Key words. boundary controllability, essential spectrum, Ingham inequality, mixed order operator

Transcription

1 EXAC BOUNDARY CONROLLABILIY OF A SYSEM OF MIXED ORDER WIH ESSENIAL SPECRUM FARID AMMAR KHODJA, KARINE MAUFFREY, AND ARNAUD MÜNCH Abstract We address in this work the exact boundary controllability of a linear hyperbolic system of the form u Au = with u = u 1, u posed in,, 1 A denotes a selfadjoint operator of mixed order, that usually appears in the modelization of linear elastic membrane shell he operator A possesses an essential spectrum which prevents the exact controllability to hold uniformly with respect to the initial data `u, u 1 We show that the exact controllability holds by a one Dirichlet control acting on the first variable u 1 for any initial data `u, u 1 generated by the eigenfunctions corresponding to the discrete part of the spectrum σa he proof relies on a suitable observability inequality obtained by the way of a full spectral analysis and the adaptation of an Ingham type inequality for the Laplacian in two space dimension his work provides a non trivial example of system controlled by a number of controls strictly lower than the number of components Some numerical experiments illustrate our study Key words boundary controllability, essential spectrum, Ingham inequality, mixed order operator AMS subject classifications 35L, 93B5, 93B7, 93B6, 93C 1 Introduction - Problem statement Let Ω =, 1 and Γ be the part of Ω defined by Γ = {x, y Ω, xy = } Let be a positive real number and a, α be two real numbers such that a > α > and a α /π / N We analyze in this work the exact boundary controllability of the following system in u = u 1, u u 1 = u 1 α x u in Q = Ω, u = α x u 1 au in Q u 1 = v 1 Γ on Σ = Ω, u,, u, = u, u 1 in Ω 11 We set H = L Ω L Ω and H 1/ = H 1 Ω L Ω Let H 1/ denotes the dual of H 1/ with respect to the pivot space H System 11 is said exactly controllable at time > if for any initial data u, u 1 H H 1/ and any target u, u1 H H 1 there exists a control function v in a suitable space, such that the unique solution u = u 1, u of system 11 satisfies u,, u, = u, u 1 in Ω We point out that the variable u in system 11 is free of any condition on the boundary Ω In particular, system 11 provides a non trivial example for which the number of controls is strictly lower than the number of components Laboratoire de Mathématiques de Besançon, UMR CNRS 663, Université de Franche-Comté, 16 route de Gray, 53 Besançon cedex, France faridammar-khodja@univ-fcomtefr Laboratoire de Mathématiques de Besançon, UMR CNRS 663, Université de Franche-Comte, 16 route de Gray, 53 Besançon cedex, France karinemauffrey@univ-fcomtefr Laboratoire de Mathématiques de Clermont-Ferrand, UMR CNRS 66, Université Blaise Pascal, Campus des Cézeaux, Aubière cedex, France arnaudmunch@mathuniv-bpclermontfr Partially supported by grants ANR-7-JCJC Agence national de la recherche, France and 87/PI/8 from Fundacíon Séneca Gobernio regional de Murcia, Spain 1

2 F AMMAR KHODJA, K MAUFFREY AND A MÜNCH As it is usual, the controllability issue is equivalent to an observability inequality that we will rigorously prove to be:, 1 H 1/ H C Γ ϕ ν αψν 1 dσdt, 1 ν = ν 1, ν denotes the unit outward normal to Γ for any initial data, 1 belonging to H 1/ H, for the homogeneous adjoint system in = ϕ, ψ : ϕ = ϕ α x ψ in Q ψ = α x ϕ aψ in Q ϕ = on Σ,,, =, 1 in Ω 13 he observation zone Γ is defined so that the triplet Ω, Γ, satisfies the geometric optic condition he natural operator we want to consider to transform 13 into a second order differential equation is the operator A = α x α x a in L Ω L Ω with domain D A = H Ω H 1 Ω D x Here x is considered as an unbounded operator in L Ω We can prove that this operator A is not closed in L Ω L Ω herefore we have to consider its closure A which is defined by : A : D A L Ω L Ω L Ω L Ω, A ϕψ = ϕ α 1 x ψ α x ϕ aψ and D A = { } ϕ, ψ H 1 Ω L Ω /ϕ α 1 x ψ H Ω see [1] he originality and difficulty of the - apparently simple - system 13 are related to the fact that A is a mixed order operator, and therefore, possesses a non-empty essential spectrum σ ess A, as shown in [6] in a general situation As a consequence, the observability does not hold uniformly with respect to the data, 1 Precisely, in [1] the authors exhibit Weil sequences, associated with some elements of σ ess A for which the observability inequality 1 is not true he observability is therefore only expected, roughly speaking, in the orthogonal of some space related to the essential spectrum o our knowledge, the only way used up to now to address this kind of problem is based on spectral analysis and Ingham type approach which allows to prove the observability for the discrete part of the spectrum, provided some spectral gap conditions see [11] In that framework, the existing literature mainly concerns the controllability of dynamical systems modeling the vibrations of elastic membrane shell, where precisely mixed order and self-adjoint operators appear we refer to [19] for a detailed spectral analysis We also mention [8, 9] where the controllability of an hemi-spherical cap is studied using a nonharmonic spectral analysis he analysis, reduced to the one space dimension by axial symmetry, exhibits the loss of uniform observability due to the essential spectrum composed of a single positive element A similar study is performed in [] for a nonuniform elliptic operator A for which σ ess A We also refer to the chapter 5 of [1] for results based on some recent extensions of Ingham type inequalities For systems of this kind, the uniform partial controllability, which consists to drive to rest only a restricted number of components is proved in [13] he observability is obtained by a so-called spectral compensation

