2.7, 2.8 and 2.9 are approxi- mately null-controllable in any positive time T > 0 from any measurable subset ω ⊂Rⁿ satisfyingLeb(ω)>0.

3. (Rapid) Stabilization of diffusive equations

The aim of this section is to prove Theorem 2.1 and Proposition 2.5 concerning the stabilization and the rapid stabilization properties of the following general control system (E_F)

(∂_tf(t, x) +F(|D_x|)f(t, x) =h(t, x)¹_ω(x), t >0, x∈Rⁿ, f(0,·) =f0∈L²(Rⁿ),

where F : [0,+∞) → R is a continuous function bounded from below and ω ⊂ Rⁿ is a measurable set.

3.1. Proof of Theorem 2.1: assertion (i). First of all, let us assume that the equation (E_F) is exponentially stabilizable from the set ω, with the additional assumption that infF ≤ 0. We aim at proving that the control support ω is then thick. To that end, we will use the following nice characterizations of exponential stabilization in terms of observability estimates:

Theorem 3.1 (Theorem 1 in [21]). The following assertions are equivalent:

(i) The evolution system (E_F) is exponentially stabilizable fromω.

(ii) For all ε∈(0,1), there exist T >0 and C >0 such that for allg∈L²(Rⁿ), e^{−T F(|Dx|)}g²

L²(Rⁿ)≤C Z T

0

e^−tF(|Dx|)g²

L²(ω) dt+εkgk²_L²₍Rⁿ). (iii) There exist ε∈(0,1), T >0 and C >0 such that for all g∈L²(Rⁿ),

e^{−T F(|Dx|)}g²

L²(Rⁿ)≤C Z T

0

e^−tF(|Dx|)g²

L²(ω) dt+εkgk²_L²₍Rⁿ).

According to the above theorem, assuming that the equation (E_F) is exponentially stabilizable fromω is equivalent to assuming that there existε∈(0,1), T >0and C >0 such that for allg∈L²(Rⁿ),

(3.1) e^{−T F(|Dx|)}g²

L²(Rⁿ)≤C Z T

0

e^−tF(|Dx|)g²

L²(ω) dt+εkgk²L²(Rⁿ).

The strategy consists in applying this observability estimate for well-chosen functions g∈L²(Rⁿ). This approach has especially been used in the works [2,3,4,9], in which expo- nential stabilization or null-controllability issues are studied for fractional heat equations or evolution equations associated with (non)-autonomous Ornstein-Uhlenbeck operators posed on the whole space Rⁿ.

Fixing x₀ ∈ Rⁿ and considering ξ₀ ∈ Rⁿ together with l ≫ 1 whose values will be adjusted later, we consider the Gaussian function g_l,ξ0 defined by

∀x∈Rⁿ, g_l,ξ0(x) = 1 lⁿexp

ix·ξ₀−|x−x₀|² 2l²

.

Classical results concerning Fourier transform of Gaussian functions show that (3.2) ∀ξ∈Rⁿ, bg_l,ξ0(ξ) = (2π)^n/2exp

−ix0·(ξ−ξ0)−l²|ξ−ξ₀|² 2

.

On the one hand, it follows from Plancherel’s theorem that the left-hand side of the in- equality (3.1) applied to the functionsg_l,ξ0 is a positive constant independent of the point x₀, denotedδ_l,ξ0 >0in the following and given by

δ_l,ξ0 =e^{−T F(|Dx|)}g_l,ξ0

²

L²(Rⁿ) = Z

Rⁿ

e^{−ix0·(ξ−ξ0)}e^{−T F(|ξ|)}e^{−l2|ξ−ξ0|2/22} dξ (3.3)

= 1 lⁿ

Z

Rⁿ

e^{−T F(|ξ/l+ξ0|)}e^−|ξ|2/22 dξ >0.

On the other hand, we get that theL²-norm of the function g_l,ξ0 also does not depend on the pointx₀ ∈Rⁿ and is given by the following Gaussian integral

(3.4) kg_l,ξ0k²L²(Rⁿ)= 1 l²ⁿ

Z

Rⁿ

e^−|x|2/l2 dx= π

l² n/2

.

