Francesco Boarotto
email: boarotto@ljll.math.upmc.fr

Mario Sigalotti
email: mario.sigalotti@inria.fr

TIME-OPTIMAL TRAJECTORIES OF GENERIC CONTROL-AFFINE SYSTEMS HAVE AT WORST ITERATED FULLER SINGULARITIES

We consider in this paper the regularity problem for time-optimal trajectories of a single-input control-affine system on a n-dimensional manifold. We prove that, under generic conditions on the drift and the controlled vector field, any control u associated with an optimal trajectory is smooth out of a countable set of times. More precisely, there exists an integer K, only depending on the dimension n, such that the non-smoothness set of u is made of isolated points, accumulations of isolated points, and so on up to K-th order iterated accumulations.

1. Introduction 1.1. Single-input systems and chattering phenomena. Let M be a smooth 1 , connected, n-dimensional manifold and denote by Vec(M ) the space of smooth vector fields on M . Consider the (single-input) control-affine system (1.1) q = f 0 (q) + uf 1 (q), q ∈ M, u ∈ [-1, 1], f 0 , f 1 ∈ Vec(M ).

An admissible trajectory of (1.1) is an absolutely continuous curve q : [0, T ] → M , T > 0, such that there exists u ∈ L ∞ ([0, T ], [-1, 1]) so that q(t) = f 0 (q(t)) + u(t)f 1 (q(t)) for almost every t ∈ [0, T ]. For any fixed initial datum q 0 ∈ M , the time-optimal control problem associated with (1.1) consists into looking for admissible trajectories q : [0, T ] → M , T > 0, that minimize the time needed to steer q 0 to q(T ) among all admissible trajectories.

A necessary (but not sufficient) condition for an admissible trajectory to be time-optimal is provided by the Pontryagin maximum principle (PMP, in short) [START_REF] Pontryagin | The mathematical theory of optimal processes[END_REF]. Introducing the controldependent Hamiltonian (1.2)

H : T * M × [-1, 1] → R, H(λ, v) = λ, (f 0 + vf 1 )(q) , q = π(λ),
the PMP states that if a trajectory q(•) associated with the control u(•) is time-optimal, then it is extremal, i.e., there exists an absolutely continuous curve t → λ(t) ∈ T * q(•) M \ {0} such that H(λ(t), u(t)) maximizes H(λ(t), •) for a.e. t ∈ [0, T ], and such that λ(t) = -→ H(λ(t), u(t)) a.e. on [0, T ]. (For the precise definition of the Hamiltonian vector field -→ H and further details see Section 2.) We call the triple (q(•), u(•), λ(•)) an extremal triple. In particular, the PMP reduces the problem of finding time-optimal trajectories to the study of extremal ones.

The kind of results we are interested in concern the regularity of time-optimal trajectories, even though our techniques handle in fact the broader class of extremal ones. Observe in any case that this is a hopeless task in full generality since, as proved by Sussmann in [START_REF] Sussmann | A weak regularity theorem for real analytic optimal control problems[END_REF], for any given measurable control t → u(t), there exist a dynamical system of the form (1.1) and an initial datum q 0 ∈ M for which the admissible trajectory driven by u and starting at q 0 is time-optimal.

It makes then sense to look for better answers imposing some genericity conditions on (f 0 , f 1 ) (with respect to the Whitney topology on the space of pairs of smooth vector fields on M ). The question we are then lead to tackle is the following: "What kind of behavior can we expect for time-optimal trajectories of a generic system?" Such a question corresponds to one of the open problems posed by A. Agrachev in [START_REF] Agrachev | Some open problems[END_REF].

The problem of the regularity of extremal trajectories for control-affine systems of the form (1.1) is known to be delicate. In his striking example, Fuller [START_REF] Fuller | Study of an optimum nonlinear system[END_REF] exhibited a polynomial system of the kind studied here, in which controls associated with optimal trajectories have a converging sequence of isolated discontinuities. Since then the phenomenon of fast oscillations (or chattering) is also called the Fuller phenomenon, and his presence has important consequences for example on the study of optimal syntheses [START_REF] Boscain | Optimal syntheses for control systems on 2-D manifolds[END_REF][START_REF] Caponigro | Regularization of chattering phenomena via bounded variation control[END_REF][START_REF]Chattering arcs and chattering controls[END_REF][START_REF] Piccoli | Regular synthesis and sufficiency conditions for optimality[END_REF][START_REF] Sussmann | Synthesis, presynthesis, sufficient conditions for optimality and subanalytic sets[END_REF]. Another striking feature of this phenomenon is its stability: if the dimension of M is sufficiently high, then chattering is structurally stable (i.e., it cannot be destroyed by a small perturbation of the initial system). The first result in this direction was presented in [16, Theorem 0] starting from dimension 6, but it was subsequently extensively explored in [START_REF] Zelikin | Theory of chattering control: with applications to astronautics[END_REF]. It is however worth mentioning the fact that, to the best of our knowledge, none of these extremal trajectories have yet been proved to be time-optimal, nor it is known in lower dimensions (already in the 3D case) whether or not the chattering appears for a generic choice of system (1.1). Finally, we remark that the absence of Fuller phenomena for (1.1) has been proved in dimension 2 for analytic systems and generic smooth systems [START_REF] Lobry | Contrôlabilité des systèmes non linéaires[END_REF][START_REF] Piccoli | Regular time-optimal syntheses for smooth planar systems[END_REF][START_REF] Sussmann | Time-optimal control in the plane[END_REF][START_REF] Sussmann | The structure of time-optimal trajectories for single-input systems in the plane: the C ∞ nonsingular case[END_REF]. A first extensive investigation of the chattering phenomenon for multi-input affine-control systems has been presented in [START_REF] Zelikin | Geometry of neighborhoods of singular trajectories in problems with multidimensional control[END_REF].

1.2. Fuller times along extremals trajectories. Many contributions have been provided to the description of the structure of optimal trajectories around a given point q ∈ M . The natural setting in which this problem is usually tackled is the study of all possible Lie bracket configurations between f 0 and f 1 at q [4, [START_REF] Agrachev | On the local structure of optimal trajectories in R 3[END_REF][START_REF] Bressan | The generic local time-optimal stabilizing controls in dimension 3[END_REF][START_REF] Krener | The structure of small time reachable sets in low dimension[END_REF][START_REF] Schättler | The local structure of time-optimal trajectories in dimension three under generic conditions[END_REF][START_REF] Schättler | On the local structure of time-optimal bang-bang trajectories in R 3[END_REF][START_REF] Sigalotti | Local regularity of optimal trajectories for control problems with general boundary conditions[END_REF][START_REF] Sussmann | Envelopes, conjugate points, and optimal bang-bang extremals[END_REF]. This approach, although very precise in its answers, has unfortunately the disadvantage of becoming computationally extremely difficult already for mildly degenerate situations in dimension 3.

Definition 1. Given an admissible trajectory q : [0, T ] → M of (1.1), we denote by O q (or simply O if no ambiguity is possible) the maximal open subset of [0, T ] such that there exists a control u : [0, T ] → [-1, 1], associated with q(•), which is smooth on O. We also define Σ q (or Σ if no ambiguity is possible) by

Σ = [0, T ] \ O.
An arc is a connected component of O. An arc ω is said to be bang if u can be chosen so that |u| ≡ 1 along ω, and singular otherwise. Two arcs are concatenated if they share one endpoint. The time-instant between two arcs is a switching time.

The set O q , defined as above, depends only on the trajectory q in the following sense: as long as f 1 (q(t)) is different from zero, the control u(t) is uniquely identified up to modification on a set of measure zero, while u can be chosen arbitrarily on {t | f 1 (q(t)) = 0}. Definition 2 (Fuller Times). Let Σ 0 be the set of isolated points in Σ and define the Fuller times as the elements of the set Σ \ Σ 0 . By recurrence, Σ k is defined as the set of isolated points of Σ \ (∪ k-1 j=0 Σ j ). If t ∈ Σ k then t is a Fuller time of order k. We say that a Fuller time is of infinite order if it belongs to

Σ ∞ = Σ \ (∪ k≥1 Σ k ).
The leading idea of this paper is to characterize the worst stable behavior for generic singleinput systems of the form (1.1), in terms of the maximal order of its Fuller times. The heuristics behind our strategy is the following: thinking of points in Σ\Σ 0 as "accumulations of switchings", points in Σ\(Σ 0 ∪Σ 1 ) as "accumulations of accumulations" and so on, then if t is a Fuller time of sufficiently high order, a large number of relations between f 0 (q(t)) and f 1 (q(t)) can be derived. The existence of such a point q(t) can then be ruled out by standard arguments based on Thom's transversality theorem (see, e.g., [START_REF] Abraham | Transversal mappings and flows. An appendix by Al Kelley[END_REF]Proposition 19.1], which can be used in combination with [14, §1.3.2] in order to the guarantee that the dense set of "good" systems can be taken open with respect to the Whitney C ∞ topology on the space of vector fields). The main result of this paper is the following. Theorem 3. Let M be a n-dimensional smooth manifold. There exists an open and dense set

V ⊂ Vec(M ) × Vec(M ) such that, if the pair (f 0 , f 1 ) is in V, then for every extremal triple (q(•), u(•), λ(•)) of the time-optimal control problem q = f 0 (q) + uf 1 (q), q ∈ M, u ∈ [-1, 1],
the trajectory q(•) has at most Fuller times of order (n -1) 2 , i.e.,

Σ = Σ 0 ∪ • • • ∪ Σ (n-1) 2 ,
where Σ and Σ j are defined as in Definition 2. Remark 4. Since each Σ i , for i = 1, . . . , (n -1) 2 , is discrete, as a consequence of Theorem 3 we deduce that the control u(•) associated with any extremal triple (q(•), u(•), λ(•)) is smooth out of a finite union of discrete sets (in particular, out of a set of measure zero).

As we already explained, deriving dependence relations directly on f 0 and f 1 is extremely complicated. The PMP naturally suggests to rather search for conditions in the cotangent space T * M , where they are more easily characterizable, and to subsequently project them down on the level of vector fields. On the other hand, the estimate (n -1) 2 on the maximal order of Fuller points obtained in this way is far from being optimal. The computation of the sharpest bound on the order of Fuller points is still an open problem. 1.3. Structure of the paper. In Section 2 we introduce the technical tools we need in the rest of the paper and we present a brief survey of related results. Section 3 is the starting point of the novel contributions of the paper: we prove that at Fuller times of order larger than zero, i.e., for t ∈ Σ \ Σ 0 , in addition to the conditions λ(t), f

1 (q(t)) = λ(t), [f 0 , f 1 ](q(t)) = 0, one also has that either λ(t), [f 0 + f 1 , [f 0 , f 1 ]](q(t)) = 0 or λ(t), [f 0 -f 1 , [f 0 , f 1 ]](q(t)) = 0.
The computations leading to this result do not require any genericity assumption. Section 4, which constitutes the technical core of this work, explains how to derive new conditions at each accumulation step and how to prove their independence. Section 5 concludes the proof of Theorem 3 and, finally, in Section 6, the case of time-optimal trajectories on three dimensional manifolds is analyzed in greater detail.

