Local (Sub)-Finslerian Geometry for the Maximum Norms in Dimension 2

Abstract

We consider specific sub-Finslerian structures in the neighborhood of 0 in $\mathbb {R}^{2}$, defined by fixing a family of vector fields (F₁,F₂) and considering the norm defined on the non-constant rank distribution Δ = vect{F₁, F₂} by

$$|G|=\inf_{u}\{\max\{|u_{1}|,|u_{2}|\} | G=u_{1} F_{1}+u_{2} F_{2}\}. $$

If F₁ and F₂ are not proportional at p, then we obtain a Finslerian structure; if not, the structure is sub-Finslerian on a distribution with non-constant rank. We are interested in the study of the local geometry of these Finslerian and sub-Finslerian structures: generic properties, normal form, short geodesics, cut locus, switching locus, and small spheres.

1 Introduction

From the 1980s, the interest for the sub-Riemannian geometry increases with a lot of contributions in several domains as PDEs, analysis, probability, geometry, and control. One of the questions was to understand the local geometry of sub-Riemannian metrics, as the singularities of small spheres, local cut locus, local conjugate locus, and so on motivated in particular by new results on the heat kernel in the sub-Riemannian context (see [9, 10, 22, 23]). The contact and the Martinet cases were deeply studied (see [1, 2, 11, 12, 19]). The quasi-contact case in dimension 4 was also studied (see [15]). These results allowed to give new results on the asymptotics of the heat kernel at cut and conjugate loci in the 3D contact and 4D quasi-contact cases [5, 7].

In this article, we start the same work for Finslerian and sub-Finslerian metrics associated with a maximum norm: consider a manifold M, a vector bundle π : E → M with fibers of same dimension as M endowed with a maximum norm, and a morphism of vector bundles f : E → TM such that the map from Γ(E) → Vec(M) defined by σ↦f ∘ σ is injective. An admissible curve is a curve γ in M such that exists a lift σ in E with $\dot \gamma (t)=f(\sigma (t))$ a.e. The length of such a curve is the infimum of the ${{\int }_{0}^{T}} |\sigma (t)| dt$ for all possible such σ and the distance between two points q₀ and q₁ is the infimum of the lengths of the curves joining q₀ and q₁. Remark that the map f itself is not assumed to be injective everywhere: at points where f is injective the structure is Finslerian when at points where it is not, it is sub-Finslerian.

Here, we concentrate our attention on the local study of such structures in dimension 2, that is when M and the fibers of E have dimension 2.

Equivalently, with a control point of view and since we are interested in local properties, we consider control systems in a neighborhood of 0 in $\mathbb {R}^{2}$ of the type

$$ \dot q=u_{1} F_{1}(q)+u_{2} F_{2}(q) $$

(1)

where F₁ and F₂ are smooth vector fields and u₁ and u₂ are control functions satisfying

$$ \left| u_{1}\right|\leq 1 \text{ and } \left| u_{2}\right|\leq 1. $$

(2)

Up to reparameterization, minimizing the distance in the geometric context is equivalent to minimizing the time of transfer in the control context.

We are interested in the study of the time optimal synthesis of such systems. Of course, the general situation cannot be completely described since singular cases may have very special behavior. For example, in the case F₁ = ∂_x and F₂ = ∂_y, any admissible trajectory with u₁ ≡ 1 and ${{\int }_{0}^{1}} u_{2}(t) dt= 0$ joins optimally (0,0) to (1,0). Hence, in the following, we will consider only “generic” situations as defined in Section 2.1.

Few works exist concerning sub-Finsler geometry since it is a new subject. Let us mention the works [17, 18] for dimension 3, considering norms which are assumed to be smooth outside the zero section. In [14], the sphere of a left invariant sub-Finsler structure associated with a maximum norm in the Heisenberg group is described. In the preprint [6], the authors describe the extremals (and discuss in particular their number of switches before the loss of optimality) for the Heisenberg, Grushin, and Martinet distributions. In the preprint [4], we describe, in the 3D generic contact case, the small spheres and the local cut locus.

The paper is organized as follows.

In Section 2, we recall Thom’s transversality theorem and some of its corollaries, define what we mean by generic, give generic properties of the couples of vector fields on two-dimensional manifolds and give a normal form for the generic couples.

In Section 3, we give first general results about the optimal synthesis; recalling classical results as Chow-Rashevski, Filippov, and Pontryagin theorems; analyzing the possibilities for extremals to switch or to be singular depending on their initial conditions; giving details on the weights of coordinates in the normal form and on the associated nilpotent approximation.

In Section 4, we present the local synthesis in all the generic cases presented in the normal form of Section 2.

2 Normal Form

In this section, the goal is to give a list of properties of generic couples (F₁,F₂) and to construct a normal form for the couple (G₁,G₂) defined by G₁ = F₁ + F₂ and G₂ = F₁ − F₂. As we will see, ± G₁ and ± G₂ are the velocities of a large class of the minimizers of the optimal control system defined by Eqs. 1 and 2.

In all this article, we will consider the following sets. We define

$$\begin{array}{@{}rcl@{}} {\Delta}_{A}&=&\{q\in M | F_{1}(q) \text{ and } F_{2}(q) \text{ are colinear}\},\\ {\Delta}_{1}&=&\{q\in M | F_{1}(q) \text{ and } [F_{1},F_{2}](q) \text{ are colinear}\},\\ {\Delta}_{2}&=&\{q\in M | F_{2}(q) \text{ and } [F_{1},F_{2}](q) \text{ are colinear}\}. \end{array} $$

In order to give the list of properties, we use the Thom’s transversality theorem and some of its corollaries.

2.1 Generic Properties of Couples of Smooth Vector Fields on 2D Manifolds

2.1.1 Thom’s Transversality Theorem

Denote J^k(M,N) the set of k-jets of maps from M to N.

Theorem 1 (Thom Transversality Theorem, 21, Page 82)

LetM,Nbe smooth manifolds andk ≥ 1 an integer. IfS₁,⋯ ,S_rare smooth submanifolds ofJ^k(M,N) then the set

$$\{f\in C^{\infty}(M,N): J^{k} f \pitchfork S_{i}~\text{for}~i = 1,2,\cdots,r\}, $$

is residual in the C^∞-Whitney topology.

Corollary 1

Assume thatcodimS_i > dimMfori = 1,⋯ ,randk ≥ 1.Then, the set

$$\{f\in C^{\infty}(M,N): J^{k} f{(M)}\cap S_{i}=\emptyset~\text{for}~i = 1,\cdots,r\}, $$

is residual in the C^∞-Whitney topology.

Corollary 2

For every f in the residual set defined in Theorem1, the inverse images$\tilde {S_{i}}:=(J^{k} f)^{-1}(S_{i})$area smooth submanifold of M and$codim\ S_{i}=codim\ \tilde {S}_{i}$fori = 1,⋯ ,r.

Remark 1

Let φ be a diffeomorphism of M and ϕ be a diffeomorphism of N. The map

$$\sigma_{\varphi,\phi} :\left\{ \begin{array}{ccc} C^{\infty}(M,N) & \longrightarrow & C^{\infty}(M,N) \\ f & \longmapsto & \varphi \circ f \circ \phi \end{array}\right. $$

induces a diffeomorphism $\sigma ^{*}_{\varphi ,\phi }$ of J^k(M,N) sending submanifolds of J^k(M,N) on submanifolds of J^k(M,N). Moreover, f is in the residual set defined in theorem 1, if and only if σ_φ,ϕ(f) is in the residual set

$$\{g\in C^{\infty}(M,N): J^{k} g \pitchfork \sigma^{*}_{\varphi,\phi}(S_{i})~\text{for}~i = 1,\cdots,r\}. $$

This remark is important to facilitate the presentation of the proofs of the generic properties given in the next section.

Definition 1

In the following, we will say that a property of maps is generic if it is true on a residual set for the C^∞-Whitney topology.

2.1.2 First Generic Properties

Here, we give a list of generic properties for couples of vector fields on 2D manifolds. In order to use Thom transversality theorem, we work locally in coordinates: we fix a point and consider the Taylor series of a couple of vector fields at this point. Locally, one can consider such a couple as the data of a map

$$g:\left\{\begin{array}{rcl} U\subset \mathbb{R}^{2} &\to& \mathbb{R}^{2}\times\mathbb{R}^{2}\\ (x,y)&\mapsto&((g_{1}(x,y),g_{2}(x,y)),(g_{3}(x,y),g_{4}(x,y))) \end{array}\right. $$

and the k-jet at q = (0,0) ∈ U of g as the data of the map

$$J^{k}g:\left\{\begin{array}{rcl} \mathbb{R}^{2} &\to& \mathbb{R}_{k}[x,y]^{4}\\ (x,y)&\mapsto&(P_{1}(x,y),\dots,P_{4}(x,y)) \end{array}\right. $$

where P_i (1 ≤ i ≤ 4) is the Taylor series of order k of g_i at q.

In order to describe submanifolds of $\mathbb {R}_{k}[x,y]^{4}$ in coordinates, we write:

$$P_{\ell}(x,y)=\sum\limits_{i = 0}^{k}\sum\limits_{j = 0}^{k-i}p_{\ell,i,j}x^{i}y^{j}, \forall \ell= 1,..,4.$$

In the following, (g₁,g₂) are the coordinates of G₁ and (g₃,g₄) the coordinates of G₂ in a local coordinate system.

Generic property 1 (GP1): :

for generic couples of vector fields (F₁,F₂) on M, the set of points where G₁ = G₂ = 0 is empty.

Indeed in coordinates, such points correspond to jets with p_1,0,0 = p_2,0,0 = p_3,0,0 = p_4,0,0 = 0 which form a submanifold of $\mathbb {R}_{k}[x,y]^{4}$ of codimension 4. Hence, thanks to corollary 1, the property is proven.

Denote ${J^{k}_{N}}$ the set of k − jets such that P₁ ≡ 1 and P₂ ≡ 0. Once assumed that we choose a coordinate system such that G₁ = (1,0), then J^kg is in ${J^{k}_{N}}$.

Assume that a set S of $J^{k}(\mathbb {R}^{2},\mathbb {R}^{4})$ is defined as the zero level of a finite number of functions h_i, i = 1…k, whose differentials form a free family when restricted to $T{J^{k}_{N}}$. Then locally, the differentials of the functions h_i form a free family and hence, close to ${J^{k}_{N}}\cap S$, the set S is locally a submanifold. In this context, the codimension of S in $J^{k}(\mathbb {R}^{2},\mathbb {R}^{4})$ is equal to the codimension of $S^{\prime }=S\cap {J^{k}_{N}}$ in ${J^{k}_{N}}$.

Thanks to remark 1, up to a permutation between ± F₁ and ± F₂ and a good choice of coordinates, we will assume in all the following that G₁ ≡ (1,0) locally. This implies that is g₁ ≡ 1 and g₂ ≡ 0 and that the jet of (G₁,G₂) is in ${J^{k}_{N}}$. As a consequence, if a set S is defined by a finite number of functions h_i, i = 1…k, whose differentials form a free family when restricted to $T{J^{k}_{N}}$, then to apply Thom’s theorem and its corollaries we are reduced to apply them to the map

$$g:\left\{\begin{array}{rcl} U\subset \mathbb{R}^{2} &\to& \mathbb{R}^{2}\\ (x,y)&\mapsto&(g_{3}(x,y),g_{4}(x,y)) \end{array}\right. $$

and the set $S^{\prime }=S\cap {J^{k}_{N}}$ seen as a submanifold of $J^{k}(\mathbb {R}^{2},\mathbb {R}^{2})$.

Generic property 2 (GP2): :

for generic couples of vector fields (F₁,F₂) on M, the set of points where G₂ = 0 is a discrete set. The same holds for the set where F₁ = 0 or the set where F₂ = 0.

Indeed, such points correspond to jets with p_3,0,0 = p_4,0,0 = 0 which is a submanifold of $\mathbb {R}_{k}[x,y]^{2}$ of codimension 2. Hence, thanks to corollary 2, the set where G₂ = 0 is generically a submanifold of M of codimension 2 that is a discrete set. For F₂ = 0, the equations are p_3,0,0 = 1 and p_4,0,0 = 0 and for F₁ = 0 the equations are p_3,0,0 = − 1 and p_4,0,0 = 0.

Generic property 3 (GP3): :

for generic couples of vector fields (F₁,F₂) on M, the set Δ_A of points where G₂ is parallel to G₁ is an embedded submanifold of codimension 1.

Indeed, Δ_A is exactly the set of points where g₄ = 0, corresponding to jets with p_4,0,0 = 0. This last set is an embedded submanifold of $\mathbb {R}_{k}[x,y]^{2}$ of codimension 1. Thanks to (GP1) and to corollary 2, we can conclude that generically Δ_A is an embedded submanifold of codimension 1.

Generic property 4 (GP4): :

for generic couples of vector fields (F₁,F₂) on M, the set Δ₁ of points where F₁ is parallel to [F₁,F₂] is an embedded submanifold of codimension 1. The same holds for Δ₂ where F₂ is parallel to [F₁,F₂].

In order to prove (GP4), compute [F₁,F₂] and describe Δ₁ in coordinates. $[F_{1},F_{2}]=-\frac 12 [G_{1},G_{2}]$ hence it has coordinates $-\frac 12 p_{3,1,0}$ and $-\frac 12 p_{4,1,0}$ and F₁ has coordinates $\frac 12 (1+p_{3,0,0})$ and $\frac 12 p_{4,0,0}$. Hence, Δ₁ corresponds to jets satisfying

$$\left|\begin{array}{lll} -\frac12 p_{3,1,0}&&\frac12 (1+p_{3,0,0})\\ -\frac12 p_{4,1,0}&&\frac12 p_{4,0,0} \end{array}\right| = 0. $$

The differential of this determinant is not degenerate hence the set of $\mathbb {R}_{k}[x,y]^{2}$ satisfying this equality is an embedded submanifold of codimension 1. Hence, generically, Δ₁ is the preimage of an immersed submanifold of codimension 1 which, thanks to corollary 2, permits to conclude that Δ₁ is an immersed submanifold of codimension 1.

Generic property 5 (GP5): :

for generic couples of vector fields (F₁,F₂) on M, the sets (Δ_A ∩Δ₁), (Δ_A ∩Δ₂) and (Δ₁ ∩Δ₂) are discrete.

Since G₁ = (1,0), the set (Δ₁ ∩Δ₂) ∖Δ_A is the set of points where (F₁,F₂) is free and [F₁,F₂] = 0 that is

$$\begin{array}{@{}rcl@{}} p_{4,0,0}&\neq&0,\\ p_{3,1,0}&=&0\\ p_{4,1,0}&=&0. \end{array} $$

This set is an immersed submanifold of codimension 2 of $\mathbb {R}_{k}[x,y]^{2}$ hence, thanks to corollary 2, the set (Δ₁ ∩Δ₂) ∖Δ_A is generically a discrete set.

The set (Δ_A ∩Δ₂) ∖Δ₁ is a set of points where F₂ = 0. By (GP2) it is a discrete set. The same holds for (Δ_A ∩Δ₁) ∖Δ₂ which is a set of points where F₁ = 0.

The set Δ_A ∩Δ₁ ∩Δ₂ is the union of the subset where F₁≠ 0 and F₁ // F₂ // [F₁,F₂] with a subset where F₁ = 0. The second is discrete. The first set is also defined by G₁ // G₂ // [G₁,G₂] that is p_4,0,0 = 0 and p_4,1,0 = 0. Hence, thanks to corollary 2, the set where F₁≠ 0 and F₁ // F₂ // [F₁,F₂] is a submanifold of codimension 2 that is a discrete set.

