NULL CONTROL AND MEASURABLE SETS

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1 NULL CONTROL AND MEASURABLE SETS J. APRAIZ AND L. ESCAURIAZA Abstract. We prove the interior and boundary null controllability of some parabolic evolutions with controls acting over measurable sets. 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 domain Ω in R n for a time interval (0, T ), T > 0 and for a distributed control f we consider u t u = f(x, t)χ ω (x), in Ω (0, T ), (1.1) u = 0, on Ω [0, T ], u(0) = u 0, in Ω. Here, ω Ω is an interior control region. The null controllability of this equation, i.e., the existence for any u 0 in L (Ω) of a control f in L (ω (0, T )) with (1.) f L (ω (0,T )) N u 0 L (Ω), such that u(t ) = 0, was proved in [16] by means of local Carleman estimates for the elliptic operator + y over 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 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 [0]. It has also been used to prove null controllability in the case of non smooth coefficients [5, 5]. 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 []. 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.)) depends on this fact. The reason for these is that the main technique used 1991 Mathematics Subject Classification. Primary: 35B37. Key words and phrases. Null-controllability. The authors are supported by the grants MTM and MTM

2 J. APRAIZ AND L. ESCAURIAZA in the arguments, Carleman inequalities, requires to construct suitable Carleman weights: a role for functions which requires smoothness (at least C ) and to have the extreme values in proper regions associated to the control region ω, the larger body Ω and possibly the value of T > 0. The construction of such functions seems to be not possible, when ω does not contain a ball. Motivated by these facts J.P. Puel and E. 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 008 meeting Control of Physical Systems and Partial Differential Equations held at the Institute Henri Poincaré. Here, we give a formal account of the results. Theorem 1. Let n. Then, t is 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 of ω. The same holds when n = 1, = 1 ρ(x) [ x (a(x) x ) + b(x) x + c(x)] and a, b, c and ρ are measurable functions in Ω = (0, 1). In regard to boundary null controllability, i.e., the existence for any u 0 in L (Ω) of a control h in L (γ (0, T )) with (1.3) h L (γ (0,T )) N u 0 L (Ω), such that the solution to u t u = 0, in Ω (0, T ), (1.4) u = h(x, t)χ γ (x), on Ω [0, T ], u(0) = u 0, in Ω, verifies u(t ) 0, we have the following result. Theorem. Let n. 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 A, V and Ω are real-analytic in an open neighborhood of γ in Ω. The results in Theorems 1 and 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 [8] (See also [3] and [4]). Theorem 3. Assume that f : B R R n R is a real-analytic function verifying (1.5) α f(x) M α! (ρr) α, when x B R, α N n,

3 NULL CONTROL AND MEASURABLE SETS 3 for some M > 0, 0 < ρ 1 and E B R is a measurable set with positive measure. Then, there are positive constants N = N(ρ, E / B R ) and θ = θ(ρ, E / B R ) such that ( θ f L (B R ) N f dx) M 1 θ. The experts will realize that the word smooth describing the global regularity of Ω, A and V (away from the measurable sets ω and γ respectively) in Theorems 1 and can be relaxed to Ω is C, A is Lipschitz in Ω and V is bounded (See [10, 16, 17, 6]). To simplify the exposition and to show the strength of Theorem 3 we give the proof of Theorems 1 and under the assumptions that Ω, A and V are globally real analytic. We do it because it makes more clear how the construction algorithm of the control function in [16] and Theorem 3 can also be applied to prove the interior null-controllability (Theorem 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 unknowns u = (u 1,..., u m ), L α u = i (a αβ ij (x) ju β ), α = 1,..., m with a αβ ij = a βα ji, for α, β = 1,..., m, i, j = 1,..., n, and with coefficients matrices verifying for some δ > 0 the strong ellipticity condition, i,j,α,β a αβ ij (x)ξα i ξ β j δ i,α E ξ α i, when ξ R nm, x R n, or the more general Legendre-Hadamard condition (1.6) a αβ ij (x)ξ iξ j η α η β δ ξ η, when ξ R n, η R m, x R n. i,j,α,β We recall that the Lamé system of elasticity (µ(x) ( u + u t)) + (λ(x) u), with µ δ, µ + λ 0 in R n, m = n and a αβ ij = µ(δ αβ δ ij + δ iβ δ jα ) + λδ jβ δ αi, are examples of systems verifying (1.6). Here, a αβ ij, µ and λ can either be constants or real analytic functions on Ω. 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 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 Theorem 3 and 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 t u + ( 1) m m u, m =,..., with Dirichlet boundary conditions on Ω, i.e., u = u = = m 1 u = 0. In section we give first the proofs of Theorems 1 and as it is explained above. We then show how to extend Theorem 1 to the evolutions (.5) and (.31) in

