CC BY
4Russian Journal of Nonlinear Dynamics, 2023, vol. 19, no. 4, pp. 559-573. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd231104
MATHEMATICAL PROBLEMS OF NONLINEARITY
MSC 2010: 53C17
Abnormal Extremals in the Sub-Riemannian Problem
We study the left-invariant sub-Riemannian problem on the free nilpotent Lie group of rank 2 and step 5. We describe some abnormal trajectories and some properties of the set filled by nice abnormal trajectories starting at the identity of the group.
Keywords: sub-Riemannian geometry, abnormal trajectories, geometric control
1. Introduction
A sub-Riemannian structure on a smooth manifold M is a vector distribution A c TM endowed with a scalar product g [1, 3]. The hardest open problems of sub-Riemannian geometry are related to abnormal trajectories:
• Are abnormal length minimizers smooth?
• Is the set filled by abnormal trajectories starting at a fixed point negligible?
It is natural to start the study of these and other questions with the most symmetric sub-Riemannian structures — left-invariant sub-Riemannian structures on Lie groups. The simplest class of these structures are left-invariant structures on free nilpotent Lie groups. Such structures are parameterized by a 2D lattice {(k, s) G N2 | k ^ 2, s ^ l}, where k is the rank of A, and s is the
Received July 30, 2023 Accepted October 28, 2023
Subsections 4.1-4.3 of the paper were written by E. Sachkova, and the rest sections were written by Yu. Sachkov. The work of E. Sachkova is supported by the Russian Science Foundation under grant 2221-00877 (https://rscf.ru/en/project/22-21-00877/) and performed at the Ailamazyan Program Systems Institute of the Russian Academy of Sciences.
Yury L. Sachkov [email protected] Elena F. Sachkova [email protected]
Program Systems Institute Russian Academy of Sciences, Pereslavl-Zalessky, Yaroslavl Region, 152020 Russia
with Growth Vector (2, 3, 5, 8, 14)
Yu. L. Sachkov, E. F. Sachkova
step (the minimal order of Lie brackets of vector fields tangent to A required to fill the tangent space TqM).
Let us consider the first nontrivial case k = 2. In the case s = 1 we get a Euclidean geometry in the plane. The case s = 2 is the Heisenberg case [3]. The case s = 3 is the Cartan case [15, 16]. Finally, the last studied case is s = 4 [6]. This paper seems to be the first paper in the literature to study the case s = 5.
The structure of this paper is the following. In Subsection 1.1 we recall some basic notions of sub-Riemannian geometry. In Subsection 1.2 we state the (2, 3, 5, 8, 14) sub-Riemannian problem studied in this paper. In Section 2 we apply the Pontryagin maximum principle to the optimal control problem for sub-Riemannian length minimizers. In Section 3 we describe abnormal trajectories for the case s = 5 inherited from the cases with s < 5. In the central Section 4 we study so-called nice abnormal extremals, which form the simplest and generic class of abnormal extremals, and for which it is possible to define a smooth Hamiltonian flow.
1.1. Sub-Riemannian structures
We recall some basic notions of sub-Riemannian geometry [3]. A sub-Riemannian structure on a smooth manifold M is a distribution on M
A = {Aq c TqM | q e M}, dim Aq = const,
endowed with a scalar product
g = {gq — scalar product in Aq | q e M}.
A horizontal (admissible) curve q(-) e Lip([0, i1], M) is a Lipschitz curve in M tangent a.e. to A:
q(t) e Aq(t) for a.e. t e [0, ti].
h
The length of a horizontal curve is l(q(-)) = /(g(q(t), q(t))i/2 dt. The sub-Riemannian distance
0
is
d(qo, qi) = inf{l(q(0) | q(-) horiz. curve, q(0) = qo, q(ti) = qi}. A sub-Riemannian length minimizer q(t), t e [0, t1] is a horizontal curve such that l(q( )) = = d(q(0), q(ti)).
