# New second-order optimality conditions in sub-Riemannian Geometry

Michał Jóźwikowski111Part of this research was conducted during the employment
of MJ at the University of Fribourg, finaced by the ERC Starting Grant
_Geometry of Metric Groups_ , grant agreement 713998 GeoMeG.
Faculty of Mathematics, Informatics and Mechanics
University of Warsaw
Department of Mathematics
University of Fribourg

###### Abstract

We study the geometry of the second-order expansion of the extended end-point
map for the sub-Riemannian geodesic problem. Translating the geometric reality
into equations we derive new second-order necessary optimality conditions in
sub-Riemannian Geometry. In particular, we find an ODE for velocity of an
abnormal sub-Riemannian geodesics. It allows to divide abnormal minimizers
into two classes, which we propose to call 2-normal and 2-abnormal extremals.
In the 2-normal case the above ODE completely determines the velocity of a
curve, while in the 2-abnormal case the velocity is undetermined at some, or
at all points. With some enhancement of the presented results it should be
possible to prove the regularity of all 2-normal extremals (the 2-abnormal
case seems to require study of higher-order conditions) thus making a step
towards solving the problem of smoothness of sub-Riemannian abnormal
geodesics.

As a by-product we present a new derivation of Goh conditions.

_Martynie i Zuzi_

##  1 Introduction

#### The problem of smoothness of abnormal geodesics.

The question weather all minimizing sub-Riemannian geodesics are smooth dates
back to the famous example of Montgomery [Mon94] of a minimizing abnormal
geodesics. It is commonly agreed [Agr14, Mon06] to be one of the main open
problem in sub-Riemannian geometry. Despite 30 years of efforts, there are not
many general results. Among the latter we must mention the derivation of the
second-order necessary conditions for optimality [AS96] (known as _Goh
conditions_) and a technical masterpiece of Hakavouri and Le Donne [HLD16] (se
also [HLD18]) which ruled out corner-like singularities (some previous results
in this direction are [LM08, Mon14b]). We are also aware of a very recent
attempt to derive third-order necessary conditions of optimality [BMP20]. We
refer to [Mon14a] for a discussion of the problem and its history.

#### Basic ideas.

In this paper we propose the following approach to the problem. Consider a
sub-Riemannian manifold
$(M,\mathcal{D},\big{\langle}\cdot,\cdot\big{\rangle})$ and the problem of
minimizing the energy by a $\mathcal{D}$-horizontal curve $q(t)$ with
$t\in[t_{0},t_{1}]$ joining given two points $q_{0}=q(t_{0})$ and
$q_{1}=q(t_{1})$ on $M$. We may (locally) reformulate this problem as an
optimal control problem in a standard way – see page $\Sigma$. Now let

${\mathbf{End}}^{[t_{0},t]}:L^{\infty}([t_{0},t],\mathbb{R}^{k})\longrightarrow
M\times\mathbb{R}\
;\qquad{\mathbf{End}}^{[t_{0},t]}[u]=(q(t),E^{[t_{0},t]}[u])$

be the extended end-point map at time $t\in[t_{0},t_{1}]$ related with this
optimal control problem. Here $q(t)$ is the sub-Riemannian trajectory
corresponding to the control $u\in L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})$
and $E^{[t_{0},t]}[u]$ is the energy of this trajectory at time $t$.

In the next step, for a distinguished trajectory of the system $q(t)\in M$
corresponding to a given control $u\in L^{\infty}([t_{0},t],\mathbb{R}^{k})$,
we study curves $s\mapsto{\mathbf{End}}^{[t_{0},t]}[u+s\cdot\Delta u]$ built
for all possible controls $\Delta u\in L^{\infty}([t_{0},t],\mathbb{R}^{k})$.
More precisely, we are interested in 2-jets of such curves at $s=0$, i.e.

${\mathbf{End}}^{[t_{0},t]}[u+s\cdot\Delta
u]=:{\mathbf{End}}^{[t_{0},t]}[u]+s\cdot\bm{b}^{(1)}(t,\Delta
u)+{s^{2}}\cdot\bm{b}^{(2)}(t,\Delta u)+o(s^{2})\ .$

One may think of the collection of pairs $(\bm{b}^{(1)}(t,\Delta
u),\bm{b}^{(2)}(t,\Delta u))$ (for all possible $\Delta u$’s) as of quadratic
approximations of the image of the end-point map around
${\mathbf{End}}^{[t_{0},t]}[u]$.

It turns out that, for a given $\Delta u\in
L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})$, curves $\bm{b}^{(1)}(t,\Delta u)$
and $\bm{b}^{(2)}(t,\Delta u)$ satisfy quite a complicated system of ODEs
(3.3)-(3.4). It is convenient to treat the latter as a control system (build
on top of a fixed trajectory $q(t)$ of the initial sub-Riemannian control
system) with $\Delta u$ playing the role of the control.

The key point now is to relate the properties of this new controls system with
optimality of a sub-Riemannian trajectory $q(t)$. To do this we rely on two
results:

  * •

Firstly, the classical theory of Agrachev and Sarychev [AS96] gives
information about the space of curves $\bm{b}^{(2)}(t,\Delta u)$ for these
controls $\Delta u\in L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})$ which satisfy
$\bm{b}^{(1)}(t,\Delta u)=\bm{0}$.

  * •

Secondly, thanks to the results of [JS21], we may change coordinates
transforming the pairs $(\bm{b}^{(1)}(t,\Delta u),\bm{b}^{(2)}(t,\Delta u))$
into pairs $(\bm{q}^{(1)}(t,\Delta u),\bm{q}^{(2)}(t,\Delta
u))\in\mathbb{R}^{n+1}\times\mathbb{R}^{n+1}$, for which the new control
system we want to analyze takes a much simpler form (3.7)–(3.8).

Using the above we are able to relate the geometry of the sets

$\\{(\bm{b}^{(1)}(t,\Delta u),\bm{b}^{(2)}(t,\Delta u))\ |\ \Delta u\in
L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})\\}\quad\text{where
$t\in[t_{0},t_{1}]$}$

with the optimality of the sub-Riemannian trajectory $q(t)$.

#### The main result.

Leaving aside some technical complications (we need to divide the curve $q(t)$
into pieces by excluding a finite number of points in the time interval
$[t_{0},t_{1}]$) it turns out that, if the curve $q(t)$ is an abnormal
minimizing geodesics, then there exists a _Pontryagin covector_
$\bm{\varphi}(t)=(\varphi(t),0)\in\mathrm{T}_{(q(t),E^{[t_{0},t]}[u])}^{\ast}(M\times\mathbb{R})$
such that for each $t\in[t_{0},t_{1}]$, the set

$S_{t}:=\\{(\bm{b}^{(1)}(t,\Delta u),\big{\langle}\bm{b}^{(2)}(t,\Delta
u),\bm{\varphi}(t)\big{\rangle})\ |\ \Delta u\in
L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})\\}\subset\mathrm{T}_{(q(t),E^{[t_{0},t]}[u])}(M\times\mathbb{R})\times\mathbb{R}$

is bounded by a graph of a quadratic map
$\bm{\Phi}_{2}(t):\mathrm{T}_{(q(t),E^{[t_{0},t]}[u])}(M\times\mathbb{R})\times\mathrm{T}_{(q(t),E^{[t_{0},t]}[u])}(M\times\mathbb{R})\rightarrow\mathbb{R}$
(see the figure below).

For a minimizing curve the second-order approximations of the image of the
extended end-pointmap (grey area) must be bounded by a graph of a quadratic
function (dashed line). Solid lines re-present different quadratic
approximations of the image of the extended end-point map
${\mathbf{End}}^{[t_{0},t]}$.${\mathbf{End}}^{[t_{0},t]}[u]$${\mathbf{End}}^{[t_{0},t]}[u]+s\cdot\bm{b}^{(1)}(t)+\frac{s^{2}}{2}\cdot\bm{b}^{(2)}(t)$$\mathbb{R}$-costs$M$-positions

Moreover, by taking this bounding map biggest possible, we may guarantee that
$\bm{\Phi}_{2}(t)$ has properties of the value function in the sense of
Bellman. In particular, it turns out that for every control $\Delta u\in
L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})$ the function

$t\longmapsto\big{\langle}\bm{b}^{(2)}(t,\Delta
u),\bm{\varphi}(t)\big{\rangle}-\bm{\Phi}_{2}(t)[\bm{b}^{(1)}(t,\Delta
u),\bm{b}^{(1)}(t,\Delta u)]$

is non-decreasing. By differentiation with respect to $t$ (we were able to
prove that $\bm{\Phi}_{2}(t)$ is differentiable almost everywhere) we get a
set of ODEs and an inequality that an abnormal minimizing SR geodesic should
satisfy. The above properties are stated as Thm 4.11, which is the main result
of this paper.

#### Discussion of the main result.

The most significant feature of Thm 4.11 is, in our opinion, the existence of
a pair of ODEs (4.8)–(4.9) that each minimizing SR geodesic should satisfy
(the well-known _Goh condition_ is a simple consequence of these equations –
see Remark 4.12). The first of these equations is of the form

(1.1) $\big{\langle}X_{i}\big{|}_{q(t)},\xi(t)\big{\rangle}+2\ a_{2}(t)\cdot
u_{i}(t)=0\ .$

It describes the controls $u_{i}(t)$ of the SR geodesic $q(t)$ in terms of
vector fields $X_{i}$ spanning the distribution $\mathcal{D}$, and two
functions $\xi(t)$, $a_{2}(t)$ related with the quadratic map
$\bm{\Phi}_{2}(t)$. Note that if $a_{2}(t)\neq 0$ we may explicitly derive
$u_{i}(t)$ as a function of other quantities. This observation is a basis of a
distinction of abnormal extremals into sub-classes of _2-normal_ and
_2-abnormal extremals_ proposed by us in Definition 4.13. The former class is
defined by the property that $a_{2}(t)\neq 0$ for every $t\in[t_{0},t_{1}]$,
hence the controls $u_{i}$ are solutions of algebraic equations. Geometrically
the 2-normal case corresponds to a situation when for every $t$ the graph of
$\bm{\Phi}_{2}(t)$ strictly separates the cost direction from the set $S_{t}$.
By contrast, in the 2-abnormal case the controls $u_{i}(t)$ are undetermined
at some times $t\in[t_{0},t_{1}]$ and the above separation is not strict.
Schematically the difference is depicted on the figure below.

$\mathbb{R}$$\mathbb{R}$$a_{2}(t)>0$$a_{2}(t)=0$In the 2-normal case (left),
the minimal 2-jet is separated from the cost direction $\mathbb{R}$,while in
the 2-abnormal case (right) there is no separation.

The presence of equation (1.1) is very important from the point of view of the
regularity problem of SR geodesics, at least in the 2-normal case. It is clear
that the regularity problem is related with the $t$-regularity of maps
$\xi(t)$ and $a_{2}(t)$ or, more generally, the whole quadratic map
$\bm{\Phi}_{2}(t)$. Unfortunately, in this paper we were only able to prove
that these objects are of bounded variation, which is not enough for our
purposes. However, I hope that other researchers, perhaps more experienced in
optimal control theory, could help improving the regularity results.

Let us note that the regularity of 2-abnormal extremals seems to be a much
more difficult task requiring, in my opinion, study of third- or higher-order
expansions of the extended end-point map.

#### Supplementary results.

The question whether the set $S_{t}$ is bounded is closely related with a
special case of a linear-quadratic optimal control problem (we speak about a
_characteristic optimal control problem_ of the SR trajectory $q(t)$). In our
main result – Thm 4.11 – we used only the existence of a bound (and the fact
that it is actually a graph of a quadratic map). An interesting problem on its
own is the question whether the boundary points on $S_{t}$ are realizable by
some controls from $L^{\infty}([t_{0},t],\mathbb{R}^{k})$, or to put it
differently, does the characteristic optimal control problem has solutions. In
general, this may not be the case as show examples studied by us in Ssec. 4.4.
However, when the solutions exists they can provide a lot of information about
the SR trajectory $q(t)$, in particular prove its regularity. We discuss this
topic in detail in Ssec. 4.5, providing both algebraic and differential (by
means of the Pontryagin Maximum Principle) criteria for the existence of
solutions of the characteristic optimal control problem.

In particular, we prove Lemma 4.22 in which we show that assumptions weaker
then these of [BMP20, Thm 1.2] imply that if the abnormal minimizing SR
geodesic $q(t)$ is a 2-normal extremal in the sense of Definition 4.13, it
must be piece-wise $C^{2}$ with a finite number of pieces.

#### Content of this paper.

In the preliminary Sec. 2 we state the sub-Riemannian geodesic problem and
reformulate it in a control-theoretic setting. We also give a brief
introduction to jets. In Sec. 3 we define the (extended) end-point map and its
second-order expansion, which we treat as a certain control system. Later in
Lem. 3.1 we introduce adapted coordinates which significantly simplifies this
system, and we make some initial observations about its properties. Sec. 4
contains main results of the paper. We begin by recalling the Agrachev-
Sarychev theory and connecting it with the control system related with the
second-order expansion of the end-point map. This leads us to a consideration
of the characteristic optimal control problem of the SR trajectory in Ssec.
4.2. In particular we prove Lemma 4.7 proving that after division of the
trajectory into a finite number of pieces, the cost function in the
characteristic optimal control problem is bounded on each piece. The latter
result is guarantees the existence of the the value function for that problem.
In Ssec. 4.3 we study basic properties of this function, which allow to
formulate or main result – Thm 4.11 – in Ssec. 4.4. Later in that subsection
we discuss consequences of our main result, introducing the notions of
2-normal and 2-abnormal extremals and studying two examples. Finally Ssec. 4.5
contains the discussion of (potential) solutions of the characteristic optimal
control problem and some consequences of their existence. We end by stating a
few hypothesis in Sec. 5.

##  2 Preliminaries

#### Regularity of curves.

Throughout this work we will talk about curves on smooth manifolds of various
classes of regularity. (Some emphasis will be put on improving the regularity
of a curve by means of a bootstrap technique.) At the bottom level we have
$L^{\infty}$-curves, i.e. _measurable and essentially bounded_. A step higher
are _ACB_ curves, i.e. _absolutely continuous with $L^{\infty}$ derivatives_.
The relation within these two is that if $\gamma(t)$ satisfies an ODE
$\dot{\gamma}(t)\overset{\text{a.e.}}{=}f(t)$, where $f(t)$ is of class
$L^{\infty}$ then $\gamma(t)$ is ACB. This resembles the standard relation
between $C^{r}$ and $C^{r+1}$ curves. Therefore it should not be very abusive
to the standard mathematical jargon to denote the class $L^{\infty}$ as
$C^{0_{-}}$, and the class ACB as $C^{1_{-}}$ (so that $C^{0_{-}+1}=C^{1_{-}}$
and also $C^{1_{-}}\subset C^{0}$).

###  2.1 The sub-Riemannian geodesic problem

#### The sub-Riemannian geodesic problem.

We shall be working within the following geometric setting. Let $M$ be a
smooth $n$-dimensional manifold and let $\mathcal{D}\subset\mathrm{T}M$ be a
smooth distribution of rank $k$ on $M$. By
$g(\cdot,\cdot):\mathcal{D}\times_{M}\mathcal{D}\rightarrow\mathbb{R}$ we
shall denote a positively-defined symmetric bilinear product on $\mathcal{D}$
(called a _sub-Riemannian metric_). The triple $(M,\mathcal{D},g)$ constitutes
a _sub-Riemannian_ _structure_ (_SR structure_ , in short). Given a pair of
points $q_{0},q_{1}\in M$ and an interval $[t_{0},t_{1}]\subset\mathbb{R}$ we
consider the following _SR geodesic problem_ :

(P) Find an absolutely continuous curve $q:[t_{0},t_{1}]\rightarrow M$
satisfying the following conditions:

  1. (P1)

$q$ is almost everywhere tangent to $\mathcal{D}$, i.e.,
$\dot{q}(t)\in\mathcal{D}_{q(t)}$ for a.e. $t\in[t_{0},t_{1}]$.

  2. (P2)

$q$ joins $q_{0}$ and $q_{1}$, that is, $q(t_{0})=q_{0}$ and $q(t_{1})=q_{1}$.