4 4 J. APRAIZ AND L. ESCAURIAZA Remark, while the problems we find to extend Theorem to these evolutions are explained in Remark 3 for the simpler case of parabolic systems. In Remark 4 we give the proof of Theorem 1 when Ω, A and V are globally smooth and A and V are real analytic in a neighborhood of the measurable set ω. An outline of a proof of Theorem, when Ω, A and V are real analytic near γ and smooth elsewhere appears in Remark 4. For the sake of completeness, we include a proof of Theorem 3 in section 3. It is built with ideas taken from [18], [3] and [8].. Proof of Theorems 1 and We begin by setting up the formal hypothesis: first we assume there is 0 < δ 1 such that δ ξ A(x)ξ ξ δ 1 ξ, for all x Ω, ξ R n, Ω is a bounded domain in R n, n, with a real analytic boundary and A, V are real analytic in Ω, i.e., there are r > 0 and 0 < δ 1 such that α A(x) + α V (x) α!δ α 1, when x Ω, α N n, and for each x Ω, there are a new coordinate system (where x = 0) and a real analytic function ϕ : B r R n 1 R verifying (.1) ϕ(0 ) = 0, α ϕ(x ) α!δ α 1, when x B r, α N n 1, For ρ > 0, we set 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 )}. Ω ρ = {x Ω : d(x, Ω) ρ}, Ω ρ = {x Ω : d(x, Ω) ρ}, Ω(ρ) = {x R n : d(x, Ω) ρ} and E denotes the Lebesgue or surface measure of a measurable set E. Proof of Theorem 1. We may assume that the eigenvalues with zero Dirichlet condition for = (A(x) ) + V (x) on Ω are all positive 0 < ω 1 < ω ω 3 ω j... and {e j } denotes the sequence of L (Ω)-normalized eigenfunctions, { e j + ωj e j = 0, in Ω, e j = 0, in Ω. When ω Ω is measurable with positive Lebesgue measure, the method in [16] shows that one can find and L (ω (0, T )) control function f verifying (1.) for the system (1.1), provided there is N = N( ω, Ω, r, δ) such that the inequality (.) a j + b j e Nµ ( aj e ωjy + b j e ωjy) e j dxdy, ω [ 1 4, 3 4 ] holds for µ ω 1 and all sequences a 1, a,... and b 1, b,... Set then (.3) u(x, y) = ( aj e ωjy + b j e ωjy) e j

