Null-control and measurable sets

DOI:10.1051/cocv/2012005 www.esaim-cocv.org

NULL-CONTROL AND MEASURABLE SETS

Jone Apraiz

and Luis Escauriaza

Abstract.We prove the interior and boundary null-controllability of some parabolic evolutions with controls acting over measurable sets.

Mathematics Subject Classification. 35B37.

Received July 20, 2011.

Published online 2 May, 2012.

1. Introduction

The control for evolution equations aims to drive the solution to a prescribed state starting from a certain initial condition. One acts on the equation through a source term, a so-called distributed control, or through a boundary condition. To achieve general results one wishes for the control to only act in part of the domain or its boundary and to have as much latitude as possible in the choice of the control region: location, size, shape.

Here, we focus on the heat equation in a smooth and bounded domainΩ in Rⁿ for a time interval (0, T), T >0 and for a distributed controlf we consider

⎧⎪

⎨

⎪⎩

u−∂_tu=f(x, t)χ_ω(x), in Ω×(0, T),

u= 0, on∂Ω×[0, T],

u(0) =u₀, in Ω.

(1.1)

Here,ω⊂Ωis an interior control region. The null controllability of this equation,i.e., the existence for anyu₀ in L²(Ω) of a controlf inL²(ω×(0, T)) with

f_L²_(ω×(0,T))≤Nu_0L²_(Ω), (1.2)

such thatu(T) = 0, was proved in [16] by means of local Carleman estimates for the elliptic operator+∂_y²over Ω×R. A second approach based on global Carleman estimates for the backward parabolic operator+∂_t[10], also led to the null controllability of the heat equation. The first approach has been used for the treatment

Keywords and phrases.Null-controllability.

∗The authors are supported by the grants MTM2004-03029 and MTM2011-24054.

1 Universidad del Pa´ıs Vasco/Euskal Herriko Unibertsitatea, Departamento de Matem´atica Aplicada, Escuela Universitaria Polit´ecnica de Donostia-San Sebasti´an, Plaza de Europa 1, 20018 Donostia-San Sebasti´an, Spain.[email protected]

2 Universidad del Pa´ıs Vasco/Euskal Herriko Unibertsitatean, Dpto. de Matem´aticas, Apto. 644, 48080 Bilbao, Spain.

[email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2012

of time-independent parabolic operators associated to self-adjoint elliptic operators, while the second allows to address time-dependent non-selfadjoint parabolic operators and semi-linear evolutions.

The method introduced in [16] was further extended to study thermoelasticity [17], thermoelastic plates [4]

and semigroups generated by fractional orders of elliptic operators [20]. It has also been used to prove null controllability in the case of non smooth coefficients [5,26]. The method of [16] has also be extended to treat some non-selfadjoint cases, e.g. non symmetric systems [15] and all 1-dimensional time-independent parabolic equations [2].

In the above works, the control region ω is always assumed to contain an open ball. Also, the cost of controllability (the smallest constant N found for the inequality (1.2)) depends on this fact. The reason for these is that the main technique used in the arguments, Carleman inequalities, requires to construct suitable Carleman weights: a role for functions which requires smoothness (at leastC²) and to have the extreme values in proper regions associated to the control regionω, the larger body Ω and possibly the value ofT > 0. The construction of such functions seems to be not possible, whenω does not contain a ball.

Motivated by these facts Puel and Zuazua raised the question wether the null controllability of the heat equation is possible when the control region is a measurable set. A positive partial answer to this question was explained by the second author at the June 2008 meetingControl of Physical Systems and Partial Differential Equations held at the Institute Henri Poincar´e. Here, we give a formal account of the results.

Theorem 1.1. Letn≥2. Then,−∂_tis null-controllable at all positive times, with distributed controls acting over a measurable set ω⊂Ω with positive Lebesgue measure, when

=∇ ·(A(x)∇ ·) +V(x),

is a self-adjoint elliptic operator, the coefficients matrix A is smooth in Ω, V is bounded in Ω and both are real-analytic in an open neighborhood ofω. The same holds whenn= 1,

= 1

ρ(x)[∂_x(a(x)∂_x) +b(x)∂_x +c(x)]

anda,b,c andρare measurable functions in Ω= (0,1).

In regard to boundary null controllability,i.e., the existence for anyu₀inL²(Ω) of a controlhinL²(γ×(0, T)) with

h_L²_(γ×(0,T)) ≤Nu0_L²_(Ω), (1.3)

such that the solution to ⎧

⎪⎨

⎪⎩

u−∂_tu= 0, inΩ×(0, T), u=h(x, t)χ_γ(x), on∂Ω×[0, T], u(0) =u₀, inΩ,

(1.4)

verifiesu(T)≡0, we have the following result.