Generic property 6 (GP6): :

for generic couples of vector fields (F₁,F₂) on M, the set of points where G₁ // G₂ // [G₁,G₂] // [G1,[G₁,G₂]] is empty.

The set where G₁ // G₂ // [G₁,G₂] // [G1,[G₁,G₂]] is such that p_4,0,0 = p_4,1,0 = p_4,2,0 = 0. Hence, thanks to corollary 2, it is a submanifold of codimension 3 that is an empty set.

Generic property 7 (GP7): :

for generic couples of vector fields (F₁,F₂) on M, at the points q where G₁(q) // G₂(q) // [G₁,G₂](q) one gets G₁(q) ∈ T_qΔ_A.

The property G₁(q) // G₂(q) // [G₁,G₂](q) implies that p_4,0,0 = p_4,1,0 = 0. If p_4,0,1≠ 0 then Δ_A can be written p_4,0,1y = o(x) that is Δ_A is tangent to the x axis and G₁ ∈ T_qΔ_A. Hence, the set of points where G₁(q) // G₂(q) // [G₁,G₂](q) and G₁(q)∉T_qΔ_A corresponds to jets with p_4,0,0 = p_4,1,0 = p_4,0,1 = 0 which is a submanifold of codimension 3. Hence, generically, at the points q where G₁(q) // G₂(q) // [G₁,G₂](q), one has G₁(q) ∈ T_qΔ_A.

One can even detail more the generic properties: using Thom transversality theorem and its corollaries, we can prove that generically

Generic property 8 (GP8): :

along Δ₁ ∖ (Δ₂ ∪Δ_A), the points where G₁ or G₂ is tangent to Δ₁ are isolated. The same holds true for Δ₂ ∖ (Δ₁ ∪Δ_A).

Generic property 9 (GP9): :

at points of (Δ₁ ∩Δ₂) ∖Δ_A, neither G₁ nor G₂ are tangent to Δ₁ or Δ₂.

Generic property 10 (GP10): :

along Δ_A ∖ (Δ₁ ∪Δ₂), the set of points where G₂ = 0 or G₂ = ±G₁ is discrete.

2.2 Normal Form

Thanks to the generic properties established in the previous section, we prove

Theorem 2 (Normal form)

For a generic couple of vector fields (F₁,F₂) on a 2d manifold M, at each point q of M, up to an exchange between± F₁and± F₂, we get thatG₁(q)≠ 0 and that it exists aunique coordinate system (x,y) centered at q such that one of the following normal form holds

• (NF₁) G₁(x,y) = ∂_x,

  G₂(x,y) = ∂_y + x(a₁₀ + a₂₀x + a₁₁y + o(x,y))∂_x

  + x(b₁₀ + b₂₀x + b₁₁y + o(x,y))∂_y,

  andq∉Δ_A.

• (NF₂) G₁(x,y) = ∂_x,

  G₂(x,y) = (a₀ + a₁₀x + a₀₁y + o(x,y))∂_x + x(1 + x(b₂₀ + O(x,y)))∂_y,

  with 0 ≤ a₀ ≤ 1,andq ∈Δ_A ∖Δ₁.

• (NF₃) G₁(x,y) = ∂_x,

  $G_{2}(x,y)=(a_{0}+o(1))\partial _{x} +(b_{01}y+\frac 12x^{2}+b_{11}xy+b_{02}y^{2}+o(x^{2},y^{2}))\partial _{y}$ ,

  withb₀₁ > 0 and 0 < a₀ < 1,q ∈Δ_A ∩Δ₁ ∩Δ₂andG₁(q) ∈ T_qΔ_A.

For (NF1) and(NF2) oneof the following subcases holds

• (NF_1a) (NF₁) holds witha₁₀ − b₁₀≠ 0 anda₁₀ + b₁₀≠ 0.It corresponds toq∉Δ_A ∪Δ₁ ∪Δ₂.

• (NF_1b) (NF₁) holds witha₁₀ − b₁₀ = 0 anda₁₀ + b₁₀≠ 0.It corresponds toq ∈Δ₁ ∖ (Δ_A ∪Δ₂).

• (NF_1c) (NF₁) holds witha₁₀ − b₁₀≠ 0 anda₁₀ + b₁₀ = 0.It corresponds toq ∈Δ₂ ∖ (Δ_A ∪Δ₁).

• (NF_1d) (NF₁) holds witha₁₀ = b₁₀ = 0.It corresponds toq ∈ (Δ₁ ∩Δ₂) ∖Δ_A.

• (NF_2a) (NF₂) holds with 0 ≤ a₀ < 1.It corresponds toq ∈Δ_A ∖(Δ₁ ∪Δ₂).

• (NF_2b) (NF₂) holds witha₀ = 1.It corresponds toq ∈ (Δ_A ∩Δ₂) ∖Δ₁that is toq ∈Δ_A ∖Δ₁such thatF₂(q) = 0.

Such coordinate system is called the normal coordinate system associated withF₁andF₂.

Proof

In order to prove this normal form, we construct in each situation a coordinate chart by mean of the flow of some linearly independent vector fields associated with the sub-Finslerian structure. More precisely, once identified such a couple of vector fields (X,Y ), we define the coordinate system by defining the map (x,y)↦e^xXe^yYq.

We assume that all the generic properties given before are satisfied. Thanks to (GP1), and thanks to the fact that we are working locally, we can assume that G₁ is not zero.

Thanks to (GP3), we know that Δ_A is a submanifold of dimension 1. Let us start by considering a point q outside Δ_A. We define the map φ which to (x,y) in a neighborhood U of (0,0) in $\mathbb {R}^{2}$ associates the point reached by starting at q and following G₂ during time y and then G₁ during time x that is

$$\varphi : \left\{ \begin{array}{rcl} U& \to& M\\ (x,y)&\mapsto&\mathrm{e}^{xG_{1}}\mathrm{e}^{yG_{2}}q \end{array} \right. $$

Since ∂_xφ(0,0) = G₁(q) and ∂_yφ(0,0) = G₂(q), φ is a local diffeomorphism hence defines a local coordinate system. One proves easily that at each point of coordinates (x,y) the vector G₁(x,y) = (1,0). Moreover, along the y axis, since $\varphi (0,y)=\mathrm {e}^{yG_{2}}q$ then G₂(0,y) = (0,1). This implies the normal form (NF₁). With the normal form (NF₁), one gets that

$$\begin{array}{@{}rcl@{}} [F_{1},F_{2}](0)&=&-\frac12[G_{1},G_{2}](0)=-\frac12(a_{10},b_{10}),\\ F_{1}(0)&=&\frac12(G_{1}(0)+G_{2}(0))=(\frac12,\frac12),\\ F_{2}(0)&=&\frac12(G_{1}(0)-G_{2}(0))=(\frac12,-\frac12) \end{array} $$

which implies that

$$[F_{1},F_{2}](0)=-\frac{a_{10}+b_{10}}{2}F_{1}(0)-\frac{a_{10}-b_{10}}{2}F_{2}(0).$$

The subcases follow immediately.

Assume now that q ∈Δ_A ∖Δ₁. Hence, G₁(q) and G₂(q) are parallel and since we assume that G₁(q) is not 0, we can assume up to a change of role that G₂(q) = αG₁(q) with α ∈ [0,1]. Since q∉Δ₁, G₁(q) and [G₁,G₂](q) are not parallel. This implies that G₁ is not tangent to Δ_A. As a consequence, one can choose a local parameterization γ(t) of Δ_A such that γ(0) = q and $\dot \gamma (t)$ has second coordinate 1 in the basis (G₁(γ(t)),[G₁,G₂](γ(t))). We can know define the map φ which to (x,y) in a neighborhood U of (0,0) in $\mathbb {R}^{2}$ associates the point reached by starting at γ(y) and following G₁ during time x that is

$$\varphi : \left\{ \begin{array}{rcl} U& \to& M\\ (x,y)&\mapsto&\mathrm{e}^{xG_{1}}\gamma(y) \end{array} \right. $$

In this coordinate system, Δ_A is the y axis, G₁(x,y) = (1,0) and the second coordinate of G₂ is null at x = 0 hence it is the product of the function (x↦x) with a smooth function g. Moreover, thanks to the property of γ, g(0,y) = 1 which implies that g(x,y) = 1 + xh(x,y) with h a smooth function. This is exactly (NF₂). If 0 ≤ a₀ < 1 then F₁(q) and F₂(q) are not null and since they are parallel but not parallel to [F₁,F₂](q) then q ∈Δ_A ∖ (Δ₁ ∪Δ₂). If a₀ = 1 then F₂(q) = 0 and q ∈ (Δ_A ∩Δ₂) ∖Δ₁.

The case where q ∈ (Δ_A ∩Δ₁) ∖Δ₂ can de treated by exchanging the roles of G₁ and G₂ since in this case G₂(q)≠ 0.

Assume finally that q ∈Δ_A ∩Δ₁ ∩Δ₂. Thanks to (GP6) and (GP7) at such a point G₁ and [G₁,[G₁,G₂]] are not parallel. Hence, we can define the map φ which to (x,y) in a neighborhood U of (0,0) in $\mathbb {R}^{2}$ associates the point reached by starting at q and following [G₁,[G₁,G₂]] during time y and then G₁ during time x that is

$$\varphi : \left\{ \begin{array}{rcl} U& \to& M\\ (x,y)&\mapsto&\mathrm{e}^{xG_{1}}\mathrm{e}^{y[G_{1},[G_{1},G_{2}]]}q \end{array} \right. $$

The fact that G₂ and [G₁,G₂] are parallel to G₁ implies b₀ = 0 and b₁₀ = 0. The fact that, along the y axis, [G₁,[G₁,G₂]] = (0,1) implies in particular that $b_{20}=\frac 12$. □

3 General Facts About the Computation of the Optimal Synthesis

3.1 Local Controllability and Existence of Minimizers

In the three cases of the normal form (NF₁), (NF₂) and (NF₃) one checks that

$$\text{span}(F_{1},F_{2},[F_{1},F_{2}],[F_{1},[F_{1},F_{2}]],[F_{2},[F_{1},F_{2}]])=\mathbb{R}^{2}.$$

Hence, as a consequence of Chow-Rashevski theorem (see [3, 16, 25]), generically such a control system is locally controllable that is locally, for any two points, always exists an admissible curve joining the two points.

Moreover, since at each point, the set of admissible velocities is convex and compact, thanks to Filippov theorem (see [3, 20]), locally for any two points, always exists at least a minimizer.

3.2 Pontryagin Maximum Principle

The Pontryagn Maximum Principle (PMP for short, see [3, 24]) gives necessary conditions for a curve to be a minimizer of a control problem. For our problem, it takes the following form.

Theorem 3 (PMP)

Define theHamiltonian

$$H(q,\lambda,u,\lambda_{0})= u_{1} \lambda.F_{1}(q)+u_{2} \lambda.F_{2}(q)+\lambda_{0}$$

where $q\in \mathbb {R}^{2}$, $\lambda \in T^{*}\mathbb {R}^{2}$, $u\in \mathbb {R}^{2}$ and $\lambda _{0}\in \mathbb {R}$. For any minimizer (q(t),u(t)), there exist a never vanishing Lipschitz covector $\lambda :t\mapsto \lambda (t)\in T^{*}_{q(t)}\mathbb {R}^{2}$ and a constant λ₀ ≤ 0 such that

• $\dot q(t)=\frac {\partial H}{\partial \lambda }(q(t),\lambda (t),u(t),\lambda _{0})$,

• $\dot \lambda (t)=-\frac {\partial H}{\partial q}(q(t),\lambda (t),u(t),\lambda _{0})$,

• 0 = H(q(t),λ(t),u(t),λ₀) = maxv{H(q,λ,v,λ₀)||v_i|≤ 1 for i = 1,2}.

A couple (q,λ) satisfying the previous conditions is called an extremal. If λ₀ = 0, it is called abnormal, if not, normal. A curve q may be associated with both abnormal and normal extremals.

Proposition 1

For a generic SF metric on a 2D manifold defined with a maximumnorm, there is no non trivial abnormal extremal. Hence, we can fixλ₀ = − 1.This is our choice in the following.

Proof

It is a classical fact that an abnormal extremal should correspond to a covector λ≠ 0 orthogonal to F₁, F₂, and [F₁,F₂]. This implies that along the trajectory the three vectors are parallel. But generically this happens only on a discrete set, which forbids to get a non trivial curve. □

3.3 Switchings

In this section, we follow the ideas of [13]. Recall that Δ_A is the set of point q where F₁(q) and F₂(q) are collinear, Δ₁ is the set of point q where F₁(q) and [F₁,F₂](q) are collinear, and Δ₂ is the set of point q where F₂(q) and [F₁,F₂](q) are collinear.

Definition 2

For an extremal triplet (q(.),λ(.),u(.)), define the switching functions

$$\phi_{i}(t)=<\lambda(t),F_{i}(q(t))> ,i = 1,2,$$

and the function ϕ₃(t) =< λ(t),[F₁,F₂](q(t)) >.

Thanks to λ₀ = − 1, the ϕ_i functions satisfy

$$u_{1}(t)\phi_{1}(t)+u_{2}(t)\phi_{2}(t)= 1, \quad\text{ for a.e. } t. $$

A direct consequence of the maximality condition is

Proposition 2

Ifϕ_i(t) > 0 (resp.ϕ_i(t) < 0)thenu_i(t) = 1 (resp.u_i(t) = − 1).

Ifϕ_i(t) = 0 and$\dot \phi _{i}(t)>0$(resp.$\dot \phi _{i}(t)<0$)thenϕ_ichanges sign at time t and the controlu_iswitches from − 1 to + 1(resp.from + 1 to − 1).

Definition 3

We call bang an extremal trajectory corresponding to constant controls with value 1 or − 1 and bang-bang an extremal which is a finite concatenation of bangs. We call u_i-singular an extremal corresponding to a null switching function ϕ_i. A time t is said to be a switching time if u is not bang in any neighborhood of t.

Definition 4

Outside Δ_A, define the functions f₁ and f₂ by

$$[F_{1},F_{2}](q)=f_{2}(q)F_{1}(q)-f_{1}(q)F_{2}(q). $$

It is clear that

$${\Delta}_{1}\setminus{\Delta}_{A}=f_{1}^{-1}(0), \quad {\Delta}_{2}\setminus{\Delta}_{A}=f_{2}^{-1}(0). $$

Proposition 3 (Switching rules)

Outside Δ_A ∪Δ₁ ∪Δ₂the possible switches of the controls are

• iff₁ > 0 thenu₁can only switch from -1 to + 1 whenϕ₁goes to 0,

• iff₁ < 0 thenu₁can only switch from + 1 to -1 whenϕ₁goes to 0,

• iff₂ > 0 thenu₂can only switch from -1 to + 1 whenϕ₂goes to 0,

• iff₂ < 0 thenu₂can only switch from + 1 to -1 whenϕ₂goes to 0.