5 NULL CONTROL AND MEASURABLE SETS 5 It satisfies (.4) { u + yu = 0, in Ω R, u = 0, on Ω R, and u is real analytic in Ω R. Moreover, given (x 0, y 0 ) in Ω R and R 1, there are N = N(r, δ) and ρ = ρ(r, δ), 0 < ρ 1, such that ( ) 1 (.5) x α y β N( α + β)! u L (B R (x 0,y 0) Ω R) u dxdy, (Rρ) α +β B R (x 0,y 0) Ω R when α N n and β 1. For the later see [, Ch. 5], [14, Ch. 3]. The orthonormality of {e j } in Ω and (.5) with R = 1 imply that (.6) 1 x α y β u L (Ω [ 5,5]) e Nµ ( α + β)!ρ α β a j + b j, for α N n, β 0, and there is C > 0 such that replacing the constants N and ρ in (.6) by CN and ρ/c respectively, u has a real analytic extension to Ω(ρ) [ 4, 4] satisfying (.7) α x β y u L (Ω(ρ) [ 4,4]) M( α + β)!(ρ) α β, for α N n, β 0, with (.8) M = e Nµ a j + b j For (x 0, y 0 ) in Ω [0, 1] with d(x 0, Ω) = ρ, we have B ρ (x 0, y 0 ) Ω(ρ) [ 4, 4], and if we apply Theorem 3 to the real analytic extension of u in B ρ (x 0, y 0 ) with E = B ρ 4 (x 0, y 0 ), (.7) implies that there are new constants N = N(ρ) > 0 and 0 < θ = θ(ρ) < 1 such that The later implies u L (B ρ(x 0,y 0)) N u θ L (B ρ 4 (x0,y0))m 1 θ. (.9) u L (Ω ρ [0,1]) N u θ L (Ω 3ρ 4 [ ρ,1+ρ]) M 1 θ, so that from (.7) with α = β = 0 and (.9) (.10) u L (Ω [0,1]) N u θ L (Ω 3ρ 4 [ 1,]) M 1 θ. Thus, from Theorem 3 and without Carleman inequalities it is possible to bound except for the factor M all the information on the size of u over Ω [0, 1] by information on the size of u over Ω 3ρ 4 [ 1, ], a region located in the interior and away from the boundary of Ω R. After choosing ρ smaller and replacing ω by a smaller measurable set if it is necessary, we can assume that ω B ρ (0), ω / B ρ (0) and B ρ(0) Ω 3ρ 4, so that E = ω ( ρ 4, +ρ 4 ) B ρ (0, 1 ) has positive Lebesgue measure inside B ρ (0, 1 ) and B ρ (0, 1 ) Ω [0, 1]. Then, a second application of Theorem 3 and (.7) gives (.11) u L (B ρ(0, 1 )) N u θ L (ω [ 1 4, 3 4 ])M 1 θ. 1.

6 6 J. APRAIZ AND L. ESCAURIAZA Also, (.7) and Theorem 3 with E = B R (x, y ) imply that 4 u L (B R (x 1,y 1)) N u θ L (B R4 (x,y )) M 1 θ, whenever B R (x 1, y 1 ) Ω [ 4, 4], (x, y ) is in B R (x 1, y 1 ) and 0 < R 1. In 4 particular, (.1) u L (B R (x 1,y 1)) N u θ L (B R (x,y )) M 1 θ, when B R (x 1, y 1 ) Ω [ 4, 4], (x, y ) is in B R (x 1, y 1 ) and 0 < R 1. 4 Because ρ is now fixed and Ω is compact there are R in (0, 1) and k, which depend on ρ and the geometry of Ω, such that for any (x 0, y 0 ) in Ω 3ρ 4 [ 1, ] there are k points (x 1, y 1 ), (x, y ),..., (x k, y k ) in Ω 3ρ 4 [ 1, ] with B R (x i, y i ) Ω [ 4, 4], (x i+1, y i+1 ) B R (x i, y i ), for i = 0,..., k 1 4 and (x k, y k ) = (0, 1 ). The later and (.1) show that (.13) u L (B R (x i,y i)) N u θ L (B R (x i+1,y i+1)) M 1 θ, for i = 0,..., k 1, while the iteration of (.13) gives (.14) u L (Ω 3ρ 4 [ 1,]) N k u θk L (B ρ(0, 1 ))M 1 θk. Combining then (.10), (.14) and (.11), one finds that Thus, u L (Ω [0,1]) N k+ u 1 θk+. θk+ L (ω [ 1 4, 3 4 ])M (.15) u L (Ω [0,1]) N u θ L (ω [ 1 4, 3 4 ])M 1 θ, for some new N and 0 < θ < 1, which depend on ρ, ω and the geometry of Ω but not on u. The fact that the inequality ( ) sinh e µ ω1 (a 1 + b ) 1 ( ae ωy + be ωy) dy, ω 1 holds, when ω 1 ω µ and a, b R and the orthonormality of {e j } in L (Ω) show that 1 (.16) a j + b j e Nµ u L (Ω [0,1]) e Nµ Ω 1 u L (Ω [0,1]), and (.) follows from (.16), (.15) and the definition of M in (.8). It is shown in [] that the null-controllability of the system (1.1) over Ω = (0, 1) with = 1 ρ(x) [ x (a(x) x ) + b(x) x + c(x)], (.17) δ a(x), ρ(x) δ 1, b(x) + c(x) δ 1, a.e. in [0, 1], 0

