Simultaneous global exact controllability in projection of infinite 1D bilinear Schrödinger equations
aa r X i v : . [ m a t h - ph ] J un Simultaneous global exact controllability inprojection
Alessandro Duca
Laboratoire de Math´ematiques de Besan¸con, Universit´e Bourgogne Franche-Comt´e16, Route de Gray, 25000 Besan¸con, France [email protected]
Dipartimento di Matematica Giuseppe Peano, Universit`a degli Studi di Torino10, Via Carlo Alberto, 10123 Torino, Italy [email protected]
Dipartimento di Scienze Matematiche Giuseppe Luigi Lagrange, Politecnico di Torino24, Corso Duca degli Abruzzi, 10129 Torino, Italy [email protected]
SPHINX team, Inria, 54600 Villers-l`es-Nancy, France [email protected]
ORCID: 0000-0001-7060-1723
Abstract
We consider an infinite number of one-dimensional bilinear Schr¨odingerequations on a segment. We prove the simultaneous local exact con-trollability in projection for any positive time and the simultaneousglobal exact controllability in projection for sufficiently large time.
AMS subject classifications:
Keywords:
Schr¨odinger equation, simultaneous control, global exact con-trollability, moment problem, perturbation theory, density matrices.
Let H be an infinite dimensional Hilbert space. In quantum mechanics,any statistical ensemble can be described by a wave function (pure state)or by a density matrix (mixed state) which is a positive operator of trace 1.For any density matrix ρ , there exists a sequence { ψ j } j ∈ N ⊂ H such that ρ = X j ∈ N l j | ψ j ih ψ j | , X j ∈ N l j = 1 , l j ≥ ∀ j ∈ N . (1) 1he sequence { ψ j } j ∈ N is a set of eigenvectors of ρ and { l j } j ∈ N are the corre-sponding eigenvalues. If there exists j ∈ N such that l j = 1 and l j = 0 foreach j = j , then the corresponding density matrix represents a pure stateup to a phase. For this reason, the density matrices formalism is said to bean extension of the common formulation of the quantum mechanics in termsof wave function.Let us consider T > H ( t )(called Hamiltonian) for t ∈ (0 , T ). The dynamics of a general densitymatrix ρ is described by the Von Neumann equation ( i dρdt ( t ) = [ H ( t ) , ρ ( t )] , t ∈ (0 , T ) ,ρ (0) = ρ , ([ H, ρ ] = Hρ − ρH ) , (2)for ρ the initial solution of the problem. The solution is ρ ( t ) = U t ρ (0) U ∗ t ,where U t is the unitary propagator generated by H ( t ), i.e. ( ddt U t = − iH ( t ) U t , t ∈ (0 , T ) ,U = Id.
In the present work, we consider H = L ((0 , , C ) and H ( t ) = A + u ( t ) B , for A = − ∆ the Dirichlet Laplacian ( i.e. D ( A ) = H ∩ H ), B abounded symmetric operator and u ∈ L ((0 , T ) , R ) control function. Fromnow on, we call Γ ut the unitary propagator U t when it is defined. Theproblem (2) is said to be globally exactly controllable if, for any coupleof unitarily equivalent density matrices ρ and ρ , there exist T > u ∈ L ((0 , T ) , R ) such that ρ = Γ uT ρ (Γ uT ) ∗ . Thanks to the decomposition(1), the controllability of (2) is equivalent (up to phases) to the simultaneouscontrollability of the Cauchy problems in H ( i∂ t ψ j ( t ) = Aψ j ( t ) + u ( t ) Bψ j ( t ) , t ∈ (0 , T ) ,ψ j (0) = ψ j , ∀ j ∈ N . (3)The state ψ j is the j-th eigenfunction of ρ corresponding to the eigenvalue λ j and ρ = P ∞ j =1 λ j | ψ j ih ψ j | . The j-th solution of (3) is ψ j ( t ) = Γ ut ψ j .To this purpose, we study the simultaneous global exact controllability ofinfinitely many problems (3) and we only rephrase the results in terms ofthe density matrices.The controllability of the bilinear Schr¨odinger equation (3) has beenwidely studied in the literature and we start by mentioning the work on thebilinear systems of Ball, Mardsen and Slemrod [BMS82]. In the frameworkof the bilinear Schr¨odinger equation, for B : D ( A ) → D ( A ), the work showsthe well-posedness of (3) in H for controls belonging to L loc ( R , R ) and an2mportant non-controllability result. In particular, let S be the unit spherein H and Z ( ψ ) := { ψ ∈ D ( A ) | ∃ T > , ∃ r > , ∃ u ∈ L r ((0 , T ) , R ) : ψ = Γ Tu ψ } . For every ψ ∈ S ∩ D ( A ), the attainable set Z ( ψ ) is contained in a countableunion of compact sets and it has dense complement in S ∩ D ( A ).Despite this non-controllability result, many authors have addressed theproblem for weaker notions of controllability. We call M µ the multiplicationoperator for a function µ ∈ H and H s (0) := D ( | A | s ) for s > . For instance in [BL10], Beauchard and Laurent improve the work [Bea05]and they prove the local exact controllability of (3) in a neighborhood of thefirst eigenfunction of A in S ∩ H when B = M µ for a suitable µ ∈ H .The global approximate controllability in a Hilbert space has been studiedby Boscain, Caponigro, Chambrion, Mason and Sigalotti in [BCCS12] and[CMSB09]. In both, simultaneous global approximate controllability resultsare provided.Morancey proves in [Mor14] the simultaneous local exact controllability in S ∩ H for at most three problems (3) and up to phases, when B = M µ forsuitable µ ∈ H .In [MN15], Morancey and Nersesyan extend the result. They provide theexistence of a residual set of functions Q in H so that, for B = M µ and µ ∈ Q , the simultaneous global exact controllability is verified for any finitenumber of (3) in H V ) := D ( | A + V | ) for V ∈ H .In the present work, we use part of the notations of [BL10], [Mor14],[MN15] and we carry on the previous results. We provide explicit conditionsin B that imply the simultaneous global exact controllability in projectionof infinitely many problems (3) in H by projecting onto suitable finitedimensional subspaces of H . Another goal of this work is to prove thesimultaneous local exact controllability in projection for any positive time T > e.g. [Mor14] and[MN15]. Indeed, in the appendix we develop a perturbation theory techniquethat we use in order to get rid of an issue appearing in the proof of thelocal controllability: the “eigenvalues resonances”. The formulation of thecontrollability for orthonormal basis allows to provide the result in terms ofdensity matrices and unitarily equivalent sets of functions.
We denote H = L ((0 , , C ), its norm k · k and its scalar product h· , ·i .The operator A is the Dirichlet Laplacian, i.e. A = − d dx and D ( A ) = H ((0 , , C ) ∩ H ((0 , , C ). The control function u belongs to L ((0 , T ) , R )and B is a bounded symmetric operator.3e consider an Hilbert basis { φ j } j ∈ N composed by eigenfunctions of A re-lated to the eigenvalues { λ j } j ∈ N and we have(4) φ j ( t ) = e − iAt φ j = e − iλ j t φ j . Let us define the spaces for s > H s (0) = H s (0) ((0 , , C ) := D ( A s ) , k · k ( s ) = k · k H s (0) = (cid:16) ∞ X k =1 | k s h· , φ k i| (cid:17) ,ℓ ∞ ( H ) = (cid:8) { ψ j } j ∈ N ⊂ H (cid:12)(cid:12) sup j ∈ N k ψ j k H < ∞ (cid:9) ,h s ( H ) = n { ψ j } j ∈ N ⊂ H (cid:12)(cid:12) ∞ X j =1 ( j s k ψ j k ) < ∞ o . We call H s := H s ((0 , , C ), H s := H s ((0 , , C ) and, for N ∈ N (5) I N := { ( j, k ) ∈ N × { , ..., N } : j = k } . Assumptions (I) . The bounded symmetric operator B satisfies the follow-ing conditions.1. For any N ∈ N , there exists C N > j ≤ N and k ∈ N |h φ k , Bφ j i| ≥ C N /k . Ran ( B | H ) ⊆ H and Ran ( B | H ) ⊆ H ∩ H .
3. For every N ∈ N and ( j, k ) , ( l, m ) ∈ I N such that ( j, k ) = ( l, m ) and j − k − l + m = 0, there holds h φ j , Bφ j i − h φ k , Bφ k i − h φ l , Bφ l i + h φ m , Bφ m i 6 = 0 . Remark 1.1.
If a bounded operator B satisfies Assumptions I, then B ∈ L ( H , H ) . Indeed, B is closed in H , so for every { u n } n ∈ N ⊂ H suchthat u n H −→ u and Bu n H −→ v , we have Bu = v . Now, for every { u n } n ∈ N ⊂ H such that u n H −→ u and Bu n H −→ v , the convergences with respect to the H -norm are implied and Bu = v . Hence, the operator B is closed in H and B ∈ L ( H , H ) . The same argument leads to B ∈ L ( H , H ∩ H ) since Ran ( B | H ) ⊆ H ∩ H . Example 1.2.
Assumptions I are satisfied for B : ψ x ψ . Indeed, thecondition 2) is trivially verified, while the first directly follows by considering |h φ j , x φ k i| = (cid:12)(cid:12)(cid:12) ( − j − k ( j − k ) π − ( − j + k ( j + k ) π (cid:12)(cid:12)(cid:12) , j = k, |h φ k , x φ k i| = (cid:12)(cid:12)(cid:12) − k π (cid:12)(cid:12)(cid:12) , k ∈ N . The point 3) holds since for ( j, k ) , ( l, m ) ∈ I N so that ( j, k ) = ( l, m ) j − k − l + m = 0 = ⇒ j − − k − − l − + m − = 0 . { ψ j } j ∈ N ⊂ H and H N (Ψ) := span { ψ j : j ≤ N } . We define π N (Ψ) the orthogonal projector onto H N (Ψ). Definition 1.3.
The problems (3) are simultaneously globally exactly con-trollable in projection in H if there exist T > { ψ j } j ∈ N ⊂ H such that the following property is verified. For every { ψ j } j ∈ N , { ψ j } j ∈ N ⊂ H unitarily equivalent, there exists u ∈ L ((0 , T ) , R ) such that(6) π N (Ψ) ψ j = π N (Ψ)Γ uT ψ j , ∀ j ∈ N . In other words, h ψ k , ψ j i = h ψ k , Γ uT ψ j i for every j, k ∈ N and k ≤ N . Definition 1.4.
Let us define O ǫ,T := n { ψ j } j ∈ N ⊂ H (cid:12)(cid:12) h ψ j , ψ k i = δ j,k ; sup j ∈ N k ψ j − φ j ( T ) k (3) < ǫ o . The problems (3) are simultaneously locally exactly controllable in pro-jection in O ǫ,T ⊂ H up to phases if there exist ǫ > T > { ψ j } j ∈ N ∈ O ǫ,T such that the following property is verified. For every { ψ j } j ∈ N ∈ O ǫ,T , there exist { θ j } j ∈ N ⊂ R and u ∈ L ((0 , T ) , R ) such that π N (Ψ) ψ j = π N (Ψ) e iθ j Γ uT ψ j , ∀ j ∈ N . In other words, h ψ k , ψ j i = e iθ j h ψ k , Γ uT ψ j i for every j, k ∈ N and k ≤ N .Let U ( H ) be the space of the unitary operators on H . We present thesimultaneous local exact controllability in projection for any T >
Theorem 1.5.
