Null-Control and Measurable Sets

NULL–CONTR OL AND MEASURABLE SETS J. APRAIZ AND L. ESCA URIAZA Abstract. W e prov e the i nterior and b oundary null–con trollability of some paraboli c evolutions with con trols acting o v er measurable sets. 1. Introduction The control for ev olution equations aims to drive t he solution to a prescr ib e d state star ting fr o m a cer tain initial condition. One acts on the equation through a source term, a s o -called distributed control, o r through a b oundary condition. T o achiev e genera l results one wishes for the co n trol to only act in pa rt of the domain or its b oundary and to have as m uch latitude as p oss ible in the choice of the control region: loca tion, size, shap e. Here, we fo cus on the heat eq uation in a smo oth a nd b ounded do ma in Ω in R n for a time interv al (0 , T ), T > 0 and for a distributed c o nt rol f w e consider (1.1)      △ u − ∂ t u = f ( x, t ) χ ω ( x ) , in Ω × (0 , T ) , u = 0 , on ∂ Ω × [0 , T ] , u (0) = u 0 , in Ω . Here, ω ⊂ Ω is a n interior co ntrol region. The null controllability of this equa tion, i.e., the existence for any u 0 in L 2 (Ω) of a co nt rol f in L 2 ( ω × (0 , T )) with (1.2) k f k L 2 ( ω × (0 ,T )) ≤ N k u 0 k L 2 (Ω) , such that u ( T ) = 0, was prov ed in [14] by means of lo c a l Ca rleman estimates for the elliptic op erator △ + ∂ 2 y ov er Ω × R . A second appr oach based on global Carleman estimates for the backw a rd par ab olic op er a tor △ + ∂ t [9], also led to the null controllability of the hea t equation. The first appr o ach has b een used for the treatment o f time-indep endent pa rab olic op erato rs a s so ciated to self-adjoint elliptic op era tors, while the second a llows to address time-dependent non-selfadjoint parab olic op era tors and s e mi-linear evolutions. The metho d intro duced in [14] w as further extended to s tudy thermo elasticity [15], ther mo e lastic plates [4 ] and s emigroups generated by fractiona l or ders of el- liptic op erator s [18]. It has a lso b een used to prove null controllabilit y in the ca se of non smo oth co efficients [5 , 2 3]. The metho d of [14] has a lso b e extended to treat some non-selfadjoint ca ses, e.g. non symmetr ic systems [13] and all 1 -dimensional time-indepe nden t pa rab olic equations [2]. In the ab ov e works, the control reg ion ω is always as s umed to co nt ain an o p e n ball. Also, the cost of co n trollability (the smallest co nstant N found for the ineq ua l- it y (1 .2)) de p ends o n this fact. The reas on for these is that the main technique us ed 1991 Mathematics Subje ct Classific ation. Primary: 35B37. Key wor ds and phr ases. Null-cont rollabili t y . The authors are suppor ted by MEC grant, MTM2004-03029. 1 2 J. APRAIZ AND L. ES CA URIAZA in the arguments, Carleman inequalities, r equires to construct suitable Carleman weigh ts: a role for functions which r equires smo o thness (at least C 2 ) and to hav e the extreme v alues in pro per regio ns a sso ciated to the co n trol region ω , the lar ger bo dy Ω and po ssibly the v alue of T > 0. The constructio n of such functions s eems to b e not p os s ible, when ω do es not contain a ball. Motiv ated b y these facts J .P . Puel a nd E. Zuazua rais ed the question wether the n ull controllability o f the hea t equation is p ossible when the control region is a measura ble s et. A p ositive pa r tial answer to this questio n was expla ined by the second author a t the June 200 8 meeting Contr ol of Physic al Systems and Partial Differ ential Equations held at the Institute Henri Poincar´ e. Her e, we g ive a for mal account of the r esults. Theorem 1. L et n ≥ 2 . Then, △ − ∂ t is nu l l-c ontr ol lable at al l p ositive times, with distribute d c ont r ols acting over a me asur able set ω ⊂ Ω with p ositive L eb esgue me asur e, when △ = ∇ · ( A ( x ) ∇ · ) + V ( x ) , is a self-adjoint el lip tic op er ator, t he c o efficients matrix A is smo oth in Ω , V is b ounde d in Ω and b oth ar e r e al-analytic in an op en neighb orho o d 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 ar e me asura ble functions in Ω = (0 , 1) . In r egard to b o undary null controllability , i.e., the exis tence for any u 0 in L 2 (Ω) of a c o nt rol h in L 2 ( γ × (0 , T )) with (1.3) k h k L 2 ( γ × (0 , T )) ≤ N k u 0 k L 2 (Ω) , such that the s olution to (1.4)      △ u − ∂ t u = 0 , in Ω × (0 , T ) , u = h ( x, t ) χ γ ( x ) , on ∂ Ω × [0 , T ] , u (0) = u 0 , in Ω , verifies u ( T ) ≡ 0, we hav e the following result. Theorem 2. L et n ≥ 2 . Then, △ − ∂ t is nul l-c ontr ol lable at al l times T > 0 with b oundary c ontr ols acting over a me asur able set γ ⊂ ∂ Ω with p ositive surfac e me asur e when △ = ∇ · ( A ( x ) ∇ · ) + V ( x ) is a self-adjoint el lip tic op er ator, t he c o efficients matrix A is smo oth in Ω , V is b ounde d in Ω and b oth ar e r e al-analytic in an op en neighb orho o d of γ in Ω . The results in Theorems 1 and 2 follow from a s traightforw ard application of the linea r co ns truction of the control function for the sys tems (1.1) and (1.4) devel- op ed in [1 4] and the following observ ability inequality or pr opagatio n of smallness estimate establishe d in [2 6] (See also [21 ] and [22]). Theorem 3. Assu me t hat f : B 2 R ⊂ R n − → R is a r e al-analytic fun ction verifying (1.5) | ∂ α f ( x ) | ≤ M | α | ! ( ρR ) | α | , when x ∈ B 2 R , α ∈ N n , NULL–CONTR OL AND MEASURABLE SETS 3 for some M > 0 and 0 < ρ ≤ 1 and E ⊂ B R 2 is a me asur able set with p ositive me a- sur e. Then, t her e ar e p ositive c onstants N = N ( ρ, | E | / | B R | ) and θ = θ ( ρ, | E | / | B R | ) such that k f k L ∞ ( B R ) ≤ N  — Z E | f | dx  θ M 1 − θ . The exp erts will realize that the word smo oth desc ribing the reg ularity (aw ay from the mea s urable set) of ∂ Ω, A and V in Theo rems 1 and 2 can b e r eplaced by ∂ Ω is C 2 , A is Lipschitz in Ω a nd V is b ounded (See [9, 14, 15, 24]). In fact, the C 2 regular ity of ∂ Ω c a n b e relaxed to r equire that either ∂ Ω is C 1 or ther e is α ∈ (0 , 1] such that ( P − Q ) · ν ( P ) ≥ − | P − Q | 1+ α , for all P, Q ∈ ∂ Ω , where ν ( P ) is the exterior unit normal vector to ∂ Ω. The la ter holds when Ω is either a conv ex doma in, a p olyhedron in R n or when ∂ Ω can be lo cally written a s the graphs of Lipschitz functions which ar e the sum of a conv ex a nd a C 1 ,α function ov er R n − 1 . T o s implify the ex po sition and to show the strength o f Theorem 3, w e give the pro of of Theorems 1 and 2 under the a s sumptions that ∂ Ω, A a nd V are glo ba lly real analytic. W e do it b ecause it makes clear how the construction a lg orithm of the co nt rol function in [14] and T he o rem 3 can also b e a pplied to prove the interior nu ll-controllability (Theorem 1) for o ther parab olic evolutions whose corresp onding observ a bilit y or sp ectra l inequalities (suitable Car leman inequalities ) are other wise unknown. Examples of these pa rab olic evolutions ar e the ones as so ciated to self- adjoint elliptic systems with unknowns u = ( u 1 , . . . , u