Singular Velocities of Even-Rank Affine Distributions

Singular curves, which are projections of singular extremals, play a special role in control theory and the theory of distributions. In this paper, we show that singular velocities, tangent to singular curves, determine affine distributions that are simultaneously of an even-rank and co-rank 2. If D is such a distribution, then its singular velocities span a distribution C D (of a rank two times smaller than that of D ) that, together with its Lie square, forms the initial distribution, i.e., We also provide canonical constructions of vector fields that are C D generators, canonical differential forms that are D cogenerators, and some functional invariants related to distribution of the considered type.


Introduction
A rank l affine distribution D on a smooth n-dimensional manifold M is a subbundle of the tangent bundle T M such that locally, around any q ∈ M, there exist smooth and linearly independent vector fields f 0 , f 1 , . . . , f l , called generators of D, such that D = f 0 + span{f 1 , . . . , f l }. (1) Thus, an affine distribution can be considered a control-affine system: seen from a geometric viewpoint. If the corank, n − l, of distribution D is smaller than its rank, then it is usually more convenient to describe the distribution in terms of its affine cogenerators, i.e., linearly independent one-forms ω 0 , ω 1 , . . . , ω n−l−1 , such that their inner products with every section of D is identically one: ∀v ∈ D q : v ω i = 1, i = 0, 1, . . . , n − l − 1.
One of the main questions addressed by control theorists is that of the feedback equivalence of two control systems. From a geometric viewpoint, if restricted to the control-affine systems, this question may be stated as follows: "Are two given affine distributions diffeomorphic or not?". In low dimensions, there are methods that allow the answer to be worked out. These methods are based on the constructions of some complete sets of invariants. Often, these invariants are obtained studying the so-called singular curves, see, e.g., [1,12,14,20]. One of the most impressive results of this kind is presented in [2]. In terms of singular velocities (i.e., velocities of singular curves), and without going into detail, we could restate that result as follows. Generic affine distributions of rank 2 on a 4-dimensional manifold M are completely determined by their singular velocities, which form rank 1 distributions that are in turn completely classified in the same paper [2]. Singular curves of affine distributions correspond to so-called singular extremals (see Section 2), which appear in the study of the time-optimal trajectories of control-affine systems (see, e.g., [3]). The fact that a singular extremal may be a non-trivial minimizer was first observed by Montgomery [16]. However, one must note that such phenomena are possible only for distributions of small rank (see [6] and [8] for the case of affine and vector distributions, respectively).
In [19], we showed that generic distributions from A(M, 2k), dim M > 2k + 2 (here and henceforth, A(M, l)stands for the class of rank l distributions on a manifold M) are also uniquely determined by their singular velocities. The distributions are affinely spanned by these velocities. This situation is quite analogous to the case of the odd-rank vector distributions of corank less than the dimension of the manifold minus 3, see [17]. Quite surprisingly, generic affine distributions of corank 2 are no longer spanned by their singular velocities, although they are still uniquely determined by them (see Theorem 2). This phenomenon is similar to that observed in the world of vector distribution and presented in [13].
Before stating a theorem that reveals relations between a corank 2 affine distribution D and its singular velocities, let us note that such a distribution can be identified with its polarizer D that is a rank 1 affine subbundle of the cotangent bundle: Throughout the paper, we will denote the vector part of an affine distribution D by D, and its linear hull by D, i.e., in the context of (1), Thus, for a distribution D of rank 2k, D and D are the vector distributions of ranks 2k and 2k + 1, respectively.
The fibers D q have an affine-line structure and hence they define pencils of skewsymmetric forms: and X, Y are germs of sections of distribution D. A * q is defined analogously: A * q (p) is an extension of A * q (p) from D q ∧ D q to D q ∧ D q such that if v and w are in D q then vector fields X and Y in Eq. 2 are sections of D, and if v and w are not in D q then X and Y in the equation are sections of the affine subbundles α X D and α Y D, where α X and α Y are constants such that Equivalently, using the well-known formula, one may define the above pencils evaluated at where v, w ∈ D q for A * and v, w ∈ D q for A * .
Let A q (n, l) denotes the class of germs at q ∈ R n of distributions D ∈ A(R n , l). According to classical theory of matrix pencils [9, chapter 12] and its applications to the case of skew-symmetric matrices [10, section 6], the following proposition holds.