3 EXAC BOUNDARY OBSERVABILIY 3 argument, remarking that the bad behavior of the part of the spectrum which accumulates to σ ess A is somehow compensated by the suitable gap of the discrete part In a different context, we also mention [3, 16, 18] for the controllability of systems with spectral accumulation point he significant novelty of this present work with respect to the literature mentioned above is that it concerns the two space dimension he proof of a positive observability result for the discrete part of the spectrum, much less straightforward than for the one-dimensional situation, is obtained by the adaptation of a recent Ingham type theorem due to Mehrenberger [17] that allows to prove the observability for the wave equation in any space dimension An other difficulty in the study of the controllability for system 11 is that the associated control operator is not bounded from the space of controls L Γ, to the state space X 1 the dual of X 1 = H 1/ H with respect to the pivot space X = H H 1/ he paper is organized as follows In Section, we first state the spectral properties of the operator A Explicit computations show that the set of the eigenvalues of A is composed of three parts: { } {} {a}, where { } is a bounded sequence which accumulates on the full interval [a α, a] = σ ess A and {} - the set of isolated eigenvalues of A of finite multiplicity - is an unbounded sequence such that p q π for p, q large In Section 3, we establish the well-posedness of the boundary value problem 11 for v and any initial data u, u 1 in suitable spaces We divide Section 4 into two parts he first part is devoted to the analysis of the adjoint system and to the formulation of the observability inequality as 1 In the second part, we prove the observability for any initial data, 1 belonging to H 1/ H, the space spanned by the eigenfunctions {e } associated with the eigenvalues {} he keypoint is that the sequence {} enjoys gap properties similar to those of the eigenvalues of with Dirichlet boundary values, used in [17] On the contrary, Section 5 exhibits the lack of observability in spaces related to the essential spectrum In particular, by numerical approximation, we check that the corresponding observability constant C is not bounded uniformly with respect to, 1 We also discuss the uniform observability with respect to the coupling parameter α as α Section 6 concludes this work with some remarks and open problems he operator A We consider the operator A defined by : A : D A L Ω L Ω = H H, A ϕψ = ϕ α 1 x ψ α x ϕ aψ, { } D A = ϕ, ψ H 1 Ω L Ω /ϕ α 1 x ψ H Ω 1 It is well-known and easy to check that the eigenvalues and normalized eigenfunctions of with domain H Ω H 1 Ω are respectively given by µ pq = p q π and ϕ pq x, y = sinpπx sinqπy for p, q in N N and x, y Ω =, 1 We

4 4 F AMMAR KHODJA, K MAUFFREY AND A MÜNCH introduce the following notations: for p, q 1 ± = 1 µ pq a ± µ pq a 4α p π, ψ pq x, y = cospπx sinqπy, e ± = ± a q ϕ ± a pq, q αpπ ψ α p π ± a pq, α p π e q x, y =, sin qπy Lemma 1 1 For all p, q N N, we have: Ae ± = ± e ±, Ae q = ae q Ker A I = {}, / {a} {} { } In other words, the sets of eigenvalues and associated eigenfunctions of A are respectively {a} {} { } and {e q } { } e { } e We have µ pq Proof Let be an eigenvalue of A and u = u 1, u be an associated eigenvector hen u is a non-zero solution of the system a u = α x u 1 u 1 α 1 x u u1 = u 1 Ω = Assuming first that = a, system implies that x u 1 =, u 1 α 1 x u = in Ω together with the boundary condition u 1 Ω = his implies that u 1 = and x u = in Ω Consequently, we can write u x, y = fy with f L, 1 It follows that = a is an eigenvalue of A and the associated eigenspace is {x, y, fy : f L, 1} Now, if a, then system may be written as u = α a xu 1 1 α a xx u 1 yy u 1 u 1 = 3 u 1 Ω = We can look for u 1 in the form u 1 = u ϕ pq with u < u 1 is a N N solution of 3 if and only if for all p, q N N, p π 1 α q π u = 4 a It follows from 4 that satisfies the eigenvalue equation a p q π a α p aq π = 5 We conclude that = 1 a p q π ± a p q π 4α p π = ± It is easily seen that Ae ± = ± e ± his completes the proof of item 1 Item is easy to check

5 EXAC BOUNDARY OBSERVABILIY 5 We refer to [5] for the definition of the essential spectrum of A, σ ess A As a consequence of the results in [6], we have: σ ess A = [ a α, a ] On the other hand, we can check that the set of accumulation points of the sequence is N N σ ess A = [a α, a] Remark 1 he asymptotic behavior of pq in Lemma 1 is in agreement with [7] where it is shown that the asymptotic behavior of σa is related to the spectrum of the principal part of A, in our case We check that a < 1,1, p, q N N hus all the are isolated eigenvalues of finite multiplicity Similarly, simple computations lead to: p 1, [a α, a] q > a α /π herefore, if a α /π < 1, then is bounded away from N N Figures 1 and illustrate these two points N N = σ ess A and σ ess A x Fig 1 Graphs of and a = q 4 4 p q [1,1] Left and of Right for α = 1 [1,1] 4 4 p mina α,π a α a 1,1 set of the accumulation points of p q π Fig Distribution of σ A along R

6 6 F AMMAR KHODJA, K MAUFFREY AND A MÜNCH 3 Using the fact that the families {ϕ pq } p 1, and {ψ pq } p, are two Hilbert bases of L Ω, we can easily check that the family B = { } e { } p 1, e {e p 1, q} forms a Hilbert basis of H = L Ω L Ω he operator A defined in 1 is self-adjoint in H, and positive if, as we have assumed it, a > α his last point comes from the formula: [ Au, u H = x u 1 αu a α u y u 1 ] dx dy, Ω where u = u 1, u D A For any δ R, we recall that the operator A δ is defined by A δ = δ, e H e δ, e H e a δ, e q H e q, D A δ = φ H, δ φ, e H δ φ, e aδ φ, e H q H < Since is bounded, D A δ = N N φ H, δ φ, e < H In the sequel, we will set H δ = D A δ, δ he operator A is a bounded operator from DA, equipped with the graph norm, to H It is well-known that A can be extended to a bounded operator from H to DA, the dual space of DA with respect to the pivot space H We continue to denote this extension by A and, thus, A can be seen as an unbounded self-adjoint operator on DA with domain H A is also a unitary operator from DA to H and from H to DA We will set in the sequel: H 1 = DA he other extension of A we will use later is the following A can also be extended as a unitary operator from DA 1/ equipped with the graph norm to DA 1/, the dual space of DA 1/ with respect to the pivot space H We will set: H 1/ = D A 1/, H 1/ = D A 1/ For details on these extensions, see for instance [] he last notations we will need in the next sections are the following: and for δ R, H ± = span { e ±, p, q 1 }, H a = span {e q, q 1} H ± δ = H δ H ±, H a δ = H δ H a 3 Well-posedness of the controlled system his section is devoted to the study of existence and uniqueness of solution for system 11