Let B satisfy Assumptions I. For every T > , there ex-ist ǫ > and Ψ := { ψ j } j ∈ N ∈ O ǫ,T such that the following holds. Forany { ψ j } j ∈ N ∈ O ǫ,T and b Γ ∈ U ( H ) such that { b Γ ψ j } j ∈ N = { φ j } j ∈ N , if (cid:8)b Γ φ j (cid:9) j ∈ N ⊂ H , then there exist { θ j } j ≤ N ⊂ R and u ∈ L ((0 , T ) , R ) suchthat ( π N (Ψ) ψ j = π N (Ψ) e iθ j Γ uT ψ j j ≤ N,π N (Ψ) ψ j = π N (Ψ)Γ uT ψ j , j > N. Proof.
See Proposition 2 . Theorem 1.6.
Let B satisfy Assumptions I and Ψ := { ψ j } j ∈ N ⊂ H be an orthonormal system. Let { ψ j } j ∈ N , { ψ j } j ∈ N , ⊂ H be complete or-thonormal systems so that there exists b Γ ∈ U ( H ) such that { b Γ ψ j } j ∈ N =5 ψ j } j ∈ N . If { b Γ ψ j } j ∈ N ⊂ H , then for any N ∈ N , there exist T > , u ∈ L ((0 , T ) , R ) and { θ k } k ≤ N ⊂ R such that e iθ k h ψ k , ψ j i = h ψ k , Γ uT ψ j i , ∀ j, k ∈ N , k ≤ N. (7) Proof.
See Section 3.In Theorem 1 .
6, if Ψ = Ψ , then b Γ ψ j ∈ H . As e iθ k h ψ k , ψ j i = e iθ k δ k,j = e iθ j h ψ k , ψ j i for every j, k ∈ N , the relation (7) becomes π N (Ψ ) e iθ j ψ j = π N (Ψ )Γ uT ψ j , j ≤ N,π N (Ψ ) ψ j = π N (Ψ )Γ uT ψ j , j > N. As Ψ is composed by orthogonal elements, then π N (Ψ ) ψ j = ψ j when j ≤ N , otherwise π N (Ψ ) ψ j = 0. Then the next corollary follows. Corollary 1.7.
Let B satisfy Assumptions I. Let Ψ := { ψ j } j ∈ N , Ψ := { ψ j } j ∈ N ⊂ H be complete orthonormal systems. For any N ∈ N , thereexist T > , u ∈ L ((0 , T ) , R ) and { θ j } j ≤ N ⊂ R such that ( Γ uT ψ j = e iθ j ψ j , j ≤ N,π N (Ψ ) Γ uT ψ j = 0 , j > N. Remark.
One can notice that Corollary . implies the simultaneous globalexact controllability (without projecting) of N bilinear Schr¨odinger equa-tions. As we have mentioned before, a similar result is proved by Moranceyand Nersesyan in [ MN15 , M ain T heorem ] . They prove the existence of aclass of multiplication operators B that guarantees the validity of the result.However, Corollary . provides a novelty as we are able to explicit condi-tions in B implying the controllability. Given any bounded operator B , onecan verify if those assumptions are satisfied, e.g. B = x . Let P ⊥ φ j be the projector onto the orthogonal space of φ j and the operator e B ( M, j ) = B (cid:0) ( λ j − A ) (cid:12)(cid:12) φ ⊥ j (cid:1) − (cid:16)(cid:0) ( λ j − A ) (cid:12)(cid:12) φ ⊥ j (cid:1) − P ⊥ φ j B (cid:17) M P ⊥ φ j B for M, j ∈ N . When ( A, B ) satisfies Assumptions I and the following as-sumptions, the phase ambiguities { θ j } j ≤ N ⊂ R appearing in Theorem 1 . n be the null vector in Q n with n ∈ N . Assumptions (A) . If for every N ∈ N there exists { r j } ≤ j ≤ N ∈ Q N +1 \ N + such that r + P Nj =1 r j λ j = 0 , then either we have P Nj =1 r j B j,j = 0,or there exists M ∈ N such that P Nj =1 r j h φ j , e B ( M, j ) φ j i 6 = 0 . emark. When the operator B is such that { B j,j } j ≤ N are rationally inde-pendent with N ∈ N , the Assumptions A are verified as, for any { r j } ≤ j ≤ N Q N +1 \ N + , there holds P Nj =1 r j B j,j = 0 . Theorem 1.8.
Let N ∈ N . Let B satisfy Assumptions I and AssumptionsA. Let Ψ := { ψ j } j ∈ N ⊂ H and { ψ j } j ∈ N , { ψ j } j ∈ N , ⊂ H such that thereexists b Γ ∈ U ( H ) such that { b Γ ψ j } j ∈ N = { ψ j } j ∈ N . If { b Γ ψ j } j ∈ N ⊂ H , thenthere exist T > and u ∈ L ((0 , T ) , R ) such that π N (Ψ ) ψ j = π N (Ψ ) Γ uT ψ j , j ∈ N . Proof.
See Paragraph 3.
Remark. If Ψ = Ψ , then the same result of Corollary . is also providedwhen B satisfies Assumptions A thanks to Theorem . . We mention now the crucial result of well-posedness for the problem in H ( i∂ t ψ ( t ) = Aψ ( t ) + u ( t ) µψ ( t ) ,ψ (0) = ψ , t ∈ (0 , T ) . (8) Proposition 1.9. [ BL10 , P roposition Let µ ∈ H , T > , ψ ∈ H and u ∈ L ((0 , T ) , R ) . There exists a unique mild solution of ( ) in H ,i.e. ψ ∈ C ([0 , T ] , H ) so that (9) ψ ( t, x ) = e − iAt ψ ( x ) − i Z t e − iA ( t − s ) ( u ( s ) µ ( x ) ψ ( s, x )) ds, ∀ t ∈ [0 , T ] . Moreover, for every
R > , there exists C = C ( T, µ, R ) > such that,if k u k L ((0 ,T ) , R ) < R , then, for every ψ ∈ H , the solution satisfies k ψ k C ([0 ,T ] ,H ) ≤ C k ψ k (3) and k ψ ( t ) k H = k ψ k H for every t ∈ [0 , T ] . The result of Proposition 1 . µ ∈ H with B ∈ L ( H , H ∩ H ). When B satisfies Assumptions I, we know that B ∈ L ( H , H ∩ H ) (see Remark 1 .
1) and there exists a unique mildsolution of (3) in H so that ψ j ( t, x ) = e − iAt ψ j ( x ) − i Z t e − iA ( t − s ) u ( s ) Bψ j ( s, x ) ds. In conclusion, for every { ψ j } j ∈ N ∈ ℓ ∞ ( H ) (respectively in h ( H )), itfollows that { Γ uT ψ j } j ∈ N ∈ ℓ ∞ ( H ) (respectively in h ( H )).7 .3 Time reversibility An important feature of the bilinear Schr¨odinger equation is the time re-versibility. If we substitute t with T − t for T > ( i∂ t Γ uT − t ψ = − A Γ uT − t ψ − u ( T − t ) B Γ uT − t ψ , t ∈ (0 , T ) , Γ uT − ψ = Γ uT ψ = ψ . We define e Γ e ut such that Γ uT − t ψ = e Γ e ut ψ for e u ( t ) := u ( T − t ) and ( i∂ t e Γ e ut ψ = ( − A − e u ( t ) B ) e Γ e ut ψ , t ∈ (0 , T ) , e Γ e u ψ = ψ . (10)Thanks to ψ = e Γ e uT Γ uT ψ and ψ = Γ uT e Γ e uT ψ , it follows e Γ e uT = (Γ uT ) − =(Γ uT ) ∗ . The operator e Γ e ut describes the reversed dynamics of Γ ut and representsthe propagator of (10) generated by the Hamiltonian ( − A − e u ( t ) B ). In Section 2, we provide Proposition 2 . . N problems (3) in Proposition 3 .
3, then the simultaneous global exactcontrollability of N (3) (Proposition 3 . . .
8, while in Section 4, we provide the mainresult in terms of density matrices.In Appendix 1 .
3, we explain the time reversibility of the (3), while in Ap-pendix A , we briefly discuss the solvability of the moment problems.In Appendix B , we develop the perturbation theory technique adopted inthe work. T > In this section, we discuss the simultaneous local exact controllability inprojection. We explain first why we modify the problem.Let Φ = { φ j } j ∈ N be an Hilbert basis composed by eigenfunctions of A .We study the local exact controllability in projection in O ǫ,T with respectto π N (Φ). Let Γ ut ψ j = P ∞ k =1 φ k ( T ) h φ k ( T ) , Γ ut φ j i be the solution of the j-th83). We consider the map α ( u ), the infinite matrix with elements α k,j ( u ) = h φ k ( T ) , Γ uT φ j i , for every k, j ∈ N and k ≤ N. Our goal is to prove theexistence of ǫ > { ψ j } j ∈ N ∈ O ǫ,T , there exists u ∈ L ((0 , T ) , R ) such that π N (Φ)Γ uT φ j = π N (Φ) ψ j , ∀ j ∈ N . This outcome is equivalent to the local surjectivity of α for T >
0. To thisend, we want to use the Generalized Inverse Function Theorem ([Lue69,Theorem 1; p. 240]) and we study the surjectivity of γ ( v ) := ( d u α (0)) · v ,the Fr´echet derivative of α the infinite matrix that, for j, k ∈ N and k ≤ N,γ k,j ( v ) : = (cid:28) φ k ( T ) , − i Z T e − iA ( T − s ) v ( s ) Be − iAs φ j ds (cid:29) = − i Z T v ( s ) e − i ( λ j − λ k ) s dsB k,j , for B k,j = h φ k , Bφ j i = h Bφ k , φ j i = B j,k . The surjectivity of γ consists inproving the solvability of the moment problem x k,j B k,j = − i Z T u ( s ) e − i ( λ j − λ k ) s ds, (11)for each infinite matrix x , with elements x k,j , belonging to a suitable space.One would use Haraux Theorem as explained in Remark A. , T heorem . j, k, n, m ∈ N , ( j, k ) = ( n, m )and k, m ≤ N , there holds λ j − λ k = λ n − λ m , which implies x k,j B k,j = − i Z T u ( s ) e − i ( λ j − λ k ) s ds = − i Z T u ( s ) e − i ( λ n − λ m ) s ds = x n,m B n,m . An example is λ − λ = λ − λ , but they also appear for all the diagonalterms of γ since λ j − λ k = 0 for j = k .We avoid the problem by adopting the following procedure. First, we de-compose A + u ( t ) B = ( A + u B ) + u ( t ) B for u ∈ R and u ∈ L ((0 , T ) , R ).We consider A + u B instead of A and we modify the eigenvalues gaps byusing u B as a perturbating term in order to remove all the non-diagonalresonances. Second, we redefine α in a map b α depending on the parameter u . We introduce α u by acting phase-shifts in order to remove the reso-nances on the diagonal terms, i.e. e ψ j ( t, x ) = b α j,j ( u ) | b α j,j ( u ) | ψ j ( t, x ), which implies α u k,j ( u ) = b α j,j ( u ) | b α j,j ( u ) | b α k,j ( u ) . Let N ∈ N and u ( t ) = u + u ( t ) , for u and u ( t ) real. We introduce thefollowing Cauchy problem ( i∂ t ψ j ( t ) = ( A + u B ) ψ j ( t ) + u ( t ) Bψ j ( t ) , t ∈ (0 , T ) , j ∈ N ,ψ j = ψ j (0) . (12) 9ts solutions are ψ j ( t ) = Γ u + u t ψ j , where Γ u + u t is the unitary propagatorof the dynamics, which is equivalent to the one of the problems (3).As B is bounded, A + u B has pure discrete spectrum. We call { λ u j } j ∈ N the eigenvalues of A + u B that correspond to an Hilbert basis composedby eigenfunctions Φ u := { φ u j } j ∈ N . We set φ u j ( T ) := e − iλ u j T φ u j and O u ǫ ,T := n { ψ j } j ∈ N ⊂ H (cid:12)(cid:12) h ψ j , ψ k i = δ j,k ; sup j ∈ N k ψ j − φ u j ( T ) k (3) < ǫ o . We choose | u | small so that λ u k = 0 for every k ∈ N (Lemma B.