An orthonormal frame is defined as follows:
Xi, ...,Xk e Vec(M), Aq = span(Xi (q),...,Xk (q)), q e M, g(Xi(q), Xj(q)) = Sij, i, j = 1, ...,k.
An optimal control problem for sub-Riemannian length minimizers is stated as
k
q = YluiXi(q), q e M, u = (ui,...,uk) e Rk,
i=i
q(0) = qo, q(ti) = q^ . i/2
/ k
l(q() = / I]u2 dt ^ min.
o Vi=i
Let M = G be a Lie group with Lie algebra g of left-invariant vector fields on G. Assume that A, g are invariant under left shifts q ^ q-q, where q, q G G. Then we get a left-invariant sub-Riemannian problem on G. In this case one can choose an orthonormal frame X1, ..., Xk G g. A Lie algebra g is a Carnot algebra if it is stratified and generated by its first layer:
g = g(1) Ф-
ф g
(s)
g(1), g«
,(¿+1)
gv" ' - ', i = 1, ..., s, g(s+1) = {0}.
Then s is called the step, and к = dim g(1) the rank of g. The Carnot group G is the corresponding connected simply connected Lie group.
Consider a left-invariant sub-Riemannian structure on a Carnot group G:
A = g(1), g — left invariant inner product in A.
By a theorem of Mitchell and Gromov, left-invariant sub-Riemannian structures on Carnot groups provide a local nilpotent approximation of generic sub-Riemannian structures in a neighborhood of generic points [3].
The growth vector of a left-invariant sub-Riemannian structure is defined as follows:
(n1, ..., ns) = (dimA1, ..., dimAs) , A = g(1), A2 = A + [A, A] = g(1) ф g(2),
A* = A*"1 + [A, A*"1] = g(1) ф ••• ф g
,(0
As = g(1) ф-.-ф g(s) =
g.
1.2. Statement of the nilpotent sub-Riemannian (2, 3, 5, 8, 14)-problem
Let g be a free nilpotent Lie algebra of rank 2, step 5. There exists a basis X1, ..., X14 of g in which the table of Lie brackets has the form [13]:
[X1, X2 ] = X3> [X11 X3] = X41 [X2) X3] = X5) [X1} X4] = X61 [X2) X5] = X8) [X1> X5] = [X2) X4] = X7)
[X1, X7] = X13, [X1, X6] = X9,
[X1> X8] = X14 > [X2) X6] = X10)
[X2, X7] = X11, [X2, X8] = X
12,
[X3, X4] = X13 " X10, [X3, X5] = X14 " X
L11.
Let G be the corresponding connected simply connected Lie group. The Lie group G is
diffeomorphic to R^ , and the basis vector fields Xi, ..., Xi4 can be realized as follows [13]:
Xi —
■X'o
d
d
dx1 2 dx
+
+
+
3
xx
12
3
720
1^4
rp rp rp
■l2 , ■ll-l2
2 12 d
2 x2
d
x
3
12
12
xx
d
dx4 12 dx5 22
' rp rp rp
_ I _1 i 13 1 2 12
2 dx.
+
240
2^5
x
3
12 720
+
12
x2x5
9 d
22
tJb2^-/4 *t/1*t/5 usy
240
d
dx6 d
rp rp rp
5 2 3
d
12 J dx-
-+
2 dx
- +
10
2 dx
+
x
2
d
11
720 dx
+
x
12
X2 =
_d_
dxc
+
x1
d
+
2 x1
d
2 y dx d
2 dx 12 dx
+
4
x
4
1
d
720 dx
9
3
ryf-> f-yi 12
240
+
14 12
14 x1 x2
"Hf
x
+
3
1*^2 720
+
xx
2*^5 12
+
x
d
2 dx
2 ) dx
3
rp"-1 rp x1 x2
12
_d_
2 ) dx d
12
2 2 2 2 2 2
360
5
rp rp rp
4 1 3 ~2 + 12
12
d dx7
d
2 dx
-+
13
x5 x2x3 T + 2
10
22
12 1 2 4 ' 15
12
360
+
240
x^A d
12
+
x
d
2 dx
11
6
dx
13
x2 x2
1^2 + a;5
720
12
(1.1)
_d_
dxs
d
dx
14
X =
d_
dx
+
x d x2 d x"2 d xx d x"2 d
3
2 dxA
+
+
1
2 dx5 12 dx
+
12
+
13
x
12
d
2 J dx
+
13
13
6
x
6 dx
+
7
12 dx.