Proposition 1 Let the pencil A
* of a distribution D ∈ A q (2k + 2, 2k) has the minimal Kronecker index equal k. Then for each p ∈ D q , the kernel V p of the form A * q (p) is onedimensional. Moreover, for a fixed affine basis (p 0 , p 1 ) in D q , one may parametrize the set of all kernels V p , p ∈ D q , by a D q -valued polynomial mapping: where the linearly independent vectors v 0 , . . . , v k span the fiber L q of a rank k + 1 distribution L ⊂ D. Conversely, if for each p ∈ D q , the kernel V p is one-dimensional and these kernels span a (k + 1)-dimensional subspace in D q , then the pencil A * has minimal Kronecker index equal k.
With a help of distribution L described above, one may define a rank k affine distribution C D ⊂ D: provided that L ⊂ D, which can be assured by regularity of pencil A * q . The regularity means that A * q (p) is non-singular for at least one p ∈ D q or, using D cogenerators ω 0 , ω 1 , that the polynomial mapping does not vanish identically.
Theorem 2 (Local determinacy theorem) Let D ∈ A q (2k + 2, 2k) satisfy the following two conditions: then the set S D of singular velocities satisfy the following relations and then C D is a unique rank k distribution satisfying Conditions (G1)-(G3) are of the open type, and they can be simultaneously satisfied (see Section 4). Therefore, one may prove (see [19] for more detail) that a pair of linearly independent 1-forms ω 0 , ω 1 on a (2k + 2)-dimensional manifold M affinely cogenerate distribution D ∈ A(M , 2k) satisfying these conditions, where M is an open and dense submanifold of M. Hence, on the basis of Theorem 2, one may claim that not only germs but also generic affine distributions of an even rank and corank 2 are completely determined by their singular velocities. A germ D ∈ A q (2k + 2, 2k) can be often determined by another set of curves. One may define a vector field f 0 as the unique intersection of D and the characteristic line field [15] of D. Then, D is uniquely determined by the integral curves of f 0 and the singular curves of D [13]. More precisely, where D is determined by its singular curves, provided that there are at least two of them passing through q (for a generic distribution D ∈ A(M, 2k), dim M = 2k + 2, there may exist an open set through which no singular curve of D passes; see [13]). In the context of Theorem 2, we must also mention the result of Bonnard [5], who proved that affine distributions are determined by singular extremals. Let us emphasize that Theorem 2 constitutes a result that is much stronger: it is enough to know singular curves, which are projections of singular extremals, to recover the distribution.
In this paper, we also reveal the structure of the bundle S D whose fiber S D q can be treated as a rational normal curve on projectivization PC D q , i.e., as a Veronese embedding of the projective line into PC D q . (1) a system of cogenerators of D, (3) a system of k − 2 real valued functions a 1 , . . . , a k−2 such that fibers S D q have parametrizations A possible construction of the above objects is described in Section 5.
, then there is a canonical way of defining the following objects: vector fieldsf 0 ,f 2 ; a coefficient α ∈ {0, 1}; and a vector fieldf 1 that is determined up to multiplication by ±1 (independently of α), such that: and the fibers S D q have parametrizations: The above theorem is illustrated in Figs. 1 and 2. The paper is organized as follows. In the next section, we study relations between affine distributions, their cogenerators, and singular velocities. In Section 3, we prove Theorem 2. An example of a distribution satisfying assumptions of this theorem is presented in Section 4. Finally, in Section 5, we construct some canonical objects related to an affine distribution. In particular, Theorems 3 and 4 are proved there.