Theorem 1.2. Let n≥2. Then, −∂_t is null-controllable at all times T > 0 with boundary controls acting over a measurable set γ⊂∂Ω with positive surface measure when

=∇ ·(A(x)∇ ·) +V(x)

is a self-adjoint elliptic operator, the coefficients matrix A is smooth in Ω, V is bounded in Ω and both are real-analytic in an open neighborhood ofγ inΩ.

The results in Theorems 1.1 and1.2 follow from a straightforward application of the linear construction of the control function for the systems (1.1) and (1.4) developed in [16] and the following observability inequality or propagation of smallness estimate established in [28] (see also [23,24]).

Theorem 1.3. Assume that f :B_2R⊂Rⁿ−→Ris a real-analytic function verifying

|∂^αf(x)| ≤ M|α|!

(ρR)^|α|, when x∈B_2R, α∈Nⁿ, (1.5)

for some M > 0, 0< ρ≤1 and E ⊂BR

2 is a measurable set with positive measure. Then, there are positive constants N =N(ρ,|E|/|B_R|)andθ=θ(ρ,|E|/|B_R|)such that

f_L^∞_(BR)≤N

—

E|f|dx _θ

M^1−θ.

The experts will realize that the word smooth describing the global regularity of∂Ω,A andV (away from the measurable sets ω andγ respectively) in Theorems1.1and1.2can be relaxed to ∂Ω isC²,Ais Lipschitz in ΩandV is bounded (See [10,16,17,25]).

To simplify the exposition and to show the strength of Theorem1.3we give the proof of Theorems1.1and1.2 under the assumptions that∂Ω,AandV areglobally real analytic. We do it because it makes more clear how the construction algorithm of the control function in [16] and Theorem1.3can also be applied to prove theinterior null-controllability (Thm. 1.1) for other parabolic evolutions whose corresponding observability or spectral inequalities (suitable Carleman inequalities) are otherwise unknown. Examples of these parabolic evolutions are the ones associated to selfadjoint elliptic systems with unknownsu= (u¹, . . . , u^m),

L_αu=∂_i(a^αβ_ij (x)∂_ju^β), α= 1, . . . , m

witha^αβ_ij =a^βα_ji , forα, β= 1, . . . , m,i, j= 1, . . . , n, and with coefficients matrices verifying for someδ >0 the strong ellipticity condition,

i,j,α,β

a^αβ_ij (x)ξ^α_iξ^β_j ≥δ

i,α

|ξ_i^α|², when ξ∈R^nm, x∈Rⁿ,

or the more general Legendre-Hadamard condition

i,j,α,β

a^αβ_ij (x)ξ_iξ_jη^αη^β≥δ|ξ|²|η|², whenξ∈Rⁿ,η∈R^m, x∈Rⁿ. (1.6) We recall that the Lam´e system of elasticity

∇ · μ(x)

∇u+∇u^t

+∇(λ(x)∇ ·u),

with μ ≥ δ, μ+λ ≥ 0 in Rⁿ, m = n and a^αβ_ij = μ(δ_αβδ_ij +δ_iβδ_jα) +λδ_jβδ_αi, are examples of systems verifying (1.6). Here,a^αβ_ij ,μandλcan either be constants or real analytic functions onΩ.

It also makes clear that under such hypothesis one may replace the Carleman inequalities used in the literature to prove the observability or propagation of smallness inequalities necessary in the process of applying the construction algorithm in [16,17] by a finite number of successive applications of Theorem 1.3. Of course, it has the drawback that it requires more smoothness on the operators and on the boundary of Ω but on the contrary, with Theorem1.3and the construction methods in [16,17] one can handle the null-controllability with distributed controls of other parabolic evolutions, which as far as the authors know were unknown: like the second order evolutions explained above or for higher order evolutions as

∂_tu+ (−1)^mmu, m= 2, . . . , with Dirichlet boundary conditions on∂Ω,i.e.,u=∇u=· · ·=∇^m−1u= 0.

In Section2 we give first the proofs of Theorems1.1and1.2as it is explained above. We then show how to extend Theorem1.1 to the evolutions (2.25) and (2.31) in Remark2.2, while the problems we find to extend Theorem 1.2 to these evolutions are explained in Remark 2.3 for the simpler case of parabolic systems. In Remark 2.4 we give the proof of Theorem1.1 when ∂Ω,A andV are globally smooth andAand V are real analytic in a neighborhood of the measurable setω. An outline of a proof of Theorem1.2, when∂Ω,AandV are real analytic nearγ and smooth elsewhere appears in Remark2.4.