Proof

The fact that $\dot \phi _{1}(t)=-u_{2}.\lambda .[F_{1},F_{2}]$ and $\dot \phi _{2}(t)=u_{1}.\lambda .[F_{1},F_{2}]$ implies that, outside Δ_A ∪Δ₁ ∪Δ₂,

$$\begin{array}{@{}rcl@{}} \dot\phi_{1}(t)&=&u_{2}(t)\left( f_{1}(q(t))\phi_{2}(t)-f_{2}(q(t))\phi_{1}(t)\right)=-u_{2}(t)\phi_{3}(t), \end{array} $$

(3)

$$\begin{array}{@{}rcl@{}} \dot\phi_{2}(t)&=&u_{1}(t)\left( f_{2}(q(t))\phi_{1}(t)-f_{1}(q(t))\phi_{2}(t)\right)=u_{1}(t)\phi_{3}(t). \end{array} $$

(4)

Now, if ϕ₁(t) = 0 then |ϕ₂(t)| = 1 which implies u₂(t)ϕ₂(t) = 1 and hence $\dot \phi _{1}(t)$ and f₁(q(t)) have same sign and the sign of f₁(q(t)) determines the switch.

The same holds true for f₂, ϕ₂ and u₂.□

As a consequence, on each connected component of the complement of Δ_A ∪Δ₁ ∪Δ₂, each control u_i can take only values − 1 and + 1 and can switch only once from − 1 to + 1 if f_i > 0 or from + 1 to - 1 if f_i < 0.

Proposition 4

At any point q outside Δ_Ait exists aτ > 0 such that for any extremal issued from q and of length less thanτ,only one of the two controls may switch.

Proof

If ϕ₁(t) = 0 then |ϕ₂(t)| = 1. Hence, if ϕ₁(t) = 0 and ϕ₂(t^′) = 0 then ϕ₁ passes from value 0 to ± 1 in time t^′− t which implies that $|\dot \phi _{1}|$ takes values larger than $\frac {1}{|t^{\prime }-t|}$. But, since $\dot \phi _{1}(t)=-u_{2}(f_{2}(q(t))\phi _{1}(q(t))-f_{1}(q(t))\phi _{2}(q(t)))$, we have $|\dot \phi _{1}(t)|\leq |f_{1}(q(t))|+|f_{2}(q(t))|$. As a consequence, if locally |f₁ + f₂| < M then |t^′− t| cannot be smaller than 1/M. □

A consequence of the previous proposition is

Proposition 5

At any point q outside Δ_A,consider the normal coordinate system centered at q.Any local extremal stays in one of the followingdomains:$\mathbb {R}_{+}\times \mathbb {R}_{+}$,$\mathbb {R}_{+}\times \mathbb {R}_{-}$,$\mathbb {R}_{-}\times \mathbb {R}_{+}$or$\mathbb {R}_{-}\times \mathbb {R}_{-}$.

Proof

Thanks to previous proposition, only one control may switch in short time. Assume that u₁ ≡ 1. Then at each time u₁F₁ + u₂F₂ = F₁ + u₂F₂ hence the dynamics takes the form αG₁ + (1 − α)G₂ with α ∈ [0,1]. This dynamics leaves invariant the set $\mathbb {R}_{+}\times \mathbb {R}_{+}$, hence the extremal does not leave this set. By the same argument one proves that if u₁ ≡− 1 then the extremal stays in $\mathbb {R}_{-}\times \mathbb {R}_{-}$, if u₂ ≡ 1 then the extremal stays in $\mathbb {R}_{+}\times \mathbb {R}_{-}$ and that if u₂ ≡− 1 then the extremal stays in $\mathbb {R}_{-}\times \mathbb {R}_{+}$. □

3.4 Initial Conditions and Their Parameterization

On proves easily that in the (NF₁) case, max(|λ_x(0)|,|λ_y(0)|) = 1. Hence, the set of initial conditions λ is compact and extremals switching in short time or singular extremals should have a ϕ_i null or close to zero. Moreover, only one control can switch in short time (see Proposition 4).

In the (NF₂) and (NF₃) cases |λ_x(0)| = 1 and there is no condition on λ_y. Hence, the set of initial condition is not compact. This allows to consider initial conditions with |λ_y| >> 1 and hence will appear optimal extremals along which the two controls switch. It is not in contradiction with the Proposition 4 since in this case the base point belongs to Δ_A.

In the (NF_2a) and (NF₃) cases, $\phi _{1}(0)=\pm \frac {1+a_{0}}2$ and $\phi _{2}(0)=\pm \frac {1-a_{0}}2$. Hence, if one considers a compact set of initial conditions, the corresponding extremals do not switch in short time. And are not singular. As a consequence, to consider the extremal switching at least once, one should consider initial conditions with |λ_y(0)| >> 1.

Let us give an idea of how to estimate the |λ_y(0)| corresponding to a u₁-switch at small time t and the consequence in terms of choice of change of coordinates.

In the (NF₂) case, $\phi _{1}(0)=\frac {1+a_{0}}2\geq \frac 12$. Hence, if along an extremal the control u₁ switches for t small hence one gets, since x(t) = O(t) and y(t) = O(t²),

$$0=\lambda(t).F_{1}(x(t),y(t))=\frac{1+a_{0}}2+\lambda_{y}(0)\frac{x(t)}{2}+O(t) $$

and it implies that if an extremal sees its control u₁ switching at τ then λ_y(0) should be like $\frac 1 \tau$. Hence, in order to make estimations of the corresponding extremals, it is natural to choose as small parameter $r_{0}=\frac {1}{\lambda _{y}(0)}$, to make the change of coordinate $r=\frac {1}{\lambda _{y}}$, the change of time $s=\frac {t}{r}$ and the change of coordinate p_x = rλ_x. This is what we do in the Sections 4.2 and 4.3.

In the (NF₃) case, $\phi _{1}(0)=\frac {1+a_{0}}2\geq \frac 12$. Hence, if along an extremal the control u₁ switches for t small hence one gets, since x(t) = O(t) and y(t) = O(t³),

$$0=\lambda(t).F_{1}(x(t),y(t))=\frac{1+a_{0}}2+\lambda_{y}(0)\frac{x^{2}(t)}{4}+O(t) $$

and it implies that if an extremal sees its control u₁ switching at τ then λ_y(0) should be like $\frac 1 {\tau ^{2}}$. Hence, in order to make estimations of the corresponding extremals, it is natural to choose as small parameter r₀ such that $\lambda _{y}(0)=\pm \frac 1 {{r_{0}^{2}}}$, to make the change of coordinate $r=\frac {\pm 1}{\sqrt {|\lambda _{y}|}}$ and the change of time $s=\frac {t}{r}$. This is what we do in the Section 4.4.

3.5 Weights, Orders and Nilpotent Approximation

Privileged coordinates and nilpotent approximations are well-known notions in SR Geometry. Their definitions being too long and classical we refer to [8]. The coordinates we constructed in the normal form are privileged coordinates.

In the (NF₁) case, x and y have weight 1 and ∂_x and ∂_y have weight − 1 as operators of derivation. In the (NF₂) case x has weight 1 and y has weight 2, ∂_x has weight − 1 and ∂_y have weight − 2. In the (NF3) case, x has weight 1 and y has weight 3, ∂_x has weight − 1 and ∂_y have weight − 3.

In privileged coordinates, one way to understand the weights of the variables naturally is to estimate how they vary with time in small time along an admissible curve. As seen before, in the (NF₁) case x and y are O(t) (and may be not o(t)), in the (NF₂) case x = O(t) and y = O(t²) and in the (NF₃) case x = O(t) and y = O(t³).

In the following, o_k(x,y) will denote a function whose valuation at 0 has order larger than k respectively to the weights of x and y. For example x⁷ has always weight 7 and y³ has weight 3 in the (NF₁) case but 9 in the (NF₃) case.

With this notion of weights, we define the nilpotent approximation of our normal forms in the three cases

$$\begin{array}{@{}rcl@{}} (NF_{1})\quad G_{1}(x,y)&=&\partial_{x}, \\ G_{2}(x,y)&=&\partial_{y},\\ (NF_{2})\quad G_{1}(x,y)&=&\partial_{x}, \\ G_{2}(x,y)&=&a_{0}\partial_{x}+x\partial_{y},\\ (NF_{3})\quad G_{1}(x,y)&=&\partial_{x}, \\ G_{2}(x,y)&=&a_{0}\partial_{x}+\frac12x^{2}\partial_{y}, \end{array} $$

which corresponds to an approximation up to order − 1. In the following, when we will compute developments with respect to the parameter r₀, that is for |λ_y(0)| >> 1, we will need the approximation up to order 0 for (NF_2a) and (NF₃), and the approximation up to order 1 for (NF_2b)

$$\begin{array}{@{}rcl@{}} (NF_{2a})\quad G_{1}(x,y)&=&\partial_{x}, \\ G_{2}(x,y)&=&(a_{0}+a_{10}x)\partial_{x}+x(1+b_{20}x)\partial_{y},\\ (NF_{2b})\quad G_{1}(x,y)&=&\partial_{x}, \\ G_{2}(x,y)&=&(1+a_{10}x+a_{01}y+a_{20}x^{2})\partial_{x}+x(1+b_{20}x+b_{30}x^{2})\partial_{y},\\ (NF_{3})\quad G_{1}(x,y)&=&\partial_{x}, \\ G_{2}(x,y)&=&(a_{0}+a_{10}x)\partial_{x}+\left( \frac{x^{2}}2+b_{01}y+b_{30}x^{3}\right)\partial_{y}, \end{array} $$

In the (NF₁) case, we will need the approximation up to order 2 in order to compute the cut locus, when present

$$\begin{array}{@{}rcl@{}} (NF_{1})\quad G_{1}(x,y)&=&\partial_{x}, \\ G_{2}(x,y)&=&x(a_{10}+a_{20} x+a_{11} y+a_{30}x^{2}+a_{21}xy+a_{12}y^{2})\partial_{x}+\\ &&+(1+x(b_{10}+b_{20} x+b_{11} y+b_{30}x^{2}+b_{21}xy+b_{12}y^{2}))\partial_{y}, \end{array} $$

3.6 Symbols of Extremals

As we will see in the following, the local extremals will be finite concatenations of bang arcs and u_i-singular arcs. In order to facilitate the presentation, a bang arc following ± G_i will be symbolized by [[±G_i]], a u₁-singular arc with control u₂ ≡ 1 will be symbolized by $[[S_{1}^{+}]]$, a u₁-singular arc with control u₂ ≡− 1 will be symbolized by $[[S_{1}^{-}]]$, and we will combine these symbols in such a way that $[[-G_{1},G_{2},S_{2}^{+}]]$ symbolizes the concatenation of a bang arc following − G₁ with a bang arc following G₂ and a u₂-singular arc with control u₁ ≡ 1.

3.7 Symmetries

One can change the roles of the vectors F₁ and F₂ and look at the effect on the functions f_i or on the invariants appearing in the normal form. For this last part, one should be careful that changing the role of F₁ and F₂ implies changing G₁ and G₂ and hence changing the coordinates x and y.

First look at the effect on the functions f_i on an example: $\bar F_{1}=-F_{1}$ and $\bar F_{2}=F_{2}$. If we define the control system with $(\bar F_{1},\bar F_{2})$, it defines the same SF structure. We compute easily that

$$[\bar F_{1},\bar F_{2}]=[-F_{1},F_{2}]=-[F_{1},F_{2}]=-(f_{2} F_{1}-f_{1} F_{2})=f_{2} \bar F_{1} - (-f_{1})\bar F_{2}$$

hence $\bar f_{1}=-f_{1}$ and $\bar f_{2}=f_{2}$. With this choice $\bar G_{1}=-G_{2}$ and $\bar G_{2}=-G_{1}$. Of course, with such a change on the vectors G₁ and G₂ the change on the invariants is not so trivial to compute.

In the following, we consider changes that send G₁ to ± G₁ and G₂ to ± G₂. These changes are interesting from a calculus point of view. Effectively, once computed the jet of a bang-bang extremals with symbol [[G₁,G₂]] and of its switching times, we are able to get the expressions for the bang-bang extremals with symbols [[±G₁,±G₂]] without new computations. For example, if one gets the expression of an extremal with symbol [[G₁,G₂]] as function of the initial conditions, one gets the expression of an extremal with symbol [[−G₁,G₂]] by respecting the effect on the coordinates and the invariants a₀, a₁₀, etc. of the corresponding change of role of F₁ and F₂.

3.7.1 $\bar G_{1}=-G_{1}$ and $\bar G_{2}=G_{2}$

Consider the change $\bar F_{1}=-F_{2}$ and $\bar F_{2}=-F_{1}$. Then $\bar G_{1}=-G_{1}$ and $\bar G_{2}=G_{2}$, $[\bar G_{1},\bar G_{2}]=-[G_{1},G_{2}]$ and $[\bar G_{1},[\bar G_{1},\bar G_{2}]]=[G_{1},[G_{1},G_{2}]]$. With this choice,

$$[\bar F_{1},\bar F_{2}]=[-F_{2},-F_{1}]=-[F_{1},F_{2}]=-(f_{2} F_{1}-f_{1} F_{2})=(-f_{1}) \bar F_{1} - (-f_{2})\bar F_{2}$$

hence $\bar f_{1}=-f_{2}$ and $\bar f_{2}=-f_{1}$.

We can know consider the effect of this change of role on the coordinates and on the invariants in the three cases of the normal form

• (NF₁) In this case, $\bar x=-x$ and $\bar y =y$, hence $\partial _{\bar x}=-\partial _{x}$ and $\partial _{\bar y}=\partial _{y}$ and

  $$\begin{array}{@{}rcl@{}} \bar G_{1} & = & \partial_{\bar x},\\ \bar G_{2} & = & (a_{10}\bar x-a_{20}\bar x^{2}+a_{11}\bar x\bar y+o_{2}(\bar x, \bar y))\partial_{\bar x}+\\ &&(1-b_{10}\bar x+b_{20}\bar x^{2}-b_{11}\bar x\bar y+o_{2}(\bar x, \bar y))\partial_{\bar y}. \end{array} $$

• (NF₂) In this case, $\bar x=-x$ and $\bar y =-y$, hence $\partial _{\bar x}=-\partial _{x}$ and $\partial _{\bar y}=-\partial _{y}$ and

  $$\begin{array}{@{}rcl@{}} \bar G_{1}&=&\partial_{\bar x}, \\ \bar G_{2}&=&(-a_{0}+a_{10}\bar x-a_{01}\bar y-a_{20}\bar x^{2}+o_{2}(\bar x, \bar y))\partial_{\bar x}+\\ && (\bar x-b_{20}\bar x^{2}+b_{30}\bar x^{3}+o_{3}(\bar x, \bar y))\partial_{\bar y}. \end{array} $$

• (NF₃) In this case, $\bar x=-x$ and $\bar y =y$, hence $\partial _{\bar x}=-\partial _{x}$ and $\partial _{\bar y}=\partial _{y}$ and

  $$\begin{array}{@{}rcl@{}} \bar G_{1}&=&\partial_{\bar x}, \\ \bar G_{2}&=&(-a_{0}+a_{10}\bar x+o_{1}(\bar x, \bar y))\partial_{\bar x}+\\&& (\bar x^{2}/2+b_{01}\bar y-b_{30}\bar x^{3}+o_{3}(\bar x, \bar y))\partial_{\bar y}. \end{array} $$

3.7.2 $\bar G_{1}=G_{1}$ and $\bar G_{2}=-G_{2}$

Consider the change $\bar F_{1}=F_{2}$ and $\bar F_{2}=F_{1}$. Then $\bar G_{1}=G_{1}$, $\bar G_{2}=-G_{2}$, $[\bar G_{1},\bar G_{2}]=-[G_{1},G_{2}]$ and $[\bar G_{1},[\bar G_{1},\bar G_{2}]]=-[G_{1},[G_{1},G_{2}]]$. With this choice,

$$[\bar F_{1},\bar F_{2}]=[F_{2},F_{1}]=-[F_{1},F_{2}]=-(f_{2} F_{1}-f_{1} F_{2})=(f_{1}) \bar F_{1} - (f_{2})\bar F_{2}$$

hence $\bar f_{1}=f_{2}$ and $\bar f_{2}=f_{1}$.

