Downloaded 05/20/16 to Redistribution subject to SIAM license or copyright; see

Transcription

1 SIAM J CONTROL OPTIM Vol 5, No 5, pp c 04 Society for Industrial and Applied Mathematics SHARP ESTIMATES OF THE ONE-DIMENSIONAL BOUNDARY CONTROL COST FOR PARABOLIC SYSTEMS AND APPLICATION TO THE N-DIMENSIONAL BOUNDARY NULL CONTROLLABILITY IN CYLINDRICAL DOMAINS ASSIA BENABDALLAH, FRANCK BOYER, MANUEL GONZÁLEZ-BURGOS, AND GUILLAUME OLIVE Abstract In this paper we consider the boundary null controllability of a system of n parabolic equations on domains of the form Ω = 0,π) Ω with Ω a smooth domain of R N, N> When the control is exerted on {0} ω with ω Ω, we obtain a necessary and sufficient condition that completely characterizes the null controllability This result is obtained through the Lebeau Robbiano strategy and requires an upper bound of the cost of the one-dimensional boundary null control on 0,π) The latter is obtained using the moment method and it is shown to be bounded by Ce C/T when T goes to 0 + Key words parabolic systems, boundary controllability, biorthogonal families, Kalman rank condition AMS subject classifications 93B05, 93C05, 35K05 DOI 037/ Introduction The controllability of systems of n partial differential equations by m<ncontrols is a relatively recent subject We can quote [LZ98], [dt00], [BN0] among the first works More recently in [AKBDGB09b], the authors have generalized the so-called Kalman rank condition using fine tools of partial differential equations This condition usually characterizes the controllability of linear ordinary differential systems see [KFA69]) In the above reference, the authors prove that a suitable generalized Kalman rank condition provides the distributed null-controllability property of some classes of linear parabolic systems See also [AKBDGB09a]wherethe authors provide a Kalman rank condition in the framework of time-dependent coupled linear parabolic systems On the other hand, while for scalar problems the boundary controllability is known to be equivalent to the distributed controllability, it has been proved in [FCGBdT0] that this is no longer the case for systems This reveals that the controllability of systems is much more subtle In [AKBGBdT3a, AKBGBdT3b], it is even shown that a minimal time of control can appear if the diffusion is different on each equation This is quite surprising for a system possessing an infinite speed of propagation It is important to emphasize that the previous quoted results concerning the boundary controllability were established in space dimension one This restriction is mainly due to the fact that they used the moment method, generalizing the works of [FR7, FR75] on the boundary controllability of the one-dimensional scalar heat equation In higher space dimensions the boundary controllability of parabolic systems remains widely open and it is the main purpose of this article to give some partial Received by the editors July 8, 03; accepted for publication in revised form) July 9, 04; published electronically September 5, 04 Aix Marseille Université, CNRS, Centrale Marseille, lm, UMR 7373, 3453 Marseille, France assiabenabdallah@univ-amufr, franckboyer@univ-amufr, guillaumeolive@univ-amufr) Dpto EDAN and IMUS, Universidad de Sevilla, Aptdo 60, 4080 Sevilla, Spain manoloburgos@uses) This author s research was supported by grant MTM of the Ministry of Science and Innovation 970

2 BOUNDARY CONTROLLABILITY FOR PARABOLIC SYSTEMS 97 answers To our knowledge, the only results on this issue are the one of [ABL] and [AB] In these articles, the results for parabolic systems are deduced from the study of the boundary control problem of two coupled wave equations using transmutation techniques As a result they rely on some geometric constraints on the control domain We will see here that those assumptions are not necessary Let us also mention [Oli3] for related questions for the approximate controllability problem We refer to [AKBGBdTb] for a more detailed survey on the controllability of parabolic systems In the present work, we focus on the boundary null controllability of the following n coupled parabolic equations by m controls in dimension N>: t y =Δy + Ay in 0,T) Ω, ) y = γ Bv on 0,T) Ω, y0) = y 0 in Ω, in the case where the domain Ω has a Cartesian product structure Ω=Ω Ω, where Ω i R Ni, i =,, are bounded open regular domains In ), T>0isthe control time, the nonempty relative subset γ Ω is the control domain, y is the state, y 0 is the initial datum, A M n C) andb M n m C) are constant matrices, and v is the boundary control Under appropriate assumptions we show that the controllability of system ) is reduced to the controllability of the same system posed in Ω see Theorem 3 below) The proof is based on the method of Lebeau Robiano [LR95] This strategy already used in a different framework in [BDR07]) requires an estimate of the cost of the N -dimensional control with respect to the control time when T 0 + In a second part, we establish that the cost of the one-dimensional null control on 0,T) is bounded by Ce C/T for some C>0, as T 0 + see Theorem 4 below) This is the second main result of this paper It shows in particular that our first result above can be applied at least in the case N = Observe that the results obtained in [AKBGBdTa] do not permit to deduce the required exponential estimate on the null-control cost The demonstration of this result follows the approach of [FR7] and [Mil04] for the scalar case) In the scalar case, [Sei84] seealso[fcz00]) gave a similar estimate of the cost of the boundary control of the heat equation, which is known to be optimal thanks to the work [Güi85] Note finally, that the extension of the present results to more general domains Ω in R N as well as the study of the case with a different diffusion coefficient in each equation, remains an open problem Reminders and notation Let us first recall that system ) is wellposed in the sense that, for every y 0 H Ω) n and v L 0,T; L Ω) m ), there exists a unique solution y C 0 [0,T]; H Ω) n ) L 0,T; L Ω) n ), defined by transposition Moreover, this solution depends continuously on the initial datum y 0 and the control v More precisely, ) y C 0 [0,T ];H Ω) n ) CeCT y 0 H Ω) n + v L 0,T ;L Ω) m ) where here and all during this work C>0 denotes a generic positive constant that may change line to line but which does not depend on T or y 0 We shall also sometimes use the notation C,C,andsoon ),

3 97 BENABDALLAH, BOYER, GONZÁLEZ-BURGOS, AND OLIVE Let us now make precise the concept of controllability which we will deal with in this paper We say that system ) is null controllable at time T if for every y 0 H Ω) n, there exists a control v L 0,T; L Ω) m ) such that the corresponding solution y satisfies yt )=0 In such a case, it is well known that there exists C T > 0 such that 3) v L 0,T ;L Ω) m ) C T y 0 H Ω) n y 0 H Ω) n The infimum of the constants C T satisfying 3) is called the cost of the null control at time T Remark Replacing yt) bye μt yt) anda by A μ, with μ>0, we can assume without loss of generality that the matrix A is stable, ie, that all its eigenvalues have a negative real part Finally, let us recall the well-known duality between controllability and observability for the general theory which relates observability and controllability, see [DR77]) Theorem Let E be a closed subspace of H 0 Ω)n and set E = ΔE H Ω) n Let us denote by Π E resp, Π E ) the orthogonal projection on E resp, E ) Let C T > 0 be fixed For every y 0 E there exists a control such that v L 0,T; L Ω) m ) { ΠE yt )=0, v L 0,T ;L Ω) m ) C T y 0 H Ω) n, where y is the corresponding solution to ), if and only if Π E z0) H 0 Ω)n C T T 0 γ B n zt) L Ω) m dt z T E, where z is the solution to the adjoint system t z =Δz + A z in 0,T) Ω, 4) z =0 on 0,T) Ω, z0) = z T in Ω Notation We gather here some standard notation that we shall use throughout this paper For any real numbers a<bwe denote a, b =[a, b] Z Forz C, Rz) and Iz) denote the real and imaginary parts of z Finally, x R x Z denotes the floor function Main results Boundary controllability for a multidimensional parabolic system The first main achievement of this work is the following Theorem Let ω Ω be a nonempty open subset and take Ω =0,π) Then, system ) is null controllable at time T on γ = {0} ω if and only if 5) rank B k A k B k A k B k A nk k B k )=nk k,