m ), L α u = ∂ i ( a αβ ij ( x ) ∂ j u β ) , α = 1 , . . . , m with a αβ ij = a β α j i , for α, β = 1 , . . . , m , i , j = 1 , . . . , n , and with co efficients matr ices verifying for some δ > 0 the strong ellipticity co ndition, X i,j,α,β a αβ ij ( x ) ξ α i ξ β j ≥ δ X i,α | ξ α i | 2 , when ξ ∈ R nm , x ∈ R n , or the more ge ne r al Legendre- Ha damard co ndition (1.6) X i,j,α,β a αβ ij ( x ) ξ i ξ j η α η β ≥ δ | ξ | 2 | η | 2 , when ξ ∈ R n , η ∈ R m , x ∈ R n . W e recall that the L a m´ e 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 j α ) + λδ j β δ αi , are examples of sys tems verifying (1.6). Here, a αβ ij , µ and λ can either b e constants or real analytic functions o n Ω. It also ma kes clea r that under suc h h yp othesis o ne may repla ce the Carlema n inequalities used in the literature, to prove the o bserv ability o r propaga tion of small- ness inequa lities necessa ry in the pr o cess o f applying the construction algor ithm in [14, 15], by the simpler application o f Theore m 3. Of course, it has the drawback that it requires more s mo othness on the oper ators and the b ounda ry of Ω but on the contrary , one can handle with Theo rem 3 and the constructio n metho ds in [1 4, 15] 4 J. APRAIZ AND L. ES CA URIAZA the null-con trollability of other parab olic evolutions with internal controls: like the second or der evolutions expla ined ab ove or fo r higher order evolutions as ∂ t u + ( − 1) m △ m u, m = 2 , . . . , with Dirichlet b oundary c o nditions on ∂ Ω, u = ∇ u = · · · = ∇ m − 1 u = 0. In sec tion 2 we g ive the pro ofs o f Theorems 1 and 2. W e explain how to ex tend Theorem 1 to the evolutions (2.1 9) and (2.21) in Remar k 2, while the pro blems we hav e to extend Theorem 2 to these evolutions ar e expla ined in Remark 3, for the simpler case of parab olic sy s tems. F or the sake of completenes s, we inc lude a pr o of of Theore m 3 in section 3 . It is built with ideas taken fr om [16], [21] and [26]. 2. P r oof of Theorems 1 and 2 W e b egin 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 all x ∈ Ω , ξ ∈ R n , Ω is a b ounded op en set in R n , n ≥ 2, with a r eal ana lytic b ounda ry and A , V ar e real analytic in Ω, i.e., there a re 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 a re a new co o r dinate system (where x = 0) a nd a real analytic function ϕ : B ′ r ⊂ R n − 1 − → R verifying (2.1) ϕ (0 ′ ) = 0 , | ∂ α ϕ ( x ′ ) | ≤ | α | ! δ −| α |− 1 , when x ′ ∈ B ′ r , α ∈ N n − 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 ′ ) } . When E is a measura ble s et, | E | will denote its Leb esg ue or sur fa ce measure . Pr o of of The or em 1. W e may assume that the eigenv alues with zero Dirichlet co n- dition for △ = ∇ · ( A ( x ) ∇ · ) + V ( x ) on Ω are a ll p ositive, 0 < ω 2 1 < ω 2 2 ≤ ω 2 3 ≤ · · · ≤ ω 2 j ≤ . . . and { e j } denotes the se q uence of L 2 (Ω)-normalized eigenfunctions, ( △ e j + ω 2 j e j = 0 , in Ω , e j = 0 , in ∂ Ω . When ω ⊂ Ω is mea surable with p ositive Leb esgue measure, the metho d in [14] shows that one find a nd L 2 ( ω × (0 , T )) control function f verifying (1.2) for the system (1.1), provided there is N = N ( | ω | , Ω , r , δ ), such that the inequality (2.2) X ω j ≤ µ a 2 j + b 2 j ≤ e N µ Z Z ω × [ 1 4 , 3 4 ] | X ω j ≤ µ  a j e ω j y + b j e − ω j y  e j | 2 dxdy , holds for µ ≥ ω 1 and a ll sequences a 1 , a 2 , . . . and b 1 , b 2 , . . . Let then, (2.3) u ( x, y ) = X ω j ≤ µ  a j e ω j y + b j e − ω j y  e j , it satisfies, △ u + ∂ 2 y u = 0, in Ω × R , u = 0 on the la teral b oundar y o f Ω × R and u is r eal a nalytic in Ω × R . Moreov er, g iven ( x 0 , y 0 ) in Ω × R and R ≤ 1, there are NULL–CONTR OL AND MEASURABLE SETS 5 N = N ( r, δ ) and ρ = ρ ( r, δ ) such tha t (2.4) k ∂ α x ∂ β y u k L ∞ ( B R ( x 0 ,y 0 ) ∩ Ω × R ) ≤ N ( | α | + β )! ( Rρ ) | α | + β — Z B 2 R ( x 0 ,y 0 ) ∩ Ω × R | u | 2 dxdy ! 