  3. (P3)

in the set of all curves satisfying conditions (P1) and (P2), curve $q$
realizes the minimum of the _energy functional_

$q\longmapsto\mathcal{E}[q]:=\frac{1}{2}\int_{t_{0}}^{t_{1}}g(\dot{q}(\tau),\dot{q}(\tau))\
\mathrm{d}\tau\ .$

Ii is well-known [Mon06] that under the assumption that $\mathcal{D}$ is
bracket-generating, and that $M$ is complete, problem (P) has a solution for
every $q_{0},q_{1}\in M$ and every $[t_{0},t_{1}]\subset\mathbb{R}$. A
solution of the problem (P) is called a _minimizing SR geodesic_.

It follows easily from the Cauchy-Schwarz inequality that the minimizing SR
geodesic has to be normalized, i.e., $g(\dot{q}(t),\dot{q}(t))$ is constant on
$[t_{0},t_{1}]$. Therefore we may actually look for the solutions of (P) in
the class of ACB curves. In the forthcoming control-theoretic formulation of
the SR geodesic problem, this simple observation will allow us to consider the
smaller class of $L^{\infty}$-controls, instead of all $L^{1}$-controls.

#### Control-theoretic formulation.

In a local version, problem (P) can be formulated as an optimal control
problem. Indeed, choose $k$ smooth, linearly independent vector fields
$X_{1},\ldots,X_{k}$ spanning locally the distribution $\mathcal{D}$. Note
that, by applying the Gram-Schmidt orthogonalization, we may assume that the
fields $\\{X_{i}\\}_{i=1,\ldots,k}$ form an orthonormal basis of
$\mathcal{D}$, i.e., $g(X_{i},X_{j})=\delta_{ij}$. Now every ACB curve $q(t)$
almost everywhere tangent to $\mathcal{D}$ may be regarded as a solution of

($\Sigma$) $\dot{q}(t)=\sum_{i=1}^{k}u^{i}(t)\cdot X_{i}\Big{|}_{q(t)}\ ,$

where $t\mapsto u(t)=(u^{1}(t),\ldots,u^{k}(t))\in\mathbb{R}^{k}$ is an
$L^{\infty}$-map, and equation ($\Sigma$) is regarded in the sense of
Caratheodory, i.e. the equality holds for almost every $t\in[t_{0},t]$. We
shall call such a $q$ a _SR trajectory_ corresponding to the _control_ $u$.

Note that under the assumption that the fields $X_{i}$ are normalized as
$g(X_{i},X_{j})=\delta_{ij}$, the energy of such a trajectory is simply
$E^{[t_{0},t_{1}]}[u]$, where we define

(E)
$E^{[t_{0},t]}[u]:=\frac{1}{2}\int_{t_{0}}^{t}\sum_{i=1}^{k}\left(u^{i}(\tau)\right)^{2}\mathrm{d}\tau=\frac{1}{2}\|u\|_{L^{2}([t_{0},t],\mathbb{R}^{k})}^{2}\
.$

From now on we shall study the SR geodesic problem (P) within such a local
framework, i.e. we take $M\overset{\text{loc.}}{=}\mathbb{R}^{n}$, and choose
$\\{X_{i}\\}_{i=1,\ldots,k}$ a global $g$-orthonormal basis of smooth vector
fields spanning $\mathcal{D}$.

######  Remark 2.1 (Notation in the extended configuration space).

It is convenient to think of both, the trajectory $q(t)$, and the energy
$E^{[t_{0},t]}$ at the same time, by introducing _the extended configuration
space_ $\bm{M}:=M\times\mathbb{R}$. For a control $u\in
L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})$ the _extended trajectory_ is defined
as $\bm{q}(t):=(q(t),E^{[t_{0},t]}[u])$, where $q(t)$ satisfies ($\Sigma$).
Similar objects, consequently denoted by bold letters, will be considered
thorough the paper.

###  2.2 On jets

In the paper we speak about second-order optimality conditions in SR geometry.
The term _second-order_ means that we will consider the second-order Taylor
expansions of various objects. Geometrically, we rather speak about _2-jets_.
Below we recall some basic definitions and constructions from jet theory
[Sau89, KMS93].

#### Jets of curves.

We say that two $C^{2}$-curves
$\gamma,\widetilde{\gamma}:(-\varepsilon,\varepsilon)\longrightarrow M$ are
_tangent of order two_ at $p=\gamma(0)=\widetilde{\gamma}(0)$ if in some (and
thus any) local coordinate system around $p$ on $M$ we have

$\gamma^{\prime}(0)=\widetilde{\gamma}^{\prime}(0)\quad\text{and}\quad\gamma^{\prime\prime}(0)=\widetilde{\gamma}^{\prime\prime}(0)\
.$

In other words, maps $s\mapsto\gamma(s)$ and $s\mapsto\widetilde{\gamma}(s)$
have the same order-two Taylor expansions at $s=0$. The relation $\sim_{2}$ of
being tangent of order two is an equivalence relation and is equivalence
classes $[\gamma]_{\sim_{2}}=[\widetilde{\gamma}]_{\sim_{2}}$ are called
_2-jets_ at $p$. We denote the set of all 2-jets at $p$ by
$\mathrm{T}^{2}_{p}M$, and call it the _second tangent space of $M$ at $p$_.
In fact the collection $\mathrm{T}^{2}M:=\bigcup_{p\in M}\mathrm{T}^{2}_{p}M$
is a locally trivial bundle over $M$. We call it the _second tangent bundle of
$M$_. It is an example of a _graded bundle_ in the sense of [GR12], with the
canonical multiplicative $\mathbb{R}$-action defined on representatives by
$\lambda\mapsto\left(s\mapsto\gamma(\lambda\cdot s)\right)$.

Given a curve $\gamma:(-\varepsilon,\varepsilon)\rightarrow M$ and its (local)
second-order Taylor expansion

$\gamma(s)=p+s\cdot\beta^{(1)}+{s^{2}}\cdot\beta^{(2)}+o(s^{2})\ ,$

we may thus identify its 2-jet with a pair
$(\beta^{(1)},\beta^{(2)})\simeq[\gamma]_{\sim_{2}}$. Although one should
remember, that only the first term $\beta^{(1)}$ is a geometric object,
representing a tangent vector at $p$, while the term $\beta^{(2)}$ on its own
(without the presence of a particular $\beta^{(1)}$) has no intrinsic
geometric meaning. In a particular situation when $M=V$ is a vector space we
may, however, canonically identify $\mathrm{T}^{2}_{p}V\simeq V\times V$, by
considering the above Taylor expansion in any linear coordinates on $V$. Thus
for a vector space, the bare term $\beta^{(2)}$ has a geometric interpretation
of an element of $V$.

#### Jets of maps.

Let $F:M\rightarrow N$ be a smooth map between manifolds. For a $C^{2}$-curve
$\gamma:(-\varepsilon,\varepsilon)\rightarrow M$ passing through
$\gamma(0)=p\in M$ consider a (local) second-order Taylor expansion

$F(\gamma(s))=F(p)+s\cdot b^{(1)}+s^{2}\cdot b^{(2)}+o(s^{2})\ .$

Then the pair $(b^{(1)},b^{(2)})$ depends only on the 2-jet
$[\gamma]_{\sim_{2}}$. Thus we may define the 2-jet of map $F$

$\mathrm{T}^{2}F:\mathrm{T}^{2}M\longrightarrow\mathrm{T}^{2}N\qquad\text{via}\qquad\mathrm{T}^{2}_{p}M\ni[\gamma]_{\sim_{2}}\longmapsto[F(\gamma)]_{\sim_{2}}\in\mathrm{T}^{2}_{F(p)}N\
.$

$\mathrm{T}^{2}F$ is a bundle map over $F:M\longrightarrow N$.

#### Pairings on jets.

As a particular example of the above consider a smooth function
$f:M\longrightarrow\mathbb{R}$ and a $C^{2}$-curve
$\gamma:(-\varepsilon,\varepsilon)\longrightarrow M$ passing through
$\gamma(0)=p\in M$. The order-two Taylor expansion of the composition
$f(\gamma):(-\varepsilon,\varepsilon)\longrightarrow\mathbb{R}$ reads as

$f(\gamma(s))=f(p)+s\cdot\operatorname{D}_{p}f[\gamma^{\prime}(0)]+s^{2}\cdot\left\\{\operatorname{D}_{p}f[\gamma^{\prime\prime}(0)]+\operatorname{D}^{2}_{p}f[\gamma^{\prime}(0),\gamma^{\prime}(0)]\right\\}+o(s^{2})\
,$

and it depends only on the 2-jet $[\gamma]_{\sim_{2}}$. Further, the resulting
2-jet $[f(\gamma)]_{\sim_{2}}$ is valued in the second tangent space of a
vector space
$\mathrm{T}^{2}_{f(p)}\mathbb{R}\simeq\mathbb{R}\times\mathbb{R}$, and so the
term of degree two

(2.1)
$\operatorname{D}_{p}f[\gamma^{\prime\prime}(0)]+\operatorname{D}^{2}_{p}f[\gamma^{\prime}(0),\gamma^{\prime}(0)]$

has an intrinsic geometric meaning.

Note that for a fixed 2-jet $[f(\gamma)]_{\sim_{2}}$, the value of (2.1)
depends only on the first and second derivatives $\operatorname{D}_{p}f$ and
$\operatorname{D}^{2}_{p}f$, i.e. the second-order Taylor expansion of $f$ at
$p$. Since we may construct smooth functions with arbitrary derivatives at a
point, we may define via

(2.2)
$\big{\langle}(\phi^{1},\phi^{2}),(b^{(1)},b^{(2)})\big{\rangle}:=\big{\langle}\phi^{1},b^{(2)}\big{\rangle}+\phi^{2}[b^{(1)},b^{(1)}]$

a natural (and geometrically meaningful) pairing between a 2-jet of a curve
$(b^{(1)},b^{(2)})\in\mathrm{T}^{2}_{p}M$ and a pair $(\phi,\Phi)$ consisting
of a linear map $\phi:\mathrm{T}_{p}M\longrightarrow\mathbb{R}$ and a
symmetric bi-linear map
$\Phi:\mathrm{T}_{p}M\times\mathrm{T}_{p}M\longrightarrow\mathbb{R}$.

######  Remark 2.2.

It is worth to mention that the pairs $(\phi,\Phi)$ as above can be identified
with the elements of $\mathrm{T}^{2\ast}_{p}M$, i.e. 2-covectors at $p\in M$
in the sense of Tulczyjew [Tul]. We define the latter as equivalence classes
of smooth functions on $M$, saying that
$f,\widetilde{f}:M\rightarrow\mathbb{R}$ are equivalent if and only if
$f(\gamma(s))$ and $\widetilde{f}(\gamma(s))$ have the same second-order
Taylor expansions at $s=0$ for any smooth curve
$\gamma:(-\varepsilon,\varepsilon)\rightarrow M$ passing through
$p=\gamma(0)$. The pairing (2.2) can be thus interpreted as a map
$\mathrm{T}^{2\ast}_{p}M\times\mathrm{T}^{2}_{p}M\longrightarrow\mathbb{R}$.
We remark that it has an elegant description in terms of the Weyl algebra of
dual numbers [KMS93].

##  3 The end-point map

###  3.1 The end-point map and its second-order expansion

#### The end-point map and its second expansion.

Denote by

$\mathrm{End}^{[t_{0},t]}_{q_{0}}:L^{\infty}([t_{0},t],\mathbb{R}^{k})\longrightarrow
M$

the _end-point map_ of the control system ($\Sigma$) at time $t$ and with the
initial condition $q_{0}$ at $t=t_{0}$. That is, for a control $u\in
L^{\infty}([t_{0},t],\mathbb{R}^{k})$, $\mathrm{End}^{[t_{0},t]}_{q_{0}}[u]$
is a solution $q(t)$ of ($\Sigma$) satisfying $q(t_{0})=q_{0}$.

We would like to study the following second-order expansion of this map
(written in some local coordinate system)

(3.1) $\mathrm{End}^{[t_{0},t]}_{q_{0}}[u+s\cdot\Delta
u]=\mathrm{End}^{[t_{0},t]}_{q_{0}}[u]+s\cdot b^{(1)}(t,\Delta u)+s^{2}\cdot
b^{(2)}(t,\Delta u)+o(s^{2})\ ,$

where $\Delta u\in\mathrm{T}_{u}L^{\infty}([t_{0},t],\mathbb{R}^{k})\simeq
L^{\infty}([t_{0},t],\mathbb{R}^{k})$ is a control, and $s$ is a real
parameter. Of course $b^{(1)}(t,\Delta
u)=\operatorname{D}_{u}\mathrm{End}^{[t_{0},t]}_{q_{0}}[\Delta u]$, and one is
tempted to write $b^{(2)}(t,\Delta u)=\frac{1}{2}\
\operatorname{D}^{2}_{u}\mathrm{End}^{[t_{0},t]}_{q_{0}}[\Delta u,\Delta u]$.
The last equality makes, however, no sense as in general the second derivative
of a manifold-valued map is not a well-defined geometric object – cf. [AS04,
Chap. 20]. Expansion (3.1) has a geometric meaning though – by our
considerations in Ssec. 2.2 it describes the second tangent map
$\mathrm{T}^{2}\mathrm{End}^{[t_{0},t]}_{q_{0}}$ evaluated on a 2-jet
$[s\mapsto u+s\cdot\Delta u+s^{2}\cdot 0]_{\sim
2}\in\mathrm{T}^{2}_{u}L^{\infty}([t_{0},t],\mathbb{R}^{k})\simeq
L^{\infty}([t_{0},t],\mathbb{R}^{k})\times
L^{\infty}([t_{0},t],\mathbb{R}^{k})$. Thus a pair $(b^{(1)}(t,\Delta
u),b^{(2)}(t,\Delta u))$ is a local representative of some element in
$\mathrm{T}^{2}_{q(t)}M$.

To calculate $b^{(1)}(t,\Delta u)$ and $b^{(2)}(t,\Delta u)$, denote by
$q_{s}(t)$ a solution of the following ODE

(3.2) $\dot{q_{s}}(t)=\sum_{i=1}^{k}\left(u^{i}(t)+s\cdot\Delta
u_{i}(t)\right)\cdot X_{i}\Big{|}_{q_{s}(t)}\qquad\text{with}\qquad
q_{s}(t_{0})=q_{0}\ .$

Obviously $q_{s}(t)=\mathrm{End}^{[t_{0},t]}_{q_{0}}[u+s\cdot\Delta u]$. What
is more, (3.2) is an ODE in the sense of Caratheodory, depending smoothly on
the parameter $s$. As such it has a solution which also smoothly depends on
$s$ – see [BP04] – and, moreover, equations for $b^{(1)}(t,\Delta
u)=\partial_{s}\big{|}_{s=0}q_{s}(t)$ and $b^{(2)}(t,\Delta
u)=\frac{1}{2}\partial_{s}^{2}\big{|}_{s=0}q_{s}(t)$ are obtained by
differentiating (3.2) with respect to $s$, leading to (for notation simplicity
below we omit the dependence of $b^{(1)}(t,\Delta u)$ and $b^{(2)}(t,\Delta
u)$ on $\Delta u$)

(3.3)
$\dot{b^{(1)}}(t)=\sum_{i=1}^{k}u_{i}(t)\cdot\operatorname{D}_{q(t)}X_{i}\left[b^{(1)}(t)\right]+\sum_{i=1}^{k}\Delta
u_{i}(t)\cdot X_{i}\Big{|}_{q(t)}$

with the initial condition $b^{(1)}(t_{0})=0$; and

(3.4)
$\dot{b^{(2)}}(t)=\sum_{i=1}^{k}u_{i}(t)\left\\{\operatorname{D}_{q(t)}X_{i}\left[b^{(2)}(t)\right]+\frac{1}{2}\operatorname{D}^{2}_{q(t)}X_{i}\left[b^{(1)}(t),b^{(1)}(t)\right]\right\\}+2\sum_{i=1}^{k}\Delta
u_{i}(t)\cdot\operatorname{D}_{q(t)}X_{i}\left[b^{(1)}(t)\right]$

where $b^{(1)}(t)$ is as above, and the initial condition is
$b^{(2)}(t_{0})=0$.

