Sub-Finsler geodesics on the Cartan group

This paper is a continuation of the work by the same authors on the Cartan group equipped with the sub-Finsler $\ell_\infty$ norm. We start by giving a detailed presentation of the structure of bang-bang extremal trajectories. Then we prove upper bounds on the number of switchings on bang-bang minimizers. We prove that any normal extremal is either bang-bang, or singular, or mixed. Consequently, we study mixed extremals. In particular, we prove that every two points can be connected by a piecewise smooth minimizer, and we give a uniform bound on the number of such pieces.


Introduction
There are several motivations for studying sub-Finsler geometry on Lie groups, especially in geometric group theory and in harmonic analysis. We only mention the prominent articles [10,6,4] and then we refer to the introductions of [15,17] for a broad explanation of the reasons and for several references of the state-of-the-art.
On the one hand, as in sub-Riemannian geometry, distributions of step 2 are easier to study and there is already some good understanding of the lower dimensional cases, see [15]. On the other hand, sub-Finsler structures defined by smooth norms have a similar theory that in the sub-Riemannian case. For these reasons the challenge is to study step-3 sub-Finsler groups with a non-strictly convex norm. The lower dimensional examples are the Engel group and the Cartan group, which both have step 3 and rank 2.
In this paper we study the Cartan group, since it is the free-nilpotent group of rank 2 and step 3 (so the Engel group is a quotient of this group), equipped with the ∞ sub-Finsler structure. In our previous paper [17], adopting the point of view of time-optimal control theory, we characterized extremal curves via Pontryagin maximum principle, we described abnormal and singular arcs, and we constructed the bang-bang flow.
The Cartan distribution can be expressed by the span of two vector fields X 1 , X 2 . We consider the ∞ norm with respect to X 1 , X 2 . Hence, every admissible trajectory is characterized by two controls. A summary of the results of this paper is given by the following statements. Theorem 1. In the ∞ sub-Finsler structure on the Cartan group the length-minimizing trajectories are of three not-mutually-exclusive types: (i) one component of the control is constantly equal to 1 or −1, (ii) bang-bang trajectory, (iii) piecewise smooth concatenation of trajectories of types (i) and (ii).
The length-minimizers that are of type (ii) but not of type (i) have at most 12 arcs. The length-minimizers of type (iii) have at most 14 arcs. All curves of type (i) are length-minimizers. Moreover, for every trajectory of type (i) there exists a piecewise-smooth length-minimizing trajectory connecting the same two points and having at most 5 smooth pieces.
As a corollary, we deduce that any pair of points can be connected by an optimal piecewise-smooth trajectory with at most 14 arcs.
The paper has the following structure. In Sec. 2 we recall the problem statement and the main results on it obtained in previous paper [17]. Section 3 is devoted to detailed study of structure of bang-bang extremal trajectories implied by Pontryagin Maximum Principle. In Sec. 4 we prove upper bounds on the number of switchings on bangbang minimizers. In Sec. 5 we prove that any normal extremal is either bang-bang, or singular, or mixed. Further, Sec. 6 is devoted to the study of mixed extremals, including upper bound on the number of switchings. Finally, in Sec. 7 we obtain a uniform bound on the number of smooth pieces on minimizers connecting arbitrary points in the Cartan group.

Problem statement and previous results
Consider the 5-dimensional free nilpotent Lie algebra with 2 generators, of step 3. There exists a basis L = span(X 1 , . . . , X 5 ) in which the product rule in L takes the form The Lie algebra L is called the Cartan algebra, and the corresponding connected simply connected Lie group M is called the Cartan group. We will use the following model: with the Lie algebra L modeled by left-invariant vector fields on R 5 The product rule in the Cartan group M in this model is given in [11]. Left-invariant ∞ sub-Finsler problem on the Cartan group is stated as the following time-optimal problem: 3) was considered first in paper [17]. We recall the main results of that paper. Existence of optimal controls follows from Rashevsky-Chow and Filippov theorem [12]. Pontryagin Maximum Principle implies that optimal abnormal controls are constant. Introduce linear-on-fibers Hamiltonians h i (λ) = λ, X i , λ ∈ T * M , i = 1, . . . , 5. A normal extremal arc λ t , t ∈ I = (α, β) ⊂ [0, T ] is called: • a singular arc if one of the condition holds: • a mixed arc if it consists of a finite number of bang-bang and singular arcs.
Singular controls have one of components constantly equal to 1 or −1, thus they are optimal. The fix-time attainable set along singular trajectories was explicitly described and was shown to be semi-algebraic.
Bang-bang extremal trajectories satisfy the Hamiltonian system with the Hamiltonian function H = |h 1 | + |h 2 |: The dual of the Lie algebra L * = T * Id M has Casimir functions h 4 , With the use of the coordinate θ ∈ S 1 = R/2πZ: the vertical part of Hamiltonian system (2.4) reduces to the following system: Thus in the study of system (2.5) we can restrict ourselves by the case h 4 ≥ h 5 ≥ 0. This case obviously decomposes into the following sub-cases:

Structure of bang-bang trajectories
In this section we consider, case by case, the structure of bang-bang trajectories implied by Pontryagin Maximum Principle.
Duration of all segments of constancy of controls (except the first and the last ones) is equal to The first and the last segments may take arbitrary values in the corresponding intervals (0, τ i ].
The subsequent analysis of the structure of bang-bang trajectories is completely analogous to the preceding one, thus we omit analogous computations and arguments in the following subsubsections. Then the control has the form Notice that despite the fact that h 2 (λ t ) vanishes when θ = π, h 3 = 0, the control u 2 (t) does not switch at such points since h 2 (λ t ) preserves sign near these points.
Then the control has the form:
-π/2 π/2 π 3π/2 Then the control has the form: , the corresponding phase portrait is shown in Fig. 12. Figure 13: (x(t), y(t)): Case 1), level line C 7 The level line C 7 is defined in the domains {θ ∈ (− π 2 , 0)} and {θ ∈ (π, 3π 2 )} by the equations Thus the level line C 7 is homeomorphic to figure 8, with self-intersection at the point (θ, are continuous curves homeomorphic to S 1 , with the only singularity -the corner point (θ, h 3 ) = ( 3π 2 , 0). Each bang-bang control is obtained by choosing a finite segment from the following infinite periodic graph: : : We have The curves (x(t), y(t)) corresponding to the curves C + 7 and C − 7 are shown in Figs. 14 and 15 respectively. An example of curve (x(t), y(t)) corresponding to two curves C + 7 and two curves C − 7 is given in Fig. 13. -π/2 π/2 π 3π/2 Figure 16: (θ(t), h 3 (t)): Case 1), domain C 8 In the case h 3 > 0 the bang-bang control has the form In the case h 3 < 0 the order of switchings is opposite. We have

Case 2), level line C 5
We have h 4 > h 5 = 0, E = h 4 , the corresponding level line is shown in Fig. 25. There are decompositions The curves C + 5 , C − 5 are homeomorphic to S 1 , with the only singularity -a corner point (θ, h 3 ) = ( 3π 2 , 0). Bang-bang controls are obtained by choosing a finite segment of the following periodic graph: We have The curves (x(t), y(t)) corresponding to the curves C + 5 and C − 5 are shown in Figs. 27 and 28 respectively. An example of curve (x(t), y(t)) corresponding to two curves C + 5 and two curves C − 5 is given in Fig. 26.
If h 3 < 0, then the order of switchings is opposite. We have
If h 3 < 0, then the order of switchings is opposite. We have
If h 3 < 0, then the order of switchings is opposite. We have On the basis of the results obtained in this section we obtain the following statement. . In all these cases θ(t) ∈ [0, π], and θ(t) takes extreme values 0, π at isolated instants of time t. Thus h 2 (λ t ) > 0 for almost all t, whence u 2 (t) ≡ 1 for almost all t. By Lemma 2 [17], the control u(t) is optimal.
The domain in the phase cylinder of system (2.5) corresponding to inequalities (4.1) is shown in Fig. 41. Remark. It is easy to see that under condition (4.1) a bang-bang trajectory is simultaneously a singular trajectory, i.e., q(t) = π(λ t ), whileλ t is an h 1 -singular extremal linearly independent of λ t . One can show that there exists also a bang-bang extremalλ t , linearly independent of λ t andλ t , such that q(t) = π(λ t ). Thus the trajectory q(t) is a projection of at least three linearly independent extremals (in this case the extremal trajectory is said to have corank not less than 3 [12]). Thus the below necessary optimality condition (Th. 3) is not applicable to a trajectory q(t) under condition (4.1).

4.2
Bound of the number of switchings on bang-bang trajectories with high energy E

Some necessary results
We obtain an upper bound on the number of switchings on optimal bang-bang trajectories via the following theorem due to A. Agrachev and R. Gamkrelidze.
We will check the sign of the quadratic form Q| W via the following test. Consider a quadratic form
Thus A 0 1 0 1 < 0, i.e., the quadratic form Q| W is not negative semidefinite. By Th. 3, the control u is not optimal.

Bound of the number of switchings in the general case of high energy E
The rest cases are considered similarly to Th. 5: In all these cases Th. 3 and Th. 4 imply that k = 12 switchings are not optimal.
Passing from the fundamental domain {h 4 ≥ h 5 ≥ 0} of the group G to the whole plane (h 4 , h 5 ), we get the following general bound of the number of switchings. Theorem 6. If E > max(−|h 4 |, −|h 5 |), then optimal bang-bang trajectories have no more than 11 switchings.

General form of normal extremals
Now we prove that the list of types of normal extremals given in Sec. 2 is complete.
Notice that singular controls that adjoin bang-bang controls are constant. Thus all mixed controls are piecewise constant, and Th. 3 can be used for bounding the number of switchings on optimal mixed trajectories.
Mixed extremals are schematically shown in Figs. 42-46. Small dashed circles near the points (θ, h 3 ) = ( 3π 2 , 0) and (θ, h 3 ) = (0, 0) denote singular arcs that adjoin bang-bang arcs. Singular arcs are shown by dashed segments.  Notice that mixed extremals λ t are not uniquely determined by the initial covector λ 0 and time t, because of arbitrary duration of singular arcs. Thus exponential mapping cannot be defined for mixed extremals, as it was defined for bang-bang ones.
7 Bound on the number of arcs of minimizers Important questions for applications of sub-Finsler geometry in metric group theory are the following: • given any pair of points in a sub-Finsler manifold, does there exist a piecewise-smooth minimizer that connects these points?