Acknowledgements. The authors have been supported by the ANR SRGI (reference ANR-15-CE40-0018) and by a public grant as part of the Investissement d'avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH, in a joint call with Programme Gaspard Monge en Optimisation et Recherche Opérationnelle. The authors also wish to thank the anonymous referee for the careful revision of the paper, and the detailed comments that let us significantly improve the quality of our exposition.

Previous results and consequences of Theorem 3

2.1. Notations. Let us introduce some technical notions which will be extensively used throughout the rest of the paper. Consider the cotangent space T * M of M , endowed with the canonical symplectic form σ. For any Hamiltonian function p : T * M → R, its Hamiltonian lift -→ p ∈ Vec(T * M ) is defined using the relation

σ λ (•, - → p ) = d λ p, • .
For all T > 0 and q 0 ∈ M we define the attainable set from q 0 at time T as A(T, q 0 ) = {q(t) | q : [0, T ] → M is an admissible trajectory of (1.1) such that q(0) = q 0 }.

The precise content of the PMP, already mentioned at the beginning of Section 1, is then recalled below (see [START_REF] Agrachev | Control theory from the geometric viewpoint[END_REF][START_REF] Pontryagin | The mathematical theory of optimal processes[END_REF]).

Theorem (PMP). Let q : [0, T ] → M be an admissible trajectory of (1.1), associated with a control u(•), such that q(T ) ∈ ∂A(T, q 0 ). Then there exists λ : [0, T ] → T * M absolutely continuous such that (q(•), u(•), λ(•)) is an extremal triple, i.e., in terms of the control-dependent Hamiltonian H introduced in (1.2),

λ(t) ∈ T * q(t) M \ {0}, ∀t ∈ [0, T ], H(λ(t), u(t)) = max v∈[-1,1] H(λ(t), v), for a.e. t ∈ [0, T ], (2.1) λ(t) = - → H(λ(t), u(t)), for a.e. t ∈ [0, T ]. (2.2)
Let (q(•), u(•), λ(•)) be an extremal triple. The curve q(•) is in particular said to be an extremal trajectory. We associate with (q(•), u(•), λ(•)) the switching function

h 1 (t) = λ(t), f 1 (q(t)) .
Differentiating a.e. on [0, T ], it follows from (2.2) that for every smooth vector field

X on M d dt λ(t), X(q(t)) = λ(t), [f 0 + u(t)f 1 , X](q(t)) , for a.e. t ∈ [0, T ].
In particular, h 1 is of class C 1 and, setting

h 01 (t) = λ(t), [f 0 , f 1 ](q(t)) , ∀t ∈ [0, T ],
we have ḣ1 (t) = h 01 (t) for every t ∈ [0, T ].

Remark 5. The maximality condition (2.1) implies that

H(λ(t), u(t)) = λ(t), f 0 (q(t)) + max v∈[-1,1] v λ(t), f 1 (q(t)) = λ(t), f 0 (q(t)) + | λ(t), f 1 (q(t)) |.
In particular, u(t) = sgn(h

1 (t)) ∈ {-1, +1} whenever h 1 (t) = 0.
Repeated differentiation shows that h 1 is smooth when the control is. In particular, in terms of the set O introduced in Definition 1,

h 1 | O ∈ C ∞ (O).
A folklore result on bang and singular arcs is the following. Recall that, for every f ∈ Vec(M ), ad f : Vec(M ) → Vec(M ) denotes the adjoint action defined by ad f g = [f, g]. Proposition 6. Assume that span{(ad k f0+f1 f 1 )(q) | k ∈ N} = T q M and span{(ad k f0-f1 f 1 )(q) | k ∈ N} = T q M for every q ∈ M . Fix an extremal triple (q(•), u(•), λ(•)) and an arc ω ⊂ O q . Then, either h 1 (t) = 0 for at most finitely many t ∈ ω and the arc is bang, or h 1 ≡ 0 on ω and the arc is singular.

Proof. Let us set Z = {τ ∈ ω | h 1 (τ ) = 0} and F ± = {τ ∈ ω | ±h 1 (τ ) > 0}.
Assume by contradiction that Z has infinitely many points and that it is different from ω. We have from Remark 5 that, up to modifying u on a set of measure zero, u ≡ 1 on F + and u ≡ -1 on F -. If Z has measure 0, then, by continuity of u| ω , u ≡ 1 or u ≡ -1 on ω. In particular, h

(k) 1 (t) = λ(t), (ad k f0+f1 f 1 )(q(t)) or h (k)
1 (t) = λ(t), (ad k f0-f1 f 1 )(q(t)) on ω. Since between any two vanishing points for h

(k-1) 1
there is a vanishing point for h (k) 1 , we deduce that at every cluster point t ∈ ω for Z (i.e., the limit of infinitely many distinct points in Z), λ(t) annihilates either (ad k f0+f1 f 1 )(q(t)) for every k ∈ N or (ad k f0-f1 f 1 )(q(t)) for every k ∈ N, leading to a contradiction. In the case where the measure of Z is positive, there exists t ∈ ω which is both a cluster point for Z and for either F + or F -. By continuity of h

(k) 1 | ω for every k ∈ N, we deduce that either h (k) 1 (t) = λ(t), (ad k f0+f1 f 1 )(q(t)) for every k ∈ N or h (k) 1 (t) = λ(t)
, (ad k f0-f1 f 1 )(q(t)) for every k ∈ N and we conclude as above.

Notice that the assumption that span{(ad k f0+f1 f 1 )(q) | k ∈ N} = T q M and span{(ad k f0-f1 f 1 )(q) | k ∈ N} = T q M for every q ∈ M holds true generically with respect to (f 0 , f 1 ) ∈ Vec(M ) 2 . From now on the term generic is used to express that a property of the pair of vector fields (f 0 , f 1 ) holds true on an open and dense subset of Vec(M ) × Vec(M ). Definition 7. Let A be the alphabet containing the letters {+, -, 0, 1}, and let

I = (i 1 • • • i d ) ∈ A d be a word of length d in A.
Then we employ the shorthand notation

f I = [f i1 , . . . , [f i d-1 , f i d ] . . . ],
with the convention that f ± = f 0 ± f 1 . Moreover, given an extremal triple (q(•), u(•), λ(•)) on [0, T ], we set

h I (t) = λ(t), f I (q(t)) , t ∈ [0, T ].
2.2. Previous results. Sussmann proved in [START_REF] Sussmann | A weak regularity theorem for real analytic optimal control problems[END_REF] that for every T > 0 and every control u ∈ L ∞ ([0, T ], [-1, 1]) there exists a control system of the type (1.1) and an initial datum q 0 such that the trajectory starting at q 0 and corresponding to u(•) is time-optimal. In generic situations, however, some further regularity can be expected, as recalled in the following three results.

Theorem 8 ([8, Theorem 0], [START_REF] Chitour | Singular trajectories of control-affine systems[END_REF]Theorem 2.6]). Generically with respect to

(f 0 , f 1 ) ∈ Vec(M ) 2 , for any extremal triple (q(•), u(•), λ(•)) on [0, T ] such that h 1 | [0,T ] ≡ 0, the set Ω = {t ∈ [0, T ] | h 101 (t) = 0} is of full measure in [0, T ] and u(t) = -h 001 (t)/h 101 (t) almost everywhere on Ω. Theorem 9 ([2, Proposition 1]). Let I(f 1 ) ⊂ Lie(f 0 , f 1 ) denote the ideal generated by f 1 . If I q (f 1 ) = T q M for every q ∈ M , then, for every extremal trajectory q : [0, T ] → M , the set O q is open and dense in [0, T ]. Theorem 10 ([7, Proposition 2]). Assume that span{(ad k f0+f1 f 1 )(q) | k ∈ N} = T q M and span{(ad k f0-f1 f 1 )(q) | k ∈ N} = T q M for
every q ∈ M . Consider an extremal trajectory q : [0, T ] → M such that the union of all bang arcs is open and dense in [0, T ]. Then either Σ = Σ 0 or there exists an infinite sequence of concatenated bang arcs.

Theorem 3 can be seen as an extension of Theorem 9 in the sense that it guarantees that, generically with respect to (f 0 , f 1 ), the open set O q is not only dense but also of countable complement and hence of full measure in [0, T ] (see Remark 4). A similar observation can be done for Theorem 10, which is generalized by Theorem 3 as follows: generically, for every k ≥ 0, either Σ = ∪ k j=0 Σ j or there exists a subinterval I of [0, T ] such that I ∩ Σ k is a converging sequence.

Concerning Theorem 8, we can strengthen its conclusion as stated in Corollary 11 below. The corollary is a direct consequence of Proposition 28, which is a step of the proof of Theorem 3 contained in Section 5.

Corollary 11. Generically with respect to the pair (f 0 , f 1 ) ∈ Vec(M ) 2 , for any extremal triple

(q(•), u(•), λ(•)) on [0, T ] such that h 1 | [0,T ] ≡ 0, the set Ω = {t ∈ [0, T ] | h 101 (t) = 0} has countable complement in [0, T ]
and u(t) = -h 001 (t)/h 101 (t) almost everywhere on Ω.

2.3.

Chattering and singular extremals. Classical instances of the chattering phenomenon occur when trying to join singular and bang arcs along time-optimal trajectories of control systems as in (1.1). Legendre condition [START_REF] Agrachev | Control theory from the geometric viewpoint[END_REF]Theorem 20.16] holds along singular extremal triples, and imposes the inequality h 101 (t) ≥ 0. If the inequality is strict, then the control u(t) is characterized as in Theorem 8, but there are significant examples of mechanical problems in which the third bracket f 101 vanishes identically (e.g. Dubin's car with acceleration [START_REF] Agrachev | Control theory from the geometric viewpoint[END_REF]Section 20.6]). This case has been intensively studied in [START_REF] Zelikin | Theory of chattering control: with applications to astronautics[END_REF], and the situation that forces the chattering can be essentially summarized as follows.

Theorem 12 [START_REF] Agrachev | Control theory from the geometric viewpoint[END_REF]Proposition 20.23]). Assume that the vector fields f 0 and f 1 satisfy the identity f 101 ≡ 0. Let q : [0, T ] → M be a time-optimal trajectory of system (1.1) which is the projection of a unique (up to a scalar factor) curve λ :

[0, T ] → T * M such that (q(•), u(•), λ(•)) is an extremal triple. Assume moreover that h 10001 (t) = 0 on [0, T ].
Then q(•) cannot contain a singular arc concatenated with a bang arc.

In particular, under the hypotheses of the theorem, the only possibility for an optimal trajectory to exit a singular arc is through chattering.

Annihilation conditions at Fuller times of an extremal trajectory

Let us fix an extremal triple (q(•), u(•), λ(•)) on [0, T ]. The goal of this section is to prove some useful annihilation conditions of functions of the form h I , with I a word in A (compare with Definition 7), at Fuller times, i.e., on Σ \ Σ 0 .

Since h 1 is (absolutely) continuous and u(t) = sgn(h 1 (t)) for almost every t such that h 1 (t) = 0, then

h 1 Σ ≡ 0.
Moreover, between two zeroes of h 1 , h

1 = h 01 has at least one zero, which yields h 01 Σ\Σ0 ≡ 0.