1 2 , when α ∈ N n and β ≥ 1 . F or the later see [20, Chapter 5], [12, Chapter 3 ]. The o rthonormality of { e j } in Ω and (2.4) with R = 1 imply (2.5) k ∂ α x ∂ β y u k L ∞ (Ω × [ − 5 , 5]) ≤ e N µ ( | α | + β )! ρ −| α |− β   X ω j ≤ µ a 2 j + b 2 j   1 2 , for α ∈ N n , β ≥ 0 , and there is C > 0 s uc h that r e placing the c o nstants N and ρ in (2.5) by C N and ρ/C respe ctively , u has a rea l analytic extension to Ω ρ × [ − 4 , 4], Ω ρ = { x ∈ R n : d ( x, Ω) ≤ ρ } , with (2.6) k ∂ α x ∂ β y u k L ∞ (Ω ρ × [ − 4 , 4]) ≤ M ( | α | + β )!(2 ρ ) −| α |− β , for α ∈ N n , β ≥ 0 , and M = e N µ   X ω j ≤ µ a 2 j + b 2 j   1 2 . F or ( x 0 , y 0 ) in Ω × [0 , 1 ] with d ( x 0 , ∂ Ω) = ρ , we hav e B 2 ρ ( x 0 , y 0 ) ⊂ Ω ρ × [ − 4 , 4], and if we a pply Theo rem 3 to the real a nalytic extension of u in B 2 ρ ( x 0 , y 0 ) with E = B ρ 4 ( x 0 , y 0 ), (2.6 ) implies there is 0 < θ 1 < 1 such that k u k L ∞ ( B ρ ( x 0 ,y 0 )) ≤ N k u k θ 1 L ∞ ( B ρ 4 ( x 0 ,y 0 )) M 1 − θ 1 , F rom this a nd a suitable covering argument we get (2.7) k u k L ∞ (Ω × [0 , 1]) ≤ N k u k θ 1 L ∞ (Ω ρ × [ − 1 , 2]) M 1 − θ 1 , with Ω ρ = { x ∈ Ω : d ( x, ∂ Ω) ≥ 3 ρ 4 } . Thu s, fr o m Theorem 3 and without Car le ma n inequalities it is po ssible to b ound except for the factor M a ll the infor mation re la ted to u ov er Ω × [0 , 1] by the information o n u over Ω ρ × [ − 1 , 2], a region lo c ate d inside Ω × R . W e may als o a ssume that E = ω × ( 1 − ρ 4 , 1+ ρ 4 ) ⊂ B ρ 2 (0 , 1 2 ) has p o sitive measur e inside B ρ 2 (0 , 1 2 ), B 8 ρ (0 , 1 2 ) ⊂ Ω ρ × [0 , 1] a nd a seco nd applicatio n of Theore m 3 gives (2.8) k u k L ∞ ( B ρ (0 , 1 2 )) ≤ N k u k θ 2 L 2 ( ω × [ 1 4 , 3 4 ]) M 1 − θ 2 . Pro ceeding as in [1 4], w e use a co vering of Ω ρ × [ − 1 , 2] and successive applica tions of Theorem 3, with E b eing a centered-mo ving ball with fixed ra dius R / 2 depending on ρ and the geometry of Ω, with the ball of the same center a nd radius 2 R contained Ω ρ × [ − 4 , 4 ], and where in the la st applications of Theo rem 3, E = B ρ (0 , 1 2 ). Th us, Theorem 3 and (2.6) imply there is 0 < θ 3 < 1 with (2.9) k u k L ∞ (Ω ρ × [ − 1 , 2]) ≤ N k u k θ 3 L ∞ ( B ρ (0 , 1 2 )) M 1 − θ 3 . 6 J. APRAIZ AND L. ES CA URIAZA The orthono rmality of { e j } in Ω and the inequality e − µ  sinh ω 1 ω 1 − 1   a 2 + b 2  ≤ Z 1 0  ae ω y + be − ω y  2 dy , when ω 1 ≤ ω ≤ µ, a, b ∈ R , give (2.10)   X ω j ≤ µ a 2 j + b 2 j   1 2 ≤ e N µ k u k L 2 (Ω × [0 , 1]) ≤ e N µ | Ω | 1 2 k u k L ∞ (Ω × [0 , 1]) and (2.2) follows fro m (2.10), (2.7), (2.9) and (2 .8). In [2] it is s hown that the null-cont rolla bility of the s ystem (1.1) ov e r Ω = (0 , 1) with △ = 1 ρ ( x ) [ ∂ x ( a ( x ) ∂ x ) + b ( x ) ∂ x + c ( x )] , (2.11) δ ≤ a ( x ) , ρ ( x ) ≤ δ − 1 , | b ( x ) | + | c ( x ) | ≤ δ − 1 , a.e. in [0 , 1 ] , is equiv a le n t to the n ull-controllability of the system (2.12)      ∂ 2 x z − ρ ( x ) ∂ t z = f χ ω , 0 < x < 1 , 0 < t < T , 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 (2.1 1) and ω a new measur able set in [0 , 1] with po sitive meas ure. The later follows from the bilipschitz change of v ariables used in [2]. Let then, 0 < ω 2 1 < ω 2 2 ≤ ω 2 3 ≤ · · · ≤ ω 2 j ≤ . . . a nd { e j } denote the sequence s of eigenv alues and L 2 (Ω)-normalized eigenfunctions verifying ( e ′′ j + ρ ( x ) ω 2 j e j = 0 , 0 < x < 1 , e j (0) = e j (1) = 0 . F rom [14], it suffices to show that (2.2) holds in or de r to find an interior n ull- control f for (2.12) verifying (1.2). Extend then e j and ρ to [ − 1 , 1 ] by o dd and even reflections re spe ctively , and to all R as p erio dic functions o f per io d 2. The extended e j is in C 1 , 1 ( R ) and verifies e ′′ j + ρ ( x ) ω j e j = 0 in R , j = 1 , 2 . . . . As befo r e, let u b e defined by (2.3), it verifies ∂ 2 x u + ∂ y ( ρ ( x ) ∂ y u ) = 0 , in R 2 . By Chebyshev’s inequality , defining  ω × [ 1 4 , 3 4 ]  \ E = { ( x, y ) ∈ ω × [ 1 4 , 3 4 ] : | u ( x, y ) | / 2 > — Z ω × [ 1 4 , 3 4 ] | u | dxdy } , we hav e (2.13) | E | ≥ 1 2 | ω × [ 1 4 , 3 4 ] | a nd k u k L ∞ ( E ) ≤ 2 — Z ω × [ 1 4 , 3 4 ] | u | dxdy . Set u ǫ ( x, y ) = u ( x ǫ , y ǫ ), it verifies ∂ 2 x u ǫ + ∂ y ( ρ ( x/ǫ ) ∂ y u ǫ ) = 0 , in R 2 , NULL–CONTR OL AND MEASURABLE SETS 7 and let v ǫ be the str e am function of u ǫ , i.e., the solution to      ∂ x v ǫ = − ρ ( x ǫ ) ∂ y u ǫ , ∂ y v ǫ = ∂ x u ǫ , v ǫ (0) = 0 , Then, f = u ǫ + iv ǫ is (1 /δ )-quasiregular, i.e., f ∈ W 1 , 2 lo c ( R 2 ) , ∂ z f = ν ( z ) ∂ z f , | ν ( z ) | ≤ 1 − δ 1 + δ , z ∈ C , and by the Ahlfors-Bers Repres en tation Theor em [1] (See also [7] or [6]), any (1 /δ )- quasireg ula r mapping f in B 4 can b e written as f = F ◦ Ψ , where F = U + iV is holo mo rphic in B 4 and Ψ : B 4 − → B 4 is a (1 /δ )-quasiconformal mapping, i.e. a (1 /δ )-qua siregular homeomo rphism from B 4 onto B 4 verifying (2.14) ∂ z Ψ = ν ( z ) ∂ z Ψ , Ψ (0) = 0 , Ψ(4) = 4 , N − 1 | z 1 − z 2 | 1 α ≤ | Ψ( z 1 ) − Ψ ( z 2 ) | ≤ N | z 1 − z 2 | α , when z 1 , z 2 ∈ B 4 , for so me 0 < α < 1 and N ≥ 1 dep ending on δ . Now, ǫE ⊂ B 2 ǫ , and from (2.14), Ψ( ǫE ) ⊂ B C ( 2 ǫ ) α . Cho ose then ǫ so that N (2 ǫ ) α = 1 2 . Thus, Ψ( ǫE ) ⊂ B 1 2 , u ǫ = U ◦ Ψ, k U k L ∞ ( B 4 ) = k u k L ∞ ( B 4 ǫ ) while the L ∞ int erior es timates for subsolutions of elliptic equations [11, § 8 .6], the per io dicity and orthogo na lit y of the eigenfunctions e j in L 2 ([0 , 1 ] , ρ dx ), gives k u k L ∞ ( B 4 ǫ ) . k u k L 2 ( B 6 ǫ ) . e 6 µ/ǫ   X ω j ≤ µ a 2 j + b 2 j   1 2 . Thu s, U is harmonic in B 4 , k U k L ∞ ( B 4 ) ≤ e N µ   X ω j ≤ µ a 2 j + b 2 j   1 2 and from (2.13) (2.15) k U k L ∞ (Ψ( ǫE )) ≤ 2 — Z ω × [ 1 4 , 3 4 ] | u | dxdy . All tog ether, U verifies the conditions in Theorem 3 in B 2 with R = 1, with the universal consta n t 0 < ρ ≤ 1 as s o ciated to the quantitativ e analyticity over B 2 of bo unded harmonic functions in B 4 . F rom (2 .1 5), (2.2) holds pr ovided we c an find a low er bo und for the Leb esgue measure of Ψ ( ǫE ) ⊂ B 1 2 . The low er b o und follows from (2.13) and the following resca le d version of [3, Theor em 1]: L et Ψ : B 4 − → B 4 b e a (1 /δ ) -quasic onformal mapping with Ψ (0 ) = 0 and E ⊂ B 4 b e a me asura ble set. Then, ther e is N = N ( δ ) such that | E | 1 δ / N ≤ | Ψ( E ) | ≤ N | E | δ .  8 J. APRAIZ AND L. ES CA URIAZA R emark 1 . Theorem 3 a lso implies the version of (2.2) a ppea ring in [15]. F or if Ω, A and V ar e as ab ov e a nd u ( x, y ) = X ω j ≤ µ a j e ω j y e j ( x ) , u verifies (2.4) and k ∂ α x u ( . , 0) k L ∞ (Ω) ≤ M | α | !