For the sake of completeness, we include a proof of Theorem1.3 in Section3. It is built with ideas taken from [18,23,28].

2. Proof of Theorems 1.1 and 1.2

We begin by setting up the formal hypothesis: first we assume there is 0< δ ≤1 such that δ|ξ|²≤A(x)ξ·ξ≤δ⁻¹|ξ|², for allx∈Ω, ξ∈Rⁿ,

Ωis a bounded domain inRⁿ,n≥2, with a real analytic boundary andA,V are real analytic inΩ,i.e., there arer >0 and 0< δ≤1 such that

|∂^αA(x)|+|∂^αV(x)| ≤ |α|!δ^−|α|−1, whenx∈Ω, α∈Nⁿ,

and for eachx∈∂Ω, there are a new coordinate system (where x= 0) and a real analytic function ϕ: B_r ⊂ Rⁿ⁻¹−→Rverifying

ϕ(0) = 0, |∂^αϕ(x)| ≤ |α|!δ^−|α|−1, whenx ∈B_r, α∈Nⁿ⁻¹, B_r∩Ω=B_r∩ {(x, x_n) :x∈B_r, x_n> ϕ(x)},

B_r∩∂Ω=B_r∩ {(x, x_n) :x ∈B_r, x_n=ϕ(x)}.

(2.1)

Forρ >0, we set

Ω_ρ={x∈Ω:d(x, ∂Ω)≤ρ}, Ω^ρ={x∈Ω:d(x, ∂Ω)≥ρ}, Ω(ρ) ={x∈Rⁿ:d(x, Ω)≤ρ}

and|E|denotes the Lebesgue or surface measure of a measurable set E.

Proof of Theorem 1.1. We may assume that the eigenvalues with zero Dirichlet condition for=∇·(A(x)∇ ·)+

V(x) onΩare all positive

0< ω₁²< ω²₂≤ω₃²≤ · · · ≤ω_j²≤. . . and{ej}denotes the sequence ofL²(Ω)-normalized eigenfunctions,

e_j+ω_j²e_j= 0, in Ω,

e_j = 0, in ∂Ω.

Whenω⊂Ωis measurable with positive Lebesgue measure, the method in [16] shows that one can find and L²(ω×(0, T)) control functionf verifying (1.2) for the system (1.1), provided there isN =N(|ω|, Ω, r, δ) such that the inequality

ωj≤μ

a²_j+b²_j ≤e^Nμ

ω×[¹₄,³₄]

_ω

j≤μ

a_je^ωjy+b_je^−ωjy e_j

2

dxdy, (2.2)

holds forμ≥ω₁ and all sequencesa₁, a₂, . . . andb₁, b₂, . . . Set then u(x, y) =

ωj≤μ

a_je^ωjy+b_je^−ωjy

e_j. (2.3)

It satisfies

u+∂_y²u= 0, inΩ×R,

u= 0, on∂Ω×R, (2.4)

and u is real analytic in Ω×R. Moreover, given (x0, y₀) in Ω×R and R ≤ 1, there are N = N(r, δ) and ρ=ρ(r, δ), 0< ρ≤1, such that

∂_x^α∂_y^βu_L^∞_{(BR(x0,y0)∩Ω×R)}≤ N(|α|+β)!

(Rρ)^|α|+β

—

B2R(x0,y0)∩Ω×R|u|²dxdy ¹₂

, (2.5)

whenα∈Nⁿ andβ≥1. For the later see [21], Chapter 5, [13], Chapter 3.

The orthonormality of{e_j}in Ωand (2.5) withR= 1 imply that

∂_x^α∂_y^βu_L^∞_{(Ω×[−5,5])}≤e^Nμ(|α|+β)!ρ^−|α|−β

⎛

⎝

ωj≤μ

a²_j+b²_j

⎞

⎠

12

, forα∈Nⁿ, β≥0, (2.6) and there isC >0 such that replacing the constants N and ρin (2.6) byCN andρ/C respectively,uhas a real analytic extension to Ω(ρ)×[−4,4] satisfying

∂_x^α∂^β_yu_L^∞(Ω(ρ)×[−4,4])≤M(|α|+β)!(2ρ)^−|α|−β, forα∈Nⁿ, β ≥0, (2.7) with

M = e^Nμ

⎛

⎝

ωj≤μ

a²_j+b²_j

⎞

⎠

12

. (2.8)

For (x₀, y₀) inΩ×[0,1] withd(x₀, ∂Ω) =ρ, we haveB_2ρ(x₀, y₀)⊂Ω(ρ)×[−4,4], and if we apply Theorem1.3 to the real analytic extension ofuinB_2ρ(x₀, y₀) withE=B^ρ