4, Appendix B ). The introduction of the new Hilbert basis imposes to define e H := D ( | A + u B | ) equipped with k · k e H = (cid:16) P ∞ k =1 (cid:12)(cid:12) | λ u k | h· , φ k i (cid:12)(cid:12) (cid:17) . However,from now on, due to Lemma B. B ), we have e H ≡ H . We define b α , the infinite matrices with elements for k ≤ N and j ∈ N such that b α k,j ( u ) = h φ u k ( T ) , Γ u + u T φ u j i and the map α u with elements ( α u k,j ( u ) = b α j,j ( u ) | b α j,j ( u ) | b α k,j ( u ) , j, k ≤ N,α u k,j ( u ) = b α k,j ( u ) , j > N, k ≤ N. (13)Now, the local surjectivity of the map α u in a suitable space is equivalentto the simultaneous local exact controllability in projection up to N phaseson O u ǫ ,T for a suitable ǫ > j ∈ N ,(14) π N (Φ u ) e iθ j Γ u + u T φ u j = N X k =1 φ u k ( T ) α u k,j ( u ) , e iθ j := b α j,j ( u ) | b α j,j ( u ) | . Let γ u ( v ) = (( d u α u )(0)) · v be the Fr´echet derivative of α u and B u k,j = h φ u k , Bφ u j i for k ≤ N and j ∈ N . Defined b γ k,j ( v ) = (( d u b α )(0)) · v , wecompute γ u ( v ) such that γ u k,j = (cid:0)b γ j,j δ k,j + b γ k,j − δ k,j ℜ ( b γ j,j ) (cid:1) when j, k ≤ N, while γ u k,j = b γ k,j when k ≤ N and j > N. Thus for k ≤ N and j ∈ N , ( γ u k,j = b γ k,j = − i R T u ( s ) e − i ( λ u j − λ u k ) s dsB u k,j , k = j,γ u k,k = ℜ ( b γ k,k ) = 0 , k = j. (15)The relation γ u k,k = 0 comes from ( i b γ k,k ) ∈ R since b γ k,j = − b γ j,k for j, k ≤ N. Due to the phase-shifts of α u , the diagonal elements of γ u are all 0. Remark. As O u ǫ ,T is composed by orthonormal elements, we have T Φ u O u ǫ ,T = (cid:8) { ψ j } j ∈ N ⊂ ℓ ∞ ( H ) (cid:12)(cid:12) h φ u k , ψ j i = −h φ u j , ψ k i (cid:9) . or every k ∈ N , from Lemma B. , there exists C > so that + ∞ X j =1 j | α u k,j | = + ∞ X j =1 j |h e Γ u + e u T φ u k , φ u j i| = k e Γ u + e u T φ u k k e H ≤ C k e Γ u + e u T φ u k k < ∞ as the propagator e Γ u + e u T (see Appendix . ) preserves H . Hence, { α u k,j } j ∈ N ∈ h ( C ) for every k ∈ N , then the maps α u and γ u take respectively valuesin Q N := (cid:8) { x k,j } k,j ∈ N k ≤ N ∈ ( h ( C )) N (cid:12)(cid:12) x k,k ∈ R , k ≤ N (cid:9) and G N := (cid:8) { x k,j } k,j ∈ N k ≤ N ∈ ( h ( C )) N (cid:12)(cid:12) x k,j = − x j,k , x k,k = 0 j, k ≤ N (cid:9) . . In the next proposition, we ensure the simultaneous local exact controllabil-ity in projection for any
T >
Proposition 2.1.
Let N ∈ N and B satisfy Assumptions I. For every T > , there exist ǫ > and u ∈ R such that, for any { ψ j } j ∈ N ∈ O ǫ,T and b Γ ∈ U ( H ) such that { b Γ ψ j } j ∈ N = { φ j } j ∈ N , if (16) (cid:8)b Γ φ j (cid:9) j ∈ N ⊂ H , then there exist a sequence of real numbers { θ j } j ∈ N = (cid:8)(cid:8)b θ j (cid:9) j ≤ N , , ... (cid:9) and u ∈ L ((0 , T ) , R ) such that π N (Φ u ) ψ j = π N (Φ u ) e iθ j Γ uT φ u j , ∀ j ∈ N . Proof. Let u in the neighborhoods defined in Appendix B by Lemma B.
4, Lemma B.
5, Lemma B. B.
9. First, the relation (16) isrequired for the following reason. Let { Γ uT φ u j } j ∈ N = { b Γ φ j } j ∈ N for T > u ∈ L ((0 , T ) , R ) and b Γ ∈ U ( H ). For | u | small enough, thanks to Lemma B. B ), there exists C > j ≤ C | λ u j | . FromLemma B. B ), there exists C > k ∈ N , (P + ∞ j =1 j |h φ k , Γ uT φ u j i| = P + ∞ j =1 j |h (Γ uT ) ∗ φ k , φ u j i| ≤ C C k e Γ e uT φ k k < ∞ , P + ∞ j =1 j |h φ k , Γ uT φ u j i| = P + ∞ j =1 j |h φ k , b Γ φ j i| = P + ∞ j =1 j |h b Γ ∗ φ k , φ j i| = k b Γ ∗ φ k k . Second, thanks to the third point of Remark B. B ), the control-lability in O u ǫ ,T implies the controllability in O ǫ,T for suitable ǫ >
0. Indeed,if | u | is small enough, then sup j ∈ N k φ j − φ u j k (3) ≤ ǫ (Remark B. { ψ j } j ∈ N ∈ O u ǫ ,T , we have { ψ j } j ∈ N ∈ O ǫ ,T sincesup j ∈ N k ψ j − φ j ( T ) k (3) ≤ sup j ∈ N k φ u j − φ j ( T ) k (3) + sup j ∈ N k ψ j − φ u j ( T ) k (3) ≤ ǫ . α u guarantees the simultaneous local exact controllability inprojection up to phases (Definition 1 .
4) of (3) with initial state { φ u j } j ∈ N on O u ǫ ,T for ǫ small enough.We consider Generalized Inverse function Theorem ([Lue69, Theorem 1; p.240]) since Q N and G N are real Banach spaces. If γ u is surjective in G N ,then the local surjectivity of α u in Q N is ensured. The map γ u is surjectivewhen the following moment problem is solvable x u k,j B u k,j = − i Z T u ( s ) e − i ( λ u j − λ u k ) s ds, j ∈ N , k ≤ N, k = j (17)for every (cid:8) x u k,j (cid:9) j,k ∈ N k ≤ N ∈ G N . The equations of (17) for k = j are redundantas γ u k,k = 0 and x u k,k = 0 for every k ≤ N and { x u k,j } k,j ∈ N k ≤ N ∈ G N . Thus, weprove the solvability of the moment problem for j = k and j = k = 1. Now, (cid:8) x u k,j (cid:9) j,k ∈ N k ≤ N ∈ ( h ) N and (cid:8) γ u k,j (cid:9) j,k ∈ N k ≤ N ∈ ( h ) N . From Lemma B. B ), it follows (cid:8) x u k,j /B u k,j (cid:9) j,k ∈ N k ≤ N ∈ ( ℓ ( C )) N and (cid:8) γ u k,j /B u k,j (cid:9) j,k ∈ N k ≤ N ∈ ( ℓ ( C )) N . Thanks to Lemma B. B ), for I N defined in (5), there exist G ′ := inf ( j,k ) , ( n,m ) ∈ IN ( j,k ) =( n,m ) | λ u j − λ u k − λ u n + λ u m | > G := sup A ⊂ I N (cid:16) inf ( j,k ) , ( n,m ) ∈ IN \ A ( j,k ) =( n,m ) | λ u j − λ u k − λ u n + λ u m | (cid:17) ≥ G ′ where A runs over the finite subsets of I N . The solvability of the momentproblem (17) is guaranteed from Remark A. { λ u j − λ u k } j,k ∈ N , k ≤ N j = k or j = k =1 . Indeed, x u , = 0 and Remark B. λ u j − λ u k = λ u l − λ u m for every j, k, l, m ∈ N . The proof is achievedsince α u is locally surjective for T > We show that the first point is valid for every
T > G = + ∞ . Let A M := { ( j, n ) ∈ N | j, n ≥ M ; j = n } for M ∈ N . Thanks to the relation(30) in the proof of Lemma B. B ), for | u | small enough and forevery K ∈ R , there exists M K > ( j,n ) ∈ A MK | λ u j − λ u n | > K. Indeed, the relation (30) implies that, for | u | small enough, | λ u j − λ u n | ≥ | λ j − λ n | − O ( | u | ) ≥ π min { λ j +1 − λ j , λ n +1 − λ n } − O ( | u | ) . Thus G ≥ sup M ∈ N (cid:0) inf ( j,n ) ∈ A M | λ u j − λ u n | − λ u N (cid:1) > . Now, for | u | smallenough, Lemma B. B ) implies the existence of C > G ≥ C (cid:0) lim M →∞ inf ( j,n ) ∈ A M | λ j − λ n | − λ N (cid:1) ≥ C lim M →∞ ( λ M +2 − λ M +1 − N π ) = + ∞ . Simultaneous global exact controllability in pro-jection
The common approach adopted in order to prove the global exact control-lability (also simultaneous) consists in gathering the global approximatecontrollability and the local exact controllability.However, this strategy can not be used to prove the controllability in pro-jection as the propagator Γ uT does not preserve the space π N (Ψ) H forany Ψ := { ψ j } j ∈ N ⊂ H , making impossible to reverse and concatenatedynamics. We adopt an alternative strategy that we call “transposition ar-gument” (see remark below). In particular, under suitable assumptions, weprove that the controllability in projection onto an N dimensional space isequivalent to the controllability of N problems (without projecting). Remark 3.1.