The following proposition states that both h 001 and h 101 vanish at every t ∈ Σ which is at positive distance from {t | h 1 (t) = 0}.

Proposition 13. Let t ∈ Σ be such that h 1 is identically equal to zero on a neighborhood of t. Then h 101 (t) = h 001 (t) = 0. Proof. Let V be a neighborhood of t such that h 1 | V ≡ 0. Therefore, the same is true for h 01 | V and (3.1) h 001 (τ ) + u(τ )h 101 (τ ) = 0 for almost every τ ∈ V .
Let us first prove that h 101 (t) = 0. By contradiction and up to reducing V , we have that h 101 (τ ) = 0 for every τ ∈ V . By (3.1), moreover, u(τ ) = -h001(τ ) h101(τ ) for almost every τ ∈ V . Notice that the differential system generated by the smooth autonomous Hamiltonian

H(p) = p, f 0 (π(p)) - p, f 001 (π(p)) p, f 101 (π(p)) p, f 1 (π(p)) ,
is well-defined on {p ∈ T * M | p, f 101 (π(p)) = 0} and all its trajectories are smooth. Since, moreover, the absolutely continuous curve (λ(t), q(t)) satisfies ṗ = -→ H (p) almost everywhere on V , we deduce that V t → (λ(t), q(t)) is a solution of the Hamiltonian system generated by H and that the control u is smooth on V , contradicting the fact that t ∈ Σ.

We conclude by showing that also h 001 (t) = 0. Following (

R) such that 1 C ≤ |h +01 (s)|, |h -01 (s)| ≤ C, ∀ s ∈ I,
for some positive constant C > 0.

By assumption, there exists a sequence of concatenated bang arcs in I, whose lengths we denote by {σ i } i∈N ∪ {τ i } i∈N ⊂ (0, +∞), with the agreement that u ≡ 1 (respectively, u ≡ -1) on the intervals of length σ i (respectively, τ i ) and that the arc of length σ i is concatenated with the arc of length τ i , which is concatenated with the arc of length σ i+1 and so on. Without loss of generality, the bang arcs converge towards t from the left, so that we can further assume that the arc of length σ i is concatenated at its right with the arc of length τ i (see Figure 1).

σ 1 = - 2h 01 (0) h +01 (0) + O(σ 2 1 ),
where the notation O(σ 2 1 ) has the following meaning: using an analogous Taylor expansion for each positive bang arc of length σ k , we obtain a reminder ρ k such that

ρ 2 k σ 2 k is uniformly bounded. We deduce from (3.2) that |h 01 (0)| ≈ σ 1 , where this notation is used to indicate that 1 c ≤ |h 01 (0)| σ 1 ≤ c
for some constant c > 0. Moreover, from the expansion h 01 (σ 1 ) = h 01 (0) + σ 1 h +01 (0) + O(σ 2 1 ), we get

h 01 (σ 1 ) = -h 01 (0) + O(σ 2 1 ).
Combining these two relations we obtain

τ 1 ≈ |h 01 (σ 1 )| ≈ σ 1 .
The same computations also imply that

σ 2 ≈ h 01 (σ 1 + τ 1 ) = -h 01 (σ 1 ) + O(τ 2 1 ) = h 01 (0) + O(σ 2 1
). In particular, the sequence σ i satisfies the relation

σ i+1 = σ i + O(σ 2 i ).
The contradiction is then a consequence of Lemma 15 below. Lemma 15. Let {t i } i∈N be a sequence of positive numbers satisfying the relation

t i+1 = t i + O(t 2 i ). Then, ∞ i=1 t i = +∞. Proof. Let c > 0 be such that (3.3) t i+1 ≥ t i (1 -ct i ), ∀ i ∈ N.
Assume by contradiction that ∞ i=1 t i < +∞. In particular, t i → 0. Up to discarding the first terms of the sequence {t i } i∈N , we can assume that 1ct i > 0 for all i ∈ N. Iterating (3.3) we deduce that

t i+1 ≥ t 1 i j=1 (1 -ct j ), ∀ i ∈ N.
Hence, for every i ∈ N,

log t i+1 ≥ log t 1 + i j=1 log(1 -ct j ) ≥ log t 1 -c i j=1 t j ,
where c > 0 is such that log(1ct j ) ≥ -c t j for all j ∈ N. The contradiction comes by noticing that the left-hand side goes to -∞ as i → ∞, while the right-hand side stays uniformly bounded.

We say that an arc is bi-concatenated if it is concatenated both at its right and at its left with other arcs. Proposition 16. Let I be a bang arc compactly contained in (0, T ) and which is not biconcatenated. Then there exists t ∈ Ī such that either h +01 (t) = 0 or h -01 (t) = 0.

Proof. Without loss of generality, assume that u ≡ 1 on I = (t 1 , t 2 ) and that I is not concatenated with any other arc at t 2 . In particular, t 2 is a cluster point for Σ ∩ (t 2 , T ]. If h 1 ≡ 0 on a right neighborhood of t 2 , then the conclusion follows from Proposition 13 and the continuity of h +01 and h -01 .

We can then assume that there exists a sequence of times converging from above to t 2 and at which h 1 is not zero. Then, necessarily, there exist a sequence of arcs I n converging to t 2 . Pick, for every n ∈ N a time τ n ∈ I n such that h 01 (τ n ) = 0. By construction, the sequence (τ k ) k∈N converges to t 2 and, by continuity, we deduce that also h 01 (t 2 ) = 0.

Since h 1 (t 1 ) = h 1 (t 2 ) = 0, then by the mean value theorem h 01 vanishes at an interior point of I, and this in turns implies that d dt h 01 | I = h +01 | I also vanishes somewhere on I. The main result of the section is the following theorem.

Theorem 17. Let t ∈ Σ \ Σ 0 . Then h 1 (t) = h 01 (t) = 0 and, in addition, either h +01 (t) = 0 or h -01 (t) = 0.

Proof. We already noticed that h 1 vanishes on Σ and h 01 on Σ \ Σ 0 . We are going to prove the theorem by showing that there exists a sequence of points converging to t at which either h +01 or h -01 vanishes.

Since t ∈ Σ 0 and thanks to Proposition 13, we can assume without loss of generality that h 1 does not vanish identically on a neighborhood of t. Hence, there exists a sequence (τ n ) n∈N ⊂ [0, T ] converging to t such that h 1 (τ n ) = 0 for every n ∈ N. Each τ n is contained in an arc ω n . If the arc is singular, then it contains a nonempty subinterval on which h 1 ≡ 0. Since moreover h 1 has either a positive maximum or a negative minimum on ω n , we deduce that there exists an inflection point of h 1 on ω n at which h +01 or h -01 vanishes.

We can then assume without loss of generality that ω n is a bang arc for every n ∈ N. Let us consider the maximal concatenation of bang arcs from ω n towards t. Three possibilities occur: (i) the concatenation is infinite and converges to a point between τ n and t, (ii) the concatenation stops with a bang arc which is not bi-concatenated, and (iii) the concatenation stops with a bang arc concatenated with a singular one. In each of the three cases, we prove that there exists a point between ω n and t at which either h +01 or h -01 vanishes. In cases (i) and (ii) the conclusion follows from Propositions 14 and 16 respectively. In the case of a bang arc concatenated with a singular one, either h 1 does not vanish everywhere on the singular arc, and we deduce as above that there exists an inflection point of h 1 on the singular arc at which h +01 or h -01 vanishes, or h 01 = 0 at the junction of the two arcs and then the bang arc contains an inflection point of h 1 at which h +01 or h -01 vanishes. This concludes the analysis in case (iii) and hence the proof of the theorem.

High-order Fuller points and genericity results

In this section we look at the new dependence conditions appearing for accumulations of Fuller points of order higher than one. We start by introducing some useful notation.

Remark 18. For any given word J = (j 1 , . . . , j r ) ∈ A r , with r ≥ 3, j r-1 = 0, j r = 1, and at least one j k in {+, -}, an easy inductive argument proves that, with the notations of Definition 7, we can decompose f J as

f J = f J1 + • • • + f J l ,
where J 1 , . . . , J l are all words of length r written only with letters in {0, 1}, ending with the string (01) and such that, if |J i | a counts the number of occurrences of the letter a in J i , then

|J 1 | 0 = max i=1,...,l |J i | 0 , and |J 2 | 1 = max i=1,...,l |J i | 1 .
Moreover, J 1 and J 2 are uniquely determined by this requirement. S I (λ, j N q (f 0 ), j N q (f 1 )) = λ, f I (q) , q = π(λ). Similarly, we call Q :

Definition 19. Let N ∈ N. A function S : T * M × J N M × J N M → R is
T * M ×J N M ×J N M → R a polynomial relation if there exist l, d 1 , . . . , d l ∈ N \ {0} and words I 1 ∈ A d1 , . . . , I l ∈ A d l such that (4.2) Q(λ, j N q (f 0 ), j N q (f 1 )) ∈ R[S I1 (λ, j N q (f 0 ), j N q (f 1 )), . . . , S I l (λ, j N q (f 0 ), j N q (f 1 ))]. Moreover, we set deg(Q) = max{d 1 , . . . , d l }.
Finally, given two simple relations S I , S J , with a slight abuse of notation we say that the Poisson bracket {S I , S J } between S I and S J is the simple relation S IJ , where IJ is defined by concatenation of words. We extend the Poisson bracket notation to polynomial relations by linearity and the Leibnitz rule.

In the following two lemmas we show how to derive new algebraic conditions on the jets of the vector fields f 0 and f 1 when increasing the order of the Fuller point.