(2 ρ ) −| α | , for α ∈ N n , with M = e N µ   X ω j ≤ µ a 2 j   1 2 and ρ a s ab ov e. Th us, u ( . , 0) has an analytic ex tension to a ρ -neighbo r ho o d of Ω, and after a finite num ber of applications of Theore m 3 and a covering arg ument , k u ( . , 0 ) k L 2 (Ω) ≤ N k u ( . , 0) k θ L 2 ( ω ) M 1 − θ . In particular , X ω j ≤ µ a 2 j ≤ e N µ Z ω | X ω j ≤ µ a j e j | 2 dx, with N = N ( | ω | , Ω , r, δ ) . Pr o of of The or em 2. Let u b e defined by (2.3). F rom [1 4], one can find a b ound- ary control h verifying (1.3) , pr ovided there is N = N ( | γ | , Ω , r, δ ) s uch that the inequality (2.16) X ω j ≤ µ a 2 j + b 2 j ≤ e N µ Z Z γ × [ 1 4 , 3 4 ] | X ω j ≤ µ  a j e ω j y + b j e − ω j y  ∂ e j ∂ n | 2 dσ dy , holds for µ ≥ ω 1 and all sequences a 1 , a 2 , . . . a nd b 1 , b 2 , . . . . H ere, ν , σ and ∂ ∂ n denote resp ectively the exterior unit no rmal vector to Ω, the surface measure o n ∂ Ω and the cono rmal de r iv ative for ∂ 2 y + △ on ∂ Ω × R , ∂ e ∂ n = A ∇ x e · ν . W e ma y also ass ume that 0 ∈ ∂ Ω, γ ⊂ B ρ 2 ∩ ∂ Ω, where B 2 ρ ∩ ∂ Ω is the regio n ab ov e the graph of a re al analytic function ϕ : B ′ ρ ⊂ R n − 1 − → R , a s in (2.1). F rom [14, § 3 (2)], there is N such that (2.17) X ω j ≤ µ a 2 j + b 2 j ≤ e N µ k ∂ u ∂ n k L ∞ ( ∂ Ω × [ − 1 , 2]) , with ∂ u ∂ n = X ω j ≤ µ  a j e ω j y + b j e − ω j y  ∂ e j ∂ n . F rom (2.5 ) and (2.1 ), there ar e N = N ( r , δ ) and ρ = ρ ( r , δ ) such that h ( x ′ , y ) = ∂ u ∂ n ( x ′ , ϕ ( x ′ ) , y ) verifies k ∂ α x ′ ∂ β y h k L ∞ ( B ′ 2 ρ × [ − 4 , 4]) ≤ M ( | α | + β )!(2 ρ ) −| α |− β , for α ∈ N n − 1 , β ∈ N , M = e N µ   X ω j ≤ µ a 2 j + b 2 j   1 2 , when B 2 ρ ∩ ∂ Ω is a co or dinate chart of ∂ Ω a s in (2.1). This fact, a suitable cov- ering arg umen t o f ∂ Ω and the thr e e-spher es ine qualities as so ciated to the obvious NULL–CONTR OL AND MEASURABLE SETS 9 extension of Theorem 3 fo r rea l analytic functions ov er a co mpact analytic sur face in R n +1 , imply there are N = N ( | γ | , r, δ ) and θ = θ ( | γ | , r, δ ) such that (2.18) k ∂ u ∂ n k L ∞ ( ∂ Ω × [ − 1 , 2]) ≤ N k ∂ u ∂ n k θ L 2 ( γ × [ 1 4 , 3 4 ]) M 1 − θ . Finally , (2.16) follows from (2.17) a nd (2.18).  R emark 2 . The extension of Theor em 1 to the para b o lic s y stem (2.19)      ∂ i ( a αβ ij ∂ j e β k ) − ∂ t u α = f α ( x, t ) χ ω ( x ) , in Ω T , α = 1 , . . . , m, u = 0 , on ∂ Ω × [0 , T ] , u (0) = u 0 , in Ω . with ∂ Ω as in (2.1), a αβ ij verifying (1 .6) a nd (2.20) | ∂ γ a αβ ij ( x ) | ≤ | α | ! δ −| α |− 1 , when x ∈ Ω , γ ∈ N n , for s ome 0 < δ ≤ 1 is now obvious: the symmetry , co erciv eness and co mpactness of the o pe rator L 2 (Ω) m − → W 1 , 2 0 (Ω) m , mapping f = ( f 1 , . . . , f m ) int o the unique solution u = ( u 1 , . . . , u m ) to ( ∂ i ( a αβ ij ∂ j u β ) − Λ u α = f α , in Ω , α = 1 , . . . , m, u = 0 , in ∂ Ω where Λ > 0 is s ufficien tly la rge [1 0, P rop. 2.1], gives the existence o f a complete system { e k } in L 2 (Ω) m , e k = ( e 1 k , . . . , e m k ), of eigenfunctions verifying ( ∂ i ( a αβ ij ∂ j e β k ) + ω 2 k e α k = 0 , in Ω , α = 1 , . . . , m, e k = 0 , in ∂ Ω with eigenv alues 0 ≤ ω 1 ≤ . . . ω k ≤ . . . and lim k → + ∞ ω k = + ∞ . By sepa ration of v ariables , the Gree n’s matrix for the system (2.1 9) ov er Ω × R is the m × m matrix Γ ( x, y , t − s ) = + ∞ X k =1 e − ω 2 k ( t − s ) e k ( x ) ⊗ e k ( y ) . Moreov er, the interior and b oundar y regularity for the e lliptic system ∂ 2 y + ∂ i ( a αβ ij ∂ j ) in Ω × R , shows that (2.4) holds for u a s in (2.3) but with e k replacing e k ([19], [10, Cha pter I I]). These and [14] suffice to find a control function f for the system (2.19) verifying (1.2). F urthermor e, if you wish to get b ounds on the reg