From time reversibility (Appendix . ), for every j, k ∈ N , h φ u k ( T ) , Γ uT φ u j i = e − iλ u k T h Γ uT φ u j , φ u k i = e − i ( λ u k + λ u j ) T h φ u j ( T ) , e Γ e uT φ u k i . Now, e − i ( λ u k + λ u j ) T does not depend on u and the last relation implies thatthe surjectivity of the two following maps is equivalent {h φ u k ( T ) , Γ uT φ u j i} j,k ∈ N k ≤ N : L ((0 , T ) , R ) −→ {{ x k,j } j,k ∈ N k ≤ N : { x k,j } j ∈ N ∈ h ( C ) , ∀ k ≤ N }{h φ u j ( T ) , e Γ e uT φ u k i} j,k ∈ N k ≤ N : L ((0 , T ) , R ) −→ {{ x j,k } j,k ∈ N k ≤ N : { x j,k } j ∈ N ∈ h ( C ) , ∀ k ≤ N } . For this reason, the simultaneous global exact controllability in projectiononto a suitable N dimensional space is equivalent to the controllability of N problems (without projection). The transposition argument is particularly important as it allows toconcatenate and reverse dynamics on ( H ) N , which is preserved by thepropagator when one wants to prove the controllability in projection.For the simultaneous local exact controllability result, we can use Proposi-tion 2 . B satisfies Assumptions A, we consider[MN15 , T heorem .
1] that requires stronger assumptions on the operator B but provides the result without phase ambiguities (as in Theorem 1 . In this section, we prove the simultaneous global approximate controllability.
Definition 3.2.
The problems (3) are said to be simultaneously globallyapproximately controllable in H s (0) if, for every N ∈ N , ψ , ...., ψ N ∈ H s (0) , b Γ ∈ U ( H ) such that b Γ ψ , ...., b Γ ψ N ∈ H s (0) and ǫ >
0, then there exist
T > u ∈ L ((0 , T ) , R ) such that k b Γ ψ k − Γ uT ψ k k H s < ǫ for every 1 ≤ k ≤ N .13 heorem 3.3. Let B satisfy Assumptions I. The problems (3) are simulta-neously globally approximately controllable in H .Proof. Let N ∈ N and u belong to the neighborhoods provided by Remark B. B. B ). We define ||| · ||| ( s ) := ||| · ||| L ( H s (0) ,H s (0) ) and k f k BV ( T ) := k f k BV ((0 ,T ) , R ) = sup { t j } ≤ j ≤ n ∈ P P nj =1 | f ( t j ) − f ( t j − ) | , where f ∈ BV ((0 , T ) , R ) and P is the set of the partitions of (0 , T ) suchthat t = 0 < t < ... < t n = T. We consider the techniques developed byChambrion in [Cha12] and we start by choosing ψ j = φ j for every j ≤ N .Now, ( A + u B, B ) admits a non-degenerate chain of connectedness (see[BdCC13 , Def inition B. B ). Up to areordering of { φ k } k ∈ N , we can assume that for every m ∈ N , the couple( π m (Φ)( A + u B ) π m (Φ) , π m (Φ) Bπ m (Φ)) admits a non-degenerate chain ofconnectedness in H m .
1) Preliminaries:Claim.
For every ǫ >
0, there exist N ∈ N and e Γ N ∈ U ( H ) suchthat π N (Φ) e Γ N π N (Φ) ∈ SU ( H N ) and(18) k e Γ N φ j − b Γ φ j k (3) < ǫ, ∀ j ≤ N. Let N ′ ∈ N be such that N ′ ≥ N . We apply the orthonormalizing Gram-Schmidt process to { π N ′ (Φ) b Γ φ j } j ≤ N and we define the sequence { e φ j } j ≤ N that we complete in { e φ j } j ≤ N ′ , an orthonormal basis of H N ′ . The operator e Γ N ′ is the unitary map such that e Γ N ′ φ j = e φ j , for every j ≤ N ′ . The provideddefinition implies lim N ′ →∞ k e Γ N ′ φ j − b Γ φ j k = 0 for every j ≤ N. Thus, forevery ǫ >
0, there exists N ′ ∈ N large enough such that(19) k e Γ N ′ φ j − b Γ φ j k (3) < ǫ, ∀ j ≤ N. We denote N the number N ′ ≥ N such that the relation (19) is verified.
2) Finite dimensional controllability:
We call T ad the set of the admis-sible transitions, i.e. the couples ( j, k ) ∈ { , ..., N } such that B j,k = 0 and | λ j − λ k | = | λ m − λ l | with m, l ∈ N implies { j, k } = { m, l } or B m,l = 0.For every ( j, k ) ∈ { , ..., N } and θ ∈ [0 , π ), we define E θj,k the N × N matrix with elements ( E θj,k ) l,m = 0 , ( E θj,k ) j,k = e iθ and ( E θj,k ) k,j = − e − iθ , for ( l, m ) ∈ { , ..., N } \ { ( j, k ) , ( k, j ) } . We call E ad = (cid:8) E θj,k : ( j, k ) ∈ T ad , θ ∈ [0 , π ) (cid:9) and we consider Lie ( E ad ). We introduce the followingfinite dimensional control system on SU ( H N ) ( ˙ x ( t ) = x ( t ) v ( t ) , t ∈ (0 , τ ) ,x (0) = Id SU ( H N ) (20)where the set of admissible controls v is the set of piecewise constant func-tions taking value in E ad and τ >
0. 14 laim. (20) is controllable, i.e. for R ∈ SU ( H N ), there exist p ∈ N , M , ..., M p ∈ E ad , α , ..., α p ∈ R + such that R = e α M ◦ ... ◦ e α p M p . For every ( j, k ) ∈ { , ..., N } , we define the N × N matrices R j,k , C j,k and D j as follow. For ( l, m ) ∈ { , ..., N } \ { ( j, k ) , ( k, j ) } , we have( R j,k ) l,m = 0 and ( R j,k ) j,k = − ( R j,k ) k,j = 1 , while ( C j,k ) l,m = 0 and( C j,k ) j,k = ( C j,k ) k,j = i. Moreover, for ( l, m ) ∈ { , ..., N } \ { (1 , , ( j, j ) } , ( D j ) l,m = 0 and ( D j ) , = − ( D j ) j,j = i. Now, e := { R j,k } j,k ≤ N ∪ { C j,k } j,k ≤ N ∪ { D j } j ≤ N is a basis of su ( H N ).Thanks to [Sac00 , T heorem . Lie ( E ad ) ⊇ su ( H N ) for su ( H N ) the Lie algebra of SU ( H N ) . The claim si valid as it is possible to obtain the matrices R j,k , C j,k and D j for every j, k ≤ N by iterated Lie brackets of elements in E ad .
3) Finite dimensional estimates:
Thanks to the previous claim and tothe fact that π N (Φ) e Γ N π N (Φ) ∈ SU ( H N ), there exist p ∈ N , M , ..., M p ∈ E ad and α , ..., α p ∈ R + such that(21) π N (Φ) e Γ N π N (Φ) = e α M ◦ ... ◦ e α p M p . Claim.
For every l ≤ p and e α l M l from (21), there exist { T ln } l ∈ N ⊂ R + and { u ln } n ∈ N such that u ln : (0 , T ln ) → R for every n ∈ N and(22) lim n →∞ k Γ u ln T ln φ k − e α l M l φ k k (3) = 0 , ∀ k ≤ N , sup n ∈ N k u ln k BV ( T n ) < ∞ , sup n ∈ N k u ln k L ∞ ((0 ,T n ) , R ) < ∞ , sup n ∈ N T n k u ln k L ∞ ((0 ,T n ) , R ) < ∞ . (23)We consider the results developed in [Cha12 , Section . Section .
2] byChambrion and leading to [Cha12 , P roposition
6] (also adopted in [Duc]).Each e α l M l is a rotation in a two dimensional space for every l ∈ { , ..., p } andthe mentioned work allows to explicit { T ln } l ∈ N ⊂ R + and { u ln } n ∈ N satisfying(23) such that u ln : (0 , T ln ) → R for every n ∈ N and(24) lim n →∞ k π N (Φ)Γ u ln T ln φ k − e α l M l φ k k = 0 , ∀ k ≤ N . As e α l M l ∈ SU ( H N ), we have lim n →∞ k Γ u ln T ln φ k − e α l M l φ k k = 0 for k ≤ N . We consider the propagation of regularity developed by Kato in [Kat53]and adopted in [Duc]. We notice that i ( A + u ( t ) B − ic ) is maximal dissipativein H for suitable c := k u k L ∞ ((0 ,T ) , R ) . Let λ > c and b H := D ( A ( iλ − A )) ≡ H . We know that B : b H ⊂ H → H and the arguments of15emark 1 . B ∈ L ( b H , H ). For T > u ∈ BV ((0 , T ) , R ),we have ||| u ( t ) B ( iλ − A ) − ||| (2) < M := sup t ∈ [0 ,T ] ||| ( iλ − A − u ( t ) B ) − ||| L ( H , b H ) ≤ sup t ∈ [0 ,T ] + ∞ X l =1 ||| ( u ( t ) B ( iλ − A ) − ) l ||| (2) < + ∞ . We know that k k + f ( · ) k BV ((0 ,T ) , R ) = k f k BV ((0 ,T ) , R ) for every f ∈ BV ((0 , T ) , R )and k ∈ R . The same idea leads to N := ||| iλ − A − u ( · ) B ||| BV (cid:0) [0 ,T ] ,L ( b H ,H ) (cid:1) = k u k BV ( T ) ||| B ||| L ( b H ,H ) < + ∞ . We call C := ||| A ( A + u ( T ) B − iλ ) − ||| (2) < ∞ and U ut the propaga-tor generated by A + uB − ic such that U ut ψ = e − ct Γ ut ψ . Thanks to[Kat53 , Section . ψ ∈ H , it follows k ( A + u ( T ) B − iλ ) U ut ψ k (2) ≤ M e MN k ( A − iλ ) ψ k (2) and k Γ uT ψ k (4) = k A Γ uT ψ k (2) ≤ C M e MN + cT k ψ k (4) as ||| ( A − iλ ) A − ||| (2) = ||| I − iλA − ||| (2) ≤ λπ . For every T > u ∈ BV ((0 , T ) , R ) and ψ ∈ H , there exists C ( K ) > K = (cid:0) k u k BV ( T ) , k u k L ∞ ((0 ,T ) , R ) , T k u k L ∞ ((0 ,T ) , R ) (cid:1) such that k Γ uT ψ k (4) ≤ C ( K ) k ψ k (4) . Then, from (23), there exists
C > ||| Γ u ln T ln ||| (4) ≤ C. For every ψ ∈ H , from the Cauchy-Schwarz inequality, we have k Aψ k ≤k A ψ kk ψ k and k A ψ k ≤ (cid:0) h A ψ, Aψ i (cid:1) ≤ k A ψ k k Aψ k , which imply(26) k ψ k ≤ k ψ k k ψ k . In conclusion, the relations (24), (25) and (26) lead to the relation (22).