We can know consider the effect of this change of role on the coordinates and on the invariants in the three cases of the normal form

• (NF₁) In this case, $\bar x=x$ and $\bar y =-y$, hence $\partial _{\bar x}=\partial _{x}$ and $\partial _{\bar y}=-\partial _{y}$ and

  $$\begin{array}{@{}rcl@{}} \bar G_{1}&=&\partial_{\bar x}, \\ \bar G_{2}&=&(-a_{10}\bar x-a_{20}\bar x^{2}+a_{11}\bar x\bar y+\bar x o(\bar x, \bar y))\partial_{\bar x}+\\&& (1+b_{10}\bar x+b_{20}\bar x^{2}-b_{11}\bar x\bar y+\bar x o(\bar x, \bar y))\partial_{\bar y}. \end{array} $$

• (NF₂) In this case, $\bar x=x$ and $\bar y =-y$, hence $\partial _{\bar x}=\partial _{x}$ and $\partial _{\bar y}=-\partial _{y}$ and

  $$\begin{array}{@{}rcl@{}} \bar G_{1}&=&\partial_{\bar x}, \\ \bar G_{2}&=&(-a_{0}-a_{10}\bar x+a_{01}\bar y-a_{20}\bar x^{2}+o_{2}(\bar x, \bar y))\partial_{\bar x}+\\&& (\bar x+b_{20}\bar x^{2}+b_{30}\bar x^{3}+o_{3}(\bar x, \bar y))\partial_{\bar y}. \end{array} $$

• (NF₃) In this case, $\bar x=x$ and $\bar y =-y$, hence $\partial _{\bar x}=\partial _{x}$ and $\partial _{\bar y}=-\partial _{y}$ and

  $$\begin{array}{@{}rcl@{}} \bar G_{1}&=&\partial_{\bar x}, \\ \bar G_{2}&=&(-a_{0}-a_{10}\bar x+o_{1}(\bar x, \bar y))\partial_{\bar x}+\\&& (\bar x^{2}/2-b_{01}\bar y+b_{30}\bar x^{3}+o_{3}(\bar x, \bar y))\partial_{\bar y}. \end{array} $$

3.7.3 $\bar G_{1}=-G_{1}$ and $\bar G_{2}=-G_{2}$

Consider the change $\bar F_{1}=-F_{1}$ and $\bar F_{2}=-F_{2}$. Then $\bar G_{1}=-G_{1}$, $\bar G_{2}=-G_{2}$, $[\bar G_{1},\bar G_{2}]=[G_{1},G_{2}]$ and $[\bar G_{1},[\bar G_{1},\bar G_{2}]]=-[G_{1},[G_{1},G_{2}]]$. With this choice,

$$[\bar F_{1},\bar F_{2}]=[-F_{1},-F_{2}]=[F_{1},F_{2}]=(f_{2} F_{1}-f_{1} F_{2})=(-f_{2}) \bar F_{1} - (-f_{1})\bar F_{2}$$

hence $\bar f_{1}=-f_{1}$ and $\bar f_{2}=-f_{2}$.

We can know consider the effect of this change of role on the coordinates and on the invariants in the three cases of the normal form

• (NF₁) In this case, $\bar x=-x$ and $\bar y =-y$, hence $\partial _{\bar x}=-\partial _{x}$ and $\partial _{\bar y}=-\partial _{y}$. Moreover

  $$\begin{array}{@{}rcl@{}} \bar G_{1}&=&\partial_{\bar x}, \\ \bar G_{2}&=&(-a_{10}\bar x+a_{20}\bar x^{2}+a_{11}\bar x\bar y+\bar x o(\bar x, \bar y))\partial_{\bar x}+\\&& (1-b_{10}\bar x+b_{20}\bar x^{2}+b_{11}\bar x\bar y+\bar x o(\bar x, \bar y))\partial_{\bar y}. \end{array} $$

• (NF₂) In this case, $\bar x=-x$ and $\bar y =y$, hence $\partial _{\bar x}=-\partial _{x}$ and $\partial _{\bar y}=\partial _{y}$. Moreover

  $$\begin{array}{@{}rcl@{}} \bar G_{1}&=&\partial_{\bar x}, \\ \bar G_{2}&=&(a_{0}-a_{10}\bar x+a_{01}\bar y+a_{20}\bar x^{2}+o_{2}(\bar x, \bar y))\partial_{\bar x}+\\&& (\bar x-b_{20}\bar x^{2}+b_{30}\bar x^{3}+o_{3}(\bar x, \bar y))\partial_{\bar y}. \end{array} $$

• (NF₃) In this case, $\bar x=-x$ and $\bar y =-y$, hence $\partial _{\bar x}=-\partial _{x}$ and $\partial _{\bar y}=-\partial _{y}$. Moreover

  $$\begin{array}{@{}rcl@{}} \bar G_{1}&=&\partial_{\bar x}, \\ \bar G_{2}&=&(a_{0}-a_{10}\bar x+o_{1}(\bar x, \bar y))\partial_{\bar x}+\\&& (\bar x^{2}/2-b_{01}\bar y-b_{30}\bar x^{3}+o_{3}(\bar x, \bar y))\partial_{\bar y}. \end{array} $$

4 The Generic Local Optimal Synthesis

We present for generic couples (F₁,F₂) the local synthesis issued from a point q. The coordinates (x,y), centered at q, are those which have been constructed in the corresponding normal form in Section 2.

4.1 (N F ₁) Case

At points q where (NF₁) holds, one can compute that

$$\begin{array}{@{}rcl@{}} f_{1}(x,y)&=& \frac12(a_{10}-b_{10})\\ &&+(2(a_{20}-b_{20})-b_{10}(a_{10}-b_{10}))\frac{x}2+(a_{11}-b_{11})\frac y2 \\ &&+ (3(a_{30}-b_{30})-b_{10}(a_{20}-b_{20})-(2b_{20}-b_{10}^{2})(a_{10}-b_{10})) \frac {x^{2}}2\\ &&+(2(a_{21}-b_{21})-b_{11}(a_{10}-b_{10})-b_{10}(a_{11}-b_{11}))\frac{xy}2\\ &&+(a_{12}-b_{12})\frac {y^{2}}2+o_{2}(x,y),\\ f_{2}(x,y)&=& -\frac12(a_{10}+b_{10})\\ &&-(2(a_{20}+b_{20})-b_{10}(a_{10}+b_{10}))\frac x2-(a_{11}+b_{11})\frac y2 \\ &&- (3(a_{30}+b_{30})-b_{10}(a_{20}+b_{20})-(2b_{20}-b_{10}^{2})(a_{10}+b_{10})) \frac {x^{2}}2\\ &&-(2(a_{21}+b_{21})-b_{11}(a_{10}+b_{10})-b_{10}(a_{11}+b_{11}))\frac{xy}2\\ &&-(a_{12}+b_{12})\frac{y^{2}}2+o_{2}(x,y). \end{array} $$

Hence, thanks to Proposition 3, if a₁₀ − b₁₀ > 0 (resp. < 0) then u₁ is bang-bang and the only possible switch is − 1 → + 1 (resp + 1 →− 1) and if a₁₀ + b₁₀ < 0 (resp. > 0) then u₂ is bang-bang and the only possible switch is − 1 → + 1 (resp + 1 →− 1).

Remark 2 (Generic invariants)

Remark that generically, in the (NF₁) case, one of the following situation occurs

• |a₁₀|≠|b₁₀| (NF_1a),

• a₁₀ = b₁₀≠ 0 and a₂₀ − b₂₀≠ 0 and a₁₁ − b₁₁≠ 0,

• a₁₀ = b₁₀≠ 0 and a₂₀ − b₂₀ = 0 and a₃₀ − b₃₀≠ 0 and a₁₁ − b₁₁≠ 0,

• a₁₀ = b₁₀≠ 0 and a₂₀ − b₂₀≠ 0 and a₁₁ − b₁₁ = 0 and a₁₂ − b₁₂≠ 0,

• a₁₀ = −b₁₀≠ 0 and a₂₀ + b₂₀≠ 0 and a₁₁ + b₁₁≠ 0,

• a₁₀ = −b₁₀≠ 0 and a₂₀ + b₂₀ = 0 and a₃₀ + b₃₀≠ 0 and a₁₁ + b₁₁≠ 0,

• a₁₀ = −b₁₀≠ 0 and a₂₀ + b₂₀≠ 0 and a₁₁ + b₁₁ = 0 and a₁₂ + b₁₂≠ 0.

• a₁₀ = b₁₀ = 0 and a₂₀ + b₂₀≠ 0 and a₁₁ + b₁₁≠ 0.

4.1.1 Singular Extremals

We consider now the properties of singular extremals and their support.

Proposition 6

Under the generic assumption that Δ_A,Δ₁andΔ₂aresubmanifolds transversal by pair then

• 1. The support of au_i-singularis included in Δ_i.

• 2. Au₁-singularextremal can follow Δ₁being optimal only if, at each pointq(t) of the singular,G₁(q(t)) andG₂(q(t)) are pointing on the same side of Δ₁(or one is tangent to Δ₁)wheref₁ > 0.

• 3. Au₂-singularextremal can follow Δ₂being optimal only if, at each pointq(t) of the singular,G₁(q(t)) and − G₂(q(t)) are pointing on the same side of Δ₂(or one is tangent to Δ₂)wheref₂ > 0.

• 4. Consider au_i-singularq(.) satisfying 2 or 3. If it does not intersectΔ_Aand if at each timeG₁(q(t)) andG₂(q(t)) are not tangent to Δ_ithenq(.) is a local minimizer that is at each time t exists𝜖such thatq(.) realizes the SF-distance betweenq(t₁) andq(t₂) for anyt₁andt₂in ]t − 𝜖,t + 𝜖[.

Proof

• 1. Outside Δ_A ∪Δ_i, ϕ_i has isolated zero hence any u_i-singular should live in Δ_A ∪Δ_i. Moreover, since generically the set of points of Δ_A where the dynamics is tangent to Δ_A is isolated, a u_i-singular crosses Δ_A only at isolated times, which are consequently also in Δ_i.

• 2. Same proof as for point 3.

• 3. If a u₂-singular q(.) has u₁ = 1 then its speed is F₁(q(t)) + u₂(t)F₂(q(t)) which is tangent to Δ₂. But u₂ ∈ [− 1,1] hence either |u₂(t)| = 1 and G₁ or G₂ are tangent to Δ₂ or |u₂(t)| < 1 and G₂(q(t)) = F₁(q(t)) − F₂(q(t)) and G₁(q(t)) = F₁(q(t)) + F₂(q(t)) point on opposite sides.

  Now, assume that Δ₂ is such that G₁ and − G₂ point in the same side where f₂ < 0 at q and that the u₂-singular is optimal. Consider the normal coordinate system centered at q and the domain $\mathbb {R}_{+}\times \mathbb {R}_{+}$. One can show, with the previous analysis, that the only possible extremals issued form q and entering the domain are the singular arc $[[S_{2}^{+}]]$ following Δ₂ and the bang-bang extremals starting with symbol [[G₁,G₂]] or [[G₂,G₁]].

  Let us prove that these last ones do not switch again before crossing Δ₂. If an extremal starts with [[G₂,G₁]], switching for the first time at t = 𝜖 and hence at y = 𝜖 then along the second bang x = t − 𝜖, y = 𝜖, λ ≡ (1,1) and one computes easily that for t > 𝜖

  $$\phi_{2}(t)=-\frac12 ((a_{20}+b_{20})(t-\epsilon)^{2}+(a_{11}+b_{11})(t-\epsilon)\epsilon+o_{2}(\epsilon, (t-\epsilon))). $$

  If (a₂₀ + b₂₀)(a₁₁ + b₁₁) < 0 then the second time of switch satisfies $t-\epsilon = -\frac {a_{11}+b_{11}}{a_{20}+b_{20}}\epsilon +o(\epsilon )$ and hence the second switching locus has the form $(-\frac {a_{11}+b_{11}}{a_{20}+b_{20}}\epsilon ,\epsilon )$. But Δ₂ satisfies that $x=-\frac 12 \frac {a_{11}+b_{11}}{a_{20}+b_{20}}y+o(y)$ and hence the second bang crosses Δ₂ before ending. In the case a₂₀ + b₂₀ = 0 hence (a₁₁ − b₁₁)(a₃₀ + b₃₀) < 0 and one shows that the second switching locus has the form $(\sqrt {-\frac {a_{11}+b_{11}}{a_{30}+b_{30}}\epsilon },\epsilon )$ and Δ₂ satisfies that $x=\sqrt {-\frac {a_{11}+b_{11}}{3(a_{30}+b_{30})}y}+o(y)$ hence again the second bang crosses Δ₂ before ending. The same kind of computations show the same result when a₁₁ + b₁₁ = 0 and (a₂₀ + b₂₀)(a₁₂ + b₁₂) < 0. The same holds for extremal starting by [[G₁,G₂]].

  Finally, the different extremals with symbol [[G₁,G₂]] do not intersect each other after their first switch hence they cannot lose optimality by crossing each other. Idem for those with symbol [[G₂,G₁]]. Hence, they can lose optimality by crossing the singular extremal or extremals with the other symbol.

  The last argument implies that optimal extremals are coming back to Δ₂ with symbol [[G₁,G₂]] or [[G₂,G₁]]. But this is not possible since in this case, since $[[S_{2}^{+}]]$ is optimal, an optimal extremal with symbol $[[G_{1},G_{2}, S_{2}^{+}]]$ would exist which is not the case since the switching is coming strictly after the crossing with Δ₂ as seen before.

  Hence, in this case, the u₂-singular is not optimal.

• 4. Assume that the u₂-singular satisfies 3, that it does not intersect Δ_A and that G₁(q) and G₂(q) are pointing on opposite sides of Δ₂. Let construct normal coordinates centered at q, then the only local extremals entering the domains {xy > 0} are the one starting by a u₂-singular and switching or not locally only once to u₂ = ± 1. For example, if we consider $\mathbb {R}_{+}\times \mathbb {R}_{+}$, since G₁(q) points in the side of the domain {q^′|f₂(q^′) > 0} then an extremal starting by G₁ enters the domain {q^′|f₂(q^′) > 0} and hence u₂ cannot switch and the extremal stays on the boundary of $\mathbb {R}_{+}\times \mathbb {R}_{+}$ and do not enter it. As a consequence the u₂-singular is locally optimal.

□

Remark 3

For what concerns the point 4, assume that q is a point where G₁ or G₂ is tangent to Δ₁ and Δ₁ ∩{xy < 0} is such that at each point G₁ and G₂ are transverse to Δ₁ and point in the domain {f₁ > 0}. Then, starting from q, a u₁-singular can run on Δ₁ ∩{xy < 0} and is locally optimal. The same arguments as those exposed at point 4 work.

