Efficient finite dimensional approximations for the bilinear Schrodinger equation with bounded variation controls
aa r X i v : . [ m a t h . A P ] J un Efficient finite dimensional approximations for the bilinear Schr ¨odingerequation with bounded variation controls
Nabile Boussa¨ıd ∗ , Marco Caponigro † and Thomas Chambrion ‡∗ Laboratoire de math´ematiques, Universit´e de Franche–Comt´e, 25030 Besanc¸on, France
[email protected] † ´Equipe M2N, Conservatoire National des Arts et M´etiers, 75003 Paris, France [email protected] ‡ Universit´e de Lorraine, Institut ´Elie Cartan de Lorraine, UMR 7502, Vandoeuvre-l`es-Nancy,F-54506, FranceCNRS, Institut ´Elie Cartan de Lorraine, UMR 7502, Vandoeuvre-l`es-Nancy, F-54506, FranceInria, CORIDA, Villers-l`es-Nancy, F-54600, France
Abstract — In the present analysis, we consider the control-lability problem of the abstract Schr¨odinger equation : ∂ t ψ = Aψ + uBψ where A is a skew-adjoint operator, B a control potential and u is the control command.We are interested by approximation of this equation by finitedimensional systems.Assuming that A has a pure discrete spectrum and B isin some sense regular with respect to A we show that suchan approximation is possible. More precisely the solutionsare approximated by their projections on finite dimensionalsubspaces spanned by the eigenvectors of A .This approximation is uniform in time and in the control u , if this control has bounded variation with a priori boundedtotal variation. Hence if these finite dimensional systems arecontrollable with a fixed bound on the total variation of u thenthe system is approximatively controllable.The main outcome of our analysis is that we can buildsolutions for low regular controls u such as bounded variationones and even Radon measures. I. E
XTENDED ABSTRACT a) The wellposedness:
Let H be a separable Hilbertspace (possibly infinite dimensional) with scalar product h· , ·i and k · k the corresponding norm, A, B be two (possiblyunbounded) skew-symmetric operators on H . We considerthe formal bilinear control system ddt ψ ( t ) = Aψ ( t ) + u ( t ) Bψ ( t ) , (1)where the scalar control u is to be chosen in a set of realfunctions.For any real interval I , we define ∆ I := { ( s, t ) ∈ I | s ≤ t } . Definition : Propagator on a Hilbert space
Let I be a real interval. A family ( s, t ) ∈ ∆ I X ( s, t ) of linear contractions, that is Lipschitz maps with Lipschitzconstant less than one, on a Hilbert space H , stronglycontinuous in t and s and such that1) for any s < r < t , X ( t, s ) = X ( t, r ) X ( r, s ) , 2) X ( t, t ) = I H ,is called a contraction propagator on H .Let us now fix some scalar function u : I R and define A ( t ) = A + u ( t ) B .Recall that a family t ∈ I U ( t ) ∈ E , E a subset of aBanach space X , is in BV ( I, E ) if there exists N ≥ suchthat n X j =1 k U ( t j ) − U ( t j − ) k X ≤ N for any partition a = t < t < . . . < t n = b of the interval ( a, b ) . The mapping U ∈ BV ( I, E ) sup a = t Let I be a real interval and D dense subset of H A ( t ) is a maximal skew-symmetric operator on H withdomain D ,2) t A ( t ) has bounded variation from I to L ( D , H ) ,where D is endowed with the graph topology associ-ated to A ( a ) for a = inf I ,3) M := sup t ∈ I (cid:13)(cid:13) (1 − A ( t )) − (cid:13)(cid:13) L ( H , D ) < ∞ , We do not assume t A ( t ) to be continuous. However asa consequence of Assumption 2 (see [Edw57, Theorem 3])it admits right and left limit in L ( D , H ) , A ( t − 0) =lim ε → + A ( t − ε ) , A ( t + 0) = lim ε → + A ( t + ε ) , for all t ∈ I , and A ( t − 0) = A ( t + 0) for all t ∈ I except acountable set.The the core of our analysis is the following result dueto Kato (see [Kat53, Theorem 2 and Theorem 3]). It givessufficient conditions for the well-posedness of the system (1). The bounded variation of t A ( t ) ensures that any choice of s ∈ I will be equivalent. heorem 1: If t ∈ I A ( t ) satisfies the above as-sumptions, then there exists a unique contraction propagator X : ∆ I → L ( H ) such that if ψ ∈ D then X ( t, s ) ψ ∈ D and for ( t, s ) ∈ ∆ I k A ( t ) X ( t, s ) ψ k ≤ M e M k A k BV ( I,L ( D , H )) k A ( s ) ψ k . and in this case X ( t, s ) ψ is strongly