#### The extended end-point map.

Now we would like to add the energy (E) to the picture. We define the
_extended end-point map_

${\mathbf{End}}^{[t_{0},t]}_{q_{0}}:L^{\infty}([t_{0},t],\mathbb{R}^{k})\longrightarrow
M\times\mathbb{R}\ ,$

simply as
${\mathbf{End}}^{[t_{0},t]}_{q_{0}}[u]:=(\mathrm{End}^{[t_{0},t]}_{q_{0}}[u],E^{[t_{0},t]}[u])$.
For any $\Delta u\in L^{\infty}([t_{0},t],\mathbb{R}^{k})$ the second
expansion of the energy term is just

(3.5) $E^{[t_{0},t]}[u+s\cdot\Delta u]=E^{[t_{0},t]}(u)+s\cdot
c^{(1)}(t,\Delta u)+s^{2}\cdot c^{(2)}(t,\Delta u)\ ,$

where $c^{(1)}(t,\Delta u)=\int_{t_{0}}^{t}\sum_{i}u_{i}(\tau)\cdot\Delta
u_{i}(\tau)\ \mathrm{d}\tau$ and $c^{(2)}(t,\Delta
u)=\int_{t_{0}}^{t}\sum_{i}\Delta u_{i}(\tau)^{2}\ \mathrm{d}\tau$. Or, in
differential terms:

(3.6) $\begin{split}\dot{c^{(1)}}(t)&=\sum_{i=1}^{k}u_{i}(t)\cdot\Delta
u_{i}(t)\\\ \dot{c^{(2)}}(t)&=\sum_{i=1}^{k}\Delta u_{i}(t)^{2}\end{split}$

with the initial conditions $c^{(1)}(t_{0})=0$ and $c^{(2)}(t_{0})=0$.

###  3.2 Adapted coordinates and the characteristic control system

#### Construction of adapted coordinates.

The set of ODEs (3.3)–(3.4) describes the 2-jet of the end-point map
$\mathrm{End}^{[t_{0},t]}_{q_{0}}$ related with a tangent element $\Delta
u\in\mathrm{T}_{u}L^{\infty}([t_{0},t],\mathbb{R}^{k})\simeq
L^{\infty}([t_{0},t],\mathbb{R}^{k})$. Fixing $u\in
L^{\infty}([t_{0},t],\mathbb{R}^{k})$ we may view these equations as a non-
autonomous control system, with $\Delta u$ playing the role of a control.
Recently in [JS21], together with Bartek Sikorski, we described a method to
simplify this control system by a construction of _adapted coordinates_. The
underlying idea is actually quite simple. For a fixed control $u$, equation
($\Sigma$) may be seen as a trajectory of a time-dependent vector field
$X_{u}(t,q)=\sum_{i=1}^{k}u_{i}(t)\cdot X_{i}(q)$. The (time-dependent) flow
$X_{u}^{\bullet\bullet}$ of this field – see [JR16a, Sec. 2] for a detailed
discussion of this concept – naturally acts on $M$ by local diffeomorphisms,
and thus its derivatives act on $\mathrm{T}M$, $\mathrm{T}^{2}M$, etc. In
particular, the second tangent map $\mathrm{T}^{2}X_{u}^{tt_{0}}$ is an
isomorphism between the fibers $\mathrm{T}^{2}_{q_{0}}M$ and
$\mathrm{T}^{2}_{q(t)}M$. Therefore we may use the inverse of
$\mathrm{T}^{2}X_{u}^{tt_{0}}$ to identify the fibre $\mathrm{T}^{2}_{q(t)}M$
(where the 2-jet represented by a pair $(b^{(1)}(t,\Delta u),b^{(2)}(t,\Delta
u))$ belongs) with $\mathrm{T}^{2}_{q_{0}}M$. Further, after identifying
$\mathrm{T}^{2}_{q_{0}}M$ with $\mathbb{R}^{n}\times\mathbb{R}^{n}$, each
fibre $\mathrm{T}^{2}_{q(t)}M$ becomes isomorphic with
$\mathbb{R}^{n}\times\mathbb{R}^{n}$ in a way naturally _adapted_ to the
evolution equation ($\Sigma$). These identifications
$\mathrm{T}^{2}_{q(t)}M\simeq\mathbb{R}^{n}\times\mathbb{R}^{n}$ are precisely
the _adapted coordinates_. A direct description reads as follows

######  Lemma 3.1 ([JS21]).

Let $u\in L^{\infty}([t_{0},t],\mathbb{R}^{k})$ be a control, $q(t)$ the
related trajectory of ($\Sigma$), and choose a basis
$(\psi_{a})_{a=1.2,\ldots,n}$ of the cotangent space
$\mathrm{T}^{\ast}_{q_{0}}M$. Define

  * •

a 1-parameter family of linear maps
$\Phi^{(1)}(t)=\left(\Phi^{(1)}_{a}(t)\right)_{a=1,2,\ldots,n}:\mathrm{T}_{q(t)}M\longrightarrow\mathbb{R}^{n}$
by setting

$\displaystyle{\dot{\Phi}}^{(1)}_{a}(t)[b]+\sum_{i}u_{i}(t)\cdot\Phi^{(1)}_{a}\left[\operatorname{D}_{q(t)}X_{i}[b]\right]=0\qquad\text{for
every $b\in\mathrm{T}_{q(t)}M$}$
$\displaystyle\Phi^{(1)}_{a}(t_{0})=\psi_{a},$

  * •

and a 1-parameter family of symmetric bi-linear maps
$\Phi^{(2)}(t)=\left(\Phi^{(2)}_{a}(t)\right)_{a=1,2,\ldots,n}:\mathrm{T}_{q(t)}M\times\mathrm{T}_{q(t)}M\longrightarrow\mathbb{R}^{n}$
by setting

$\displaystyle{\dot{\Phi}}^{(2)}_{a}(t)[b,b]+\sum_{i}u_{i}(t)\cdot\left\\{\Phi^{(1)}_{a}(t)\left[\operatorname{D}^{2}_{q(t)}X_{i}[b,b]\right]+2\Phi^{(2)}_{a}(t)\left[\operatorname{D}_{q(t)}X_{i}[b],b\right]\right\\}=0\qquad\text{for
every $b\in\mathrm{T}_{q(t)}M$}$ $\displaystyle\Phi^{(2)}_{a}(t_{0})=0\ .$

Then $\Phi^{(1)}(t)$ is reversible for every $t\in[t_{0},t_{1}]$. It follows
that for every $t\in[t_{0},t_{1}]$ the assignment
$\mathrm{T}^{2}_{q(t)}M\ni(b^{(1)}(t),b^{(2)}(t))\longmapsto(q^{(1)}(t),q^{(2)}(t))\in\mathbb{R}^{n}\times\mathbb{R}^{n}$,
were $q^{(1)}(t)=(q^{(1)}_{a}(t))_{a=1,2,\ldots,n}\in\mathbb{R}^{n}$ and
$q^{(2)}(t)=(q^{(2)}_{a}(t))_{a=1,2,\ldots,n}\in\mathbb{R}^{n}$ are defined by

(3.7) $\displaystyle q^{(1)}_{a}(t):=$
$\displaystyle\Phi^{(1)}_{a}(t)\left[b^{(1)}(t)\right]$ (3.8) $\displaystyle
q^{(2)}_{a}(t):=$
$\displaystyle\Phi^{(1)}_{a}(t)\left[b^{(2)}(t)\right]+\frac{1}{2!}\Phi^{(2)}_{a}(t)[b^{(1)}(t),b^{(1)}(t)]$

is an isomorphism.

Moreover, if $(b^{(1)}(t),b^{(2)}(t))$ satisfies (3.3)–(3.4) then the above
$(q^{(1)}(t),q^{(2)}(t))$ are subject to the following evolution equations

(3.9) $\displaystyle{\dot{q}}^{(1)}(t)$ $\displaystyle=\sum_{i}\Delta
u_{i}(t)\cdot Y_{i}^{(1)}(t)$ (3.10) $\displaystyle{\dot{q}}^{(2)}(t)$
$\displaystyle=\sum_{i}\Delta u_{i}(t)\cdot Y_{i}^{(2)}(t,q^{(1)})\ ,$

where $Y_{i}^{(1)}(t):=\Phi^{(1)}(t)\left[X_{i}\big{|}_{q(t)}\right]$, and
after denoting the inverse of $\Phi^{(1)}(t)$ by $A(t)$, we have
$Y_{i}^{(2)}(t,q^{(1)}):=\left\\{\Phi^{(1)}(t)\left[\operatorname{D}_{q(t)}X_{i}[A(t)q^{(1)}]\right]+\Phi^{(2)}(t)\left[X_{i},A(t)q^{(1)}\right]\right\\}$.

Finally, if the control $u$ is of class $C^{r}$, with
$r=0_{-},1_{-},1,2,3,\ldots$, then $\Phi^{(1)}(t)$, the inverse of
$\Phi^{(1)}(t)$, and $\Phi^{(2)}(t)$ are of class $C^{r+1}$ with respect to
$t$.

The above result can be checked by a direct calculation. The
$C^{r+1}$-regularity follows easily from the evolution equations for
$\Phi^{(1)}(t)$ and $\Phi^{(2)}(t)$. For details and discussion, we refer to
the original publication [JS21].

#### The characteristic control system and its reachable sets.

Since maps $\Phi^{(1)}(t)$ constructed in Lemma 3.1 are reversible, a pair
$(q^{(1)}(t),c^{(1)}(t))$ contains the same information as a pair
$(b^{(1)}(t),c^{(1)}(t))$, describing the first term of the 2-jet expansion of
the extended end-point map ${\mathbf{End}}^{[t_{0},t]}_{q_{0}}$. Thanks to the
simple form of equation (3.7) the evolution of the former pair is, however,
much easier to analyse. Therefore to understand the geometry of
${\mathbf{End}}^{[t_{0},t]}_{q_{0}}$, we shall now study the pairs
$(q^{(1)}(t),c^{(1)}(t))\in\mathbb{R}^{n}\times\mathbb{R}$. In accordance with
Rem. 2.1 denote $\bm{q}^{(1)}(t):=(q^{(1)}(t),c^{(1)}(t))$ and
$\bm{0}:=(0_{n},0)\in\mathbb{R}^{n}\times\mathbb{R}$.

######  Definition 3.2.

By a _characteristic control system_ of the trajectory $q(t)$, with
$t\in[t_{0},t_{1}]$, we will understand the following control system in
$\mathbb{R}^{n+1}\ni\bm{q}^{(1)}=(q^{(1)},c^{(1)})$:

($\Lambda_{1}$) $\begin{split}{\dot{q}}^{(1)}(t)&=\sum_{i}\Delta u_{i}(t)\cdot
Y_{i}^{(1)}(t)\\\ \dot{c^{(1)}}(t)&=\sum_{i=1}^{k}\Delta u_{i}(t)\cdot
u_{i}(t)\end{split}$

Here $Y_{i}^{(1)}(t)=\Phi^{(1)}(t)\left[X_{i}\big{|}_{q(t)}\right]$, and we
treat $\Delta u\in L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})$ as a control. By
$\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},\tau)$ we shall denote _time- $\tau$
reachable set of the system ($\Lambda_{1}$)_ for the initial conditions
$\bm{q}^{(1)}(t_{0})=\bm{0}$. That is

$\displaystyle\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},\tau):=\\{\bm{q}^{(1)}(\tau)\
|\ $ $\displaystyle\text{$\bm{q}^{(1)}(t)$ satisfy \eqref{eqn:CS_deg_1}}\text{
for some $\Delta u\in L^{\infty}([t_{0},\tau],\mathbb{R}^{k})$ with
$\bm{q}^{(1)}(t_{0})=\bm{0}$.}\\}$

######  Proposition 3.3.

The reachable sets $\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},\tau)$ have the
following properties:

  1. (i)

for each $\tau\in(t_{0},t_{1}]$, the set
$\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},\tau)$ is a linear subspace of
$\mathbb{R}^{n+1}$;

  2. (ii)

if $\tau<\tau^{\prime}$ then
$\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},\tau)\subseteq\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},\tau^{\prime})$;

  3. (iii)

there exists $t_{1}^{\prime}\in(t_{0},t_{1})$ such that
$\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},\tau)=\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},t_{1})$
for every $\tau\in(t_{1}^{\prime},t_{1}]$.

###### Proof.

The first point is obvious as the system is linear. So is the second, since
each control $u$ defined on $[t_{0},\tau]$ can be extended to a control on
$[t_{0},\tau^{\prime}]$ by setting $u(t)=0$ on $(\tau,\tau^{\prime}]$.

The third point follows from (ii) and the fact that the monotonous integer-
valued function $\tau\mapsto\dim\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},\tau)$
is left semi-continuous. Indeed, assume that for some
$\tau_{0}\in[t_{0},t_{1}]$ we have
$\dim\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},\tau_{0})>\dim\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},\tau)$
for all $\tau<\tau_{0}$. If this is the case then there exist a covector
$\bm{\psi}\in(\mathbb{R}^{n+1})^{\ast}$ annihilating
$\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},\tau)$ for every $\tau<t_{0}$, but
not-vanishing on $\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},\tau_{0})$. If so,
choose a control $\Delta u_{0}$ such that

$\big{\langle}\bm{\psi},\bm{q}^{(1)}(\tau_{0},\Delta u_{0})\big{\rangle}\neq
0\ .$

The map $\tau\mapsto\big{\langle}\bm{\psi},\bm{q}^{(1)}(\tau,\Delta
u_{0})\big{\rangle}$ is, however, defined by an integral of a MB function over
$[t_{0},\tau]$ and thus is continuous with respect to $\tau$, so necessarily
$\big{\langle}\bm{\psi},\bm{q}^{(1)}(\tau,\Delta u_{0})\big{\rangle}\neq 0$
for some $\tau<\tau_{0}$. On the other hand, $\bm{q}^{(1)}(\tau,\Delta
u_{0})\in\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},\tau)\subset\ker\bm{\psi}$.
The contradiction ends the proof. ∎

The dimension of the reachable set $\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},t)$
grows along the trajectory
$q(t)$.$q(t)$$\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},t)$

Choose $\tau\in[t_{0},t_{1}]$. We will say that a vector
$\bm{v}\in\mathbb{R}^{n+1}$ is _controllable to zero from time $\tau$_ if
there exists a trajectory $\bm{q}^{(1)}(t)$ of the control system
($\Lambda_{1}$) with the initial condition $\bm{q}^{(1)}(\tau)=\bm{v}$ such
that $\bm{q}^{(1)}(t_{1})=\bm{0}$. It turns out that after cutting the
trajectory to a finite number of pieces all elements of the reachable sets are
controllable to zero.

######  Lemma 3.4.

There exist at most $n+1$-points
$t_{0}=\tau_{0}<\tau_{1}<\ldots<\tau_{s}=t_{1}$ such that for each
$\tau\in[\tau_{i},\tau_{i+1})$ all points of the reachable set
$\bm{\mathcal{R}}_{(\Lambda_{1})}(\tau_{i},\tau)$ are controllable to zero
from time $\tau$.

###### Proof.

Denote by $(\overline{\Lambda_{1}})$ the control system ($\Lambda_{1}$) with
reversed time (i.e we control from $t_{1}$ to $t_{0}$), and let
$\bm{\mathcal{R}}_{(\overline{\Lambda_{1}})}(t_{1},\tau)$ denote its time
$\tau$-reachable set. Obviously, a vector $\bm{v}\in\mathbb{R}^{n+1}$ is
controllable to zero from time $\tau\in[t_{0},t_{1}]$ if and only if
$\bm{v}\in\bm{\mathcal{R}}_{(\overline{\Lambda_{1}})}(t_{1},\tau)$. Observe
that
$\bm{\mathcal{R}}_{(\overline{\Lambda_{1}})}(t_{1},t_{0})=\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},t_{1})$
as both linear spaces are defined by the same integrals differing only by the
direction of integration. System $(\overline{\Lambda_{1}})$ has properties
analogous to those of ($\Lambda_{1}$) stated in Prop. 3.3. Therefore there
exists a maximal $\tau_{1}>t_{0}$ such that for every
$\tau\in[t_{0},\tau_{1})$

$\bm{\mathcal{R}}_{(\overline{\Lambda_{1}})}(t_{1},\tau)=\bm{\mathcal{R}}_{(\overline{\Lambda_{1}})}(t_{1},t_{0})=\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},t_{1})\overset{\text{Prop.~{}\ref{prop:reach_set}
\eqref{point:2}}}{\supset}\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},\tau)\ ,$

Proving the assertion for $i=0$. Now repeat the argument to the system on the
new interval $[\tau_{1},t_{1}]$. Note that by choosing $\tau_{1}$ maximal we
have
$\dim\bm{\mathcal{R}}_{(\Lambda_{1})}(\tau_{1},t_{1})=\dim\bm{\mathcal{R}}_{(\overline{\Lambda_{1}})}(t_{1},\tau_{1})<\dim\bm{\mathcal{R}}_{(\overline{\Lambda_{1}})}(t_{1},t_{0})=\dim\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},t_{1})$,
so the procedure will end in at most $n$ steps when the dimension of the
reachable set falls to zero. ∎

#### Normal and abnormal extremals

Observe that by construction, for each $t\in[t_{0},t_{1}]$, the reachable set
$\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},t)$ equals, up to the linear
isomorphism
$(\Phi^{(1)}(t),\operatorname{id}_{\mathbb{R}}):\mathrm{T}_{q(t)}M\times\mathrm{T}_{E^{[t_{0},t]}(u)}\mathbb{R}\rightarrow\mathbb{R}^{n}\times\mathbb{R}$,
to the image of the differential of the extended end-point map
$\operatorname{Im}\operatorname{D}_{u}{\mathbf{End}}^{[t_{0},t]}_{q_{0}}$.
Therefore the standard classification of SR trajectories by means of of the
properties of he latter space can be easily translated to the language of the
characteristic control system ($\Lambda_{1}$).