Lemma 20. Let l, d 1 , . . . , d l ∈ N \ {0} and consider l words I 1 ∈ A d1 , . . . , I l ∈ A d l with d j < d l for every j < l and I l = (+I l-1 ), where we denote by (+I l-1 ) the concatenation of the letter + and the word I l-1 . Fix an integer N > d l and consider the family of simple relations S j = S Ij , 1 ≤ j ≤ l, using the notation introduced in (4.1). Define the set

B ⊂ T * M × J N M × J N M by B = (λ, j N q (f 0 ), j N q (f 1 )) | q = π(λ), (f 0 , f 1 ) ∈ Vec(M ) 2 , S 1 (λ, j N q (f 0 ), j N q (f 1 )) = • • • = S l (λ, j N q (f 0 ), j N q (f 1 )) = 0 . If (q(•), u(•), λ(•)
) is an extremal triple on [0, T ] for the time-optimal control problem (1.1) associated with the pair (f 0 , f 1 ), and if the sequence

{t i } i∈N ⊂ [0, T ] is such that i) (λ(t i ), j N q(ti) (f 0 ), j N q(ti) (f 1 )) ∈ B for every i ∈ N, ii) there exists t ∞ = lim i→∞ t i ,
then there exists a further simple relation

S l+1 ∈ S (-I l-1 ) , S (-I l ) , S (+I l )
such that (λ(t ∞ ), j N q(t∞) (f 0 ), j N q(t∞) (f 1 )) ∈ B ∩ {S l+1 = 0}. Finally, defining for every q ∈ M the set B q ⊂ T * q M × J N q M × J N q M by B q = (λ, j N q (g 0 ), j N q (g 1 )) | λ ∈ T * q M \ {0}, (g 0 , g 1 ) ∈ Vec(M ) 2 , g 0 (q) ∧ g 1 (q) = 0,

S 1 (λ, j N q (g 0 ), j N q (g 1 )) = • • • = S l (λ, j N q (g 0 ), j N q (g 1 )) = 0 , if the codimension of B q in T * q M × J N q M × J N q M is equal to l, then codim T * q M ×J N q M ×J N q M (B q ∩ {S l+1 = 0}) = l + 1. Proof. Let (q(•), u(•), λ(•)
) be an extremal triple defined on [0, T ] and {t i } i∈N ⊂ [0, T ] be a sequence of points satisfying i) and ii) in the statement. Then, since for every word J ∈ {I 1 , . . . , I l } we have that h J (t i ) = λ(t i ), f J (q(t i )) vanishes for every i ∈ N, by continuity the same is also true for h J (t ∞ ), which implies that the point (λ(t ∞ ), j N q(t∞) (f 0 ), j N q(t∞) (f 1 )) belongs to B. Now, up to the choice of a suitable subsequence of {t i } i∈N , we infer the identity

0 = lim i→∞ h J (t ∞ ) -h J (t i ) t ∞ -t i = lim i→∞ 1 t ∞ -t i t∞ ti (h 0J (τ ) + u(τ )h 1J (τ ))dτ (4.3) = h 0J (t ∞ ) + ūh 1J (t ∞ ), ū = lim i→∞ 1 t ∞ -t i t∞ ti u(τ )dτ ∈ [-1, 1],
which is valid for every J ∈ {I 1 , . . . , I l }. The first of our claims is then proved. Indeed, if ū = ±1 we use (4.3) with J = I l to deduce that λ(t ∞ ), f (±I l ) (q(t ∞ )) = 0, so that S l+1 is in the form S (±I l ) , and we are done. If, on the other hand, ū ∈ (-1, 1) we apply (4.3) with J = I l-1 , and we deduce that

S (ū,l-1) (λ(t ∞ ), j N q(t∞) (f 0 ), j N q(t∞) (f 1 )) := λ(t ∞ ), f (0I l-1 ) (q(t ∞ )) + ūf (1I l-1 ) (q(t ∞ )) = 0. The combination of the relations S (ū,l-1) = S l = 0 at the point (λ(t ∞ ), j N q(t∞) (f 0 ), j N q(t∞) (f 1 )) yields λ(t ∞ ), f (0I l-1 ) (q(t ∞ )) = 0 and λ(t ∞ ), f (1I l-1 ) (q(t ∞ )) = 0,
which in turn implies that λ(t ∞ ), f (-I l-1 ) (q(t ∞ )) = 0, so that we conclude by taking S l+1 = S (-I l-1 ) .

To prove the second claim of the statement, it is not restrictive to work within a coordinate neighborhood (U, x) ⊂ R n centered at the origin (identified with q), the whole argument being local. Then g i (x) = n j=1 α j i (x)∂ xj on U , for i = 0, 1. On J N 0 M × J N 0 M , J N 0 g 0 and J N 0 g 1 are given in local coordinates respectively by (α j 0 (0), ∇α j 0 (0), . . . , ∇ (N ) α j 0 (0), 0, . . . , 0

) n j=1 ∈ R × R n × • • • × R n N 2×n and (0, . . . , 0, α j 1 (0), ∇α j 1 (0), . . . , ∇ (N ) α j 1 (0)) n j=1 ∈ R × R n × • • • × R n N 2×n .
Moreover, since g 0 (q) ∧ g 1 (q) = 0, without loss of generality we can assume that

α 1 0 (0) = α 2 1 (0) = 1, α j 0 (0) = 0 if j = 1, α j 1 (0) = 0 if j = 2. Let the codimension of B 0 in T * 0 M × J N 0 M × J N
0 M be equal to l, and assume that S l+1 is of the form S (±I l ) . In particular, the degree of S l+1 is maximal among {deg(S 1 ), . . . , deg(S l+1 )}. Following Remark 18, let us write the decomposition

g I l+1 = g I l+1,1 + • • • + g I l+1,k ,
where we recall that I l+1,1 is uniquely identified by the requirement that it contains the maximal number, say s, of occurrences of the letter 0. Writing the analogous decomposition for simple relations

S I l+1 = S I l+1,1 + • • • + S I l+1,k ,
we see that the coordinate expression of S I l+1,1 at (λ, j N 0 (g 0 ),

j N 0 (g 1 )) ∈ T * 0 M \{0}×J N 0 M ×J N 0 M takes the form 0 = λ, g I l+1,1 (0) = n j=1 λ j ∂ s x1 ∂ deg(S l+1 )-s-1 x2 α j 1 (0) + P I l+1,1 (λ, j N 0 (g 0 ), j N 0 (g 1 )),
where P I l+1,1 is a polynomial expression in the coordinates of λ, j N 0 (g 0 ) and j N 0 (g 1 ) that does not contain any term of the form

∂ s x1 ∂ deg(S l+1 )-s-1 x2
α j 1 (0), for 1 ≤ j ≤ n. By construction, these terms do not appear in any of the other summands λ, g I l+1,i , for i = 1, neither among all other simple relations S 1 , . . . , S l . Therefore, as λ = 0, we infer the existence of a further independent relation, and we conclude that

codim T * 0 M ×J N 0 M ×J N 0 M (B 0 ∩ {S l+1 = 0}) = l + 1.
The case in which S l+1 = S (-I l-1 ) can be tackled similarly. In this situation deg(S l ) = deg(S l+1 ) > deg(S i ) for every i < l. We may again exploit Remark 18, and isolate the terms g I l,1 , g I l,2 and g I l+1,1 , g I l+1,2 in the decompositions of g I l and g I l+1 respectively. Observe that, by definition of I l and I l+1 , one has g I l+1,1 = g I l,1 and g I l+1,2 = -g I l,2 . Moreover, 0 appears s times in I l,1 , while 1 appears t times in I l,2 , and both s and t are maximal among their corresponding decompositions, so that we can write 0 = λ, g I l (0) = λ, g I l,1 (0) + λ, g I l,2 (0) + P I l (λ, j N 0 (g 0 ), j N 0 (g 1 ))

=

n j=1 λ j ∂ s x1 ∂ deg(S l )-s-1 x2 α j 1 (0) + ∂ deg(S l )-t-1 x1 ∂ t x2 α j 0 (0) + Q I l (λ, j N 0 (g 0 ), j N 0 (g 1 )), 0 = λ, g I l+1 (0) = λ, g I l,1 (0) -λ, g I l,2 (0) + P I l+1 (λ, j N 0 (g 0 ), j N 0 (g 1 )) = n j=1 λ j ∂ s x1 ∂ deg(S l )-s-1 x2 α j 1 (0) -∂ deg(S l )-t-1 x1 ∂ t x2 α j 0 (0) + Q I l+1 (λ, j N 0 (g 0 ), j N 0 (g 1 )),
where P I l , P I l+1 , Q I l , Q I l+1 are polynomial expressions in the coordinates of λ, j N 0 (g 0 ) and j N 0 (g 1 ) that do not contain any term of the form

∂ s x1 ∂ deg(S l )-s-1 x2 α j 1 (0) and ∂ deg(S l )-t-1 x1
∂ t x2 α j 0 (0), for 1 ≤ j ≤ n. In addition, these two terms are neither found among all other simple relations S 1 , . . . , S l-1 . Thus, as λ = 0, the relations λ, g I l (0) = 0 and λ, g I l+1 (0) = 0 are mutually independent (since their gradients are not parallel) and also independent from λ, g I k (0) = 0, k = 1, . . . , l -1.

Lemma 21. Let l, d 1 , . . . , d l ∈ N\{0} and consider l words I 1 ∈ A d1 , . . . , I l ∈ A d l with d j < d l-1 for every j < l -1 and d l-1 = d l . Suppose that there exists j < l -1 such that I l-1 = (0 I j ) and I l = (1 I j ). Using the notations introduced in (4.1) and (4.2), consider the family of polynomial relations Q r , r ∈ N\{0}, constructed inductively using the simple relations S I1 , . . . , S I l as follows

Q 1 = det {S 0 , S I l } {S 1 , S I l } {S 0 , S I l-1 } {S 1 , S I l-1 } , Q r = det {S 0 , S I l } {S 1 , S I l } {S 0 , Q r-1 } {S 1 , Q r-1 } for r ≥ 2.
Fix h ∈ N, an integer N > d l + h, and define the set

B ⊂ T * M × J N M × J N M by B = (λ, j N q (f 0 ), j N q (f 1 )) q = π(λ), S I1 (λ, j N q (f 0 ), j N q (f 1 )) = • • • = S I l (λ, j N q (f 0 ), j N q (f 1 )) = 0 Q 1 (λ, j N q (f 0 ), j N q (f 1 )) = • • • = Q h (λ, j N q (f 0 ), j N q (f 1 )) = 0 . If (q(•), u(•), λ(•)
) is an extremal triple on [0, T ], and if the sequence

{t i } i∈N ⊂ [0, T ] is such that i) (λ(t i ), j N q(ti) (f 0 ), j N q(ti) (f 1 )) ∈ B for every i ∈ N, ii) there exists t ∞ = lim i→∞ t i , then, setting I l+1 = (0I l ), I l+2 = (1I l ), either (λ(t ∞ ), j N q(t∞) (f 0 ), j N q(t∞) (f 1 )) ∈ B ∩ {S I l+2 = 0} ∩ {Q h+1 = 0}, or (λ(t ∞ ), j N q(t∞) (f 0 ), j N q(t∞) (f 1 )) ∈ B ∩ {S I l+1 = 0} ∩ {S I l+2 = 0}. Finally, defining for every q ∈ M the set B q ⊂ T * q M × J N q M × J N q M by B q = (λ, j N q (g 0 ), j N q (g 1 )) | λ ∈ T * q M \ {0}, (g 0 , g 1 ) ∈ Vec(M ) 2 , g 0 (q) ∧ g 1 (q) = 0, S I1 (λ, j N q (g 0 ), j N q (g 1 )) = • • • = S I l (λ, j N q (g 0 ), j N q (g 1 )) = 0 , if the codimension of B q in T * q M × J N q M × J N q M is equal to l, then codim T * q M ×J N q M ×J N q M (B q ∩ {S I l+2 = 0} ∩ {Q 1 = 0} ∩ • • • ∩ {Q h+1 = 0}) = l + h + 1, codim T * q M ×J N q M ×J N q M (B q ∩ {S I l+1 = 0} ∩ {S I l+2 = 0}) = l + 2.
Proof. The proof of the first part of the statement follows along the same lines of Lemma 20, using equation (4.3) both on S I l and on Q h , with the convention that Q 0 = S I l-1 . We prove in this way that the relations

(4.4) {S 0 , S I l } + ū{S 1 , S I l } = 0 and {S 0 , Q h } + ū{S 1 , Q h } = 0 hold at (λ(t ∞ ), j N q(t∞) (f 0 ), j N q(t∞) (f 1 )
), where the value ū is the same in both identities, since it is computed as the limit of a common sequence. If S I l+2 = {S 1 , S I l } vanishes on the triple (λ(t ∞ ), j N q(t∞) (f 0 ), j N q(t∞) (f 1 )), then so does S I l+1 = {S 0 , S I l }. From equation (4.4) we also deduce that (1, ū) is in the kernel of

{S 0 , S I l } {S 1 , S I l } {S 0 , Q h } {S 1 , Q h } ,
and therefore that its determinant Q h+1 vanishes at (λ(t ∞ ), j N q(t∞) (f 0 ), j N q(t∞) (f 1 )). In order to prove the second part of the statement, as in Lemma 21 the idea is to express all relations in local coordinates around q on the product space T * q M × J N q M × J N q M , with the non-restrictive hypothesis that g 0 (0) = ∂ x1 and g 1 (0) = ∂ x2 . Notice that for what concerns the codimension of B q ∩ {S I l+1 = 0} ∩ {S I l+2 = 0} we can reason exactly as in Lemma 21, since we deal in fact only with simple relations. Thus we are left with the task of proving that, if S I l+2 = 0, each polynomial relation Q r provides a condition independent from S I1 , . . . , S I l and Q 1 , . . . , Q r-1 .