4) Infinite dimensional estimates:Claim.
There exist K , K , K > ǫ >
0, thereexist
T > u ∈ L ((0 , T ) , R ) such that k Γ uT φ k − b Γ φ k k (3) ≤ ǫ forevery k ≤ N and k u k BV ( T ) ≤ K , k u k L ∞ ((0 ,T ) , R ) ≤ K , T k u k L ∞ ((0 ,T ) , R ) ≤ K . Let us assume p = 2. The following result is valid for any p ∈ N . Thanksto (22) and to the propagation of regularity from [Kat53], for every ǫ > N ∈ N , there exists n ∈ N large enough such that, for every k ≤ N , k Γ u n T n Γ u n T n φ k − e α M e α M φ k k (3) ≤ ||| Γ u n T n ||| (3) k Γ u n T n φ k − e α M φ k k (3) + N X l =1 k (cid:0) Γ u n T n φ l − e α M φ l (cid:1) h φ l , e α M φ k ik (3) ≤ ||| Γ u n T n ||| (3) k Γ u n T n φ k − e α M φ k k (3) + k e α M φ k k (cid:16) N X l =1 k (cid:0) Γ u n T n φ l − e α M φ l (cid:1) k (cid:17) ≤ ǫ. In the previous inequality, we considered that e α M φ k ∈ H N and that ||| Γ u n T n ||| (3) is uniformly bounded in n ∈ N thanks to the propagation ofregularity from [Kat53] and to (23). The identity (21) leads to the existenceof K , K , K > ǫ >
0, there exist
T > u ∈ L ((0 , T ) , R ) such that k Γ uT φ k − e Γ N φ k k (3) < ǫ for every k ≤ N and k u k BV ( T ) ≤ K , k u k L ∞ ((0 ,T ) , R ) ≤ K , T k u k L ∞ ((0 ,T ) , R ) ≤ K . (27)The relation (18) and the triangular inequality achieve the claim.
5) Conclusion:
For every { ψ j } j ≤ N ⊂ H , b Γ ∈ U ( H ) such that { b Γ ψ j } j ≤ N ⊂ H and ǫ >
0, there exists a natural number M ∈ N such that, for ev-ery l ≤ N , it follows k ψ l k (3) ≤ (cid:13)(cid:13) P Mk =1 φ k h φ k , ψ l i (cid:13)(cid:13) + ǫ and k b Γ ψ l k (3) ≤ (cid:13)(cid:13) P Mk =1 b Γ φ k h φ k , ψ l i (cid:13)(cid:13) + ǫ. The proof is achieved by simultaneously driving { φ k } k ≤ M close enough to { b Γ φ k } k ≤ M since, for every l ≤ N , T > u ∈ L ((0 , T ) , R ) satisfying (27), k Γ uT ψ l − b Γ ψ l k (3) ≤ k ψ l k (cid:16) M X k =1 k Γ uT φ k − b Γ φ k k (cid:17) + ( ||| Γ uT ||| (3) + 1) ǫ. . and Theorem . In the current section, we provide the proofs of Theorem 1 . .
8, which require the following proposition.
Proposition 3.4.
Let N ∈ N and B satisfy Assumptions I.1. For any { ψ k } k ≤ N , { ψ k } k ≤ N ⊂ H orthonormal systems, there exist T > , u ∈ L ((0 , T ) , R ) and { θ k } k ≤ N ⊂ R such that e iθ k ψ k = e Γ uT ψ k for every k ≤ N.
2. If B satisfies Assumptions A, then for any { ψ k } k ≤ N , { ψ k } k ≤ N ⊂ H orthonormal systems, there exist T > and u ∈ L ((0 , T ) , R ) so that ψ k = e Γ uT ψ k for every k ≤ N. roof. Let N ∈ N and let u ∈ R belong to the neighborhoods provided byLemma B.
5, Lemma B. B. B ). Let e α u be the map with elements ( b α j,j ( u ) | b α j,j ( u ) | b α k,j ( u ) , j, k ≤ N, b α k,j ( u ) , k > N, j ≤ N. The proof of Proposition 2 . e α u for every T >
0, instead of α u introduced in (13). Asexplained in Remark 3 .
1, this result corresponds to the simultaneous localexact controllability up to phases of N problems (3) in a neighborhood O Nǫ,T := n { ψ j } j ≤ N ⊂ H (cid:12)(cid:12) h ψ j , ψ k i = δ j,k ; N X j =1 k ψ j − φ u j k (3) < ǫ o with ǫ > { ψ k } k ≤ N ∈ O Nǫ,T , there exist u ∈ L ((0 , T ) , R ) and { θ j } j ≤ N ⊂ R so that Γ uT φ u j = e iθ j ψ j for any j ≤ N. Theorem 3 . N problems. For any { ψ j } j ≤ N ⊂ H composed by orthonormalelements, there exist T > u ∈ L ((0 , T ) , R ) such that k Γ u T ψ j − φ u j k (3) < ǫN , ∀ j ≤ N, = ⇒ { Γ u T ψ j } j ≤ N ∈ O Nǫ,T . The local controllability is also valid for the reversed dynamics of (10), forevery
T >
0, there exist u ∈ L ((0 , T ) , R ) and { θ j } j ≤ N ⊂ R so that { Γ u T ψ j } j ≤ N = { e iθ j e Γ uT φ u j } j ≤ N = ⇒ { e − iθ j Γ e uT Γ u T ψ j } j ≤ N = { φ u j } j ≤ N . Then, there exist T > u ∈ L ((0 , T ) , R ) such that { e − iθ j Γ u T ψ j } j ≤ N = { φ u j } j ≤ N . Now, the same property is valid for the reversed dynamics of (10)and, for every { ψ j } j ≤ N ⊂ H composed by orthonormal elements, there ex-ist T > u ∈ L ((0 , T ) , R ) and { θ ′ j } j ≤ N ⊂ R such that { e − iθ ′ j e Γ u T ψ j } j ≤ N = { φ u j } j ≤ N . In conclusion, for e u ( · ) = u ( T − · ), the proof is achieved as { e − i ( θ j − θ ′ j ) Γ e u T Γ u T ψ j } j ≤ N = { ψ j } j ≤ N . The proof of the second claim follows as in , with the difference thatif B satisfies Assumptions A, then Remark B.
10 provides the validity of asimultaneous local exact controllability without phase ambiguities.Indeed, keeping in mind our notation, let H V ) be the space defined in[MN15]. We know that H V ) corresponds to e H when V = u B and B is a suitable multiplication operator. We consider the assumptions ( C C
4) and ( C
5) introduced in [MN15 , p. V with u B and µ by − B , then the statement of [MN15 , T heorem .
1] is still valid.The condition ( C
3) is ensured by Lemma B. B ), while theassumptions ( C
4) and ( C
5) respectively follow from the first point of Remark B. B.
10 (Appendix B ). The result of [MN15 , T heorem . O Nǫ,T ⊂ H for suitable ǫ > T > .
1. For every { ψ k } k ≤ N ∈ O Nǫ,T , there exists u ∈ L ((0 , T ) , R )such that ψ k = Γ uT φ u k for every k ≤ N. The remaining part of the proof isachieved as in . Proof of Theorem . . Let N ∈ N and u ∈ R belong to the neighborhoodsprovided by Lemma B.
5, Lemma B. B. B ). LetΨ := { ψ j } j ∈ N ∈ H be an orthonormal systems. We consider { ψ j } j ∈ N , { ψ j } j ∈ N ⊂ H complete orthonormal systems and b Γ ∈ U ( H ) such that b Γ ψ j = ψ j and b Γ ∗ ψ j ∈ H for every j ∈ N . Then, for every k ≤ N , e ψ k := ∞ X j =1 ψ j h ψ j , ψ k i = ∞ X j =1 ψ j h b Γ ψ j , ψ k i = ∞ X j =1 ψ j h ψ j , b Γ ∗ ψ k i = b Γ ∗ ψ k ∈ H . Thanks to the first point of Proposition 3 .
4, there exist
T > u ∈ L ((0 , T ) , R )and { θ k } k ≤ N ⊂ R such that e iθ k e ψ k = e Γ uT ψ k for each k ≤ N . Hence h ψ j , e Γ uT ψ k i = h e iθ j ψ j , e iθ k e ψ k i = h ψ j , e iθ k ψ k i , ∀ j, k ∈ N , k ≤ N. Thanks to the time reversibility (Appendix 1 . h Γ e uT ψ j , ψ k i = h ψ j , e Γ uT ψ k i = h ψ j , e iθ k ψ k i , ∀ j, k ∈ N , k ≤ N. Proof of Theorem . . Let N ∈ N and let u ∈ R belong to the neighbor-hoods provided by Lemma B.
5, Lemma B.
6, Remark B. B. B ).
1) Controllability in projection of orthonormal systems:
Let Ψ := { ψ j } j ∈ N ∈ H be an orthonormal system. Let us consider { ψ j } j ∈ N , { ψ j } j ∈ N ⊂ H be complete orthonormal systems and b Γ ∈ U ( H ) be such that b Γ ψ j = ψ j and b Γ ∗ ψ j ∈ H for every j ∈ N . As in the proof of Theorem 1 . k ≤ N , we define e ψ k := P ∞ j =1 ψ j h ψ j , ψ k i . Thanks to the secondpoint of Proposition 3 .
4, there exist
T > u ∈ L ((0 , T ) , R ) such that e ψ k = e Γ uT ψ k for each k ≤ N . Hence h ψ j , e Γ uT ψ k i = h ψ j , e ψ k i = h ψ j , ψ k i , ∀ j, k ∈ N , k ≤ N. Thanks to Appendix 1 .
3, we have h Γ e uT ψ j , ψ k i = h ψ j , e Γ uT ψ k i = h ψ j , ψ k i andthen π N (Ψ ) ψ j = π N (Ψ )Γ e uT ψ j for every j ∈ N .19 ) Controllability in projection of unitarily equivalent functions: Let us consider { ψ j } j ∈ N , { ψ j } j ∈ N ⊂ H unitarily equivalent. Let Ψ := { ψ j } j ∈ N be an orthonormal system. We suppose the existence of b Γ ∈ U ( H )such that b Γ ψ j = ψ j and b Γ ∗ ψ j ∈ H for every j ∈ N . One knows that,for every j ∈ N , there exists { a jk } k ∈ N ∈ ℓ ( C ) such that ψ j = P k ∈ N a jk ψ k . However, { b Γ ψ j } j ∈ N is an Hilbert basis of H and ψ j = b Γ ψ j = P k ∈ N a jk b Γ ψ k . The point 2) implies that there exist
T > u ∈ L ((0 , T ) , R ) such that π N (Ψ ) Γ uT ψ k = π N (Ψ ) b Γ ψ k for every k ∈ N , and then for any j ∈ N , π N (Ψ ) Γ uT ψ j = X k ∈ N a jk (cid:0) π N (Ψ ) Γ uT ψ k (cid:1) = π N (Ψ ) X k ∈ N a jk b Γ ψ k = π N (Ψ ) ψ j .