-+
12
d
2 7 dx
14
X4 =
d . x1 d x2 d x2 d
X =
dx4 _d_
dx
+
+
2 <9x6 ^ 2 <9x x d x2 d
+
7
12 dxc
+
1 x2
d
6 dx
+
8
x2 d
5
2 dx.
+
+
x
d
+
x1 x2 12
7
x + f
2 dx8 12 dx
d
+
12
10 d
12 dx
10
6 dx
+
x
11 d
rp rp rp
, ti/lit/O Juo
+ + —
1 12 2
d
dx
13
11
12 dx
+
x
d
12
12 dx
- +
13
dx
X6 =
d x, d x2
14
x2 d
X14 —
(1.2)
(1.3)
(1.4)
(1.5)
(1.6)
(1.7)
(1.8)
yi4 dxii
The sub-Riemannian structure with the growth vector (2, 3, 5, 8, 14) is defined as follows:
A = span(Xi, X2), g(Xi, Xj) = 5ij. Consider the optimal control problem for sub-Riemannian minimizers:
x = uiXi(x) + u2X2(x), x e G = Ri4, u = (ui, u2) e R2,
X—
dx6 JL
dx-,
2 dxa 2 dx
+
x
d
2 dx
+
10 d
10
2 dx
+
x
d
11
2 dx
X —
13
d_
dxa
+
d
2 dx
+
x
d
11
2 dx
+
d
X9 — o )
dx9
d_
x9
d dx1
X10 —
d
dx
X11 —
d
10
dx
X12 —
d
11
dx
X13 —
d
12
dx
+
12
d
2 dx
14
13
dx
10
+
d
dx.
4
«(0) = x0 = Id = 0, x(t1) = xl,
h
Kx(')) = j \JU1 + u2 dt —min •
Let us call it the (2, 3, 5, 8, l4)-problem.
Existence of sub-Riemannian minimizers in this problem follows from the Rashevskii - Chow and Filippov theorems [3].
2. Pontryagin's maximum principle
Denote by T*G the cotangent space of G at a point x G G, and by T*G = [J T*G the
xec
cotangent bundle of G. Introduce Hamiltonians linear on fibers T*G and corresponding to the basis vector fields Xt: ht(X) = (X, XJ, X G T*G, i = l, ..., l4. Let hvu(X) = u1h1 (A) + + u2h2(X) + | (it'l + be the Hamiltonian of Pontryagin's maximum principle. Finally, denote by h G Vec (T*G) the Hamiltonian field with a Hamiltonian h G C~ (T*G).
Theorem 1 (Pontryagin's maximum principle [2, 4, 5]). If x(t), t G [0, t1], is a sub-Riemannian length minimizer for the (2, 3, 5, 8, 14)-problem, then there exist X(t) G T^G and v G {—1, 0} for which:
(1) A(t) = u1(t)h1 (X(t))+ U2(t)h2(X(t)),
(2) hu(t)(X(t))=maxhvv(X(t)),
(3) (v, X(t)) = (0, 0).
2.1. Normal case
Let v = —1. Then
Ui(t) = hi(X(t)), i = 1, 2;
X(t) = H (X(t)); (2.1)
H=1-(h:i + hi).
Theorem 2 (L. Lokutsievsky, Yu. Sachkov [14]). The Hamiltonian system (2.1) is not
Liouville integrable.
Thus, we address the abnormal case.