Let us check that the pointξ₀ ∈Rⁿand the large positive parameterl≫1can be adjusted so thatδ_l,ξ0 −εkg_l,ξ0k²_L²₍Rⁿ)>0, that is, by (3.3) and (3.4),

(3.5)

Z

Rⁿ

e^{−T F(|ξ/l+ξ0|)}e^−|ξ|2/22 dξ > επ^n/2.

Since ε ∈ (0,1) and the function F satisfies infF ≤ 0, we can assume that the point ξ₀∈Rⁿis chosen in order to satisfye^{−2T F(|ξ0|)} > ε. Since the functionF is bounded from below, the dominated convergence theorem then implies that

l→+∞lim Z

Rⁿ

e^{−T F(|ξ/l+ξ0|)}e^−|ξ|2/22 dξ =e^{−2T F(|ξ0|)} Z

Rⁿ

e^−|ξ|2/22 dξ

=e^{−2T F(|ξ0|)}π^n/2 > επ^n/2.

The parameterl≫1 can therefore be adjusted so that (3.5) holds. The values of ξ₀ ∈Rⁿ andl≫1are now fixed. We therefore deduce from (3.1) and (3.5) that

(3.6) M_l,ξ0 ≤C Z T

0

e^−tF(|Dx|)g_l,ξ0

²

L²(ω) dt with M_l,ξ0 =δ_l,ξ0 −εkg_l,ξ0k²L²(Rⁿ) >0.

Moreover, by introducing F⁻¹

ξ the partial inverse Fourier transform with respect to the variable ξ∈Rⁿand using (3.2), the right-hand side of this inequality (up to the constant C) writes as

Z T 0

e^−tF(|Dx|)g_l,ξ0

²_L2(ω) dt= (2π)ⁿ Z T

0

Z

ω

F⁻¹

ξ (e^{−ix0·(ξ−ξ0)}e^−tF(|ξ|)e^{−l2|ξ−ξ0|2/2})(x)² dxdt

= (2π)ⁿ Z T

0

Z

ω

F⁻¹

ξ (e^−tF(|ξ|)e^{−l2|ξ−ξ0|2/2})(x−x₀)² dxdt

= (2π)ⁿ Z T

0

Z

ω−x⁰

F⁻¹

ξ (e^−tF(|ξ|)e^{−l2|ξ−ξ0|2/2})(x)² dxdt.

Givenr >0a positive radius whose value will be chosen later, we split the previous integral in two parts and obtain the following estimate:

(3.7) Z T

0

e^−tF(|Dx|)g_l,ξ0

²

L²(ω) dt

≤(2π)ⁿ Z T

0

Z

(ω−x⁰)∩[−r,r]ⁿ

F⁻¹

ξ (e^−tF(|ξ|)e^{−l2|ξ−ξ0|2/2})(x)² dxdt + (2π)ⁿ

Z T 0

Z

|x|>r

F⁻¹

ξ (e^−tF(|ξ|)e^{−l2|ξ−ξ0|2/2})(x)² dxdt.

Now, we study one by one the two integrals appearing in the right-hand side of (3.7). First, notice that for all0≤t≤T,

F⁻¹

ξ (e^−tF(|ξ|)e^{−l2|ξ−ξ0|2/2})

L^∞(Rⁿ)≤ 1 (2π)ⁿ

e^−tF(|ξ|)e^{−l2|ξ−ξ0|2/2}

L¹(Rⁿ)

≤ e^−TinfF (2π)ⁿ

e^{−l2|ξ−ξ0|2/2}

L¹(Rⁿ)

= e^−TinfF (2π)ⁿ

2π l²

n/2

.