3) Controllability in projection with generic projector:
Let Ψ = { ψ j } j ∈ N ⊂ H be a sequence of linearly independent elements. For every N ∈ N , thanks the Gram-Schmidt orthonormalization process, there existsan orthonormal system e Ψ := {{ e ψ j } j ≤ N , , ... } such that span { ψ j : j ≤ N } = span { e ψ j : j ≤ N } . The claim follows as π N (Ψ ) ≡ π N ( e Ψ ) . IfΨ = { ψ j } j ∈ N ⊂ H is a generic sequence of functions, then we extract fromΨ a subsequence of linearly independent elements and repeat as above. Let ψ , ψ ∈ H . We define the rank one operator | ψ ih ψ | such that | ψ ih ψ | ψ = ψ h ψ , ψ i for every ψ ∈ H . For any b Γ ∈ U ( H ), we have b Γ | ψ ih ψ | = | b Γ ψ ih ψ | , | ψ ih ψ | b Γ ∗ = | ψ ih b Γ ψ | . Corollary 4.1.
Let B satisfy Assumptions I and Assumptions A. Let ρ , ρ ∈ T ( H ) be two density matrices so that Ran ( ρ ) , Ran ( ρ ) ⊆ H . We sup-pose the existence of b Γ ∈ U ( H ) so that ρ = b Γ ρ b Γ ∗ . Let Ψ := { ψ j } j ∈ N ⊂ H be such that { b Γ ψ j } j ∈ N ⊂ H , for every j ∈ N . For any N ∈ N , thereexist T > and a control function u ∈ L ((0 , T ) , R ) such that π N (Ψ ) Γ uT ρ (Γ uT ) ∗ π N (Ψ ) = π N (Ψ ) ρ π N (Ψ ) . Proof.
Let
T > := { ψ j } j ∈ N ∈ H . Let ρ , ρ ∈ T ( H ) betwo unitarily equivalent density matrices such that Ran ( ρ ) , Ran ( ρ ) ⊆ H . We suppose that the unitary operator b Γ ∈ U ( H ) such that ρ = b Γ ρ b Γ satisfies the condition b Γ ∗ ψ j ∈ H for every j ∈ N . One can en-sure the existence of two complete orthonormal systems Ψ := { ψ j } j ∈ N , := { ψ j } j ∈ N ∈ H respectively composed by eigenfunctions of ρ and ρ such that ρ = P ∞ j =1 l j | ψ j ih ψ j | and ρ = P ∞ j =1 l j | ψ j ih ψ j | . The sequence { l j } j ∈ N ⊂ R + corresponds to the spectrum of ρ and ρ . Now, thanks toTheorem 1 .
8, there exists a control function u ∈ L ((0 , T ) , R ) such that π N (Ψ ) Γ uT ψ j = π N (Ψ ) ψ j . Thus π N (Ψ ) Γ uT ρ (Γ uT ) ∗ π N (Ψ ) = X ∞ j =1 l j | π N (Ψ ) Γ uT ψ j ih ψ j Γ uT π N (Ψ ) | = X ∞ j =1 l j π N (Ψ ) | ψ j ih ψ j | π N (Ψ ) = π N (Ψ ) ρ π N (Ψ ) . Acknowledgments.
The author thanks Thomas Chambrion for sug-gesting him the problem and Nabile Boussa¨ıd for the periodic discussions.He is also grateful to Morgan Morancey for the explanation about the works[Mor14] and [MN15].
A Moment problem
In this appendix, we briefly adapt some results concerning the solvabilityof the moment problems (as (11) and (17)). Let [BL10 , P roposition
19; 2)]be satisfied and { f k } k ∈ Z be a Riesz basis (see [BL10 , Def inition X = span { f k : k ∈ Z } H ⊆ H , with H and Hilbert space. For { v k } k ∈ Z theunique biorthogonal family to { f k } k ∈ Z ([BL10 , Remark { v k } k ∈ Z is also aRiesz basis of X ([BL10 , Remark , P roposition
19; 2)],there exist C , C > C P k ∈ Z | x k | ≤ k u k H ≤ C P k ∈ Z | x k | forevery u ( t ) = P k ∈ Z x k v k ( t ) with { x k } k ∈ N ∈ ℓ ( C ). Moreover, for every u ∈ X , we know that u = P k ∈ Z v k h f k , u i H since { f k } k ∈ Z and { v k } k ∈ Z arereciprocally biorthonoromal (see [BL10 , Remark C X k ∈ Z |h f k , u i H | ≤ k u k H ≤ C X k ∈ Z |h f k , u i H | . When Haraux’s Thoerem [KL05 , T heorem .
6] is verified, for
T > { e iλ k ( · ) } k ∈ Z is a Riesz basis in X = span { e iλ k ( · ) : k ∈ Z } L ⊆ L ((0 , T ) , C ). The relation (28) is satisfied and F : u ∈ X (cid:8) h e iλ k ( · ) , u i H (cid:9) k ∈ Z ∈ ℓ ( C ) is invertible. For every sequence { x k } k ∈ Z ∈ ℓ ( C ), there exists u ∈ X such that x k = R T u ( s ) e − iλ k s ds for every k ∈ Z . Remark A.1.
Let { λ k } k ∈ N be an ordered sequence of real numbers suchthat λ k = − λ l for every k, l ∈ N . Let G := inf k = j | λ k − λ j | > and G ′ :=sup K ⊂ N inf k = j k,j ∈ N \ K | λ k − λ j | , where K runs over the finite subsets of Z . For k > , we call ω k = − λ k , while we impose ω k = λ − k for k < and k = − l .We call Z ∗ = Z \ { } . The sequence { ω k } k ∈ Z ∗ \{− l } satisfies the hypothesesof [ KL05 , T heorem . for sup K ⊂ Z ∗ \{− l } inf k = j k,j ∈ N \ K | ω k − ω j | = G ′ . Given x k } k ∈ N ∈ ℓ ( C ) , we introduce { e x k } k ∈ Z ∗ \{− l } ∈ ℓ ( C ) such that e x k = x k for k > , while e x k = x − k for k < and k = − l . For T > π/G , there exists u ∈ L ((0 , T ) , C ) such that e x k = R T u ( s ) e − iω k s ds for every k ∈ Z ∗ \ {− l } .Then ( x k = R T u ( s ) e iλ k s ds = R T u ( s ) e iλ k s ds, k ∈ N \ { l } ,x k = R T u ( s ) ds, k = l, which implies that, if x l ∈ R , then u is real. B Analytic Perturbation
Let us consider the problem (12) and the eigenvalues { λ u j } j ∈ N of the opera-tor A + u B . When B is a bounded symmetric operator satisfying Assump-tions I and A = − ∆ is the Laplacian with Dirichlet type boundary conditions D ( A ) = H ((0 , , C ) ∩ H ((0 , , C ) , thanks to [Kat95 , T heorem V II. . , T heorem V II. . Proposition B.1.
Let B be a bounded symmetric operator satisfying As-sumptions I. There exists a neighborhood D of u = 0 in R small enoughwhere the maps u λ uj are analytic for every j ∈ N . The next lemma proves the existence of perturbations, which do notshrink the eigenvalues gaps.
Lemma B.2.
Let B be a bounded symmetric operator satisfying Assump-tions I. There exists a neighborhood U (0) in R of u = 0 such that, for each u ∈ U (0) , there exists r > such that, for every j ∈ N , µ j := λ j + λ j +1 ∈ ρ ( A + u B ) , ||| ( A + u B − µ j ) − ||| ≤ r. Proof.
Let D be the neighborhood provided by Proposition B.
1. We know( A − µ j ) is invertible in a bounded operator and µ j ∈ ρ ( A ) (resolvent setof A ). Let δ := min j ∈ N {| λ j +1 − λ j |} . We know that ||| ( A − µ j ) − ||| ≤ sup k ∈ N | µ j − λ k | = | λ j +1 − λ j | ≤ δ . Thus ||| ( A − µ j ) − u B ||| ≤ | u | ||| ( A − µ j ) − ||| ||| B ||| ≤ δ | u | ||| B ||| and if | u | ≤ δ (1 − ǫ )2 ||| B ||| for ǫ ∈ (0 , , then ||| ( A − µ j ) − u B ||| ≤ − ǫ. Theoperator ( A + u B − µ j ) is invertible and ||| ( A + u B − µ j ) − ||| ≤ δǫ as k ( A + u B − µ j ) ψ k ≥ k ( A − µ j ) ψ k − k u Bψ k ≥ δ k ψ k − δ (1 − ǫ )2 k ψ k for every ψ ∈ D ( A ). 22 emma B.3. Let B be a bounded symmetric operator satisfying Assump-tions I. There exists a neighborhood U (0) of in R such that, for every u ∈ U (0) , ( A + u P ⊥ φ k B − λ u k ) is invertible with bounded inverse from D ( A ) ∩ φ ⊥ k to φ ⊥ k , for every k ∈ N and P ⊥ φ k is the projector onto the orthogonal space of φ k .Proof. Let D be the neighborhood provided by Lemma B.
2. For any u ∈ D ,one can consider the decomposition ( A + u P ⊥ φ k B − λ u k ) = ( A − λ u k )+ u P ⊥ φ k B. The operator A − λ u k is invertible with bounded inverse when it acts onthe orthogonal space of φ k and we estimate ||| (( A − λ u k ) (cid:12)(cid:12) φ ⊥ k ) − u P ⊥ φ k B ||| . However, for every ψ ∈ D ( A ) ∩ Ran ( P ⊥ φ k ) such that k ψ k = 1, we have k ( A − λ u k ) ψ k ≥ min {| λ k +1 − λ u k | , | λ u k − λ k − |}k ψ k . Let δ k := min (cid:8) | λ k +1 − λ u k | , | λ u k − λ k − | (cid:9) . Thanks to Lemma B.