7 NULL CONTROL AND MEASURABLE SETS 7 is equivalent to the null-controllability of the system xz ρ(x) t z = fχ ω, 0 < x < 1, 0 < t < T, (.18) z(0, t) = z(1, t) = 0, 0 t T, z(x, 0) = z 0, 0 x 1, where ρ is a new function verifying (.17) and ω a new measurable set in [0, 1] with positive measure. The later follows from the bilipschitz change of variables used in []. Let then, 0 < ω1 < ω ω3 ωj... and {e j} denote the sequences of eigenvalues and L (Ω)-normalized eigenfunctions [9, Ch.VII] verifying { e j + ρ(x)ω j e j = 0, 0 < x < 1, e j (0) = e j (1) = 0. From [16] it suffices to show that (.) holds in order to find an interior nullcontrol f for (.18) verifying (1.). Extend then e j and ρ to [ 1, 1] by odd and even reflections respectively, and to all R as periodic functions of period. The extended e j is in C 1,1 (R) and verifies e j +ρ(x)ω je j = 0 in R, j = 1,.... As before, let u be defined by (.3), it verifies xu + y (ρ(x) y u) = 0, in R. By Chebyshev s inequality and defining E as ( ω [ 1 4, 3 4 ]) \ E = {(x, y) ω [ 1 4, 3 4 ] : u(x, y) / > ω [ 1 4, 3 4 ] u dxdy}, we have (.19) E 1 ω [ 1 4, 3 4 ] and u L (E) u dxdy. ω [ 1 4, 3 4 ] Set then u ɛ (x, y) = u( x ɛ, y ɛ ), it verifies xu ɛ + y (ρ(x/ɛ) y u ɛ ) = 0, in R and let v ɛ be the stream function of u ɛ, i.e., the solution to x v ɛ = ρ( x ɛ ) yu ɛ, y v ɛ = x u ɛ, v ɛ (0, 0) = 0, Then, f = u ɛ + iv ɛ is (1/δ)-quasiregular, i.e., f W 1, loc (R ), z f = ν(z) z f, ν(z) 1 δ 1 + δ, z C, and by the Ahlfors-Bers Representation Theorem [1] (See [7] or [6]) all the (1/δ)- quasiregular mappings f in B 4 can be written as f = F Ψ, where F = U +iv is holomorphic in B 4 and Ψ : B 4 B 4 is a (1/δ)-quasiconformal mapping, i.e., a (1/δ)-quasiregular homeomorphism from B 4 onto B 4 verifying (.0) z Ψ = ν(z) z Ψ, Ψ(0) = 0, Ψ(4) = 4, N 1 z 1 z 1 α Ψ(z1 ) Ψ(z ) N z 1 z α, when z 1, z B 4,

8 8 J. APRAIZ AND L. ESCAURIAZA for some 0 < α < 1 and N 1 depending on δ. Now, ɛe B ɛ, and from (.0), Ψ(ɛE) B N(ɛ) α. Choose then ɛ so that N(ɛ) α = 1. Thus, Ψ(ɛE) B 1, u ɛ = U Ψ, U L (B 4) = u L (B 4ɛ ) while the L -interior estimates for subsolutions of elliptic equations [1, 8.6], the periodicity and orthogonality of the eigenfunctions e j in L ([0, 1], ρ(x) dx), gives 1 u L (B 4ɛ ) u L (B 6ɛ ) e 6µ/ɛ a j + b j. Thus, U is harmonic in B 4, U L (B 4) e Nµ a j + b j and from (.19) (.1) U L (Ψ(ɛE)) u dxdy. ω [ 1 4, 3 4 ] All together, U verifies the conditions in Theorem 3 in B with R = 1 and the universal constant 0 < ρ 1 associated to the quantitative property of analyticity over B for bounded harmonic functions on B 4 [9, Ch. ]. From (.1), (.) holds provided we can find a lower bound for the Lebesgue measure of Ψ(ɛE) B 1. The lower bound follows from (.19) and the following rescaled version of [3, Theorem 1]: Let Ψ : B 4 B 4 be a (1/δ)-quasiconformal mapping with Ψ(0) = 0 and E B 4 be a measurable set. Then, there is N = N(δ) such that E 1 δ /N Ψ(E) N E δ. Remark 1. Theorem 3 also implies the version of (.) appearing in [17]. For if Ω, A and V are as above and u(x, y) = a j e ωjy e j (x), u verifies (.5) and x α u(., 0) L (Ω) M α!(ρ) α, for α N n, with M = e Nµ 1 Thus, u(., 0) has an analytic extension to a ρ-neighborhood of Ω and after a finite number of applications of Theorem 3 and a suitable covering argument we get In particular, u(., 0) L (Ω) N u(., 0) θ L (ω) M 1 θ. a j e Nµ ω a j e j dx, with N = N( ω, Ω, r, δ). a j 1