######  Definition 3.5.

Consider a sub-Riemannian trajectory $q(t)$ with $t\in[t_{0},t_{1}]$ and its
characteristic control system ($\Lambda_{1}$). Curve $q(t)$ is called an
_extremal_ if the reachable set
$\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},t_{1})$ is a proper subspace of
$\mathbb{R}^{n+1}$. An extremal is called _normal_ if the reachable set
$\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},t_{1})$ does not contain the subspace
$\\{0_{n}\\}\times\mathbb{R}$. If the reachable set
$\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},t_{1})$ is contained in a space
$V\times\mathbb{R}$, where $V\subset\mathbb{R}^{n}$ is some proper subspace,
$q(t)$ is called _abnormal_. We will say an abnormal extremal has _co-rank
$r$_ if $\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},t_{1})$ is a co-rank
$r$-subspace of $\mathbb{R}^{n+1}$.

_Strictly abnormal_ extremals are abnormal extremals which are not normal,
that is $\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},t_{1})=V\times\mathbb{R}$ for
some proper subspace $V\subset\mathbb{R}^{n}$.

It is instructive to see that the notions from Definition 3.5 give the well-
known first order optimality condition. For example if $q(t)$ is an abnormal
extremal then there exists a non-zero covector
$\varphi_{0}\in(\mathbb{R}^{n})^{\ast}$ such that
$\varphi_{0}+0\cdot\mathrm{d}r\in(\mathbb{R}^{n+1})^{\ast}$ annihilates the
reachable set $\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},t_{1})$. In more
concrete terms $\big{\langle}q^{(1)}(t_{1},\Delta
u),\varphi_{0}\big{\rangle}=0$ for every $\Delta u\in
L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})$. By (3.9) this gives us

$\displaystyle 0=\big{\langle}\int_{t_{0}}^{t_{1}}\sum_{i}\Delta u_{i}(t)\cdot
Y_{i}^{(1)}(t)\
\mathrm{d}t,\varphi_{0}\big{\rangle}=\int_{t_{0}}^{t_{1}}\sum_{i}\Delta
u_{i}(t)\cdot\big{\langle}Y_{i}^{(1)}(t),\varphi_{0}\big{\rangle}\
\mathrm{d}t$

and so for every $t\in[t_{0},t_{1}]$ and every $i=1,2\ldots,k$

$\displaystyle
0=\big{\langle}Y_{i}^{(1)}(t),\varphi_{0}\big{\rangle}\overset{\text{Lem.~{}\ref{lem:adapted}}}{=}\big{\langle}\Phi^{(1)}(t)\left[X_{i}(q(t))\right],\varphi_{0}\big{\rangle}=\big{\langle}X_{i}(q(t)),\Phi^{(1)}(t)^{\ast}\varphi_{0}\big{\rangle}\
.$

As we see
$\varphi(t):=\Phi^{(1)}(t)^{\ast}\varphi_{0}\in\mathrm{T}^{\ast}_{q(t)}M$
annihilates the whole distribution $\mathcal{D}$ along the trajectory $q(t)$.
Note that thanks to the evolution equations of $\Phi^{(1)}(t)$, $\varphi(t)$
defined in such a way is a _Pontryagin covector_ , i.e. it is subject to an
ODE

$\big{\langle}b,\dot{\varphi}(t)\big{\rangle}+\big{\langle}\sum_{i}u_{i}(t)\cdot\operatorname{D}_{q(t)}X_{i}[b],\varphi(t)\big{\rangle}=0\quad\text{for
every $b\in\mathrm{T}_{q(t)}M$. }$

At the end of the day we arrived at the standard conditions for an abnormal
extremal [Mon06].

##  4 Optimality conditions of degree two

So far we studied only the first-order part of the expansion (3.1). Now it is
time to bring the second term $b^{(2)}(t,\Delta u)$ (or, actually,
$q^{(2)}(t,\Delta u)$ which we prefer to analyze) into play. In general, we
would like to relate non-controllability of the system
$(\bm{q}^{(1)}(t),q^{(2)}(t))$ evolving according to ($\Lambda_{1}$) and
(3.10) with the optimality of an abnormal trajectory $q(t)$. This relation
will be clarified through Agrachev-Sarychev theory [AS96].

###  4.1 The Agrachev-Sarychev Index Lemma

#### Statement of the Index Lemma

Recall that by a _negative index_ of a quadratic form we understand the
maximal dimension of a subspace on which this form is negatively-defined.

######  Lemma 4.1 (Index Lemma [AS96]).

Let $q(t)$, with $t\in[t_{0},t_{1}]$, be an abnormal minimizing SR geodesic of
co-rank $r$, corresponding to the control $u\in
L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})$. Then there exists a non-zero
covector $\varphi_{0}\in\mathrm{T}^{\ast}_{q(t_{1})}M$ such that
$\operatorname{Im}{\operatorname{D}_{u}{\mathbf{End}}^{[t_{0},t_{1}]}_{q_{0}}}\subset\ker\varphi_{0}$,
and that the negative index of the quadratic form

$\big{\langle}\operatorname{D}^{2}_{u}{\mathbf{End}}^{[t_{0},t_{1}]}_{q_{0}}[\cdot,\cdot],\varphi_{0}\big{\rangle}:\ker\operatorname{D}_{u}{\mathbf{End}}^{[t_{0},t_{1}]}_{q_{0}}\times\ker\operatorname{D}_{u}{\mathbf{End}}^{[t_{0},t_{1}]}_{q_{0}}\longrightarrow\mathbb{R}$

is smaller than $r$.

Above we identify $\varphi_{0}\in\mathrm{T}^{\ast}M$ with
$\varphi_{0}+0\cdot\mathrm{d}r\in\mathrm{T}^{\ast}M\times\mathrm{T}^{\ast}\mathbb{R}$,
so that the pairing between $\varphi_{0}$ and the element
$\operatorname{D}_{u}{\mathbf{End}}^{[t_{0},t_{1}]}_{q_{0}}[\Delta u]$ or
$\operatorname{D}^{2}_{u}{\mathbf{End}}^{[t_{0},t_{1}]}_{q_{0}}[\Delta
u,\Delta u]$ makes sense.

For a concise proof we refer to the recent monograph [Rif14, Appendix B.].

#### Index lemma in adapted coordinates

Now we would like translate the results of Lem. 4.1 to the language of adapted
coordinates $(q^{(1)},q^{(2)})$. First of all, let us introduce a new notation
for $\ker\operatorname{D}_{u}{\mathbf{End}}^{[t_{0},t_{1}]}_{q_{0}}$, which
will be more suitable in later application. Denote

$\mathcal{U}_{0}(t_{0},t_{1}):=\\{\Delta u\in
L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})\ |\ b^{(1)}(t_{1},\Delta
u)=0,c^{(1)}(t_{1},\Delta u)=0\\}\ .$

Now, note that since $\varphi_{0}$ above acts trivially on the
$\mathrm{T}\mathbb{R}$-direction, then

$\big{\langle}\operatorname{D}^{2}_{u}{\mathbf{End}}^{[t_{0},t_{1}]}_{q_{0}}[\cdot,\cdot],\varphi_{0}\big{\rangle}=\big{\langle}\operatorname{D}^{2}_{u}\mathrm{End}^{[t_{0},t_{1}]}_{q_{0}}[\cdot,\cdot],\varphi_{0}\big{\rangle}\
.$

Now,
$\frac{1}{2}\operatorname{D}^{2}_{u}\mathrm{End}^{[t_{0},t_{1}]}_{q_{0}}[\Delta
u,\Delta u]$ is just the second term of the expansion (3.1), i.e.
$b^{(2)}(t_{1},\Delta u)$.222In this particular situation, since we restrict
our attention to
$\ker\operatorname{D}_{u}{\mathbf{End}}^{[t_{0},t_{1}]}_{q_{0}}$, i.e. these
controls $\Delta u\in L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})$ which give
$b^{(1)}(t_{1},\Delta u)=0$ (and also $c^{(1)}(t_{1},\Delta u)=0$), the bare
term $b^{(2)}(t_{1},\Delta u)$ makes sense, and so does
$\operatorname{D}^{2}_{u}\mathrm{End}^{[t_{0},t_{1}]}_{q_{0}}[\cdot,\cdot]$
(the latter after dividing by the image of
$\operatorname{D}_{u}\mathrm{End}^{[t_{0},t_{1}]}_{q_{0}}$, which happens
thanks to the pairing with $\varphi_{0}$ which annihilates
$\operatorname{Im}\operatorname{D}_{u}\mathrm{End}^{[t_{0},t_{1}]}_{q_{0}}$).
For a detailed discussion see [AS04, Chap. 20].  Therefore we translate the
Index Lemma 4.1 in the following way

######  Proposition 4.2.

Under the assumptions of Lemma 4.1, there exists a non-zero covector
$\varphi_{0}\in\mathrm{T}^{\ast}_{q(t_{1})}M$ such that

  * •

$\big{\langle}b^{(1)}(t_{1},\Delta u),\varphi_{0}\big{\rangle}=0$ for every
$\Delta u\in L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})$ and

  * •

the quadratic map

$\mathcal{Q}_{AS}:\mathcal{U}_{0}(t_{0},t_{1})\longrightarrow\mathbb{R};\qquad\mathcal{Q}_{AS}:\Delta
u\longmapsto\big{\langle}b^{(2)}(t_{1},\Delta u),\varphi_{0}\big{\rangle}$

has a negative index smaller than $r$.

Equivalently, in adapted coordinates: $\mathcal{U}_{0}(t_{0},t_{1})=\\{\Delta
u\in L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})\ |\ \bm{q}^{(1)}(t_{1},\Delta
u)=\bm{0}\\}$ and there exists a non-zero covector
$\psi_{0}\in(\mathbb{R}^{n})^{\ast}$ such that

  * •

$\big{\langle}q^{(1)}(t_{1},\Delta u),\psi_{0}\big{\rangle}=0$ for every
$\Delta u\in L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})$ and

  * •

the quadratic map

(4.1)
$\mathcal{Q}_{AS}:\mathcal{U}_{0}(t_{0},t_{1})\longrightarrow\mathbb{R};\qquad\mathcal{Q}_{AS}:\Delta
u\longmapsto\big{\langle}q^{(2)}(t_{1},\Delta u),\psi_{0}\big{\rangle}$

has a negative index smaller than $r$.

The first part of the assertion is just a re-statement of Lem. 4.1 in terms of
the expansion (3.1). For the second part take
$\psi_{0}:=\Phi^{(1)}(t_{1})^{\ast}\varphi_{0}$ and recall the definition of
adapted coordinates (3.7)–(3.8).

#### Setting the index to zero

Given a covector $\psi_{0}$ as above we may repeat the construction of the set
$\mathcal{U}_{0}(t_{0},t_{1})$ and a map
$\mathcal{Q}_{AS}=\mathcal{Q}_{[t_{0},t_{1}]}$ on subintervals
$[t_{0}^{\prime},t_{1}^{\prime}]\subset[t_{0},t_{1}]$ to produce analogous
objects: $\mathcal{U}_{0}(t_{0}^{\prime},t_{1}^{\prime})$ and
$\mathcal{Q}_{[t_{0}^{\prime},t_{1}^{\prime}]}$, respectively. Actually, the
extension of an $L^{\infty}$-map $\Delta
u:[t_{0}^{\prime},t_{1}^{\prime}]\rightarrow\mathbb{R}^{k}$ by zero on
$[t_{0},t_{0}^{\prime})\cup(t_{1}^{\prime},t_{1}]$ gives us a canonical
inclusion $L^{\infty}([t_{0}^{\prime},t_{1}^{\prime}],\mathbb{R}^{k})\subset
L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})$, which allows to interpret each
$\mathcal{U}_{0}(t_{0}^{\prime},t_{1}^{\prime})$ as a subset of
$\mathcal{U}_{0}(t_{0},t_{1})$ and, consequently,
$\mathcal{Q}_{[t_{0}^{\prime},t_{1}^{\prime}]}$ as a restriction of
$\mathcal{Q}_{AS}=\mathcal{Q}_{[t_{0},t_{1}]}$ to
$\mathcal{U}_{0}(t_{0}^{\prime},t_{1}^{\prime})$. With this simple observation
the below result is now easy to prove. It states that the time interval
$[t_{0},t_{1}]$ can be divided into at most $r$ pieces on which the quadratic
map $\mathcal{Q}_{AS}$ is non-negative.

######  Lemma 4.3.

Under the assumptions of Lemma 4.1, there exist at most $r+1$ points
$\tau_{0}=t_{0}\leq\tau_{1}\leq\tau_{2}\leq\ldots\leq\tau_{r^{\prime}}=t_{1}$
such that for each subinterval
$[t_{0}^{\prime},t_{1}^{\prime}]\subset(\tau_{j},\tau_{j+1}]$, the quadratic
map

$\mathcal{Q}_{AS}:\mathcal{U}_{0}(t_{0}^{\prime},t_{1}^{\prime})\longrightarrow\mathbb{R}$

is non-negatively defined.

###### Proof.

Since for $t<t^{\prime}$ we have a canonical inclusion
$\mathcal{U}_{0}(t_{0},t)\subset\mathcal{U}_{0}(t_{0},t^{\prime})$, the
integer-valued function

$I:[t_{0},t_{1}]\ni t\longmapsto\text{the negative index of $\mathcal{Q}_{AS}$
on $\mathcal{U}_{0}(t_{0},t)$}$

is non-decreasing. Further, as being negatively defined on a subspace is an
open property, $I$ is left semi-continuous. By Prop. 4.2 the map $I$ has at
most $r-1$ points of growth. Together with $t_{0}$ and $t_{1}$ this gives us
at most $r+1$ points
$\tau_{0}=t_{0}\leq\tau_{1}\leq\tau_{2}\leq\ldots\leq\tau_{r^{\prime}}=t_{1}$,
with the property that $I$ is constant on $(\tau_{j},\tau_{j+1}]$ Take now any
$[t_{0}^{\prime},t_{1}^{\prime}]\subset(\tau_{j},\tau_{j+1}]$ and assume that
$\mathcal{Q}_{AS}$ is not non-negatively defined on
$\mathcal{U}_{0}(t_{0}^{\prime},t_{1}^{\prime})$. For
$\tau_{j}+\varepsilon<t_{0}^{\prime}$ the sets
$\mathcal{U}_{0}(t_{0},\tau_{j}+\varepsilon)$ and
$\mathcal{U}_{0}(t_{0}^{\prime},t_{1}^{\prime})$ are
$\mathcal{Q}_{AS}$-orthogonal (consisting of controls with disjoints supports,
for which $q^{(1)}$ vanishes at the end-points), it follows that the negative
index of $\mathcal{Q}_{AS}$ on
$\mathcal{U}_{0}(t_{0},\tau_{j}+\varepsilon)\cup\mathcal{U}_{0}(t_{0}^{\prime},t_{1}^{\prime})\subset\mathcal{U}_{0}(t_{0},t_{1}^{\prime})$
is greater than the negative index of $\mathcal{Q}_{AS}$ on
$\mathcal{U}_{0}(t_{0},\tau_{j}+\varepsilon)$. This is, however, impossible as
the index function $I$ had no growth-points between $\tau_{j}+\varepsilon$ and
$\tau_{j+1}\geq t_{1}^{\prime}$. ∎

###  4.2 The characteristic optimal control problem.

#### An optimal control problem involving 2-jets

Motivated by our considerations about the index of (4.1), we will study the
geometry of the second-order expansion of the extended end-point map
${\mathbf{End}}^{[t_{0},t_{1}]}_{q_{0}}$ through the following optimization
problem.