4(x₀, y₀), (2.7) implies that there are new constants N =N(ρ)>0 and 0< θ=θ(ρ)<1 such that

u_L^∞_(Bρ(x0,y0))≤Nu^θ_L∞(B^ρ

4(x0,y0))M^1−θ. The later implies

u_L^∞_{(Ωρ×[0,1])}≤Nu^θ

L^∞(Ω^3ρ4×[−ρ,1+ρ])M^1−θ, (2.9) so that from (2.7) withα=β = 0 and (2.9)

u_L^∞_(Ω×[0,1])≤Nu^θ

L^∞(Ω^3ρ4 ×[−1,2])M^1−θ. (2.10) Thus, from Theorem 3 and without Carleman inequalities it is possible to bound except for the factorM all the information on the size ofuoverΩ×[0,1] by information on the size ofuoverΩ^3ρ4 ×[−1,2], a region located in the interior and away from the boundary ofΩ×R.

After choosing ρsmaller and replacingω by a smaller measurable set if it is necessary, we can assume that ω⊂B^ρ

4(0),|ω|/|B^ρ₄(0)| ≥ ¹₂ and B_2ρ(0)⊂Ω^3ρ4, so that E=ω×(^2−ρ₄ ,^2+ρ₄ )⊂B^ρ

2(0,¹₂) has positive Lebesgue measure insideB^ρ₂(0,¹₂) andB_2ρ(0,¹₂)⊂Ω×[0,1]. Then, a second application of Theorem1.3and (2.7) gives

u_L∞(Bρ(0,¹₂))≤Nu^θ_L2(ω×[¹₄,³₄])M^1−θ. (2.11)

Also, (2.7) and Theorem1.3withE=BR

4(x₂, y₂) imply that u_L^∞_(BR(x1,y1))≤Nu^θ_L∞(BR

4(x2,y2))M^1−θ, wheneverB_2R(x₁, y₁)⊂Ω×[−4,4], (x₂, y₂) is in BR

4(x₁, y₁) and 0< R≤1. In particular,

u_L^∞_(BR(x1,y1)) ≤Nu^θ_L∞(BR(x2,y2))M^1−θ, (2.12) whenB_2R(x₁, y₁)⊂Ω×[−4,4], (x₂, y₂) is inBR

4(x₁, y₁) and 0< R≤1.

Becauseρis now fixed andΩis compact there areRin (0,1) andk≥2, which depend onρand the geometry ofΩ, such that for any (x₀, y₀) inΩ^3ρ4 ×[−1,2] there arekpoints (x₁, y₁),(x₂, y₂), . . . ,(x_k, y_k) inΩ^3ρ4 ×[−1,2]

with

B_2R(x_i, y_i)⊂Ω×[−4,4], (x_i+1, y_i+1)∈BR

4(x_i, y_i), fori= 0, . . . , k−1 and (x_k, y_k) = (0,¹₂). The later and (2.12) show that

u_L^∞_(BR(xi,yi))≤Nu^θ_L∞(BR(xi+1,yi+1))M^1−θ, fori= 0, . . . , k−1, (2.13) while the iteration of (2.13) gives

uL^∞(Ω^3ρ4 ×[−1,2])≤N^ku^θ_L^k∞(Bρ(0,¹₂))M^1−θk. (2.14) Combining then (2.10), (2.14) and (2.11), one finds that

u_L^∞_(Ω×[0,1])≤N^k+2u^θ_L^k+22(ω×[¹₄,³₄])M^1−θk+2. Thus,

u_L^∞_(Ω×[0,1])≤Nu^θ_L2(ω×[¹₄,³₄])M^1−θ, (2.15) for some newN and 0< θ <1, which depend onρ,|ω| and the geometry ofΩbut not onu.

The fact that the inequality e^−μ

sinhω₁

ω₁ −1 a²+b²

≤ ₁

0

ae^ωy+be^−ωy₂ dy, holds, whenω₁≤ω≤μanda, b∈Rand the orthonormality of{e_j}in L²(Ω) show that

⎛

⎝

ωj≤μ

a²_j+b²_j

⎞

⎠

12

≤e^Nμu_L²_(Ω×[0,1])≤e^Nμ|Ω|¹²u_L^∞_(Ω×[0,1]), (2.16)

and (2.2) follows from (2.16), (2.15) and the definition ofM in (2.8).