9 NULL CONTROL AND MEASURABLE SETS 9 Proof of Theorem. Let u be defined by (.3). From [16], one can find a boundary control h verifying (1.3) provided there is N = N( γ, Ω, r, δ) such that the inequality (.) a j + b j e Nµ γ [ 1 4, 3 4 ] ( aj e ωjy + b j e ωjy) e j ν dσdy, holds for µ ω 1 and all sequences a 1, a,... and b 1, b,.... Here, ν, σ and denote respectively the exterior unit normal vector to Ω, the surface measure on Ω and the conormal derivative for y e + on Ω R, ν = A e ν. We may also assume that 0 Ω, γ B ρ Ω, where B ρ Ω is the region above the graph of a real analytic function, ϕ : B ρ R n 1 R, as in (.1). From [16, 3 ()], there is N such that (.3) with a j + b j u ν = 1 e Nµ u ν L ( Ω [ 1,]), ( aj e ωjy + b j e ωjy) e j ν. From (.6) and (.1), there are N = N(r, δ) and ρ = ρ(r, δ) such that h(x, y) = u n (x, ϕ(x ), y) verifies α x β y h L (B ρ [ 4,4]) M ( α + β)!(ρ) α β, for α N n 1, β N, with M as in (.8) and provided that B ρ Ω is a coordinate chart of Ω as in (.1). This fact, a suitable covering argument of Ω and the successive iteration of a finite number of applications of the three-spheres type inequalities associated to the obvious extension of Theorem 3 for real analytic functions defined over real analytic hypersurfaces in R n+1 (See [13, pp ] for a quantitative version of the real analyticity of composite functions) imply that there are N = N( γ, r, δ) and θ = θ( γ, r, δ) such that (.4) u ν L ( Ω [ 1,]) N u ν θ L (γ [ 1 4, 3 4 ])M 1 θ. Finally, (.) follows from (.3) and (.4). Remark. The extension of Theorem 1 to 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, (.5) u = 0, on Ω [0, T ], u(0) = u 0, in Ω. with Ω as in (.1), a αβ ij verifying (1.6) and (.6) γ a αβ ij (x) γ!δ γ 1, when x Ω, γ N n, for some 0 < δ 1 is now obvious: the symmetry, coerciveness and compactness of the operator L (Ω) m W 1, 0 (Ω) m, mapping f = (f 1,..., f m ) into the unique solution u = (u 1,..., u m ) to { i (a αβ ij ju β ) Λu α = f α, in Ω, α = 1,..., m, u = 0, in Ω ν