Definition 5

If a connected part of Δ₁ (resp. Δ₂) is such that at each point G₁ and G₂ (resp. G₁ and − G₂) point on the same side where f₁ > 0 (resp. f₂ > 0), it is called a turnpike. If it does not at each point, it is called an anti-turnpike (see [13]).

Remark 4

Along a u_i-singular extremal the control u_i is completely determined by the fact that the dynamics should be tangent to Δ_i.

4.1.2 Optimal Synthesis in the Domain $\mathbb {R}_{+}\times \mathbb {R}_{+}$

Consider a point q and the normal coordinate system (x,y) centered at q. The dynamics entering $\mathbb {R}_{+}^{*}\times \mathbb {R}_{+}^{*}$ is with u₁ ≡ 1 since u₂ switches (Propositions 4 and 5). Three different cases can be identified.

1st. case. :

${\Delta }_2\cap (\mathbb {R}_{+}\times \mathbb {R}_{+}\setminus \{0\})$ is empty locally. Thanks to proposition 6, no u₂-singular enters the domain. It corresponds to the case (NF_1a) where |a₁₀|≠|b₁₀| and to the cases (NF_1c) and (NF_1d) where a₁₀ + b₁₀ = 0 and

• (a₂₀ + b₂₀)(a₁₁ + b₁₁) > 0,

• or a₂₀ + b₂₀ = 0 and (a₃₀ + b₃₀)(a₁₁ + b₁₁) > 0,

• or a₁₁ + b₁₁ = 0 and (a₂₀ + b₂₀)(a₁₂ + b₁₂) > 0.

Only one u₂-switch can occur along the extremal. One has f₂ > 0 in the domain if

• a₁₀ + b₁₀ < 0,

• or a₁₀ + b₁₀ = 0 and a₂₀ + b₂₀ < 0,

• or a₁₀ + b₁₀ = 0 and a₂₀ + b₂₀ = 0 and a₁₁ + b₁₁ < 0,

and in this case the possible extremals of the domain have symbol [[G₁]] or [[G₂]] or [[G₂,G₁]]. One has f₂ < 0 in the domain if

• a₁₀ + b₁₀ > 0,

• or a₁₀ + b₁₀ = 0 and a₂₀ + b₂₀ > 0,

• or a₁₀ + b₁₀ = 0 and a₂₀ + b₂₀ = 0 and a₁₁ + b₁₁ > 0.

and in this case, the possible extremals of the domain have symbol [[G₁]] or [[G₂]] or [[G₁,G₂]].

In this case 1, the picture of the synthesis is given in Fig. 1.

Fig. 1

The syntheses when f₂≠ 0 in $(\mathbb {R}_{+}\times \mathbb {R}_{+})\setminus \{0\}$

2nd. case. :

${\Delta }_2\cap (\mathbb {R}_{+}^{*}\times \mathbb {R}_{+}^{*})$ is not empty locally and is a turnpike. We are in the context of point 4 of proposition 6. It corresponds to the cases where a₁₀ + b₁₀ = 0 and

• a₂₀ + b₂₀ < 0 and a₁₁ + b₁₁ > 0,

• or a₂₀ + b₂₀ = 0 and a₁₁ + b₁₁ > 0 and a₃₀ + b₃₀ < 0,

• or a₁₁ + b₁₁ = 0 and a₂₀ + b₂₀ < 0 and a₁₂ + b₁₂ > 0.

Then f₂ > 0 locally along {x > 0,y = 0} and f₂ < 0 along {x = 0,y > 0}. Hence, no bang-bang extremal with symbol [[G₁,G₂]] or [[G₂,G₁]] exists and any extremal entering the domain starts with a u₂-singular arc. If it switches to G₁ then it enters the domain $(\mathbb {R}_{+}^{*}\times \mathbb {R}_{+}^{*})\cap \{f_{2}>0\}$ which is invariant by G₁ hence it does not switch anymore. If it switches to G₂ it enters the domain $(\mathbb {R}_{+}^{*}\times \mathbb {R}_{+}^{*})\cap \{f_{2}<0\}$ which is invariant by G₂ hence it does not switch anymore.

As a consequence, the only possible symbols for extremals are [[G₁]], [[G₂]], $[[S_{2}^{+},G_{1}]]$ and $[[S_{2}^{+},G_{2}]]$.

In this case 2, the picture of the synthesis is given in Fig. 2.

Fig. 2

The syntheses when a₁₀ + b₁₀ = 0 and Δ₂ is a turnpike

3rd. case. :

${\Delta }_2\cap (\mathbb {R}_{+}^{*}\times \mathbb {R}_{+}^{*})$ is not empty locally and is a anti-turnpike. Then, thanks to proposition 6, no singular can enter the domain. It corresponds to the cases where a₁₀ + b₁₀ = 0 and

• a₂₀ + b₂₀ > 0 and a₁₁ + b₁₁ < 0,

• or a₂₀ + b₂₀ = 0 and a₁₁ + b₁₁ < 0 and a₃₀ + b₃₀ > 0,

• or a₁₁ + b₁₁ = 0 and a₂₀ + b₂₀ > 0 and a₁₂ + b₁₂ < 0.

Then, as seen in Proposition 6, no u₂-singular is extremal. Hence, the possible beginning of symbols entering the domain are [[G₁,G₂]] and [[G₂,G₁]]. In order to complete the synthesis in this case, we have to compute the cut time and cut locus. In fact, the two kinds of extremals intersect before their second switching time. Let us prove it.

Fix an 𝜖₂ > 0 and consider at time t > 𝜖₂ the extremal with symbol [[G₂,G₁]] switching at time 𝜖₂. One computes easily that x(t) = t − 𝜖₂ and y(t) = 𝜖₂. For an 𝜖₁ > 0 and the extremal with symbol [[G₁,G₂]] switching at time 𝜖₁, one gets by integrating the equations

$$\begin{array}{@{}rcl@{}} x(t)&=&\epsilon_{1}+a_{10}\epsilon_{1} (t-\epsilon_{1}) + a_{20} {\epsilon_{1}^{2}} (t-\epsilon_{1}) + \frac12 (a_{10}^{2} + a_{11}) \epsilon_{1} (t-\epsilon_{1})^{2} \\&&+ a_{30} {\epsilon_{1}^{3}} (t-\epsilon_{1}) + \frac12 (3 a_{10} a_{20} + a_{21} + a_{11} b_{10}) {\epsilon_{1}^{2}} (t-\epsilon_{1})^{2} \\&&+ \frac13 (\frac12 a_{10}^{3} + \frac32 a_{10} a_{11} + a_{12}) \epsilon_{1} (t-\epsilon_{1})^{3}\\ y(t)&=&(t-\epsilon_{1}) + b_{10} \epsilon_{1} (t-\epsilon_{1}) + b_{20} {\epsilon_{1}^{2}} (t-\epsilon_{1}) + \frac12 (a_{10} b_{10} + b_{11}) \epsilon_{1} (t-\epsilon_{1})^{2} \\&&+ b_{30} {\epsilon_{1}^{3}} (t-\epsilon_{1}) +\frac12 (a_{20} b_{10} + b_{10} b_{11} + 2 a_{10} b_{20} + b_{21}) {\epsilon_{1}^{2}} (t-\epsilon_{1})^{2} \\&&+\frac13 (\frac12 (a_{10}^{2} + a_{11}) b_{10} + a_{10} b_{11} + b_{12}) \epsilon_{1} (t-\epsilon_{1})^{3} \end{array} $$

Assume first that a₂₀ + b₂₀ > 0 and a₁₁ + b₁₁ < 0. Along the first front (depending on 𝜖₂) x + y = t when along the second $x+y=t+\epsilon _{1}(t-\epsilon _{1})((a_{20}+b_{20})\epsilon _{1}+\frac 12(a_{11}+b_{11}),$ hence, they are transverse at

$$\epsilon_{1}=\frac{t}{1-\frac{2(a_{20}+b_{20})}{a_{11}+b_{11}}}$$

and they intersect at a point such that $y=-2\frac {a_{20}-b_{20}}{a_{11}-b_{11}}x+o(x)$. As seen previously, the switching locus for extremals with symbol [[G₂,G₁]] satisfies $y=-\frac {a_{20}-b_{20}}{a_{11}-b_{11}}x+o(x)$ hence it stops to be optimal before switching. The same holds true for the extremals with symbol [[G₁,G₂]]. Finally, the cut locus satisfies

$$y_{cut}=-2\frac{a_{20}-b_{20}}{a_{11}-b_{11}}x_{cut}+o(x_{cut}) $$

and is tangent to Δ₂.

The same computations can be done when G₁ or G₂ is tangent to Δ₂. Then one computes that the extremals lose optimality by crossing the cut before the second switch and that

• if a₂₀ + b₂₀ = 0 then

  $$y_{cut}=-3\frac{a_{30}+b_{30}}{a_{11}+b_{11}}x_{cut}^{2}+o(x_{cut}^{2}), $$

• if a₁₁ + b₁₁ = 0 then

  $$x_{cut}=-\frac12\frac{a_{12}+b_{12}}{a_{20}+b_{20}}y_{cut}^{2}+o(y_{cut}^{2}). $$

In all cases, the cut is tangent to Δ₂ and the contact is of order 2 when (a₂₀ + b₂₀)(a₁₁ + b₁₁) = 0.

In this case 3, the picture of the synthesis is given in Fig. 3.

Fig. 3

The syntheses when a₁₀ + b₁₀ = 0 and Δ₂ is not a turnpike

Remark 5

Using the symmetries presented in Section 3.7, one can obtain from the optimal synthesis in the domain $\mathbb {R}_{+}\times \mathbb {R}_{+}$ the optimal synthesis in the three other domains.

4.1.3 Optimal Synthesis in the Domain $\mathbb {R}_{-}\times \mathbb {R}_{-}$

The dynamics entering $\mathbb {R}_{-}^{*}\times \mathbb {R}_{-}^{*}$ is with u₁ ≡− 1 since u₂ switches (Propositions 4 and 5). Three different cases can be identified.

1st. case. :

${\Delta }_2\cap (\mathbb {R}_{-}\times \mathbb {R}_{-}\setminus \{0\})$ is empty locally. No u₂-singular enters the domain. It corresponds to the case (NF_1a) where |a₁₀|≠|b₁₀| and to the cases (NF_1c) and (NF_1d) where a₁₀ + b₁₀ = 0 and

• (a₂₀ + b₂₀)(a₁₁ + b₁₁) > 0,

• or a₂₀ + b₂₀ = 0 and (a₃₀ + b₃₀)(a₁₁ + b₁₁) < 0,

• or a₁₁ + b₁₁ = 0 and (a₂₀ + b₂₀)(a₁₂ + b₁₂) < 0.

Only one u₂-switch can occur along the extremal. One has f₂ > 0 in the domain if

• a₁₀ + b₁₀ < 0,

• or a₁₀ + b₁₀ = 0 and a₂₀ + b₂₀ > 0,

• or a₁₀ + b₁₀ = 0 and a₂₀ + b₂₀ = 0 and a₁₁ + b₁₁ > 0,

and in this case the possible extremals of the domain have symbol [[−G₁]] or [[−G₂]] or [[−G₁,−G₂]]. One has f₂ < 0 in the domain if

• a₁₀ + b₁₀ > 0,

• or a₁₀ + b₁₀ = 0 and a₂₀ + b₂₀ < 0,

• or a₁₀ + b₁₀ = 0 and a₂₀ + b₂₀ = 0 and a₁₁ + b₁₁ < 0.

and in this case, the possible extremals of the domain have symbol [[−G₁]] or [[−G₂]] or [[−G₂,−G₁]].

2nd. case. :

${\Delta }_2\cap (\mathbb {R}_{-}^{*}\times \mathbb {R}_{-}^{*})$ is not empty locally and is a turnpike. It corresponds to the cases where a₁₀ + b₁₀ = 0 and

• a₂₀ + b₂₀ < 0 and a₁₁ + b₁₁ > 0,

• or a₂₀ + b₂₀ = 0 and a₁₁ + b₁₁ > 0 and a₃₀ + b₃₀ > 0,

• or a₁₁ + b₁₁ = 0 and a₂₀ + b₂₀ < 0 and a₁₂ + b₁₂ < 0.

In this case, the possible symbols for extremals are [[−G₁]], [[−G₂]], $[[S_{2}^{-},-G_{1}]]$ and $[[S_{2}^{-},-G_{2}]]$.

3rd. case. :

${\Delta }_2\cap (\mathbb {R}_{-}^{*}\times \mathbb {R}_{-}^{*})$ is not empty locally and is a anti-turnpike. It corresponds to the cases where a₁₀ + b₁₀ = 0 and

• a₂₀ + b₂₀ > 0 and a₁₁ + b₁₁ < 0,

• or a₂₀ + b₂₀ = 0 and a₁₁ + b₁₁ < 0 and a₃₀ + b₃₀ < 0,

• or a₁₁ + b₁₁ = 0 and a₂₀ + b₂₀ > 0 and a₁₂ + b₁₂ > 0.

The only optimal symbols are [[−G₁]], [[−G₂]], [[−G₁,−G₂]], and [[−G₂,−G₁]]. Moreover

• if a₂₀ + b₂₀ > 0 and a₁₁ + b₁₁ < 0, the cut locus satisfies

  $$y_{cut}=-2\frac{a_{20}+b_{20}}{a_{11}+b_{11}}x_{cut}+o(x_{cut}), $$

• if a₂₀ + b₂₀ = 0 then

  $$y_{cut}=-3\frac{a_{30}+b_{30}}{a_{11}+b_{11}}x_{cut}^{2}+o(x_{cut}^{2}), $$

• if a₁₁ + b₁₁ = 0 then

  $$x_{cut}=-\frac12\frac{a_{12}+b_{12}}{a_{20}+b_{20}}y_{cut}^{2}+o(y_{cut}^{2}). $$

In all cases, the cut is tangent to Δ₂ and the contact is of order 2 when (a₂₀ + b₂₀)(a₁₁ + b₁₁) = 0.

4.1.4 Optimal Synthesis in the Domain $\mathbb {R}_{+}\times \mathbb {R}_{-}$

The dynamics entering $\mathbb {R}_{+}^{*}\times \mathbb {R}_{-}^{*}$ is with u₂ ≡ 1 since u₁ switches (Propositions 4 and 5). Three different cases can be identified.

1st. case. :

${\Delta }_1\cap (\mathbb {R}_{+}\times \mathbb {R}_{-}\setminus \{0\})$ is empty locally. No u₁-singular enters the domain. It corresponds to the case (NF_1a) where |a₁₀|≠|b₁₀| and to the cases (NF_1b) and (NF_1d) where a₁₀ − b₁₀ = 0 and

• (a₂₀ − b₂₀)(a₁₁ − b₁₁) < 0,

• or a₂₀ − b₂₀ = 0 and (a₃₀ − b₃₀)(a₁₁ − b₁₁) < 0,

• or a₁₁ − b₁₁ = 0 and (a₂₀ − b₂₀)(a₁₂ − b₁₂) > 0.

Only one u₁-switch can occur along the extremal. One has f₁ > 0 in the domain if

• a₁₀ − b₁₀ > 0,

• or a₁₀ − b₁₀ = 0 and a₂₀ − b₂₀ > 0,

• or a₁₀ − b₁₀ = 0 and a₂₀ − b₂₀ = 0 and a₁₁ + b₁₁ < 0,

and in this case, the possible extremals of the domain have symbol [[G₁]] or [[−G₂]] or [[−G₂,G₁]]. One has f₁ < 0 in the domain if

• a₁₀ − b₁₀ < 0,

• or a₁₀ − b₁₀ = 0 and a₂₀ − b₂₀ < 0,

• or a₁₀ − b₁₀ = 0 and a₂₀ − b₂₀ = 0 and a₁₁ − b₁₁ > 0.

and in this case, the possible extremals of the domain have symbol [[G₁]] or [[−G₂]] or [[G₁,−G₂]].