7 EXAC BOUNDARY OBSERVABILIY 7 31 A Dirichlet map In this part, we introduce and analyze the Dirichlet map D : D D L Γ H corresponding to system 11 defined as follows: D D = { v L Γ, Dv H } and for v L Γ, we denote by Dv the solution θ = θ 1, θ of the abstract elliptic problem { Λθ = in Ω θ 1 = v 1 Γ on Ω, Λ = α x α x a 31 he map D satisfies the following continuity property: Proposition 31 For every ɛ 1/, 1], D L H ɛ Γ, H Moreover for every v D D we have Dv, e ± H = pπ ± aα qπ ± ± a a v,q α p π ± a v 1,p 3 α p π ± Dv, e q H = α a v,q 33 with v 1,p = 1 vx, sin pπx dx and v,q = 1 v, y sin qπy dy Proof Let ɛ 1/, 1] and v H ɛ Γ Suppose that 31 has a solution θ H Using integrations by parts, we have = Λθ, e ± = θ, Ae ± ϕ ± H αψ ± H Γ ν ν 1 v dσdt where e ± = ϕ ±, ψ ± Since Ae ± = ± e ±, this yields θ, e ± H = pπ ± a α qπ ± ± a a v,q α p π ± a v 1,p 34 α p π ± By the same arguments, we obtain θ, e q H = ± ± α a v,q 35 his proves that if 31 has a solution θ, then this solution is unique and it writes θ = θ, e H e θ, e H e θ, e q H e q, q with θ, e ± and θ, e H q H given by 34 and 35 Now, we have to check that such a θ is an element of H Formula 34 gives θ, e p ɛ1 π a α v,q q ɛ1 π H a a v α p π p ɛ 1,p a α p π q ɛ From the asymptotic property π a α a α p π µ pq, we have π a a α p π π µ pq

8 8 F AMMAR KHODJA, K MAUFFREY AND A MÜNCH Consequently, there exists a positive constant c 1, independant of v, such that for every p, q N N, π a π a α a c 1 and α p π µ pq a α p π c 1 µ pq Hence p ɛ1 π aα p ɛ1 p a c 1 c 1 α p π µ pq µ pq c 1 and q ɛ1 π a a c 1 α p π Finally θ, e H v,q c 1 p ɛ v 1,p q ɛ his gives θ, e H 1 c1 r ɛ v,q v 1,p = 4c 1 1 r ɛ v L Γ r 1 q N p N r 1 36 From 34, we also have θ,e p ɛ1 π aα H q ɛ a q ɛ v,q q α p π p ɛ ɛ1 π a p ɛ a p ɛ v 1,p α p π q ɛ Using the definition of, we can prove that as p, q π aα a α q α p π p p q, π a a α p π α p p q Since is bounded away from, there exists a constant c N N > such that for every p, q N N : π aα a α p π c q p p q, π a a α p π c p p q his implies that: p ɛ1 π aα q ɛ a α p π c p ɛ q ɛ p q, q ɛ1 π a p ɛ a α p π c p 1 ɛ q ɛ1 p q We have pɛ q ɛ p q 1 and p1 ɛ q ɛ1 p q 1 since ɛ 1 We conclude that θ, e H q ɛ v,q c p ɛ pɛ v 1,p q ɛ

9 EXAC BOUNDARY OBSERVABILIY 9 Since v H ɛ Γ, we can write from this inequality θ, e H 1 c r ɛ q ɛ v,q p ɛ v 1,p 4c 1 r ɛ v H ɛ Γ r 1 q N p N r 1 37 Besides, formula 35 clearly gives θ, e q H l q N N, with α θ, eq H = a v,q α a v L Γ 38 q N q N Combining 36, 37 and 38, we obtain θ H, with θ H c c ɛ = max π r, c ɛ r, α ɛ a r 1 r 1 c ɛ v Hɛ Γ and Proposition 31 gives that H ɛ Γ D D for every ɛ 1/, 1] his implies that D : D D L Γ H is an unbounded operator with dense domain in L Γ Consequently, the adjoint operator D of D is well-defined as an unbounded operator D : D D H L Γ Proposition 3 D is given by D D = D g = { A 1 g 1 g H, Γ A 1 g 1 ν α A 1 g ν 1 Proof For v D D and g H, we have Dv,g H = Dv,e H By 3 and 33, we have ν α A 1 g ν 1 dσ < A where A 1 1 g g = 1 Γ A 1 g g,e Dv,e H H g,e H }, g D D Dv,e q H g,e q H Dv,g H = v 1,p qπ a g,e H p 1 a qπ a g,e H α p π a α p π,q v pπ aα g,e H p 1 a pπ aα g,e H α p π a α p π α a g, e q H 39 By definition of the adjoint of an unbounded operator, D D is the set of the elements g H such that v Dv, g H is a continuous linear form on L Γ Using 39, we see that D D is the set of the elements g H such that the two following sequences are in l N qπ : a g,e q H qπ a g,e q H, a α p π a α p π p 1