It is shown in [2] that the null-controllability of the system (1.1) overΩ= (0,1) with = 1

ρ(x)[∂_x(a(x)∂_x) +b(x)∂_x +c(x)],

δ≤a(x), ρ(x)≤δ⁻¹, |b(x)|+|c(x)| ≤δ⁻¹ , a.e. in [0,1], (2.17)

is equivalent to the null-controllability of the system

⎧⎪

⎨

⎪⎩

∂_x²z−ρ(x)∂_tz=f χ_ω, 0< x <1 , 0< t < T, z(0, t) =z(1, t) = 0, 0≤t≤T,

z(x,0) =z₀, 0≤x≤1,

(2.18)

whereρis a new function verifying (2.17) andω a new measurable set in [0,1] with positive measure. The later follows from the bilipschitz change of variables used in [2]. Let then, 0< ω²₁< ω₂²≤ω₃²≤ · · · ≤ω_j²≤. . . and {e_j}denote the sequences of eigenvalues andL²(Ω)-normalized eigenfunctions [29], Chapter VII, verifying

e_j +ρ(x)ω_j²e_j= 0, 0< x <1, e_j(0) =e_j(1) = 0.

From [16] it suffices to show that (2.2) holds in order to find an interior null-controlf for (2.18) verifying (1.2).

Extend thene_j andρto [−1,1] by odd and even reflections respectively, and to allRas periodic functions of period 2. The extendede_j is inC^1,1(R) and verifiese_j +ρ(x)ω_je_j = 0 inR, j = 1,2. . .. As before, let u be defined by (2.3), it verifies

∂_x²u+∂_y(ρ(x)∂_yu) = 0, inR². By Chebyshev’s inequality and definingEas

ω×

1 4,3

4

\E=

(x, y)∈ω× 1

4,3 4

:|u(x, y)|/2>—

ω×[¹₄,³₄]|u|dxdy

,

we have

|E| ≥ 1 2

ω× 1

4,3 4

and u_L^∞_(E)≤2 —

ω×[¹₄,³₄]|u|dxdy. (2.19) Set thenu(x, y) =u(^x,^y), it verifies

∂_x²u +∂_y(ρ(x/)∂_yu) = 0, inR² and letv be thestreamfunction ofu,i.e., the solution to

⎧⎪

⎨

⎪⎩

∂_xv =−ρ(^x)∂_yu,

∂_yv =∂_xu, v (0,0) = 0.

Then,f =u +iv is (1/δ)-quasiregular,i.e.,

f ∈W_loc^1,2(R²), ∂_zf =ν(z)∂_zf, |ν(z)| ≤ 1−δ

1 +δ, z∈C,

and by the Ahlfors-Bers representation theorem [1] (see [6] or [7]) all the (1/δ)-quasiregular mappingsf in B₄ can be written as

f =F◦Ψ,

where F =U +iV is holomorphic inB₄ andΨ :B₄−→B₄ is a (1/δ)-quasiconformal mapping,i.e., a (1/δ)- quasiregular homeomorphism fromB₄ ontoB₄ verifying

∂_zΨ =ν(z)∂_zΨ, Ψ(0) = 0, Ψ(4) = 4,

N⁻¹|z₁−z₂|^α1 ≤ |Ψ(z₁)−Ψ(z₂)| ≤N|z₁−z₂|^α, whenz₁, z₂∈B₄, (2.20)

for some 0 < α < 1 and N ≥1 depending onδ. Now, E ⊂B₂ , and from (2.20), Ψ(E) ⊂B_{N(2 )}α. Choose thenso thatN(2)^α= ¹₂. Thus,Ψ(E)⊂B1

2,u =U◦Ψ, U_L^∞_(B4)=u

L^∞

B4

while the L^∞-interior estimates for subsolutions of elliptic equations [12], Section 8.6, the periodicity and orthogonality of the eigenfunctionse_j inL²([0,1], ρ(x) dx), gives

u_L^∞_(B4

)u_L²_(B6

)e^6μ/

⎛

⎝

ωj≤μ

a²_j+b²_j

⎞

⎠

12

.

Thus,U is harmonic inB₄,

U_L^∞_(B4)≤e^Nμ

⎛

⎝

ωj≤μ

a²_j+b²_j

⎞

⎠

12

and from (2.19)

U_L^∞_{(Ψ( E))}≤2 —

ω×[¹₄,³₄]|u|dxdy. (2.21)

All together, U verifies the conditions in Theorem 1.3 in B₂ with R = 1 and the universal constant 0 <

ρ≤1 associated to the quantitative property of analyticity overB₂ for bounded harmonic functions onB₄ [9], Chapter 2. From (2.21), (2.2) holds provided we can find a lower bound for the Lebesgue measure ofΨ(E)⊂B1

2. The lower bound follows from (2.19) and the following rescaled version of [3], Theorem 1:

Let Ψ :B₄−→B₄ be a(1/δ)-quasiconformal mapping withΨ(0) = 0andE ⊂B₄be a measurable set. Then, there isN =N(δ)such that

|E|^1δ/N≤ |Ψ(E)| ≤N|E|^δ.