2.2. PMP in the abnormal case
Let v = 0. Then Pontryagin's maximum principle implies the following:
hi(X(t)) = h2(X(t)) = ha (X(t)) = 0, X(t) = 0, (2.2)
u1h4 + u2h5 = 0, (2.3)
h4 = u1h6 + u2h7, (2.4)
0
h5 = u1h7 + u2h8, (2.5)
h6 = u1h9 + u2h10, (2.6)
h7 = u1h13 + u2h11, (2.7)
h8 = u1h14 + u2h12, (2.8)
h 9 = ... = h 14 = 0. (2.9)
Let us call the system of ODEs (2.3)-(2.9) the vertical abnormal system.
3. Inherited abnormal trajectories
If h9 = ... = h14 = 0, then the vertical abnormal system (2.3)-(2.9) reduces to the system
h4 = —h5 h6 + h4h7, h 5 = —h5 h7 + h4h8,
h 6 = h 7 = h 8 = 0,
which is the vertical subsystem of the Hamiltonian system for abnormal extremals in the nilpotent (2, 3, 5, 8)-problem [6]. Thus, projections of all abnormal trajectories (x1(t), x2(t)) of the
nilpotent (2, 3, 5, 8)-problem are inherited to the (2, 3, 5, 8, 14)-problem:
• straight lines (inherited in fact from the sub-Riemannian (2, 3, 5)-problem [7]),
• second-order curves (ellipses, hyperbolas, parabolas, pairs of intersecting lines).
4. Nice abnormal trajectories
4.1. Definition of nice abnormal trajectories
Consider the increasing chain of subspaces of the Lie algebra g defined by the distribution A:
A = g(1) =span(X1 ,X2), A2 = g(1) + g(2) =span(X1, X2, X3), A3 = g(1) + g(2) + g(3) = span(X1, ..., X5), A4 = g(1) + ••• + g(4) = span(X1, ..., X8), A5 = g(1) + ••• + g(5) = g = span(X1, ..., X14), A c A2 c-^c A5 = g.
Accordingly, a decreasing chain of annihilators, subspaces in the Lie coalgebra g*, is defined:
AiX = {A £ g* I (A, A*) = 0}, Ax d A2± D A3± D A4± D A5± = {0},
= {h1(A) = h2(A) = 0},
A2± = {h1(A) = ... = h3 (A) = 0},
A3± = {h1(A) = ... = h5 (A) = 0},
a4± = {h1(A) = ... = h8 (A) = 0}.
Identities (2.2) mean that an abnormal extremal X(t) is contained in A2±. Consider a decomposition
A2± = (a2± \ A3±) u A3±
and denote
E = A2± \ A3± = {X G g* | h1 = h2 = h3 = 0, h4 + h25 = 0}. (4.1) Proposition 1.
(1) E is a smooth submanifold of g* diffeomorphic to R10 x S1.
(2) The set E is semialgebraic and thus semianalytic.
Proof. Follows immediately from the definition (4.1) of the set E. □
Definition 1 ([3]). An abnormal extremal X(t), t G [0, t1], is called nice if X(t) G E for all t G [0, t1].
Proposition 2 ([3, Th. 12.30]). If the abnormal extremal X(t) is nice, then, up to time reparameterization,
u (t) = —h5(X(t)), u2(t) = h4(X(t)),
h 4 = —h5 h6 + h4h7, (4.2)
h5 = —h5 h7 + h4h8, (4.3)
h6 = —h5h9 + h4h10, (4.4)
h7 = —h5h13 + h4h11, (4.5)
h 8 = —h5h14 + h4h12, (4.6)
hg = ... = h 14 = 0, (4.7)
X = —h5X1 + h4X2. (4.8)
In other words, X(t) is a reparameterization of a solution of the Hamiltonian system
X(t) = H)(X(t)), Xo G E, (4.9)
where H0 = —h5h1 + h4h2.
Denote the vertical part of the Hamiltonian vector field H0 as
d d d A = (-Me + M<7 + (~h5h7 + + (~h5hg + h4h10)—+
d d
+ (~h5h13 + h4hn)— + (~h5hu + h4h12) — , (4.10)
see (4.2)-(4.7). The main goal of this paper is the study of the vector fields A and H0. _RUSSIAN JOURNAL OF NONLINEAR DYNAMICS, 2023, 19(4), 559 573
4.2. Properties of nice abnormal extremals
Proposition 3 ([3]). Nice abnormal extremals are real-analytic.