10 1 F AMMAR KHODJA, K MAUFFREY AND A MÜNCH p 1 pπ aα g,e q H pπ aα g,e H a α p π q α a α p π a g, e q H It is easily seen that A 1 g 1 α A 1 g ν ν 1 L Γ = pπ aα g,e H p 1 a pπ aα g,e H α p π a α α p π a g,e q H qπ a g,e H p 1 a qπ a g,e H α p π a α p π { his allows to conclude that D D = g H, Now we suppose that g D D Dv,g H = Γ v A 1 g 1 ν α A 1 g ν 1 ν α A 1 g ν 1 A 1 g 1 L Γ< hen we can easily check that 39 leads to dσ and then D A g = 1 g 1 ν α A 1 g ν 1 Proposition 33 D : D D L Γ H is a closed operator on L Γ Proof Let v n n N be a sequence in D D which converges to a certain v in L Γ and such that Dv n converges in H to an element θ Let v1,p n and v,q n denote v n 1,p = 1 v n x, sin pπx dx, v n,q = 1 v n, y sin qπy dy Similarly, let v 1,p = 1 vx, sin pπx dx and v,q = 1 v, y sin qπy dy Since v n v L Γ = 1 v n 1,p v 1,p v n,q v,q, p 1 we clearly have: v1,p n v 1,p for any p N and v,q n v,q for any q N n n From 3 and 33 we deduce that for every p, q N N : Dvn, e ± H n ± α n a v,q pπ ± aα q ± a α p π v,q ± qπ ± a q v ± a 1,p α p π Dv n, e q H Besides, the convergence of Dv n to θ in H implies the convergence of Dv n, e ± H resp Dv n, e q H to θ, e ± resp θ, e H q H for every p, q N N We deduce that, for all p, q N N θ, e ± H = pπ ± aα q v ± a,q α p π θ, e q H = ± α a v,q ± qπ ± a q v ± a 1,p α p π From 3 and 33, it means that θ = Dv Since θ H, this implies that v D D We have thus proved that D is closed } Γ

11 EXAC BOUNDARY OBSERVABILIY 11 3 oward an internal control problem We introduce the following notations: X = H H 1/, X 1 = H 1/ H 1, X 1 = H 1/ H, and L : D L X 1 X 1, L = I A, D L = X Note that the operator occuring in the definition of L is the extension of A from H to H 1 In this part we transform the boundary control problem 11 into the familiar form of an internal control problem: { Z = LZ Bv in Q Z = u, u 1 31 in Ω he originality of this problem is that B is an unbounded control operator from L Γ to X 1 Assume that v is an element of H 1 [, ], D D For the moment we denote by Z the vector Z = u, u Dv, where u is solution of 11 hen Z is solution of Z = I Λ Z Dv Since v H 1 [, ], D D, we have that Dv L [, ], H so that Dv, is an element of L [, ], D L herefore Z is solution of Z = LZ Dv in X 1 and the semigroup theory gives Zt = StZ t St s Dv s ds 311 where St t is the C -semigroup associated with the maximal and dissipative operator L Integrating by parts in 311 and using Dvs D L, we obtain Zt = StZ Dvt St Dv t St sl Dvs ds Replacing Zt by its definition, we obtain the following expression for ut, u t : ut u t u = St t u 1 St sbvsds 31 where B = AD : DB L Γ X 1 313

12 1 F AMMAR KHODJA, K MAUFFREY AND A MÜNCH is an unbounded operator with dense domain DB = D D Formula 31 means that u, u is the mild solution Z of the internal control system 31 heorem 34 For every u, u 1 X and every v H 1 [, ], D D, system 11 has a unique solution u in X 1 Moreover, u C [, ], H C 1 [, ], H 1/ C [, ], H 1 Proof Let u, u 1 X and v H 1 [, ], D D Set Z = u, u 1 We have to solve system 31 in X 1 Let us prove that Bv H 1 [, ], X 1 : Bv H 1 [, ],X 1 = Bv L [, ],X 1 Bv L [, ],X 1 = ADv L [, ],H 1 ADv L [, ],H 1 = Dv L [, ],H Dv L [, ],H = Dv H 1 [, ],H v H 1 [, ],DD < Since Bv H 1 [, ], X 1 and Z X, the semigroup theory see [, heorem 416 page 113] ensures that system 31 has a unique solution Z in X 1 and that Z C [, ], X C 1 [, ], X 1 he existence and uniqueness of the solution u of 11 with the regularity u C [, ], H C 1 [, ], H 1/ C [, ], H 1 follow 4 Controllability and observability 41 Formulation of the observability inequality We introduce the control operator L : D L L Γ, X defined by L v = S tbvtdt, 41 with domain D L = { v L Γ,, L v X } he exact boundary controllability problem for system 11 is the following: given > large enough, initial data u, u 1 X and final data u, u1 X, to find a control function v in D L, such that the solution u of system 11 satisfies u, = u, u, = u 1 in Ω herefore, system 11 is exactly controllable at time if and only if the operator L is onto he following proposition may be proved using similar arguments as in Proposition 33 Proposition 41 L : D L L Γ, X is a closed operator By heorem 34, DL contains H 1 [, ],DD, which is dense in L Γ, since DD is dense in L Γ Hence DL is dense in L Γ, his allows to compute the adjoint L of L Since L is closed, the surjectivity of L is equivalent see for instance [4, heorem II19 page 9] to the existence of a positive constant c such that L Ψ Ψ 1 L Σ c Ψ Ψ 1 X, Ψ Ψ 1 D L his last inequality is also equivalent to: L L 1 c 1, 1 D L L 4 X 1 L Σ

13 EXAC BOUNDARY OBSERVABILIY 13 Consequently, the exact controllability problem for system 11 relies on the observability inequality 4 In what follows, we compute L L to translate 4 in terms of the adjoint system 13 of system 11 By the definition of L in 41 we have L = B S So we have to compute the adjoint of B Lemma 4 he adjoint B : D B X 1 L Γ of the operator B defined in 313 is given by: Moreover for every { D B =, 1 } X1 / 1 D D B 1 = D 1, 1 D B L we have 1 D B 43 B L 1 = 1 ν α ν 1, where = 1, 44 Γ Proof Let v D B = D D and 1 X 1 We recall that since A is self-adjoint in H, L is skew-adjoint in X 1 ie L = L his allows to write 1, Bv = 1, LL 1 Bv X 1,X 1 = It is easily seen that L 1 = that L his gives 43 Now suppose that 1, L 1 Bv A 1 I X 1,X 1 = X = so that L 1 Bv = 1, Bv = 1, Dv H X 1,X 1 1 D B L hen B L 1 = B 1 A = D A L 1, L 1 Bv 1 A, L 1 Bv Dv X X It follows Proposition 3 easily gives 44 Proposition 43 he operator L L : D L L X 1 L Γ, is given by: D L L = { 1 L L 1 = X 1 / ϕ ϕ ν αψν 1 } ν αψν 1 L Γ, Γ 45