10 10 J. APRAIZ AND L. ESCAURIAZA where Λ > 0 is sufficiently large [11, Prop..1], gives the existence of a complete system {e k } in L (Ω) m, e k = (e 1 k,..., em k ), of eigenfunctions verifying { i (a αβ ij je β k ) + ω k eα k = 0, in Ω, α = 1,..., m, e k = 0, in Ω with eigenvalues 0 ω 1... ω k... and lim k + ω k = +. By separation of variables, the Green s matrix for the system (.5) over Ω R is the m m matrix (.7) Γ(x, y, t s) = + k=1 e ω k (t s) e k (x) e k (y). Moreover, the interior and boundary regularity for the elliptic system y+ i (a αβ ij j ) in Ω R, shows that (.5) holds for u as in (.3) but with e k replacing e k ([1], [11, Ch. II]). These and [16] suffice to find a control function f for the system (.5) verifying (1.). Furthermore, if you wish to get bounds on the regularity of the control function f, [16] shows that suffices to know that and e k Hs (Ω) N s (1 + ω k ) s, when k, s 0 (.8) #{k 1 : 0 ω k µ} Nµ n, when µ 1, for some N which does not depend on µ 1. The first holds from elliptic regularity and (.6), while (.8) follows from the Gaussian estimates verified by Γ; i.e., there are N and κ [8, Corollary 4.14] such that (.9) Γ(x, y, t) N(1 t) n e Λt κ x y /t, for x, y R n and t > 0. To verify the last claim, observe that (.7), (.9) and the orthonormality of the eigenfunctions e k give (.30) Γ(x, y, t) dxdy = e ω k t Ne Λt Ω t n, k 1 Ω Ω and it suffices to choose t = 1/µ in (.30) to get (.8). In particular, the methods in [16] also generate a control function f in C0 ((0, T ), C (Ω)). The null-controllability of the system t u + ( 1) m m u = f(x, t)χω, in Ω (0, T ], (.31) u = u = = m 1 u = 0, in Ω (0, T ), u(0) = u 0, in Ω, m, is better managed with the approach in [17]: if {e j } and 0 ω1 m ωk m... are the eigenvectors and eigenvalues for m in W m, 0 (Ω), { ( 1) m m e j ωj m e j = 0, in Ω, e j = e j = = m 1 e j = 0, in Ω, define u(x, y) = e ωjy, a j X j (y)e j (x), with X j (y) = wj m µ πi e ωje m y, for m odd, for m even.

11 NULL CONTROL AND MEASURABLE SETS 11 It verifies { y m u + m u = 0, in Ω R, u = u = = m 1 u = 0, in Ω R. Again, u verifies (.5) [1] and from Theorem 3 applied to u(., 0) as in Remark 1, a j e Nµ 1 m a j e j dx, with N = N( ω, Ω, r, δ, m). ωj m µ ω µ ω m j From [17], the last inequality suffices to find a control f in L (ω (0, T )) for (.31). Remark 3. The proof of Theorem for the scalar case is based on the bound (.3). In the literature, it is is obtained from (.16) and the interpolation inequality below which has been proved which is proved with Carleman inequalities: there are N and θ such that the inequality u L (Ω [0,1]) N u ν θ L ( Ω [ 1,]) u 1 θ L (Ω [ 3,3]), holds when u verifies (.4). However, the authors are not aware wether the interpolation inequality u L (Ω [0,1]) N u ν θ L ( Ω [ 1,]) u 1 θ L (Ω [ 3,3]), with ( u ν )α = a αβ ij ju β ν i, α = 1,..., m, holds for solutions u to the corresponding analog elliptic system and lateral boundary conditions: { i (a αβ ij ju β ) + yu 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 Theorem to general parabolic systems with analytic coefficients and also analytic lateral boundaries. In spite of that, the inverse mapping theorem shows that there is ρ > 0 such that the mapping Ω (0, ρ) U ρ, (Q, t) Q + tν(q), is an analytic diffeomorphism onto U ρ = {x R n \ Ω : 0 < d(x, Ω) < ρ} and the null-controllability for parabolic systems with analytic coefficients over Ω U ρ and with controls acting over ω = U 3ρ \ U ρ has been stablished in Remark. From 4 these, standard arguments show that at 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) = u 0, in Ω, can be null-controlled with controls g acting over the full lateral boundary of Ω. Remark 4. To prove Theorem 1 when Ω, A and V are globally smooth and A and V are only real analytic in a neighborhood of ω, one may assume that (.3) (.33) ω B R, ω / B R 1/, B 4R Ω 4 α A(x) + α V (x) α!δ α 1, when x B 4R, α N n,