4 BOUNDARY CONTROLLABILITY FOR PARABOLIC SYSTEMS 973 Ω Ω ω Fig Typical geometric situation where we have introduced the following notation recall that λ j = j ): 6) λ + A λ + A A k = M nk C), λ k + A B B B k = M nk m C) B One may think of a cylindrical domain where the control domain is a subset of the top or bottom face see Figure ) This result will be obtained as a corollary of some other theorems that are important results too The first one is the following and should be connected with [Fat75] and [Mil05] Theorem 3 Let γ Ω be a nonempty relative subset Assume that the following N -dimensional system t y =Δ x y + Ay in 0,T) Ω, 7) y = γ Bv on 0,T) Ω, y 0) = y0 in Ω, is null controllable for any time T>0 with, in addition, the following bound for the control cost C Ω T : 8) C Ω T CeC/T T>0

5 974 BENABDALLAH, BOYER, GONZÁLEZ-BURGOS, AND OLIVE Then, for any nonempty open set ω Ω the N-dimensional system ) is null controllable at any time T>0 on the control domain γ = γ ω Remark The converse of Theorem 3 also holds More precisely, if the N- dimensional system ) is null controllable at time T, then the N -dimensional system 7) is also null controllable at time T This can be proved using a Fourier decomposition in the direction of Ω It is worth mentioning that such a decomposition also shows that, when ω =Ω, the proof of Theorem3 is much simpler and it does not need the control cost estimate 8) Moreover, the domain Ω can even be unbounded in this case Estimate of the control cost for a one-dimensional boundary controllability problem The third result of this paper provides an important example where Theorem 3 can be successfully applied More precisely, we show that the assumption 8) on the short time behavior of the control cost actually holds in the one-dimensional case for the following system if we assume the rank condition 5) 9) t y = xx y + Ay yt, 0) = Bvt), yt, π) =0 y0) = y 0 in 0,T) 0,π), in0,t), in 0,π) We recall that it has been established in [AKBGBdTa] thatsystem9) isnull controllable at time T>0 if and only if the rank condition 5) holds However, in the abovementioned reference, no estimate on the control cost is provided This is the next goal of the present paper: to give a more precise insight into the proof of the controllability result for system 9) that allows a precise estimate of the control cost as a function of T Theorem 4 Assume that the rank condition 5) holds Then, for every T>0 and y 0 H 0,π) n there exists a null control v L 0,T) m for system 9) which, in addition, satisfies v L 0,T ) m CeC/T y 0 H 0,π) n This theorem, combined with Theorem 3 and Remark, gives a proof of Theorem 3 Bounds on biorthogonal families of exponentials The proof of Theorem 4 is mainly based on the existence of a suitable biorthogonal family of timedependent exponential functions The construction provided in [AKBGBdTa] does not allow us to estimate the control cost That is the reason why we propose here a slightly different approach which is the key to obtaining the factor e C/T This abstract result, which is interesting in itself and potentially useful in other situations, can be formulated as follows Theorem 5 Let {Λ k } k C be a sequence of complex numbers fulfilling the following assumptions: Λ k Λ n for all k, n N with k n; RΛ k ) > 0 for every k ; 3 for some β>0, IΛ k ) β RΛ k ) k ;

6 BOUNDARY CONTROLLABILITY FOR PARABOLIC SYSTEMS {Λ k } k is nondecreasing in modulus, Λ k Λ k+ k ; 5 {Λ k } k satisfies the following gap condition: for some ρ, q > 0, Λ k Λ n ρ k n k, n : k n q, inf Λ k Λ n > 0; k n: k n <q 6 for some p, α > 0, 0) p r Nr) α r >0, where N is the counting function associated with the sequence {Λ k } k, that is the function defined by ) N r) =#{k : Λ k r} r>0 Then, there exists T 0 > 0 such that, for every η and 0 <T <T 0, we can find a family of C-valued functions {ϕ k,j } k,j 0,η L T/,T/) biorthogonal to {e k,j } k,j 0,η, where for every t T/,T/), with, in addition, e k,j t) =t j e Λ kt ) ϕ k,j L T/,T/) CeC RΛ k )+ C T for any k, j 0,η Boundary null controllability on product domains This section is devoted to the proof of Theorem 3 Settings and preliminary remarks Let λ Ω j Dirichlet eigenvalues of the Laplacian on Ω resp, Ω ), and let φ Ω resp, λ Ω j ), j, be the j resp, φ Ω j )be the corresponding normalized eigenfunctions Let us introduce the closed) subspaces of H0 Ω) n on which we will establish the partial observability later on section ): J E J = u, φ Ω j j= L Ω φω j ) u H 0 Ω)n H 0 Ω)n, J, where the notation J j= u, φω j L Ω )φ Ω j is used to mean the function x,x ) Ω J ux, ),φ Ω j L Ω )φ Ω j x ) j= That is ϕ k,j,e l,ν L T/,T/) = T/ T/ ϕ k,jt)e l,ν t) dt = δ kl δ jν

7 976 BENABDALLAH, BOYER, GONZÁLEZ-BURGOS, AND OLIVE We then define the dual spaces of E J E J = ΔE J H Ω) n, J Let us recall that we denote by Π EJ resp, Π E ) the orthogonal projection in H0 Ω)n J resp, H Ω) n )ontoe J resp, E J ) It is not difficult to see that we have the relation Π E Δu) = ΔΠ EJ u for any u H0 Ω)n J Lemma For any u H0 Ω)n we have u = + j= u, φ Ω j L Ω )φ Ω j It follows from this lemma that Π EJ u = J j= u, φω j L Ω )φ Ω j H 0 Ω)n Proof of Lemma Let us show that the sequence {S J u} J defined by S J u = J u, φ Ω j L Ω )φ Ω j j= is a Cauchy sequence of H 0 Ω)n For any J>K wehave S J u S K u H 0 Ω)n J = u, φ Ω j j=k+ J = u, φ Ω j j=k+ L Ω φω j ) L Ω ) H 0 Ω)n H 0 Ω)n + J j=k+ λ Ω j u, φ Ω j L Ω ) for any u L Ω ) n Using Lebesgue s dominated convergence theorem it is not difficult to see that these H termsgotozeroasj, K + AsaresultS J u 0 v for some v J + H 0 Ω) n In particular, v, φ Ω k φω j L Ω) = u, φ Ω k φω j L Ω) for every j, k, and it follows that v = u Partial observability One of the key points in making use of the Lebeau Robbiano strategy is the estimate of the cost of the partial observabilities on the approximation subspaces This will be used for the active control phase Proposition Let Ω be of class C Assume that system 7) is controllable at time T with cost C Ω T Then, ) T Π EJ z0) H CC 0 Ω Ω)n T ) e C λ Ω J γ ω B n zt) L Ω) dt z m T E J, 0 where z is the solution to the adjoint system 4) By Theorem we deduce the following Corollary 3 For every J and y 0 E J there exists a control vy 0 ) L 0,T; L Ω) m ) with ) vy 0 ) L 0,T ;L Ω) m ) C CΩ T )ec λ Ω J y0 H Ω) n,