14 14 F AMMAR KHODJA, K MAUFFREY AND A MÜNCH where = ϕ, ψ is the solution of the backward adjoint system of 11: Proof Let { A = in Q, =,, = 1 in Ω 1 X 1 such that 1 D L L hen L L 1 = B S L 1 = B LS 1 = B L where is the solution of = L = L = 1 in [, ] 46 By 44 we have B L = ϕ ν αψν 1, where = ϕ, ψ his proves Γ the proposition hanks to Proposition 43, the observability inequality 4 consists in the following inequality: 1 D L L Γ ϕ ν αψν 1 dσdt c 1 where = ϕ, ψ is the solution of the backward adjoint system 46 Remark that this is also equivalent to the same inequality when = ϕ, ψ denotes the solution of the forward adjoint system 13 Consequently we have the following characterization of the controllability: Corollary 44 System 11 is exactly controllable at time if and only if there exists a constant C > such that for all initial data, 1 D L L, the solution = ϕ, ψ of the adjoint system 13 with initial data, 1 satisfies the following observability inequality, 1 X 1 C Γ X 1 ϕ ν αψν 1 dσdt 47 4 Observability inequality in H 1/ H From now on, we assume that a π he main result of this paper is the following uniform observability of 13 for initial data in D L L H 1/ H Our proof adapts an Ingham type theorem due to Mehrenberger see [17] which allows to prove, by a spectral method, the observability inequality for the wave equation he crucial point here is that the discrete part {} of σa has the same asymptotic behavior as σ

15 EXAC BOUNDARY OBSERVABILIY 15 heorem 45 Let γ = π π and = π 4 π α γ 1 1,1 aα If a π, 1,1 a then for any > there exists a positive constant C such that for all initial data, 1 in D L L H 1/ H the solution of 13 satisfies the following observability inequality:, 1 ϕ C X 1 ν αψν 1 dσdt 48 Γ 41 he observability inequality in terms of Fourier series Fix, 1 D L L H 1/ H By definition of H 1/ and H, and 1 may be written as = e, 1 = with < and system 13 with initial data, 1 is given by t = 1 i 1 e i t 1 e 1 < he solution = ϕ, ψ of i 1 e i t e Using the definition of e and the Parseval equality, we can easily prove that ϕ Γ ν αψν 1 dσdt = 1 pπ i 1 e i t p 1 1 qπ i 1 e i t p 1 i 1 e i t i 1 e i t Notation 46 1 For every p, q N N we set a = i 1 For p Z and q N we define πa q aα if p 1 x = a α p π x if p 1 3 For p N and q Z we define πa q a if q 1 y = a α p π y p, q if q 1 a α a dt α p π a a dt α p π 49

16 16 F AMMAR KHODJA, K MAUFFREY AND A MÜNCH he left-hand side of 48 is given by, 1 = X 1 1 = a Furthermore, using Notation 46, we can write 49 in the form Γ ϕ ν αψν 1 dσdt = q N p N p p N q N q x e i t x e i t y e i t y e i t dt 41 Consequently, the observability inequality 48 is equivalent to the following inequality C a q N p N p p N q N q x e i t x e i t y e i t y e i t dt 411 where C is a positive constant which does not depend on a N N We recall below the main theorem of [17] for the observability of the wave equation in two space dimension: heorem 47 Mehrenberger [17] We assume the existence of γ 1 > and γ > such that for every p, p, q and q in N p max q, q µ pq ± µ pq γ1 q ± q q max p, p µ pq ± µ p q γ p ± p 41 hen for any > π 1 1, there exists a constant C > such that γ1 γ C p q x p N q N q x e i µ pqt x e i µ pqt dt q N x e i µ pqt x e i µ pqt dt 413 p N p for every complex sequence x N N such that the sums involved are finite he use of Ingham type methods see [11, 1] is based on some gap properties here given by 41 Our observability inequality 411 is similar to 413 since µ pq as p, q he difference is the presence of an other sequence y N N which is different from the sequence x N N Consequently, we cannot apply directly heorem 47 to obtain the observability inequality 411 However, from the definition of x and y we can prove that both x and y dt dt

17 EXAC BOUNDARY OBSERVABILIY 17 are equivalent to the same term a = i 1 / his allows us to adapt the proof of heorem 47 he keypoint is to prove that we have some good gap properties o do this, we consider the sequence Λ Z Z defined below Notation 48 For p Z and q Z let Λ denote if p 1 and q 1 Λ = p, q if p 1 and q 1 if p 1 and q 1 With Notation 46 we deduce from 41 that Γ ϕ ν αψν 1 dσdt = px e iλt dt p Z p 1 qy e iλt dt q Z hus the observability inequality 48 that we have to prove is the following inequality C a px e iλt dt p Z p 1 qy e iλt dt 414 q Z where C is a positive constant which does not depend on the sequence a 4 Some gap properties o adapt the proof of Mehrenberger see [17] we prove the following gap properties for the sequence Λ N N Proposition 49 Let γ be as in heorem 45 1 For all p N and all q, q Z Z such that p max q, q, Λ Λ γ q q For all q N and all p, p Z Z such that q max p, p, Λ Λ p,q γ p p Proof 1 According to the definition of Λ, it is sufficient to show that for all p N and all q, q N N such that p max q, q, we have γ q q for all p, q and q in N, we have γ q q Let us first consider p N and q, q N N such that p max q, q We can easily check that 1 = q q p π 1 q π a p q π a δ δ with δ = p q π a 4α p π 415

