Ugo Boscain
email: ugo.boscain@polytechnique.edu

Marco Caponigro
email: marco.caponigro@cnam.fr

Mario Sigalotti
email: mario.sigalotti@inria.fr

Multi-input Schrödinger equation: controllability

Keywords: controllability, tracking, and application to the quantum angular momentum, tracking, and application to the quantum angular momentum Quantum control, bilinear Schrödinger equation, Galerkin approximations, quantum angular momentum

published or not. The documents may come

Introduction

In this paper we study the controllability and the tracking problem for the multi-input bilinear Schrödinger equation

i dψ dt (t) = (H 0 + u 1 (t)H 1 + . . . + u p (t)H p )ψ(t) (1)
where H 0 , . . . , H p are self-adjoint operators on a Hilbert space H and the drift Schrödinger operator H 0 (the internal Hamiltonian) has discrete spectrum. The control functions u 1 (•), . . . , u p (•), representing external fields, are real-valued and ψ(•) takes values in the unit sphere of H. The controllability of system (1) is a well-established topic when the state space H is finite-dimensional (see for instance [D'A08] and reference therein), thanks to general controllability methods for left-invariant control systems on compact Lie groups ([Bro72, JS72, JK81, GB82, EAGK96]).

When H is infinite-dimensional and the operators H 1 , . . . , H p are bounded, it is known that the bilinear Schrödinger equation is not exactly controllable (see [START_REF] Ball | Controllability for distributed bilinear systems[END_REF][START_REF] Turinici | On the controllability of bilinear quantum systems[END_REF]). Nevertheless, weaker controllability properties, such as approximate controllability or controllability between eigenstates of the internal Hamiltonian H 0 (which are the most relevant physical states), may hold. In certain cases, when H is a function space on a subset of R, a description of reachable sets has been provided (see [BC06,[START_REF] Beauchard | Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control[END_REF]). In R d , d > 1, or for more general situations, the exact description of the reachable set seems a difficult task and at the moment only approximate controllability results are available. Most of them have been proved in the single-input case (see, in particular, [CMSB09, Mir09, Ner09, Ner10, BN10, BCCS12, NN12]). They are based on sufficient conditions for controllability that are generic [PS10, [START_REF] Mason | Generic controllability properties for the bilinear Schrödinger equation[END_REF][START_REF] Vahagn Nersesyan | Global approximate controllability for Schrödinger equation in higher Sobolev norms and applications[END_REF] even in the case p = 1. Nevertheless, in many examples these conditions cannot be directly applied or controllability fails to hold, as a consequence of the symmetries of the system. Symmetries can induce degeneracies in the spectrum (e.g. multiple eigenvalues or presence of identical spectral gaps) and reduce the coupling of eigenstates via the control. This happens, for instance, for a planar rotating molecule controlled by one external field only [BCCS12, Section 8], which is not (approximately) controllable.

Using more than one input opens new possibility for control. Multi-input controllability results have been obtained for specific systems [EP09, BBR10] and some general approximate controllability results between eigenfunctions have been proved via adiabatic methods [AB05, BCMS12]. The first multi-input result via Lie-algebraic methods is given in [START_REF] Boscain | A weak spectral condition for the controllability of the bilinear Schrödinger equation with application to the control of a rotating planar molecule[END_REF]Section 8], where the spectral degeneracies of the planar rotating molecule have been tackled by associating with every 1-dimensional slice of the set of admissible controls an invariant subspace of the state space H on which the single-input controllability result applies. However, such a technique does not apply in more general cases. In the case of a rotating rigid symmetric 3D molecule, the application of this method is obstructed by the fact that eigenspaces may have arbitrarily large dimension.

In this paper, we present a sufficient condition for controllability of the discrete-spectrum bilinear Schrödinger equation which applies even when the spectrum of the internal Hamiltonian H 0 is very degenerate. The results fully exploit the presence of more than one control and extend to simultaneous controllability, approximate controllability in H s , and tracking in modulus (or mtracking; for precise definitions see Section 2). Proving that a system is an m-tracker is a crucial issue when dealing with dissipative levels. A common strategy is to neglect the dissipativity of the level in the mathematical model and to keep its population as low as possible during the transition (see for instance the STIRAP model [CH90, VHBB01, BCG + 02]).

The result presented in this paper is more general than those in the literature even in the single-input case. Consider, for instance, the Laplace-Dirichlet operator on a compact interval Ω of R with a control term of the type (H 1 ψ)(x) = xψ(x), x ∈ Ω (see [BC06]): in [START_REF] Boscain | A weak spectral condition for the controllability of the bilinear Schrödinger equation with application to the control of a rotating planar molecule[END_REF], approximate simultaneous controllability of this model has been proved by breaking the degeneracies between spectral gaps through perturbation techniques. Here we prove the approximate simultaneous controllability and m-tracking without perturbation arguments. The advantage is that the constructive proof of the main result translates into an explicit motion-planning algorithm [START_REF] Caponigro | Software registered at[END_REF].

Brief description of the general results

The main result of the paper is a sufficient condition for approximate simultaneous controllability which we call the Lie-Galerkin Control Condition (see Definition 2.5).

Roughly speaking, both the sufficient condition proposed in [START_REF] Boscain | A weak spectral condition for the controllability of the bilinear Schrödinger equation with application to the control of a rotating planar molecule[END_REF] and the one presented here are based on the idea of driving the system with control laws that are in resonance with spectral gaps of the internal Hamiltonian H 0 . However, while in [START_REF] Boscain | A weak spectral condition for the controllability of the bilinear Schrödinger equation with application to the control of a rotating planar molecule[END_REF] the only actions on the system obtained by resonance that are exploited for the controllability are those corresponding to elementary transitions between two eigenstates, no such a restriction is imposed in the Lie-Galerkin Control Condition (see Section 2.5).

The Lie-Galerkin Control Condition ensures strong controllability properties for the Galerkin approximations: it provides controllability for a fixed Galerkin approximation while avoiding the transfer of population to higher energy levels for higher-order Galerkin approximations. This yields estimates on the difference between the dynamics of the finite-dimensional Galerkin approximation and the original infinite-dimensional system. The Lie-Galerkin Control Condition also ensures a bound on the L 1 norm of the control achieving controllability between finite combinations of eigenstates, which is uniform with respect to the prescribed tolerance.

Under the Lie-Galerkin Tracking Condition, a slight modification of the Lie-Galerkin Control Condition, we can prove that any trajectory can be tracked in modulus (see Theorem 2.8).

Following [START_REF] Boussaïd | Weakly-coupled systems in quantum control[END_REF], the Lie-Galerkin Control Condition, under the additional assumption that the system is s-weakly coupled (see Definition 2.11), implies that the system is approximately controllable in H s/2 (see Theorem 2.12).

Application to the quantum angular momentum

Rotational molecular dynamics constitute one of the most important examples of quantum systems on an infinite-dimensional Hilbert space and with discrete spectrum. Molecular orientation and alignment are well-established topics in the quantum control of molecular dynamics both from the experimental and the theoretical point of view (see [SS03, SKA + 04, SH06] and references therein). For linear molecules driven by linearly polarized laser fields in gas phase, alignment means an increased probability direction along the polarization axis whereas orientation requires in addition the same direction as the polarization vector. A large amount of numerical simulations have been carried on in this area but the mathematical part is not yet fully understood.

We focus in this paper on the control of the orientation of a rigid linear molecule in R 3 by external fields. The corresponding controlled Schrödinger equation is defined on the unit sphere S 2 . We show that the system driven by three fields along the three axes is approximately controllable for arbitrarily small controls.

Up to normalization of physical quantities (in particular, in units such that = 1), the dynamics are modeled by the equation

i ∂ψ(θ, ϕ, t) ∂t = -∆ψ(θ, ϕ, t)+ + (u 1 (t) sin θ cos ϕ + u 2 (t) sin θ sin ϕ + u 3 (t) cos θ)ψ(θ, ϕ, t), (2)
where θ, ϕ are the spherical coordinates, which are related to the Euclidean coordinates through the identities x = sin θ cos ϕ, y = sin θ sin ϕ, z = cos θ, while ∆ is the Laplace-Beltrami operator on the sphere S 2 (called in this context the angular momentum operator ), i.e.,

∆ = 1 sin θ ∂ ∂θ sin θ ∂ ∂θ + 1 sin 2 θ ∂ 2 ∂ϕ 2 .
The wavefunction ψ evolves in the unit sphere S of H = L 2 (S 2 , C). As a consequence of the general multi-input result presented in Section 2 we have that (2) is approximately controllable with arbitrarily small controls. A stronger statement, including simultaneous controllability in H s and tracking in modulus, is given in Section 3.

Theorem 1.1. For every ψ 0 , ψ 1 belonging to S and every ε, δ > 0, there exist T > 0 and u ∈ L ∞ ([0, T ], [0, δ] 3 ), such that the solution ψ(•) of equation (2), corresponding to the control u and with initial condition ψ 0 , satisfies ψ 1ψ(T ) < ε.

There are two main difficulties preventing the application of previous results in the literature to this system. Firstly, we deal here with several control parameters, while those general results were specific to the single-input case. Notice that, because of symmetry obstructions, equation (2) is not controllable with only two of the three controls u 1 , u 2 , u 3 . Secondly, the general theory developed in [START_REF] Chambrion | Controllability of the discrete-spectrum Schrödinger equation driven by an external field[END_REF][START_REF] Vahagn Nersesyan | Global approximate controllability for Schrödinger equation in higher Sobolev norms and applications[END_REF][START_REF] Boscain | A weak spectral condition for the controllability of the bilinear Schrödinger equation with application to the control of a rotating planar molecule[END_REF] is based on nonresonance conditions on the spectrum of the internal Hamiltonian. The Laplace-Beltrami operator on S 2 , however, has a severely degenerate spectrum, since the -th eigenvalue -i ( + 1) has multiplicity 2 + 1. A perturbation technique has been proposed in [START_REF] Chambrion | Controllability of the discrete-spectrum Schrödinger equation driven by an external field[END_REF], in order to overcome resonance relations in the spectrum of the drift. The technique was applied in [BCM + 09] to the case of the orientation of a molecule confined in a plane driven by one control. The planar case is already technically challenging and a generalization of the same technique to the case of three controls in 3D seems hard to achieve.

Structure of the paper

The paper is organized as follows: in the next section we present the general multi-input abstract framework and the main abstract results (Theorems 2.6, 2.8, and2.12). In Section 3 we apply them to system (2). The proofs of Theorems 2.6, 2.8, and 2.12 are contained, respectively, in Sections 4, 5, and 6.

Framework and main results

Let p ∈ N, δ > 0, and

U = U 1 ו • •×U p with either U j = [0, δ] or U j = [-δ, δ].
Definition 2.1. Let H be an infinite-dimensional Hilbert space with scalar product •, • and A, B 1 , . . . , B p be (possibly unbounded) skew-adjoint operators on H, with domains D(A), D(B 1 ), . . . , D(B p ). Let us introduce the controlled equation

dψ dt (t) = (A + u 1 (t)B 1 + • • • + u p (t)B p )ψ(t), u(t) ∈ U. (3)
We say that A satisfies (A1) if the following assumption is true:

(A1) A has discrete spectrum with infinitely many distinct eigenvalues (possibly degenerate).