It therefore follows from the invariance by translation of the Lebesgue measure that (3.8) (2π)ⁿ

Z T 0

Z

(ω−x⁰)∩[−r,r]ⁿ

F⁻¹

ξ (e^−tF(|ξ|)e^{−l2|ξ−ξ0|2/2})(x)² dxdt

≤ e^−2TinfF l²ⁿ

Z T 0

Leb (ω−x₀)∩[−r, r]ⁿ

dt= T e^−2TinfF

l²ⁿ Leb ω∩(x₀+ [−r, r]ⁿ) .

In order to control the second integral, we use the dominated convergence theorem which justifies the following convergence

Z T 0

Z

|x|>r

F⁻¹

ξ (e^−tF(|ξ|)e^{−l2|ξ−ξ0|2/2})(x)² dxdt_r→+∞→ 0, since

F⁻¹

ξ (e^−tF(|ξ|)e^{−l2|ξ−ξ0|2/2})∈L²([0, T]×Rⁿ).

Thus, we can choose the radius r≫1large enough so that (3.9) (2π)ⁿC

Z T 0

Z

|x|>r

F⁻¹

ξ (e^−tF(|ξ|)e^{−l2|ξ−ξ0|2/2})(x)² dxdt≤ M_l,ξ0

2 . Gathering (3.6), (3.7), (3.8) and (3.9), we obtain the following estimate

∀x₀ ∈Rⁿ, M_l,ξ0

2 ≤ C

l²ⁿT e^−2TinfFLeb ω∩(x₀+ [−r, r]ⁿ) . This proves that the control supportω is actually a thick set.

3.2. Proof of Theorem 2.1: assertion (ii). In this second subsection, assuming that ω is a thick set and that lim inf_+∞F > infF, we establish the estimate (2.2). The strategy consists in constructing an adapted Lyapunov function, inspired by the proof of [23] (Theorem 1.1) in which the author studies the stabilization properties of the heat equation posed on bounded domains of Rⁿ.

We consider the function G=F −infF. Since lim inf_+∞G > 0 by assumption, there exists a positive constantR₀ >0 such that

(3.10) ∀R≥R0, α˜R= inf

r≥RG(r)>0.

Let us fix R ≥R₀ and consider two positive real numbers λ_R, µ_R>0 to be chosen later.

For all initial data f₀ ∈ L²(Rⁿ) and all time t ≥ 0, we define f as the mild solution on [0,+∞)of the control system (E_G) with feedbackλ_RK_R at time t, that is

(3.11) ∀t≥0, f(t) =e^{−t(G(|Dx|)+λR1ωKR)}f₀, where K_R stands for the following orthogonal projection

(3.12) K_R:L²(Rⁿ)→

g∈L²(Rⁿ) : Suppbg⊂B(0, R) .

We want to prove that the two constantsλ_Randµ_Rcan be adjusted so that the following estimate holds for allt≥0,

(3.13) kf(t)k²L²(Rⁿ)≤µ_Re^−α˜Rtkf₀k²L²(Rⁿ). To that end, we consider the following Lyapunov function

(3.14) V(y) =µ_RkK_Ryk²L²(Rⁿ)+k(1−K_R)yk²L²(Rⁿ), y∈L²(Rⁿ).

Notice that the function V ◦f is differentiable on(0,+∞), with

∀t >0, d

dtV(f(t)) =µ_Rd

dtkK_Rf(t)k²_L²₍Rⁿ)+ d

dtk(1−K_R)f(t)k²_L²₍Rⁿ).

We shall need to estimate each term of the right-hand side of this equality. On the one hand, we have that for allt >0,

µ_R d

dtkK_Rf(t)k²_L²₍Rⁿ)= 2µ_RRehK_Rf(t), K_Rf^′(t)iL²(Rⁿ)

=−2µ_RhK_Rf(t), G(|D_x|)K_Rf(t)iL²(Rⁿ)−2λ_Rµ_RkK_Rf(t)k²_L²_(ω). The operator G(|D_x|) being accretive, we get that for all t >0,

hK_Rf(t), G(|D_x|)K_Rf(t)iL²(Rⁿ)≥0, and as a consequence,

µ_Rd

dtkK_Rf(t)k²_L²₍Rⁿ) ≤ −2λ_Rµ_RkK_Rf(t)k²_L²_(ω).