2nd. case. :

${\Delta }_1\cap (\mathbb {R}_{+}^{*}\times \mathbb {R}_{-}^{*})$ is not empty locally and is a turnpike. It corresponds to the cases where a₁₀ − b₁₀ = 0 and

• a₂₀ − b₂₀ > 0 and a₁₁ − b₁₁ > 0,

• or a₂₀ − b₂₀ = 0 and a₁₁ − b₁₁ > 0 and a₃₀ − b₃₀ > 0,

• or a₁₁ − b₁₁ = 0 and a₂₀ − b₂₀ > 0 and a₁₂ − b₁₂ < 0.

In this case, the possible symbols for extremals are [[G₁]], [[−G₂]], $[[S_{1}^{+},G_{1}]]$, and $[[S_{1}^{+},-G_{2}]]$.

3rd. case. :

${\Delta }_1\cap (\mathbb {R}_{+}^{*}\times \mathbb {R}_{-}^{*})$ is not empty locally and is a anti-turnpike. It corresponds to the cases where a₁₀ − b₁₀ = 0 and

• a₂₀ − b₂₀ < 0 and a₁₁ − b₁₁ < 0,

• or a₂₀ − b₂₀ = 0 and a₁₁ − b₁₁ < 0 and a₃₀ − b₃₀ < 0,

• or a₁₁ − b₁₁ = 0 and a₂₀ − b₂₀ < 0 and a₁₂ − b₁₂ > 0.

Then, the only optimal symbols are [[G₁]], [[−G₂]], [[G₁,−G₂]], and [[−G₂,G₁]]. Moreover,

• if a₂₀ − b₂₀ < 0 and a₁₁ + b₁₁ < 0, the cut locus satisfies

  $$y_{cut}=-2\frac{a_{20}-b_{20}}{a_{11}-b_{11}}x_{cut}+o(x_{cut}), $$

• if a₂₀ − b₂₀ = 0 then

  $$y_{cut}=-3\frac{a_{30}-b_{30}}{a_{11}-b_{11}}x_{cut}^{2}+o(x_{cut}^{2}), $$

• if a₁₁ − b₁₁ = 0 then

  $$x_{cut}=-\frac12\frac{a_{12}-b_{12}}{a_{20}-b_{20}}y_{cut}^{2}+o(y_{cut}^{2}). $$

In all cases, the cut is tangent to Δ₁ and the contact is of order 2 when (a₂₀ − b₂₀)(a₁₁ − b₁₁) = 0.

4.1.5 Optimal Synthesis in the Domain $\mathbb {R}_{-}\times \mathbb {R}_{+}$

The dynamics entering $\mathbb {R}_{-}^{*}\times \mathbb {R}_{+}^{*}$ is with u₂ ≡− 1 since u₁ switches (Propositions 4 and 5). Three different cases can be identified.

1st. case. :

${\Delta }_1\cap (\mathbb {R}_{-}\times \mathbb {R}_{+}\setminus \{0\})$ is empty locally. No u₁-singular enters the domain. It corresponds to the case (NF_1a) where |a₁₀|≠|b₁₀| and to the cases (NF_1b) and (NF_1d) where a₁₀ − b₁₀ = 0 and

• (a₂₀ − b₂₀)(a₁₁ − b₁₁) < 0,

• or a₂₀ − b₂₀ = 0 and (a₃₀ − b₃₀)(a₁₁ − b₁₁) > 0,

• or a₁₁ − b₁₁ = 0 and (a₂₀ − b₂₀)(a₁₂ − b₁₂) < 0.

Only one u₁-switch can occur along the extremal. One has f₁ > 0 in the domain if

• a₁₀ − b₁₀ > 0,

• or a₁₀ − b₁₀ = 0 and a₂₀ − b₂₀ < 0,

• or a₁₀ − b₁₀ = 0 and a₂₀ − b₂₀ = 0 and a₁₁ + b₁₁ > 0,

and in this case, the possible extremals of the domain have symbol [[−G₁]] or [[G₂]] or [[−G₁,G₂]]. One has f₁ < 0 in the domain if

• a₁₀ − b₁₀ < 0,

• or a₁₀ − b₁₀ = 0 and a₂₀ − b₂₀ > 0,

• or a₁₀ − b₁₀ = 0 and a₂₀ − b₂₀ = 0 and a₁₁ − b₁₁ < 0,

and in this case, the possible extremals of the domain have symbol [[−G₁]] or [[G₂]] or [[G₂,−G₁]].

2nd. case. :

${\Delta }_1\cap (\mathbb {R}_{-}^{*}\times \mathbb {R}_{+}^{*})$ is not empty locally and is a turnpike. It corresponds to the cases where a₁₀ − b₁₀ = 0 and

• a₂₀ − b₂₀ > 0 and a₁₁ − b₁₁ > 0,

• or a₂₀ − b₂₀ = 0 and a₁₁ − b₁₁ > 0 and a₃₀ − b₃₀ < 0,

• or a₁₁ − b₁₁ = 0 and a₂₀ − b₂₀ > 0 and a₁₂ − b₁₂ > 0.

In this case, the possible symbols for extremals are [[−G₁]], [[G₂]], $[[S_{1}^{-},-G_{1}]]$, and $[[S_{1}^{-},G_{2}]]$.

3rd. case. :

${\Delta }_1\cap (\mathbb {R}_{-}^{*}\times \mathbb {R}_{+}^{*})$ is not empty locally and is a anti-turnpike. It corresponds to the cases where a₁₀ − b₁₀ = 0 and

• a₂₀ − b₂₀ < 0 and a₁₁ − b₁₁ < 0,

• or a₂₀ − b₂₀ = 0 and a₁₁ − b₁₁ < 0 and a₃₀ − b₃₀ > 0,

• or a₁₁ − b₁₁ = 0 and a₂₀ − b₂₀ < 0 and a₁₂ − b₁₂ < 0.

Then the only optimal symbols are [[−G₁]], [[G₂]], [[−G₁,G₂]], and [[G₂,−G₁]]. Moreover

• if a₂₀ − b₂₀ < 0 and a₁₁ + b₁₁ < 0, the cut locus satisfies

  $$y_{cut}=-2\frac{a_{20}-b_{20}}{a_{11}-b_{11}}x_{cut}+o(x_{cut}), $$

• if a₂₀ − b₂₀ = 0 then

  $$y_{cut}=-3\frac{a_{30}-b_{30}}{a_{11}-b_{11}}x_{cut}^{2}+o(x_{cut}^{2}), $$

• if a₁₁ − b₁₁ = 0 then

  $$x_{cut}=-\frac12\frac{a_{12}-b_{12}}{a_{20}-b_{20}}y_{cut}^{2}+o(y_{cut}^{2}). $$

In all cases, the cut is tangent to Δ₁ and the contact is of order 2 when (a₂₀ − b₂₀)(a₁₁ − b₁₁) = 0.

4.2 (N F _2a) Case

Recall that the normal form (NF_2a) gives

$$G_{1}(x,y)=\partial_{x}, \quad G_{2}(x,y)=(a_{0}+a_{10}x+o_{1}(x,y))\partial_{x} +(x+b_{20}x^{2}+o(x,y))\partial_{y}, $$

with 0 ≤ a₀ < 1.

A point where the normal form is given by (NF_2a) is neither in Δ₁ nor in Δ₂. Hence, no singular extremal will appear in the study of the local synthesis.

One can compute easily that, for any extremal starting at 0, $\phi _{1}(0)=\frac 12\lambda _{x}(0) (1+a_{0})$ and $\phi _{2}(0)=\frac 12\lambda _{x}(0) (1-a_{0})$. With H = 0 it gives |λ_x(0)| = 1. Hence, since $\dot \phi _{1}=-u_{2}\phi _{3}$ and $\dot \phi _{2}=u_{1}\phi _{3}$, if we want to study extremals that switch in short time, we need to consider ϕ₃ large that is |λ_y| large.

Moreover, since along an extremal issued from 0 $|\dot x(t)|\leq 1$ for t small, one gets easily that |x(t)|≤ t and |y(t)|≤ t² for t small enough. Hence $\phi _{1}(t)=\frac {1+a_{0}}2 \lambda _{x}(0)+x(t)\lambda _{y}(0)+o(t,x(t)\lambda _{y}(0))$ and $\phi _{2}(t)=\frac {1-a_{0}}2 \lambda _{x}(0)+x(t)\lambda _{y}(t)+o(t,x(t)\lambda _{y}(t))$. This implies that if one wants to consider an extremal switching at time τ small, he should consider initial conditions $\lambda _{y}(0)\sim \frac {1}{\tau }$. Inversing the point of view, if we consider an initial condition $\lambda _{y}(0)=\frac 1{r_{0}}$ with r₀ small, the switching time should be of order 1 in r₀. This motivates the following change of coordinates on the fibers of the cotangent: $r=\frac {1}{\lambda _{y}}$, p = rλ_x and the change of time s = t/r.

4.2.1 Equations of the Dynamics

With the new variables (x,y,p,r) and the new time s, the Hamiltonian equations become

$$\begin{array}{rclcrcl} x^{\prime}&=&r\frac{\partial H}{\partial \lambda_{x}}\left( x,y,p,-1\right),&&p^{\prime}&=&-r\frac{\partial H}{\partial x}\left( x,y,p,-1\right)+rp\frac{\partial H}{\partial y}\left( x,y,p,-1\right),\\ y^{\prime}&=&r\frac{\partial H}{\partial \lambda_{y}}\left( x,y,p,-1\right), &&r^{\prime}&=&r^{2}\frac{\partial H}{\partial y}\left( x,y,p,-1\right). \end{array} $$

Now, looking for the solutions as taylor series in r₀, that is under the form

$$\begin{array}{rclcrcl} x(r_{0},s)&=& x_{1}(s)r_{0}+ x_{2}(s){r_{0}^{2}} + o({r_{0}^{2}}),&&p(r_{0},s)&=&p_{1}(s)r_{0} +p_{2}(s) {r_{0}^{2}} + o({r_{0}^{2}}),\\ y(r_{0},s)&=&y_{2}(s){r_{0}^{2}} + y_{3}(s){r_{0}^{3}} + o({r_{0}^{3}}),&&r(r_{0},s)&=&r_{0}+r_{2}(s){r_{0}^{2}}+o({r_{0}^{2}}), \end{array} $$

one finds the equations

$$\begin{array}{rclcrcl} x_{1}^{\prime}&=&\frac{u_{1}+u_{2}}{2} +\frac{u_{1}-u_{2}}{2}a_{0},&&x_{2}^{\prime}&=&\frac{u_{1}-u_{2}}2a_{10}x_{1},\\ y_{2}^{\prime}&=&\frac{u_{1}-u_{2}}{4} x_{1},&&y_{3}^{\prime}&=&\frac{u_{1}-u_{2}}2(b_{20} {x_{1}^{2}} + x_{2}),\\ p_{1}^{\prime}&=&-\frac{u_{1}-u_{2}}{2} x_{1},&&p_{2}^{\prime}&=&-\frac{u_{1}-u_{2}}2(a_{10} p_{1} + 2 b_{20} x_{1}),\\ r_{2}^{\prime}&=&0,&&&& \end{array} $$

with the initial conditions x₁(0) = x₂(0) = y₂(0) = y₃(0) = p₂(0) = r₂(0) = 0 and p₁(0) = ± 1.

4.2.2 Computation of the Jets

Using these equations, we are able to compute the jets with respect to r₀ of four types of extremals: depending on the sign of p₁(0) = ± 1 and of r₀. For each of these types, we can compute the functions x₁, x₂, y₂, y₃, p₁, p₂ of the variable s for the first bang. We can then compute the jets of ϕ₁ and ϕ₂ for the first bang and look for the first switching time under the form s₁ = s₁₀ + s₁₁r₀ and then repeat the procedure for the second bang and so on. Finally, if we denote δ_p = sign(p(0)) and δ_r = sign(r₀) then the controls during the first bang are u₁ = u₂ = δ_p. The first time of switch is

$$s_{1}=\delta_{r}(1-\delta_{r} a_{0})-\delta_{p}(1 -\delta_{r} a_{0}) (\delta_{r} a_{10} + b_{20} -\delta_{r} a_{0} b_{20})r_{0}+o(r_{0})$$

and corresponds to ϕ₂(s₁) = 0 if δ_r = 1 or ϕ₁(s₁) = 0 if δ_r = − 1. The second bang corresponds to u₁ = δ_pδ_r and u₂ = −δ_pδ_r and the second switch is at

$$s_{2}=\delta_{r}(3 -\delta_{r} a_{0}) -\delta_{p} ((1 -\delta_{r} a_{0}) (\delta_{r} a_{10} + b_{20} -\delta_{r} a_{0} b_{20}) + 4 b_{20}) r_{0}+o(r_{0})$$

where ϕ₁(s₂) = 0 if δ_r = 1 and ϕ₂(s₂) = 0 if δ_r = − 1. At this time

$$\begin{array}{@{}rcl@{}} x(s_{2})&=&\delta_{p}(\delta_{r} + a_{0})r_{0}-\delta_{r}(\delta_{r} + a_{0}) (-\delta_{r} a_{10} + b_{20} +\delta_{r} a_{0} b_{20}){r_{0}^{2}}+o({r_{0}^{2}}),\\ y(s_{2})&=&2\delta_{r} {r_{0}^{2}}-\delta_{p}\frac43 (-a_{0} a_{10} + 3 b_{20} + {a_{0}^{2}} b_{20}){r_{0}^{3}}+o({r_{0}^{3}}). \end{array} $$

The third bang corresponds to u₁ = u₂ = − 1 if δ_p = 1 and to u₁ = u₂ = 1 if δ_p = − 1. The third switching time satisfies s₃ = δ_r(5 − δ_ra₀) + O(r₀) and the corresponding time t₃ is larger than the cut time as we will see later.

Let us analyze a little the situation in terms of cut locus for these extremals: if we consider the extremals with δ_p = δ_r = 1, they all start following G₁, without loosing optimality. Then they switch to G₂ at t = r₀(1 − a₀) + o(r₀). During this second bang, they do not intersect one each other since they are all following G₂ with a different initial condition on {x > 0,y = 0}. Then they switch to − G₁ but at a different y hence again they cannot intersect. The loss of optimality cannot come from an intersection with extremals with δ_r = − 1 since these last one live in {y ≤ 0}. As we will see in the following, the loss of optimality will come from the intersection with an extremal with − δ_p = δ_r = 1 during the third bang. Of course, the same occurs for extremals with δ_r = − 1.

Fix a small parameter ρ > 0. Since the dynamics during the third bang of all the extremals is given by ± G₁ = ±∂_x, y is constant during these third bangs. Hence, for the extremals with δ_r = 1, we can look for the r₀, as a jet in ρ, such that y = 2ρ² during the third bang, and for the extremals with δ_r = − 1, we can look for the r₀, as a jet in ρ, such that y = − 2ρ² during the third bang. The result is

$$r_{0}=\delta_{r}\rho+\delta_{r}\delta_{p}\frac13 (-a_{0} a_{10} + 3 b_{20} + {a_{0}^{2}} b_{20})\rho^{2}+o(\rho^{2})$$

which allows to compute

$$t_{2}=(3 -\delta_{r} a_{0}) \rho -\delta_{r}\delta_{p} \frac{3 a_{10} - {a_{0}^{2}} a_{10} +\delta_{r} 6 b_{20} - 3 a_{0} b_{20} + {a_{0}^{3}} b_{20}}{3} \rho^{2}+o(\rho^{2}). $$

Hence, we can compute x(t) = x(t₂) + (t − t₂) for this r₀ that is

$$x(t)=-\delta_{p} t+\delta_{p} 4 \rho - \frac23 (-a_{0} a_{10} + 3 b_{20} + {a_{0}^{2}} b_{20}) \rho^{2} +o(\rho^{2}).$$

We are now in situation to complete the computation of the jet of the cut locus: an extremal intersects an extremal of same length at the time t_cut = 4ρ + o(ρ²) which is less than t₃ = (5 − δ_ra₀)ρ hence t_cut is the cut time. When δ_r = 1 the cut point satisfies

$$x_{cut}=-\frac23 (-a_{0} a_{10} + 3 b_{20} + {a_{0}^{2}} b_{20}) \rho^{2}+o(\rho^{2}),\quad y_{cut}= 2\rho^{2}, $$

and when δ_r = − 1 the cut point satisfies

$$x_{cut}=-\frac23 (-a_{0} a_{10} + 3 b_{20} + {a_{0}^{2}} b_{20}) \rho^{2}+o(\rho^{2}),\quad y_{cut}=-2\rho^{2}. $$

Finally, if one wants to describe the sphere at time t small, one have that the first switching time is

$$t_{1}=\delta_{r}(1-\delta_{r} a_{0})r_{0}-\delta_{p}(1 -\delta_{r} a_{0}) (\delta_{r} a_{10} + b_{20} -\delta_{r} a_{0} b_{20}){r_{0}^{2}}+o({r_{0}^{2}}) $$

and hence, at t small, the r₀ corresponding to a first switching point is

$$r_{1}=\frac{t}{\delta_{r}(1 -\delta_{r} a_{0}) }+\delta_{r}\delta_{p} \frac{\delta_{r} a_{10} +b_{20}(1-\delta_{r} a_{0})}{(1 - a_{0})^{2}} t^{2}+o(t^{2}). $$

The second switching time is

$$t_{2}=\delta_{r}(3 -\delta_{r} a_{0})r_{0} -\delta_{p} ((1 -\delta_{r} a_{0}) (\delta_{r} a_{10} + b_{20} -\delta_{r} a_{0} b_{20}) + 4 b_{20}) {r_{0}^{2}}+o({r_{0}^{2}}) $$

which implies that, at t small, the r₀ corresponding to a second switching point is

$$r_{2}=\frac{t}{\delta_{r}(3 -\delta_{r} a_{0})} +\delta_{p} \frac{(1 -\delta_{r} a_{0}) (\delta_{r} a_{10} + b_{20} -\delta_{r} a_{0} b_{20}) + 4 b_{20}}{\delta_{r}(3 -\delta_{r} a_{0})^{3}}t^{2}+o(t^{2}). $$

And the cut time is

$$t_{cut}= 4\delta_{r} (r_{0} -\delta_{r}\delta_{p} \frac13 (-a_{0} a_{10} + 3 b_{20} + {a_{0}^{2}} b_{20}) {r_{0}^{2}})+o({r_{0}^{2}}) $$

which implies that at t small the r₀ corresponding to a cut point is

$$r_{cut}=\frac{\delta_{r}}4(t + \frac{\delta_{p}}{12} (-a_{0} a_{10} + 3 b_{20} + {a_{0}^{2}} b_{20}) t^{2})+o(t^{2}). $$

4.3 (N F _2b) Case

Recall that the normal form (NF_2b) gives G₁(x,y) = ∂_x, and

$$G_{2}(x,y)=(1+a_{10}x+a_{01}y+a_{20}x^{2}+o_{2}(x,y))\partial_{x} +(x+b_{20}x^{2}+b_{30}x^{3}+o_{3}(x,y))\partial_{y}. $$

In this case, the extremals with initial condition |λ_y(0)| >> 1 are the limit when a₀ goes to 1 of the extremal presented in the case (NF_2a). If λ_y(0) >> 1 then the symbol starts with [[G₂,−G₁]] or with [[−G₂,G₁]] and if − λ_y(0) >> 1 then the symbol starts with [[G₁,−G₂]] or with [[−G₁,G₂]] (Fig. 4).

Fig. 4

The optimal synthesis in the (NF_2a) case

But F₂(0) = 0 then for all extremals ϕ₂(0) = 0. Hence, an extremal may also, depending on the invariants, have symbol starting by [[G₂,G₁]], [[G₁,G₂]], $[[S_{2}^{+},G_{1}]]$ or $[[S_{2}^{+},G_{2}]]$ if λ_x(0) = 1, or starting by [[−G₂,−G₁]], [[−G₁,−G₂]], $[[S_{2}^{-},-G_{1}]]$ or $[[S_{2}^{-},-G_{2}]]$ if λ_x(0) = − 1.

If λ_x(0) = 1 then at least for small time u₁(t) = 1 and x(t) = t + o(t) and y(t) = o(t). Then, computing ϕ₂ one finds $\phi _{2}(t)=-\lambda _{x}(t)\frac {a_{10}}{2}-\lambda _{y}(t) \frac {x(t)}2+o(t) =-(\frac {a_{10}+\lambda _{y}(0)}{2})t+o(t)$. Hence if λ_y(0) > −a₁₀ then, since ϕ₂(0) < 0 for small time, the extremal starts by a bang following G₂. If λ_y(0) < −a₁₀ then ϕ₂(0) > 0 for small time and the extremal starts by a bang following G₁.

If λ_x(0) = − 1 then at least for small time u₁(t) = − 1 and x(t) = −t + o(t) and y(t) = o(t). Then $\phi _{2}(t)=(\frac {a_{10}-\lambda _{y}(0)}{2})t+o(t)$. Hence if λ_y(0) > a₁₀ then, since ϕ₂(0) < 0 for small time, the extremal starts by a bang following − G₁. If λ_y(0) < a₁₀ then ϕ₂(0) > 0 for small time and the extremal starts by a bang following − G₂.

In coordinates, one can compute that

$$\det(F_{2},[F_{1},F_{2}])(x,y)=\frac14 ((a_{10} b_{20}-a_{20} )x^{2} + a_{01} y)+o_{2}(x,y) $$

where x has weight 1 and y has weight 2. Since generically at such points (which are isolated points) a₀₁≠ 0 then an equation for Δ₂ is given by

$$y=\frac{a_{20} - a_{10} b_{20}}{a_{01} }x^{2} +o(x^{2}). $$

Remark that generically $\frac {a_{20} - a_{10} b_{20}}{a_{01} }$ is neither 0 nor $\frac 12$. Moreover

$$f_{2}(x,y)=\frac{\det(F_{2},[F_{1},F_{2}])(x,y)}{\det(F_{2},F_{1})(x,y)} =\frac{((a_{10} b_{20}-a_{20} ) x^{2} + a_{01} y)+o_{2}(x,y)}{2x}. $$

Recall that an equation of the support of the integral curve of G₁ passing by 0 is y = 0 and that an equation for the support of the integral curve of G₂ passing by 0 is $y=\frac {x^{2}}{2}+o(x^{2})$.

If $\frac {a_{20} - a_{10} b_{20}}{a_{01} }<0$ or if $\frac {a_{20} - a_{10} b_{20}}{a_{01} }>\frac 12$ then Δ₂ does not enter the domain ${\mathcal {D}}=\{x>0, 0<y<\frac {x^{2}}2 \}$ and along it G₁ and G₂ point on the same side of Δ₂ hence Δ₂ is not a turnpike. In these cases

• if a₁₀b₂₀ − a₂₀ > 0 then f₂ > 0 in $\mathcal {D}$ and the new extremals, that are not described as limit of the case NF_2a, have symbol [[G₂,G₁]].

• if a₁₀b₂₀ − a₂₀ < 0 then f₂ < 0 in $\mathcal {D}$ and the new extremals, that are not described as limit of the case NF_2a, have symbol [[G₁,G₂]].

If $0<\frac {a_{20} - a_{10} b_{20}}{a_{01} }<\frac 12$ then Δ₂ enters $\mathcal {D}$ and along it G₁ and G₂ point on opposite sides of Δ₂. In this case:

• if a₁₀b₂₀ − a₂₀ > 0 then, along ${\Delta }_2\cap {\mathcal {D}}$, G₁ points in direction of f₂ > 0 and Δ₂ is a turnpike. Then, the only extremals entering the domain $\mathcal {D}$ start with a singular arc and have symbols $[[S_{2}^{+}]]$, $[[S_{2}^{+},G_{1}]]$ or $[[S_{2}^{+},G_{2}]]$.

• if a₁₀b₂₀ − a₂₀ < 0 then, along ${\Delta }_2\cap {\mathcal {D}}$, G₁ points in direction of f₂ < 0 and Δ₂ is not a turnpike. In this case, the symbols start with [[G₁,G₂]] and [[G₂,G₁]]. One can compute, with the same techniques that in Section 4.2.2, the switching times and the second switching locus for extremals that enter the domain $\mathcal {D}$, that is for extremal with initial condition λ_y(0) = −a₁₀ + δ𝜖 with 𝜖 > 0 small and δ = ± 1. If δ < 0 then the symbol is [[G₁,G₂,G₁]] and the switching times are $t_{1}=\frac {\epsilon }{a_{20} - a_{10} b_{20}}$ and $t_{2}=t_{1}+\frac {2\epsilon }{a_{01} - 2 a_{20} + 2 a_{10} b_{20}}$, the second switching locus being

  $$\begin{array}{@{}rcl@{}} x(\epsilon)&=&\frac{a_{01}\epsilon}{(a_{20} - a_{10} b_{20}) (a_{01} - 2 a_{20} + 2 a_{10} b_{20})},\\ y(\epsilon)&=&\frac{2 (a_{01} - a_{20} + a_{10} b_{20})\epsilon^{2}}{(a_{20} - a_{10} b_{20}) (a_{01} - 2 a_{20} + 2 a_{10} b_{20})^{2}}. \end{array} $$

  If δ > 0 then the symbol is [[G₂,G₁,G₂]] and the switching times are $t_{1}=\frac {2\epsilon }{a_{01} - 2 a_{20} + 2 a_{10} b_{20}}$ and $t_{2}=t_{1}+\frac {\epsilon }{a_{20} - a_{10} b_{20}}$, the second switching locus being

  $$\begin{array}{@{}rcl@{}} x(\epsilon)&=&\frac{a_{01}\epsilon}{(a_{20} - a_{10} b_{20}) (a_{01} - 2 a_{20} + 2 a_{10} b_{20})}, \\ y(\epsilon)&=&\frac{2 \epsilon^{2}}{ (a_{01} - 2 a_{20} + 2 a_{10} b_{20})^{2}}. \end{array} $$

  In fact, these extremals lose optimality before the second switching. Effectively, the two fronts intersect before creating cut locus. In order to compute this cut locus, one can compute the jets of the two corresponding families of curves: the first one following G₁ during time 𝜖₁₁ then following G₂ during time 𝜖₁₂, with 𝜖₁₁ + 𝜖₁₂ = t; the second one following G₂ during time 𝜖₂₁ then following G₁ during time 𝜖₂₂, with 𝜖₂₁ + 𝜖₂₂ = t. Writing 𝜖₁₁ = t − 𝜖₁₂ and 𝜖₂₂ = t − 𝜖₂₁ and then 𝜖₁₂ = s₁t + s₂t² + o(t²) and 𝜖₂₁ = t₁t + t₂t² + o(t²), one can compute the jets with respect to t of both families and compute the cut locus. On finds

  $$t_{1}=\frac{a_{01} - 6 (\alpha - a_{10} \beta) - 2 (a_{01} -3 (\alpha - a_{10} \beta)) s_{1} + (a_{01} - 2 (\alpha - a_{10} \beta)) {s_{1}^{2}}}{2 (\alpha - a_{10} \beta) (-2 + s_{1})} $$

  and

  $$s_{1}=\frac{1 - 2 \gamma + 8 \gamma^{2} - 2 \sqrt{\gamma}}{1 + 4 \gamma^{2}} $$

  where $\alpha =\frac {a_{01}+ 2a_{20}+a_{10}^{2}}{6}$, $\beta =\frac {2b_{20}+a_{10}}{6}$, $\gamma =\frac {\alpha -a_{10}\beta }{a_{01}-2(\alpha -a_{10}\beta )}$. Under the hypotheses of this case, one proves easily that $\frac 14\leq \gamma \leq 1$ which allows to prove that the expression in the formula of s₁ varies between 0 and 1. Finally, one gets the formula for the cut locus

  $$x(t)=t +\frac12 (2a_{10} s_{1} - a_{10} {s_{1}^{2}}) t^{2} +o(t^{2}), y(t)= \frac12(2s_{1} - {s_{1}^{2}}) t^{2} +o(t^{2}). $$

  Hence, the only optimal symbols entering the domain $\mathcal {D}$ are [[G₁,G₂]] and [[G₂,G₁]].

Pictures for the (NF_2b) case are in Figs. 5 and 6.

Fig. 5

(NF_2b) case: when $0<\frac {a_{20} - a_{10} b_{20}}{a_{01} }<\frac 12$

Fig. 6

(NF_2b) case: when $\frac {a_{20} - a_{10} b_{20}}{a_{01} }<0$ or $\frac {a_{20} - a_{10} b_{20}}{a_{01} }>\frac 12$

4.4 (N F ₃) Case

Recall that in the (NF₃) case, x has weight 1 and y has weight 3. Hence, we can write

$$\begin{array}{@{}rcl@{}} G_{1}(x,y)&=&\partial_{x} \\ G_{2}(x,y)&=&(a_{0}+a_{10}x+o(x,y))\partial_{x} +\left( \frac{x^{2}}2+b_{01}y+b_{30}x^{3}+o_{3}(x,y)\right)\partial_{y} \end{array} $$

with b_0,1≠ 0 and 0 < a₀ < 1, where o_k(x,y) has the meaning given in Section 3.5. As in the (NF_2b) case, for any extremal starting at 0,

$$\phi_{1}(0)=\frac12\lambda_{x}(0) (1+a_{0}) \quad \text{and} \quad \phi_{2}(0)=\frac12\lambda_{x}(0) (1-a_{0}).$$

And for the same reasons, if we want to study extremals that switch in short time, we need to consider |λ_y| large.

The set of initial condition is {(λ_x(0),λ_y(0))|λ_x(0) = ± 1}. We parameterize the upper part of this set by setting $\lambda _{y}(0)=\frac 1{{r_{0}^{2}}}$ and the lower part by $\lambda _{y}(0)=-\frac 1{{r_{0}^{2}}}$.

As explained in Section 3.4, in order to compute extremals with λ_y(0) >> 1 we make the change of coordinates $r=\frac {1}{\sqrt {\lambda _{y}}}$, $X=\frac x r$, $Y=\frac y {r^{3}}$ and the change of time $s=\frac t r$.