18 18 F AMMAR KHODJA, K MAUFFREY AND A MÜNCH From a π, we have p q π a π a herefore, Writing = 1 q q π π q, we obtain q q q q 416 Consequently, we are reduced to bound from below the quantity can remark that hus qq = 1 p q π a p q π a 4α p π q q 1 p q π a p q π a α pπ = π p q π α p 1 q We q q π p q π α p p q π α p 417 By assumption, p maxq, q Without loss of generality we can assume that q q From 417 it follows that q q 1 1 q q π p q q p q π α q p q 1 π α Combining this inequality with 416 we conclude that π π 4 π α q q = γ q q Now, consider any p, q and q in N Writing = 1 p q π a δ π q with δ defined by 415, we obviously obtain π q q γ q q p q 1 1 π π α his completes the proof of item 1 he second assertion of Proposition 49 can be proved in the same way by interchanging p, p and q, q

19 EXAC BOUNDARY OBSERVABILIY Proof of the observability inequality Notation 41 Given >, we denote by k the function { sin πt kt := if t else and we define the following quantities: I 1 := p 1 I 3 := p 1 I 4 := kt qy e iλt dt, I := kt px e iλt dt q p p q kt qy e iλt qy e iλt dt q Z q Z q<p q p kt px e iλt px e iλt dt p Z p<q p Z p q It is easily seen that the Fourier transform of k is given by kξ = e i ξ cos ξ π π, ξ R 418 ξ In particular, k is an even function and k = π π is real In the following lemma we bound from below the right-hand side of the observability inequality 414, using the quantities I j for j = 1,, 4 Lemma 411 px e iλt dt p Z p 1 qy e iλt dt I 1 I I 3 I 4 q Z Proof If we prove that qy e iλt dt I 1 I 3, then, replacing p p 1 q Z by q and y by x, we deduce that px e iλt dt I I 4 p Z

20 F AMMAR KHODJA, K MAUFFREY AND A MÜNCH p 1 Since kt 1 for all t [, ], it follows that qy e iλt dt kt qy q Z p 1 e iλt dt q Z = kt qy e iλt qy e iλt dt p 1 q Z q Z q p q<p kt qy e iλt dt kt qy e iλt dt p 1 q Z p 1 q Z q p q<p kt qy e iλt qy e iλt dt p 1 q Z q Z q p q<p =I 1 kt qy p 1 e iλt dt I 3 I 1 I 3 q Z q<p As a consequence of Lemma 411, the observability inequality will be established if we bound from below I 1 and I and bound from above I 3 and I 4 Lemma 41 [Lower bound of I 1 ] I 1 π 1 π q y p 1 q p Proof First, we remark that: I 1 = kt qy p 1 e iλt dt q p = kt e p 1 q pqy iλt q y e iλ t dt q p = π qy q y k Λ Λ p 1 q p q p π k q y q y q y k Λ Λ p 1 q p q p q p q q π k q y q y q y k Λ Λ 419 p 1 q p q p q p q q

21 Fix p N From parity of k it follows EXAC BOUNDARY OBSERVABILIY 1 q y q y k Λ Λ = q y k Λ Λ 4 q p q p q p q q q p q q Now, consider q Z and q Z such that q p, q p and q q Proposition 49 yields Since > = π γ Λ Λ γ q q > γ ,1 aα 1,1 a π γ Using 418, 4 and then 41, we get k Λ Λ Suming over q p, we obtain, we obtain Λ Λ > γ > π 4 π Λ Λ π = k π k π 1 4 q q 1 k Λ Λ k q p q q By 4, we can assert that k π π q p q q q y q y k Λ Λ k q p q p q q 1 Λ Λ q q 1 1 4r 1 = k r Z Combining this inequality with 419, we conclude that I 1 π π k 1 q y = 1 π q p p 1 γ π π π q y 43 q p π q p which proves the lemma Interchanging p and q and replacing y by x, we also get I π 1 p x π p q p 1 q y,

22 F AMMAR KHODJA, K MAUFFREY AND A MÜNCH Since > a, it is easily seen that x y see Notation 46 It follows that I π 1 p y 44 π p q Combining Lemma 41 with 44, we obtain I 1 I π 1 π p 1 q pq y p y p q Remarking that we get p q y p 1 I 1 I π 1 p 1 π q pq y p y p q p q y Replacing y by its definition gives π I 1 I π 1 p q a a γ 1 a α p π 45 Now, let us bound from above I 3 Lemma 413 [Upper bound of I 3 ] π I 3 π q a a a α p π 46 p 1 Proof By definition of I 3 see Notation 41, we have I 3 = kt qy e iλt q y e iλ t dt p 1 q Z q Z q<p q p = π qy q y k Λ Λ p 1 q Z q p q<p π q y q y k Λ Λ p 1 q Z q p q<p Analysis similar to that in the proof of Lemma 41 shows that for all p N and all q Z such that q < p we have k Λ Λ k π 1 4 q q 1 q p q p k π 1 4r 1 = k π r 1

23 EXAC BOUNDARY OBSERVABILIY 3 herefore q y k Λ Λ k q Z q p q<p π q y q Z q<p Similarly we can prove that q y k Λ Λ k q p q Z q<p Adding these last two inequalities we obtain: I 3 π 4 k Replacing y by its definition gives the desired inequality Similarly, I 4 π Adding 46 and 47 gives I 3 I 4 π From 45 and 48 it follows that π q y q p π p 1 q Z q y π p a a α a α p π 47 π p q a a α a α p π 48 I 1 I I 3 I 4 π where b N N is defined by b = 1 p q a b, 49 π a π a α p π a α a α p π o bound from below I 1 I I 3 I 4 by a, it suffices to prove that p the sequence q b is bounded from below Actually b = 1 and it is easy to check that Since > = π γ N N a π a a α 1 α p π γ a sup N N a α 1 1,1 aα a = N N Besides, from the asymptotic property 1,1 a α 1,1 a 1,1 a, this implies that b > for any p, q in µ pq = p q π