Proposition 4 ([3]). If an abnormal extremal is nice, then its small arcs are optimal, that is, it is a sub-Riemannian geodesic.
Lemma 1. The vector fields A and H0 are divergence-free, thus their flows preserve natural volumes in g* and in T*G, respectively.
Proof. Immediate computations on the basis of (4.10) and (4.2)-(4.8), (1.1)—(1.8). □
4.3. Projections of nice extremals onto the plane (xir x2) 4.3.1. Euler elasticae
Euler elasticae are stationary configurations of planar homogeneous elastic rods with fixed endpoints and tangents at endpoints [8, 9], see Fig. 1. An elastica has different shapes depending on the first integrals E = ^ — r cos(0 + 0o) G r> +oo) and r ^ 0 of ODEs (4.11): it, is a straight, line for E = -r> 0, it has inflection points for -r<E<r> 0, it is a critical elastica drawn as a dashed curve for E = r > 0, it has no inflection points for E > r > 0, and it is a circle for r = 0, E > 0. Elasticae (xi(t), x2(t)) satisfy the system of ODEs
xi = cos 9, x2 = sin 9, 9 = —r sin(9 + d0), r = const ^ 0. (4.11)
Fig. 1. Euler elasticae
Proposition 5. Any Euler elastica can be realized as a projection onto the plane (x1, x2) of a nice abnormal trajectory.
Proof. The Hamiltonian system (4.9) for h7 = h10 = hn = h13 = 0, h9 = h14 = 1, h6 = h8 reduces to the following system:
h 4 = —h^hg, h 5 = h 4 hg,
h6 =
x = -h5X1 + h4X2.
Introduce the polar coordinates: h4 = r cos d, h5 = r sin d, then we get ODEs of elasticae (4.11) with 90 = 0, r = sJKl + /?,§. □
Corollary 1. The flow of the field A (4.10) is not integrable in elementary functions. Proof. The ODEs of elasticae (4.11) are integrable in elliptic functions. □
4.3.2. Hyperbolic elasticae
Similarly to Euler elasticae (4.11), let us call a curve (xl(t),x2(t)) a hyperbolic elastica if it satisfies the ODEs
x 1 = cosh z, x2 = sinh z, z = —r sinh(z — z0), r = const ^ 0. (4.12)
. 2
A hyperbolic elastica has different shapes depending on the first integrals I = + rcosh(z — — z0) G [r, +rc>) and r ^ 0 of ODEs (4.12). See Figs. 2-5 for several plots of the hyperbolic elasticae.
Fig. 2. Line as a hyperbolic elastica, I = 1, r = 1 Fig. 3. Hyperbolic elastica, I = 1.1, r = 1
Fig. 4. Hyperbolic elastica, I = 2.1, r =1 Fig. 5. Hyperbolic elastica, I = 9.5, r =1
Proposition 6. Any hyperbolic elastica can be realized as a projection onto the plane (xl, x2) of a nice abnormal trajectory.
Proof. The Hamiltonian system (4.9) for h6 = h8 = h9 = h10 = hl2 = h14 = 0 reduces to the following system:
h4 = h4 h7, h5 = -h5^ h6 = h5h13 + h4h11,
X = -h5X1 + h4X2.
Let h4(0) = h5 (0) = 1. Introduce a variable z such that h4 = ez, h5 = e-z and denote h7 = c. Then we get the ODEs
z = c,
C = (h11 — h13) cosh z + (h11 + h13) sinh z.
After a change of variables hn — h13 = r sinhz0, hn + h13 = — rcoshz0, yi = Xl 2'T2, y2 = Xl+2X2 we get the ODEs
yi = cosh z, i)2 = sinh z, Z = c,
c = (hii — hi3) cosh z + (hii + hi3) sinh z,
which coincide with (4.12) up to hyperbolic rotation and homothety of the plane (xi, x2), with r = -\f¡3'2 — ei2, z0 = arcsinh(^), a = hn — h13, ¡3 = hn + h13. But these mappings
are symmetries of the Hamiltonian system (4.9), and the statement of this proposition follows.