Now, looking for the solutions as taylor series in r₀, that is under the form

$$\begin{array}{rclcrcl} X(r_{0},s)&=&X_{0}(s) + r_{0} X_{1}(s) + o(r_{0}),&&\lambda_{x}(r_{0},s)&=&\lambda_{x0}(s) + r_{0} \lambda_{x1}(s) + o(r_{0}),\\ Y(r_{0},s)&=&Y_{0}(s) + r_{0} Y_{1}(s)+ o(r_{0}),&&r(r_{0},s)&=&r_{0}+{r_{0}^{2}} r_{2}(s)+o_{2}(r_{0}) \end{array} $$

one finds the equations

$$\begin{array}{@{}rcl@{}} X_{0}^{\prime}(s)&=&\frac{1}{2} (u_{1}+u_{2})+\frac{a_{0}}{2}(u_{1}-u_{2}),\\ X_{1}^{\prime}(s)&=&\frac{(u_{1}-u_{2})}4(2 a_{10} - b_{01})X_{0}(s),\\ Y_{0}^{\prime}(s)&=&\frac{1}{4} (u_{1}-u_{2}) {X_{0}^{2}}(s),\\ Y_{1}^{\prime}(s)&=&\frac{(u_{1}-u_{2})}4(2 b_{30} {X_{0}^{3}}(s) + 2 X_{0}(s) X_{1}(s) - b_{01} Y_{0}(s)),\\ \lambda_{x0}^{\prime}(s)&=&-\frac{1}{2} (u_{1}-u_{2}) X_{0}(s),\\ \lambda_{x1}^{\prime}(s)&=&-\frac{(u_{1}-u_{2})}2(a_{10} \lambda_{x0}(s) + 3 b_{30} {X_{0}^{2}}(s) + X_{1}(s)),\\ r_{2}^{\prime}(s)&=&\frac{b_{01}}{4} (u_{1}-u_{2}), \end{array} $$

For an initial condition λ_x(0) = 1, one find ϕ₁(0) > 0 and ϕ₂(0) > 0, hence u₁(0) = u₂(0) = 1. One can integrate the equations and look for the first switching time as a Taylor series $s^{1}={s^{1}_{0}}+r_{0} {s^{1}_{1}}+o(r_{0})$ and compute $\phi _{2}(r_{0},{s^{1}_{0}}+r_{0} {s^{1}_{1}}+o(r_{0}))$ in order to compute

$${s^{1}_{0}}=\sqrt{2} \sqrt{1 - a_{0}} \quad \text{and}\quad {s^{1}_{1}}=-a_{10} - 2 b_{30}(1-a_{0}).$$

At the switching time

$$\begin{array}{rclcrcl} X(s^{1})&=&\sqrt{2} \sqrt{1 - a0}-(a_{10} + 2 b_{30})(1-a_{0})r_{0},&&\lambda_{x}(s^{1})&=&1,\\ Y(s^{1})&=&0,&&r(s^{1})&=&r_{0}. \end{array} $$

After this first switch ϕ₁(0) > 0 and ϕ₂(0) < 0, hence u₁(0) = 1 and u₂(0) = − 1. We can compute and look for the next switching time and one finds that ϕ₁ goes to 0 at $s^{2}={s^{2}_{0}}+r_{0} {s^{2}_{1}}+o(r_{0})$ with

$$\begin{array}{@{}rcl@{}} {s^{2}_{0}}&=&{s^{1}_{0}}+\sqrt 2\frac{\sqrt{1 + a_{0}}-\sqrt{1 - a_{0}}}{a_{0}}, \\ {s^{2}_{1}}&=&{s^{1}_{1}}+\frac{b_{01}((1 - a_{0})^{\frac32} - \sqrt{1 + a_{0}}(1-2a_{0})) - 12b_{30} {a_{0}^{2}} \sqrt{1 + a_{0}} }{3 {a_{0}^{2}} \sqrt{ 1 + a_{0}}}. \end{array} $$

At the second switching time

$$\begin{array}{@{}rcl@{}} X(s^{2})&=&\sqrt{2} \sqrt{1 + a_{0}}\\&+&\frac{3 a_{10}a_{0} \sqrt{1 + a_{0}}+b_{01}((1-a_{0})^{\frac32}-(1+a_{0})^{\frac32}) -6b_{30}a_{0}(1+a_{0})^{\frac32}}{3 a_{0} \sqrt{1 + a_{0}}} r_{0},\\ Y(s^{2})&=&\frac{\sqrt{2} ((1 + a_{0})^{\frac{3}{2}} - (1 - a_{0})^{\frac{3}{2}})}{3 a_{0}}\\&-&\frac{2 b_{01}(1-a_{0}+{a_{0}^{2}}-(1 - a_{0})^{\frac32} \sqrt{1 + a_{0}} )+ 12 {a_{0}^{2}} b_{30} }{3 {a_{0}^{2}} } r_{0},\\ \lambda_{x}(s^{2})&=&-1,\\ r(s^{2})&=&r_{0}+\frac{(\sqrt{1 + a_{0}}-\sqrt{1 - a_{0}}) b_{01}}{\sqrt{2} a_{0}} {r_{0}^{2}}. \end{array} $$

After this second switch, ϕ₁(0) < 0 and ϕ₂(0) < 0, hence u₁(0) = u₂(0) = − 1. One can compute the third switch as being $s^{3}={s^{3}_{0}}+r_{0} {s^{3}_{1}}+o(r_{0})$ with

$${s^{3}_{0}}={s^{2}_{0}}+ 2 \sqrt{2} \sqrt{1 + a0}, \quad {s^{3}_{1}}={s^{2}_{1}}-\frac{2 ((1+a_{0})^{\frac32}-(1 - a_{0})^{\frac32} ) b_{01}}{3 a_{0} \sqrt{1 + a_{0}}}. $$

At this time $X(s^{3})=-\sqrt {2} \sqrt {1 + a_{0}}+O(r_{0})$ and we will see that this third switching time comes after the cut time.

The same computations can be done for the extremals starting with λ_x(0) = − 1. We use the notation $\bar z$ for variables z corresponding to these extremals. During the first bang the controls are $\bar u_{1}=\bar u_{2}=-1$, during the second $\bar u_{1}= 1$ and $\bar u_{2}=-1$ and during the third one $\bar u_{1}=\bar u_{2}= 1$. The switching times are ${\bar s}^{1}$ and ${\bar s}^{2}$ satisfying

$$\begin{array}{rclcrcl} {\bar s}^{1}_{0}&=&\sqrt{2} \sqrt{1 + a_{0}}, && {\bar s}^{1}_{1}&=&-a_{10} + 2 b_{30}(1+a_{0}),\\ {\bar s}^{2}_{0}&=&{\bar s}^{1}_{0}+\sqrt 2\frac{\sqrt{1 + a_{0}}-\sqrt{1 - a_{0}}}{a_{0}}, && {\bar s}^{2}_{1}&=&{\bar s}^{1}_{1}+\frac{b_{01}((1 + a_{0})^{\frac32} - \sqrt{1 - a_{0}}(1 + 2a_{0})) + 12b_{30} {a_{0}^{2}} \sqrt{1 - a_{0}} }{3 {a_{0}^{2}} \sqrt{1 - a_{0}}}. \end{array} $$

And at the second switching time

$$\begin{array}{@{}rcl@{}} {\bar X}({\bar s}^{2})&=&-\sqrt{2} \sqrt{1 - a_{0}}\\&+&\frac{-3 a_{10}a_{0} \sqrt{1 - a_{0}}+b_{01}((1+a_{0})^{\frac32}-(1-a_{0})^{\frac32}) -6b_{30}a_{0}(1-a_{0})^{\frac32}}{3 a_{0} \sqrt{1 - a_{0}}} \bar r_{0},\\ \bar Y({\bar s}^{2})&=&\frac{\sqrt{2} ((1 + a_{0})^{\frac32} - (1 - a_{0})^{\frac{3}{2}})}{3 a_{0}}\\&-& \frac{2 b_{01}(1+a_{0}+{a_{0}^{2}}-\sqrt{1 - a_{0}}(1+a_{0})^{\frac32})- 12 {a_{0}^{2}} b_{30} }{3 {a_{0}^{2}} } \bar r_{0},\\ \bar \lambda_{x}({\bar s}^{2})&=&-1,\\ \bar r({\bar s}^{2})&=&\bar r_{0}+\frac{(\sqrt{1 + a_{0}}-\sqrt{1 - a_{0}}) b_{01}}{\sqrt{2} a_{0}} \bar {r_{0}^{2}}. \end{array} $$

One can compute that at the third switching time $\bar X({\bar s}^{3})=\sqrt {2} \sqrt {1 - a_{0}}+O(r_{0})$.

We are now ready to compute the cut locus. As one can estimate easily, an extremal starting with λ_x(0) > 0 intersects an extremal starting with λ_x(0) < 0, both during their third bang. Moreover, since $Y(s^{2})=\bar Y(\bar s^{2})+o(r_{0})$ one have that $\bar r_{0}=r_{0} +o(r_{0})$.

Fix ρ and look for the extremals intersecting at $y=\frac {\sqrt {2} ((1 + a_{0})^{\frac 32} - (1 - a_{0})^{\frac {3}{2}})}{3 a_{0}}\rho ^{3}$. We write r₀ = ρ + R_cutρ² + o(ρ²) and look for R_cut such that $r_{0} Y(s^{2})=\frac {\sqrt {2} ((1 + a_{0})^{\frac 32} - (1 - a_{0})^{\frac {3}{2}})}{3 a_{0}}\rho ^{3}+o(\rho ^{4})$. We find

$$R_{cut}=\frac{\sqrt2 ((-2 {a_{0}^{2}} + (2+a_{0}) (-1 + \sqrt{1 - {a_{0}^{2}}}) ) b_{01} + 6 {a_{0}^{2}} b_{30})}{3 a_{0} ((1 + a_{0})^{\frac32} - (1 - a_{0})^{\frac32})}.$$

For $\bar r_{0}=\rho +\bar R_{cut} \rho ^{2}+o(\rho ^{2})$ one finds

$$\bar R_{cut}=\frac{\sqrt2 ((-2 {a_{0}^{2}} + (2-a_{0}) (-1 + \sqrt{1 - {a_{0}^{2}}}) ) b_{01} - 6 {a_{0}^{2}} b_{30})}{3 a_{0} ((1 + a_{0})^{\frac32} - (1 - a_{0})^{\frac32})}.$$

With these values, we can compute the second switching times $t^{2}=r s^{2}={t^{2}_{1}} \rho +{t^{2}_{2}} \rho ^{2}+o(\rho ^{3})$ and $\bar t_{2}=\bar r \bar s^{2}=\bar {t^{2}_{1}} \rho +\bar {t^{2}_{2}} \rho ^{2}+o(\rho ^{3})$ with

$$\begin{array}{@{}rcl@{}} {t^{2}_{1}}&=&\sqrt2 \left( \sqrt{1 - a_{0}} + \frac{\sqrt{1 + a_{0}} - \sqrt{1 - a_{0}}}{a_{0}}\right)\\ {t^{2}_{2}}&=&-a_{10}+\frac{2 (-4 + a_{0} - 2 {a_{0}^{2}} + {a_{0}^{3}} + (-1 + a_{0}) \sqrt{1 - {a_{0}^{2}}}) }{3 + {a_{0}^{2}} }b_{30}\\ &&+\frac{(-5 + 2 a_{0} - 6 {a_{0}^{2}} + {a_{0}^{3}}) \sqrt{1 + a_{0}} - (-5 - 3 a_{0} + {a_{0}^{2}} + 3 {a_{0}^{3}})\sqrt{1 - a_{0}} }{ 3 {a_{0}^{2}} \sqrt{1 + a_{0}} (2 + \sqrt{1 - {a_{0}^{2}}})} b_{01}\\ \bar {t^{2}_{1}}&=&\sqrt2 \left( \sqrt{1 + a_{0}} + \frac{\sqrt{1 + a_{0}} - \sqrt{1 - a_{0}}}{a_{0}}\right)\\ \bar {t^{2}_{2}}&=&-a_{10}+\frac{2 (4 +a_{0} + 2{a_{0}^{2}} + {a_{0}^{3}})+(1 +a_{0}) \sqrt{1 - {a_{0}^{2}}} ) }{3 + {a_{0}^{2}}}b_{30}\\ &&-\frac{ (-5 + 3 a_{0} + {a_{0}^{2}} - 3 {a_{0}^{3}}) \sqrt{1 + a_{0}}+ (5 + 2 a_{0} + 6 {a_{0}^{2}} + {a_{0}^{3}})\sqrt{1 - a_{0}} } {3 {a_{0}^{2}} \sqrt{1 - a_{0}} (2 + \sqrt{1 - {a_{0}^{2}}})} b_{01} \end{array} $$

and the x coordinates of the point of second switching under the form x = x₁ρ + x₂ρ² + o(ρ³) and $\bar x=\bar x_{1} \rho +\bar x_{2}\rho ^{2}+o(\rho ^{3})$ with

$$\begin{array}{@{}rcl@{}} x_{1}&=&\frac{2 \sqrt{2} (1 + 3 {a_{0}^{2}} - (1 - {a_{0}^{2}})^{\frac32})}{a_{0} ((1 + a_{0})^{\frac32} -(1 - a_{0})^{\frac32} )},\\ x_{2}&=&-\frac{5+a_{0}+ 5{a_{0}^{2}}-(5+a_{0})\sqrt{1-{a_{0}^{2}}} }{3 {a_{0}^{2}} }b_{01}-4b_{30},\\ \bar x_{1}&=&-\frac{2 \sqrt{2} (1 + 3 {a_{0}^{2}} - (1 - {a_{0}^{2}})^{\frac32})}{a_{0} ((1 + a_{0})^{\frac32} -(1 - a_{0})^{\frac32} )},\\ \bar x_{2}&=&\frac{ 5-a_{0}+ 5{a_{0}^{2}}+(-5+a_{0})\sqrt{1-{a_{0}^{2}}}}{3 {a_{0}^{2}}}b_{01}-4b_{30}. \end{array} $$

One find easily that the cut locus is at $x_{c}=\frac {x_{1}+\bar x_{1}}2\rho +\frac {x_{2}+\bar x_{2}}2\rho ^{2}+o(\rho ^{2})$ that is

$$\begin{array}{@{}rcl@{}} x_{cut}^{+}&=&-\left( \frac{a_{0}}{3 (1 + \sqrt{1 - {a_{0}^{2}}})}b_{01}+ 4 b_{30}\right)\rho^{2}+o(\rho^{2}),\\ y_{cut}^{+}&=&\frac{\sqrt{2} ((1 + a_{0})^{\frac32} - (1 - a_{0})^{\frac{3}{2}})}{3 a_{0}}\rho^{3}. \end{array} $$

When − λ_y(0) >> 1, then we set $r=\frac {1}{\sqrt {-\lambda _{y}}}$. Equations are changed but the final result is very similar

$$\begin{array}{@{}rcl@{}} x_{cut}^{-}&=&-\left( \frac{a_{0}}{3 (1 + \sqrt{1 - {a_{0}^{2}}})}b_{01}+ 4 b_{30}\right)\rho^{2}+o(\rho^{2}),\\ y_{cut}^{-}&=&-\frac{\sqrt{2} ((1 + a_{0})^{\frac{3}{2}} - (1 - a_{0})^{\frac{3}{2}})}{3 a_{0}}\rho^{3}. \end{array} $$

Finally, the cut locus appears to be a cusp whose tangent at the singular point is the tangent to Δ_A, see Fig. 7.

Fig. 7

The synthesis in the (NF₃) case