Singular Velocities
A singular curve of a distribution D ∈ A(n, l) is an absolutely continuous curve that is D-horizontal, i.e., almost everywhere tangential to D, and that is a singular point of the endpoint mapping that maps horizontal curves (with fixed T ) to their endpoints. We refer the reader to [4] and [17,Appendix D] for details on the differential structure of the space of horizontal curves. The initial velocity of a singular curve will be called a singular velocity. The set of such velocities over a point q ∈ M is denoted by S D q . The main tool of the analysis of singular velocities is the following theorem due to Hsu [11] (a version of this theorem that is particularly adapted for affine distributions may be found in [7]). This theorem constitutes a precise formulation of the already mentioned relation: a singular curve is a projection of a singular extremal, and it is based on the classical Pontryagin Maximum Principle [18].

Theorem 5 A vector v ∈ T q 0 M is a singular velocity for a distribution D ∈ A(M, l) if and only if there exists a D-horizontal curve
with a lift to the cotangent bundle p t ∈ T * q t M satisfying the following conditions:

an integral curve of a Hamiltonian system with Hamiltonian of the form
where, for a fixed system (ii) p t does not vanish, i.e., p t = 0, for t ∈ I.
We say that a 1-form ω polarizes a distribution D, if it satisfies the following condition: where (D) denotes a section of D (i.e., a vector field). Similarly, we will write that a For brevity's sake, we will write ω ∈ D = to state that a 1-form ω either annihilates or polarizes distribution D.
The above corollary will be proved with a help of the following lemma, which is a direct consequence of a well-known formula:

Lemma 7
Let ω be a differential 1-form and X 1 and X 2 be the two vector fields such that the inner products X 1 ω and X 2 ω are constant functions. Then, Proof of Corollary 6 Let λ t = (q t , p t ) be as in Theorem 5 and let ω ∈ D = be a form that coincides with p 0 at q 0 . Differentiating (5) along λ t , we get The proof is completed by invoking Lemma 7.
If the assumptions of the above proposition are satisfied, then by extending (8) to nearby points we obtain a germ of vector field X ω that consists of singular velocities of distribution D: D q X ω (q) dω(q) = 0, X ω (q) ∈ D q .

Proof of Proposition 8
The existence and uniqueness of v come from the facts that D q is odd-dimensional and dω k (q 0 )| D = 0. It remains to prove that v ∈ S D q 0 . Let Using Lemma 7, we get Consider the equations where, for brevity, we write M for R 2k+2 . By the implicit function theorem, there exist unique functions U 1 , . . . , U 2k such that and (9) holds in a neighborhood of is of full rank, which is implied by the fact ω ∈ D = reg . Let λ t = (q t , p t ) be the integral curve of the Hamiltonian system with Hamiltonian This curve is also an integral curve of the Hamiltonian system with Hamiltonian where u i , i = 1, . . . , 2k represent any functions satisfying equations By Theorem 5, v =q 0 ∈ S D q 0 .

Proof of the Local Determinacy Theorem
Propositions 1 and 8 prove the first part of Theorem 2, i.e., that Conditions (G2) and (G2) imply C D = L ∩ D = aff-span S D ∈ A q (2k + 2, k) The following two lemmas complete the proof of Theorem 2.

Lemma 10
If D ∈ A q 0 (2k+2, 2k) satisfies Conditions (G1)-(G3), then C D = aff-span S D is the unique rank k distribution satisfying equality Proof of Lemma 9 Let p 0 , p 1 constitute an affine basis of D q . By Propositions 1 and 8, there exist k + 1 numbers t 0 , . . . , t k and k + 1 independent vectors v 0 , . . . , v k ∈ D q such that each v i belongs to the kernel of Let X i be a vector field tangent to C and coinciding with v i at q. By definition of A * q , we have Proof of Lemma 10 Conditions (G1) and (G2) imply (see the discussion following Proposition 1) Thus, it is enough to show that S D q ⊂ C. Let (f 0 , f 1 , . . . , f 2k ) be a system of generators of D such that C = f 0 + span{f 1 , . . . , f k }.