Denote by Φ a Hilbert basis (φ k ) k∈N of H made of eigenvectors of A associated with the family of eigenvalues (iλ k ) k∈N and let L be the set of finite linear combinations of eigenstates, that is,

L = k∈N span{φ 1 , . . . , φ k }.
We say that (A, B 1 , . . . , B p , U, Φ) satisfies (A) if A satisfies (A1) and the following assumptions hold:

(A2) φ k ∈ D(B j ) for every k ∈ N, j = 1, . . . , p; (A3) A + u 1 B 1 + • • • + u p B p : L → H is essentially skew-adjoint for every u ∈ U . If (A, B 1 , . . . , B p , U, Φ) satisfies (A) then, for every (u 1 , . . . , u p ) ∈ U , A + u 1 B 1 + • • • + u p B p
generates a subgroup e t(A+u1B1+•••+upBp) of the group of unitary operators U(H). It is therefore possible to define the propagator Υ u T at time T of system (2) associated with a p-uple of piecewise constant controls u(•) = (u 1 (•), . . . , u p (•)) by composition of flows of the type e t(A+u1B1+•••+upBp) . If, moreover, B 1 , . . . , B p are bounded operators then the definition can be extended by continuity to every L ∞ control law (see [BMS82, Theorem 2.5]). Definition 2.2. Let (A, B 1 , . . . , B p , U, Φ) satisfy (A). We say that (3) is approximately controllable if for every ψ 0 , ψ 1 in the unit sphere of H and every ε > 0 there exists a piecewise constant control function u : [0, T ] → U such that ψ 1 -Υ u T (ψ 0 ) < ε. Definition 2.3. Let (A, B 1 , . . . , B p , U, Φ) satisfy (A). We say that (3) is approximately simultaneously controllable if for every r in N, ψ 1 , . . . , ψ r in H, Υ in U(H), and ε > 0 there exists a piecewise constant control u :

[0, T ] → U such that Υψ k -Υ u T ψ k < ε, k = 1, . . . , r.
If, moreover, for every ψ 1 , . . . , ψ r ∈ L and Υ ∈ U(H) such that Υψ 1 , . . . , Υψ r ∈ L, there exists K > 0 (not depending on ε) such that u can be chosen to satisfy, in addition, u L 1 ≤ K, we say that (3) is L 1 -bounded approximately simultaneously controllable.

This last definition of controllability with a priori bound on the L 1 -norm of the control achieving controllability has been observed in preceding works [START_REF] Boscain | A weak spectral condition for the controllability of the bilinear Schrödinger equation with application to the control of a rotating planar molecule[END_REF][START_REF] Chambrion | Periodic excitations of bilinear quantum systems[END_REF]. It implies a stronger controllability property as discussed in Section 2.4.

Because of the presence of a drift (the internal Hamiltonian) and of the boundedness of the controls, it is not possible in general to track, with arbitrarily precision, an unfeasible curve in S. We introduce the notion of an m-tracker, that is, a system for which any given curve can be tracked up to phases (both for a single initial condition and in the spirit of simultaneous control). This definition makes sense from the physical point of view, since tracking up to phases means imposing the population of all energy levels of H 0 along the evolution.

The identification up to phases of elements of H in the basis Φ = (φ k ) k∈N can be accomplished by the projection

M : ψ → k∈N | φ k , ψ |φ k .
Definition 2.4. Let (A, B 1 , . . . , B p , U, Φ) satisfy (A). We say that (3) is a mtracker if, for every r in N, ψ 1 , . . . , ψ r in H, Υ : [0, T ] → U(H) continuous with Υ0 = Id H , and ε > 0, there exist an invertible increasing continuous function

τ : [0, T ] → [0, T τ ] and a piecewise constant control u : [0, T τ ] → U such that M( Υt ψ k ) -M(Υ u τ (t) ψ k ) < ε, k = 1, . . . , r, for every t ∈ [0, T τ ].

Notation

For every n in N, define the orthogonal projection

π n : H ψ → n k=1 φ k , ψ φ k ∈ H.
Given a linear operator Q on H we identify the linear operator π n Qπ n preserving span{φ 1 , . . . , φ n } with its n × n complex matrix representation with respect to the basis (φ 1 , . . . , φ n ). We define

A (n) = π n Aπ n and B (n) j = π n B j π n ,
for every j = 1, . . . , p.

Let us introduce the set Σ n of spectral gaps associated with the n-dimensional Galerkin approximation as

Σ n = {|λ l -λ k | | l, k = 1, . . . , n}.
For every σ ≥ 0, every m ∈ N, and every m × m matrix M , let

E σ (M ) = (M l,k δ σ,|λ l -λ k | ) m l,k=1 ,
where

δ j,k = 1 if and only if j = k and δ j,k = 0 otherwise. The n × n matrix E σ (B (n) j ), j = 1, . . . , p, corresponds to the "activation" in B (n) j of the spectral gap σ: every element is 0 except the (l, k)-elements such that |λ l -λ k | = σ.
(It reflects the action of the convexification procedure detailed in Section 4.4, which sets to zero all the matrix elements (B

(n) j ) l,k for which |λ l -λ k | = σ.) Define Ξ n = (σ, j) ∈ Σ n × {1, . . . , p} | ∃M ∈ u(n) s.t. E σ (B (N ) j ) = M 0 0 * for every N > n . (4)
The matrices E σ (B (n) j ) for (σ, j) ∈ Ξ n correspond to "compatible dynamics" for the n-dimensional Galerkin approximation (compatible, that is, with higherdimensional Galerkin approximations).

Controllability results

Let

V 0 n = A (n) ∪ E σ (B (n) j ) | (σ, j) ∈ Ξ n and j is such that (0, j) ∈ Ξ n ∪ E σ (B (n) j ) | (σ, j) ∈ Ξ n , σ = 0, U j = [-δ, δ] .
The family V 0 n is obtained by collecting compatible dynamics coming from the convexification procedure mentioned in the previous section. It also contains matrix A (n) obtained by the truncation of the drift A, which is itself the drift of the Galerkin approximation of order n. The collections of the "compatible dynamics" E σ (B (n) j ) is restricted to the indices j such that (0, j) ∈ Ξ n or such that U j = [-δ, δ]. These conditions allow to decouple the contribution of the drift in the convexification procedure, as detailed in Section 4.6.

Definition 2.5. Let (A, B 1 , . . . , B p , U, Φ) satisfy (A). We say that the Lie-Galerkin Control Condition holds if for every n 0 ∈ N there exists n > n 0 such that LieV 0 n ⊇ su(n).

(5)

Theorem 2.6 (Abstract multi-input controllability result). Assume that (A) holds true. If the Lie-Galerkin Control Condition holds then the system

ẋ = (A + u 1 B 1 + • • • + u p B p )x, u ∈ U,
is L 1 -bounded approximately simultaneously controllable.

Tracking results

For every ξ ∈ S 1 ⊂ C, consider the matrix operator J ξ such that

(J ξ (M )) j,k =      ξM j,k if λ j < λ k , 0 if λ j = λ k , ξM j,k if λ j > λ k . Let V n = J ξ (E σ (B (n) j )) | (σ, j) ∈ Ξ n , σ = 0, ξ ∈ S 1 . (6)
Notice that V n ⊂ su(n).

Definition 2.7. Let (A, B 1 , . . . , B p , U, Φ) satisfy (A). We say that the Lie-Galerkin Tracking Condition holds if for every n 0 ∈ N there exists n > n 0 such that

LieV n = su(n). (7)
Theorem 2.8 (Abstract multi-input tracking result). Let U j = [-δ, δ] for some δ > 0 and every j = 1, . . . , p. Assume that (A) holds true. If the Lie-Galerkin Tracking Condition holds then the system

ẋ = (A + u 1 B 1 + • • • + u p B p )x, u ∈ U, is a m-tracker.
Remark 2.9. If U j = [-δ, δ] for every j = 1, . . . , p, then the Lie-Galerkin Tracking Condition implies the Lie-Galerkin Control Condition, as it follows from the relations

A (n) , E σ (B (n) j ) = σJ i (E σ (B (n) j )) and J 1 (E σ (B (n) j )) = E σ (B (n) j ),
for σ = 0.

Controllability in higher norms

We define for s > 0,

|A| s ψ = n∈N |λ n | s φ n , ψ φ n ,
for every ψ belonging to

D(|A| s ) = ψ ∈ H | n∈N |λ n | 2s | φ n , ψ | 2 < +∞ .
For

ε : [0, T ε ] → R such that Υψ j -Υ uε Tε ψ j s < ε,
for every j = 1, . . . , r.

We say that (A, B 1 , . . . , B p , U, Φ) satisfies (A ) if it satisfies (A) and the following additional assumptions hold:

(A4) the operator i(A + u 1 B 1 + • • • + u p B p ) is bounded from below for every u ∈ R p ;
(A5) the sequence (λ k ) k∈N is non-increasing and unbounded.

Definition 2.11. Let (A, B 1 , . . . , B p , U, Φ) satisfy Assumption (A ) and let s > 0.

Then (A, B 1 , . . . , B p ) is s-weakly-coupled if D(|A + u 1 B 1 + • • • + u p B p | s/2 ) = D(|A| s/2
) for every u ∈ R p and there exists C such that

| |A| s ψ, B j ψ | ≤ C| |A| s ψ, ψ |, for every j = 1, . . . , p, ψ ∈ D(|A| s ).
The following result is a consequence of [BCC13, Proposition 2] and can be obtained by adapting the arguments of [BCC13, Proposition 5]. We provide a proof in Section 6.

Theorem 2.12. Let (A, B 1 , . . . , B p , U, Φ) satisfy Assumption (A ) and (A, B 1 , . . . , B p ) be s-weakly coupled for some s > 0. If (3) is L 1 -bounded approximately simultaneously controllable then it is approximately simultaneously controllable for the s/2-norm.

As a direct consequence we have the following result generalizing [BCC13, Proposition 5].

Corollary 2.13. Let (A, B 1 , . . . , B p , U, Φ) satisfy Assumption (A ) and (A, B 1 , . . . , B p ) be s-weakly coupled for some s > 0. If the Lie-Galerkin Control Condition holds then system (3) is approximately simultaneously controllable for the s/2-norm.

Example: the infinite potential well

We present the case of a particle confined in the interval (-π/2, π/2) as a toy model to compare the results in [START_REF] Boscain | A weak spectral condition for the controllability of the bilinear Schrödinger equation with application to the control of a rotating planar molecule[END_REF] and Theorem 2.8 in the case of a single-input system. The model was extensively studied by several authors in the last decade and it has been the first quantum system for which a positive controllability result was obtained (see [BC06]). In [START_REF] Boscain | A weak spectral condition for the controllability of the bilinear Schrödinger equation with application to the control of a rotating planar molecule[END_REF] an approximate simultaneous controllability result was obtained via geometric methods and using perturbations techniques.

The Schrödinger equation reads

i ∂ψ ∂t = - ∂ 2 ψ ∂x 2 -u(t)xψ(x, t), (8)
with the boundary conditions ψ(-π/2, t) = ψ(π/2, t) = 0 for every t ∈ R. The controls u(•) are piecewise constant with values in U = [-δ, δ] for some δ > 0.