□
After time rescaling and application of hyperbolic rotations of the plane (xi, x2), the system of ODEs (4.12) reduces to the following one:
x i = cosh z, x 2 = sinh z, (4.13)
z = c, C = — sinh z. (4.14)
2
This system has an integral I = y + coshz ^ 1.
Proposition 7. The system of ODEs (4.13), (4.14) with the initial condition (xi, x2)(0) = = (0, 0), (z, c)(0) = (z0, c0) has the following solutions:
(1) If z,o) = (0, 0), then
xi(t) = t, x2(t) = z(t) = c(t) = 0.
(2) If z,c0) = (0, 0), then
xi
i(i) = 72(7+1) E(r) - t, x2(t) = \/2(I — 1) sn r, (4.15)
where r = \J^-t; snr = sn(r, k), cnr = cn(r, k) are Jacobian elliptic functions with
the modulus k = ^Jj^j G (0, 1); E(r) = E(r, k) = Z?(am(r, k), k), where E(ip, k) is the elliptic integral of the second kind, and am(r, k) is the Jacobi amplitude [17].
Proof. Introduce the coordinates (<p, k) in the punctured plane (z, c), z2 + c2 = 0, by
formulas (4.16), where r = \J^f-tp, k = \Jj^j G (0, 1). In these coordinates the system of
ODEs (4.14) rectifies: Lp = 1, k = 0, so the expressions (4.16) follow. Then formulas (4.15) are obtained by integration of (4.13). □
Remark 1. Trajectories c){t) of system (4.14) are periodic of period T = 4K(k)y j^-j, except the equilibrium (z, c) = 0. See Fig. 6 for the phase portrait of this system.
Fig. 6. Phase portrait of system (4.14)
Remark 2. The hyperbolic elasticae are graphs of smooth periodic functions x2(x1), of the same period T.
As k ^ +0, these curves look like sinusoids of small amplitude
x1(t) = t + O (k2), x2(t) = 2ksint + O (k3),
see Fig. 3 for k = 0.2.
On the other hand, as k ^ 1 — 0, the hyperbolic elasticae look like a saw with big equilateral teeth of right angles, see Fig. 5 for k = 0.9.
4.4. Properties of the vector field A
4.4.1. Incompleteness
Proposition 8. The vector field A is incomplete, that is, there exist T > 0 and a nice abnormal extremal A(t) = etH°(A0), t G [0, T), such that l™0 A(t) = to.
Proof. Let h6 = h8 = h9 = h10 = h12 = h13 = h14 = 0, h11 = 1, then system (4.2)-(4.7) reduces to the system
h4 = h4h7, h5 = —h5h7, h7 = h4. If we choose for this system the initial conditions h4(0) > 0, h5(0) = h7(0) = \/'2h4(0),
then its solution h4(t) = (t3ryi ~>ooasi—>T — 0, T = oy-
Other examples of extremals tending to infinity for a final time are given in Subsection 4.4.2.
□
4.4.2. Special case h4 = h6 = h13 = 0
Let h4 = h6 = h13 = 0. Then the ODEs (4.2)-(4.7) for nice abnormal extremals yield the following system:
h 5 = —h5h7, (4.17)
h 7 = —h5 h13, (4.18)
h 13 = 0, (4.19)
(h5 ,h7)(0) = (h0,h0). (4.20)
This system has an integral C = h5h13 — On this basis the Cauchy problem (4.17)-(4.20) is explicitly integrated in Propositions 9 and 10 (see below). The formulas obtained in these propositions show that the system of ODEs (4.17)-(4.19) is incomplete, which proves once more incompleteness of the field A.
Proposition 9. Let hi3 > 0.
(1) Let C< 0.