According to Corollary 6 and Proposition 1, if
i=1 v i f i (q 0 ) ∈ S D q , then there exists ω ∈ D = reg such that the following equations hold: Therefore, vector V = (v 1 , . . . , v 2k ) T must satisfy the equation Due to the regularity of ω, det A(q 0 ) = 0 and thus v i = 0 for i = k + 1, . . . , 2k, which means that v ∈ C q 0 .
In what follows, we will use the following technical lemma. We leave its proof to the reader (one may also find it in [19]).

Lemma 11
Consider the following differential forms defined on R n = R 2k+2 : where /dx 1 /dx a denotes the product dx 2 ∧ dx 3 ∧ · · · ∧ dx n with factor dx a omitted.
By Lemma 11(a): For each i = 0, 1, . . . , k − 1, the form can be written as a linear combination of (n − 2)-forms: If a is even, then /dx 1 /dx a can result only from dω i 0 ∧(•) k−i . If a is odd, then /dx 1 /dx a can result only from those summands of dω i 0 ∧ dω k−i 1 in which there is exactly one factor from the part of dω 1 denoted by + . In other words, for odd a, the form /dx 1 /dx a can result only from products Due to Lemma 11, we thus have, for i = 0, 1, . . . , k − 1, Using equalities equations (12), (13), and (15), we get that the forms where = dx 1 ∧ · · · ∧ dx n . Condition (G1) is satisfied because all the coefficients of the polynomial mapping (3) are non-zero.

Condition (G2)
As in the previous subsection, we may show that the forms and for 1 ≤ i ≤ k − 1 they equal where /dx i = dx 1 ∧ · · · ∧ dx i−1 ∧ dx i+1 ∧ · · · ∧ dx n . Clearly, the above forms are linearly independent at the origin and Condition (G2) is satisfied due to the following lemma.

Lemma 12
Let ω 0 and ω 1 be affine cogenerators of a distribution D ∈ A q (2k + 2, 2k). Condition (G2) is equivalent to the linear independence (at q) of the following forms: Proof D q may be parametrized as: The kernel V p t of A * q (p t ) is the kernel of the 2-form tdω 1 + (1 − t)dω 0 evaluated at q and restricted to D q ∧ D q . Thus, the kernel V p t is exactly the kernel of the following (2k + 1)-form evaluated at q: Let be a local volume form and let vector fields Z i , i = 0, . . . , k be defined as follows: Then, the kernel V p t is linearly spanned by and the assertion of the lemma is proved by invoking Proposition 1.

Condition (G3)
Computations performed in the two previous subsections show that vector fields Y i , i = 0, . . . , k, defined by the relations are linearly independent and span distribution L introduced in Proposition 1. These vector fields also are sections of the affine distribution D, for if we multiply (from the left) both sides of Eq. (24) by ω r , r = 0, 1, we get and thus Y i ω r = 1. Hence, vector fields Y i , i = 0, . . . , k, affinely span distribution C D . By Eqs. 17-22, we have: and, for i = 1, 2, . . . , k − 1, In particular, and therefore dim In fact, the above inequality must be strict because

Normalization
In this section, we provide results that form Theorems 3 and 4. Let D ∈ A q (2k+2, 2k) be a distribution satisfying Conditions (G1) and (G2) and let K = (ω 0 , ω 1 ) be its cogenerators. The Pfaffian of the pencil A * can be viewed as a polynomial in one variable: where R ω is the determinant of the matrix R appearing in Eq. (11). By R K (t), we will denote this Pfaffian normalized to a monic polynomial (i.e., with the leading coefficient equal to 1). This normalization can be always performed by an appropriate choice of D generators. The polynomial R K is uniquely determined by the distribution D up to an affine change in variable t (which comes from the same freedom in parameterizing the affine line D ). We will refer to this polynomial as R D whenever the choice of cogenerators is irrelevant. In particular, for a distribution D ∈ A q (2k + 2, 2k), the following conditions make sense independently of the freedom above: the arithmetic mean of the roots of R D is not a root of R D , (G5) the mean of the roots of R D is not a root of R D 's first derivative. (G6)

Remark 13
One can show that Condition (G4) is equivalent to the fact that the characteristic line field (see [15]) of the corank 1 distribution D is not parallel to the corank 2 distribution D.