By construction, Q r is a polynomial relation in the variables S (AB) , where (AB) is the concatenation of a word A of length at most r with letters in {0, 1} and a word B equal either to I l-1 or I l . It is not hard to show, by induction, that

Q r = (-1) r (S I l+2 ) r ad r S0 (S I l-1 ) + Q r = (-1) r S r (1I l ) S (0•••0I l-1 ) + Q r ,
where ad r S0 denotes the iterated Poisson bracket with S 0 and Q r is a polynomial relation in the same variables as Q r except for ad r S0 (S I l-1 ). Following Remark 18, we further decompose

f I l-1 as f I l-1 ,1 + • • • + f I l-1 ,k
, where the letter 0 appears in I l-1,1 the maximal number of times, say s, among the collection {I l-1,i } k i=1 . In coordinates we then write

ad r S0 (S I l-1 )(λ(t ∞ )) = n j=1 λ j ∂ r+s x1 ∂ deg(S I l-1 )-s-1 x2 α j 1 (0) + P I l-1 (λ, j N 0 (g 0 ), j N 0 (g 1 )),
where P I l-1 is a polynomial expression in λ, j N 0 (g 0 ) and j N 0 (g 1 ) that does not contain any term of the form

∂ r+s x1 ∂ deg(S I l-1 )-s-1 x2
α j 1 (0). Since λ = 0 and the above is true for any r ∈ N, we conclude that, as soon as S I l+2 = 0, each Q r gives a new independent condition, and the claim on the codimension follows.

4.1. Collinear case. The computation of the codimension of the sets B q identified in Lemmas 20 and 21 relies on the linear independence at q of f 0 (q) and f 1 (q). We study in this section what happens when the condition f 0 (q) ∧ f 1 (q) = 0 fails to hold.

We associate with the pair (f 0 , f 1 ) ∈ Vec(M ) 2 the collinearity set

(4.5) C = {q ∈ M | f 0 (q) ∧ f 1 (q) = 0}. Lemma 22. Let u ∈ L ∞ ([0, T ], [-1, 1]
) and q : [0, T ] → M be a trajectory of the control system (1.1) associated with the control u.

Assume that t ∞ ∈ [0, T ] is such that q(t ∞ ) ∈ {q ∈ M | f 1 (q) ∧ [f 0 , f 1 ](q) = 0}
and that there exists a sequence {t i } i∈N ⊂ [0, T ] converging to t ∞ such that q(t i ) ∈ C for every i ∈ N. Then there exists

ū := lim i→∞ 1 t ∞ -t i t∞ ti u(τ )dτ ∈ [-1, 1] and f 0 (q(t ∞ )) + ūf 1 (q(t ∞ )) = 0. Proof. First notice that, by continuity, f 0 (q(t ∞ )) ∧ f 1 (q(t ∞ )) = 0. Moreover, since f 1 (q(t ∞ )) ∧ [f 0 , f 1 ](q(t ∞ )) = 0,
the set C is, locally around q(t ∞ ), contained in an embedded (n -1)dimensional manifold Ĉ transversal to the vector field f 1 . This can be seen, for instance, by choosing a local system of coordinates (x 1 , . . . , x n ) such that

f 1 = ∂ x1 near q(t ∞ ). Write f 0 (x) = n i=1 a i (x)∂ xi . Then C is locally described by the conditions a 2 (x) = • • • = a n (x) = 0.
Furthermore, up to restricting the coordinate chart, the condition f 1 (q(t ∞ )) ∧ [f 0 , f 1 ](q(t ∞ )) = 0 implies that there exists j ∈ {2, . . . , n} such that ∂ x1 a j is nowhere vanishing. In particular, C is locally contained in the manifold Ĉ = {x | a j (x) = 0}, which is transversal to f 1 .

Let us take any coordinate system around q(t ∞ ). Notice that any converging subsequence of q(t∞)-q(ti) t∞-ti is tangent to Ĉ. Writing

q(t ∞ ) -q(t i ) t ∞ -t i = 1 t ∞ -t i t∞ ti (f 0 (q(τ )) + u(τ )f 1 (q(τ )))dτ,
we deduce that for every converging subsequence of { 1 t∞-ti t∞ ti u(τ )dτ } i∈N , its limit ũ is such that f 0 (q(t ∞ )) + ũf 1 (q(t ∞ )) is tangent to Ĉ. The proof is concluded by noticing that, by transversality of Ĉ and f 1 , the only vector of the form f 0 (q(t ∞ )) + ũf 1 (q(t ∞ )) ∈ span(f 1 (q(t ∞ ))) which is tangent to Ĉ is zero.

Remark 23. The lemma says in particular that for every (f 0 , f 1 ) ∈ Vec(M ) 2 and every trajectory q : [0, T ] → M of (1.1), almost everywhere on {t ∈ [0, T ] | q(t) ∈ C, f 1 (q) ∧ [f 0 , f 1 ](q) = 0} we have q = 0. This result is in the same spirit as [START_REF] Chitour | Singular trajectories of control-affine systems[END_REF]Theorem 2.1], where the multi-input case in considered.

Definition 24. For any extremal triple (q(•), u(•), λ(•)) on [0, T ] of the time-optimal control problem (1.1), we call Ω = {t ∈ [0, T ] | q(t) ∈ C, h 1 (t) = 0}. Moreover, we denote by Ω 0 the set of all isolated points in Ω, and inductively we declare Ω k to be the set of isolated points in Ω \ ( 2 and let (q(•), u(•), λ(•)) be any extremal trajectory on [0, T ] of the time-optimal control problem (1.1). Assume that there exist a sequence {t i } i∈N ⊂ [0, T ] and an integer k ≥ 0 such that

k-1 j=0 Ω j ). Theorem 25. Let (f 0 , f 1 ) ∈ Vec(M )
a) t i ∈ Ω \ ( k j=0 Ω j ) for every i ∈ N, b) there exists t ∞ = lim i→∞ t i and q(t ∞ ) ∈ {q ∈ M | f 1 (q) ∧ [f 0 , f 1 ](q) = 0}.
Then there exists a ∈ [-1, 1] such that, with the notation s a (λ) := λ, (f 0 + af 1 )(π(λ)) , we have ad j sa (s 1 )(λ(t ∞ )) = 0, for every 0 ≤ j ≤ k + 2. Proof. We proceed by induction on k, and we begin with the case k = 0. First notice that for t ∈ Ω both h 0 (t) = 0 and h 1 (t) = 0. Hence, also s 1 (λ(t)) = 0. By continuity and by Rolle's theorem, {s 0 , s 1 }(λ(t)) = h 01 (t) = 0 for every t ∈ Ω \ Ω 0 . Also notice that {s 0 , s 1 } = ad sa s 1 for every a ∈ [-1, 1]. Moreover, by item b) and Lemma 22, there exists

a = lim i→∞ 1 t ∞ -t i t∞ ti u(τ )dτ ∈ [-1, 1]
and f 0 (q(t ∞ )) + af 1 (q(t ∞ )) = 0.

From the identity

0 = 1 t ∞ -t i t∞ ti d dτ {s 0 , s 1 }(λ(τ ))dτ = 1 t ∞ -t i t∞ ti
({s 0 , {s 0 , s 1 }}(λ(τ )) + u(τ ){s 1 , {s 0 , s 1 }}(λ(τ ))) dτ, which is valid for every i ∈ N, passing to the limit as i → ∞ we deduce the further relation ad 2 sa s 1 (λ(t ∞ )) = ad sa {s 0 , s 1 }(λ(t ∞ )) = 0. Assume now that the theorem holds for some k ∈ N, and consider any sequence of points {t i } i∈N ∈ Ω \ ( k+1 j=0 Ω j ) satisfying items a) and b). Apply Lemma 22 and define a as above. The conclusion comes from noticing that

0 = 1 t ∞ -t i t∞ ti d dτ ad k+1 sa (s 1 )(λ(τ ))dτ = 1 t ∞ -t i t∞ ti (ad s0 ad k+1 sa (s 1 ) + u(τ )ad s1 ad k+1 sa (s 1 ))(λ(τ ))dτ → ad k+2 sa (s 1 )(λ(t ∞ )) as i → ∞.
Inspired by the arguments of [8, Definition 4 and Lemma 4], we are now in the position of deducing quantitative estimates on the possible accumulations of points of Ω within the collinearity set C. Lemma 26. Let q ∈ M and N = n-1. Let us define the following two subsets of J N q M ×J N q M :

L = (j N q (f 0 ), j N q (f 1 )) ∈ J N q M × J N q M | dim (span{f 0 (q), f 1 (q), [f 0 , f 1 ](q)}) ≤ 1 , L = (j N q (f 0 ), j N q (f 1 )) ∈ J N q M × J N q M | f 1 (q) = 0, ∃ a ∈ R such that f 0 (q) = af 1 (q) and dim span{ad i f0+af1 (f 1 )(q) | 0 ≤ i ≤ n -1} < n . Then codim J N q M ×J N q M L = 2n -2 and codim J N q M ×J N q M L = n. Proof.
The first assertion is clear. For the second one just notice that for every a ∈ R, the dimension of span{ad i f0+af1 (f 1 )(q) | 0 ≤ i ≤ n-1} is smaller than n if and only if, in coordinates, det(H) = 0, with H = f 1 , . . . , ad n-1 f0+af1 (f 1 ) .