In this case H is the space L 2 ((-π/2, π/2), C) endowed with the Hermitian product ψ 1 , ψ 2 = π/2 -π/2 ψ 1 (x)ψ 2 (x)dx. The operators A and B = B 1 are defined by Aψ = i ∂ 2 ψ ∂x 2 for every ψ in D(A) = (H 2 ∩ H 1 0 ) ((-π/2, π/2), C) and Bψ = ixψ. A complete set of eigenfunctions of A associated with the eigenvalues iλ k = -ik 2 , k ∈ N is given by

φ k (x) =    2 π cos(kx) when k is odd, 2 π sin(kx) when k is even, k ∈ N. Notice that φ j , Bφ k = 0, if and only if j + k is odd. In particular J ξ (E σ (B (n) )) = E σ (B (n) ) for every ξ ∈ S 1 , n ∈ N, and σ ∈ Σ n .
We prove by induction on n ≥ 2 that LieV n = su(n), and hence that the Lie-Galerkin Tracking Condition is fulfilled. Notice that, for every k = 2, . . . , n, the spectral gap |λ k -λ k-1 | = 2k -1 belongs to Σ n and that the (j, l)-th element of the matrix E 2k-1 (B (n) ) is zero for j ≤ n and l ≥ n + 1, since

l 2 -j 2 ≥ (n + 1) 2 -n 2 = 2n + 1 > 2k -1. Hence E 2k-1 (B (n) ) ∈ V n for k = 2, . . . , n.
We prove the claim by showing that

Lie {E 2k-1 (B (n) ) | k = 1, . . . , n} = su(n). (9)
For n = 2, the matrices

E 3 (B (2) ) = 0 b 12 -b12 0 and J i (E 3 (B (2) )) generate su(2) because b 12 = 0. Now assume that Lie J ξ ({E 2k-1 (B (n-1) )) | ξ ∈ S 1 , k = 2, . . . , n -1} = su(n -1),
and let us prove (9). The matrices

J ξ (E 2k-1 (B (n) )), ξ ∈ S 1 , k = 2, . . . , n -1,
generate the subalgebra of matrices in su(n) with zero elements in the n-th row and n-th column. In particular, there exists

M ∈ Lie {J ξ (E 2k-1 (B (n) )) | ξ ∈ S 1 , k = 2, . . . , n -1} such that M + E 2n-1 (B (n)
) has the only two nonzero elements in the positions

(n -1, n) and (n, n -1). So Lie({J ξ (E 2k-1 (B (n) )) | ξ ∈ S 1 , k = 2, . . . , n}) ⊃ Lie {J ξ (E 2k-1 (B (n) )) | ξ ∈ S 1 , k = 2, . . . , n -1} ∪ {M + E 2n-1 (B (n) )} = su(n).
Therefore, thanks to Theorems 2.6 and 2.8, system ( 8) is approximately simultaneously controllable and an m-tracker.

The 3D molecule

Let us go back to the system for the orientation of a linear molecule presented in the introduction, that is,

i ψ = -∆ψ + (u 1 cos θ + u 2 cos ϕ sin θ + u 3 sin ϕ sin θ)ψ, (10)
where

ψ(t) ∈ H = L 2 (S 2 , C).
A basis of eigenvectors of the Laplace-Beltrami operator ∆ is given by the spherical harmonics Y m (θ, ϕ), which satisfy

∆Y m (θ, ϕ) = -( + 1)Y m (θ, ϕ). The spectrum of A = i∆ is {-i ( + 1) | ∈ N}. Each eigenvalue -i ( + 1
) is of finite multiplicity 2 + 1. Therefore A satisfies Assumptions (A1) and (A5). Using the notations of the preceding sections, we set B 1 , B 2 , B 3 to be the multiplication operators by -i cos ϕ sin θ, -i sin ϕ sin θ, -i cos θ, respectively. Since B 1 , B 2 , B 3 are bounded, conditions (A2), (A3), and (A4) hold. Hence (A ) is satisfied. Moreover, as proved in [BCC13, Proposition 8], (10) is sweakly coupled for every s > 0. The main goal of this section is to prove that system (10) satisfies the Lie-Galerkin Tracking Condition. As a consequence, we obtain the following result, whose corollary is Theorem 1.1.

{φ k | k = 1, . . . , 4 + 4} = {Y -, . . . , Y , Y --1 +1 , . . . , Y +1 +1 },
and it remains to prove that

LieV 4 +4 = su(4 + 4). (11)
The characteristic spectral gap of the space

H = span{Y -, . . . , Y , Y --1 +1 , . . . , Y +1 +1 }
is ( + 1)( + 2) -( + 1) = 2( + 1). In particular, (2( + 1), 1), (2( + 1), 2), and (2( + 1), 3) are in Ξ 4 +4 . The rest of Section 3 is devoted to showing that J ξ (E 2 +1 (B (4 +4) j

), for ξ ∈ S 1 and j = 1, 2, 3, generate su(4 + 4), proving (11).

Matrix representations

Denote by J the set of integer pairs {(r, m) | r = , + 1, m = -r, . . . , r}. Consider the lexicographic ordering : {1, . . . , 4 + 4} → J . For j, k = 1, . . . , 4 + 4, let e j,k be the (4 + 4)-square matrix whose entries are all zero, except the one at row j and column k, which is equal to 1. Define

E j,k = e j,k -e k,j , F j,k = ie j,k + ie k,j , D j,k = ie j,j -ie k,k .
By a slight abuse of notation, also set e (j), (k) = e j,k . The analogous identification can be used to define E (j), (k) , F (j), (k) , D (j), (k) . Note that

J i (E ( ,m),( +1,n) ) = -F ( ,m),( +1,n) and J i (F ( ,m),( +1,n) ) = E ( ,m),( +1,n) .
Thanks to this notation, we can conveniently represent the matrices corresponding to the controlled vector field (projected onto H ). A computation shows that the control potentials in the x and y directions, B 1 and B 2 respectively, projected onto H , have the matrix representations

B (4 +4) 1 = m=- (-q ,m F ( ,m),( +1,m-1) + q ,-m F ( ,m),( +1,m+1) ), B (4 +4) 2 = m=- (q ,m E ( ,m),( +1,m-1) + q ,-m E ( ,m),( +1,m+1) ),
where

q ,m = ( -m + 2)( -m + 1) 4(2 + 1)(2 + 3) .
Similarly, we associate with the control potential B 3 in the z direction the matrix representation

B (4 +4) 3 = m=- p ,m F ( ,m),( +1,m) , with p ,m = - ( + 1) 2 -m 2 (2 + 1)(2 + 3) .

Useful bracket relations

From the identity [e j,k , e n,m ] = δ kn e j,m -δ jm e n,k

we get the relations

[E j,k , E k,n ] = E j,n , [F j,k , F k,n ] = -E j,n , [E j,k , F k,n ] = F j,n , (12)
and

[E j,k , F j,k ] = 2D j,k , [F j,k , D j,k ] = 2E j,k . (13)
The relations above can be interpreted following a "triangle rule": the bracket between an operator coupling the states j and k and an operator coupling the states k and n couples the states j and n. On the other hand, the bracket is zero if two operators couple no common states, that is,

[Y j,k , Z j ,k ] = 0 if {j, k} ∩ {j , k } = ∅, (14)
with Y, Z ∈ {E, F, D}.

Controllability in su(4 + 4)

Lemma 3.2. The Lie algebra L generated by B

(4 +4) 1 , B (4 +4) 2 , B (4 +4) 3 , J i (B (4 +4) 1
),

J i (B (4 +4) 2 ), J i (B (4 +4) 3
) is equal to su(4 + 4).

Proof. The first step of the proof consists in showing that the Lie algebra L contains the elementary matrices

E ( ,k),( +1,k+j) for k = -, . . . , , j = -1, 0, 1. (15)
With a slight abuse of notation and for the sake of readability, let us write

B j = B (4 +4) j , j = 1, 2, 3. Let us also write ad α β for [α, β] and ad j+1 α β for [α, ad j α β]. Notice that J i (B 3 ) = m =-m p ,m J i (F ( ,m),( +1,m) ) = m =-m p ,m E ( ,m),( +1,m) .
By induction on j ≥ 0 and using the bracket relations (13), we have

ad 2j B3 J i (B 3 ) = [B 3 , [B 3 , ad 2j-2 B3 J i (B 3 )]] = (-1) j 2 2j m=- p 2j+1 ,m E ( ,m),( +1,m) .
By invertibility of the Vandermonde matrix and since p ,m = p ,n for every n = m, -m, it follows that E ( ,-m),( +1,-m) + E ( ,m),( +1,m) ∈ L, for m = 0, . . . , .

In particular, E ( ,0),( +1,0) ∈ L. The double bracket of

B 2 -J i (B 1 ) 2 = m=- q ,m E ( ,m),( +1,m-1) ∈ L (17)
with E ( ,0),( +1,0) is easily computed using (12) and ( 14) and gives m=-q ,m E ( ,m),( +1,m-1) , E ( ,0),( +1,0) , E ( ,0),( +1,0) = = -q ,1 [E ( ,0),( ,1) , E ( ,0),( +1,0) ] -q ,0 [E ( +1,-1),( +1,0) , E ( ,0),( +1,0) ] = q ,0 E ( ,0),( +1,-1) + q ,1 E ( ,1),( +1,0) ∈ L.

Define

Q m =     
q ,0 E ( ,0),( +1,-1) + q ,1 E ( ,1),( +1,0) for m = 0, q ,-m E ( ,-m),( +1,-m-1) + q ,m+1 E ( ,m+1),( +1,m) for 0 < m < , q ,-E ( ,-),( +1,--1) for m = .

In particular, B 2 -J i (B 1 ) = 2 m=0 Q m . Using again ( 12) and ( 14), we have

m=k Q m , E ( ,-k),( +1,-k) + E ( ,k),( +1,k) , E ( ,-k),( +1,-k) + E ( ,k),( +1,k) = Q k ,
for k = 1, . . . , . By interation on k and because of (16), it follows that Q k ∈ L for k = 0, . . . , . Now, since Q /q ,-= E ( ,-),( +1,--1) is in L, then

ad 2 E ( ,-),( +1,--1) (E ( ,-),( +1,-) + E ( , ),( +1, ) ) = -E ( ,-),( +1,-) ∈ L,
which, in turns, implies that

ad 2 E ( ,-),( +1,-) (Q -1 ) = -q ,-E ( ,-+1),( +1,-) ∈ L.
Iterating the argument, E ( ,m),( +1,m) and E ( ,m),( +1,m-1) are in L for every m = -, . . . , . By the same argument as above with (17) replaced by

B 2 + J i (B 1 ) 2 = m=- q ,-m E ( ,m),( +1,m+1) ∈ L,
we also have that E ( ,m),( +1,m+1) is in L for every m = -, . . . , , proving (15). It then follows from (12) that each E j,k is in L.

If we now replace (17) with

B 1 + J i (B 2 ) 2 = - m=- q ,m F ( ,m),( +1,m-1) ∈ L, or B 1 -J i (B 2 ) 2 = m=- q ,-m F ( ,m),( +1,m+1) ∈ L,
the arguments above prove that F ( ,m),( +1,m-1) and F ( ,m),( +1,m+1) are in L for every m = -, . . . , . The relations (12) and (13) then imply that L = su(4 + 4).

Proof of Theorem 2.6

The outline of the proof is the following:

Step 1. Thanks to a change of coordinates in time and within H, we restate the control problem in the so-called interaction framework, to handle the drift term in the Schrödinger operator. This is also useful to study, in Step 2, the controllability properties of the system via convexification techniques.

(See Sections 4.1 and 4.2.)

Step 2. We develop the convexification procedure. We design the control law in order to excite a given spectral gap and set to zero all the elements of the control Hamiltonian corresponding to other spectral gaps. (See Sections 4.4 and 4.6.)

Step 3. The Lie-Galerkin Control Condition is used to prove that the dynamics obtained in Step 2 generate every unitary transformation of the finite dimensional Galerkin approximation. (See Section 4.5.)

Step 4. The sparsity structure of the dynamics obtained by convexification guarantees that the flows of the infinite-dimensional system and of the Galerkin approximations are as close as desired. (See Section 4.7.)

Time-reparametrization

Up to replacing each B j with δB j , we can assume that δ = 1.

For every piecewise constant function z such that z(t) ≥ 1 for every t, we consider the time-reparametrization

dψ dt (t) = (z(t)A + u 1 (t)z(t)B 1 + • • • + u p (t)z(t)B p )ψ(t) (18)
of system (3). Each u j (t)z(t) belongs to the time-varying set z(t)U j .

If u 1 , . . . , u p are control laws in (18) then the corresponding controls in (3) are their time-reparametrizations ũj (s) = u j (t(s)) with t(s) = s 0 z(τ )dτ , j = 1, . . . , p. By restricting the range of available controls and setting v j (t) = u j (t)z(t), we can focus our attention on trajectories of

dψ dt (t) = (z(t)A + v 1 (t)B 1 + • • • + v p (t)B p )ψ(t), (19)
with z(t) ≥ 1 and v(t) = (v 1 (t), . . . , v p (t)) ∈ U . Each solution of (19) with z and v piecewise constant is the time-reparametrization of a solution of (3) with piecewise constant controls (but the converse is not necessarily true, since we restricted the set of admissible controls). Hence, the approximate simultaneous controllability of (19) implies the approximate simultaneous controllability of (3). Moreover

ũj L 1 = t -1 (T ) 0 |ũ j (τ )|dτ = T 0 |u j (t)|z(t)dt = T 0 |v j (t)|dt ≤ T,
for j = 1, . . . , p. The last inequality holds since either

U j = [0, 1] or U j = [-1, 1].
The L 1 -bounded approximate simultaneous controllability of (3) is proved in the next sections by showing the approximate simultaneous controllability in L of (19) with a bound on the controllability time which is uniform with respect to the tolerance.

Interaction framework

Given a solution ψ(•) of ( 19) with controls z(•), v 1 (•), . . . , v p (•) and a piecewise constant function α(•) with values in {0, 1}, let us define

ω(t) = t 0 (z(s) -α(s))ds and y(t) = e -ω(t)A ψ(t).
In particular

| φ k , y(t) | = | φ k , ψ(t) |, k ∈ N, (20)
for every t. For ω, v 1 , . . . , v p ∈ R set Θ(ω, v 1 , . . . , v p ) = e -ωA (v 1 B 1 + • • • + v p B p )e ωA . Note that Θ(ω, v 1 , . . . , v p ) jk = φ k , Θ(ω, v 1 , . . . , v p )φ j = e i(λ k -λj )ω (v 1 (B 1 ) jk + • • • + v p (B p ) jk ) , (21)
and that y(•) satisfies

ẏ(t) = (α(t)A + Θ(ω(t), v 1 (t), . . . , v p (t)))y(t), α ∈ {0, 1}, v ∈ U, ω + α ≥ 1. (22)
Conversely, each solution of ( 22) with α ∈ {0, 1} and v ∈ U piecewise constant and ω continuous and piecewise affine, with ω + α = z ≥ 1 almost everywhere, is, up to a time-dependent change of coordinates preserving the modulus of each component with respect to the basis Φ, a solution of (19) with u piecewise constant. In particular, each solution of

ẏ(t) = (α(t)A + Θ(ω(t), v 1 (t), . . . , v p (t)))y(t), α ∈ {0, 1}, v ∈ U, ω ≥ 1, (23)
with α, v piecewise constant and ω continuous and piecewise affine is, up to a time-dependent change of coordinates preserving the modulus of each component, a solution of (19) with u piecewise constant (but the converse is not necessarily true).

Proposition 4.1. Approximate simultaneous controllability of (23) implies approximate simultaneous controllability of (3). If, moreover, approximate simultaneous controllability in L = k∈N span{φ 1 , . . . , φ k } of (23) is achieved with a uniform bound on time then (3) is L 1 -bounded approximately simultaneous controllable.

Proof. The strategy of the proof follows [BCCS12, Proposition 6.1].

Approximate simultaneous controllability of (23) implies approximate simultaneous controllability of (19) in modulus thanks to (20).

Moreover, because of the unitarity of the evolution, the approximate simultaneous controllability of ( 23) is equivalent to the approximate simultaneous controllability of the system

ẏ(t) = -(α(t)A + Θ(ω(t), v 1 (t), . . . , v p (t)))y(t), α ∈ {0, 1}, v ∈ U, ω ≥ 1,
which implies approximate simultaneous controllability in modulus of the timereversed version of (19). Take r orthonormal initial conditions ψ 1 0 , . . . , ψ r 0 and r orthonormal final conditions ψ 1 1 , . . . , ψ r 1 . Since the the spectrum of A is infinite by Assumption (A1) we can apply [BCCS12, Lemma 6.3] so that for every tolerance η > 0 there exist k 1 , . . . , k r ∈ N pairwise distinct such that

C = {e tA φ k1 + • • • + e tA φ kr | t ∈ R}.
is η-dense in the torus

T = {e θ1A φ k1 + • • • + e θrA φ kr | θ 1 , . . . , θ r ∈ R}.
By approximate simultaneous controllability in modulus of (19), it follows that there exists an admissible control (z, v) steering simultaneously each ψ j 0 , for j = 1, . . . , r, η-close to e θj A φ kj for some θ 1 , . . . , θ r ∈ R.

Similarly, by approximate simultaneous controllability in modulus of the time-reversed equation of ( 19) there exists an admissible control (z, ṽ) steering system (19) simultaneously, for some θ1 , . . . , θr ∈ R, from e θ1A φ k1 , . . . , e θrA φ kr to an η-neighborhood of ψ 1 1 , . . . , ψ r 1 . The concatenation of the control (z, v), of a control constantly equal to (1, 0) on a time interval of suitable length, and of (z, ṽ) steers system (19) simultaneously from ψ 1 0 , . . . , ψ r 0 to a 3η-neighborhood of ψ 1 1 , . . . , ψ r 1 . Finally, according to the conclusion of Section 4.1, the approximate simultaneous controllability of (19) implies approximate simultaneous controllability of (3).

Galerkin approximation

Definition 4.2. Let N ∈ N. The Galerkin approximation of (23) of order N is the system ẋ = (αA

(N ) + Θ (N ) (ω, v 1 , . . . , v p ))x, x ∈ H, (24)
where

Θ (N ) (ω, v 1 , . . . , v p ) = π N Θ(ω, v 1 , . . . , v p )π N .
Recall that the elements of Θ (N ) (ω, v 1 , . . . , v p ) are as in (21). The controls v are piecewise constant with values in U , while ω is continuous and piecewise affine, with ω ≥ 1 almost everywhere.

In the following section we recall a convexification result whose role it is to identify the matrices that can be obtained by convexification of matrices of the form Θ (N ) (ω, v 1 , . . . , v p ).

Convexification

The following technical result has been proved in [BCCS12, Lemma 4.3].

Lemma 4.3. Let κ be a positive integer and γ 1 , . . . , γ κ ∈ R \ {0} be such that

|γ 1 | = |γ j | for j = 2, . . . , κ. Let ϕ(t) = (e itγ1 , . . . , e itγκ ).
Then, for every τ 0 ∈ R, we have

convϕ([τ 0 , ∞)) ⊇ S 1 × {(0, . . . , 0)} , where = ∞ k=2 cos π 2k > 0. (25)
Moreover, for every R > 0 and ξ ∈ S 1 there exists a sequence (τ k ) ∞ k=1 such that τ 1 ≥ τ 0 , τ k+1 -τ k > R, and

lim K→∞ 1 K K k=1 ϕ(τ k ) = ( ξ, 0, . . . , 0) .

Control in SU (n)

Let n be such that hypothesis (5) holds true. Define the set of matrices

W n = A (n) ∪ E 0 (B (n) j ) | (0, j) ∈ Ξ n ∪ E 0 (B (n) j ) + E σ (B (n) j ) | (σ, j) ∈ Ξ n and σ, j are such that (0, j) ∈ Ξ n , σ = 0 ∪ E σ (B (n) j ) | (σ, j) ∈ Ξ n , σ = 0, and U j = [-1, 1] ,
where Ξ n and are defined as in (4) and (25), respectively. (Recall that by rescaling we are assuming δ = 1.)

Notice that Lie(W n ) = Lie(V 0 n ). Consider the auxiliary control system

ẋ = M (t)x, M (t) ∈ W n , (26)
where M plays the role of control. It follows from (5) and standard controllability results on compact Lie groups (see [START_REF] Jurdjevic | Control systems on Lie groups[END_REF]) that for every g ∈ SU (n) there exists a piecewise constant function

M : [0, T ] → W n such that -→ exp T 0 M (s) ds = g,
where the chronological notation -→ exp t 0 V s ds is used for the flow from time 0 to time t of the time-varying equation q = V s (q), q ∈ C n (see [AS04]).

We stress that T can be bounded from above by a constant depending on n and W n but not on g.

System reduction by convexification

Let n be fixed as in the previous section. For every N ≥ n let

W n,N = A (N ) ∪ E 0 (B (N ) j ) | (0, j) ∈ Ξ n ∪ E 0 (B (N ) j ) + E σ (B (N ) j ) | (σ, j) ∈ Ξ n and σ, j are such that (0, j) ∈ Ξ n , σ = 0 ∪ E σ (B (N ) j ) | (σ, j) ∈ Ξ n , σ = 0, and U j = [-1, 1] .
Lemma 4.4. For every N ≥ n and for every piecewise constant M : [0, T ] → W n,N there exist α : [0, T ] → {0, 1}, v : [0, T ] → U piecewise constant and a sequence (ω h (•)) h∈N of continuous and piecewise affine functions from [0, T ] to [0, ∞) with ωh ≥ 1 almost everywhere, such that

t 0 (α(s)A (N ) + Θ (N ) (ω h (s), v 1 (s), . . . , v p (s)))ds - t 0 M (s)ds → 0
uniformly with respect to t ∈ [0, T ] as h tends to infinity.

Proof. Let N ≥ n. Define α(t) and v 1 (t), . . . , v p (t) at each t ∈ [0, T ] as follows: if M (t) = A (N ) then α(t) = 1 and v 1 (t) = • • • = v p (t) = 0; otherwise, if M (t) = E 0 (B (N ) j ), M (t) = E 0 (B (N ) j ) + E σ (B (N ) j ), or M (t) = E σ (B (N ) j
) for some j, then take such a j minimal and set v j (t) = 1 and α(t) = v k (t) = 0 for k = j.

We are going to apply Lemma 4.3 for every interval on which M (•) is constant. Fix ω h (0) = 0 for every h. Take an interval (t 0 , t 1 ) on which M (•) is constant and assume that ω h (t 0 ) has been computed. We next extend ω h on (t 0 , t 1 ).

If α = 1 on (t 0 , t 1 ) then take ω h (τ ) = ω h (t 0 ) + τ -t 0 for every τ ∈ (t 0 , t 1 ). Otherwise, let v j = 1 on (t 0 , t 1 ) and assume for now that M (t) = E 0 (B

(N ) j )+ E σ (B (N ) j
). Apply Lemma 4.3 with γ 1 = σ, {γ 2 , . . . , γ κ } = Σ N \ {σ}, ξ = 1, R = T , and τ 0 = ω h (t 0 ). Then there exists a sequence (τ k ) ∞ k=1 such that τ 1 ≥ ω h (t 0 ), τ k+1 -τ k > T , and

lim K→∞ 1 K K k=1
(e iτ k γ1 , . . . , e iτ k γκ ) = ( , 0, . . . , 0).

In particular there exists K = K(h) such that

1 K K k=1 e i(λ l -λm)τ k b (j) ml -E 0 (B (N ) j ) + E σ (B (N ) j ) m,l < 1 h , for every 1 ≤ l, m ≤ N .
Consider the piecewise constant function Y : (t 0 , t 1 ) → R defined as follows: set

s α = t 1 + (t 1 -t 0 ) α K , α = 0, . . . , K,
and let

Y (t) = ω h (t 0 ) + K α=1 τ α χ [sα-1,sα) (t), t ∈ (t 0 , t 1 ).
By a simple smoothing procedure (see for instance [BCCS12, Proposition 5.5]) one can construct a continuous piecewise affine approximation

ω h : [t 0 , t 1 ] → R of Y with ωh ≥ 1 almost everywhere such that t t0 (Θ (N ) (ω h (s), v 1 (s), . . . , v p (s)))ds - t t0 M (s)ds → 0 (27)
uniformly with respect to t ∈ [t 0 , t 1 ] as h tends to infinity. The same argument can be carried out in the case in which

M (t) = E 0 (B (N ) j
) by applying Lemma 4.3 with γ 1 in (0, ∞)\Σ N , {γ 2 , . . . , γ κ } = Σ N , ξ = 1, R = T , and τ 0 = ω h (t 0 ).

The final case to be considered is when

M (t) = E σ (B (N ) j
) with σ = 0, (σ, j) ∈ Ξ n , and

U j = [-1, 1]. Notice that E σ (B (N ) j ) = (E 0 (B (N ) j ) + E σ (B (N ) j )) -(E 0 (B (N ) j ) + J -1 (E σ (B (N ) j
))) 2 .

(28) The argument above can be easily adapted to matrices M (t) of the type v j (E 0 (B

(N ) j ) + J ξ (E σ (B (N ) j
))), with v j ∈ U j , ξ ∈ S 1 (not assuming ξ = 1 when applying Lemma 4.3), and in particular to -(E 0 (B

(N ) j ) + J -1 (E σ (B (N ) j ))).
It suffices then to introduce a sequence (M h ) h∈N of piecewise constant functions with values in

{v j (E 0 (B (N ) j ) + J ξ (E σ (B (N ) j ))) | v j ∈ U j , ξ ∈ S 1 } such that t t0 M h (
s)ds converges uniformly for t ∈ [t 0 , t 1 ] to t t0 M (s)ds as h tends to infinity and to apply a diagonal procedure based on the approximation introduced above.

As a consequence of the lemma above and thanks to [AS04, Lemma 8.2], we have

-→ exp t 0 α(s)A (N ) + Θ (N ) (ω h (s), v 1 (s), . . . , v p (s)) ds- -→ exp t 0 M (s) ds → 0 (29)
uniformly with respect to t ∈ [0, T ] as h tends to infinity.

Control of the infinite-dimensional system

Proposition 4.5 below states that we can pass to the limit in (29) as N tends to infinity. Its proof is based on the sparsity structure of the matrices in W n,N , guaranteeing that the dynamics of the infinite-dimensional system and the one of the Galerkin approximations are as close as desired.

We introduce the following notation: given n ∈ N and a bounded linear transformation L of H, let Crop n (L) be the n × n matrix ( φ j , Lφ k ) n j,k=1 . We use the same symbol Crop n also to denote the similar cropping operation acting on the space of N × N matrices, with N ≥ n. Proof. Consider µ > 0 to be fixed later. For every j ∈ N the hypothesis that φ j belongs to D(B l ) implies that the sequence ((B l ) jk ) k∈N is in 2 for every l = 1, . . . , p. It is therefore possible to choose N ≥ n in such a way that ((B l ) jk ) k>N 2 < µ for every j = 1, . . . , n and l = 1, . . . , p.

Let M be a piecewise constant function from By Lemma 4.4, for every η > 0, there exist piecewise constant controls α : [0, T ] → {0, 1}, v : [0, T ] → U and a continuous piecewise affine function ω with ω ≥ 1 such that

-→ exp t s α(τ )A (N ) + Θ (N ) (ω(τ ), v 1 (τ ), . . . , v p (τ )) dτ - -→ exp t s M (τ ) dτ < η, for every s, t in [0, T ].
Consider the solution Ψ of (23) associated with α, ω and v. Set, for k ∈ N,

Q (k) t = Crop k Ψ t . Now Q(N) t = αA (N ) + Θ (N ) (ω, v 1 , . . . , v p ) Q (N ) t + R (N ) t
, and the choice of N is such that

|(R (N ) t ) j,k | ≤ µ, (30)
for every j = 1, . . . , n and k = 1, . . . , N . Notice, moreover, that the norm of R

(N ) t can be uniformly bounded by a positive constant C independent of η (possibly depending on N and hence on µ).

By the variation formula and since Q

(n) t = Crop n (Q (N ) t
) we have

Q (n) t = Crop n -→ exp t 0 α(τ )A (N ) + Θ (N ) (ω(τ ), v 1 (τ ), . . . , v p (τ )) dτ + t 0 -→ exp t s α(τ )A (N ) + Θ (N ) (ω(τ ), v 1 (τ ), . . . , v p (τ )) dτ R (N ) s ds , so that Crop n Ψ t - -→ exp t 0 α(τ )A (N ) + Θ (N ) (ω(τ ), v 1 (τ ), . . . , v p (τ )) dτ ≤ tηC + t 0 -→ exp t s M (τ ) dτ Crop n R (N ) s ds .
The norm of the matrix product

-→ exp t s M (τ ) dτ n R (N ) s is equal to Crop n R (N ) s .
The max norm of Crop n R (N ) s is smaller than µ by (30). Hence

Crop n Ψ t - -→ exp t 0 α(τ )A (N ) + Θ (N ) (ω(τ ), v 1 (τ ), . . . , v p (τ )) dτ ≤ T (ηC + √ nµ).
The constant T (ηC + √ nµ) can be made arbitrarily small by choosing µ small with respect to n and T and then η small with respect to C = C(µ) and T .

Based on the previous results, we can now easily complete the proof of Theorem 2.6.

Proof of Theorem 2.6. Let r in N, ψ 1 , . . . , ψ r in H, Υ in U(H), and ε > 0. Let n 0 be large enough so that there exists g ∈ SU (n 0 ) for which

| φ j , Υψ k -π n0 φ j , gπ n0 ψ k | < ε,
for every 1 ≤ k ≤ r and j ∈ N.

Let n ≥ n 0 be such that hypothesis (5) is satisfied. Notice that if ψ 1 , . . . , ψ r and Υ(ψ 1 ), . . . , Υ(ψ r ) are in L then n can be taken independently of ε.

From Section 4.5, there exists M :

[0, T ] → W n such that -→ exp T 0 M (s) ds = g,
where g is seen as an element of SU (n). Proposition 4.5 ensures the existence of two piecewise constant functions z and v and of a continuous piecewise affine function ω with ω ≥ 1 almost everywhere such that the associated propagator Ψ of (23) satisfies

-→ exp T 0 M (s) ds -Crop n (Ψ T ) < ε.
If ψ 1 , . . . , ψ r , Υ(ψ 1 ), . . . , Υ(ψ r ) are in L then T is independent of ε. By Proposition 4.1 system (3) is L 1 -bounded approximately simultaneously controllable.

Proof of Theorem 2.8

The proof of Theorem 2.8 follows the same scheme as the one of Theorem 2.6.

The key new argument is the following: it has been proved in Proposition 4.5 that system (19) can track every trajectory of (26). The idea is to replace (26) with a system which can track with arbitrary precision every trajectory in SU (n). The crucial property, beyond the Lie bracket generating condition, that the new version of (26) should satisfy in order to achieve this goal is to be driftless (i.e., the time-reversal of each admissible trajectory is itself admissible).

The same time-reparameterization and time-dependent change of coordinates as in Section 4.1 allows one to consider the tracking problem for system (23) instead of system (3). As in the previous section we consider δ to be renormalized to 1.

We can then base our argument on the following analogue of Proposition 4.1.

Proposition 5.1. If (23) is a m-tracker then (3) is a m-tracker as well.

The following proposition allows one to reduce a tracking problem in the space of unitary operators of H to a tracking problem in SU (n) for n large enough. Its proof can be found in [BCCS12, Proposition 5.7].

Proposition 5.2. Let Υ : [0, T ] → U(H) be a continuous curve. Take ε > 0 and m ∈ N. Then for n ≥ m sufficiently large there exists a continuous curve g : [0, T ] → SU (n) such that | φ j , Υ(t)φ k -e j , g(t)e k | < ε for every t in [0, T ], 1 ≤ k ≤ m, and j = 1, . . . , n, where (e 1 , . . . , e n ) denotes the canonical basis of R n .

Let n be chosen as in Proposition 5.2. According to the Lie-Galerkin Tracking Condition, we can assume, without loss of generality, that Lie(V n ) = su(n).

The roles played in Sections 4.5 and 4.6 by W n and W n,N are now played by V n and V n,N , where V n is defined as in (6) and

V n,N = J ξ (E σ (B (N ) j )) | (σ, j) ∈ Ξ n , σ = 0, ξ ∈ S 1 .
In particular, we consider the auxiliary control system

ẋ = M (t)x, M (t) ∈ V n , (31)
with M being the matrix-valued control parameter. It follows from the equality Lie(V n ) = su(n) and Rashevski-Chow's theorem that every trajectory on SU (n) can be tracked with arbitrary precision (up to time-reparameterization) by a trajectory of (31).

The relation between the trajectories of (24) and those of (31) (or, more precisely, ẋ = M (t)x, M (t) ∈ V n,N ), is described by the following lemma. ). The only difference is in replacing (28) with

J ξ (E σ (B (N ) j )) = (E 0 (B (N ) j ) + J ξ (E σ (B (N ) j ))) -(E 0 (B (N ) j ) + J -ξ (E σ (B (N ) j
))) 2 .

We then apply the same convexification argument.

As in the previous section, the lemma above and [AS04, Lemma 8.2] imply that

-→ exp t 0 α(s)A (N ) + Θ (N ) (ω h (s), v 1 (s), . . . , v p (s)) ds - -→ exp t 0 M (s) ds → 0 (32)
uniformly with respect to t ∈ [0, T ] as h tends to infinity. In analogy with Section 4.7 we can conclude the proof of Theorem 2.8 thanks to the proposition below, which states that we can pass to the limit in (32) as N tends to infinity. Its proof is basically the same as the one of Proposition 4.5. Remark 5.5. The hypothesis that each U j contains 0 in its interior can be relaxed. Indeed, up to reordering, let p be such that U j = [0, δ] if j = 1, . . . , p and U j = [-δ, δ] for j > p . Assume that, for every j ∈ {1, . . . , p }, if l = k are such that λ l = λ k , then φ l , B j φ k = 0. Assume, moreover, that the Lie-Galerkin Control Condition is satisfied with V n replaced by the set of all matrices J ξ (E σ (B (n) j )) with (σ, j) ∈ Ξ n , ξ ∈ S 1 , σ = 0, and either j > p or the following holds: if l, k, l , k ∈ {1, . . . , n} satisfy λ l -λ k = λ l -λ k = σ and φ l , B j φ k = 0 = φ l , B j φ k , then φ l , B j φ l -φ k , B j φ k = φ l , B j φ l -φ k , B j φ k .

In this case the proof of Lemma 5.3 becomes more technically involved. The point is that, even if E 0 (B (N ) j

) cannot be eliminated by convexification, it is a diagonal matrix by hypothesis. Hence, it can be used to define a new interaction framework. The sequence (ω h (•)) h can then be constructed following [BCCS12, Proposition 5.5].

Proof of Theorem 2.12

First, let us prove L 1 -bounded approximate simultaneous controllability for system (3) in s/2-norm for initial and final data in L. Namely, we want to prove that, for r ∈ N, ψ 1 , . . . , ψ r ∈ L, and Υ ∈ U(H) with Υψ 1 , . . . , Υψ r ∈ L, there exists K > 0 such that the following holds: For every ε > 0 there exists a control u, with u L 1 ≤ K, such that ψ j -Υ u T ψ j s/2 < ε, j = 1, . . . , r.

Let N be such that ψ 1 , . . . , ψ r and Υψ 1 , . . . , Υψ r are in span{φ 1 , . . . , φ N }. Note that on span{φ 1 , . . . , φ N } we have

ψ s/2 = N k=1 |λ k | s | φ k , ψ | 2 1/2 ≤ (max{|λ 1 |, |λ N |}) s/2 ψ .
(34)

Since system (3) is L 1 -bounded approximately simultaneously controllable, there exists K > 0 such that for every ε > 0 there exists a piecewise constant control u with u L 1 ≤ K and ψ j -Υ u T ψ j < ε, j = 1, . . . , r.

Hence (34) implies L 1 -bounded approximate simultaneous controllability in s/2norm in L. Now let ψ 1 , . . . , ψ r ∈ D(|A| s/2 ) and Υ ∈ U(H) be such that Υψ 1 , . . . , Υψ r ∈ D(|A| s/2 ). Let ε > 0 and consider ψ 0 1 , . . . , ψ 0 r and ψ 1 1 , . . . , ψ 1 r in L such that ψ 0 j -ψ j s/2 < ε and ψ 1 j -Υψ j s/2 < ε, for j = 1, . . . , r. As proved above (see (33)), there exist K, independent on ε, and a piecewise constant control u with u L 1 ≤ K such that ψ 1 j -Υ u T ψ 0 j s/2 < ε, j = 1, . . . , r.

By [BCC13, Proposition 2] and since system (3) is s-weakly coupled, there exists a constant C depending only on s and A, B 1 , . . . , B p such that

Υ u T ψ s/2 ≤ C K ψ s/2 ,
for every ψ ∈ D(|A| s/2 ). Therefore Υ u T (ψ j ) -Υ(ψ j ) s/2 ≤ Υ u T (ψ j -ψ 0 j ) s/2 + Υ u T ψ 0 j -ψ 1 j s/2 + ψ 1 j -Υψ j s/2 ≤ (C K + 2)ε, for j = 1, . . . , r. Remark 6.1. Using arguments similar to those of the proof of Theorems 2.6 and 2.12, it is possible to prove a finer statement than Corollary 2.13. Indeed it is possible to prove that a system that satisfies Assumptions (A ), the Lie-Galerkin Tracking Condition, and is s-weakly coupled is an m-tracker for the s/2-norm. This is due to the fact that the Lie-Galerkin Tracking Condition actually implies m-tracking in L with a uniform bound on the L 1 norm of the control.

[AB05] Riccardo Adami and Ugo Boscain

  Theorem 3.1. System (10) is:(i) L 1 -bounded approximately simultaneously controllable, (ii) approximately simultaneously controllable in H s for every s > 0, (iii) a m-tracker.Using classical identities for Legendre polynomials and trigonometric relations, one can prove that Y m , B j Y m = 0 for every j = 1, 2, 3, and m, m ∈ {--1, . . . , + 1}.Moreover, Y m , B j Y m = 0 with | -| ≥ 2 for every m ∈ {--1, . . . , + 1}, m ∈ {--1, . . . , + 1}, j = 1, 2, 3.In order to prove that the Lie-Galerkin Tracking Condition is satisfied, we choose a reordering (φ k ) k∈N of the spherical harmonics in such a way that

Proposition 4. 5 . 0 M

 50 Let n ∈ N and M : [0, T ] → W n be piecewise constant. Then, for every ε > 0, there exist piecewise constant controls z : [0, T ] → [1, ∞) and v : [0, T ] → U , and a continuous piecewise affine function ω with ω ≥ 1 such that the propagator Ψ of (23) satisfies-→ exp t (s) ds -Crop n (Ψ t ) < ε, for every t ∈ [0, T ].

  [0, T ] to W n,N such that Crop n M (t) = M (t) for every t in [0, T ]. Because of the definition of Ξ n and of the classes W n,N and W n we have

Lemma 5. 3 . 0 M

 30 For every N ≥ n and for every piecewise constant M : [0, T ] → V n,N there exist α : [0, T ] → {0, 1}, v : [0, T ] → U piecewise constant, and a sequence (ω h (•)) h∈N of continuous and piecewise affine functions from [0, T ] to [0, ∞) with ωh ≥ 1 almost everywhere such that t 0 α(s)A (N ) + Θ (N ) (ω h (s), v 1 (s), . . . , v p (s)) ds -t (s)ds → 0 uniformly with respect to t ∈ [0, T ] as h tends to infinity. Proof. The proof is almost identical to that of Lemma 4.4 in the case M (t) = E σ (B (N ) j

Proposition 5. 4 . 0 M

 40 Let n ∈ N and M : [0, T ] → V n be piecewise constant. Then, for every ε > 0, there exist piecewise constant controls z : [0, T ] → [1, ∞) and v : [0, T ] → U , and a continuous piecewise affine function ω with ω ≥ 1 almost everywhere such that the propagator Ψ of (23) satisfies -→ exp t (s) ds -Crop n (Ψ t ) < ε, for every t ∈ [0, T ].

  . Controllability of the Schrödinger equation via intersection of eigenvalues. In Proceedings of the 44th IEEE Conference on Decision and Control, pages 1080-1085, 2005. [AS04] Andrei A. Agrachev and Yuri L. Sachkov. Control theory from the geometric viewpoint, volume 87 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, 2004. Control Theory and Optimization, II. [BBR10] Anthony M. Bloch, Roger W. Brockett, and Chitra Rangan. Finite controllability of infinite-dimensional quantum systems. IEEE Trans. Automat. Control, 55(8):1797-1805, 2010. [BC06] Karine Beauchard and Jean-Michel Coron. Controllability of a quantum particle in a moving potential well. J. Funct. Anal., 232(2):328-389, 2006.

$ This research has been supported by the European Research Council, ERC StG 2009 "GeCoMethods", contract number 239748, by the ANR project GCM, program "Blanc", project number NT09-504490


represented by various matrices for a number of bases for the corresponding vector spaces.

Additional reason to avoid the use of affine and metric structures in the formulation of stress theory originates from the various applications of the continuum mechanics methods to generalized continua. For such continua (e.g. [START_REF] Mermin | The topological theory of defects in ordered media[END_REF], [START_REF] Capriz | Continua with Microstructure[END_REF], [START_REF] Capriz | Interactions in general continua with microstructure[END_REF], the state of the body is described by internal state variables, or order parameters, for which we might not have affine or metric properties.

Over the years, research work contributed to the subject by weakening the postulates needed to prove Cauchy's theorem, by making the proof more rigorous and establishing the existence of stresses using alternative methods and approaches (e.g., [START_REF] Gurtin | An axiomatic foundation for continuum thermodynamics[END_REF][START_REF] Gurtin | Cauchy's theorem in classical physics[END_REF], [START_REF] Marsden | Mathematical Foundations of Elasticity[END_REF], [START_REF] Fosdick | A variational proof of the stress theorem of Cauchy[END_REF], [START_REF] Segev | Forces and the existence of stresses in invariant continuum mechanics[END_REF][START_REF] Segev | On the consistency conditions for force systems[END_REF]). This paper presents the generalization of Cauchy's theory of forces and stresses, to general differentiable manifolds. The terminology of differential forms is used to formulate Cauchy's postulates and Cauchy's theorem.

In a previous paper, [START_REF] Segev | On the geometry of Cauchy's theory for balance of scalar valued properties[END_REF], the corresponding theory was introduced for balance laws of real valued properties (such as mass). Here, the theory is extended to forces valued in the dual of a vector bundle W over an m-dimensional material manifold S . Thus, forces are presented as entities that produce power densities when acting on generalized velocity fields-sections of W . The usual assumption that forces consist of body forces and surface forces is retained. Thus, following [START_REF] Epstein | Differentiable manifolds and the principle of virtual work in continuum mechanics[END_REF], a body force β R on a body R is a section of L W, m (T R) and a surface force τ R is a section of L W, m-1 (T ∂ R) . Using integration of differential forms, the power densities of the body force and surface force are combined to generate the force functional F R as

F R (w) = R β R (w) + ∂ R τ R (w).
These and other preliminary topics are considered in Section 2.

As a unit normal is not available in the case of general manifolds, the local dependence of the surface force on the normal is replaced by dependence on the tangent space to the boundary-an element of the Grassmann manifold of hyperplanes in the tangent space to S . In addition, the outwards direction of the normal is replaced by the outwards inclination or outer orientation of the tangent space to the boundary.

In the setting presented here, stresses are (m -1)-forms, valued in W * , defined on the material manifold. The analog of the operation S T (n) that appears in Cauchy's theorem, is the inclined restriction of forms. Instead of "taking the normal component" of the stress tensor, the stress form is restricted to the tangent space of the boundary. However, while a simple restriction is not sensitive to the outwards inclination (outer orientation) of the tangent space at the point on the boundary under consideration, the inclined restriction is an odd function of the inclination. This is done using a sign rule that is induced by an orientation of the material manifold. While the orientation is assumed to be given, odd forms can be used to render the presentation independent of the arbitrary choice of orientation (see [START_REF] Segev | Interactions on Manifolds and the Construction of Material Structure[END_REF]). Since the orientation is needed only locally, there is no need to require of orientability of the material manifold.

One additional aspect of the theory that needs attention is the continuous dependence of the surface force on both the point on the boundary and the hyperplane. Since, in the case of manifolds, it is not possible to compare hyperplanes at distinct points, the mapping that assigns the surface force to the inclined tangent space is presented as a continuous section-the Cauchy section-of a fiber bundle, the bundle of interaction densities, over the bundle of inclined hyperplanes. Following [START_REF] Segev | Interactions on Manifolds and the Construction of Material Structure[END_REF], inclined hyperplanes, Cauchy sections and inclined restrictions are considered in Section 3.

Section 4 presents the generalized Cauchy postulates and theorem. The only aspect of the theory not mentioned above is the balance equation. The balance equation is formulated in terms of the total expanded power in the form

F R (w) = R β (w) + ∂ R τ R (w) ≤ R ς (w),
where ς is some m-form that we assume to exist and that satisfies this inequality for any body R. This form of the balance equation emphasizes the fact that we regard the balance law as a regularity assumption on the force fields for the various bodies, or alternatively, as a boundedness assumption on the collection of force functionals for the various bodies (c.f. [START_REF] Segev | On the consistency conditions for force systems[END_REF]).

Section 5 considers the situation where a volume element is given on the material manifold S (e.g., a mass density). In such a case the stress form can be represented by a tensor-a vector bundle morphism W → T S . It also considers a somewhat generalized version of the normal projection of the stress. Finally, Section 6 assumes that a Riemannian metric is given on S and the classical Cauchy formula is recovered. As a final remark it is noted that in the current framework, the Cauchy stress and the first Piola-Kirchhoff stresses are simply two representations of the stress form under two different Riemannian metrics.