2, for | u | small enough, λ u k ∈ (cid:16) λ k − + λ k , λ k + λ k +1 (cid:17) and then δ k ≥ min n(cid:12)(cid:12)(cid:12) λ k +1 − λ k + λ k +1 (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) λ k − + λ k − λ k − (cid:12)(cid:12)(cid:12)o ≥ (2 k − π > k. Afterwards, ||| (( A − λ u k ) (cid:12)(cid:12) φ ⊥ k ) − u P ⊥ φ k B ||| ≤ δ k | u | ||| B ||| and, if | u | ≤ (1 − r ) δ k ||| B ||| for r ∈ (0 , ||| (( A − λ u k ) (cid:12)(cid:12) φ ⊥ k ) − u P ⊥ φ k B ||| ≤ (1 − r ) < . The operator A k := ( A − λ u k + u P ⊥ φ k B ) is invertible when it acts on theorthogonal space of φ k and, for every ψ ∈ D ( A ), k A k ψ k ≥ k ( A − λ u k ) ψ k − ||| u P ⊥ φ k Bψ k ≥ δ k k ψ k − ||| u P ⊥ φ k B ||| k ψ k = rδ k k ψ k . In conclusion, ||| (( A − λ u k + u P ⊥ φ k B ) (cid:12)(cid:12) φ ⊥ k ) − ||| ≤ rk for every k ∈ N . Lemma B.4.
Let B be satisfy Assumptions I. There exists a neighborhood U (0) of in R such that, for any u ∈ U (0) , we have λ u j = 0 and λ u j ≍ λ j for every j ∈ N . In other words, there exist two constants C , C > suchthat, for each j ∈ N , C λ j ≤ λ u j ≤ C λ j . Proof.
Let u ∈ D for D the neighborhood provided by Lemma B.
3. We de-compose the eigenfunction φ u j = a j φ j + η j , where a j is an orthonormalizingconstant and η j is orthogonal to φ j . Hence λ u k φ u k = ( A + u B )( a k φ k + η k )and λ u k a k φ k + λ u k η k = Aa k φ k + Aη k + u Ba k φ k + u Bη k . By projectingonto the orthogonal space of φ k , λ u k η k = Aη k + u P ⊥ φ k Ba k φ k + u P ⊥ φ k Bη k . B. A + u P ⊥ φ k B − λ u k is invertible withbounded inverse when it acts on the orthogonal space of φ k and then(29) η k = − a k (( A + u P ⊥ φ k B − λ u k ) (cid:12)(cid:12) φ ⊥ k ) − u P ⊥ φ k Bφ k , = ⇒ λ u j = h a j φ j + η j , ( A + u B )( a j φ j + η j ) i = | a j | λ j + u h a j φ j , Ba j φ j i + h a j φ j , ( A + u B ) η j i + h η j , ( A + u B ) a j φ j i + h η j , ( A + u B ) η j i . By using the relation (29), h η j , ( A + u B ) η j i = h η j , ( A + u P ⊥ φ k B − λ u j ) η j i + λ u j k η j k = λ u j k η j k + D η j , − a j ( A + u P ⊥ φ j B − λ u j )(( A + u P ⊥ φ j B − λ u j ) (cid:12)(cid:12) φ ⊥ j ) − u P ⊥ φ j Bφ j E . However, ( A + u P ⊥ φ j B − λ u j )(( A + u P ⊥ φ j ) B − λ u j ) (cid:12)(cid:12) φ ⊥ j ) − = Id and h η j , ( A + u B ) η j i = λ u j k η j k − u a j h η j , P ⊥ φ j Bφ j i . Moreover, we have h φ j , ( A + u B ) η j i = u h φ j , Bη j i = u h P ⊥ φ j Bφ j , η j i and h η j , ( A + u B ) φ j i = u h η j , P ⊥ φ j Bφ j i . Thus λ u j = | a j | λ j + u | a j | B j,j + λ u j k η j k + u a j h P ⊥ φ j Bφ j , η j i . (30)One can notice that | a j | ∈ [0 ,
1] and k η j k are uniformly bounded in j . Weshow that the first accumulates at 1 and the second at 0. Indeed, from theproof of Lemma ( B.
3) and the relation (29), there exists C > k η j k ≤ | u | ||| (( A + u P ⊥ φ j B − λ u j ) (cid:12)(cid:12) φ ⊥ j ) − ||| | a j | k Bφ j k ≤ C j (31)for r ∈ (0 , j →∞ k η j k = 0 . Afterwards, by contradic-tion, if | a j | does not converge to 1, then there exists { a j k } k ∈ N a subsequenceof { a j } j ∈ N such that | a j ∞ | := lim k →∞ | a j k | ∈ [0 , k →∞ k φ u j k k ≤ lim k →∞ | a j k |k φ j k k + k η j k k = lim k →∞ | a j k | + k η j k k = | a j ∞ | < j →∞ | a j | = 1. From (30), it follows λ u j ≍ λ j for | u | small enough. The relation also implies that λ u j = 0 for every j ∈ N and | u | small enough. Lemma B.5.
Let B be a bounded symmetric operator satisfying Assump-tions I. For every N ∈ N , there exists a neighborhood U (0) of in R such that there exists e C N > such that, for any u ∈ U (0) , we have |h φ u k , Bφ u j i| ≥ e C N k for every k, j ∈ N and j ≤ N .Proof. We start by choosing k ∈ N such that k = j and u ∈ D for D theneighborhood provided by Lemma B.
4. Thanks to Assumptions II, we have |h φ u k , Bφ u j i| = |h a k φ k + η k , B ( a j φ j + η j ) i|≥ C N a k a j k − (cid:12)(cid:12) a k h φ k , Bη j i + a j h η k , Bφ j i + h η k , Bη j i (cid:12)(cid:12) . (32) 24 ) Expansion of h η k , Bφ j i , h φ k , Bη j i , h η k , Bη j i : Thanks to (29), we have h η k , Bφ j i = h− a k (( A + u P ⊥ φ k B − λ u k ) (cid:12)(cid:12) φ ⊥ k ) − u P ⊥ φ k Bφ k , P ⊥ φ k Bφ j i for every k ∈ N and j ≤ N , while (cid:0) ( A + u P ⊥ φ k B − λ u k ) (cid:12)(cid:12) φ ⊥ k (cid:1) − corresponds to(( A − λ u k ) P ⊥ φ k ) − ∞ X n =0 (cid:0) u (( A − λ u k ) P ⊥ φ k ) − P ⊥ φ k BP ⊥ φ k (cid:1) n for | u | small enough. For M k := P ∞ n =0 (cid:0) u (( A − λ u k ) P ⊥ φ k ) − P ⊥ φ k B (cid:1) n P ⊥ φ k , h η k , Bφ j i = − u h a k M k Bφ k , (( A − λ u k ) P ⊥ φ k ) − P ⊥ φ k Bφ j i . (33)Thanks to B : D ( A ) → D ( A ), for every k ∈ N and j ≤ N ,(( A − λ u k ) P ⊥ φ k ) − P ⊥ φ k Bφ j = P ⊥ φ k B (( A − λ u k ) P ⊥ φ k ) − φ j − (cid:2) P ⊥ φ k B, (( A − λ u k ) P ⊥ φ k ) − P ⊥ φ k (cid:3) φ j = P ⊥ φ k B (( A − λ u k ) P ⊥ φ k ) − φ j − (( A − λ u k ) P ⊥ φ k ) − P ⊥ φ k [ B, A ](( A − λ u k ) P ⊥ φ k ) − φ j . For e B k := (( A − λ u k ) P ⊥ φ k ) − P ⊥ φ k [ B, A ] , we have (( A − λ u k ) P ⊥ φ k ) − P ⊥ φ k Bφ j = P ⊥ φ k ( B + e B k )( λ j − λ u k ) − φ j . and, for every k ∈ N and j ≤ N , h η k , Bφ j i = − u λ j − λ u k h a k M k Bφ k , ( B + e B k ) φ j i . (34)For every k ∈ N and j ≤ N , we obtain |h η k , Bη j i| = |h Bη k , η j i| = |h u a k B (( A − λ u k ) P ⊥ φ k ) − M k Bφ k ,u a j (( A − λ u j ) P ⊥ φ j ) − M j Bφ j i (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) a j a k u λ k − λ u j (cid:10) φ k , L k,j φ j (cid:11)(cid:12)(cid:12)(cid:12) (35)with L k,j := ( A − λ u j ) BM k (( A − λ u k ) P ⊥ φ k ) − P ⊥ φ k B (( A − λ u j ) P ⊥ φ j ) − M j B. Now, there exists ǫ > | a l | ∈ ( ǫ,
1) for every l ∈ N . Thanks to(34), (35) and (32), there exists b C N such that |h φ u k , Bφ u j i| ≥ b C N k − (cid:12)(cid:12)(cid:12) u λ j − λ u k h M k Bφ k , ( B + e B k ) φ j i (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) u λ k − λ u j h ( B + e B j ) φ k , M j Bφ j i (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) u λ k − λ u j (cid:10) φ k , L k,j φ j (cid:11)(cid:12)(cid:12)(cid:12) . (36)
2) Features of the operators M k , e B k , L k,j : Each M k for k ∈ N is uni-formly bounded in L ( H , H ) when | u | is small enough so that ||| u (( A − λ u k ) P ⊥ φ k ) − P ⊥ φ k BP ⊥ φ k ||| L ( H ) < . The definition of e B k implies that e B k P ⊥ φ k =(( A − λ u k ) P ⊥ φ k ) − P ⊥ φ k B ( A − λ u k ) P ⊥ φ k − P ⊥ φ k BP ⊥ φ k . Hence, the operators e B k areuniformly bounded in k in L (cid:0) H ∩ Ran ( P ⊥ φ k ) , H ∩ Ran ( P ⊥ φ k ) (cid:1) . Third, one25an notice that B (( A − λ u j ) P ⊥ φ j ) − M j B ∈ L ( H , H ) for every j ∈ N . Then, for every k ∈ N and j ≤ N ,( A − λ u j ) BM k (( A − λ u k ) P ⊥ φ k ) − P ⊥ φ k = ( A − λ u j ) B (( A − λ u k ) P ⊥ φ k ) − ∞ X n =0 (cid:0) u P ⊥ φ k B (( A − λ u k ) P ⊥ φ k ) − (cid:1) n P ⊥ φ k = ( A − λ u j )(( A − λ u k ) P ⊥ φ k ) − P ⊥ φ k ( e B k + B ) f M k with f M k := P ∞ n =0 (cid:0) u P ⊥ φ k B (( A − λ u k ) P ⊥ φ k ) − (cid:1) n P ⊥ φ k . Now, the operators f M k are uniformly bounded in L ( H , H ) as M k . Hence L k,j are uniformlybounded in L ( H , H ).Let { F l } l ∈ N be an infinite uniformly bounded family of operators in L ( H , H ).For every l, j ∈ N , there exists c l,j > P ∞ k =1 | k h φ k , F l φ j i| < ∞ ,which implies |h φ k , F l φ j i| ≤ c l,j k for every k ∈ N . Now, the constant c l,j canbe assumed uniformly bounded in l since, for every k, j ∈ N ,sup l ∈ N | k h φ k , F l φ j i| ≤ sup l ∈ N X m ∈ N | m h φ m , F l φ j i| ≤ sup l ∈ N k F l φ j k < ∞ . Thus, for every infinite uniformly bounded family of operators { F l } l ∈ N in L ( H , H ) and for every j ∈ N , there exists a constant c j such that(37) |h φ k , F l φ j i| ≤ c j k , ∀ k, l ∈ N .