######  Definition 4.4.

Let $q(t)$, with $t\in[t_{0},t_{1}]$, be a SR trajectory corresponding to the
control $u\in L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})$. Choose
$\psi_{0}\in(\mathbb{R}^{n})^{\ast}$, $\tau\in[t_{0},t_{1}]$ and
$\bm{v}\in\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},\tau)$. By a _characteristic
optimal control problem_ (_characteristic OCP_ , in short) of the trajectory
$q(t)$ we understand a problem of minimizing the functional

$Q_{\tau}[\Delta u]:=\big{\langle}q^{(2)}(\tau,\Delta
u),\psi_{0}\big{\rangle}$

in the class of controls $\Delta u\in L^{\infty}([t_{0},\tau],\mathbb{R}^{k})$
such that $\bm{q}^{(1)}(t_{0},\Delta u)=\bm{0}$ and $\bm{q}^{(1)}(\tau,\Delta
u)=\bm{v}$. (In other words, $\Delta u$ steers ($\Lambda_{1}$) from $\bm{0}$
to $\bm{v}$ on $[t_{0},\tau]$.) Note that by (3.10)

$Q_{\tau}[\Delta u]=\int_{t_{0}}^{\tau}\sum_{i}\Delta
u_{i}(t)\cdot\big{\langle}Y_{i}^{(2)}(t,q^{(1)}(t,\Delta
u)),\psi_{0}\big{\rangle}\ \mathrm{d}t\ ,$

hence we are actually solving a Larange-type optimal control problem.

Putting for the moment the question of the existence of solutions aside,
define _the minimal 2-jet_ to be

$\widehat{Q}_{\tau}(\bm{v}):=\inf\\{Q_{\tau}[\Delta u]\ |\ \text{$\Delta u\in
L^{\infty}([t_{0},\tau],\mathbb{R}^{k})$ is a control steering
\eqref{eqn:CS_deg_1} from $\bm{0}$ to $\bm{v}$ on $[t_{0},\tau]$}\\}\ .$

We admit a possibility that $\widehat{Q}_{\tau}(\bm{v})$ equals to $-\infty$.

Finally, note that the cost functional $Q_{\tau}[\cdot]$ is quadratic, in the
sense that $Q_{\tau}[\Delta u]=B_{\tau}[\Delta u,\Delta u]$, where
$B_{\tau}[\cdot,\cdot]:L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})\times
L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})\longrightarrow\mathbb{R}$ is a
symmetric bilinear map defined by

$\displaystyle 2\ B_{\tau}[\Delta u,\Delta
v]:=\int_{t_{0}}^{\tau}\sum_{i}\big{\langle}\Delta u_{i}(t)\cdot
Y_{i}^{(2)}(t,q^{(1)}(t,\Delta v))+\Delta v_{i}(t)\cdot
Y_{i}^{(2)}(t,q^{(1)}(t,\Delta u)),\psi_{0}\big{\rangle}\ \mathrm{d}t\ .$

#### Bellman-like inequalities

The minimal 2-jet $\widehat{Q}_{t}(\cdot)$ is defined analogously to the
Bellman’s value function for an optimal control problem. Thus it should not be
surprising that it shares its basic properties. Namely, it turns out, that for
a fixed control $\Delta u\in L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})$, the
distance between $Q_{t}[\Delta u]$ and the infimum value
$\widehat{Q}_{t}(\bm{v}=\bm{q}^{(1)}(t,\Delta u))$ is a non-decreasing
function of time $t$. This property will be of crucial importance later.

######  Lemma 4.5 (Bellman’s inequality).

For every $\Delta u\in L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})$, function

$t\longmapsto Q_{t}[\Delta u]-\widehat{Q}_{t}(\bm{q}^{(1)}(t,\Delta u))$

is non-decreasing.

###### Proof.

Fix a control $\Delta u\in L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})$, and
assume for simplicity that $Q_{t}[\Delta
u]-\widehat{Q}_{t}(\bm{q}^{(1)}(t,\Delta u))=5$, while $Q_{t^{\prime}}[\Delta
u]-\widehat{Q}_{t^{\prime}}(\bm{q}^{(1)}(t^{\prime},\Delta u))=1$ for some
$t_{0}\leq t<t^{\prime}\leq t_{1}$. Since $\widehat{Q}_{t}(\cdot)$ is defined
as the infimum, there is some control $\Delta u^{\prime}\in
L^{\infty}([t_{0},t],\mathbb{R}^{k})$ such that $\bm{q}^{(1)}(t,\Delta
u)=\bm{q}^{(1)}(t,\Delta u^{\prime})$, while $Q_{t}[\Delta
u^{\prime}]-\widehat{Q}_{t}(\bm{q}^{(1)}(t,\Delta u))<5-1=4$. Thus

(4.2) $Q_{t}[\Delta u^{\prime}]-Q_{t}[\Delta u]<-1\ .$

Now consider a concatenated control $\widetilde{\Delta u}:=\Delta
u^{\prime}|_{[t_{0},t]}\circ\Delta u|_{[t,t^{\prime}]}\in
L^{\infty}([t_{0},t^{\prime}],\mathbb{R}^{k})$. Obviously,
$\bm{q}^{(1)}(t,\widetilde{\Delta u})=\bm{q}^{(1)}(t,\Delta
u^{\prime})=\bm{q}^{(1)}(t,\Delta u)$; and
$\bm{q}^{(1)}(t^{\prime},\widetilde{\Delta u})=\bm{q}^{(1)}(t^{\prime},\Delta
u)$ as both controls $\widetilde{\Delta u}$ and $\Delta u$ equal on
$[t,t^{\prime}]$ and share the same initial point $\bm{q}^{(1)}(t,\Delta u)$.
On the other hand,

$\displaystyle Q_{t^{\prime}}[\widetilde{\Delta u}]=$ $\displaystyle
Q_{t}[\Delta u^{\prime}]+\int_{t}^{t^{\prime}}\sum\Delta
u_{i}(\tau)\cdot\big{\langle}Y_{i}^{(2)}(\tau,q^{(1)}(\tau,\Delta
u)),\psi_{0}\big{\rangle}\ \mathrm{d}\tau=$ $\displaystyle Q_{t}[\Delta
u^{\prime}]+Q_{t^{\prime}}[\Delta u]-Q_{t}[\Delta
u]\overset{\eqref{eqn:inequality}}{<}Q_{t^{\prime}}[\Delta u]-1\ .$

It follows that

$Q_{t^{\prime}}[\widetilde{\Delta
u}]-\widehat{Q}_{t^{\prime}}(\bm{q}^{(1)}(t^{\prime},\widetilde{\Delta
u}))=Q_{t^{\prime}}[\widetilde{\Delta
u}]-\widehat{Q}_{t^{\prime}}(\bm{q}^{(1)}(t^{\prime},\Delta
u))<Q_{t^{\prime}}[\Delta
u]-\widehat{Q}_{t^{\prime}}(\bm{q}^{(1)}(t^{\prime},\Delta u))-1=0\ .$

This is a contradiction, with the definition of
$\widehat{Q}_{t^{\prime}}(\cdot)$ as the infimum.

∎

#### A division of the trajectory

As a corollary we get a result stating that if we can somehow guarantee that
$\widehat{Q}_{t_{1}}(\bm{0})$ is finite, then, after a division of the curve
into a finite number of pieces, we may guarantee the finiteness of
$\widehat{Q}_{\tau}(\bm{v})$ for every $\bm{v}$ belonging to the reachable set
at time $\tau$.

######  Lemma 4.6.

Let $q(t)$, with $t\in[t_{0},t_{1}]$ be an abnormal SR trajectory. Assume that
$\widehat{Q}_{t_{1}}(\bm{0})$ is finite. Then there exist at most $n+1$ points
$\tau_{0}=t_{0}<\tau_{1}<\tau_{2}<\ldots<\tau_{s}=t_{1}$ such that for each
$\tau\in[\tau_{i},\tau_{i+1})$ the value of $\widehat{Q}_{\tau}(\bm{v})$ (for
the characteristic OCP on a shorter interval $q\big{|}_{[\tau_{i},t_{1}]}$) is
finite at all points $\bm{v}$ in the reachable set
$\bm{\mathcal{R}}_{(\Lambda_{1})}(\tau_{i},\tau)$.

###### Proof.

Let points $\tau_{i}$, with $i=0,1,\ldots,s$, be as in the assertion of Lem.
3.4. Choose $\tau\in[\tau_{i},\tau_{i+1})$ and
$\bm{v}\in\bm{\mathcal{R}}_{(\Lambda_{1})}(\tau_{i},\tau)$. On the one hand we
know that $\bm{v}=\bm{q}^{(1)}(\tau,\Delta u)$ for some control $\Delta u\in
L^{\infty}([\tau_{i},\tau],\mathbb{R}^{k})$. On the other hand, by Lem. 3.4,
we know that $\bm{v}$ is controllable to zero from $\tau$, thus there exists a
control $\Delta u^{\prime}\in L^{\infty}([\tau,t_{1}],\mathbb{R}^{k})$ which
steers ($\Lambda_{1}$) from $\bm{v}$ to $\bm{0}$ on $[\tau,t_{1}]$. We
conclude that the concatenated control $\widetilde{\Delta}u:=\Delta
u|_{[\tau_{i},\tau]}\circ\Delta u^{\prime}|_{[\tau,t_{1}]}\in
L^{\infty}([\tau_{i},t_{1}],\mathbb{R}^{k})$ steers ($\Lambda_{1}$) from
$\bm{0}$ to $\bm{0}$ on $[\tau_{i},t_{1}]$, taking value $\bm{v}$ at time
$\tau$. By Lemma 4.5 for the characteristic OCP on $[\tau_{i},t_{1}]$ we have

$\displaystyle Q_{\tau}[\widetilde{\Delta u}]-\widehat{Q}_{\tau}(\bm{v})=$
$\displaystyle Q_{\tau}[\widetilde{\Delta
u}]-\widehat{Q}_{\tau}(\bm{q}^{(1)}(\tau,\widetilde{\Delta u}))\leq$
$\displaystyle Q_{t_{1}}[\widetilde{\Delta
u}]-\widehat{Q}_{t_{1}}(\bm{q}^{(1)}(t_{1},\widetilde{\Delta
u}))=Q_{t_{1}}[\widetilde{\Delta u}]-\widehat{Q}_{t_{1}}(\bm{0})<+\infty$

(note that $\widehat{Q}_{t_{1}}(\bm{0})$ is finite also for the characteristic
OCP on the shorter interval $[\tau_{i},t_{1}]\subset[t_{0},t_{1}]$ as
decreasing the set cannot increase the set of values of the cost function).
This ends the proof. ∎

Now note that since for controls $\Delta u\in\mathcal{U}_{0}(t_{0},t_{1})$, we
obviously have $Q_{t_{1}}[\Delta u]=\mathcal{Q}_{AS}[\Delta u]$, we conclude
that if $\mathcal{Q}_{AS}$ is non-negatively defined on
$\mathcal{U}_{0}(t_{0},t_{1})$ then $\widehat{Q}_{t_{1}}(\bm{0})\geq 0$. (In
fact in that case the trivial control $\Delta u\equiv 0$ gives us
$\widehat{Q}_{t_{1}}(\bm{0})=0$.) On the other hand, by Lemma 4.3, we may
guarantee that $\mathcal{Q}_{AS}$ is non-negatively defined by cutting the
interval into at most $r$-pieces. This, together with the above Lemma 4.6
gives us

######  Lemma 4.7.

Assume that $q(t)$, with $t\in[t_{0},t_{1}]$, is a minimizing abnormal SR
geodesic of co-rank $r$. Then there exists a non-zero covector
$\psi_{0}\in(\mathbb{R}^{n})^{\ast}$, and at most $r(n-r-k+1)+1$ points
$\tau_{0}=t_{0}<\tau_{1}<\ldots<\tau_{N}=t_{1}$ such that on each sub-interval
$[t_{0}^{\prime},t_{1}^{\prime}]\subset(\tau_{i},\tau_{i+1})$ the value of
$\widehat{Q}_{\tau}(\bm{v})$ (for the characteristic OCP on a shorter curve
$q\big{|}_{[t_{0}^{\prime},t_{1}^{\prime}]}$) is finite for every
$\tau\in[t_{0}^{\prime},t_{1}^{\prime}]$ and every
$\bm{v}\in\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0}^{\prime},\tau)$.

###### Proof.

Choose $\psi_{0}$ as in Prop. 4.2. Then, by Lemma 4.3 we may divide
$[t_{0},t_{1}]$ into at most $r$ pieces of the form $(\tau_{j},\tau_{j+1}]$
such that the quadratic map $\mathcal{Q}_{AS}$ is non-negatively defined on
every $[t_{0}^{\prime},\tau_{j+1}]\subset(\tau_{j},\tau_{j+1}]$ . In
particular this implies that $\widehat{Q}_{\tau_{j+1}}(\bm{0})=0$ (for the
characteristic OCP on $q\big{|}_{[t_{0}^{\prime},\tau_{j+1}]}$). Now by Lemma
4.6 we divide $[t_{0}^{\prime},\tau_{j+1}]$ into at most $n$ pieces to
guarantee that on each of these $\widehat{Q}_{\tau}(\bm{v})$ is finite for all
elements $\bm{v}$ of the reachable set. Actually, it follows from the proof of
Lemma 3.4 that those new division points are the points at which the dimension
of the reachable set
$\bm{\mathcal{R}}_{(\overline{\Lambda_{1}})}(\tau_{j+1},\tau)$ (for system
($\Lambda_{1}$) with reversed time) grows. Thus, first of all, these new
division points do not depend on the choice of $t_{0}^{\prime}$, hence in fact
we have a division of the whole $[\tau_{j},\tau_{j+1}]$. Secondly, the number
of pieces of the new division is actually not greater than $n-r-k+1$, as the
dimension of $\bm{\mathcal{R}}_{(\overline{\Lambda_{1}})}(\tau_{j+1},\tau)$ is
at most $n+1-r$ (by the assumption that $q(t)$ is of co-rank $r$) and at least
$k$ (as ($\Lambda_{1}$) is controlled by $k$ independent vector fields). This
ends the proof. ∎

###  4.3 The shape of the minimal 2-jet

Now let us address the situation when the minimal 2-jet
$\widehat{Q}_{\tau}(\bm{v})$ is finite for all possible choices of
$\tau\in[t_{0},t_{1}]$ and
$\bm{v}\in\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},\tau)$. By the results of
Lemma 4.7 such a situation can be achieved by dividing a minimizing abnormal
geodesics into a finite number of pieces. Our goal is to derive the properties
of $\widehat{Q}_{\tau}(\bm{v})$ in such a situation. We will begin by proving
the following simple result

######  Proposition 4.8.

If $\widehat{Q}_{t}(\bm{0})$ is finite then it equals to $0$.

###### Proof.

Of course taking a trivial control $\Delta u\equiv 0$ gives us $Q_{t}[\Delta
u]=0$, and hence $\widehat{Q}_{t}(\bm{0})\leq 0$. Assume that there exists
some $\Delta v\in\mathcal{U}_{0}(t_{0},t)$ such that $Q_{t}[\Delta v]=-c<0$.
Note that, since $\mathcal{U}_{0}(t_{0},t)$ is a linear space, and since the
functional $Q_{t}[\cdot]$ is quadratic, for every $\lambda\in\mathbb{R}$ we
have $\lambda\cdot\Delta v\in\mathcal{U}_{0}(t_{0},t)$ and
$Q_{t}[\lambda\cdot\Delta v]=-\lambda^{2}c$. Taking $\lambda\to\infty$ gives
us $\widehat{Q}_{t}(\bm{0})=-\infty$. We get a contradiction with the
assumption that $\widehat{Q}_{t}(\bm{0})$ was finite. ∎

######  Theorem 4.9.

Consider the characteristic OCP of the trajectory $q(t)$ corresponding to the
control $u\in L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})$. Assume that for some
covector $\psi_{0}\in(\mathbb{R}^{n})^{\ast}$ the map
$\widehat{Q}_{\tau}(\bm{v})$ is finite for all possible choices of
$\tau\in[t_{0},t_{1}]$ and
$\bm{v}\in\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},\tau)$. Then

  * •

$\widehat{Q}_{\tau}(\bm{v})$ is quadratic with respect to $\bm{v}$.

  * •

for a fixed $\bm{v}$, function $\tau\mapsto\widehat{Q}_{\tau}(\bm{v})$ is non-
increasing.