8 BOUNDARY CONTROLLABILITY FOR PARABOLIC SYSTEMS 977 such that the solution y to system ) satisfies Π E yt )=0 J Proof of Proposition Letz T E J so that z T x,x )= J j= z j T x )φ Ω j x ) for some z j T H 0 Ω ) n Let z be the solution of 4), the adjoint system of ), associated with z T Thus, where z j is the solution to t z j = zt, x,x )= Δ x λ Ω j J j= z j t, x )φ Ω j x ), ) z j + A z j in 0,T) Ω, z j =0 on0,t) Ω, z j T )=z j T in Ω Note that Π EJ z0) = z0) A computation of z0) H gives 0 Ω)n J z0) H = z j 0) + 0 Ω)n H0 Ω)n j= Using the Poincaré inequality we obtain 3) z0) H 0 CλΩ Ω)n J J j= λ Ω j z j 0) L Ω ) n J z j 0) H0 Ω)n Observe now that z j t) =e T t)λω j ψt), where ψ is the solution to the adjoint system of 7) associated with z j T Thus, using the assumption that 7) is controllable with cost C Ω T, we obtain by Theorem that z j 0) H 0 Ω)n j= T C Ω T ) γ B n z j t) L Ω ) dt, m 0 where n denotes the unit outward normal vector of Ω Combined with 3), this gives z0) H 0 Ω)n CCΩ T ) λ Ω J T 0 J γ B n z j t) L Ω ) dt m Let us denote by B k the kth column of B Applying the Lebeau Robbiano s spectral inequality [LR95] seealso[lr07, section 3A] ) J a j Ce C λ Ω J J a j φ Ω j x ) dx ω j= j= j= See also [TT, Theorem 5] when Ω is a rectangular domain

9 978 BENABDALLAH, BOYER, GONZÁLEZ-BURGOS, AND OLIVE to the sequence of scalars a j = Bk n z j t, σ ), σ Ω being fixed, and summing over k m, thisgives J B n z j t, σ ) Ce C λ Ω J C n J B n z j t, σ )φ Ω j x ) dx j= ω j= C n To conclude it only remains to integrate over γ and to observe that ) n σ nσ) = ) for σ =σ 0,x ) Ω Ω 3 Dissipation along the direction Ω The other point of the Lebeau Robbiano strategy relies on the natural dissipation of the system when no control is exerted the passive phase) For our purpose, we need an exponential dissipation in the direction Ω Proposition 4 If there is no control on t 0,t ) ie, v =0on t 0,t ))and the corresponding solution y of system ) satisfies Π E yt 0 )=0, J then we have the following dissipation estimate yt) H Ω) n Ce λω J+ t t0) yt 0 ) H Ω) n t t 0,t ) Proof Let yt 0 )= Δỹ 0,ỹ 0 H0 Ω)n The assumption Π E yt 0 )=0translates into Π EJ ỹ 0 =0 J Let ỹ be the solution in H0 Ω) n to t ỹ =Δỹ + Aỹ in t 0,t ) Ω, ỹ =0 ont 0,t ) Ω, ỹt 0 )=ỹ 0 in Ω Since the matrix A is constant, we can check that and thus y = Δỹ in t 0,t ) Ω, yt) H Ω) n = ỹt) H 0 Ω)n, yt 0) H Ω) n = ỹ 0 H 0 Ω) n As a consequence it only remains to prove the dissipation for regular data, namely, ỹt) H 0 Ω) n Ce λω J+ t t0) ỹ 0 H 0 Ω) n t t 0,t ) for ỹ 0 such that Π EJ ỹ 0 = 0, ie, of the form see Lemma ) ỹ 0 = + j=j+ ỹ 0,j φ Ω j, ỹ 0,j = ỹ 0,φ Ω j L Ω H ) n 0 Ω ) n Since Π EJ ỹ 0 =0andA is constant, we have Π EJ ỹt) = 0 for every t t 0,t )andas a result the following inequalities hold: λ Ω J+ ỹt) L Ω) n ỹt) L Ω) n t t 0,t ), λ Ω J+ ỹt) L Ω) n Δỹt) L Ω) n for ae t t 0,t )

10 BOUNDARY CONTROLLABILITY FOR PARABOLIC SYSTEMS 979 Using now standard energy estimates and the fact that the matrix A is constant and stable see Remark ), we finally obtain the desired dissipation ỹt) H 0 Ω) n Ce λω J+ t t0) ỹ 0 H 0 Ω) n 4 Lebeau Robbiano time procedure We are now ready to prove Theorem 3 Let y 0 H Ω) n be fixed Let us decompose the interval [0,T) as follows: with [0,T)= + k=0 [a k,a k+ ] a 0 =0, a k+ = a k +T k, T k = M kρ, where ρ 0, N )andm = T ρ ) has been determined to ensure that + k=0 T k = T 0 a k control e ck ) N dissipation e ck ) N ρ Π E k y =0 a k+ We define the control v and the corresponding solution y piecewise and by induction as follows: { ) v Π vt) = E ya k ) t) if t a k,a k + T k ), k 0 if t a k + T k,a k+ ) Let us show that v belongs to L 0,T; L Ω) m ) and steers y to 0 at time T Step : Estimate on the interval [a k,a k + T k ] From the continuous dependence with respect to the data ) and since T k T we know that ) 4) ya k + T k ) H Ω) C ya n k ) H Ω) + v n L a k,a k +T k ;L Ω) m ) Using the estimate of the cost of the control ) wehave v L a k,a k +T k ;L Ω) m ) CCΩ T k e C λ Ω ΠE k ya k ) k and since Π E k LH ), this gives H Ω) n v L a k,a k +T k ;L Ω) m ) CCΩ T k e C λ Ω k ya k ) H Ω) n T

11 980 BENABDALLAH, BOYER, GONZÁLEZ-BURGOS, AND OLIVE Using now the estimate of C Ω T with respect to T assumption 8)), this leads to v L a k,a k +T k ;L Ω) m ) cec T + λ Ω k ) ya k k ) H Ω) n On the other hand, Weyl s asymptotic formula states that λ Ω C k) N k + and by the choice of ρ) so that T k = M kρ C k N, 5) v L a k,a k +T k ;L Ω) m ) CeC k N ya k ) H Ω) n Combined with 4) this yields 6) ya k + T k ) H Ω) n C +e C k N ) ya k ) H Ω) n Ce C k N ya k ) H Ω) n Step : Estimate on the interval [a k + T k,a k+ ] Since Π E k ya k + T k )=0,the dissipation Proposition 4) gives 7) ya k+ ) H Ω) n Ce λω k T k + ya k + T k ) H Ω) n Step 3: Final estimate From 7) and6) we deduce ya k+ ) H Ω) n Ce λω k T N k+c k + ya k ) H Ω) n By induction we obtain Since we obtain ya k+ ) H Ω) n Ce k p=0 λ Ω p + Tp+C p N ) λ Ω p + T p C p +) N pρ C p ) N ρ, + ya k+ ) H Ω) n Ce k p=0 Since ρ< N, there exists a p 0 such that ) C p ) N ρ +C p ) N 8) C p ) N ρ + C p ) N C p ) N ρ It follows that, for k p 0,wehave k p=0 C p ) N ρ + C p ) N ) C C So that, finally, y 0 H Ω) n y 0 H Ω) n p p 0 k p ) N ρ C C k) p=p 0 9) ya k+ ) H Ω) n Ce Ck ) N ρ y0 H Ω) n N ρ