Remark 2.1. Theorem 1.3also implies the version of (2.2) appearing in [17]. For ifΩ,AandV are as above and

u(x, y) =

ωj≤μ

a_je^ωjye_j(x), uverifies (2.5) and

∂_x^αu(. ,0)_L^∞_(Ω)≤M|α|!(2ρ)^−|α|, forα∈Nⁿ, withM = e^Nμ

⎛

⎝

ωj≤μ

a²_j

⎞

⎠

12

.

Thus, u(. ,0) has an analytic extension to a ρ-neighborhood ofΩ and after a finite number of applications of Theorem1.3and a suitable covering argument we get

u(. ,0)_L²_(Ω)≤Nu(. ,0)^θ_L2(ω)M^1−θ. In particular,

ωj≤μ

a²_j ≤e^Nμ

ω

_ωj≤μa_je_j

2

dx, withN =N(|ω|, Ω, r, δ).

Proof of Theorem 1.2. Letube defined by (2.3). From [16], one can find a boundary controlhverifying (1.3) provided there isN =N(|γ|, Ω, r, δ) such that the inequality

ωj≤μ

a²_j+b²_j≤e^Nμ

γ×[¹₄,³₄]

ωj≤μ

a_je^ωjy+b_je^−ωjy ∂e_j

∂ν

2

dσdy, (2.22)

holds forμ≥ω₁ and all sequencesa₁, a₂, . . . andb₁, b₂, . . .. Here,ν,σand _∂ν^∂ denote respectively the exterior unit normal vector to Ω, the surface measure on ∂Ω and the conormal derivative for ∂_y²+ on ∂Ω ×R,

∂ν∂e =A∇e·ν. We may also assume that 0∈∂Ω,γ⊂B^ρ

2 ∩∂Ω, whereB_2ρ∩∂Ω is the region above the graph of a real analytic function,ϕ:B_ρ ⊂Rⁿ⁻¹−→R, as in (2.1). From [16], Section 3(2), there isN such that

⎛

⎝

ωj≤μ

a²_j+b²_j

⎞

⎠

12

≤e^Nμ

∂u

∂ν

L^∞(∂Ω×[−1,2])

, (2.23)

with ∂u

∂ν =

ωj≤μ

a_je^ωjy+b_je^−ωjy ∂e_j

∂ν ·

From (2.6) and (2.1), there areN =N(r, δ) andρ=ρ(r, δ) such that h(x, y) = _∂n^∂u(x, ϕ(x), y) verifies ∂_x^α∂_y^βhL^∞(B_2ρ×[−4,4])≤M(|α|+β)!(2ρ)^−|α|−β, forα∈Nⁿ⁻¹, β∈N,

with M as in (2.8) and provided that B_2ρ∩∂Ω is a coordinate chart of ∂Ω as in (2.1). This fact, a suitable covering argument of ∂Ω and the successive iteration of a finite number of applications of the three-spheres type inequalitiesassociated to the obvious extension of Theorem1.3for real analytic functions defined over real analytic hypersurfaces inRⁿ⁺¹(See [14], pp. 67–69 for a quantitative version of the real analyticity of composite functions) imply that there areN =N(|γ|, r, δ) andθ=θ(|γ|, r, δ) such that

∂u

∂ν

L^∞(∂Ω×[−1,2])

≤N

∂u

∂ν

θ

L²(γ×[¹₄,³₄])

M^1−θ. (2.24)

Finally, (2.22) follows from (2.23) and (2.24).

Remark 2.2. The extension of Theorem1.1to the parabolic system

⎧⎪

⎨

⎪⎩

∂_i(a^αβ_ij ∂_je^β_k)−∂_tu^α=f^α(x, t)χ_ω(x), in Ω×(0, T), α= 1, . . . , m,

u= 0, on∂Ω×[0, T],

u(0) =u₀, in Ω,

(2.25)

with∂Ω as in (2.1),a^αβ_ij verifying (1.6) and

|∂^γa^αβ_ij (x)| ≤ |γ|!δ^−|γ|−1, whenx∈Ω, γ∈Nⁿ, (2.26) for some 0< δ≤1 is now obvious: the symmetry, coerciveness and compactness of the operatorL²(Ω)^m −→

W₀^1,2(Ω)^m, mappingf = (f¹, . . . , f^m) into the unique solutionu= (u¹, . . . , u^m) to