24 4 F AMMAR KHODJA, K MAUFFREY AND A MÜNCH we deduce that p q b 1 π b It is easily seen that lim b π = 1 3 Since > > 3 π γ, it follows that lim b p > Consequently, the sequence q b N N which is positive with a positive limit is uniformly bounded from below by a positive constant, denoted by c It follows from 49 that I 1 I I 3 I 4 πc a, which gives inequality 414 according to Lemma 411 and then the observability inequality Uniform controllability in H H 1/ heorem 45 implies, by usual duality arguments see [14, ] and the references therein the following controllability result π heorem 414 Let γ = π and = π 4 π α γ 1 1,1 aα Assume 1,1 a that a π For any >, any initial data u, u 1 H H 1/ and any target u, u 1 H H 1/, there exist control functions v D L such that the unique solution u of system 11 satisfies u, = u and u, = u 1 in Ω 5 Lack of controllability and dependance with respect to α 51 Controllability in H a H 1/ a We also obtain the observability of the adjoint system in D L L Ha 1/ Ha, the eigenspace associated with the eigenvalue a of infinite multiplicity In that case, the observability inequality is simply, ψ,, ψ 1 X 1 C 1 α ψ y, tdydt 51 with ψy, t = q cos at 1 sin at q a e q y Simple computations permit to show that 51 is true for any > π Remark that, in that specific case, the a observation is simply from Γ = {} [, 1] he corresponding controllability result is the following: Proposition 51 If > π a, then for every u, u 1 H a H 1/ a and any u, u 1 H a H 1/ a, there exists v D L such that the solution u of 11 satisfies u, = u, u, = u 1 in Ω Proof Let, 1 D L L H a1/ Ha We write =, ψ and 1 =, ψ 1 in the form = qe q and 1 = 1 qe q, with, 1 = X 1 a q 1 q he solution = ϕ, ψ of system 13 with initial data, 1 is clearly given by t = 1 q i 1 q e iat a q i 1 q e i at e q a

25 EXAC BOUNDARY OBSERVABILIY 5 Set c q = q i 1 q a his gives for every x, y, t Ω [, ], ϕ x, y, t = and ψ x, y, t = 1 c q e iat c q e at i sin qπy Consequently, Γ ϕ ν αψν 1 dσdt = α 4 Since kt 1 for all t [, ], we obtain Γ c q e i at c q e i at dt ϕ ν αψν 1 dσdt α kt c q e iat c q e i at dt 4 = α 4 kt c q e iat c q e i at dt 5 Let us study the integral dt kt c q e iat c q e at i Using the parity of k we can easily prove that dt kt c q e iat c q e i at = cq k c q c q k a hus kt c q e iat c q e i at dt = cq k R cq k a k c q k a 53 If > π a k, then, using 418, we obtain a π π 4a < π π From 5, 53 and c q =, 1 X 1, we finally obtain = k ϕ Γ ν αψν 1 dσdt k α k a, 1 with k k a > his gives 51 5 Lack of controllability in H H 1/ In agreement with the general result [1] we pointed out in the introduction, the lack of observability is related to essential spectrum Proposition 5 For any > and any ɛ >, there exist initial data, 1 H 1/ H for which the solution = ϕ, ψ of 13 satisfies, 1 X 1 Γ ϕ ν αψν 1 dσdt < ɛ 54 Proof We consider the two sequences p n n N and q n n N of positive integers given by p n = nn 1/ and q n = n Let n = ϕ n, ψ n be the solution of the adjoint system 13 with initial data n, n 1 = e pn,q n, H 1/ H he norm of the initial data is given by n, 1 n = e X 1 p n,q n = H 1/ p n,q n We can X 1

26 6 F AMMAR KHODJA, K MAUFFREY AND A MÜNCH write p n, q n in the polar coordinates p n, q n = r n cosθ n, sinθ n It is easily seen that lim r n = and lim θ n = his readily implies lim n n n p n,q n = a α Since a > α, we obtain lim n, n 1 n X 1 > On the other hand, n t = cos p n,q n t implies that e p n,q n ϕn Γ ν αψ nν 1 dσdt sin p n,q n = π p n,q n Letting n, we obtain the uniform observability since cos θ n pn,q n a α sin θ n pn,q n a α cos θ n π lim n lim n pn,qn a r n ϕn dσdt Γ ν αψ nν 1 = his contradicts n, n 1 > X 1 Remark that the counterexample is obtained for, 1 composed of only one eigenfunction for which the limit of the associated eigenvalue is = a α his value is very particular because any other datum, 1 composed of one eigenfunction associated with a α, a] does not contradict the uniform observability for instance, we refer to previous section for = a he loss of observability may be exhibited by considering a non trivial combination of such modes as done in [1] using Weil sequence, in order to enhance the lack of spectral gap Simpler, this phenomenon may be observed numerically as follows Let H ± N be the space of the initial data, 1 in H ± 1/ H± spanned by {e ± } 1 N If we denote by ± R N the components of, 1 in the basis {e ± } 1 N, then we can write, 1 X 1 = A ± ±, ± R N, ϕ ν αψν 1 dσdt = B ± ±, ± RN Γ for all, 1 in H ± N, where A±, B ± R N N denote real symmetric matrices and, R N denotes the scalar product in RN A ± is diagonal On H ± N, the observability inequality formally writes A ± ±, ± R N C± N B± ±, ± R N he observability constant C ± N whose behavior allows to detect the lack of observability, is then solution of the generalized eigenvalue problem C ± N = max{ >, A± = B ±, R N \ {}} 55 In practice, since A ± is diagonal, it is easier to evaluate C ± N 1 equal to the lowest eigenvalue of B ± A ± 1 able 51 gives the value of C N for various values of N and clearly exhibits the non-uniform boundedness with respect to N, in constrast to C N his is in agreement with heorem 414