12 BOUNDARY CONTROLLABILITY FOR PARABOLIC SYSTEMS 98 Step 4: The function v is a control Estimates 5) and9) show that the function v is in L 0,T; L Ω)): + v L 0,T :L Ω) m ) = v L a k,a k +T k :L Ω) m ) C k=0 + k=0 ) e C N k C k N ) ρ y 0 H Ω) n } {{ } <+ by 8) Moreover, estimate 9) also shows that the function v is indeed a control: ya k+ ) H Ω) n k + 0= yt ) H Ω) n 3 Cost of the one-dimensional boundary null control We prove here Theorem 4 assuming Theorem 5 is proved see the next section) All along this part we shall use the notation of [AKBGBdTa] 3 Arrangement and properties of the eigenvalues Let us first recall that the Dirichlet eigenvalues of the Laplacian xx on 0,π)withdomainH 0,π) H0 0,π)) are λ k = k, k We denote by {μ l } l,p C the set of distinct eigenvalues of A Forl,p, we denote the dimension of the eigenspace of A associated with μ l by n l and the size of its Jordan chains by τ l,j, j,n l In [AKBGBdTa, Case, p 583], it is shown that we can always assume that τ l,j = τ l is independent of j Finally, we set n =max l,p n l We assume that the set {μ l } l,p is arranged in the following nonunique) way, 3) l,p { Rμl ) Rμ l+ ), μ l μ l+ if Rμ l )=Rμ l+ ) We should point out that in [AKBGBdTa, p 56], it is assumed that {μ l } l,p is ordered in such a way that n = n Actually, this is only used for convenience and the same reasoning holds if we take n instead of n Let us now recall that the eigenvalues of the operator xx + A with domain H 0,π) n H 0 0,π) n )aregivenby λ k + μ i, k andi,p Moreover, there exists k 0 such that 3) λ k + μ i λ l + μ j for every k k 0, l, l k, andi, j,p with i j see [AKBGBdTa, Proposition 3]) From 3), we see that there exists k large enough so that λ k Rμ l ) Rμ l+ )) + μ l+ μ l 0 for every l,p Therefore, we deduce that 33) λ k μ l λ k μ l+ for every k k and l,p

13 98 BENABDALLAH, BOYER, GONZÁLEZ-BURGOS, AND OLIVE Finally, let k be large enough so that 34) + λ k μ i λ k+ μ j for every k k and i, j,p with i j, which is always possible since λ k = k We set K 0 =max{k 0,k,k } With this K 0 we associate p, the number of distinct eigenvalues of the matrix A K 0 defined in 6) Let {γ l } l, p { λ k + μ l } k,k0,l,p be the set of distinct eigenvalues of A K 0 arranged in such a way that γ l γ l+ for every l, p For l, p, the dimension of the eigenspace of A K 0 associated with γ l is denoted by N l, and the size of its Jordan chains by τ l,j, j,n l Since we assumed that τ l,j = τ l it follows that τ l,j = τ l is also independent of j Finally, we set N =max l, p N l We choose to arrange the eigenvalues {Λ k } k C of the operator Δ + A )as follows: { Λl = γ l for l, p, i i Λ p+i = λ K0+j μ l with j = p +andl = i p p for i Observe that the sequence {Λ k } k satisfies the assumptions -5 of Theorem 5: follows from 3); holds because the matrix A is stable see Remark ); 3 is clear since IΛ k ) max l,p Iμ l ) and RΛ k ) λ max l,p Rμ l ) which is positive since A is stable); 4 is a consequence of 33) and34); finally, let us show that 5 holds for q large enough Let k = p + i k and n = p + i n the case k p or n p is simpler) Let j k,j n and l k,l n be such that Λ k = λ K0+j k μ lk and Λ n = λ K0+j n μ ln Wehave Λ n Λ k = λ K0+j k λ K0+j n + μ ln μ lk λ K0+j k λ K0+j n μ ln μ lk λ K0+j k λ K0+j n λ K0+j k λ K0+j n μ ln μ lk + μ ln μ lk μ l μ l, d = Let us denote m = min l,l p μ l μ l, M =max l,l p l l l l j k j n, s = j k + j n,andx = ds +K 0 ) Thus, Λ n Λ k x Mx+ m On the other hand, since i k i n <p j k j n +) and i k +i n pj k +j n )+, we have k n = i k i n i k + i n + p) p d +) sp ++ p) By assumption d, s +, sothat k n Cd s +K 0 ) = Cx Taking, for instance, ρ = / C and x large enough we obtain the first property of 5 The second property is actually satisfied for any q

14 BOUNDARY CONTROLLABILITY FOR PARABOLIC SYSTEMS 983 The counting function We recall that the counting function N associated with the sequence {Λ k } k is given by N r) =#{k : Λ k r} r>0 This function N is piecewise constant and nondecreasing on the interval [0, + ) Thanks to 5 we have lim k + Λ k =+, sothatn r) < + for every r [0, + ) and lim r + N r) =+ Moreover,4 shows that, for every r>0, we have 35) N r) =n Λ n r and Λ n+ >r), so that, in particular, we have ΛN r) r< ΛN r)+ On the other hand, from the very definition of Λ k for k> p, wehave ) N r) + p K 0 M ) ΛN r) N r) + K0 + M for any r st N r) > p, p where M =max l,p μ l, K 0 = K 0 p+ p +, and K0 = K 0 + Combining the two previous estimates, it is not difficult to obtain the last assumption 6 of Theorem 5 3 The moment problem In [AKBGBdTa, Proposition 5] it has been proved that, under the assumption 5), system 9) is null controllable at time T if for every q, N there exists a solution u q L 0,T) to the moments problem 36) T t ν 0 ν! eγ lt u q t) dt = c l,ν,q y 0 ; T ) l, p, ν 0, τ l, T t σ σ! e λ k+μ l )t u q t) dt = d k l,σ,qy 0 ; T ) k >K 0, l,p, σ 0,τ l, 0 where c l,ν,q and d k l,σ,q are given in [AKBGBdTa, Proposition 5] The precise definition of those terms is not really important here, however, we recall that they satisfy the following estimates see [AKBGBdTa, equations 49) and 5)]): c l,ν,q y 0 ; T ) C e A K 0 T MnK0 0 37) C) y H 0,π) n Ce CT y 0 H 0,π) n and 38) d k l,σ,q y 0; T ) C k e λ k+a )T MnC) C y 0,φ k H,H0 0,π) n Ce CT λk k e λ kt y 0 H 0,π) n The control vt) is then given as a linear combination of u q T t), q, N, and as a result satisfies 39) v L 0,T ) C m max u q L 0,T ) q, N