Remark 14
In what follows, we use Conditions (G4)-(G6) to ensure the existence of canonical cogenerators of D and canonical generators of C D . As the anonymous reviewer notice, if we were satisfied with cogenerators and generators defined canonically up to a specific action of Z 2 , we could drop Condition (G6) and replace Conditions (G4) and (G5) with their weaken versions: the arithmetic mean of the roots of R D is not a root of R D or it is a root of constant multiplicity m < d, Before going into further analysis, let us present an example of distributions satisfying and not satisfying Conditions (G4)-(G6).
Example 2 Let D ∈ A 0 (2k + 2, 2k), n = 2k + 2 ≥ 4, be a distribution with cogenerators K = (ω 0 , ω 1 ) defined in Section 4. We have (see Section 4): The matrices R ω i , i = 0, 1 (constructed at q = 0) can be computed using the formula: .., f 2k (0) = − ∂ ∂x n−1 . Thus, R ω 0 becomes the k by k identity matrix and matrices R ω 1 and tR ω 1 + (1 − t)R ω 0 are, respectively, ⎛ The determinant of the latter matrix can be computed using Laplace expansion with respect to the last column. In this way, we obtain, up to normalization to a monic polynomial: In particular, R K (t) is of degree k (Condition (G4) is satisfied) if k is even, and it is of degree k − 1 ((G4) fails) if k is odd. Moreover, R K is invariant with respect to the change of variable t → 1 − t, and thus the average of all R K roots is 1/2. Since Condition (G5) is satisfied whereas Condition (G6) is not.
We can fix the "origin" (one-form ω 0 ) on D by making the sum of R K roots equal to zero or equivalently by making a k−1 = 0. Then, we may fix the "scale" (one-form ω 1 − ω 0 ) on D by making a 1 = a 0 for k > 2 (this is possible if Conditions (G5) and (G6) are satisfied) and by making a 0 = ±1 for k = 2 (provided that (G5) holds). Thus, we get the following two propositions.

Proposition 16
Let D ∈ A 0 (6, 4) satisfy Conditions (G1) and (G2) together with Conditions (G4) and (G5). Then, there exist exactly two systems of cogenerators, K 1 and K 2 , such that These systems are of the form: whereω annihilates D (the choice of ±ω does not affect the sign of the free term of R K 1 ).
The above cogenerators differ from that given in Section 4 by the underlined coefficient 2 (in place of 1). We may repeat computations from that section to verify that Conditions (G1)-(G2) are satisfied and and C D 0 + [C D , C D ] 0 ⊂ D 0 . As in Example 2, we compute the polynomial R K (at the origin), up to a factor making this polynomial monic: The coefficient of the term t k equals and that of term t k−1 : Thus, for an even k, the arithmetic mean of the roots of R K equals t = − a k−1 ka k = 0.

Canonical Generators of C D
Let a distribution D ∈ A q (2k + 2, 2k) has affine cogenerators K = (ω 0 ,ω 1 ) introduced in Proposition 15 and 16 for k > 2 and k = 2, respectively. System K defines a parametrization of the fiber S D q : t → ω = tω 1 + (1 − t)ω 0 → (t) = X ω , defined for all t at which R K (t) = 0 (vector field X ω is defined after the formulation of Proposition 8). The mapping : t → R K (t) ( (t) − (0)) ∈ C D is polynomial. Since the image of must be (k+1)-dimensional, we thus have the following sequence of linearly independent vectors: Since the construction of these vectors depends smoothly on the point q, we may write about smooth vector fieldsf i , i = 0, 1, . . . defined in a neighborhood of that point. The construction of these vector fields implies that the mapping gets the form presented in Theorems 3 and 4: