SYMMETRIES AND PARAMETERIZATION OF ABNORMAL EXTREMALS IN THE SUB-RIEMANNIAN PROBLEM WITH THE GROWTH VECTOR (2, 3, 5, 8) Текст научной статьи по специальности «Математика»

Russian Journal of Nonlinear Dynamics, 2019, vol. 15, no. 4, pp. 577-585. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd190417

MSC 2010: 93C15

Symmetries and Parameterization of Abnormal Extremals in the Sub-Riemannian Problem with the Growth Vector (2, 3, 5, 8)

Yu. L.Sachkov, E. F.Sachkova

The left-invariant sub-Riemannian problem with the growth vector (2,3, 5,8) is considered. A two-parameter group of infinitesimal symmetries consisting of rotations and dilations is described. The abnormal geodesic flow is factorized modulo the group of symmetries. A parameterization of the vertical part of abnormal geodesic flow is obtained.

Keywords: sub-Riemannian geometry, abnormal extremals, symmetries

1. Problem statement

Let L be the free nilpotent Lie algebra with 2 generators of step 4. There exists a basis L = span(Xi, ... , X8) in which the product table in L reads as follows:

[Xi ,X2]= X3, [Xi,X3 ]= X4, [X2 ,Xa]= X5, (1.1)

[Xi, X4] = Xe, [X2, X4] = [Xi, X5] = X7, [X2, X5] = X8. (1.2)

Let G be the connected simply connected Lie group with the Lie algebra L. Consider the left-invariant sub-Riemannian structure [1, 2] on G defined by (Xi ,X2) as an orthonormal frame. The corresponding optimal control problem reads as follows:

x = uiXi(x) + u2X2(x), x E G, u = (ui,u2) E R2, (1.3)

x(0) = xo = Id, x(ti) = xi, (1.4) ti

J = i J(n'l + n'l) dt ->• min. (1.5)

Received May 30, 2019 Accepted October 01, 2019

The work of Yu. L. Sachkov and E. F. Sachkova was supported by the Russian Science Foundation under grant 17-11-01387 and performed at the Ailamazyan Program Systems Institute of the Russian Academy of Sciences.

Yury L. Sachkov yusachkov@gmail.com Elena F. Sachkova efsachkova@mail.ru

Ailamazyan Program Systems Institute of RAS Pereslavl-Zalessky, Yaroslavl Region, 152020 Russia

A symmetric model of this problem is the following one [4]:

G = M^ ...x8, (1.6)

^ d X2 d x\+x?2 d xix'2 d x'2 d

dx\ 2 dx¿ 2 dx5 4 dxj 6 dxg' Y ^ Xl ® ^x\+x2 d a;f d ^x\x2 d dx2 2 dx¿ 2 dxi 6 Oxq 4 dxj'

In this paper we continue the study of abnormal extremals in problem (1.3)-(1.5) started in [3, 5]. Notice that the normal geodesic flow in problem (1.3)-(1.5) is not Liouville integrable [6].

Denote by D the distribution spanned by the vector fields X1, X2, and by g the inner product in D determined by (Xi , X2) as an orthonormal frame. Then (D,g) is the sub-Riemannian structure given by (X1 ,X2) as an orthonormal frame.

This work has the following structure. In Section 2 we describe some infinitesimal symmetries of the sub-Riemannian structure (D,g) and the distribution D — rotation Xo and dilation Y. In Section 3 we lift these symmetries to the cotangent bundle T*G. In Section 4 we describe the action of these lifted symmetries on the Hamiltonian system for abnormal extremals. In particular, we show that initial conditions for abnormal extremals can be factorized via the rotations to a fundamental domain {h7 = 0}. In Section 5, we give an explicit parameterization of rotations. In Section 6 we present an explicit parameterization of the vertical part of abnormal extremals with initial conditions in the fundamental domain {h7 = 0}. Finally, in Section 7 we conclude on final parametrization of abnormal extremals for arbitrary initial conditions.

2. Infinitesimal symmetries of (D,g) and D

Definition 1. A vector field V G Vec G is called an infinitesimal symmetry of a distribution D if its flow etV preserves D, i.e., etVD = D.

A vector field V G Vec G is called an infinitesimal symmetry of a sub-Riemannian structure (D,g) if its flow preserves both the distribution D and the inner product g, i.e., et^VD = D and (etV)*g = g.

The Lie algebras of symmetries of a distribution D (a sub-Riemannian structure (D,g)) will be denoted by Sym(D) (respectively Sym(D,g)).

Symmetries of distributions and sub-Riemannian structures may be computed via the following

Proposition [7]. Let X G Vec(G).

(1) X G Sym(D) iff ad X(D) c D, or, equivalently, ad X G gl(Dx) for all x G G, i.e.,

2

[X, Xi] =J2 aij Xj, aij G C~ (G).

i,j=1

(2) X G Sym(D, g) iff ad X g so(Dx) for all x G G, i.e.,

2

[X,Xi ] = ^^ aij Xj, aji = -aij, aij G C00 (G). i,j=1

Theorem 1.

(1) There exists a vector field Y G Sym(D) such that

[Y, Xi] = -X!, [Y, X2] = -X2, Y(0) = 0. In model (1.6)-(1.8) this vector field reads

d d d d d t d t d t d

=Xi ---h X2 --h 2x3 --b 3.T4 --b 3.T5 --b 4x6 ---b 4x7---b 4r8 -—.

dx1 dx2 dx3 dx4 dx5 dx6 dx7 dx8

(2) There exists a vector field X0 G Sym(D,g) such that

[Xo ,Xi ] = -X2, [Xo, X2] = X!, Xo (0) = 0.

In model (1.6)-(1.8) this vector field reads

d d d d ^ d ^ d ^ d A0 = -.T27T--h.Ti---—--------h RT~,

dx1 dx2 dx4 dx5 dx6 dx7 dx8 4 22

-J - 1 f-fi^ f-fi^

• " 1 • " 1 ■' ■)

3 3

ry> ry> W* W* r\

Q = -^-^ + 2x6-2xs,

2 2 4

>Xj 1 >XJ O «X/Q

(3) The vector fields Y and X0 commute: [Y, X0] = 0.

Proof. Follows from Theorem 2 [4]. □

The product table given by Eqs. (1.1), (1.2) yields the following statement. Corollary 1.

(1) The symmetry Y has the following Lie brackets with the basis vector fields in Lie algebra L:

[Y, X3] = -2X3, [Y, X4] = -3X4, [Y, X5] = -3X5, [Y, Xe ] = -4X6, [Y, X7 ] = -4X7, [Y, X8 ] = -4XS.

(2) The symmetry Xo has the following Lie brackets with the basis vector fields in Lie algebra L:

[X0,X3] = ° [X0, X4] = -X5, [X0,X5] = X4,

[Xo, Xe] = 2X7, [X0,X7] = X8 - Xe, [X0, X8] = -2X7. 3. Lift of symmetries to T*G

Introduce Hamiltonians linear on fibers and corresponding to the vector fields Xi, Y:

hi(X) = (X,Xi(x)), i = 0, ...,8,

hy(A) = (X,Y(x)), x = n(A), A G T*G,

where n: T*G ^ G is the canonical projection. Consider the corresponding Hamiltonian vector fields on T*G

hi (A), i = 0,..., 8, hy (A), A e T *G.

The vertical part of these vector fields reads in the coordinates (hi, rotation ho

h i = -h2, h 2 = hi, h3 = 0,

h 4 = —h5, h 5 = h4,

h6 = —2h7, h 7 = h6 — h8, h8 = 2h7,

the dilation hy

,h8) as follows: the

(3.1)

h i = —hi, h 2 = - -h2, h 3 = —2h3,

h4 = —3h4, h 5 = —3h5,

h 6 = —4he, h 7 = —4h7, h 8 = —4h8

The phase flow of rotations is visible via the Casimir A = h6h8 — h7: we have

ho A = ho (he + hg) = 0.

The vertical part of the field h0 is tangent to the closed curves {A = const, h6 + h8 = const}, thus it is periodic. An explicit parameterization of the flow of ODE (3.1) is given in Section 7.

4. Canonical abnormal flow and its symmetries

We described in [5] the structure of abnormal extremals for the sub-Riemannian structure (D,g) in terms of the Casimir A and an integral of abnormal extremals I = hgh4 — — 2h7 h4 h5 + heh2.

In the (asymptotic) case A < 0, I = 0 projections of abnormal extremals to the plane (h4,h5) are straight lines or broken lines.

In the complementary (main) case A ^ 0 or I = 0 projections of abnormal extremals to the plane (h4, h5) are first- or second-order curves (straight lines, ellipses, hyperbolas, parabolas). In this case extremals are reparameterization of trajectories of the canonical Hamiltonian system

A = —h5hi + h4h2, A e (A2)

2

(4.1)

where (A2)± = {A e T*G | hi(A) = h2(A) = h3(A) = 0}. The vertical part of system (4.1) reads as follows:

C =

hi = h2 = h3 = 0,

(hi)=eft) •

he = h7 = h g = 0.

Following [5], we call system (4.2)-(4.4) the canonical system for abnormal extremals.

h7 —h6 hg —h7

(4.2)

(4.3)

(4.4)

The symmetries h0 and hy act on the canonical abnormal Hamiltonian vector field A = -h5h1 + h4h2 defined by system (4.1) as follows:

[ho ,A ] =0, [hy ,A] = -4 A.

We get from the Lie brackets (4.5), (4.6) the following statement. Proposition 1. For any t,s,r G R we have

(4.5)

(4.6)

etA o esh0 = esh0 o etA, etÂQ erhY = erhY o etIA,

t' = te

4r

(4.7)

(4.8)

Consequently, we can find the vertical part of canonical abnormal extremals as follows:

etA (Ac) For r = 0 we get:

= e~sh° o e-rhY

etA (Ac)

A ),

_ —sho

At' = etJA o esh0 o erhY (Ac), t' = te

4r

(At),

At = etA o esho(Ac).

(4.9)

(4.10)

It is obvious from (3.1) that the space h hs factorizes by the flow of the rotation h0 to the half-plane {(h6, h7, h8) G R3 | h7 = 0, h6 - h8 ^ 0}. Thus, we can take in (4.10)

A0 G Q = {h1 = h2 = h3 = 0, h7 = 0, h6 - h8 ^ 0}.

We call the previous set the fundamental domain of the rotation h0.

5. Explicit parameterization of rotations

Denote % = (h6,h7,h8) G Rh6 hr hs. Then ODE (3.1) defines in Rh6 hr hs a linear system

/0-2 0 \

x = Bx, B

1 0 -1 0 2 0

/

System (5.1) has the solutions %(s) = eBs\°, explicitly

M«) = 2 + /?«) + {he ~ cos 2s - 2h°7 sin2s),

h"(s) = - h°8) sin 2s + 2h°7 cos 2s),

hs(s) = + h°8) - (h°6 - h°8) cos 2s + 2/?° sin2s).

In the coordinates

ihl\

h*7

Vh8/

(h6-h8\

V2h7 \h6 + h8y

(5.1)

(5.2)

we have

h6 (s) = P cos(2s + ^o),

Kis) = -4= sin(2s + <p o), (5.3)

hg (s) = hg0, hf

where p2 = (h*®)2 + 2(/?|0)2, cosyo = ——, sinyo = -— • It is visible from formulas (5.3)

PP

that the flow of ODE (5.1) defines motion along ellipses

{hg = const, 2(hg)2 + (h6)2 = const} = {A = const, h6 + h8 = const}. Consequently, the fundamental set of rotation is

F = {Xf = (h6, 0, hg) | h6 ^ hg}. (5.4)

6. Solution to the canonical system (4.3), (4.4) in the fundamental case h7 = 0

In this section we consider the case h7 = 0 and describe a solution to the canonical system (4.3), (4.4) with an initial condition (h4,h5,h6,h7)(0) = (h°°, h0,h6,0,hg). If h.7 = 0, then A = h6hg. Denote 5 = s/\A\.

6.1. Elliptic case A > 0

6.1.1. Subcase I = 0

In this case the fundamental set of rotation is

F = {xf = (h6,0, hg) | h6 ^ hg > 0 and h6 ^ hg, h6 < 0,hg < 0}. Then system (4.3) has solution for parameters Xf as follows:

/h4(t)\ = /h0 —h6h0\ I cos 0t \ = (acos(St + <p)\ \h5(t)J \h°5 h8hl J \lSmSt) \bsm(5t + <p))'

where

b ^fhe'

,0/ hgh4 h6h5 ,0/7 w \

cosy? = n4/a = , smy = —— = %/6, y £ (—7r, 7r). oo oa

6.1.2. Subcase I = 0

If I = 0, then system (4.3) has solutions

h4 = 0, h5 = 0.

6.2. The hyperbolic case A < 0

In this case the fundamental set of rotation is

F = {Xf = (h6, 0, hg) | h6 > 0,hg < 0}.