15 984 BENABDALLAH, BOYER, GONZÁLEZ-BURGOS, AND OLIVE Assume for the moment that Theorem 5 is proved Let T 0 > 0bethetime given by Theorem 5 and set T η =max{τ l, τ l, l,p, l, p } For T < T 0 we can then introduce the biorthogonal family {ϕ k,j } k,j 0,η L T/,T/) associated with the sequence {Λ k } k As we need to work on the interval T/,T/), we perform the change of variable s = t T in 36) andobtain T s + T ) ν e γls u q s + T ) ds ν! = e T γ l c l,ν,q y 0 ; T ) l, p, ν 0, τ l, s + T ) σ e λ k+μ l )s u q s + T ) ds σ! T T = e λ k+μ l ) T d k l,σ,q y 0 ; T ) k >K 0, l,p, σ 0,τ l Using the binomial formula s + T ) J = J j=0 J j) s J j T ) j we finally have with 30) ν j=0 j=0 ) ) j T ν T s ν j e γls u q s + T ) ds j T = ĉ l,ν,q y 0 ; T ) l, p, ν 0, τ l, σ ) ) j T σ T s σ j e λ k+μ l )s u q s + T ) ds j = d k l,σ,q y 0; T ) T k >K 0, l,p, σ 0,τ l ĉ l,ν,q y 0 ; T )=ν!e T γ l c l,ν,q y 0 ; T ), dk l,σ,q y 0 ; T )=σ!e λ k+μ l ) T d k l,σ,q y 0 ; T ) For T < T 0, a solution to the moments problem 36) isthengivenforevery t 0,T) by note that λ k + μ l = Λ p+k K0 )p+l for k>k 0 ) u q t) = p τ l l= ν=0 ĉ l,ν,q y 0 ; T )ϕ l,ν t T ) + p τ l k>k 0 l= σ=0 d k l,σ,q y 0; T )ϕ p+k K0 )p+l,σ t T ), provided that u q lies in L 0,T)seebelow),andwhereĉ l,ν,q and dk l,σ,q solve the

16 BOUNDARY CONTROLLABILITY FOR PARABOLIC SYSTEMS 985 triangular systems P T ) ĉ l,0,q c l, τl,q = ĉ l,0,q c l, τl,q, QT ) d k l,0,q d k l,τ l,q = d k l,0,q d k l,τ l,q where the coefficients of P T )andqt) are, respectively, given for i j by p ij T )= i T ) i j, i j) qij T )= i T ) i j, i j) andpij T )=q ij T ) = 0 otherwise Observe that P T ) M τl C) CT τ l, QT ) Mτl C) CTτ l From this, the definition 30) ofĉ l,ν,q and d k l,σ,q, and the estimates 37) and38) of c l,ν,q and d k l,σ,q,weobtain 3) ĉ l,ν,q y 0 ; T ) CT τ l e T γ l e CT y 0 H 0,π) n CeCT y 0 H 0,π) n and 3) d k l,σ,q y 0; T ) CTτ l e λ k+μ l ) T λk k ect e λ kt y 0 H 0,π) n Ce CT λk k e λ T k y0 H 0,π) n It remains to prove that u q L 0,T) and to estimate its norm with respect to T and y 0 It is actually thanks to the estimate ) that this can be achieved Indeed, using also 3) and3) wehave 33) u q L 0,T ) CeCT p l= + Ce CT e C Rγ l )+ C T y0 H 0,π) n k>k 0 λk k Ce CT+ C T + k>k 0 e λ T k λk Let us now estimate the series Young s inequality gives C T λ k λ k 4 + C T for every k andt>0, so that T λ k + C T λ k λ k 4 + C T Thus, using also that λ k = k,weobtain λk k e λ T k +C λ k e C T e k T 4 k>k 0 k, p e C λ k Rμ l )+ C T y0 H 0,π) n l= ) e λ T k +C λ k y 0 H 0,π) n k 0

17 986 BENABDALLAH, BOYER, GONZÁLEZ-BURGOS, AND OLIVE A comparison with the Gauss integral gives e k T 4 k 0 Coming back to 33) wethenhave Finally, 39) gives, for every T<T 0, 4π T CeC T u q L 0,T ) CeCT+ C T y0 H 0,π) n v L 0,T ) CeC T y0 H 0,π) n Thus, when T<T 0 we have obtained a null control to system 9) which satisfies the desired estimate The case T T 0 is actually reduced to the previous one Indeed, any continuation by zero of a control on 0,T 0 /) is a control on 0,T) and the estimate follows from the decrease of the cost with respect to the time 4 Biorthogonal families to complex matrix exponentials This section is devoted to the proof of Theorem 5 4 Idea of the proof For T small enough and any η, we have to construct a family {ϕ k,j } k,j 0,η in L T/,T/) such that T T ϕ k,j t)t ν e Λ lt dt = δ kl δ jν for every k, l andj, ν 0,η, with in addition the following bound, ϕ k,j L T, T ) CeC RΛ k )+ C T for any k andj 0,η The idea is to use the Fourier transform with the help of the Paley Wiener theorem see [Rud74, Theorem 93]) that we recall here Theorem 4 Let Φ be an entire function of exponential type T/ that is 3 Φz) Ce T z for all z C) such that + Φ L,+ ) = Φx) dx < + Then, there exists ϕ L T/,T/) such that 4) Φz) = T π Moreover, the Plancherel theorem gives T ϕt)e itz dt z C ϕ L T, T ) = Φ L,+ ) 3 Here and only here, C may even depend on T without affecting the result

18 BOUNDARY CONTROLLABILITY FOR PARABOLIC SYSTEMS 987 Observe that the function in 4) is infinitely differentiable on C with, for every ν 0,η, Φ ν) z) = T iν π T ϕt)t ν e itz dt z C Thus, Theorem 5 will be proved if we manage to build suitable entire functions as stated in the following result Theorem 4 Assume that the sequence {Λ k } k C satisfies the assumptions 6 There exists T 0 > 0 such that, for any η and 0 <T <T 0,thereexistsa family {Φ k,j } k,j 0,η of entire functions of exponential type T/ satisfying 4) Φ ν) k,j iλ l)= and iν π δ kl δ jν k, l, j, ν 0,η, 43) Φ k,j L,+ ) CeC RΛ k )+ C T for any k and j 0,η Remark 3 A sequence {Λ k } k C satisfies the assumptions 6 if and only if so does the sequence { Λ k For this reason, we will prove Theorem 4 for the }k sequence { Λ k }k 4 Proof of Theorem 4 Some preliminary remarks It is interesting to point out some properties of the sequence {Λ k } k which can be deduced from assumptions 3, 4, and6 First, under assumptions 4 and 6 we have that 44) k Λ k < + Indeed, using that N is piecewise constant and nondecreasing on the interval [0, + ), we can write + Λ k = dn r) = Λ r k + Λ + Λ N r) dr r α + p r r dr = α Λ + p Λ < + Then, from assumption 3 we can also deduce the following behavior of the sequence {Λ k } k : 45) Λ k RΛ k ) β RΛ k ) and Λ k CRΛ k ) k Indeed, one has Λ k = RΛ k ) + IΛ k ) RΛ k ) + β RΛ k ) RΛ k )+β RΛ k ))