###### Proof.

To prove the first part of the assertion, fix $\tau\in[t_{0},t_{1}]$ choose
$\varepsilon>0$, and choose any basis $\\{\bm{v}_{1},\ldots,\bm{v}_{s}\\}$ of
$\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},\tau)$. For $i=1,2,\ldots,s$ denote
$\bm{v}_{s+i}:=-\bm{v}_{i}$ and let $P$ be the convex polygon with vertices
$\pm\bm{v}_{i}$. Since $\widehat{Q}_{\tau}(\cdot)$ is defined as an infimum,
for every $j=1,2,\ldots,2s$ we may find a control $\Delta u_{j}\in
L^{\infty}([t_{0},\tau],\mathbb{R}^{k})$ such that $\bm{b}^{(1)}(\tau,\Delta
u_{j})=\bm{v}_{j}$ and $Q_{\tau}[\Delta
u_{j}]\leq\widehat{Q}_{\tau}(\bm{v}_{j})+\varepsilon$. Since for every $\Delta
v\in\mathcal{U}_{0}(t_{0},\tau)$ we have

$Q_{\tau}[\Delta u_{j}+\Delta v]=Q_{\tau}[\Delta u_{j}]+2B_{\tau}[\Delta
u_{j},\Delta v]+Q_{\tau}[\Delta
v]\geq\widehat{Q}_{\tau}(\bm{v}_{j}=\bm{b}^{(1)}(\tau,\Delta u_{j}+\Delta v))\
,$

the condition $Q_{\tau}[\Delta
u_{j}]\leq\widehat{Q}_{\tau}(\bm{v}_{j})+\varepsilon$ is equivalent to the
following one:

$2B_{\tau}[\Delta u_{j},\Delta v]+Q_{\tau}[\Delta
v]\geq-\varepsilon\qquad\text{for every $\Delta
v\in\mathcal{U}_{0}(t_{0},\tau)$.}$

Now for every convex combination $\sum_{j=1}^{2s}\lambda_{j}\cdot\Delta u_{j}$
consider
$F(\sum_{j}\lambda_{j}\cdot\bm{v}_{j}):=Q_{\tau}[\sum_{j}\lambda_{j}\cdot\Delta
u_{j}]$. Obviously, as $\bm{b}^{(1)}(\tau,\sum_{j}\lambda_{j}\cdot\Delta
u_{j})=\sum_{j}\lambda_{j}\cdot\bm{b}^{(1)}(\tau,\Delta
u_{j})=\sum_{j}\lambda_{j}\cdot\bm{v}_{j}$, and $Q_{\tau}[\cdot]$ is a
quadratic function, then $F$ is a quadratic function on the polygon $P$. We
further claim that $\widehat{Q}_{\tau}(\bm{v})\leq
F(\bm{v})\leq\widehat{Q}_{\tau}(\bm{v})+\varepsilon$. The first inequality is
obvious as $\widehat{Q}_{\tau}(\bm{v})$ is the infimum value of the functional
$Q_{\tau}[\cdot]$ and $F(\bm{v})=Q_{\tau}[\sum_{j}\lambda_{j}\cdot\Delta
u_{j}]$ a particular one. The second inequality follows easily by an addition
of analogous inequalities for particular $\Delta u_{j}$’s. Namely for every
$\Delta v\in\mathcal{U}_{0}(t_{0},\tau)$ we have

$\displaystyle 2B_{\tau}[\sum_{j}\lambda_{j}\cdot\Delta u_{j},\Delta
v]+Q_{\tau}[\Delta v]=2B_{\tau}[\sum_{j}\lambda_{j}\cdot\Delta u_{j},\Delta
v]+(\sum_{j}\lambda_{j})\cdot Q_{\tau}[\Delta v]=$
$\displaystyle\sum_{j=1}^{2s}\lambda_{j}\cdot\left(2B_{\tau}[\Delta
u_{j},\Delta v]+Q_{\tau}[\Delta
v]\right)\geq\sum_{j=1}^{2s}\lambda_{j}\cdot(-\varepsilon)=-\varepsilon\ .$

At the end of the day we showed that the function $\widehat{Q}_{\tau}(\bm{v})$
can be $\varepsilon$-uniformly approximated by a quadratic function
$F(\bm{v})$ on the polygon $P\ni\bm{v}$. Note that a uniform limit of a
sequence of quadratic maps is also a quadratic map, thus
$\widehat{Q}_{\tau}(\cdot)$ is quadratic on $P$. Since $P$ was arbitrary the
assertion follows.

The second part of the assertion is straightforward. As extending a given
control $\Delta u\in L^{\infty}([t_{0},\tau],\mathbb{R}^{k})$ to a control
$\Delta u^{\prime}\in L^{\infty}([t_{0},\tau^{\prime}],\mathbb{R}^{k})$ by a
concatenation with the zero control, i.e. $\Delta u^{\prime}=\Delta
u|_{[t_{0},\tau]}\circ 0|_{[\tau,\tau^{\prime}]}$, gives $Q_{\tau}[\Delta
u]=Q_{\tau^{\prime}}[\Delta u^{\prime}]$, we have
$\widehat{Q}_{\tau}(\bm{v})\geq\widehat{Q}_{\tau^{\prime}}(\bm{v})$ for every
$\tau^{\prime}>\tau$ and
$\bm{v}\in\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},\tau)$. ∎

Theorem 4.9, in combination with Lemma 4.5, has some important consequences.

######  Theorem 4.10.

Under the assumptions of Theorem 4.9, for each $t\in[t_{0},t_{1}]$ there
exists a family of symmetric bi-linear maps
$\bm{\Psi}_{2}(t):(\mathbb{R}^{n}\times\mathbb{R})\times(\mathbb{R}^{n}\times\mathbb{R})\longrightarrow\mathbb{R}$
such that

$\widehat{Q}_{t}(q^{(1)},c^{(1)})=\bm{\Psi}_{2}(t)[(q^{(1)},c^{(1)}),(q^{(1)},c^{(1)})]=\Psi_{2}(t)[q^{(1)},q^{(1)}]+\big{\langle}q^{(1)},\zeta(t)\big{\rangle}c^{(1)}+a_{2}(t)(c^{(1)})^{2}\
,$

where $\Psi_{2}(t):\mathbb{R}^{n}\times\mathbb{R}^{n}$ is a symmetric bi-
linear map, $\zeta(t):\mathbb{R}^{n}\rightarrow\mathbb{R}$ a linear map, and
$a_{2}(t)\in\mathbb{R}$ a number. These objects satisfy the following
conditions:

  * •

for very $t\in[t_{0},t_{1}]$ and every control $\Delta u\in
L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})$

(4.3) $\big{\langle}\bm{q}^{(2)}(t,\Delta
u),\psi_{0}\big{\rangle}-\bm{\Psi}_{2}(t)[\bm{q}^{(1)}(t,\Delta
u),\bm{q}^{(1)}(t,\Delta u)]\geq 0\ ,$

  * •

$\Psi_{2}(t)$, $\zeta(t)$ and $a_{2}(t)$ are maps of bounded variation with
respect to $t$, and as such are differentiable almost everywhere;

  * •

for almost every $t\in[t_{0},t_{1}]$ the quadratic form

$(q^{(1)},c^{(1)})\longmapsto\dot{\Psi}_{2}(t)[q^{(1)},q^{(1)}]+\big{\langle}q^{(1)},\dot{\zeta}(t)\big{\rangle}c^{(1)}+\dot{a_{2}}(t)(c^{(1)})^{2}$

is non-negatively defined on
$\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},t)\ni(q^{(1)},c^{(1)})$;

  * •

for almost every $t\in[t_{0},t_{1}]$ the following equations are satisfied for
every $i=1,2,\ldots,k$, and every $q^{(1)}$ such that
$(q^{(1)},c^{(1)})\in\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},t)$ for some
$c^{(1)}\in\mathbb{R}$

(4.4) $\displaystyle 0=$
$\displaystyle\big{\langle}Y_{i}^{(2)}(t,q^{(1)}),\psi_{0}\big{\rangle}-2\
\Psi_{2}(t)\left[q^{(1)},Y_{i}^{(1)}(t)\right]-\big{\langle}q^{(1)},\zeta(t)\big{\rangle}u_{i}(t)$
(4.5) $\displaystyle 0=$
$\displaystyle\big{\langle}Y_{i}^{(1)}(t),\zeta(t)\big{\rangle}+2a_{2}(t)\cdot
u_{i}(t)$

###### Proof.

By the results of Theorem 4.9 the map $\widehat{Q}_{t}(\bm{v})$ is quadratic
in $\bm{v}$ and so we may present it in the form

$\widehat{Q}_{t}(q^{(1)},c^{(1)})=\bm{\Psi}_{2}(t)[(q^{(1)},c^{(1)}),(q^{(1)},c^{(1)})]=\Psi_{2}(t)[q^{(1)},q^{(1)}]+\big{\langle}q^{(1)},\zeta(t)\big{\rangle}c^{(1)}+a_{2}(t)(c^{(1)})^{2}\
.$

Further this quadratic form is non-increasing as a function of $t$, and so its
coefficients are either non-increasing functions themselves of differences of
two such functions. As such they are functions of bounded variation, and thus
are differentiable almost everywhere. Further, by Lemma 4.5, for every control
$\Delta u\in L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})$ the map
$t\mapsto\big{\langle}\psi_{0},q^{(2)}(t,\Delta
u)\big{\rangle}-\widehat{Q}_{t}(q^{(1)}(t,\Delta u),c^{(1)}(t,\Delta u))$ is
non-decreasing (and consequently always non-negative as at $t=t_{0}$ it equals
zero). Thus its derivative (defined almost everywhere) must be non-negative
for every such $\Delta u$. By differentiation we get333For conciseness we do
not write the dependence of $q^{(1)}$, $q^{(2)}$ and $c^{(1)}$ on $(t,\Delta
u)$ in the calculations below.

$\displaystyle 0\leq$
$\displaystyle\big{\langle}{\dot{q}}^{(2)},\psi_{0}\big{\rangle}-2\Psi_{2}(t)[q^{(1)},{\dot{q}}^{(1)}]-\big{\langle}q^{(1)},\zeta(t)\big{\rangle}\dot{c_{1}}-\big{\langle}{\dot{q}}^{(1)},\zeta(t)\big{\rangle}c^{(1)}-2a_{2}(t)c^{(1)}\dot{c^{(1)}}+$
$\displaystyle-\left\\{\dot{\Psi}_{2}(t)[q^{(1)},q^{(1)}]+\big{\langle}q^{(1)},\dot{\zeta}(t)\big{\rangle}c^{(1)}+\dot{a_{1}}(t)(c^{(1)})^{2}\right\\}=$
$\displaystyle\big{\langle}\sum_{i}\Delta u_{i}(t)\cdot
Y_{i}^{(2)}(t,q^{(1)}),\psi_{0}\big{\rangle}-2\Psi_{2}(t)\left[q^{(1)},\sum_{i}\Delta
u_{i}(t)\cdot Y_{i}^{(1)}(t)\right]+$
$\displaystyle-\big{\langle}q^{(1)},\zeta(t)\big{\rangle}\sum_{i}u_{i}(t)\cdot\Delta
u_{i}(t)-\big{\langle}\sum_{i}\Delta u_{i}(t)\cdot
Y_{i}^{(1)}(t),\zeta(t)\big{\rangle}c^{(1)}-2a_{2}(t)c^{(1)}\sum_{i}u_{i}(t)\cdot\Delta
u_{i}(t)+$
$\displaystyle-\left\\{\dot{\Psi}_{2}(t)[q^{(1)},q^{(1)}]+\big{\langle}q^{(1)},\dot{\zeta}(t)\big{\rangle}c^{(1)}+\dot{a_{1}}(t)(c^{(1)})^{2}\right\\}.$

Note that the right-hand side of the above inequality consists of two parts.
The first depends linearly on $\Delta u_{i}$’s, while the second is quadratic
in $(q^{(1)},c^{(1)})$ and has no $\Delta u_{i}$-dependence. If the linear
part would be non-vanishing then, by a suitable choice of $\Delta u_{i}$’s we
would be able to make the whole expression negative. This is, however,
impossible. We conclude that the linear part vanishes, and that the remaining
quadratic part must be non-negative. The vanishing of the linear part leads to

$\displaystyle 0=$
$\displaystyle\big{\langle}Y_{i}^{(2)}(t,q^{(1)}),\psi_{0}\big{\rangle}-2\Psi_{2}(t)\left[q^{(1)},Y_{i}^{(1)}(t)\right]-\big{\langle}q^{(1)},\zeta(t)\big{\rangle}u_{i}(t)+$
$\displaystyle-\big{\langle}Y_{i}^{(1)}(t),\zeta(t)\big{\rangle}c^{(1)}-2a_{2}(t)c^{(1)}u_{i}(t)$

for every $i=1,2,\ldots,k$ and every
$(q^{(1)},c^{(1)})\in\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},t)$. Now note that
the first part of the latter expression is linear in $q^{(1)}$, while the
second part is linear in $c^{(1)}$. Thus both parts separately need to vanish.
This ends the proof. ∎

We will discuss some consequences of the above result in the next subsection.
For now let us observe how equations (4.4)–(4.5) imply _Goh conditions_.
Namely, set $q^{(1)}=Y_{j}^{(1)}(t)$ into the first equation to get the
following expression, symmetric in $i$ and $j$:

$\displaystyle\big{\langle}Y_{i}^{(2)}(t,Y_{j}^{(1)}),\psi_{0}\big{\rangle}=$
$\displaystyle 2\
\Psi_{2}(t)\left[Y_{j}^{(1)},Y_{i}^{(1)}(t)\right]+\big{\langle}Y_{j}^{(1)},\zeta(t)\big{\rangle}u_{i}(t)\overset{\eqref{eqn:second_in_q}}{=}$
$\displaystyle 2\
\Psi_{2}(t)\left[Y_{j}^{(1)},Y_{i}^{(1)}(t)\right]-2a_{2}(t)u_{j}(t)u_{i}(t)\
.$

However, by definition of the fields $Y_{i}^{(2)}$ and $Y_{j}^{(1)}$ given in
Lemma 3.1 we have

$Y_{i}^{(2)}(t,Y_{j}^{(1)}):=\left\\{\Phi^{(1)}(t)\left[\operatorname{D}_{q(t)}X_{i}[X_{j}]\right]+\Phi^{(2)}(t)\left[X_{i},X_{j}\right]\right\\}\
,$

thus anti-symmetrization of
$\big{\langle}Y_{i}^{(2)}(t,Y_{j}^{(1)}),\psi_{0}\big{\rangle}$ in $i$ and $j$
gives us

$0=\big{\langle}\Phi^{(1)}(t)\left[\operatorname{D}_{q(t)}X_{i}[X_{j}]-\operatorname{D}_{q(t)}X_{j}[X_{i}]\right],\psi_{0}\big{\rangle}=\big{\langle}\Phi^{(1)}(t)\left[X_{i},X_{j}\right],\psi_{0}\big{\rangle}\
.$

###  4.4 Optimality conditions of degree two

#### The main result

The results of Thm 4.10 can be easily translated to the language of 2-jets
$(\bm{b}^{(1)}(t,\Delta u),$ $\bm{b}^{(2)}(t,\Delta u))$ by reversing the
transformation to adapted coordinates $(\bm{q}^{(1)}(t,\Delta
u),\bm{q}^{(2)}(t,\Delta u))$ described in Lemma 3.1. This, in combination
with the results on dividing the trajectory, gives us our main result.