27 EXAC BOUNDARY OBSERVABILIY 7 N = 5 N = 1 N = N = 4 N = 8 C N C N able 51 Evolution of the observability constant C ± N vs N for a, α, = 4, 1, 3 53 Controllability with respect to α and If α =, then system 11 degenerates into an uncoupled system and the control only acts on the variable u 1 However, we observe from the proof of heorem 45, that the minimal time as well as the observability constant C are uniformly bounded with respect to α herefore, the controllability holds uniformly wrt α in H H 1/ and by classical arguments, the corresponding controls converge toward controls for u 1, the solution of the wave equation with initial condition u 1, u1 1 his property is related to the fact that the second component of {e } degenerates as α goes to zero Numerically, we observe that the variation of C N with respect to α is very low For = 3, N = 5 and a = 4, we obtain C N = for α = and C N = for α = /1 Remark that the uniform controllability with respect to α does not hold in H H 1/, nor in Ha H 1/ a the right term of 51 going to zero with α he eigenvalue problem 55 allows also to estimate numerically the minimal controllability time for a, α, N fixed Figure 51 depicts the evolution of C N with respect to for a, α, N = 4, 1, 5 and suggests that the minimal controllability time is about 5 he lower bound time in heorem 414 leading for a, α, N = 4, 1, 5 to 196 is thus not sharp Fig 51 Evolution of C N with respect to for a, α, N = 4, 1, 5 6 Concluding remarks and comments 1 Characterization of the controllable data - We can sum up the different controllability results we have obtained in the following theorem: heorem 61 For every N N, let us denote by H N resp H N 1/ the Hilbert subspace of H resp H 1/ spanned by the e for 1 p, q N If a π, then for any > max data u, u1, π a in H a H H N, any N N, any initial data u, u 1 and final H a H H N 1 1 1/ there exists a control

28 8 F AMMAR KHODJA, K MAUFFREY AND A MÜNCH function v in D L such that the solution u of 11 satisfies u, = u, u, = u 1 in Ω Partial controllability - Following [13], one may analyze the uniform partial controllability which consists in controlling to rest only the first component u 1 In that weaker situation, the controllability is uniform with respect to the data u, u 1 From the second equation of 11, we express the component u in terms of u 1 as follows u, t = α t xu 1, s sin at sds in Q, assuming for simplicity that u, u 1 =, he variable u1 to be controlled is then solution of u 1 = u 1 α t xxu 1, s sin at sds in Q, u 1 = v 1 Γ on Σ, u 1,, u 1, = u 1, u 1 1 in Ω he corresponding spectrum is { } {} with corresponding eigenfunctions {e } and {e } he difference with respect to the full controllability problem, is that the Fourier coefficients in ϕ 1, the adjoint solution of u 1, are all connected to each other his allows a compensation of the modes {e } by the modes {e } we refer to [13] for the analysis on a similar system he analysis remains to be fully written 3 Null boundary controllability of a cylindrical membrane shell - he operator which describes a membrane cylindrical elastic shell is as follows see [19] A = a xx c yy b c xy ar 1 x b c xy c xx a yy br 1 y r 1 a x r 1 b y r a with a = 8µ µ/ µ, b = 4µ/ µ and c = µ, µ > denote the Lamé coefficients r 1 > denotes the curvature of the cylinder and is the coupling parameter between the tangential displacement u 1, u and the normal displacement u 3 of the shell Ω is still equal to, 1 his mixed order and self-adjoint operator enters in the framework of [6] so that we can compute σ ess A using [7] We obtain σ ess A = [, r 3 µ/ µ ] he spectrum of A is therefore composed of two distinct parts, the essential spectrum plus a discrete spectrum with asymptotic behavior equal, up to some constant, to σ he difficulty here is that the discrete spectrum is not known explicitly REFERENCES [1] F V Atkinson, H Langer, R Mennicken and A A Shkalikov, he essential spectrum of some matrix operators, Math Nachr, , pp 5 [] F Ammar-Khodja, G Geymonat and AMünch, On the exact controllability of a system of mixed order with essential spectrum, C R Math Acad Sci Paris, , pp [3] F D Araruna and E Zuazua, Controllability of the Kirchhoff system for beams as a limit of the Mindlin-imoshenko system, SIAM J Control Optim, 477 8, pp [4] H Brezis, Analyse fonctionnelle héorie et applications, Collection Mathématiques Appliquées pour la Maîtrise Masson, Paris 1983 [5] E B Davies, Spectral theory and differential operators, Cambridge Studies in Advanced Mathematics, vol 4, Cambridge University Press, Cambridge 1995

29 EXAC BOUNDARY OBSERVABILIY 9 [6] G Geymonat and G Grubb, he essential spectrum of elliptic systems of mixed order, Math Ann, , pp [7] G Geymonat and G Grubb, Eigenvalue asymptotics for selfadjoint elliptic mixed order systems with nonempty essential spectrum, Bolletino UMI, , pp [8] G Geymonat, P Loreti and V Valente, Contrôlabilité exacte d un modèle de coque mince, C R Acad Sci Paris Sér I Math, , pp [9] G Geymonat, P Loreti and V Valente, Exact controllability of thin elastic hemispherical shell via harmonic analysis, Boundary value problems for partial differential equations and applications, RMA Res Notes Appl Math, 9, Masson, Paris 1993, pp [1] G Geymonat and V Valente, A noncontrollability result for systems of mixed order, SIAM J Control Optim, 393, pp [11] A E Ingham, Some trigonometrical inequalities with applications to the theory of series, Math Z, , pp [1] V Komornik and P Loreti, Fourier series in control theory, Springer Monographs in Mathematics, Springer-Verlag, New York 4 [13] P Loreti and V Valente, Partial exact controllability for spherical membranes, SIAM J Control Optim, , pp [14] J E Lagnese and J L Lions, Modelling analysis and control of thin plates, Masson, RMA 6, Paris 1988 [15] J L Lions and E Magenes, Problèmes aux limites non homogènes et applications, vol 1, Dunod, Paris 1968 [16] Z Liu, B Rao, A spectral approach to the indirect boundary control of a system of weakly coupled wave equations, Discrete Contin Dyn Syst, 31-9, pp [17] M Mehrenberger, An Ingham type proof for the boundary observability of a N-d wave equation, C R Math Acad Sci Paris, , pp [18] L Rosier and P Rouchon, On the controllability of a wave equation with structural damping, Int J omogr Stat, 5W7 7, pp [19] J Sanchez Hubert and E Sanchez-Palencia, Vibration and coupling of continuous systems Asymptotic theory, Springer, Berlin 1989 [] M ucsnak and G Weiss, Observation and control for operator semigroups, Birkhäuser Verlag, Basel 9