19 988 BENABDALLAH, BOYER, GONZÁLEZ-BURGOS, AND OLIVE Let us now introduce the complex functions given, for every z C, by 46) fz) = ) zλk, f n z) = ) zλk k k k n Thanks to 44), the previous products are uniformly convergent on compact sets of C and therefore f and f n are entire functions Moreover, the zeros of f and f n are exactly {Λ k } k and {Λ k } k n and they are zeros of multiplicity recall that the Λ k are distinct by ) For a proof of these facts we refer to [Rud74, Theorem 54] On the other hand, let us fix d = pπ + For any τ>0such that τ<d /we define the real positive sequence {a n } n 0 given by 47) a n = d τ + 4 n ) d n 0 With this sequence we associate a complex function M defined by 48) Mz) = sinz/a n ) z C z/a n n Since sinz) z e z z C, and a n Cn, the previous product is uniformly convergent on compact sets of C + and M is an entire function of exponential type τ M > 0, where 49) τ M = n More precisely, M satisfies a n < + 40) Mz) e τm z z C Observe that there is no constant in front of the term e τm z This point will be very important in what follows see the proof of Proposition 43 in Appendix A) to obtain estimates with constants C that do not depend on τ which will play the role of T ; see below) Note also that M has only real zeros since {a n } n is a real sequence Finally, we will often use that τ M <τ ThisfactisprovedinLemmaA in Appendix A Proof of Theorem 4 We follow some techniques developed in [AKBGBdTa] see, in particular, Lemma 44 in this reference) Set T 0 = d and, for any 0 <T <T 0,setτ = T η in such a way that the condition τ<d / holds The function M defined above will then correspond to this value of τ Let us consider the functions Φ k z) = η! [W kz)] η, W k z) = f iz) Mz + IΛ k )), 4) Φ k z) = [ Wk z) η! defined for every z C and k if Λ k ) ] η, Wk z) = f k iz) if Λ k ) MiRΛ k )) Mz + IΛ k )), MiRΛ k ))

20 BOUNDARY CONTROLLABILITY FOR PARABOLIC SYSTEMS 989 Let us first give some estimates for the functions W k, W k and, as a result, also for Φ k and Φ k ) that will be used later Proposition 43 Assume that the sequence {Λ k } k satisfies the assumptions 6, and let τ<d / Then, for any k and z C, 4) W k z) + W k z) e C z +τ M z RΛ k ))+C RΛ k )+ C τ On the other hand, for any k and x R, 43) W k x) + W k x) e x +C RΛ k )+ C τ The proof of this rather technical proposition is given in Appendix A For now, let us continue with the proof of Theorem 4 Since the function M only has real zeros, all the functions introduced in 4) are well-defined and they are entire functions For every l, iλ l is a simple zero of the function W k since Λ l is a simple zero of f and iλ l + IΛ k ) is not a zero of M I[iΛ l + IΛ k )] = RΛ l ) 0 by assumption ) Thus, we deduce that, for every l, iλ l is a zero of Φ k with exact multiplicity η, ie, Φ η) k iλ l)=[w kiλ l )] η 0 and Φ ν) k iλ l)=0 k, l, ν 0,η Observe that, in particular Φ η) k iλ k) = At this point, the function Φ k,j =Φ k then satisfies 4) forl k For any k, j 0,η, andz C, let us now set ) η Φ k z) f k,j z) = z iλ k ) η j = Φ k z)z iλ k ) j iλ k Note that, for x R, we deduce from 43), assumption 4, and45), that 44) f k,j x) Ce η x +C RΛk )+ C τ From the properties of the function Φ k,weget f ν) k,j iλ l)=0 l with l k, ν 0,η, 45) f ν) k,j iλ k)=0 ν 0,j, f j+r) j + r)! k,j iλ k )= η + r)! Φη+r) k iλ k ) r 0 We look now for Φ k,j in the following form: Φ k,j z) =pz)f k,j z) with p a polynomial function of degree η j which depends on k, j for simplicity, this dependence is omitted in the notation) As a consequence of inequality 4) andthefactthatτ M <τ, the function Φ k,j is an entire function of exponential type ητ = T/ 4 4 The constant C such that Φ k,j z) Ce ητ z for every z C depends on k, j, τ, etc, but this is not important as mentioned earlier

21 990 BENABDALLAH, BOYER, GONZÁLEZ-BURGOS, AND OLIVE In view of 45), if we simply take p =, then the relations 4) are satisfied for l k and l = k if ν<j Thus, in order to get 4), we have to choose p such that Φ j) k,j iλ k)= ij π and Φ j+r) k,j iλ k )=0forr,η j, thatis, 46) where piλ k )= ij π f j) k,j iλ k) = ij η! π j!, r a rl p l) iλ k )+p r) iλ k )=0 l=0 47) a rl = j+r l j+r r ) ) f j+r l) k,j iλ k ) f j) k,j iλ = k) r,η j, r!η! l!η + r l)! Φη+r l) k iλ k ) for every r,η j and l 0,r they are well-defined since f j) k,j iλ k) 0) These relations allow us to compute p r) iλ k ) for every r 0,η j and thus completely determine p which is then given by pz) = η j r=0 p r) iλ k ) z iλ k ) r r! In order to get the bound 43) forφ k,j, let us prove some estimates of the polynomial p previously constructed If we set P = p r) iλ k ) ) r 0,η j Cη j, then we can rewrite the identities in 46) as a linear system of the form AP = B with 0 0 a 0 A = a 0 a M η j C), 0 a η j,0 a η j, a η j,η j B = i j π η! j! 0 0 and a rl given in 47) possible to show C η j, 48) P C η j C Again, following [AKBGBdTa, eq 3), p 570], it is r,η j l 0,r Φ η+r l) k iλ k ) η j

22 BOUNDARY CONTROLLABILITY FOR PARABOLIC SYSTEMS 99 Finally, let us estimate Φ η+r l) k iλ k ), forr,η j and l 0,r Since Φ k is an entire function, we can write Φ m) k iλ k )= m! Φ k z) dz m 0, iπ z iλ k ) m+ so that z iλ k = Φ m) k iλ k ) C sup z: z iλ k = Φ k z) Using inequality 4), the fact that z + Λ k for z such that z iλ k =, inequalities 45), and the fact that τ M <d /, we obtain Φ m) k iλ k ) Ce C RΛ k )+ C τ k, m 0 Goingbackto48), we get P C η j Ce C RΛ k )+ C τ Recall that the vector P contains the coefficients p r) iλ k ) of the polynomial p Thus, using that z r /r! Ce η 4 z for any r 0,η, and using 45), we obtain pz) Ce η 4 z +C RΛk )+ C τ z C Combining the previous estimate, written for x R, and44) we deduce the expected bound 43) forφ k,j = pf k,j Appendix A Proof of Proposition 43 We start with another property satisfied by the sequence {Λ k } k, namely, that it behaves as k Lemma A Under assumptions 4, 5, and6, we have A) Ck Λ k C k k The second lemma was often used Lemma A Let τ<d / ForthefunctionM given by 48) we have τ M <τ where τ M is given in 49)) The next lemmas are devoted to giving bounds of every term involved in the definitions 4) ofw k and W k Lemma A3 Under assumption 6 we have, for every z C and n, log fz) d ) z + C, log f n z) d ) z + C, where f and f n are defined in 46) Lemma A4 Under assumptions 4, 5, and6 we have, for every n, log f Λ n ) C Λ n, where f is defined in 46) Lemma A5 Let τ<d / The function M given by 48) satisfies A) M0) =, log Mx) d x + C τ x R