######  Theorem 4.11.

Let $q(t)$, with $t\in[t_{0},t_{1}]$, be a strictly abnormal minimizing SR
geodesic of co-rank $r$, corresponding to the control $u\in
L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})$. Then there exists a Pontryagin
covector $\varphi(t)\in\mathrm{T}_{q(t)}^{\ast}M$ and a division of the
interval $[t_{0},t_{1}]$ by at most $r(n-r-k+1)+1$ points
$\tau_{0}=t_{0}<\tau_{1}<\ldots<\tau_{N}=t_{1}$ such that on each sub-interval
$[t_{0}^{\prime},t_{1}^{\prime}]\subset(\tau_{i},\tau_{i+1})$ there exists a
family of symmetric bi-linear maps
$\bm{\Phi}_{2}(t):\mathrm{T}_{(q(t),E^{[t_{0}^{\prime},t]}[u])}(M\times\mathbb{R})\times\mathrm{T}_{(q(t),E^{[t_{0}^{\prime},t]}[u])}(M\times\mathbb{R})\longrightarrow\mathbb{R}$
parameterized by $t\in[t_{0}^{\prime},t_{1}^{\prime}]$, which we decompose as

$\bm{\Phi}_{2}(t)[(b^{(1)},c^{(1)}),(b^{(1)},c^{(1)})]=\Phi_{2}(t)[b^{(1)},b^{(1)}]+\big{\langle}b^{(1)},\xi(t)\big{\rangle}c^{(1)}+a_{2}(t)(c^{(1)})^{2}\
,$

where
$\Phi_{2}(t):\mathrm{T}_{q(t)}M\times\mathrm{T}_{q(t)}M\rightarrow\mathbb{R}$
is a symmetric bi-linear map,
$\xi(t):\mathrm{T}_{(q(t)}M\rightarrow\mathbb{R}$ a linear map, and
$a_{2}(t)\in\mathbb{R}$ a number.

These objects satisfy the following conditions:

  * •

for very $t\in[t_{0}^{\prime},t_{1}^{\prime}]$ and every control $\Delta u\in
L^{\infty}([t_{0}^{\prime},t_{1}^{\prime}],\mathbb{R}^{k})$

(4.6) $\big{\langle}\bm{b}^{(2)}(t,\Delta
u),\varphi(t)\big{\rangle}-\bm{\Phi}_{2}(t)[\bm{b}^{(1)}(t,\Delta
u),\bm{b}^{(1)}(t,\Delta u)]\geq 0\ ,$

  * •

maps $\Phi_{2}(t)$, $\xi(t)$ and $a_{2}(t)$ are of bounded variation, hence
differentiable almost everywhere.

  * •

for almost every $t\in[t_{0}^{\prime},t_{1}^{\prime}]$ the quadratic form

(4.7)
$\begin{split}\big{\langle}\sum_{i=1}^{k}u_{i}(t)\operatorname{D}^{2}_{q(t)}X_{i}[b^{(1)},b^{(1)}],\varphi(t)\big{\rangle}-\dot{\Phi}_{2}(t)[b^{(1)},b^{(1)}]-2\Phi_{2}(t)\left[\sum_{i=1}^{k}u_{i}(t)\operatorname{D}_{q(t)}X_{i}[b^{(1)}],b^{(1)}\right]+\\\
-\left(\big{\langle}b^{(1)},\dot{\xi}(t)\big{\rangle}+\big{\langle}\sum_{i=1}^{k}u_{i}(t)\operatorname{D}_{q(t)}X_{i}[b^{(1)}],\xi(t)\big{\rangle}\right)\cdot
c^{(1)}-\dot{a_{2}}(t)\cdot(c^{(1)})^{2}\end{split}$

is non-negative for every $(b^{(1)},c^{(1)})=\bm{b}^{(1)}(t,\Delta u)$.

  * •

for almost every $t\in[t_{0}^{\prime},t_{1}^{\prime}]$ the following equations
are satisfied for every
$b^{(1)}\in\operatorname{Im}\left(\operatorname{D}_{u}\mathrm{End}^{[t_{0},t]}_{q_{0}}\right)$
and each $i=1,2,\ldots,k$

(4.8) $\displaystyle\big{\langle}X_{i}\big{|}_{q(t)},\xi(t)\big{\rangle}+2\
a_{2}(t)\cdot u_{i}(t)=$ $\displaystyle 0$ (4.9) $\displaystyle
2\big{\langle}\operatorname{D}_{q(t)}X_{i}[b^{(1)}],\varphi(t)\big{\rangle}+2\Phi_{2}(t)\left[X_{i}\big{|}_{q(t)},b^{(1)}\right]+\big{\langle}b^{(1)},\xi(t)\big{\rangle}\cdot
u_{i}(t)=$ $\displaystyle 0\ .$

###### Proof.

With the results of previous subsections the proof is now very simple. First,
pass to the adapted coordinates, where by Proposition 4.2 (the restatement of
Agrachev-Sarychev lemma) we can find a certain covector
$\psi_{0}\in(\mathbb{R}^{n})^{\ast}$. Later we use Lemma 4.7 to construct the
division points $\tau_{i}$ such that within each sub-interval
$[t_{0}^{\prime},t_{1}^{\prime}]\subset(\tau_{i},\tau_{i+1})$ the
characteristic OCP related with the abnormal trajectory $q(t)$ has a finite
minimal 2-jet $\widehat{Q}_{\cdot}(\cdot)$ (for the above choice of
$\psi_{0}$). By Thm 4.10 we prove the existence of a quadratic maps
$\bm{\Psi}_{2}(t):\mathbb{R}^{n+1}\times\mathbb{R}^{n+1}\rightarrow\mathbb{R}$
for $t\in[t_{0}^{\prime},t_{1}^{\prime}]$ with properties analogous to the
desired properties of $\bm{\Phi}_{2}(t)$’s.

To end the proof it suffices to use the inverse of the transformation of
$(\bm{b}^{(1)},\bm{b}^{(2)})$ into adapted coordinates
$(\bm{q}^{(1)},\bm{q}^{(2)})$ – see Lemma 3.1. We would get

$\big{\langle}\bm{q}^{(2)}(t,\Delta
u),\psi_{0}\big{\rangle}-\bm{\Psi}_{2}(t)[\bm{q}^{(1)}(t,\Delta
u),\bm{q}^{(1)}(t,\Delta u)]=\big{\langle}\bm{b}^{(2)}(t,\Delta
u),\varphi(t)\big{\rangle}-\bm{\Phi}_{2}(t)[\bm{b}^{(1)}(t,\Delta
u),\bm{b}^{(1)}(t,\Delta u)]\ ,$

where $\varphi(t)=\Phi^{(1)}(t)^{\ast}\phi_{0}$ for
$\psi_{0}=\Phi^{(1)}(t_{1})^{\ast}\phi_{0}$, and $\bm{\Phi}_{2}(t)$ is a
symmetric bi-linear map on $\mathrm{T}(M\times\mathbb{R})$. The latter
statement follows from the fact that the transformation of adapted coordinates
preserves the natural grading associated with $(\bm{b}^{(1)},\bm{b}^{(2)})$.
The regularity of the new object $\bm{\Phi}_{2}(t)$ is the same as the
regularity of $\bm{\Psi}_{2}(t)$, since we pass from one to another by means
of maps $\Phi^{(1)}(t)$ and $\Phi^{(2)}(t)$ which, by Lemma 3.1, are at least
absolutely continuous maps.

Finally note that in order to prove the assertion of Thm 4.10 we only needed
to know (4.3) and the fact that the left hand-side of this inequality is a
non-decreasing function of $t$. In non-adapted coordinates (4.3) translates as
(4.6) and the non-decreasing property remains true. A differentiation
analogous to the one in the proof of Thm 4.10 gives us the assertion. ∎

#### Discussion of the main result

######  Remark 4.12.

Let us briefly discuss some consequences of Thm 4.11.

  1. (i)

If $a_{2}(t)\neq 0$ then (4.8) gives us an algebraic equation for the controls
$u_{i}(t)$ (and thus an ODE for the trajectory). Since both $a_{2}(t)$, and
$\xi(t)$ are maps of bounded variation, and $X_{i}\big{|}_{q(t)}$ shares the
same regularity as $q(t)$, we can only conclude that $u_{i}(t)$’s are
functions of bounded variation. If, however, the time-regularity of the
minimal 2-jet in Thm 4.9 could be improved, we could also improve the
regularity of the controls, and thus the SR geodesic.

  2. (ii)

Regardless of the value of $a_{2}(t)$, equations (4.8)-(4.9) imply _Goh
conditions_. The derivation is done by the same argument as in remark
following the proof of Thm 4.10. Put $b^{(1)}=X_{j}$ in (4.9) ($b^{(1)}=X_{j}$
can always be) to get

$\displaystyle
2\big{\langle}\operatorname{D}X_{i}[X_{j}],\varphi\big{\rangle}\overset{\eqref{eqn:Phi_b_1}}{=}-2\Phi_{2}\left[X_{i},X_{j}\right]-\big{\langle}X_{j},\xi\big{\rangle}u_{i}(t)\overset{\eqref{eqn:xi_u_1}}{=}-2\Phi_{2}\left[X_{i},X_{j}\right]+a_{2}\cdot
u_{j}(t)u_{i}(t)\ .$

The right-hand side of the above expression is symmetric in $i$ and $j$, hence

$\big{\langle}[X_{i},X_{j}],\varphi\big{\rangle}=\big{\langle}\operatorname{D}X_{i}[X_{j}],\varphi\big{\rangle}-\big{\langle}\operatorname{D}X_{j}[X_{i}],\varphi\big{\rangle}=0\
.$

The Goh condition may be seen as a vanishing of a non-symmetric part of the
derivatives (4.8)–(4.9) due to the symmetric nature of the quadratic form
$\bm{\Phi}_{2}(t)$.

  3. (iii)

If $a_{2}(t)\equiv 0$ (or more generally if $\dot{a}_{2}(t)=0$), then the bi-
linear form (4.7) contains a bare term linear in $c^{(1)}$. This term should
vanish for every possible choice of $b^{(1)}$, since otherwise the form would
not be positively-defined. This reasoning gives the evolution equation for
$\xi(t)$:

(4.10)
$\big{\langle}b^{(1)},\dot{\xi}(t)\big{\rangle}+\big{\langle}\sum_{i=1}^{k}u_{i}(t)\cdot\operatorname{D}_{q(t)}X_{i}[b^{(1)}],\xi(t)\big{\rangle}=0\qquad\text{for
every
$b^{(1)}\in\operatorname{Im}\left(\operatorname{D}_{u}\mathrm{End}^{[t_{0},t]}_{q_{0}}\right)$}$

And so, $\xi(t)$ is a (possibly trivial) _Pontryagin covector_ – see page 3.2.

  4. (iv)

The question whether the bi-linear form (4.7) is strictly positive or has some
null-directions is closely related to the question of existence of solutions
of the characteristic OCP of the abnormal trajectory. We will discuss it in
more detail in Ssec. 4.5. At this point note that if the bi-linear form (4.7)
is constantly zero, and if the objects $\Phi_{2}(t)$, $\xi(t)$ and $a_{2}(t)$
are absolutely continuous w.r.t. $t$ then, in addition to the above equation
(4.10), we get $\dot{a_{2}}(t)\equiv 0$ and finally evolution equation for
$\Phi_{2}(2)$:

(4.11)
$\big{\langle}\sum_{i=1}^{k}u_{i}(t)\operatorname{D}^{2}_{q(t)}X_{i}[b^{(1)},b^{(1)}],\varphi\big{\rangle}-\dot{\Phi}_{2}[b^{(1)},b^{(1)}]-2\Phi_{2}\left[\sum_{i=1}^{k}u_{i}(t)\operatorname{D}_{q(t)}X_{i}[b^{(1)}],b^{(1)}\right]=0$

wchich should hold for every
$b^{(1)}\in\operatorname{Im}\left(\operatorname{D}_{u}\mathrm{End}^{[t_{0},t]}_{q_{0}}\right)$.

Observe that if $a_{2}(t)\neq 0$, equation (4.8) provides information about
the controls $u_{i}(t)$. Basing on this fact we propose the following
definition:

######  Definition 4.13.

Let $q(t)$, with $t\in[t_{0},t_{1}]$ be a strictly abnormal sub-Riemannian
trajectory such that the pair $(\varphi(t),\bm{\Phi}_{2}(t))$ described by Thm
4.11 exists for every $t\in(t_{0},t_{1}]$. If for every $t\in(t_{0},t_{1}]$ we
have $\bm{\Phi}_{2}(t)[(0,1),(0,1)]=a_{2}(t)\neq 0$ we call $q(t)$ a _2-normal
extremal_. Otherwise we say that $q(t)$ is an _2-abnormal extremal_.

By the results of Thm 4.11 every strictly abnormal SR trajectory is either
non-minimizing, or can be divided into a finite number of pieces, each of them
being either 2-normal or 2-abnormal extremal.

The division of abnormal trajectories into the classes of 2-normal and
2-abnormal extremals proposed above shares some natural similarities with an
analogous distinction between normal and abnormal trajectories in degree one.
Namely, 2-normal extremals are characterized by a property that the second-
order approximation of their extended end-point map can be strictly separated
from the direction of the decreasing cost (see fig. on page 1). Further they
satisfy an ODE (although we were not able to prove their regularity from this
property). On the other hand, for 2-abnormal extremals this separation may not
be strict, and equation (4.8) gives us no information about the control $u(t)$
at the points where $a_{2}(t)=0$.

Note that if $a_{2}(t)\equiv 0$, equations (4.10) and (4.8) are the same as
equations for an abnormal extremal given at page 3.2. Thus we have two
abnormal Pontryagin covectors $\varphi(t)$ and $\xi(t)$ in that case (although
the latter may be trivial).

Finally, note that the classes of 2-normal and 2-abnormal extremals are, in
general, not disjoint as an abnormal trajectory of co-rank greater than one
may admit several distinct Pontryagin covectors.

#### Example - Martinet system

Consider $M=\mathbb{R}^{3}\ni(x,y,z)$ and a distribution $\mathcal{D}$ spanned
by orthonormal vector fields $X_{1}=\partial_{x}$, and
$X_{2}=(1-x)\partial_{y}+\frac{x^{2}}{2}\partial_{z}$. Let $q(t)=(t,0,0)$ be a
trajectory corresponding to the control $(u_{1}(t)\equiv 0,u_{2}(t)\equiv 1)$.
It is well-known that $q(t)$ is an abnormal minimizing geodesic, and that
$\phi(t)=\mathrm{d}z$ is the related adjoint curve [Mon06]. We want to see how
Thm. 4.11 works in this case.

First we would like to derive equations (3.3) and (3.4) for the considered
situation. Let us decompose

$b^{(1)}(t)=b^{(1)}_{x}(t)\cdot\partial_{x}+b^{(1)}_{y}(t)\cdot\partial_{y}+b^{(1)}_{z}(t)\cdot\partial_{z}\quad\text{and}\quad
b^{(2)}(t)=b^{(2)}_{x}(t)\cdot\partial_{x}+b^{(2)}_{y}(t)\cdot\partial_{y}+b^{(2)}_{z}(t)\cdot\partial_{z}\
.$

Now note that

$\sum_{i=1}^{2}u_{i}(t)\operatorname{D}_{q(t)}X_{i}[b^{(1)}]=-\mathrm{d}x[b^{(1)}]\cdot\partial_{y}\quad\text{and}\quad\sum_{i=1}^{2}u_{i}(t)\operatorname{D}^{2}_{q(t)}X_{i}[b^{(1)},b^{(1)}]=\mathrm{d}x\otimes\mathrm{d}x[b^{(1)},b^{(1)}]\cdot\partial_{z}\
.$

From the above observation equation (3.3) reads as

$\dot{b^{(1)}_{x}}(t)=\Delta
u_{1}(t),\quad\dot{b^{(1)}_{y}}(t)=-b^{(1)}_{x}(t)+\Delta
u_{2}(t),\quad\dot{b^{(1)}_{z}}(t)=0,\quad\text{and}\quad\dot{c^{(1)}}(t)=\Delta
u_{2}(t)\ .$

In particular,

(4.12) $b^{(1)}_{y}(t)=-\int_{0}^{t}b^{(1)}_{x}(\tau)\
\mathrm{d}\tau+\int_{0}^{t}\Delta u_{2}(\tau)\
\mathrm{d}\tau=-\int_{0}^{t}b^{(1)}_{x}(\tau)\ \mathrm{d}\tau+c^{(1)}(t)\ .$

Similarly, (3.4) gives us

$\dot{b^{(2)}_{x}}(t)=0,\quad\dot{b^{(2)}_{y}}(t)=-b^{(2)}_{x}(t)-\Delta
u_{2}(t)b^{(1)}_{x}(t),\quad\dot{b^{(1)}_{z}}(t)=b^{(1)}_{x}(t)^{2},\quad\text{and}\quad\dot{c^{(2)}}(t)=\Delta
u_{1}(t)^{2}+\Delta u_{2}(t)^{2}\ .$

In consequence, by the Cauchy-Schwarz inequality we can estimate

$\displaystyle\big{\langle}b^{(2)}(t),\phi(t)\big{\rangle}=b^{(2)}_{z}(t)=\int_{0}^{t}b^{(1)}_{x}(\tau)^{2}\
\mathrm{d}\tau\overset{\text{C-S}}{\geq}\frac{1}{t}\cdot\left(\int_{0}^{t}b^{(1)}_{x}(\tau)\
\mathrm{d}\tau\right)^{2}=\frac{1}{t}\cdot\left[c^{(1)}(t)-b^{(1)}_{y}(t)\right]^{2}\
.$

It is easy to see that the above inequality is always strict, yet on the other
hand, it can be satisfied with an arbitrary accuracy. It follows that every
minimal 2-jet is finite, but the minimum is never attained. We conclude that
$\widehat{Q}_{t}(b^{(1)},c^{(1)})=\frac{1}{t}\cdot\left[c^{(1)}(t)-b^{(1)}_{y}(t)\right]^{2}$
and, consequently,

$\Phi_{2}(t)=\frac{1}{t}\mathrm{d}y\otimes\mathrm{d}y,\quad\xi(t)=-\frac{2}{t}\mathrm{d}y\quad\text{and}\quad
a_{2}(t)=\frac{1}{t}\ .$

Since $a_{2}(t)\neq 0$ the considered trajectory is a _2-normal extremal_ in
the sense of Definition 4.13. We leave to the Reader checking that conditions
(4.8)-(4.9) hold for the above data. Further, condition (4.7) reads as

$\left(b^{(1)}_{x}(t)+\frac{1}{t}b^{(1)}_{y}(t)\right)^{2}-\left(\frac{2}{t^{2}}b^{(1)}_{y}(t)+\frac{2}{t}b^{(1)}_{x}(t)\right)c^{(1)}(t)+\frac{1}{t^{2}}c^{(1)}(t)^{2}=\left(b^{(1)}_{x}(t)+\frac{1}{t}b^{(1)}_{y}(t)-\frac{1}{t}c^{(1)}(t)\right)^{2}\geq
0$

and of course holds.