12 1 J. APRAIZ AND L. ESCAURIAZA for some fixed 0 < R, δ 1 and the goal is to show that (.) holds. The Carleman and interpolation inequalities stablished in [16, 3 (1)] for solutions to elliptic operators with smooth coefficients and for Ω smooth show that there is N = N(A, V, Ω, R) such that the inequalities 1 (.34) a j + b j e Nµ u L (B R (0, 1 )), hold when µ ω 1, for all sequences a 1, a,..., b 1, b,... and for u as in (.3). Because u verifies u + yu = 0, in B 4R (0, 1 ) and (.33) holds, u is a local solution to an elliptic equation with local real analytic coefficients and there are N = N(δ) and ρ = ρ(δ), 0 < ρ 1, such that ( ) 1 (.35) x α y β N( α + β)! u L (B R (0, 1 )) u dxdy, (Rρ) α +β B 4R (0, 1 ) when α N n and β 1. The later follows from the corresponding result for R = 1 and rescaling ([, Ch. 5], [14, Ch. 3]). The orthonormality of {e j } in Ω and (.35) show that (.36) α x β y u L (B R (0, 1 )) M( α + β)!(rρ) α β, for α N n, β 0, with M as in (.8). Finally, Theorem 3 in B R (0, 1 ) with E = ω ( R 4, ) +R 4 B R (0, 1 ), (.36) and (.3) show that (.37) u L (B R (0, 1 )) N u θ L (ω [ 1 4, 3 4 ])M 1 θ, and (.) follows from (.34), (.37) and (.8). To prove Theorem with Ω, A and V real analytic near γ and smooth elsewhere, one may assume that 0 is in Ω and that there are 0 < R, δ 1 and ϕ : B R Rn 1 R such that (.38) γ B R Ω, γ / B R Ω 1 4, ϕ(0 ) = 0, α ϕ(x ) α!δ α 1, when x B 4R, α N n 1, (.39) α A(x) + α V (x) α!δ α 1, when x B 4R Ω, α N n, 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 )} and the goal is to show that (.) holds. The Carleman and interpolation inequalities stablished in [16, 3 ()] for solutions to elliptic operators with smooth coefficients and with Ω smooth show that there is N = N(A, V, Ω, R) such that the inequalities 1 (.40) a j + b j e Nµ u ν L (B R (0, 1 ) Ω R),

13 NULL CONTROL AND MEASURABLE SETS 13 hold when µ ω 1, for all sequences a 1, a,..., b 1, b,... and for u as in (.3). Because u verifies { u + yu = 0, in B 4R (0, 1 ) Ω R, u = 0, in B 4R (0, 1 ) Ω R, and (.39) holds there are N = N(δ) and ρ = ρ(δ), 0 < ρ 1, such that ( (.41) x α y β N( α + β)! u L (B R (0, 1 ) Ω R) u dxdy (Rρ) α +β B 4R (0, 1 ) Ω R when α N n and β 1. The later follows from the corresponding result for R = 1 and rescaling ([, Ch. 5], [14, Ch. 3]). The orthonormality of {e j } in Ω and (.41) show that (.4) α x β y u L (B R (0, 1 ) Ω R) M( α + β)!(rρ) α β, for α N n, β 0, with M as in (.8). Finally, the obvious extension of Theorem 3 for real analytic functions over B R (0, 1 ) Ω R [13, pp ] with E = γ ( R 4, ) +R 4 B R (0, 1 ) Ω R, (.4) and (.38) show that (.43) u ν L (B R (0, 1 ) Ω R) N u ν θ L (γ [ 1 4, 3 4 ])M 1 θ, and (.) follows from (.40), (.43) and (.8). 3. Proof of Theorem 3 First we recall Hadamard s three-circle theorem [19] and prove two Lemmas before the proof of Theorem 3. Theorem 4. Let F be a holomorphic function of a complex variable in the ball B r. Then, the following is valid for 0 < r 1 r r, r F L (B r) F θ log r L (B r1 ) F 1 θ L (B r ), θ =. log r r 1 Lemma 1. Let f be holomorphic in B 1, f(z) 1 in B 1 and E be a measurable set in [ 1 5, 1 5 ]. Then, there are N = N( E ) and γ = γ( E ) such that f L (B 1 ) N f γ L (E) Proof. For n 1, there are n + 1 points with 1 5 x 0 < x 1 < < x n 1 5, with x i E, i = 0,..., n and x i x i 1 E n+1, i = 1,..., n. For example, x 0 = inf E, ( x i = inf E [x i 1 + E n+1, 1 5 ). ] Let n j i P n (z) = f(x i ) (z x j) j i (x i x j ) Then, P n (z) f L (E) E n n i=0 i=0 ) 1 (n + 1) n ( ) n 3 i!(n i)! f L (E), for z 1 E.,