∂_i(a^αβ_ij ∂_ju^β)−Λu^α=f^α, in Ω, α= 1, . . . , m,

u= 0, in ∂Ω

whereΛ >0 is sufficiently large [11], Proposition 2.1, gives the existence of a complete system{e_k}inL²(Ω)^m, e_k= (e¹_k, . . . , e^m_k), of eigenfunctions verifying

∂_i(a^αβ_ij ∂_je^β_k) +ω²_ke^α_k = 0, inΩ, α= 1, . . . , m,

e_k= 0, in∂Ω

with eigenvalues 0≤ω₁≤. . . ω_k≤. . . and lim_k→+∞ω_k= +∞. By separation of variables, the Green’s matrix for the system (2.25) overΩ×Ris them×mmatrix

Γ(x, y, t−s) =

+∞

k=1

e^{−ωk2(t−s)}e_k(x)⊗e_k(y). (2.27)

Moreover, the interior and boundary regularity for the elliptic system∂_y²+∂_i(a^αβ_ij ∂_j ) inΩ×R, shows that (2.5) holds for u as in (2.3) but with e_k replacing e_k ([11,22], Chap. II). These and [16] suffice to find a control functionf for the system (2.25) verifying (1.2). Furthermore, if you wish to get bounds on the regularity of the control functionf, [16] shows that suffices to know that

e_kH^s_(Ω)≤N_s(1 +ω_k)^s, whenk, s≥0 and

#{k≥1 : 0≤ω_k≤μ} ≤N μⁿ, whenμ≥1, (2.28) for some N which does not depend on μ≥1. The first holds from elliptic regularity and (2.26), while (2.28) follows from the Gaussian estimates verified byΓ; i.e., there areN andκ[8], Corollary 4.14, such that

|Γ(x, y, t)| ≤N(1∧t)⁻ⁿ²e^{Λt−κ|x−y|2/t}, forx, y∈Rⁿ andt >0. (2.29) To verify the last claim, observe that (2.27), (2.29) and the orthonormality of the eigenfunctionse_k give

Ω

Ω|Γ(x, y, t)|²dxdy=

k≥1

e^−2ω2kt≤Ne^2Λt|Ω|t⁻ⁿ², (2.30)

and it suffices to choose t = 1/μ² in (2.30) to get (2.28). In particular, the methods in [16] also generate a control functionf inC₀^∞((0, T), C^∞(Ω)).

The null-controllability of the system

⎧⎪

⎨

⎪⎩

∂_tu+ (−1)^mmu=f(x, t)χω, in Ω×(0, T], u=∇u=· · ·=∇^m−1u= 0, in ∂Ω×(0, T),

u(0) =u₀, in Ω,

(2.31)

m≥2, is better managed with the approach in [17]: if{e_j}and 0≤ω₁^2m≤ · · · ≤ω^2m_k ≤. . . are the eigenvectors and eigenvalues for^minW₀^m,2(Ω),

(−1)^mme_j−ω^2m_j e_j= 0, in Ω, e_j =∇ej=· · ·=∇^m−1e_j= 0, in ∂Ω, define

u(x, y) =

w^m_j ≤μ

a_jX_j(y)e_j(x), with X_j(y) =

⎧⎨

⎩

e^ωjy, formodd, e^ωje

2m yπi , formeven.

It verifies

∂_y^2mu+^mu= 0, inΩ×R, u=∇u=· · ·=∇^m−1u= 0, in∂Ω×R.

Again,uverifies (2.5) [22] and from Theorem1.3applied tou(. ,0) as in Remark2.1,

ω^m_j ≤μ

a²_j ≤e^Nμ

m1

ω

ω^m_j ≤μ

a_je_j

2

dx, withN =N(|ω|, Ω, r, δ, m).

From [17], the last inequality suffices to find a controlf inL²(ω×(0, T)) for (2.31).

Remark 2.3. The proof of Theorem1.2for the scalar case is based on the bound (2.23). In the literature, it is is obtained from (2.16) and the interpolation inequality below which has been proved with Carleman inequalities:

there areN andθsuch that the inequality u_L²_(Ω×[0,1])≤N

∂u

∂ν

θ

L²(∂Ω×[−1,2])

u^1−θ_L2(Ω×[−3,3]),

holds whenuverifies (2.4). However, the authors are not aware wether the interpolation inequality u_L²_(Ω×[0,1])≤N

∂u

∂ν

θ

L²(∂Ω×[−1,2])

u^1−θ_L2(Ω×[−3,3]),

with

∂u

∂ν _α

=a^αβ_ij ∂_ju^βν_i, α= 1, . . . , m,

holds for solutionsuto the corresponding analog elliptic system and lateral boundary conditions:

∂_i(a^αβ_ij ∂_ju^β) +∂_y²u^α= 0, in Ω×R, α= 1, . . . , m,

u= 0, in ∂Ω×R.

