Periodic controls in step 2 sub-Finsler problems

We consider control-linear left-invariant time-optimal problems on step 2 Carnot groups with strictly convex set of control parameters (in particular, sub-Finsler problems). We describe all linear-in-momenta Casimirs on the dual of the Lie algebra. In the case of rank 3 Lie groups we describe the symplectic foliation on the dual of the Lie algebra. On this basis we show that extremal controls are either constant or periodic. Some related results for other Carnot groups are presented.

It is enough to consider the free-nilpotent cases since any step-2 Carnot group is a quotient of a step-2 free-nilpotent Lie group (with the same number of generators) and, moreover, every minimizing curve lifts to a minimizing curve. Indeed, see Theorem 4.2 in [11] for the existence of free Carnot groups; see Corollary 2.11 in [12] for the fact that the quotient is a submetry and therefore geodesics lift to geodesics.
We describe linear Casimirs on the dual of the Lie algebra. As a consequence, in the rank 3 case we characterize the symplectic foliation. Further, we apply Pontryagin maximum principle, and show that in the rank 3 case the extremal controls are either constant or periodic.

Problem statement
Let L be the step 2 free-nilpotent Lie algebra with k ≥ 2 generators: Let G be the connected simply connected Lie group with the Lie algebra L. We will think of X i , X ij as left-invariant vector fields on G.
A model of vector fields X i , X ij on is given by here we follow Section 2.2 in [11]. Let U ⊂ R k be a compact convex set containing the origin in its interior. We consider the following time-optimal problem: If U = −U, we obtain a sub-Finsler problem, and if U is an ellipsoid centered at the origin, we obtain a sub-Riemannian problem.
The sub-Riemannian case U = { k i=1 u 2 i ≤ 1} was first considered by R.Brockett [9], and was completely solved for k = 3 by O.Myasnichenko [10]. Some partial results for k = 4 were obtained by L. Rizzi and U. Serres [14].
We consider in greater detail the case k = 3, although some results concern the general case k ≥ 2.
Existence of optimal solutions in problem (2)-(4) follows in a standard way from the Rashevsky-Chow and Filippov theorems [2].

Linear Casimirs and symplectic foliation
Before our study of extremals for the problem (2)-(4), we consider Casimirs and symplectic foliation (decomposition into coadjoint orbits) on the dual L * of the Lie algebra L [15]. This is important for our study of extremals for the problem.
Introduce linear on fibers of the cotangent bundle T * G Hamiltonians corresponding to the basic left-invariant vector fields on G: Product rule for Lie bracket (1) implies the following multiplication table for Poisson bracket: The Hamiltonians h i , h ij can be considered as coordinates on the dual L * of the Lie algebra L.
Notice that the Poisson bivector (i.e., the matrix of pairwise Poisson brackets of the basis Hamiltonians h i , h ij ) is determined by the skew-symmetric matrix For a vector a = (a 1 , . . . , a k ) ∈ R k , consider a linear function The next lemma gives conditions for a linear function I a to be a Casimir on L * .
By virtue of (5), h ij are Casimirs on L * . By Lemma 1, on each kdimensional subspace {h ij = const} ⊂ L * there are N linear in h i Casimirs, where N = dim ker M, and M is given by (6). This observation yields the whole symplectic foliation on L * in the case k = 3. Notice that in the Heisenberg case k = 2, the symplectic foliation consists of 2-dimensional leaves {h 12 = const = 0} and 0-dimensional leaves {h 12 = 0, (h 1 , h 2 ) = const}. Theorem 1. If the number k of generators equals 3, then the symplectic foliation on L * consists of the following leaves: Proof. In the case k = 3 equality (6) reduces to There are two possibilities:

Pontryagin maximum principle
We apply Pontryagin maximum principle (PMP) in invariant form [2] to problem (2)-(4). The control-dependent Hamiltonian for this problem is and the maximality condition of PMP is where is the support function of the set U [16]. H is convex, positive homogeneous, and continuous. Along extremal trajectories we have H ≡ const ≥ 0. The abnormal case can be omitted since the distribution ∆ = span(X 1 , . . . , X k ) satisfies the condition ∆ 2 = ∆ + [∆, ∆] = T G, thus by Goh condition [2] all locally optimal abnormal trajectories are simultaneously normal. So we consider the normal case: H ≡ const > 0. In view of homogeneity of the vertical part (7), (8) of the Hamiltonian system of PMP, we will assume that H ≡ 1 along extremal trajectories.
From now on we suppose additionally that the set U is strictly convex. Then the maximized Hamiltonian H is C 1 -smooth on R k \{0}, and maximum in (10) is attained at the control u = ∇H = (∂H/∂h 1 , . . . , ∂H/∂h k ) [16]. Denote H i = ∂H/∂h i , i = 1, . . . , k. Then the vertical subsystem of the Hamiltonian system reads as follows: In addition to obvious integrals h ij , the system (11) has also the integral H and the linear integrals The last claim follows from Lemma 1.

Extremals in the case k = 3
Let k = 3. Then the skew-symmetric matrix M = (h ij ) has a nonzero kernel, this allows us to characterize solutions to system (11) as follows.
If M = 0, then all solutions to system (11) are constant. If M = 0, then dim ker M = 1, and we have the following description.
Theorem 2. Let L be a step-2 Carnot algebra with 3 generators. Let 0 = M = (h ij ) ∈ so (3). Suppose that U is strictly convex and compact, and contains the origin in its interior. Let ker M = Ra, 0 = a ∈ R 3 . Then for any h 0 ∈ H −1 (1) the solution h(t) to system (11) with the initial condition h(0) = h 0 is unique and C 2 -smooth. Moreover: is a regular periodic planar curve.
Proof. As we noticed in Sec. 1, we can assume that L is the step-2 freenilpotent Lie algebra with 3 generators. Consider the constrained optimization problem Since the polar set U • = {H ≤ 1} is compact and 0 ∈ int U • , this problem has solutions I max The condition ∇H(h) a is a necessary and sufficient condition for global extremum in the convex optimization problem (12).
Then the point h 0 is not a solution to problem (12), thus I a (h 0 ) ∈ (I min a , I max a ). The curve is compact and planar. Any h ∈ Γ satisfies the inclusion I a (h) ∈ (I min a , I max a ), thus it is not a solution to problem (12), so ∇H(h) ∦ a. Consequently, Γ is a C 1 -regular curve diffeomorphic to S 1 .
Corollary 1. Let L be a step-2 Carnot algebra with 3 generators. Suppose that U is strictly convex and compact, and contains the origin in its interior. Then all normal extremal controls are constant or periodic, and continuous.
It is known that in the sub-Riemannian case U = { 3 i=1 u 2 i ≤ 1} the extremal controls are given by constant or periodic trigonometric functions [10].
In a recent paper [17], E. Hakavuori proved that for step 2 sub-Finsler Carnot groups with strictly convex norms, only lines are infinite geodesics. Thus in the case k = 3, in view of Th. 2, we have corollaries: 1. constant controls are optimal (trivial), 2. for a periodic nonconstant control u(t) there exists t 1 > 0 such that u| [0,t 1 ] is not optimal (nontrivial).

General free-nilpotent Lie groups
It is interesting to look for a generalization of Cor. 1 in the perspective of arbitrary free-nilpotent Lie groups with k ≥ 2 generators and s ≥ 1 steps.
In the Abelian case s = 1, k ≥ 2 we have the probleṁ which obviously has only constant extremal controls.
It would be interesting to characterize similarly optimal controls in the cases s = 3, k ≥ 3 and s ≥ 4. In these cases, if U = { k i=1 u 2 i ≤ 1}, the normal Hamiltonian system of Pontryagin maximum principle is not Liouville integrable [19,20].