(1.1) If \h7\ > y/2\C\, then:
h7(t) = y/2\C\ 1 + ■
2
/2\C\(t-2@) _ i
Mf) = P(i + -
2
ICI
h
13
/2\C\(t-2P) _i
13
P =
1
2.y2|C|
ln
h°7 - ./2\C\
h°7 + y/2\C\
Xi(t) =
2y/2\C\ ( 1___1
h13 I eS/2\C\(t-2l3) _ 1 e-2Py/m - 1
X2(t) = 0.
(1.2) If \h7\ < v^ÎCÎ, then:
h7(t)=vm i-
2
xi(t) = -
eV2\c\(t-2^ +1 )
i Vrn (_l
h
h5(t) = Pll-
h
13 \ ee 1
13 e
/2\c\(t-m +1 e-2N2\C\ +1
ICI
/2\c\(t-2p) + 1j h13 '
, X2(t) = 0.
(1.3) If C = 0, then :
h7(t) =
t - 2£'
h5 (t) =
hi3 (t - 2{)
2
e = -
h7'
xi(t) =
11
+ 7T7 > x2(i)=0.
hi3 \t - 2e 2e
(2) If C> 0, then :
h7(t) = -V^Ctan ( \Hr(t - 2a) ), h5(t) = y— cos"2 ( - 2a)
h
i3
= —
h
a
■ tan
: arctan
h°7
V2C'
i3
IT \ h°
l^(t-2a)\-^-, x2(t) = 0.
2
e
e
2
2
2
2
1
2
1
Proposition 10. Let hi3 = 0.
(1) Let h° = 0, then :
h°(t) = h°7, h5 (t) = h°5e-h7t, hg
®1(t) = ^(e-fc?t-l)> a:2(i) = 0.
(2) Let h° = 0, then :
h° (t) = 0, h5 (t) = hg, (t) = —hgt, (t) = 0.
Remark 3. The case h13 < 0 reduces to the case h13 > 0 considered in Proposition 9 by the change of variables
h7 i—y — h7, h5 i—y — h 5, xi i—y —Xi.
4.5. Nice abnormal set
The abnormal set is defined as
Abn = {x(t) | x(-) is an abnormal trajectory, x(0) = Id = 0, t G R}.
One of the most important open questions of sub-Riemannian geometry is whether the abnormal set has zero Lebesgue measure (sub-Riemannian Sard's property [18, 19]). Let us consider a related — nice abnormal — set:
Abnnice = {x(t) | x(-) is a nice abnormal trajectory, x(0) = Id = 0, t G R}. Define the nice exponential mapping as follows:
Expnice: (A, t) ^ x(t) = n ◦ etH0(A), A g E, t G R, (4.21)
where n: T*G —y G, T*G 3 A ^ x G G, is the natural projection. We have
Abnnice = Expnice(E X R). (4.22)
Theorem 3.
(1) The mapping Expnice is real-analytic.
(2) The mapping Expnice is proper, that is, the preimage of a compact is compact. Proof. Item (1) follows since the vector field H0 is real-analytic.
Item (2) follows from the definition (4.22) of the nice abnormal set and the coordinate expansion (4.2)-(4.8) of the vector field HH0. □
Theorem 4.
(1) Abnnice is a stratified Whitney space.
(2) codim(Abnnice) = 3.
(3) ^(Abnnice) =
Proof. Item (1). The set Abnnice = Expnice(E) is subanalytic since the mapping Expnice is real-analytic and proper, and the set E is semianalytic. Further, by Hironaka's theorem [10], a subanalytic set has a Whitney stratification [11, 12].
Item (2) follows since the manifold E has codimension 3 in g*, and the nice abnormal exponential mapping Expnice has maximal rank.
Item (3) follows immediately from item (2) of this theorem. □
Remark 4. Item (3) of Theorem 4 is related to Sard's sub-Riemannian hypothesis: the required equality ^(Abn) =0 is equivalent to the equality ^(Abn \ Abnnice) = 0. That is, there remains an open question: is the set filled by non-nice abnormal trajectories negligible?
Acknowledgments
The authors thank anonymous reviewers for helpful comments on exposition in the paper.