Moreover, O. Kovrijkine established in [14] (Theorem 3) a quantitative version of the Logvinenko-Sereda theorem for thick sets which will allow us to control the right-hand side of the above estimate. Precisely, this result is the following:

Theorem 3.2 (Theorem 3 in [14]). There exists a universal positive constant Cn ≥ e depending only on the dimension n ≥ 1 such that for all γ-thick at scale L > 0 subset ω⊂Rⁿ,

∀R >0,∀f ∈L²(Rⁿ), Suppfb⊂[−R, R]ⁿ, kfkL²(Rⁿ)≤C_n γ

Cn(1+LR)

kfkL²(ω). We therefore deduce from this theorem and the definition (3.12) of the orthogonal projec- tionK_R that there exists a positive constant C =C(ω)≥1 only depending on the thick setω (and not on the positive real number R) such that for allt >0,

(3.15) µ_Rd

dtkK_Rf(t)k²L²(Rⁿ)≤ −2λ_Rµ_RC⁻¹e^−CRkK_Rf(t)k²L²(Rⁿ).

On the other hand, the second term we aim at controlling is given for allt >0 by (3.16) d

dtk(1−K_R)f(t)k²_L²₍Rⁿ)= 2 Reh(1−K_R)f(t),(1−K_R)f^′(t)iL²(Rⁿ)

=−2h(1−KR)f(t), G(|Dx|)(1−KR)f(t)iL²(Rⁿ)−2λRReh(1−KR)f(t),(1−KR)¹ωKRf(t)iL²(Rⁿ). We notice that by definition of the orthogonal projection K_R, the Fourier transforms of the functions(1−K_R)f(t)are supported in Rⁿ\B(0, R), which implies that for allt >0, (3.17) 2h(1−K_R)f(t), G(|D_x|)(1−K_R)f(t)iL²(Rⁿ)≥2˜α_Rk(1−K_R)f(t)k²L²(Rⁿ). By using in addition Cauchy-Schwarz’ and Young’s inequalities, we obtain

(3.18)

−2λ_RReh(1−K_R)f(t),(1−K_R)¹_ωK_Rf(t)iL²(Rⁿ)≤2λ_Rk(1−K_R)f(t)kL²(Rⁿ)kK_Rf(t)kL²(Rⁿ)

≤ λ²_R

˜

αRkK_Rf(t)k²_L²₍Rⁿ)+ ˜α_Rk(1−K_R)f(t)k²_L²₍Rⁿ). Combining the estimates (3.16), (3.17) and (3.18), we obtain that for all t >0,

d

dtk(1−K_R)f(t)k²L²(Rⁿ)≤ λ²_R

˜

α_RkK_Rf(t)k²L²(Rⁿ)−α˜_Rk(1−K_R)f(t)k²L²(Rⁿ). This inequality and (3.15) then imply that for allt >0,

d

dtV(f(t))≤ −2

λRµRC⁻¹e^−CR−λ²_R

˜ α_R

K_Rf(t)²_L2(Rⁿ)−α˜R(1−KR)f(t)²_L2(Rⁿ). By making the following choices for the constantsµ_R andλ_R,

µ_R= 2C²e^2CR, λ_R=Ce^CRα˜_R, we get that for allt >0,

d

dtV(f(t))≤ −α˜_RV(f(t)).

This latest estimate and Grönwall’s inequality readily imply that for allt≥0, V(f(t))≤e^−˜αRtV(f(0)),

and then, by Pythagore’s theorem, sinceµ_R≥1, we obtain kf(t)k²L²(Rⁿ)≤µRe^−α˜Rtkf0k²L²(Rⁿ).

The estimate (3.13) therefore holds. Recalling the definitions (3.10) and (3.11) of α˜_R and f(t) respectively, and also recalling that G = F−infF, we have established that for all R≥R₀ and t≥0,

e^{−t(F(|Dx|)+CeCR(αR−infF)1ωKR)2}_L(L2(Rⁿ))≤2C²e^2CRe^{−(αR+infF)t}, withαR= infr≥RF(r). This ends the proof of assertion(ii) in Theorem 2.1.

3.3. Proof of Proposition 2.5. In this last subsection, we prove Proposition 2.5which provides a negative result for the rapid stabilization of the evolution equation (E_F). We assume that ω is not dense in Rⁿ, and also that lim_+∞F exists and is a non-negative real number L ≥ 0. Since the function F is continuous, this implies that F is bounded, i.e. supF < +∞. We aim at proving that if α > supF, then the equation (E_F) is not exponentially stabilizable from ω at rate α. To that end, we will use the following interpretation of exponential stabilization at rate α >0in terms of observability.

Proposition 3.3 (Theorem 1.1 in [17]). If the evolution system (E_F) is exponentially stabilizable from ω at rate α > 0, then there exists a positive constant A_α >0 such that for all T >0, there exists a positive constant C_α,T >0 satisfying that for allg∈L²(Rⁿ),

e^{−T F(|Dx|)}g²

L²(Rⁿ)≤C_α,T Z T

0

e^−tF(|Dx|)g²

L²(ω) dt+A_αe^−2αTkgk²_L²₍Rⁿ).

The proof of Proposition 3.3is contained in the proof of Theorem 1.1 in [17], although [17] (Theorem 1.1) only states characterizations of complete stabilization. For the sake of completeness, we recall the arguments given by the authors of [17] in Section 6.

Proceeding by contradiction, we considerα >supF and assume that the equation (E_F) is exponentially stabilizable at rateα fromω. According to Proposition3.3, there exists a positive constantA_α>0such that for allT >0, there exists a positive constant C_α,T >0 such that for allg∈L²(Rⁿ),

(3.19) e^{−T F(|Dx|)}g²

L²(Rⁿ)≤C_α,T Z T

0

e^−tF(|Dx|)g²

L²(ω) dt+A_αe^−2αTkgk²_L²₍Rⁿ). Since we get that for allT >0 and g∈L²(Rⁿ),

e^{−T F(|Dx|)}g

L²(Rⁿ)≥e^−TsupFkgkL²(Rⁿ), it follows from (3.19) that for allT >0 andg∈L²(Rⁿ),

(3.20) kgk²L²(Rⁿ) ≤C_α,Te^2TsupF Z T

0

e^−tF(|Dx|)g²

L²(ω) dt+A_αe^{2(supF−α)T}kgk²L²(Rⁿ). Notice that since supF −α < 0, we have lim_T→+∞e^{2(supF−α)T} = 0. Therefore, there existsT₀ >0 such that A_αe^{2(supF−α)T0} <1/2. This fact, together with (3.20), imply that for allg∈L²(Rⁿ),

(3.21) kgk²L²(Rⁿ)≤2C_α,T0e^2T0supF Z T0

0

e^−tF(|Dx|)g²

L²(ω) dt.

Sinceω is not dense in Rⁿ and that the evolution equation (E_F) is invariant under trans- lations, we can assume that there exists a positive radiusr >0such thatB(0, r)⊂Rⁿ\ω.

Let us fix a non-zeroL²-function ψand define the function g_h for all h >0by

∀x∈Rⁿ, g_h(x) =ψx h

. On the one hand, we have that for allh >0,

(3.22) kg_hk²L²(Rⁿ)=hⁿkψk²L²(Rⁿ). On the other hand, we get that for allh >0,

Z T0

0

e^−tF(|Dx|)g_h²

L²(ω) dt≤ Z T0

0

e^−tF(|Dx|)g_h²

L²(Rⁿ\B(0,r)) dt (3.23)

=hⁿ Z T0

0

e^{−tF(|Dx|/h)}ψ²

L²(Rⁿ\B(0,r/h)) dt.

It follows from (3.21), (3.22) and (3.23) that for all h >0, kψk²L²(Rⁿ) ≤2C_α,T0e^2T0supF

Z T⁰ 0

e^{−tF(|Dx|/h)}ψ²

L²(Rⁿ\B(0,r/h)) dt.