left differentiable in t and right differentiable in s with derivative (when t = s ) A ( t + 0) ψ and − A ( t − ψ respectively.In the case in which t A ( t ) is continuous and skew-adjoint, if ψ ∈ D then t ∈ ( s, + ∞ ) X ( t, s ) ψ isstrongly continuously differentiable in H with derivative A ( t ) X ( t, s ) ψ .This theorem addresses the problem of existence of solutionfor the kind of non-autonomous system we consider hereunder very mild assumptions on the control command u .For instance we consider bounded variation controls. Undersome additional assumptions such as the boundedness of thecontrol potential, we can also consider Radon measures.Let us insist on the quantitative aspect of the theorem asit provides an estimate on the growth of the solution in thenorm associated with A . This is quantitative aspect is thestarting point of the subsequent comments. b) Some supplementary regularity: The regularity ofthe solution with respect to the natural structure of theproblem is considered now. In that respect we can adopttwo complementary strategies :1) the regularity of the input-output mapping, the flow ofthe problem, is obtained by proving that the controlpotential if it is regular enough do not alter theregularity properties of the uncontrolled problem.2) the regularity can be added to the functional setting ofthe wellposedness problem; namely we solve the prob-lem imposing regularity to the constructed solution.For the first strategy we introduce the following definition. Definition : Weakly coupled Let k be a non negative real. A couple of skew-adjoint operators ( A, B ) is k -weakly coupled if1) A is invertible with bounded inverse from D ( A ) to H ,2) for any real t , e tB D ( | A | k/ ) ⊂ D ( | A | k/ ) ,3) there exists c ≥ and c ′ ≥ such that B − c and − B − c ′ generate contraction semigroups on D ( | A | k/ ) for the norm ψ 7→ k| A | k/ u k .We set, for every positive real k , k ψ k k/ = q h| A | k ψ, ψ i . The optimal exponential growth is defined by c k ( A, B ) := sup t ∈ R log k e tB k L ( D ( | A | k/ ) ,D ( | A | k/ ) | t | . Theorem 2: Let k be a non negative real. Let ( A, B ) be k -weakly coupled .For any u ∈ BV ([0 , T ] , R ) ∩ B L ∞ ([0 ,T ]) (0 , / k B k A ) ,there exists a family of contraction propagators in H thatextends uniquely as contraction propagators to D ( | A | k/ ) : Υ u : ∆ [0 ,T ] → L ( D ( | A | k/ )) such that 1) for any t ∈ [0 , T ] , for any ψ ∈ D ( | A | k/ ) k Υ t ( ψ ) k k/ ≤ e c k ( A,B ) | R t u |k ψ k k/ 2) for any t ∈ [0 , T ] , for any ψ ∈ D ( | A | k/ ) for any u ∈ BV ([0 , T ] , R ) ∩ B L ∞ ([0 ,T ]) (0 , / k B k A )) , thereexists m (depending only on A , B and k u k L ∞ ([0 ,T ]) ) k Υ t ( ψ ) k k/ ≤ me m k u k BV ([0 ,T ] , R ) ×× e c k ( A,B ) | R t u |k ψ k k/ Moreover, for every ψ in D ( | A | k/ ) , the end-point mapping Υ( ψ ) : BV ([0 , T ] , K ) → D ( | A | k/ ) u Υ u (0 , T )( ψ ) is continuous.As announced before these theorem the two strategies wereused in complement to establish the second point of thetheorem.There is several outcomes to our analysis. First eachattainable target from an initial state has to be as regularas the initial state and the control potential allows it to be.For instance if we consider the harmonic potential for theShr¨odinger equation and we try to control it by a smoothbounded potential then from any initial eigenvector onecannot attain non-smooth non exponentially decaying states. c) A negative result: An auxiliary result of our analysisis an immediate generalisation of the famous negative resultby Ball, Marsden and Slemrod [BMS82] that the attainableset is included in a countable union of compact sets for theinitial Hilbert setting of the problem for integrable controllaws. We can show that this still holds for much smallerspaces than the initial Hilbert space, for instance the domainsand iterated domains of the uncontrolled problem, for muchless regular controls such as bounded variation function oreven Radon measures. Theorem 3: Let k be a non negative real. Let ( A, B ) be k -weakly coupled . Let ψ ∈ D ( | A | k/ ) .Then [ L,T,a> (cid:8) α Υ u ( ψ ) , k u k BV ([0 ,T ] , R ) ≤ L, t ∈ [0 , T ] , | α | ≤ a (cid:9) is a meagre set (in the sense of Baire) in L ∞ ( I, D ( | A | k/ ) as a union of relatively compact subsets. d) Gallerkin approximation for bounded control poten-tial: In a practical setting it is clear that this negative resultis useless. Nonetheless we consider that such a negativeresult has a philosophical consequence, natural systems forwhich regularity of the bounded potential is natural cannotbe exactly controllable. such an observation is even moredramatic if one considers systems with continuous spectrum.Once such a comment is made, practical questions re-mains, one of them is the question of the approximatecontrollability (see [BCC13, Definition 1]), we choose hereto give a corollary of our analysis that can be helpfull whenone want to tackle this issue.or every Hilbert basis Φ = ( φ k ) k ∈ N of H , we define,for every N in N , π Φ N : H → H ψ P j ≤ N h φ j , ψ i φ j . Definition Let ( A, B ) be a couple of unbounded operators and Φ =( φ n ) n ∈ N be an Hilbert basis of H . Let N ∈ N and denote L Φ N = span( φ , . . . , φ N ) . The Galerkin approximation oforder N of system (1), when it makes sense, is the system ˙ x = ( A (Φ ,N ) + uB (Φ ,N ) ) x ( Σ Φ N )where A (Φ ,N ) and B (Φ ,N ) , defined by A (Φ ,N ) = π Φ N A ↾ L N and B (Φ ,N ) = π Φ N B ↾ L N , are the compressions of A and B (respectively) associatedwith L N .Notice that if A is skew-adjoint and Φ an Hilbert basismade of eigenvectors of A then ( A (Φ ,N ) , B (Φ ,N ) ) satisfiesthe same assumptions as ( A, B ) . We can therefore definethe contraction propagator X u (Φ ,N ) ( t, of ( Σ Φ N ) associatedwith bounded variation control u . We can also write that ( A (Φ ,N ) , B (Φ ,N ) ) is k -weakly coupled for any prositive real k as the weak coupling is actually invariant or at least doesnot deteriorate by compression with respect to a basis ofeigenvectors of A .In the case of bounded potentials we can state the fol-lowing proposition. Below we consider B be bounded in D ( | A | k/ ) which implies that ( A, B ) is k -weakly coupled . Theorem 4: Let k be a positive real. Let A with domain D ( A ) be the generator of a contraction semigroup and let B be bounded in H and D ( | A | k/ ) with B (1 − A ) − compact.Let s be non-negative numbers with ≤ s < k . Then forevery ε > , L ≥ , n ∈ N , and ( ψ j ) ≤ j ≤ n in D ( | A | k/ ) n there exists N ∈ N such that for any u ∈ R ((0 , T ]) , | u | ([0 , T ]) < L ⇒ k Υ ut ( ψ j ) − X u ( N ) ( t, π N ψ j k s/ < ε, for every t ≥ and j = 1 , . . . , n .Hence if these finite dimensional systems are controllablewith a fixed bound on the total variation of u then the systemis approximatively controllable. e) An example Smooth potentials on compact mani-folds: This example motivated the present analysis becauseof its physical importance. We consider Ω a compact Rie-mannian manifold endowed with the associated Laplace-Beltrami operator ∆ and the associated measure µ , V, W :Ω → R two smooth functions and the bilinear quantumsystem i ∂ψ∂t = ∆ ψ + V ψ + u ( t ) W ψ. (2)With the previous notations, H = L (Ω , C ) endowed withthe Hilbert product h f, g i = R Ω ¯ f g d µ , A = − i(∆ + V ) and B = − i W . For every r ≥ , D ( | A | r ) = H r (Ω , C ) . Thereexists a Hilbert basis ( φ k ) k ∈ N of H made of eigenvectors of A . Each eigenvalue of A has finite multiplicity. For every k , there exists λ k in R such that Aφ k = i λ k φ k . The sequence ( λ k ) k tends to + ∞ and, up to a reordering, is non decreasing.Since B is bounded from D ( | A | k ) to D ( | A | k ) , ( A, B, R ) satisfies our assumptions and ( A, B ) is k -weakly coupled fornon negative real k . R EFERENCES[BCC13] N. Boussaid, M. Caponigro, and T. Chambrion. Total variation ofthe control and energy of bilinear quantum systems. In Decisionand Control (CDC), 2013 IEEE 52nd Annual Conference on ,pages 3714–3719, Dec 2013.[BMS82] J. M. Ball, J. E. Marsden, and M. Slemrod. Controllability fordistributed bilinear systems. SIAM J. Control Optim. , 20(4):575–597, 1982.[Edw57] D. A. Edwards. On the continuity properties of functionssatisfying a condition of Sirvint’s. Quart. J. Math. Oxford Ser.(2) , 8:58–67, 1957.[Kat53] Tosio Kato. Integration of the equation of evolution in a Banachspace.