The latter condition, taking as a the unique scalar such that f 0 (q) + af 1 (q) = 0, identifies a set of codimension one inside

D := (j N q (f 0 ), j N q (f 1 )) ∈ J N q M × J N q M | f 1 (q) = 0, f 0 (q) ∧ f 1 (q) = 0 .
Summing it up, we deduce that

codim J N q M ×J N q M L = codim J N q M ×J N q M D + codim D (j N q (f 0 ), j N q (f 1 )) ∈ J N q M × J N q M | det(H) = 0 = (n -1) + 1 = n.
Corollary 27. Let n ≥ 2. For a generic pair (f 0 , f 1 ) ∈ Vec(M ) 2 and for every extremal trajectory of the time-optimal control problem (1.1), we have

Ω = Ω 0 ∪ • • • ∪ Ω n-2 ,
where Ω and Ω j are defined as in Definition 24.

Proof. If along an extremal triple (q(•), u(•), λ(•)) there exists t ∈ Ω \ ( n-3 j=0 Ω j ), which is not isolated in this set and such that q(t) ∈ {q ∈ M | f 1 (q) ∧ f 01 (q) = 0}, then by Theorem 25 λ(t) annihilates ad i f0+af1 (f 1 )(q(t)) for every 0 ≤ i ≤ n -1, where a is the proportionality coefficient between -f 0 (q(t)) and f 1 (q(t)). By Lemma 26 and Thom's transversality theorem (see, e.g., [START_REF] Abraham | Transversal mappings and flows. An appendix by Al Kelley[END_REF][START_REF] Goresky | Stratified Morse Theory[END_REF]), for a generic pair (f 0 , f 1 ) ∈ Vec(M ) 2 this is possible only at isolated points of M . Equivalently, for a generic pair (f 0 , f 1 ) ∈ Vec(M ) 2 the set Ω is equal to n-2 j=0 Ω j . On the other hand, another application of Thom's transversality theorem says that, for a generic choice of (f 0 , f 1 ), the points q ∈ M such that f 0 (q) ∧ f 1 (q) = 0 and f 1 (q) ∧ f 01 (q) = 0 are isolated (since 2n -2 ≥ n when n ≥ 2). This concludes the proof.

Proof of Theorem 3

Theorem 3 directly follows from Theorem 17 and Proposition 28 below.

Proposition 28.

There exists an open and dense set V ⊂ Vec(M ) 2 such that, for any pair (f 0 , f 1 ) ∈ V and for any extremal triple (q(•), u(•), λ(•)) on [0, T ] of the time-optimal control problem (1.1), the set 2 , where Ξ 1 denotes the set of isolated points of Ξ and Ξ j+1 denotes the set of isolated points of Ξ \ ∪ j i=1 Ξ i for j ≥ 1. Proof. Let k ∈ N, (f 0 , f 1 ) ∈ Vec(M ) 2 and (q(•), u(•), λ(•)) be a time-extremal trajectory of the time-optimal control problem (1.1). Let t ∈ Ξ \ ( k j=1 Ξ j ) and assume for now that f 0 (q(t)) ∧ f 1 (q(t)) = 0. Owing to the fact that t is an accumulation point for Ξ \ ( Using repeatedly Lemmas 20 and 21 and exploiting the fact that each application of one of the two lemmas yields a finite number of alternatives, we deduce from a diagonal extraction argument that, up to taking suitable subsequences, i) There exist k + 1 sets B 0 , . . . , B k ⊂ T * M × J k+2 M × J k+2 M such that (λ(t), j k+2 q(t) (f 0 ), j k+2 q(t) (f 1 )) ∈ B k and (λ(t n1,...,nr ), j k+2 q(tn 1 ,...,nr ) (f 0 ), j k+2 q(tn 1 ,...,nr ) (f 1 )) ∈ B k-r for every r = 1, . . . , k, n 1 , . . . , n r ∈ N. ii) For every 0 ≤ r ≤ k, B r is defined by the vanishing of, say, l r simple relations and m r polynomial relations (using the terminology of Definition 19). Moreover, denoting

Ξ = {t ∈ [0, T ] | h 1 (t) = h 01 (t) = h +01 (t) = 0 or h 1 (t) = h 01 (t) = h -01 (t) = 0} satisfies Ξ = Ξ 1 ∪ • • • ∪ Ξ (n-1)
B r (q) = B r ∩ T * q M × J k+2 q M × J k+2 q M , we have codim T * q(t) M ×J k+2 q(t) M ×J k+2 q(t) M B k (q(t)) = l k + m k , codim T * q(tn 1
,...,nr ) M ×J k+2 q(tn 1 ,...,nr ) M ×J k+2 q(tn 1 ,...,nr ) M B k-r (q(t n1,...,nr )) = l k-r + m k-r , for every r = 1, . . . , k, n 1 , . . . , n r ∈ N. By construction, the set B k (q) is homogeneous with respect to the first component. To prove the proposition in the set {q ∈ M | f 0 (q)∧f 1 (q) = 0} it is then sufficient to show that there exists K ≤ (n-1) 2 such that if k ≥ K, then there exists r ∈ {0, . . . , k} such that the codimension l k-r +m k-r of B k-r (q(t n1,...,nr )) in T * q(tn 1 ,...,nr ) M ×J k+2 q(tn 1 ,...,nr ) M ×J k+2 q(tn 1 ,...,nr ) M is strictly larger than 2n-1. Indeed, if this were true, then denoting by π : T * M × J k+2 M × J k+2 M → J k+2 M × J k+2 M the canonical projection, we would conclude by standard transversality arguments [START_REF] Goresky | Stratified Morse Theory[END_REF] combined with the inequality codim J k+2 q(tn 1 ,...,nr ) M ×J k+2 q(tn 1 ,...,nr ) M π (B k-r (q(t n1,...,nr ))) ≥ codim T * q(tn 1 ,...,nr ) M ×J k+2 q(tn 1 ,...,nr ) M ×J k+2 q(tn 1 ,...,nr ) M B k-r (q(t n1,...,nr ))n + 1 > n, where the term +1 is due to the homogeneity of B k-r (q) with respect to the first component.

We introduce now a discrete dynamics on N 2 , which describes the admissible patterns of r → (l r , m r ). Define three mappings F 0 , F 1 , F 2 :

N 2 → N 2 by F 0 (x 1 , x 2 ) = (x 1 , x 2 ) + (1, 0), F 1 (x 1 , x 2 ) = (x 1 , x 2 ) + (0, 1), F 2 (x 1 , x 2 ) = (x 1 , 0) + (2, 0).
We say that an admissible curve γ of length p ∈ N for this dynamical system is a map γ : {0, . . . , p} → N 2 such that i) γ(0) = (3, 0), ii) there exists j ∈ {1, . . . , p} such that γ(i) = F 0 (γ(i -1)) for i = 1, . . . , j and γ(i) = F σi (γ(i -1)), with σ i ∈ {1, 2}, for i = j + 1, . . . , p. Observe that the initial condition fixed in i) reflects the definition of Ξ, F 0 describes the creation of a new simple relation (Lemma 20), while F 1 and F 2 encode the occurrence of, respectively, a new polynomial relation and two new simple relations (Lemma 21).

We are going to compute the minimal K so that, for k ≥ K, any admissible curve γ of length k exits the region T := {(x 1 , x 2 ) ∈ N 2 | x 1 + x 2 ≤ 2n -1}. It is not difficult to see that the longest admissible curve γ staying in T is as indicated in Figure 2, that is, we apply once F 0 , then 2n -5 times F 1 , then once F 2 , then 2n -7 times F 1 , once F 2 , and so on. The length of such curve γ is equal to

length(γ) = 1 + (2n -5) + 1 + (2n -7) + 1 + • • • + (2n -(2n -1)) = (n -2)(n -1), which implies that K = 1 + (n -2)(n -1).
It just remains to explain what can happen inside the collinearity set C introduced in (4.5): for a generic choice of (f 0 , f 1 ), along any extremal trajectory the points of Ω can accumulate (3, 0) (4, 0) 6. Time-optimal trajectories in dimension n = 3

x 1 + x 2 = 2n -1
We devote this section to a more careful analysis of Fuller times for time-optimal (and not only extremal) trajectories, in the case of a three dimensional manifold M = M 3 . In fact, for a time-optimal trajectory there are powerful second-order techniques [START_REF] Agrachev | Symplectic geometry for optimal control[END_REF] that permit us to be a bit sharper in our estimate on the maximal order of Fuller points, at least if we just focus on this smaller class of curves. By Theorem 3, we already know the upper bound (3 -1) 2 = 4. The main result of this section is the following.

Theorem 29. For a generic pair (f 0 , f 1 ) ∈ Vec(M ) 2 , none of the time-optimal trajectories of the control system (1.1) has Fuller times of order greater than two.

For the rest of this section we adopt the following convention: for any subset Θ ⊂ [0, T ], we denote by q(Θ) its image along the trajectory q(•).

Let us fix then a time-optimal trajectory. We collect previous results from [START_REF] Agrachev | On the local structure of optimal trajectories in R 3[END_REF][START_REF] Krener | The structure of small time reachable sets in low dimension[END_REF][START_REF] Sigalotti | Regularity properties of optimal trajectories of control systems with one input in dimension three[END_REF] in the following statement. Proposition 30. Let (f 0 , f 1 ) ∈ Vec(M ) 2 and q(•) be any time-optimal trajectory of the control system (1.1). Let us consider, with the notations of Definition 7, the subsets A 1 = {q ∈ M | f 1 (q) ∧ f 01 (q) ∧ f +01 (q) = 0, f 1 (q) ∧ f 01 (q) ∧ f -01 (q) = 0}, A 2 = {q ∈ M | f 1 (q) ∧ f 01 (q) ∧ f +01 (q) = 0, f 1 (q) ∧ f 01 (q) ∧ f ++01 (q) = 0, f 1 (q) ∧ f 01 (q) ∧ f -01 (q) = 0}, A 3 = {q ∈ M | f 1 (q) ∧ f 01 (q) ∧ f -01 (q) = 0, f 1 (q) ∧ f 01 (q) ∧ f --01 (q) = 0, f 1 (q) ∧ f 01 (q) ∧ f +01 (q) = 0}, A 4 = {q ∈ M | f 1 (q) ∧ f 01 (q) ∧ f +01 (q) = 0, f 1 (q) ∧ f 01 (q) ∧ f ++01 (q) = 0, f 1 (q) ∧ f 01 (q) ∧ f +++01 (q) = 0, f 1 (q) ∧ f 01 (q) ∧ f -01 (q) = 0}, A 5 = {q ∈ M | f 1 (q) ∧ f 01 (q) ∧ f -01 (q) = 0, f 1 (q) ∧ f 01 (q) ∧ f --01 (q) = 0, f 1 (q) ∧ f 01 (q) ∧ f ---01 (q) = 0, f 1 (q) ∧ f 01 (q) ∧ f +01 (q) = 0}, A 6 = {q ∈ M | f 1 (q) ∧ f 01 (q) = 0, f 1 (q) ∧ f +01 (q) ∧ f -01 (q) = 0, f 1 (q) ∧ f +01 (q) ∧ f ++01 (q) = 0, f 1 (q) ∧ f -01 (q) ∧ f --01 (q) = 0}.

If q(t) ∈ 6 i=1 A i , then t ∈ Σ \ Σ 0 . Define now the set W = {q ∈ M | f 1 (q) ∧ f 01 (q) ∧ f +01 (q) = 0, f 1 (q) ∧ f 01 (q) ∧ f -01 (q) = 0, f 1 (q) ∧ f 01 (q) = 0}.