23 99 BENABDALLAH, BOYER, GONZÁLEZ-BURGOS, AND OLIVE Lemma A6 Let τ<d / The function M given by 48) satisfies A3) log Miy) 0 y R, and also A4) log Miy) τ M y C y C y R τ Proof of Proposition 43 Let us recall the definition of W k : W k z) = f iz) if Λ k ) Mz + IΛ k )) MiRΛ k )) From Lemmas A3 and A4 and Λ k CRΛ k )see45)) we deduce that A5) f iz) if Λ k ) ed ) z +C RΛ k ) On the other hand, from inequality A4) of Lemma A6 and using 40) wecan also infer Mz + IΛ k )) MiRΛ k )) eτm z +τm IΛ k) RΛ k ))+C RΛ k )+ C τ Note that τ M IΛ k ) C RΛ k ) thanks to assumption 3 and τ M <d / Thus, putting both inequalities together we deduce estimate 4) for the function W k Let us now take x R Applying inequality A) of Lemma A5 and, this time, inequality A3) of Lemma A6, we arrive at Mx + IΛ k )) MiRΛ k )) e d x +d IΛ k ) + C τ Note that IΛ k ) C RΛ k )by45) Thus, the previous inequality together with A5) written for x R) providetheestimate43) forw k x), with x real The same reasoning provides the estimate for W k ProofofLemmaA The lower bound easily follows from assumption 5 by taking n = To prove the upper bound, let us first observe that, for any k and n such that Λ k = Λ n, we have, using assumption 4, RΛk ) RΛ n ) = IΛk ) IΛ n ) β RΛ k )+RΛ n )), so that It follows that using assumption 4 again) RΛ k ) RΛ n ) β Λ k Λ n RΛ k ) RΛ n ) + IΛ k ) IΛ n ) β +β Λ k By using assumption 5, and the fact that k + n k, weobtain { k n max q, β +β } Λ k ρk

24 BOUNDARY CONTROLLABILITY FOR PARABOLIC SYSTEMS 993 Note that if k is such that β +β Λ k ρk q then Λ k qρ β )k and we are done Let us then deal with the k such that β +β Λ k ρk >q Applying the previous estimate with n = N Λ k ) which indeed satisfies Λ n = Λ k by 35) and assumption 4), we deduce that N Λ k ) k + N Λ k ) k k + β +β Λ k, ρk and by assumption 6 we finally obtain For k large enough, we obtain p Λ k α + N Λ k ) k + β +β Λ k ρk p Λk k + β ρk ) + β k, ρ and the lemma is proved ProofofLemmaA For the proof we will follow some ideas from [FR7] and [Mil04] see also[red77]) Let us consider the counting function N associated with the sequence {a n } n given by 47): Nr) =#{n :a n r} Observe that the sequence {a n } n 0 can be written as a n = a 0 + n A n with A = d and a 0 = d τ 4 d, and that a 0 > 0 since we assumed that τ<d / Thus, Nr) =0forr<a and Nr) = A r a 0 r a, where we recall that is the floor function Note that These remarks in mind, we have A r A a 0 Nr) A r r 0 τ M = + = a n n a r dnr) = + r A <A a r dr = = τ, a + where the last inequality is strict since a 0 0 Proof of Lemma A3 Givenz C, one has log fz) k a Nr) r dr + a A r a0 r dr log + z ) + = log + z ) dn t) Λ k Λ t

25 994 BENABDALLAH, BOYER, GONZÁLEZ-BURGOS, AND OLIVE Taking into account lim t + N t)/t = 0 consequence of assumption 6) anintegration by parts gives + log + z ) + z dn t) = N t) dt Λ t Λ t z + t) After the change of variable t = z s, weobtain + z N t) dt = Λ t z + t) From assumption 6, we conclude that + Λ / z N z s) + ss +) ds p z Λ / z pπ z + α log + Λ / z N z s) ss +) ds + ds + α ss +) + z ) Λ Λ / z ss +) ds Since the function z C α log + z / Λ ) z is bounded on C, the lemma is proved Repeating the arguments, we obtain the same estimate for f n ProofofLemmaA4 To prove the result we are going to follow some ideas from [LK7] and[fr75] seealso[fcgbdt0]) First, note that A6) f Λ n )= Λ n k n Given n, let us introduce the sets Λ ) n Λ k n S n) ={k n : Λ k Λ n } and S n) ={k : Λ k > Λ n } and the infinite product A7) P n = Λ n Λ k k n Let us give a lower bound for the product P n To this end, we split this product into two parts using the sets S n) ands n) From the definition of S n) and using 5, we can write where k S n) Λ n = Λ k A = k S n) k n q k S n) k n q Λ k Λ n Λ k k S n) k n <q ρ k n k + n) Λ n inf k n: k n <q Λ k Λ n > 0 Λ k Λ n Λ k k S n) k n <q A Λ n,

26 BOUNDARY CONTROLLABILITY FOR PARABOLIC SYSTEMS 995 It follows that Λ n Since k S n) k n <q k S n) we deduce that Λ k A ρ k n k S n) ρ k n k + n) Λ n ) q A ρq k S n) k n <q k S n) k n <q k S n) k n <q A ρ k n k + n) k + n, n, n + q ) q A k n k + n) C n + q ) q As Λ n Cn for every n seea)), we obtain A k n k + n) C Λ n q k S n) k n <q Let us define r n =#{k S n) :k<n} and s n =#{k S n) :k>n} From A), we deduce that k + n C Λ n for any n, k Thus, Λ ) n rn ) sn Λ k C Λ n q ργ ργ r n! s Λ n / n! A8) Λ n / k S n) and = C Λ n q P ) n P ) n n Let us argue with P ) n A similar reasoning will provide a lower bound for P ) n Observe that there exist two constants c 0,c > 0 such that r! c 0 r e ) r r c =infslog s) s>0 We can then write ) rn ) rn P n ) ργ ργ r n = r n! c Λ n / 0 e Λ n / [ e Λn / ) )] ργ r n ργ r n = c 0 exp log c ργ e Λ n / e Λ n / 0 exp ec ) Λ n / ργ Putting this inequality and the similar one for the product P n ) )ina8) weobtain A9) Λ n e C Λ n C n k S n) Λ k

27 996 BENABDALLAH, BOYER, GONZÁLEZ-BURGOS, AND OLIVE Let us now estimate the product A7) fork S n) which we denote by P 3) n Let c > 0 be such that A0) log s) c s s [0, /] Observe that, for k S n) one has Λ n / Λ k /, so that we can use A0) to obtain log P n 3) log Λ ) n c Λ n Λ k Λ k = c Λ n dn r) k S n) k S Λ n) n r = c Λ n N Λ ) n ) N r) N r) + Λ n Λ n r dr c Λ n Λ n r dr ) α + p r α c Λ n Λ n r dr = c Λ n Λ n + p Λn = αc pc Λ n / Putting A9) and this last inequality into A7), we deduce P n = Λ n Λ k e C Λ n C n k n Since Λ n Λ for every n see assumption 4) we finally have P n e C Λ n n This inequality and formula A6) provide the desired estimate This ends the proof ProofofLemmaA5 For the proof we will follow some ideas from [FR7] and [Mil04] seealso[red77]) Let us first consider again the counting function N associated with the sequence {a n } n given by 47): Nr) =#{n :a n r} Observe again that the sequence {a n } n 0 can be written as A) a n = a 0 + n A n with A = d and a 0 = d τ 4 d, and that a 0 > 0 since we assumed that τ<d / Thus, Nr) =0forr<a and A) Nr) = A r a 0 r a, We will often use that A r A a 0 Nr) A r r 0 Let us prove the inequality A) Observe that M is an even function So, we will show A) forx 0, + ) From the definition 48) ofm, one has log Mx) = log sinx/a n ) + x ) x/a n = g dnr), n a r