Conflict of interest
The authors declare that they have no conflict of interest.
References
[1] Montgomery, R., A Tour of Subriemannian Geometries, Their Geodesics and Applications, Math. Surveys Monogr., vol. 91, Providence, R.I.: AMS, 2002.
[2] Agrachev, A. A. and Sachkov, Yu. L., Control Theory from the Geometric Viewpoint, Encyclopaedia Math. Sci., vol. 87, Berlin: Springer, 2004.
[3] Agrachev, A., Barilari, D., and Boscain, U., A Comprehensive Introduction to Sub-Riemannian Geometry: From the Hamiltonian Viewpoint, Cambridge Stud. Adv. Math., vol. 181, Cambridge: Cambridge Univ. Press, 2020.
[4] Sachkov, Yu., Introduction to Geometric Control, Springer Optim. Appl., vol. 192, Cham: Springer, 2022.
[5] Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V., and Mishchenko, E.F., The Mathematical Theory of Optimal Processes, L. W. Neustadt (Ed.), New York: Wiley, 1962.
[6] Sachkov, Yu.L. and Sachkova, E. F., The Structure of Abnormal Extremals in a Sub-Riemannian Problem with Growth Vector (2, 3, 5, 8), Sb. Math., 2020, vol. 211, no. 10, pp. 1460-1485; see also: Mat. Sb., 2020, vol. 211, no. 10, pp. 112-138.
[7] Sachkov, Yu. L., An Exponential Mapping in the Generalized Dido Problem, Sb. Math., 2003, vol. 194, nos. 9-10, pp. 1331-1359; see also: Mat. Sb, 2003, vol. 194, no. 9, pp. 63-90.
[8] Euler, L., Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimitrici latissimo sensu accepti, Lausanne: Bousquet, 1744.
[9] Love, A. E.H., A Treatise on the Mathematical Theory of Elasticity, 4th ed., New York: Dover, 1944.
[10] Hironaka, H., Subanalytic Sets, in Number Theory, Algebraic Geometry and Commutative Algebra: In Honor of Yasuo Akizuki, Tokyo: Kinokuniya, 1973, pp. 453-493.
[11] Goresky, M. and MacPherson, R., Stratified Morse Theory, Ergeb. Math. Grenzgeb. (3), vol. 14, Berlin: Springer, 1988.
[12] Bierstone, E. and Milman, P.D., Semianalytic and Subanalytic Sets, Inst. Hautes Études Sci. Publ. Math., 1988, No. 67, pp. 5-42.
[13] Le Donne, E. and Tripaldi, F., A Cornucopia of Carnot Groups in Low Dimensions, Anal. Geom. Metr. Spaces, 2022, vol. 10, no. 1, pp. 155-289.
[14] Lokutsievskii, L. V. and Sachkov, Yu. L., On the Liouville Integrability of Sub-Riemannian Problems on Carnot Groups of Step 4 and Higher, Sb. Math., 2018, vol. 209, no. 5, pp. 672-713; see also: Mat. Sb, 2018, vol. 209, no. 5, pp. 74-119.
[15] Sachkov, Yu. L., Conjugate Time in the Sub-Riemannian Problem on the Cartan Group, J. Dyn. Control Syst., 2021, vol. 27, no. 4, pp. 709-751.
[16] Ardentov, A. and Hakavuori, E., Cut Time in the Sub-Riemannian Problem on the Cartan Group, ESAIM Control Optim. Calc. Var, 2022, vol. 28, Paper No. 12, 19 pp.
[17] Whittaker, E. T. and Watson, G. N., A Course of Modern Analysis, Cambridge: Cambridge Univ. Press, 1996.
[18] Rifford, L. and Trélat, E., Morse-Sard Type Results in Sub-Riemannian Geometry, Math. Ann., 2005, vol. 332, no. 1, pp. 145-159.
[19] Belotto da Silva, A. and Rifford, L., The Sard Conjecture on Martinet Surfaces, Duke Math. J., 2018, vol. 167, no. 8, pp. 1433-1471.