PRELIMINARY DEFINITIONS AND ASSUMPTIONS

2.1. Kinematics. It is assumed that there is a material manifold S of dimension m containing all bodies as subsets. A body is an m-dimensional compact submanifold with corners of S .

The physical space is modeled as a manifold without boundary M , and a configuration of a body R in space is a smooth mapping κ : R → M . One usually requires that the configuration is an embedding although this will not have any affect on the following discussion.

A generalized velocity field at a given configuration κ is a mapping w : R → T M such that τ M • w = κ. Here T M is the tangent bundle of the space manifold and τ M : T M → M is the tangent bundle projection.

For a given configuration κ, consider the pull-back

κ * (τ M ) : κ * (T M ) → R,
of T M by κ. A velocity field at the configuration κ can be regarded also as the section w : R → κ * (T M ), where w (x) is identified with the element w(x) ∈ T κ(x) M . Since the configuration κ will be fixed during the following discussion, the second point of view is preferable as we can regard the velocity field as a section of a fixed vector bundle. Thus, in the sequel, we will consider a fixed vector bundle W → S over the material manifold. We can restrict this vector bundle naturally to any body R ⊂ S and we will abuse the notation by writing W → R for the restricted bundle. Thus, a velocity field over a body R will be a section w : R → W .

2.2. Forces. For a general manifold geometry, forces may be defined invariantly as covectors or linear functionals. For example, a concentrated force f acting on the material point x ∈ S at the configuration κ is an element of the cotangent space T * κ(x) M . Such a force may be evaluated for a velocity w(x) and the value f (w(x)) is interpreted as the power that the force f expends while the material point x travels at the velocity w(x). Clearly, using the pullback of T M by κ the velocity becomes a member of W and the force will be a member of W * .

Force densities operate on the velocity fields to produce power densities that may be integrated. It is assumed henceforth that force densities (and the resulting power densities) are continuous. Thus, a force density on an n-dimensional submanifold P of S is a continuous section of L W, n (T P) . Here n (T P) is the bundle of n-forms on P, and for two vector bundles U and V over S , L(U,V ) is the vector bundle whose fiber at x is the vector space of linear mappings U x → V x . The power the force density γ produces for the generalized velocity w is

P γ(w) = P γ(x) w(x) .
In the sequel we will often abuse notation by using the same symbol for both a differential form and its value at a point.

It is usually assumed in continuum mechanics that the force on a body consists of a body force distributed over the volume of the body and a surface force distributed over the boundary. Hence, the force F on a body R is represented by two force densities, the body force β , a section of L W, m (T S) , and a boundary or surface force τ, a section of L W, m-1 (T ∂ R) . Thus, for x ∈ R

β (x) : W x → m T x S
is a linear mapping, and so is

τ(y) : W y → m-1 T y ∂ R.
for some y ∈ ∂ R.

We conclude that the power expended by a combination of a body force and a surface force is of the form

R β (w) + ∂ R τ(w).
It is note that the last expression may be conceived as the evaluation of a linear functional F on the space of velocity fields on a particular field w to produce the global power.

Thus, we will refer to such a linear functional F as a force acting on the body R represented by {β , τ}, the body force and the surface force. The power that the force F produces for the generalized velocity field w is represented in the form

F(w) = R β (w) + ∂ R τ(w).
So far we were concerned with one particular body R. When the family of all bodies within the material manifold and the force operating on each is considered, we use the notation F R to denote the force acting on the body R. Similarly, β R and τ R will denote the corresponding body force and boundary force. Thus, we have

F R (w) = R β R (w) + ∂ R τ R (w).
We will refer to the collection {(β R , τ R )} for the various bodies, as a force system on S . A force system may be regarded as a set function assigning to each body R the corresponding two force fields (β R , τ R ). Usually, it is assumed that the value of the body force at a point does not depend on the particular body containing that point. Thus, in fact, only the surface force field τ R depends on R. It is the stress that allows one to specify the fields τ R for the various bodies with a single field on S . 2.3. Vector Valued Forms. Mathematically, the body force β and the surface force τ are similar-at each x ∈ R they are linear mappings from W x into a space of alternating multi-linear mappings. We will identify such objects with (co-)vector valued forms as follows.

Let γ x ∈ L W x , k (T x P) , be a linear mapping, where P is an arbitrary submanifold of S and k ≤ dim(P). For a fixed collection of k vectors v 1 , . . . ,v k ,

γ x (u)(v 1 , . . . ,v k ) ∈ R is linear in u. Hence, there is an element ψ ∈ W * x such that ψ(u) = γ x (u)(v 1 , . . . ,v k ).
Since ψ depends on the collection v 1 , . . . ,v k multi-linearly and in alternating way, there is an alternating mapping, a W * x -valued form γx :

(T x P) n → W * x , such that γx (v 1 , . . . ,v k )(u) = γ x (u)(v 1 , . . . ,v k ).
It is noted that the same relation can be used to define a unique element of L W x , k (T x P) once a W * x vector valued k-form is given. (The isomorphism is of course a particular case of isomorphisms of the type L E, L(E, F) → L 2 (E, F) for two vector spaces E and F.)

Clearly, this can be globalized and in the sequel we will use

k T P,W * for the bundle of W * -valued k-forms. Because of the isomorphisms L W, m (T R) → m T R,W *
we will often view body forces as sections of either bundle and the meaning will be clear from the context. A similar convention will apply to surface forces as a result of the isomorphism

L W, m-1 (T ∂ R) → m-1 T ∂ R,W * .

HYPERPLANES, ORIENTATION AND INCLINATION

One expects that a theory of stresses will enable the formulation and proof of an analogue of Newton's third law of action and reaction. In order to do that, a notion of a "sense" for a tangent hyperplane should be defined so we can distinguish between the interactions on two bodies that are tangent at a point but are situated at opposite sides of the common tangent hyperplane.

In the following, a hyperplane H at a point x means an (m -1)-dimensional subspace of T x S . The collection of hyperplanes at x, a Grassmann manifold, will be denoted by G m-1 (T x S ). Clearly, a hyperplane H is uniquely defined by an arbitrary form φ in the one-dimensional subspace

H + ⊂ T * x S of annihilators of H. The union G m-1 (T S ) = x∈S G m-1 (T x S )
is the Grassmann bundle of hyperplanes.

Inclinations of Hyperplanes.

Let H be a hyperplane at x. For two vectors v 1 , v 2 ∈ T x S that are transversal to H, one says that v 1 and v 2 are on the same side of H if for some φ ∈ H + , φ (v 1 ) and φ (v 2 ) in R have the same sign. Otherwise, one says that the two vectors are on opposite sides of H. We note that if φ (v 1 ) and φ (v 2 ) have the same sign, then, φ (v 1 ) and φ (v 2 ) will have the same sign for any other φ ∈ H + . Clearly, -v and v are on opposite sides. In addition, the relation "v 1 and v 2 are on the same side of H" is an equivalence relation. The quotient space of T x S -H by this equivalence relation contains two elements-the two sides of H. An inclined hyperplane is a hyperplane H together with a side-the positive inclination of H.

In [START_REF] Schouten | Pfaff's Problem and its Generalizations[END_REF] the term outer orientation is used for what is referred to here as "inclination". We chose to use the latter for the sake of brevity.

For an inclined hyperplane h, there exists an inclined hyperplane -h of the opposite inclination. A vector v ∈ T x S is positively inclined if it belongs to the equivalence class corresponding to to the positive inclination. The collection of inclined hyperplanes at x ∈ S will be denoted by G ⊥ m-1 (T x S ).

Given an inclined hyperplane, there is a subset H ⊥ of H + containing the forms φ such that φ (v) > 0 for any positively inclined vector v. Consider the equivalence relation by which φ 1 ∼ φ 2 if φ 2 = aφ 1 for some a > 0. Clearly, any two elements of H ⊥ are so related. Thus, H ⊥ is an element of the quotient space T *

x S /∼. Since any covector φ ∈ T *

x S determines a unique inclined hyperplane that it annihilates, we can identify G ⊥ m-1 (T x S ) with this quotient space. Clearly, the operation h → -h, is a natural involution on G ⊥ m-1 (T x S ). In addition, we have a natural mapping

p x : G ⊥ m-1 (T x S ) → G m-1 (T x S
) that ignores the inclination of the hyperplanes. For a given inclined hyperplane h, we will often use H to denote its non-inclined counterpart p(h).

Using a metric to identify H ⊥ with a unit normal vector to h, G ⊥ m-1 (T x S ) can be given the differentiable structure of the (m -1)-sphere. Thus, the construction can be globalized to the entire S manifold to generate the fiber bundle

G ⊥ m-1 (T S ) = x∈S G ⊥ m-1 (T x S ) → S
of inclined hyperplanes, and a fiber bundle morphism

p : G ⊥ m-1 (T S ) → G m-1 (T S ). It is noted that if x ∈ S is
on an (m -1)-dimensional face on the boundary of an m-dimensional submanifold with corners R of S , then, outward pointing vectors are defined invariantly as tangents to curves in S through x that lie outside R for 0 < t < ε, and some ε > 0. This allows one to assign a specific inclination to T x ∂ R.

Interaction Densities and Cauchy Sections.

The basic Cauchy postulate for force systems, states that the surface force at point x ∈ ∂ R depends on the outwards pointing normal to the boundary at that point. In the current framework, the dependence on the normal is replaced by dependence on the inclined hyperplane h, T x ∂ R with the outwards inclination.

For an inclined hyperplane h associated with T x (∂ R) , τ R (x) belongs to the vector space m-1 (H,W * x ), where, H = p(h). Consider the vector bundle

m-1 G m-1 (T S ),W * → G m-1 (T S ) whose fiber at H ∈ G m-1 (T S ), with H ⊂ T x S , is m-1 (H,W * x ).
One can pull back this vector bundle using p :

G ⊥ m-1 (T S ) → G m-1 (T S ) to obtain D = p * m-1 G m-1 (T S ),W * → G ⊥ m-1 (T S ),
to which we will refer as the interaction density bundle. Thus, mathematically, the analog of Cauchy's postulate is the assumption that there is a section, the Cauchy section

τ : G ⊥ m-1 (T S ) → D, such that τ R (x) = τ(h),
where h is the outwards inclined tangent space to ∂ R at x.

Inclined Restrictions of Forms.

In order to formulate the generalized version of Cauchy's formula t = S T (n) we will need the notion of inclined restriction. Consider an oriented hyperplane h at x whose inclination is specified by the transversal vector v. We will use I : H = p(h) → T x S to denote the inclusion mapping so

I * : k (T x S ,W * ) → k (H,W * ),
given by

I * (γ)(v 1 , . . . ,v k ) = γ I (v 1 ), . . . , I (v k ) ,
is the restriction of forms.

For the definition, it is assumed that an orientation of S is given in a neighborhood of x. (See [START_REF] Segev | Interactions on Manifolds and the Construction of Material Structure[END_REF] for the required generalization in case an orientation is not specified.) For an (m -1),

W * -valued form σ ∈ m-1 (T x S ,W * x ), the inclined restriction ι * h (σ ) ∈ m-1 p(h) is defined by the requirement that ι * h (σ )(v 1 , . . . ,v m-1 ) = I * (σ )(v 1 , . . . ,v m-1 ) if (v, v 1 , . . . ,v m-1
) are positively oriented and

ι * h (σ )(v 1 , . . . ,v m-1 ) = -I * (σ )(v 1 , . . . ,v m-1 ) if (v, v 1 , . . . ,v m-1
) are negatively oriented relative to the given orientation. It follows from the definition that

ι * h (σ ) = -ι * -h (σ ).
Let σ be a W * -valued, (m -1)-form. Then, the mapping

h → ι * h (σ ) ∈ m-1 p(h),W *
is clearly a Cauchy section. We will refer to the form σ as a stress form. Indeed, Cauchy's postulates guarantee that the surface forces are generated by a unique stress form.