28 BOUNDARY CONTROLLABILITY FOR PARABOLIC SYSTEMS 997 where gs) = log sin s s, s R Since, g is nonincreasing on [0, ), for any x [0,a ]wehave log Mx) log M0) =0 d x + d a d x + d τ, which gives the claim in that case Assume now that x>a Wewrite log Mx) = gx/a n )+ gx/a n ) I + J a n x a n>x Since g is negative and nonincreasing on [0, ], the second sum J can be bounded as follows: J gx/a n ) g/) Nx) Nx)) x a n>x g/) A x a 0 A x a 0 ) x = g/) A g/) x a0 + x a 0 g/) A g/) + x In the first sum I, we use the inequality gs) log s for any s 0toget I x r x Nr) logx/a n )= log dnr) = dr a a x) n x a r x A r a0 dr a r x x ) = logx/a ) A dr a 0 a r a0 a r dr r a 0 logx/a ) A x a 0 +A a a 0 + A a 0 + A x + c A a 0 + logx)+ r r dr with c =+ + r dr r Combining the two estimates gives log Mx) A + g/) ) x + +logx + c A a 0 ++ g/) Observe now that a 0 d /τ,thata = d, and that the function x [0, + [ A g/) + x + logx)++ g/)

29 998 BENABDALLAH, BOYER, GONZÁLEZ-BURGOS, AND OLIVE is bounded by some number c > 0 depending only on A = d/ We finally get the inequality log Mx) d x + c d which gives the claim by using that d τ Proof of Lemma A6 We start by observing that sin iy iy = sinh y y y R τ + c, As a consequence, we obtain Miy) for any y R Thus, we immediately get A3) We will now obtain the proof of A4) by adapting the proof of Lemma 63 of [FR7] to the sequence {a n } n given by 47) We set c 0 =log 3 > 0 Assume first that y /c 0 a Then, by using A3), we get log Miy) 0 τ M y τ M c 0 a = τ M y τ M c 0 d d τ M y c 0 y τ c 0 and the claim is proved in that case Assume now that y /c 0 a Observethat siniy) iy = e y e y ) = e y e y y y a = d τ, y e y = e y Thus, using the definitions 48) and49) ofm and τ M,wehave A3) τ y 0 log Miy) = y + e y /a n ) log = τ M y + I, a n y /a n n n where the sequence {a n } n is given by 47) In order to bound the series I, we will use the inequalities e y y e y y>0 and So, for y R with y /c 0 a, one has A4) I = e y /a n ) log y /a n n n a n> y /c 0 e y y y a n + 3y log n a n y /c 0 y log 3=c 0 ) an I +I 3 y LetusfirstboundfrombelowI in the expression A4) One has I = n a n> y /c 0 y a n + dnr) + = y y a n 0 r y /c 0 dnr), r

30 BOUNDARY CONTROLLABILITY FOR PARABOLIC SYSTEMS 999 where n 0 is the smallest integer such that a n0 > y /c 0 and N ) is the counting function associated to the sequence {a n } n see A) and A)) Integrating by parts, we obtain that is to say, I y [ r Nr) + y A A y y /c y /c 0 y /c 0 r a0 dr r y /c 0 r 3/ dr, Nr) r A5) I 4c / 0 A y y c 0 >a Let us deal with the second term I in A4) for y satisfying a < y /c 0 Usingthatforanyr [a, y /c 0 ] one has r<3 y c 0 =log 3), we can write ) an ) an r I = = log dnr) 3 y a 3 y log n a n y /c 0 y /c0 a ) r log dnr), 3 y where n is the largest integer such that a n y /c 0 Again, integrating by parts, we deduce ) r y /c 0 y /c0 Nr) I Nr)log dr 3 y a a r y /c0 r a0 = log3c 0 )N y /c 0 ) A dr a r y /c0 log3c 0 )N y /c 0 ) A dr r A +log3c 0 )) c / 0 0 y In view of A3) anda4), this last inequality together with A5) provides c 0 =log 3) log Miy) τ M y c d y with c =+c 0 + log3c 0 )/)c / 0 Owing to the previous calculations, we finally obtain the inequality A4) This ends the proof dr ]

31 3000 BENABDALLAH, BOYER, GONZÁLEZ-BURGOS, AND OLIVE REFERENCES [AB] [ABL] [AKBDGB09a] [AKBDGB09b] F Alabau-Boussouira, Controllability of cascade coupled systems of multidimensional evolution PDEs by a reduced number of controls, CRAcad Sci Paris Sér I Math, 350 0), pp F Alabau-Boussouira and M Léautaud, Indirect stabilization of locally coupled wave-type systems, ESAIM Control Optim Calc Var, 8 0), pp F Ammar-Khodja, A Benabdallah, C Dupaix, and M González-Burgos, A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems, Differential Equations Appl, 009), pp 39 5 F Ammar-Khodja, A Benabdallah, C Dupaix, and M González-Burgos, A Kalman rank condition for the localized distributed controllability of a class of linear parabolic systems, J Evol Equations, 9 009), pp 67 9 [AKBGBdTa] F Ammar-Khodja, A Benabdallah, M González-Burgos, and L de Teresa, The Kalman condition for the boundary controllability of coupled parabolic systems Bounds on biorthogonal families to complex matrix exponentials, J Math Pures Appl 9), 96 0), pp [AKBGBdTb] F Ammar-Khodja, A Benabdallah, M González-Burgos, and L de Teresa, Recent results on the controllability of linear coupled parabolic problems: A survey, Math Control Related Fields, 0), pp [AKBGBdT3a] F Ammar-Khodja, A Benabdallah, M González-Burgos, and L de Teresa, A new relation between the condensation index of complex sequences and the null controllability of parabolic systems, C R Acad Sci Paris Sér I Math, 35 03), pp [AKBGBdT3b] F Ammar-Khodja, A Benabdallah, M González-Burgos, and L de Teresa, Minimal time for the null controllability of parabolic systems: The effect of the condensation index of complex sequences, J Funct Anal, 67 04), pp [BDR07] [BN0] [DR77] [Fat75] [FCGBdT0] [FCZ00] [FR7] [FR75] [Güi85] [KFA69] [LK7] [LR07] A Benabdallah, Y Dermenjian, and J Le Rousseau, On the controllability of linear parabolic equations with an arbitrary control location for stratified media, C R Acad Sci ParisSér I Math, ), pp A Benabdallah and M G Naso, Null controllability of a thermoelastic plate, Abstr Appl Anal, 7 00), pp S Dolecki and D L Russell, A general theory of observation and control, SIAM J Control Optim, 5 977), pp 85 0 H O Fattorini, Boundary control of temperature distributions in a parallelepipedon, SIAM J Control, 3 975), pp 3 E Fernández-Cara, M González-Burgos, and L de Teresa, Boundary controllability of parabolic coupled equations, J Funct Anal, 59 00), pp E Fernández-Cara and E Zuazua, The cost of approximate controllability for heat equations: The linear case, Adv Differential Equations, 5 000), pp H O Fattorini and D L Russell, Exact controllability theorems for linear parabolic equations in one space dimension, Arch Ration Mech Anal, 43 97), pp 7 9 H O Fattorini and D L Russell, Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations, Quart Appl Math, 3 974), pp E N Güichal, A lower bound of the norm of the control operator for the heat equation, J Math Anal Appl, 0 985), pp R E Kalman, P L Falb, and M A Arbib, Topics in Mathematical Control Theory, McGraw-Hill, New York, 969 W A J Luxemburg and J Korevaar, Entire functions and Müntz-Szász type approximation, Trans Amer Math Soc, 57 97), pp 3 37 J Le Rousseau, Représentation Microlocale de Solutions de Systèmes Hyperboliques, Application à l Imagerie, et Contributions au Contrôle et aux Problèmes Inverses pour des Equations Paraboliques, Habilitation thesis, Aix-Marseille Université, 007)