As a consequence of Proposition 30, we can infer the following result.

Lemma 31. For a generic pair (f 0 , f 1 ) ∈ Vec(M ) 2 and for every time-optimal trajectory q(•) of the control system (1.1), q(Σ \ Σ 0 ) \ W is made of isolated points only.

Proof. The result is proved by using the same computational approach based on transversality theory as in the proof of Lemma 20. Instead of working in T * M as in Lemma 20, it is actually sufficient to prove that

codim J N q M ×J N q M 6 i=1 A c i ∩ W c ≥ 3, q ∈ M,
where A 1 , . . . , A 6 and W are the subsets of J N M × J N M defined implicitly by the relations A i = {q ∈ M | (j N q (f 0 ), j N q (f 1 )) ∈ A i }, i = 1, . . . , 6, W = {q ∈ M | (j N q (f 0 ), j N q (f 1 )) ∈ W}. Pick then any point q ∈ W c that satisfies f 1 (q) ∧ f 01 (q) = 0. Then W ∩ J N q M × J N q M is already a set of codimension two in J N q M × J N q M . Moreover, if q ∈ A c 6 , then necessarily the jets of f 0 , f 1 at q satisfy another nontrivial dependence relation, and we can conclude.

On the other hand, suppose that q ∈ ∩ 6 i=1 A c i and that f 1 (q) ∧ f 01 (q) ∧ f +01 (q) = 0, the remaining case being identical. Then since q ∈ A c 1 we infer the relation f 1 (q)∧f 01 (q)∧f -01 (q) = 0. We pass now to the condition q ∈ A c 3 , and we see that this obliges f 1 (q) ∧ f 01 (q) ∧ f --01 (1) = 0. Finally, the relation q ∈ A c 5 forces f 1 (q) ∧ f 01 (q) ∧ f ---01 (q) = 0, which in turn provides us with a third dependence relation at q, and therefore once again we conclude.

Proof of Theorem 29. Lemma 31 states, in particular, that for a generic choice of the pair (f 0 , f 1 ) and for every time-optimal trajectory q(•) we have that q(Σ \ Σ 0 ) \ W ⊂ q(Σ 1 ), or equivalently that q(Σ \ (Σ 0 ∪ Σ 1 )) ⊂ W.

We are left to prove that the density points of q(Σ \ (Σ 0 ∪ Σ 1 )) = q(Σ \ (Σ 0 ∪ Σ 1 )) ∩ W are isolated.

We have already shown that along any time-extremal (q(•), u(•), λ(•)), whenever t ∈ Σ \ Σ 0 the relations h 1 (λ(t)) = λ(t), f 1 (q(t)) = 0 and h 01 (λ(t)) = λ(t), f 01 (q(t)) = 0 hold true. Since, by definition, for every point q ∈ W both f +01 (q) and f -01 (q) belong to the two-dimensional space span{f 1 (q), f 01 (q)}, then for every t ∈ Σ \ (Σ 0 ∪ Σ 1 ) also h +01 (λ(t)) = h -01 (λ(t)) = 0. If t ∞ is an accumulation point of Σ \ (Σ 0 ∪ Σ 1 ), then, by Lemma 21 and using the Jacobi identity, either h 0101 (t ∞ ) = 0 or h 0101 (t ∞ ) = 0 and (6.1) h 0001 (t ∞ )h 1101 (t ∞ )h 0101 (t ∞ ) 2 = 0.

When h 0101 (t ∞ ) = 0, we conclude by transversality, noticing that f 0101 (q(t ∞ )) ∈ λ(t ∞ ) ⊥ = span{f 1 (q(t ∞ )), f 01 (q(t ∞ ))} provides a third independent condition on the jet of the pair (f 0 , f 1 ) at q(t ∞ ). In the case h 0101 (t ∞ ) = 0, let us define in a neighborhood of q(t ∞ ) a system of coordinates (x 1 , x 2 , x 3 ) so that (dx 1 , dx 2 , dx 3 ) is dual to (f 1 , f 01 , f 0101 ). Then (6.1) says that the product of the third components of f 0001 (q(t ∞ )) and f 1101 (q(t ∞ )) is equal to one, which corresponds to a third independent condition on the jet of the pair (f 0 , f 1 ) at q(t ∞ ).

2 Figure 1 .

 21 Figure 1. A concatenation of bang arcs

  said to be a simple relation of degree d ≤ N if there exists a word I ∈ A d of length d such that S = S I , where (4.1)

k- 1 j=1

 1 Ξ j ) and reasoning iteratively, we identify a set {t n1,...,nr | r = 1, . . . , k, n 1 , . . . , n r ∈ N} such that lim n1→∞ t n1 = t, lim nr→∞ t n1,...,nr = t n1,...,nr-1 , for r = 2, . . . , k and n 1 , . . . , n r-1 ∈ N, t n1,...,nr ∈ Ξ \ k-r j=1 Ξ j , for r = 2, . . . , k and n 1 , . . . , n r ∈ N, t n1,...,n k ∈ Ξ, for n 1 , . . . , n k ∈ N.

Figure 2 .

 2 Figure 2. The longest admissible curve γ

  3.1), we have |h 001 (τ )| = |u(τ )||h 101 (τ )| ≤ |h 101 (τ )| for almost every τ ∈ V and then we conclude by continuity of h 101 and h 001 . Proposition 14. Assume that there exists an infinite sequence of concatenated bang arcs converging to t ∈ [0, T ]. Then either h +01 (t) = 0 or h -01 (t) = 0.Proof. First notice that t ∈ Σ \ Σ 0 . Assume by contradiction that neither h +01 (t) nor h -01 (t) is equal to zero. Consider a neighborhood I of t in [0, T ] (with respect to the topology induced by


Ivan Madjarov
email: ivan.madjarov@lsis.org

CLOUD-BASED MASH-UP AUTHORING TOOLS FOR E-LEARNING

Keywords: Cloud computing, Authoring tools, e-Learning, m-Learning, SaaS, Web services

The objective of an e-Learning authoring tool is the creation of an accessible and reusable pedagogical content that is conforms to the existing e-Learning standards. We argue that this objective can be extended when the learning content is created on a multimodal SaaS authoring and publishing platform with an XMLbased language for content and metadata description to satisfy instructional designers for a responsive interactivity coupled with rich and informative substance. This position paper presents a Cloud-services based solution for building a virtual and personal learning environment that combines a wide range of technology and tools for learning-content mash-up authoring and diffusion. The proposed service-based framework is intended to support lifelong learning content creation and to enable mash-up of various learning services and applications by adapting learning objects on desktop and small handheld devices.

INTRODUCTION

The current e-learning applications required large investments in infrastructure systems and professional staff to maintain and upgrade systems. It is envisioned that, in the near future, cloud computing will have a significant impact on the educational and learning environment, enabling their own users to perform their tasks effectively with less cost by utilizing the available applications offered by the cloud service providers [START_REF] Anwar Hossain Masud | A Novel Approach for Adopting Cloud-based E-learning System[END_REF]. Interest in cloud computing in the e-learning area is growing due to potential greater cost savings from scalable architectures and open source products, and the possibility of higher learning outcomes [START_REF] Fernández | An Overview of E-Learning in Cloud Computing, Workshop on Learning Technology for Education in Cloud[END_REF].

Software-as-a-Service (SaaS) is a cloud service which provides software functionality through Internet and can help efficiently the management of webbased applications and pedagogical data in an elearning and m-learning context. Recently, more and more e-learning tools are migrating to the cloud. For instance, e-learning developers access the content authoring tools over the Internet via a secure hosted system without thinking about IT configurations, software set-ups and licenses. Cloud-based pedagogical content authoring is an e-learning process that is free from the constraints of typical desktop solutions and offers many advantages. In this position paper, we introduce the concept of e-Learning-Software-asa-Service for e-learning and m-learning contentediting, storing, content-adaptation and diffusion. All aspects of an e-learning or m-learning solution can be delivered using the Cloud-SaaS model, including Learning Course Management Systems (LCMS), authoring tools, and collaboration tools like webcasting and white boarding [START_REF] Basal | A Saas-Based Approach in an E-Learning System[END_REF].

In this position paper, we present our research in an ongoing project organized as follows: in Section 2 cloud-based content authoring tools and evaluation criteria are discussed. Section 3 presents the developed cloud-based e-learning semantic authoring suite. Finally, Section 4 ends the paper with conclusion and future work.

CLOUD-BASED COURSE DEVELOPMENT TOOLS

The objective of an e-learning authoring tool is the creation of an accessible and reusable pedagogical content that is conforms to the existing e-learning standards. We argue that this objective can be done if the learning content is created on a multimodal SaaS authoring and publishing platform with an XML-based language for content and metadata description to satisfy instructional designers for a responsive interactivity coupled with rich and informative substance.

Authoring process is an essential part of learning process and is supported by a suite of authoring tools for text, hypertext, graphics, mathematics, chart, and questionnaire content creation; i.e., Learning Objects (LOs). The purpose of an LCMS is to define the learning process by using and reusing created and existing LOs. This is a design process that includes: determination of learning objectives; planning how to assess learner comprehension, and identifying content needed so the learner can meet objectives.

We advocate for an approach where the learning process and the creation process are managed separately and independently of each other, i.e.: (1) a LOs authoring process managed by a suite of authoring tools; (2) a pedagogical process of course building, assessment and content adaptation managed by a LCMS.

E-Learning Authoring

The course development authoring tools are complex programs (e.g. Lectora (Lectora, 2013), Captivate (Adobe, 2014[START_REF] Sumtotal | [END_REF][START_REF] Smartbuilder | [END_REF], StoryLine (Articulate, 2014), etc.) that allows the integration of media objects and the creation of interactive learning materials. So many authors started e-learning content creation with off-the-shelf software e.g. word processors, spread-sheets, desktop publishing packages, graphics packages etc.

The most famous and readily available of these is PowerPoint, which is powerful and complex software for slides and multimedia presentations but with some limitations concerning different versions and their use in the e-learning domain. For instance, PowerPoint's interface is somewhat cumbersome when creating pedagogical sequences that integrate multimedia components (e.g. text, images, audio and video). A course is designed to be integrated in a LCMS. To be imported or exported into a LCMS an external interface-program (plug-in) should manage the PowerPoint non ASCII file format which is pro-prietary and different from one version to another (e.g. Microsoft Office, Open Office, Documents To Go, Kingsoft Office, etc.). This lack of internal format of PowerPoint does not contribute to the mobility and interoperability of pedagogical content. In this case, the learning object discovery process (LODP) is hampered by the lack of metadata. The reuse of a course or selected parts of a course by others is practically compromised because this authoring tool is not SCORM or LOs standards compliant.

Except PowerPoint, for course development, there is a list (Tools, 2012) thoring and publishing platform. This authoring tool allows developers and nonprogrammers alike to create interactive applications quickly, as well as share, publish, and sell objects or entire applications. Some activities in these software tools seem "easy-to-use" by limiting development options. For example, they might provide a set of predefined interaction templates for developers to fill with content. This is really useful if those templates reflect exactly the interactions that the author want to create, but if the author want to customize the output these tools appear very restrictive. In addition, these programs have a common drawback. The authors edit a full course in a proprietary format whose components are not metadata described and hard to discover thereafter. The publication in SCORM format in a LCMS dissolves completely the created content in the learning process.