6.2.1. The nonasymptotic subcase I = 0

In this subcase the initial point does not belong to eigenspaces of the matrix C:

Introduce the next value: a = sign \h°A\-

\h0\ . For hyperbolic Xf it is easy to

he h8

prove that Ih8 > 0 if a = 1 and Ih6 > 0 if a = -1. Introduce the next parameters for fundamental set of rotation in hyperbolic case:

ahs' b V G he

1. Let a = 1. Then system (4.3) has solution for parameters Xf as follows:

h(t)\ h -heh0\ ( ch öt \ / sign h0a ch(öt - sign h°4a) \ [h5(t)J = ^o hshl ) (jsh St) = [-sign h°4b sh(öt-sign h^a))'

where

ch« = |/7°|/a. = sh a = h°5/b=}-^, ae K.

öb öa

2. Let a = -1.

Then system (4.3) has solution for parameters Xf as follows:

fh4(t)\ (h4 -heh0Ä ( chöt \ /- signh0ash(öt - signh°ß)\ \h5(t)) ~ \h°5 hsh\ ) \JshSt) ~ V signh°5bch(öt - signh°5ß) )'

where

chß = \h05\/b=^ß-, shß = h°Ja = -^, ßeR.

öa öb

6.2.2. The asymptotic subcase I = 0

In this case the initial point belongs to eigenspaces of the matrix C, and we have an equality in (6.2).

Introduce the parameter p =

hs he

. In the upper case (6.2) we have

h4(t) = h0 est, h5(t) = -h0 peSt, and in the lower case (6.2) we have

h0

/74 (t) = h5(t) = h°5e-M.

p5

6.3. The parabolic case A = 0

6.3.1. Subcase C = 0

In this case the fundamental set of rotation is

F = {xf = (h6, 0,hg) | h6 > 0,hg = 0and h6 = 0,hg < 0}.

1. Let Ch0 = 0. If h6 = 0, then

h4(t) = h6 h51 + h0, h5(t) = h0.

If hg = 0, then

h4(t) = h4, h5(t) = hg h0t + h0.

2. Let Ch0 = 0. If h6 = 0, then

h4(t) = h°4, h5 (t) = 0.

If hg = 0, then

h4(t) = 0, h5 (t) = h0.

6.3.2. Subcase C = 0

We have

h(t) = h0.

7. Conclusion on parameterization of extremals

On the basis of results of the previous sections, we can get a parameterization of abnormal extremals (4.10), with explicit parameterization of At for the case A0 e 0 c {h7 = 0} given in Section 6, and explicit parameterization of the flow esho as follows:

h4 (s) N h5(s) ) 1

cos s — sin s sin s cos s

h04 h05

h(s) = + h°8) + (h°6 - h°8) cos 2s - 2h°7 sin2s),

h"{s) = ~ h°8) sin 2s + 2h°7 cos 2s),

hs(s) = + - (h% - cos 2s + 2/?° sin2s).

An explicit parameterization of abnormal extremal trajectories will be performed similarly in a forthcoming paper.

References

[1] Montgomery, R., A Tour of Subriemannian Geometries, Their Geodesies 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] Sachkov, Yu. L. and Sachkova, E. F., Degenerate Abnormal Trajectories in a Sub-Riemannian Problem with Growth Vector (2, 3, 5, 8), Differ. Equ., 2017, vol. 53, no. 3, pp. 352-365; see also: Differ. Uravn., 2017, no. 3, pp. 362-374.

[4] Gauthier, J.-P. and Sachkov, Yu. L., On the Free Carnot (2, 3, 5, 8) Group, Program Systems: Theory and Applications, 2015, vol.6, no. 2(25), pp. 45-61.

[5] Sachkov, Yu.L. and Sachkova, E.F., Structure of Abnormal Extremals in Sub-Riemannian Problem with the Growth Vector (2, 3, 5, 8), submitted for publication (2019).

[6] Lokutsievsky, L.V. and Sachkov, Yu.L., On Liouville Integrability of Sub-Riemannian Problems on Carnot Groups of Step 4 and More, Sb. Math., 2018, vol. 209, no. 5, pp. 672-713; see also: Mat. Sb, 2018, vol.209, no. 5, pp. 74-119.

[7] Sachkov, Yu. L., Symmetries of Flat Rank Two Distributions and Sub-Riemannian Structures, Trans. Amer. Math.. Soe, 2004, vol. 356, no. 2, pp. 457-494.