The later explains why we can not extend the boundary null-controllability results in Theorem1.2 to general parabolic systems with analytic coefficients and also analytic lateral boundaries. In spite of that, the inverse mapping theorem shows that there isρ >0 such that the mapping

∂Ω×(0, ρ)−→U_ρ, (Q, t)Q+tν(Q),

is an analytic diffeomorphism onto U_ρ = {x ∈ Rⁿ \Ω : 0 < d(x, ∂Ω) < ρ} and the null-controllability for parabolic systems with analytic coefficients over Ω∪U_ρ and with controls acting overω =U3ρ

4\U^ρ

2 has been stablished in Remark2.2. From these, standard arguments show thatat least the parabolic system

⎧⎪

⎨

⎪⎩

∂_i(a^αβ_ij ∂_je^β_k)−∂_tu^α= 0, in Ω×(0, T), α= 1, . . . , m,

u=g on∂Ω×[0, T],

u(0) =u₀, in Ω,

can be null-controlled with controlsgacting over thefull lateral boundaryofΩ.

Remark 2.4. To prove Theorem 1.1 when ∂Ω, A and V are globally smooth and A and V are only real analytic in a neighborhood ofω, one may assume that

ω⊂BR

4, |ω|/|BR| ≥1/2, B_4R⊂Ω (2.32)

|∂^αA(x)|+|∂^αV(x)| ≤ |α|!δ^−|α|−1, when x∈B_4R, α∈Nⁿ, (2.33) for some fixed 0< R, δ≤1 and the goal is to show that (2.2) holds. The Carleman and interpolation inequalities stablished in [16], Section 3(1). for solutions to elliptic operators with smooth coefficients and for∂Ω smooth show that there isN =N(A, V, Ω, R) such that the inequalities

⎛

⎝

ωj≤μ

a²_j+b²_j

⎞

⎠

12

≤e^Nμu_L∞(BR(0,¹₂)), (2.34)

hold whenμ≥ω₁, for all sequencesa₁, a₂, . . .,b₁, b₂, . . . and foruas in (2.3). Becauseuverifies Δu+∂_y²u= 0, inB_4R

0,1

2

and (2.33) holds, uis a local solution to an elliptic equation with local real analytic coefficients and there are N =N(δ) andρ=ρ(δ), 0< ρ≤1, such that

∂_x^α∂_y^βu_L∞(B2R(0,¹₂))≤N(|α|+β)!

(Rρ)^|α|+β

—

B4R(0,¹₂)|u|²dxdy ¹₂

, (2.35)

whenα∈Nⁿ andβ≥1. The later follows from the corresponding result forR= 1 and rescaling ([21], Chap. 5, [13], Chap. 3). The orthonormality of {e_j} inΩ and (2.35) show that

∂_x^α∂_y^βu_L∞(B2R(0,¹₂))≤M(|α|+β)!(Rρ)^−|α|−β, forα∈Nⁿ, β ≥0, (2.36) withM as in (2.8). Finally, Theorem1.3inB_2R(0,¹₂) with

E=ω×

2−R 4 ,2 +R

4

⊂BR 2

0,1

2

, (2.36) and (2.32) show that

u_L∞(BR(0,¹₂))≤Nu^θ_L2(ω×[¹₄,³₄])M^1−θ, (2.37) and (2.2) follows from (2.34), (2.37) and (2.8).

To prove Theorem1.2with ∂Ω,Aand V real analytic nearγand smooth elsewhere, one may assume that 0 is in∂Ω and that there are 0< R, δ ≤1 andϕ:B_R ⊂Rⁿ⁻¹−→Rsuch that

γ⊂BR

4 ∩∂Ω, |γ|/|B_R∩∂Ω| ≥ 1

2, (2.38)

ϕ(0) = 0, |∂^αϕ(x)| ≤ |α|!δ^−|α|−1, whenx∈B_4R , α∈Nⁿ⁻¹,

|∂^αA(x)|+|∂^αV(x)| ≤ |α|!δ^−|α|−1, when x∈B_4R∩Ω, α∈Nⁿ, B_4R∩Ω=B_4R∩ {(x, x_n) :x ∈B_4R , x_n> ϕ(x)},

B_4R∩∂Ω =B_4R∩ {(x, x_n) :x ∈B_4R , x_n=ϕ(x)}

(2.39)