THE GENERALIZED CAUCHY'S POSTULATES AND THEOREM

The following are the generalized Cauchy postulates for a force system {(β R , τ R ); R ⊂ S }.

(i) For every x ∈ S and every body R, β R (x) = β (x), that is, the value of the body force at a point is independent of the body containing it. Accordingly, we will omit the subscript R.

(ii) There is a Cauchy section

τ : G ⊥ m-1 (T S ) → D, such that for any body R, τ R (x) = τ(h),
where h is the tangent hyperplane to R at x with the outwards pointing inclination. (iii) The Cauchy section τ is continuous. (iv) There is a W * -valued m-form ς on S such that

F R (w) = R β (w) + ∂ R τ R (w) ≤ R ς (w)
for every body R.

Remark 4.1. It is noted that the last Cauchy postulate implies that F R is a linear functional on the space of vector fields over R which is continuous with respect to the C 0 -topology. In fact, it is a statement of absolute continuity relative to ς .

Remark 4.2. If we consider the balance of a real valued property, or any other situation where the vector bundle W is trivial, one can consider the power for constant velocity fields. In the case of a balance of a real valued property, it is natural to define the total (resultant) force FR = F R (1). In this case, the fourth postulate implies that

FR = R β + ∂ R τ R ≤ R ς .
This is the relevant Cauchy postulate in the case of balance laws for real valued properties considered in [START_REF] Segev | On the geometry of Cauchy's theory for balance of scalar valued properties[END_REF].

Remark 4.3. Let {F R = (β R , τ R )} be a force system that satisfies Cauchy's postulates and let w be a vector field on S . Then,

{F R (w) = β R (w), τ R (w) },
is a force system for a balance of a real valued property on S for which the corresponding Cauchy postulates are satisfied.

Proposition 4.4 (The generalized Cauchy theorem). Assume that an orientation is given on S and let {F R = (β R , τ R )} be a force system that satisfies the generalized Cauchy postulates. Then, there exists a unique differential stress form σ , a section of m-1 (T S ,W * ), such that for any body R and

x ∈ ∂ R, τ R (x) = ι * h σ (x)
, where, h is the tangent space to ∂ R at x equipped with the outwards pointing inclination.

Proof. The sketch of proof described below is based on the corresponding detailed proof for the real valued case in [START_REF] Segev | On the geometry of Cauchy's theory for balance of scalar valued properties[END_REF]. Another simple method for the construction of the proof is implied by the discussion pertaining to Riemannian manifolds (see Remark 6.2).

Since the expression in the assertion of the proposition is invariant and local, it is enough to prove that it holds in a vector bundle chart on W . As any vector field w can be written locally as a linear combination w = w i e i , of the µ = dimW x base vector fields e i corresponding to the chart, we may consider only the scalar valued surface forces τ Ri = τ R (e i ). However, by Cauchy's theorem for scalar valued balance laws (see [START_REF] Segev | On the geometry of Cauchy's theory for balance of scalar valued properties[END_REF]), for each i = 1, . . . , m there is a unique (m -1)-form σ i , such that

τ Ri (x) = ι * h σ i (x)
. Clearly, the forms σ i transform as components of an element of W * -the required (m -1), W * -valued form σ .

Remark 4.5. In the case where an orientation is not given on S , the Proposition still holds if one replaces the form σ by an odd form (see [START_REF] Segev | Interactions on Manifolds and the Construction of Material Structure[END_REF] for the details of the required generalization). Orientation is assumed here merely to make the presentation clearer.

THE CASE OF VOLUME MANIFOLDS

5.1. The Stress Tensor. In this Section we restrict ourselves to the situation where a volume element θ is given on the material manifold S . Evidently, such a volume element induces an orientation on S . We will first make a straightforward extension of the contraction of a form by a vector. If we regard S φ as a vector valued form in k-1 (T x S ,W * x ), then S φ (v 1 , . . . ,v k-1 )(w) = φ (S(w), v 1 , . . . ,v k-1 ). Proposition 5.2. Given a volume element θ ∈ m T S and a stress form σ ∈ m-1 (T x S ,W * x ) there is a unique linear mapping S ∈ L(W x , T x S )

such that σ = S θ .

Proof. The mapping

L(W, T S ) → L W, m-1 (T S )
given by S → S θ is an isomorphism due to the fact that for S = 0, S θ (w)(v 1 , . . . ,v m-1 ) = θ S(w), v 1 , . . . ,v m-1 = 0 for some choice of w, v 1 , . . . ,v m-1 .

Thus, when a volume element is given, we obtain the representation of stresses by linear mappings as in traditional continuum mechanics. Given a volume element θ and a stress σ , we will refer to the unique linear mapping S such that S θ = σ as the stress tensor representing σ relative to θ . Using the orientation induced by the volume element θ we can define the mapping or :

G ⊥ m-1 (T x S ) → F (T x S ) m-1 , {+1, -1} , where F (T x S ) m-1 , {+1, -1} is the set of functions from (T x S ) m-1 to {+1, -1}, by or(h)(v 1 , . . . ,v m-1 ) = +1 if v, v 1 , . . . ,v m-1 are postively oriented, -1 if v, v 1 , . . . ,v m-1 are negatively oriented.
Here, v is any vector representing the inclination of h. Clearly, this mapping is odd. Next, consider the linear mapping

r H (θ ) : T x S → m-1 H,
given by r H (θ )(u) = I * H (u θ ), where I * H is the restriction of u θ to H induced by the inclusion I H . With this notation, the expression

τ R (x)(w)(v 1 , . . . ,v m-1 ) = or(h)(v 1 , . . . ,v m-1 ) θ (S(w), v 1 , . . . ,v m-1 ),
where v is any vector specifying the inclination of H = T x ∂ R, may be written in the form τ R (w) = or(h) r H θ )(S(w) .

Remark 5.3. Let u be any vector in H. Clearly, r H (θ )(u) = 0 and in fact Kernel(r H (θ )) = H. In the sequel will use the following notation. For a hyperplane H at x ∈ S , the one dimensional space of layers at x is Λ = T x S /H, i.e., a layer λ ∈ Λ is an equivalence class of vectors such that the difference between any two belongs to H. We will denote by π Λ : T x S → Λ the natural projection onto the quotient space.

Thus, there is a linear mapping rH :

Λ → m-1 H such that rH (θ )(λ ) = rH (θ ) π Λ (u) = r H (θ )(u),
where u is any element of the equivalence class λ = [u]. Thus, the relation between the surface force and the stress tensor assumes the form τ R (w) = or(h) rH (θ ) π Λ • S(w) .

Remark 5.4. An annihilator φ of H induces an isomorphism I φ : Λ → R by λ → φ (v), where v is any vector in the equivalence class λ = [v]. Thus,

I φ (π Λ • S) = φ • S = S * (φ ),
where S * : (T x S ) * → W * is the dual (adjoint) mapping to the stress tensor.

Also, I φ induces the mapping

ρ φ (θ ) : R → m-1 T x S , ρ φ (θ ) = rH (θ ) • I -1 φ , so we have τ R (w) = or(h) ρ φ (θ ) S * (φ )(w) .
As ρ φ (θ ) is represented by the unique (m -1)-form ρ φ = ρ φ (θ )(1) we may write τ R (w) = or(h) ρ φ S * (φ )(w).

Remark 5.5. Consider an additional body R such that x ∈ ∂ R and the inclined hyperplane tangent to R at

x is h = -h. Then τ R = or(h ) ρ φ S * (φ ) = -or(h) ρ φ S * (φ )
If we require that φ is compatible with the inclination of h so that φ (v) > 0 for every vector v specifying the inclination of h, then, the sign or(h) is included in the choice of φ and τ R = ρ φ S * (φ ) holds for both R and R .

THE CASE OF RIEMANNIAN MANIFOLDS

We now assume that a Riemannian metric is given on S . Hence, even if the volume element θ is not the one induced by the metric, we have a natural volume element n θ on ∂ R, where n is the unit outwards pointing normal vector. Thus, the surface force τ is represented by a W * -valued field t R , the component with respect to the basis element n θ , such that τ R = t R (n θ ).

On the other hand, with the metric structure there is a natural form φ n , φ n (u) = n • u, compatible with the inclination of h. Thus we have Hence, in the metric case, choosing the natural normal form, we have

ρ φ n = n θ .
In addition, S * (φ n )(w) = φ n S(w)

= n • S(w)

= S T (n)(w),
where S T : T x S → W *

x is defined by the condition v • S(w) = S T (v)(w).

We conclude that in the situation where S is a metric manifold, t R = S T (n)

-the classical Cauchy formula.

Remark 6.1. It is noted that in the reduction above it is not required that the volume element is the one induced by the Riemannian metric. Also there is no need for any additional structure on the vector bundle W .

Remark 6.2. The reduction to the classical Cauchy formula indicates another method of proof of Cauchy's theorem. Since the theorem is formulated without any reference to a Riemannian metric and since one can always assign a Riemannian metric to a manifold, Cauchy's theorem in the framework of Riemannian Geometry (e.g., [START_REF] Fosdick | A variational proof of the stress theorem of Cauchy[END_REF] or [START_REF] Marsden | Mathematical Foundations of Elasticity[END_REF] for the three dimensional scalar valued case) implies the validity of the generalized version of Cauchy's theorem given here.

Remark 6.3. The First Piola-Kirchhoff Stress Tensor, used frequently in continuum mechanics, is usually defined in term of the Cauchy stress tensor but is introduced as a different entity. (See for example [START_REF] Marsden | Mathematical Foundations of Elasticity[END_REF], p. 135, or [START_REF] Gurtin | An Introduction to Continuum Mechanics[END_REF] p. 178.) The foregoing formulation implies that the Cauchy stress and first Piola-Kirchhoff stress are simply two representations of the same mathematical entity under different metric tensors (in space and the body reference configuration, respectively). In other words, the Piola transformation of [START_REF] Marsden | Mathematical Foundations of Elasticity[END_REF] is simply the transformation rule for the stress tensor representing a stress form, under a change of a Riemannian metric.

Definition 5 . 1 .

 51 Let φ ∈ k (T x S ) be a form and let S ∈ L(W x , T x S ) be a linear mapping. The contraction S φ ∈ L(W x , k-1 T x S ) of the form φ with the mapping S is defined by (S φ )(w) = S(w) φ , for every w ∈ W x .

5. 2 .

 2 The Orthogonal Projection of the Stress Tensor. Consider a body R and a point x ∈ ∂ R. Let h be the inclined hyperplane tangent to ∂ R at x and let H = p(h) be T x ∂ R. In this paragraph we are concerned with the form that the relation τ R (x) = ι * h (σ ) assumes when a volume element is given.

1 =

 1 ρ φ n (v 1 , . . . ,v m-1 ) = ρ φ n (θ )(1)(v 1 , . . . ,v m-1 ) = θ (u, v 1 , . . . ,v m-1 ), φ n (u) = n • u = θ (n, v 1 , . . . ,v m-1), for u = n in particular,= n θ (v 1 , . . . ,v m-1 ).

Acknowledgements. The research was partially supported by the Paul Ivanier Center for Robotics Research and Production Management at Ben-Gurion University.