14 14 J. APRAIZ AND L. ESCAURIAZA By Cauchy s formula, 1 f(z) P n (z) = πi ξ =1 The last two inequalities give (3.1) f L (B 1 ) f L (E) f(ξ)(z x 0 )... (z x n ) ( ) n (ξ z)(ξ x 0 )... (ξ x n ) dξ 7, for z 1 8. ( ) n ( ) n 3 7 +, for all n 1, E 8 and the minimization in the n-variable of the right hand side of (3.1) implies Lemma 1. Lemma. Let f be analytic in [0, 1], E be a measurable set in [0, 1] and assume there are positive constants M and ρ such that (3.) f (k) (x) Mk!(ρ) k, for k 0, x [0, 1]. Then, there are N = N(ρ, E ) and γ = γ(ρ, E ) such that f L ([0,1]) N f γ L (E) M 1 γ. Proof. (3.) implies that f has a holomorphic extension to D ρ = 0 x 1 B(x, ρ), 5 with f M in D ρ. Write [0, 1] as a disjoint union of ρ non-overlapping closed intervals of length ρ 5. Among them there is at least one, I = [x 0 δ 5, x 0 + δ 5 ], such that E I δ E 5. Then, g(z) = f(x 0 + δz)/m is holomorphic in B 1, E x0,ρ = ρ 1 (E I x 0 ) is measurable in [ 1 5, 1 5 ] with measure bounded from below by E 5, g L (E x0,ρ) f L (E) and applying Lemma 1 to g, we have (3.3) f L (B ρ (x0)) N f γ L (E) M 1 γ, for some 0 x 0 1. By making successive iterations of Hadamard s three-circle theorem (a finite number which depends only ρ) with a suitable chain of threecircles contained in D ρ of radius comparable to ρ and with centers at points in [0, 1], while recalling that on the largest ball f M, we get that (3.4) f L (0,1) N f θ L (B ρ (x0))m 1 θ, θ = θ(ρ). Finally, Lemma follows from (3.4) and (3.3). Proof of Theorem 3. We may assume R = 1. Let x B 1. Using spherical coordinates centered at x, E {t [0, 1] : x + tz E} dz, S n 1 and there is at least one z S n 1 with {t [0, 1] : x + tz E} E /(ω n ), with ω n the surface measure on S n 1. Set ϕ(t) = f(x + tz). From (1.5), ϕ satisfies (3.), ϕ L (E z) f L (E) and Lemma gives (3.5) f L (B 1 ) N f γ L (E) M 1 γ. Finally, setting Ẽ = {x E : f(x) / f dx}, E

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16 16 J. APRAIZ AND L. ESCAURIAZA [4] N. S. Nadirashvili, Estimation of the solutions of elliptic equations with analytic coefficients which are bounded on some set. Moscow Univ. Math. Bull. 34, (1979) [5] J. Le Rousseau, L. Robbiano, Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces. Inventiones mathematicae 183, (011) [6] J. Le Rousseau, G. Lebeau, On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations. ESAIM Control Optim. Calc. Var., doi: /cocv/ [7] D.L. Russel, A unified boundary controllability theory for hyperbolic and parabolic partial differential equations. Studies in Appl. Math. 5 (1973) [8] S. Vessella, A continuous dependence result in the analytic continuation problem. Forum Math. 11,6 (1999) [9] H.F. Weinberger, A first course in partial differential equations with complex variables and transform methods. New York: Dover Publications (1995). (J. Apraiz) Universidad del País Vasco/Euskal Herriko Unibertsitatea, Departamento de Matemática Aplicada, Escuela Universitaria Politécnica de Donostia-San Sebastián, Plaza de Europa 1, 0018 Donostia-San Sebastián, Spain. address: jone.apraiz@ehu.es (L. Escauriaza) Universidad del País Vasco/Euskal Herriko Unibertsitatea, Dpto. de Matemáticas, Apto. 644, Bilbao, Spain. address: luis.escauriaza@ehu.es