3) Conclusion:
We know that | λ j − λ u k | − and | λ k − λ u j | − asymptoti-cally behave as k − thanks to Lemma B.
4. From the previous point, thefamilies of operators { BM k ( B + e B k ) } k ∈ N , { L k,j } k ∈ N are uniformly boundedin L ( H , H ) and BM j ( B + e B j ) ∈ L ( H , H ) for every 1 ≤ j ≤ N .Hence, we use the relation (37) in (36) and there exist C , C , C , C > j ∈ N such that, for | u | small enough and k ∈ N large enough, |h φ u k , Bφ u j i| ≥ b C N k − C | u || λ j − λ u k | k − C | u || λ k − λ u j | k − C | u | | λ k − λ u j | k ≥ C k . (38)Let K ∈ N be so that |h φ u k ( T ) , Bφ u j ( T ) i| ≥ C k for every k > K. For j ∈ N , the zeros of the analytic map u
7→ {|h φ u k ( T ) , Bφ u j ( T ) i|} k ≤ K ∈ R K are discrete. Then, for | u | small enough, |h φ u k ( T ) , Bφ u j ( T ) i| 6 = 0 for every k ≤ K. Thus, for every j ∈ N and | u | small enough, there exists C j > |h φ u k ( T ) , Bφ u j ( T ) i| ≥ C j k for every k ∈ N . In conclusion, the claimis achieved for every k ∈ N and j ≤ N with e C N = min { C j : j ≤ N } .26 emma B.6. Let B be a bounded symmetric operator satisfying Assump-tions I. There exists a neighborhood U (0) of in R contained in the oneintroduced in Lemma B. such that, for any u ∈ U (0) , (cid:16) ∞ X j =1 (cid:12)(cid:12) | λ u j | h φ u j , ·i (cid:12)(cid:12) (cid:17) ≍ (cid:16) ∞ X j =1 | j h φ j , ·i| (cid:17) . Proof.
Let D be the neighborhood provided by Lemma B.
4. For | u | smallenough, we prove that there exist C > k| A + u B | s ψ k ≤ C k| A | s ψ k for s = 3. We start with s = 4 and we recall that B ∈ L ( H )thanks to Remark 1 .
1. For any ψ ∈ H , there exists C > k ( A + u B ) ψ k ≤ k A ψ k + | u | k B ψ k + | u |k Aψ k ( ||| B ||| (2) + ||| B ||| ) ≤ C k| A | ψ k . The proof of [BdCC13 , Lemma
1] implies the validity of the relation alsofor s = 3. There exists C > k ψ k e H = k| A + u B | ψ k ≤ C k| A | ψ k = C k ψ k H for every ψ ∈ H . Now, H = D ( | A | ) = D ( | A + u B | ) = e H and B preserves e H since B : H −→ H . The argumentsof Remark 1 . B ∈ L ( e H ) and the opposite inequality followsas above thanks to the identity A = ( A + u B ) − u B . Remark B.7.
Let B be a bounded symmetric operator satisfying Assump-tions I. The techniques of the proof of Lemma B. also allow to prove that,for s ∈ (0 , , there exists a neighborhood U (0) of in R such that, for any u ∈ U (0) , it follows (cid:16) P ∞ j =1 (cid:12)(cid:12) ( λ u j ) s h φ u j , ·i (cid:12)(cid:12) (cid:17) ≍ (cid:16) P ∞ j =1 | j s h φ j , ·i| (cid:17) . Lemma B.8.
Let B be a bounded symmetric operator satisfying Assump-tions I and N ∈ N . Let ǫ > small enough and I N be the set defined in (5) .There exists a U ǫ ⊂ R \ { } such that, for each u ∈ U ǫ , inf ( j,k ) , ( n,m ) ∈ IN ( j,k ) =( n,m ) | λ u j − λ u k − λ u n + λ u m | > ǫ. Moreover, for every δ > small there exists ǫ > such that dist ( U ǫ , < δ. Proof.
Let us consider the neighborhood D provided by Lemma B.
3. Themaps λ uj − λ uk − λ un + λ um are analytic for each j, k, n, m ∈ N and u ∈ D . Onecan notice that the number of elements such that(39) λ j − λ k − λ n + λ m = 0 , j, n ∈ N , k, m ≤ N is finite. Indeed λ k = k π and (39) corresponds to j − k = n − m . Wehave | j − n | = | k − m | ≤ N − , which is satisfied for a finite numberof elements. Thus, for I N (defined in (5), the following set is finite R := { (( j, k ) , ( n, m )) ∈ ( I N ) : ( j, k ) = ( n, m ); λ j − λ k − λ n + λ m = 0 } . ) Let (( j, k ) , ( n, m )) ∈ R , the set V ( j,k,n,m ) = { u ∈ D (cid:12)(cid:12) λ uj − λ uk − λ un + λ um = 0 } is a discrete subset of D or equal to D . Thanks to the relation (30), λ uj − λ uk − λ un + λ um = | a j | λ j + u | a j | B j,j + λ uj k η j k + ua j h P ⊥ φ j Bφ j , η j i − | a k | λ k − u | a k | B k,k − λ uk k η k k − ua k h P ⊥ φ k Bφ k , η k i − | a n | λ n − u | a n | B n,n − λ un k η n k − ua n h P ⊥ φ n Bφ n , η n i + | a m | λ m + u | a m | B m,m + λ um k η m k + ua m h P ⊥ φ m Bφ m , η m i = ⇒ λ uj − λ uk − λ un + λ um = | a j | λ j − | a k | λ k − | a n | λ n + | a m | λ m + ( | a j | B j,j − | a k | B k,k − | a n | B n,n + | a m | B m,m ) u + o ( u ) . (40)For | u | small enough, thanks to lim | u |→ | a j | = 1 and to the third point ofAssumptions I, λ uj − λ uk − λ un + λ um can not be constantly equal to 0. Then, V ( j,k,n,m ) is discrete and V = { u ∈ D (cid:12)(cid:12) ∃ ( j, k, n, m ) ∈ R : λ uj − λ uk − λ un + λ um =0 } is a discrete subset of D . As R is a finite set e U ǫ := { u ∈ D : ∀ ( j, k, n, m ) ∈ R (cid:12)(cid:12) | λ uj − λ uk − λ un + λ um | ≥ ǫ } has positive measure for ǫ > δ > ǫ > dist (0 , e U ǫ ) < δ. Let (( j, k ) , ( n, m )) ∈ ( I N ) \ R be different numbers. We know that | λ j − λ k − λ n + λ m | = π | j − k − n + m | > π . First, thanks to (30), wehave λ uj ≤ | a j | λ j + | u | C and λ uj ≥ | a j | λ j − | u | C for suitable constants C , C > j . Thus | λ uj − λ uk − λ un + λ um | ≥ || a j | λ j − | a k | λ k − | a n | λ n + | a m | λ m | − | u | (2 C + 2 C ) . Now, lim k →∞ | a k | = 1. For any u in D and ǫ small enough, there exists M ǫ ∈ N such that || a j | λ j − | a k | λ k − | a n | λ n + | a m | λ m | ≥ π − ǫ forevery (( j, k ) , ( n, m )) ∈ R C := ( I N ) \ R and j, k, n, m ≥ M ǫ . Howeverlim | u |→ | a k | = 1 uniformly in k thanks to (31) and then there exists aneighborhood W ǫ ⊆ D such that, for each u ∈ W ǫ , it follows || a j | λ j −| a k | λ k − | a n | λ n + | a m | λ m | ≥ π − ǫ for every (( j, k ) , ( n, m )) ∈ R C and1 ≤ j, k, n, m < M ǫ . Thus, for each u ∈ W ǫ and (( j, k ) , ( n, m )) ∈ R C suchthat ( j, k ) = ( n, m ), we have | λ uj − λ uk − λ un + λ um | ≥ π − ǫ. The proof is achieved since, for ǫ > e U ǫ ∩ W ǫ is a non-zero measure subset of D . For any u ∈ e U ǫ ∩ W ǫ and for any (( j, k ) , ( n, m )) ∈ ( I N ) such that ( j, k ) = ( n, m ), we have | λ uj − λ uk − λ un + λ um | ≥ min { π − ǫ, ǫ } . Remark B.9.
Let B be a bounded symmetric operator satisfying Assump-tions I. By using the techniques of the proofs of Lemma B. and Lemma B. , one can ensure the existence of a neighborhood U of u in R and U ,a countable subset of R such that, for any u ∈ U (0) := ( U \ U ) \ { } , wehave:1. For every N ∈ N , ( j, k ) , ( n, m ) ∈ I N (see (5) ) such that ( j, k ) = ( n, m ) ,there holds λ u j − λ u k − λ u n + λ u m = 0 . . B u j,k = h ψ u j ( T ) , Bφ u k ( T ) i 6 = 0 for every j, k ∈ N .
3. For ǫ > , if | u | is small enough, then sup j ∈ N k φ j − φ u j k (3) ≤ ǫ. Remark B.10.
Let B be a bounded symmetric operator satisfying Assump-tions II and Assumptions A. As Remark B. , there exists a neighborhood U of u in R and U , a countable subset of R containing u = 0 such that, forany u ∈ U (0) := ( U \ U ) \ { } and N ∈ N , the numbers { } ∪ (cid:8) λ u j (cid:9) j ≤ N are rationally independent. Indeed, we denote x u j,M := B (cid:0) ( λ u j − A ) (cid:12)(cid:12) φ ⊥ j (cid:1) − (cid:16)(cid:0) ( λ u j − A ) (cid:12)(cid:12) φ ⊥ j (cid:1) − P ⊥ φ j B (cid:17) M P ⊥ φ j B, ∀ j, M ∈ N . As (1 − k η j k ) = | α j | for every j ∈ N , by using (29) in (30) , for | u | small, λ u j = | α j | λ j − k η j k + u | α j | B j,j − k η j k − u | α j | − k η j k D P ⊥ φ j Bφ j , (( A + u P ⊥ φ j B − λ u j ) (cid:12)(cid:12) φ ⊥ j ) − u P ⊥ φ j Bφ j E = λ j + u B j,j + u D φ j , + ∞ X M =0 (cid:0) u M x u j,M (cid:1) φ j E . Let x j,M = h φ j , e B ( M, j ) φ j i with e B ( M, j ) defined in Assumptions A and j, M ∈ N . We have lim | u |→ x u j,M = x j,M . Let M ≤ N and r := { r j } j ≤ M ∈ Q M \ M . Thanks to Assumptions A, the map u r + P Mj =2 r j λ uj isnon-constant and analytic. The set V r of its positive zeros is discrete. Theproperty is valid for U := ∪ M ≤ N ∪ r ∈ Q M \ M V r that is discrete. References [BCCS12] U. Boscain, M. Caponigro, T. Chambrion, and M. Sigalotti. Aweak spectral condition for the controllability of the bilinearSchr¨odinger equation with application to the control of a ro-tating planar molecule.
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