ularity o f f , [1 4] shows it suffices to know that, k e k k H s (Ω) ≤ C s (1 + ω k ) s , for s ≥ 0, and that the num b er of 0 ≤ ω k ≤ µ , is b ounded by N µ n , when µ ≥ 1.The first ho lds from elliptic r egularity and (2.20), while the second follows from the Gaussia n estimates verified by Γ , i.e., there are N and κ [8, Corolla ry 4.14] such that | Γ ( x, y , t ) | ≤ N (1 ∧ t ) − n 2 e Λ t − κ | x − y | 2 /t , for x, y ∈ R n , t > 0 . It implies, Z Ω Z Ω | Γ ( x, y , t ) | 2 dxdy = X k ≥ 1 e − 2 ω 2 k t ≤ N e 2Λ t | Ω | t − n 2 , and suffices to cho ose t = 1 µ 2 . In par ticula r, f ∈ C ∞ 0 ((0 , T ) , C ∞ ( Ω)). 10 J. APRAIZ AND L. ES CA URIAZA The null-controllabilit y of the system (2.21)      ∂ t u + ( − 1) m △ m u = f ( x, t ) χω , in Ω × (0 , T ] , u = ∇ u = · · · = ∇ m − 1 u = 0 , in ∂ Ω × (0 , T ) , u (0) = u 0 , in Ω , m ≥ 2, is b etter ma naged with the approa ch in [15]. If { e j } and 0 ≤ ω 2 m 1 ≤ · · · ≤ ω 2 m k ≤ . . . are the eigenvectors and eigenv a lues for △ m in W 2 ,m 0 (Ω), ( ( − 1) m △ m e j − ω 2 m j e j = 0 , in Ω , e j = ∇ e j = · · · = ∇ m − 1 e j = 0 , in ∂ Ω , u ( x, y ) = X w m j ≤ µ a j X j ( y ) e j ( x ) , verifies, ∂ 2 m y u + △ m u = 0 in Ω × R , u = ∇ u = · · · = ∇ m − 1 u = 0 in ∂ Ω × R , when X j ( y ) =    e ω j y , for m o dd , e ω j e π i 2 m y , for m even . Again, u verifies (2.4) [19] and fro m Theorem 3 applied to u ( . , 0) as in Remar k 1, X ω m j ≤ µ a 2 j ≤ e N µ 1 m Z ω | X ω m j ≤ µ a j e j | 2 dx, with N = N ( | ω | , Ω , r, δ, m ) . F rom [1 5], the last inequalit y s uffices to find a control f in L 2 (0 , T , L 2 (Ω)) for (2.21). R emark 3 . The pro of of Theorem 2 for the scalar ca s e is based on the b ound (2.17). In the litera ture it is obta ined from (2.10) and the interpola tion inequa lit y b elow, prov ed v ia Carle ma n inequa lities: there ar e N and θ such that k u k L 2 (Ω × [0 , 1]) ≤ N k ∂ u ∂ n k θ L 2 ( ∂ Ω × [ − 1 , 2]) k u k 1 − θ L 2 (Ω × [ − 3 , 3]) , holds whenever u verifies △ u + ∂ 2 y u = 0 in Ω × R and u = 0 on ∂ Ω × R . Howev er , the authors are not aware wether the corresp onding in terp olation inequa lit y for solutions u of the elliptic system ( ∂ i ( a αβ ij ∂ j u β ) + ∂ 2 y u α = 0 , in Ω × R , α = 1 , . . . , m, u = 0 , in ∂ Ω × R , k u k L 2 (Ω × [0 , 1]) ≤ N k ∂ u ∂ n k θ L 2 ( ∂ Ω × [ − 1 , 2]) k u k 1 − θ L 2 (Ω × [ − 3 , 3]) , holds. Here, ( ∂ u ∂ n ) α = a αβ ij ∂ j u β ν i , α = 1 , . . . , m , and fo r this reason we can not extend Theor em 2 to parab olic systems. O n the o ther ha nd, ther e is ρ > 0 suc h that the mapping ∂ Ω × (0 , ρ ) − → U ρ , ( Q, t ) Q + tν ( Q ), is an analytic diffeomor phism onto U ρ = { x ∈ R n \ Ω : 0 < d ( x, ∂ Ω) < ρ } . Because the n ull-control of the parab olic system ov er Ω ∪ U ρ and with c ontrols acting ov e r ω = U 3 ρ 4 \ U ρ 2 is p ossible, standar d arguments show that the system      ∂ i ( a αβ ij ∂ j e β k ) − ∂ t u α = 0 , in Ω T , α = 1 , . . . , m, u = g on ∂ Ω × [0 , T ] , u (0) = u 0 , in Ω . can b e null-con trolled with co nt rols g acting ov er the full later al b o undary o f Ω. NULL–CONTR OL AND MEASURABLE SETS 11 3. P r oof of Theorem 3 First we r ecall Hadamard’s three-cir cle theo rem [17] and prove tw o Lemmas befo re the pro of o f Theorem 3 . Theorem 4. L et F b e a holomorphic funct ion of a c omplex variab le in t he b al l B r 2 . Then, the fol lowing is valid for 0 < r 1 ≤ r ≤ r 2 , k F k L ∞ ( B r ) ≤ k F k θ L ∞ ( B r 1 ) k F k 1 − θ L ∞ ( B r 2 ) , θ = log r 2 r log r 2 r 1 . Lemma 1. L et f b e holomorphic in B 1 , | f ( z ) | ≤ 1 in B 1 and E b e a me asur able set in [ − 1 5 , 1 5 ] . Then, ther e ar e N = N ( | E | ) and γ = γ ( | E | ) such that k f k L ∞ ( B 1 2 ) ≤ N k f k γ L ∞ ( E ) Pr o of. F or n ≥ 1, there a re n + 1 p oints 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 . F o r ex a mple, x 0 = inf E , x i = inf  E ∩ [ x i − 1 + | E | n +1 , 1 5 ]  . Let P n ( z ) = n X i =0 f ( x i ) Q j 6 = i ( z − x j ) Q j 6 = i ( x i − x j ) Then, | P n ( z ) | ≤ k f k L ∞ ( E ) | E | − n n X i =0 ( n + 1 ) n i !