In this paper, we propose a solution with twofold purpose: on the one hand, it aims at presenting an XML-based language for semantic description of a course. On the other hand, it proposes a cloud-based authoring tool to build learning resources applying the defined language. The proposed solution stores away natively courses and facilitates the reuse of parts of a course in other courses because these parts can be easy discovered by an XML-based search method.

Authoring Tools Evaluation Criteria

The main function of an e-learning authoring tool is to integrate different media and create interactivities required in a learning program. As mentioned above, a number of e-learning authoring tools have been developed with a wide range of functionalities: from simple (e.g. template-based) to very complex with its own programming language and multimodal user interface. These software programs are too complex to be used by an author who is not an IT or an elearning specialist. Subsequently a large number of users expressed a real difficulty in using these tools.

It becomes clear that e-learning authors need simpler but powerful authoring tools that could lower the skill barrier and allow more actors (i.e. teachers, academic authors, e-learning system administrators) to participate in the development and customization process. Moreover, to be really useful these tools should be able to reduce development time, effort, and cost, by allowing the reuse, enrichment and customization of available learning contents [START_REF] Nicola | A mash-up authoring tool for elearning based on pedagogical templates[END_REF]. This necessity has motivated research and development of MOOCs (Massive Open Online Course) making use of cloud-based learning tools and online tools as well as learning support specifically for LOs creation and sharing [START_REF] Rizzardini | MOOCs Concept and Design using Cloud-based Tools: Spanish MOOCs Learning Experiences[END_REF]. MOOC is an online course aimed at unlimited participation and access via the web.

In this paper we introduce some evaluation criteria essential for the appropriate operation of an elearning authoring tool:

 The authoring tool should be easy to learn and use because, in general, the author is not a professional e-learning developer and his experience in multimedia application is limited.

 The authoring tool should have the capacity to integrate media objects in various formats in the learning content with maximum flexibility.  For educational subjects such as mathematics, informatics and science in general, some of the contents involve mathematics equations, graphs and diagrams. The authoring tool should provide the functionalities for such content creation.  The authoring tool should provide the possibility to create quizzes for learners' assessment.  An authoring tool should not require authors any programming skills.  The authoring tool should provide deployment method for a LCMS with SCORM compliance integration.  The authoring tool should also provide many alternative deployment methods for standalone (XML, HML or PDF), or for the web publication without any content reproduction during the deployment stage.  The authoring tool should avoid the proprietary format for e-learning content for a maximum interoperability and reusability of LOs.

Otherwise the LOs discovery and composition process are doomed to failure.  The authoring tool should be accessible as service with standardized web browser. To these technical criteria, we add a key technological one: cloud service-based e-learning authoring process. The cloud authoring is free from the constraints of typical desktop solutions. Advances in internet technology make appear web-based tools with collaborative capabilities, but also with some limitations than advantages. Some web-based tools are cumbersome with an input process using templates and forms, which results in a limited creativity and flexibility. The pivotal advantage of cloud-based content authoring is not the number of connected users but saving time, reducing rework, and sharing learning content across all projects. Authors access authoring software over the Internet via a secure, affordable hosted system, and regardless of location they can make updates and reusing content to complete a course. The adaptation of a cloud-based approach brings many benefits and relieves the author from embarrassing task [START_REF] Fernández | An Overview of E-Learning in Cloud Computing, Workshop on Learning Technology for Education in Cloud[END_REF].

We argue that an effective solution for a successful cloud-based authoring process passes by: (1) the choice of a common format for the totality of the course content, and (2) the choice of an accessible and reachable storage format for the course. We suggest a non-proprietary format for better LOs reuse and to facilitate the LOs discovery and assembly, i.e. XML-based. Once created, a course can be saved in a single XML document or as a collection of XML documents. According to the evolution of the course, authors may need to modify its content. Therefore, the correct operation of a collaborative authoring system imposes the storage of learning collections, possibly in an appropriate database, for a better reuse and diffusion of these documents. We suggest the choice of a native database (NXD) which allows the storage of XML documents in their native format. This choice, in opposition to that of a relational database, is explained by the nature of pedagogical documents which are of narrative type, i.e. document-centric and not data-centric that is typical for relational tables. NXD and RDB follow two very different approaches to managing data. The concept of XML documents is to keep all attributes for a given LO together as one entity so that it can easily retrieve or insert all the relevant information for an LO in one shot. This is mostly useful for complex LOs whose attributes would otherwise be spread over many relational tables in the RDB model. XML is also suitable for LO whose set of attributes can evolve over time, since XML offers schema flexibility. Although relational database products today provide built-in XML document and query support, NXDs are arguably the best choice for learning content and metadata storage. As far as the query language is concerned, XQuery is recommended. Interoperability and data integration with existing e-learning systems and LCMS can be done by the means of cloud-based services. This solution simplifies the LOs reuse and the process of LOs discovery. As side effect of this approach, we can see a marked improvement of interoperability between e-learning systems.

E-LEARNING AUTHORING AS A SERVICE

This section presents a solution for building a virtual and personalized learning environment which utilizes a cloud-based technology to create a serviceoriented model for e-learning and m-learning application service providers and learners. The concept of e-Learning Services-as-a-Software is introduced as software distribution model in which applications are hosted by a service provider and distributed via the web. In this environment we can easily combine semi-structured data, stored in a native XML database (NXD), with structured data stored in a relational database (RDB) through web services (WS).

The objective is to provide direct data and application integration, located at distributed sites in order to improve the achievement of learning outcomes, i.e. integrate authoring process managed by a cloudservice management system (CSMS) [START_REF] Fernández | An Overview of E-Learning in Cloud Computing, Workshop on Learning Technology for Education in Cloud[END_REF] with pedagogical process managed by a LCMS. This approach promotes a deviceindependent gateway between different units and the huge number of learning resources available on a plethora of LCMSs. A semantic content adaptation service is plugged for content standardization. It becomes possible by combining the web-based open semantic editor suite (WOSES) [START_REF] Madjarov | Multimodality and Context-adaptation for Mobile Learning, Social Media Tools and Platforms in Learning Environments[END_REF] with a set of additional cloud-services to allow different mobile units a direct access to LOs customarily designed for desktop web browsers. An alternative service is available for a speech solution, which allows learners to turn any written text into natural speech files, when using standard voices. This approach allows the generation and the progressive downloading of text and audio based learning material dynamically for m-learning and ubiquitous access. As result, we cover the essential part of introduced criteria in section 2.2. Details are presented in the following section.

E-Learning Authoring Tools

The tools we developed are inspired by mashing-up principles allowing an easy and fast integration, over defined schemas, of local and remote resources.

Xesop is an open source authoring software [START_REF] Madjarov | XESOP: a Content-Adaptive m-Learning Environment[END_REF]) that provides a flexible XMLbased suite of tools for author customization, editing, storage and publication of LOs compliant with existing e-learning standards. The content aggregation specification comprises two models: a metadata model specifying the metadata elements of learning resources, and a content packaging model representing content structure. Both are hierarchical, which is convenient for representing data consisting of many elements and sub-elements. XML is perfectly suited for representing hierarchical models, as exemplified by the LOM and content packaging XML binding specifications published by IMS, both of which are adopted in SCORM.

In Figure 1 the structure of the created course with the Xesop project's semantic tools is presented. The original LOM structure is extended with new elements that seem important for more homogeneous and identifiable pedagogical content. For encoding textual information and content assembly, an XML semantic editor is developed (Figure 2) and a tree structure of a generic learning document is generated, while a validation grammar of XML schema type is used. Depending on course specificity, (mathematics or informatics course and science in general), the author can represent texts, diagrams, mathematical formulas or data in tables. A MathML editor was created for mathematical expressions, a SVG editor for vector graphics creation (Figure 3), a QTI editor for student's progression evaluation, a schema for table generation and a chart editor for data presentation. In this case, XML is used for encoding non-textual information such as vector graphics and mathematical expressions. In our e-learning authoring suite, binary data of multimedia content is embedded directly into XML course content. During the editing process, if the author inserts an image or any binary data into the edited content, the semantic editor will encode it using the Base64 encoding method. As a result, the course collection can be managed easily since all materials relating to the course are stored in a single XML collection (Figure 4). For more flexibility and content reuse, author can separately save different elements of the course as LOs in the NXD as XML collection with multiple files.

A large number of media-rich content can be stored using the abundant set of available XML schemas. By providing proper XSLT transformation files, the XML content can be presented in many forms, such as HTML5 for web-based desktop or mobile users. If necessary, formatted HTML and PDF versions of extracted learning content can be published in a LCMS via web services. Existing and old pedagogical documents can be adapted in the compliant XML format via importing web services as external input in the XML semantic editor. M-Learning pedagogical content can be given in the form of a visual presentation as text, pictures, tables in XML, responsive HTML5 format or as PDF data. Optionally, m-learning content can be given as sound data in the form of an acoustic presentation in an MP3 or WAV format [START_REF] Madjarov | XESOP: a Content-Adaptive m-Learning Environment[END_REF].

Implementation Scenario

The implementation scenario highlights the WOSES cloud-based application integration with a LCMS. The interconnection is carried out by a web services management system (WSMS). In our scenario, the learning-centric data and the management-centric data are clearly separated. Pedagogical documents are developed in WOSES framework of the e-Learning Services-as-a-Software based Xesop system and thereafter are stored in a NXD. The information relevant to learner personal data, learner profiles, course maps, LOs sequencing, data presentation and general user data is stored in the RDB of the LCMS. The publication process of learning content is carried out by WSMS. This allows integration existing LCMS systems via a cloud-based service. In the discussed case, web service-based modules make the bridge in a simple and effective way through Apache Libcloud, an open source library that provides a system-neutral interface to cloud provider APIs. The Java version supports Amazon EC2.

For system deployment, we used AmeTice (Moodle-based) e-Learning system deployed at Aix-Marseille University. Our system is based on the Apache containers suite for data storage and service management. We integrated the PHP-based LCMS interface via web services. For services deployment we used Apache Axis. For storing and managing LOs, we used eXist running in the Apache Tomcat Servlet engine as a web application and invoked via REST-style web services API. To integrate with other e-Learning and/or m-Learning systems we implemented an Apache jUDDI registry.

CONCLUSIONS

The developed Cloud-SaaS solution makes possible to create adaptive and responsive e-learning courses. Authors need to follow, in general, four phases of course development. To satisfy the phases of analysis, design, development and updating a free XML authoring language is used. The developed editor suite provides high-level WYSIWYM visual tools and enables a complete course to be designed without writing any programming code. We believe that a future cloud-based e-learning system should consist of a set of independent but cooperating nonmonolithic services-based applications that integrate pedagogical data between common LCMS.

Figure 1 .

 1 Figure 1. An extract of the Xesop course semantic

Figure 3 .

 3 Figure 3. MathML and SVG plug-ins