( n − i )! ≤ k f k L ∞ ( E )  3 | E |  n , for | z | ≤ 1 2 . By Cauch y’s formula, | f ( z ) − P n ( z ) | =      1 2 π i Z | ξ | =1 f ( ξ )( z − x 0 ) . . . ( z − x n ) ( ξ − z )( ξ − x 0 ) . . . ( ξ − x n ) dξ      ≤ 2  7 8  n , for | z | ≤ 1 2 . The last tw o ineq ua lities g ive (3.1) k f k L ∞ ( B 1 2 ) ≤ k f k L ∞ ( E )  3 | E |  n + 2  7 8  n , for all n ≥ 1 , and the minimizatio n in the n -v ariable o f the right hand side o f (3.1 ) implies Lemma 1.  Lemma 2. L et f b e analytic in [0 , 1 ] , E b e a me asu ra ble set in [0 , 1 ] and assum e ther e ar e p ositive c onstants M and ρ such that (3.2) | f ( k ) ( x ) | ≤ M k !(2 ρ ) − k , for k ≥ 0 , x ∈ [0 , 1] . Then, ther e ar e N = N ( ρ, | E | ) and γ = γ ( ρ, | E | ) such that k f k L ∞ ([0 , 1]) ≤ N k f k γ L ∞ ( E ) M 1 − γ . Pr o of. (3.2) implies tha t f has a holomo rphic extension to D ρ = ∪ 0 ≤ x ≤ 1 B ( x, ρ ), with | f | ≤ 2 M in D ρ . W rite [0 , 1] as a disjoint union of 5 2 ρ non-ov erlapping closed int erv a ls of length 2 ρ 5 . Among them there is at lea st o ne, I = [ x 0 − δ 5 , x 0 + δ 5 ], such that | E ∩ I | ≥ 2 δ | E | 5 . Then, g ( z ) = f ( x 0 + δ z ) / 2 M is holomor phic in B 1 , 12 J. APRAIZ AND L. ES CA URIAZA E x 0 ,ρ = ρ − 1 ( E ∩ I − x 0 ) is mea surable in [ − 1 5 , 1 5 ] with measure b ounded fro m b elow by 2 | E | 5 , k g k L ∞ ( E x 0 ,ρ ) ≤ k f k L ∞ ( E ) and a pplying Lemma 1 to g (3.3) k f k L ∞ ( B ρ 2 ( x 0 )) ≤ N k f k γ L ∞ ( E ) M 1 − γ , with 0 ≤ x 0 ≤ 1. Finally , ma ke s uccessive applicatio ns of Hadamard’s three - circle theorem (a finite num b er dep ending on ρ ) with a suitable c hain of thr e e-circles of radius compar able to ρ and with c e nter at p oints x in [0 , 1] contained in D ρ , while recalling that on the larg est ball | f | ≤ 2 M , to get that (3.4) k f k L ∞ ([0 , 1]) ≤ N k f k θ L ∞ ( B ρ 2 ( x 0 )) M 1 − θ , θ = θ ( ρ ) , and Lemma 2 follows from (3.4 ) and (3.3).  Pr o of of The or em 3. W e may assume R = 1. Let x ∈ B 1 2 . Using s pherical co ordi- nates centered at x , | E | ≤ Z S n − 1 |{ t ∈ [0 , 1 ] : x + tz ∈ E }| dz , and there is at lea st one z ∈ S n − 1 with |{ t ∈ [0 , 1] : x + tz ∈ E }| ≥ | E | / (2 ω n ), with ω n the sur face measur e on S n − 1 . Set ϕ ( t ) = f ( x + tz ). F rom (1.5), ϕ s atisfies (3.2), k ϕ k L ∞ ( E z ) ≤ k f k L ∞ ( E ) and Lemma 2 g ives (3.5) k f k L ∞ ( B 1 2 ) ≤ N k f k γ L ∞ ( E ) M 1 − γ . Finally , setting e E = { x ∈ E : | f ( x ) | / 2 ≤ — Z E | f | dx } , Chebyshev’s inequality shows that | e E | ≥ | E | / 2 , k f k L ∞ ( e E ) ≤ 2 — Z E | f | dx, and Theor e m 3 follows after replacing E b y e E in (3.5).  A cknow le dgement : The authors wis h to thank S. V essella for s ha ring his res ults in [26]. References [1] L. A hlfors, L. Bers, Riemann ’s mapping the or em for v ariable metrics. Ann. Math. 72 (1960) , 265–296. [2] G. Al essandrini, L. Escauriaza, Nul l-Contr ol lability of One -Dimensional Par ab olic Equations. ESAIM Contr. Op. Ca. 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