#### Example – a corner on a free Carnot group

The example below was studied by Hakavouri and Le Donne as communicated to us
by [LD21]. It turns out to be a 2-abnormal extremal.

Consider a free Carnot group $G$ or rank 2 and step 4. That is, the Lie
algebra of left-invariant vector fields on $G$ is generated by two fields
$X_{1}$, $X_{2}$, whose Lie brackets of order higher than 4 vanish, and Lie
brackets of order less or equal 4 have no relations apart from the obvious
ones arising either from the skew-symmetry, or from the Jacobi identity. It
follows that all left-invariant vector fields on $G$ are spanned (over
$\mathbb{R}$) by fields: $X_{1}$, $X_{2}$, $Y:=[X_{1},X_{2}]$,
$Z_{1}:=[X_{1},Y]=[X_{1},[X_{1},X_{2}]]$,
$Z_{2}:=[X_{2},-Y]=[X_{2},[X_{2}.X_{1}]]$,
$W_{1}:=[X_{1},Z_{1}]=[X_{1},[X_{1},[X_{1},X_{2}]]]$,
$W_{2}:=[X_{2},Z_{2}]=[X_{2},[X_{2},[X_{2},X_{1}]]]$, and
$W_{3}:=[X_{2},Z_{1}]=[X_{1},-Z_{2}]=[X_{1},[X_{2},[X_{1},X_{2}]]]$.Let us
consider a left-invariant SR geodesic problem on $G$, constituted by setting
$\mathcal{D}_{q}=\operatorname{span}_{\mathbb{R}}\\{X_{1},X_{2}\\}\big{|}_{q}$
and the assumption that the fields $X_{1}$ and $X_{2}$ are orthonormal.

In such a setting let $q:[-1,1]\rightarrow G$ be the _corner_ controlled by
$u(t)=(\chi_{[-1,0]}(t),\chi_{[0,1]}(t))$, i.e.

$q(t)=\begin{cases}\exp(t\cdot X_{1})&\text{for $t\in[-1,0]$}\\\ \exp(t\cdot
X_{2})&\text{for $t\in[0,1]$}\end{cases}\ .$

It turns out that $q(t)$ is a strictly abnormal SR extremal, with

$\operatorname{Im}\operatorname{D}_{u}{\mathbf{End}}^{[-1,1]}=\operatorname{span}_{\mathbb{R}}\\{X_{1},X_{2},Y,Z_{1},Z_{2},W_{1},W_{2}\\}\big{|}_{q(1)}\oplus\mathbb{R}\
.$

In consequence, there is only one (up to rescaling) candidate for the
Pontryagin covector $\varphi(t)\in\mathrm{T}^{\ast}_{q(t)}G$ uniquely defined
by conditions

$\big{\langle}W_{3}\big{|}_{q(t)},\varphi(t)\big{\rangle}=1,\ \text{and}\
\big{\langle}\cdot,\varphi(t)\big{\rangle}=0\ \text{on the space}\
\operatorname{span}_{\mathbb{R}}\\{X_{1},X_{2},Y,Z_{1},Z_{2},W_{1},W_{2}\\}\big{|}_{q(t)}\subset\mathrm{T}_{q(t)}G\
.$

In this particular situation we were able to calculate
$\big{\langle}\bm{b}^{(2)}(1,\Delta u),\varphi(1)\big{\rangle}$ for every
control $\Delta u=(\Delta u_{1},\Delta u_{2})\in
L^{\infty}([-1,1],\mathbb{R}^{2})$. We skip the details of this computation,
as it is rather long and would require introduction of some additional theory
regarding invariant SR structures on Lie groups. To state our result we need
to introduce the function $f:G\rightarrow\mathbb{R}$ defined by

$f\left(\exp(w_{3}W_{3})\exp(w_{2}W_{2})\exp(w_{1}W_{1})\exp(z_{2}Z_{2})\exp(z_{1}Z_{1})\exp(yY)\exp(x_{2}X_{2})\exp(x_{1}X_{1})q(1)\right):=w_{3}\
$

(obviously $\operatorname{D}_{q(1)}f=\varphi(1)$), and a covector
$\rho\in\mathrm{T}^{\ast}_{q(1)}G$ defined by conditions

$\big{\langle}X_{1}\big{|}_{q(1)},\rho\big{\rangle}=1,\ \text{and}\
\big{\langle}\cdot,\rho\big{\rangle}=0\ \text{on the space}\
\operatorname{span}_{\mathbb{R}}\\{X_{2},Y,Z_{1},Z_{2},W_{1},W_{2},W_{3}\\}\big{|}_{q(1)}\
,$

i.e. $\big{\langle}b^{(1)}(1,\Delta u),\rho\big{\rangle}$ is the
$X_{1}$-component of the vector $b^{(1)}(1,\Delta u)\in\mathrm{T}_{q(1)}G$.

It turns out that for every $\Delta u\in L^{\infty}([-1,1],\mathbb{R}^{2})$
the pair $(\bm{b}^{(1)}(1,\Delta u),\bm{b}^{(2)}(1,\Delta u))$ is subject to
the following equality

$\displaystyle\big{\langle}\bm{b}^{(2)}(1,\Delta
u),\varphi(1)\big{\rangle}+\operatorname{D}^{2}_{q(1)}f[b^{(1)}(1,\Delta
u),b^{(1)}(1,\Delta u)]+\frac{1}{2}\big{\langle}b^{(1)}(1,\Delta
u),\rho\big{\rangle}^{2}=$
$\displaystyle\int_{0}^{1}t\left(\int_{-1}^{t}\Delta u_{1}(s)\
\mathrm{d}s\right)^{2}\ \mathrm{d}t-\int_{-1}^{0}t\left(\int_{-1}^{t}\Delta
u_{2}(s)\ \mathrm{d}s\right)^{2}\ \mathrm{d}t\ .$

Clearly the right-hand side is positively defined. We conclude that in the
considered case inequality (4.6) holds with
$\bm{\Phi}_{2}(1)[(b^{(1)},c^{(1)}),(b^{(1)},c^{(1)})]=-\operatorname{D}^{2}_{q(1)}f[b^{(1)},b^{(1)}]-\frac{1}{2}\big{\langle}b^{(1)},\rho\big{\rangle}^{2}$.
As the the dependence on $c^{(1)}$ is trivial, $a_{2}(1)=0$, and hence we deal
with a _2-abnormal extremal_ in the sense of Definition 4.13. It is worth to
mention that by the results of [HLD18] no corner curve can be a SR geodesic.

###  4.5 On solutions of the characteristic OCP

In Ssec. 4.3 we were able to describe the shape of the minimal 2-jet
$\widehat{Q}_{t}(\bm{v})$, i.e the infimum of the set of values of
$Q_{t}[\Delta u]$ under the condition that $\bm{q}^{(1)}(t,\Delta u)=\bm{v}$.
This, in turn, allowed us to prove Thm 4.11 describing sub-Riemannian
optimality conditions of degree two. So far, however, we didn’t address the
question whether these infimum values are realized by some controls, i.e. if
the characteristic OCP actually has solutions. In general this will not
happen, as may be seen in the examples discussed in the previous subsection.
The reason is that $\widehat{Q}_{t}(\bm{v})$ is defined as an infimum taken
over elements $\Delta u$ belonging to an affine subspace of finite codimension
in $L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})$. And we lack any sort of
universal (weakly) compactness results in such situations. Below we will
discuss some conditions that can guarantee the existence of the infimum. They
are related with the assumptions used in [BMP20] to derive third-order analog
of Goh conditions. Later we will show that the existence of solutions of the
characteristic OCP imposes strong conditions on underlying the sub-Riemannian
trajectory.

#### Algebraic criteria for the existence of solutions

We begin our discussion with a simple algebraic criteria for a control to be a
solution of the characteristic OCP.

######  Proposition 4.14.

Let $\Delta u\in L^{\infty}([t_{0},t],\mathbb{R}^{k})$ be such that
$\bm{q}^{(1)}(t,\Delta u)=\bm{v}$. Then $\Delta u$ is a solution of the
characteristic OCP if and only if

  1. (i)

$B_{t}[\Delta u,\Delta v]=0$ for every $\Delta v\in\mathcal{U}_{0}(t_{0},t)$
and

  2. (ii)

$\widehat{Q}_{t}(\bm{0})$ is finite (hence equal to zero by Prop. 4.8).

###### Proof.

First let us prove that conditions (i)–(ii) imply optimality of the control
$\Delta u$. Note that controls $\Delta u,\Delta u^{\prime}\in
L^{\infty}([t_{0},t],\mathbb{R}^{k})$ are such that $\bm{q}^{(1)}(t,\Delta
u)=\bm{q}^{(1)}(t,\Delta u^{\prime})=\bm{v}$ if and only if $\Delta v:=\Delta
u^{\prime}-\Delta u$ belongs to $\mathcal{U}_{0}(t_{0},t)$. Now by our
assumptions $Q_{t}[\Delta v]\geq\widehat{Q}_{t}(\bm{0})=0$ and $B_{t}[\Delta
u,\Delta v]=0$. Therefore we have

$Q_{t}[\Delta u^{\prime}]=Q_{t}[\Delta u+\Delta v]=Q_{t}[\Delta
u]+2B_{t}[\Delta u,\Delta v]+Q_{t}[\Delta v]=Q_{t}[\Delta u]+Q_{t}[\Delta
v]\geq Q_{t}[\Delta u]\ ,$

Since $\Delta u^{\prime}$ was arbitrary, $\Delta u$ is indeed a solution of
the characteristic OCP.

Assume now that $\Delta u$ is a solution of the characteristic OCP. This
implies, in particular, that $\widehat{Q}_{t}(\bm{v})$ is finite for
$\bm{v}=\bm{q}^{(1)}(t,\Delta u)$. The latter condition suffices to prove
(ii). Indeed, take any $\Delta v\in\mathcal{U}_{0}(t_{0},t)$ and any
$\lambda\in\mathbb{R}$. Then

$Q_{t}[\Delta u+\lambda\Delta v]=Q_{t}[\Delta u]+2B_{t}[\Delta u,\lambda\Delta
v]+Q_{t}[\lambda\Delta v]=Q_{t}[\Delta u]+2\lambda\cdot B_{t}[\Delta u,\Delta
v]+\lambda^{2}\cdot Q_{t}[\Delta v]\ .$

Thus if $Q_{t}[\Delta v]$ would be negative, we could make $Q_{t}[\Delta
u+\lambda\Delta v]$ as low as possible by taking $\lambda$ big enough.
Therefore the assumption of finteness of $\widehat{Q}_{t}(\bm{v})$ requires
$Q_{t}[\Delta v]\geq 0$ for every $\Delta v\in\mathcal{U}_{0}(t_{0},t)$. In
consequence, $\widehat{Q}_{t}(\bm{0})\geq 0$ and thus
$\widehat{Q}_{t}(\bm{0})=0$ by Prop. 4.8.

To prove point (i) take $\Delta v\in\mathcal{U}_{0}(t_{0},t)$. Then for each
$\lambda\in\mathbb{R}$ the control $\Delta u+\lambda\Delta v$ belongs to
$L^{\infty}([t_{0},t],\mathbb{R}^{k})$, and ends at $\bm{q}^{(1)}(t,\Delta
u+\lambda\cdot\Delta v)=\bm{v}$. Thus the map

$\mathbb{R}\ni\lambda\longmapsto Q_{t}[\Delta u+\lambda\Delta v]=Q_{t}[\Delta
u]+2\lambda\cdot B_{t}[\Delta u,\Delta v]+\lambda^{2}\cdot Q_{t}[\Delta v]$

has a minimum at $\lambda=0$, which implies $B_{t}[\Delta u,\Delta v]=0$. ∎

We conclude that the space of solutions of the characteristic OCP has a simple
structure.

######  Proposition 4.15.

The set of solutions of the characteristic OCP

$\operatorname{Sol}_{t}{}:=\\{\Delta u\in
L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})\ |\ Q_{t}[\Delta
u]=\widehat{Q}_{t}(\bm{q}^{(1)}(t,\Delta u))\\}$

is a vector space. Further the space of solutions satisfying
$\bm{q}^{(1)}(t,\Delta u)=\bm{v}$

$\operatorname{Sol}_{t}{(\bm{v})}:=\\{\Delta u\in
L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})\ |\ Q_{t}[\Delta
u]=\widehat{Q}_{t}(\bm{v}),\bm{q}^{(1)}(t,\Delta u)=\bm{v}\\}$

is an affine space modelled on

$\operatorname{Sol}_{t}{(\bm{0})}=\\{\Delta u\in
L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})\ |\ Q_{t}[\Delta
u]=\widehat{Q}_{t}(\bm{0}),\bm{q}^{(1)}(t,\Delta u)=\bm{0}\\}\ .$

###### Proof.

By Prop. 4.14, each solution $\Delta u\in
L^{\infty}([t_{0},t],\mathbb{R}^{k})$ is characterized by the condition that
the map

$B_{t}[\Delta u,\cdot]:\mathcal{U}_{0}(t_{0},t)\longrightarrow\mathbb{R}$

is null. Note that $B_{t}[\Delta u,\cdot]$ is linear in $\Delta u$. This ends
the first part of the proof.

Let now $\Delta u,\Delta u^{\prime}$ be two elements of
$\operatorname{Sol}_{t}{(}\bm{v})$. Then, of course $\Delta v:=\Delta
u^{\prime}-\Delta u$ satisfies $\bm{q}^{(1)}(t,\Delta v)=\bm{0}$. Further

$\displaystyle 0=\widehat{Q}_{t}(\bm{v})-\widehat{Q}_{t}(\bm{v})=Q_{t}[\Delta
u^{\prime}]-Q_{t}[\Delta u]=Q_{t}[\Delta u+\Delta v]-Q_{t}[\Delta
u]=2B_{t}[\Delta u,\Delta v]+Q_{t}[\Delta v]\ .$

By the results of Prop. 4.14 the bilinear term vanishes, and so $Q_{t}[\Delta
v]=0=\widehat{Q}_{t}(\bm{0})$, i.e. $\Delta u^{\prime}-\Delta
u\in\operatorname{Sol}_{t}{(\bm{0})}$. Finally if put $\bm{v}=\bm{0}$ in the
above calculation, we would get that a difference of two elements of
$\operatorname{Sol}_{t}{(\bm{0})}$ lies in $\operatorname{Sol}_{t}{(\bm{0})}$.
We conclude that the latter is a vector space. This ends the proof. ∎

The space $\operatorname{Sol}_{\tau}{(\bm{0})}$ has some interesting
properties.

######  Proposition 4.16.

Assume that the minimal 2-jet $\widehat{Q}_{\tau}(\bm{v})$ is finite for every
$\bm{v}\in\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},\tau)$. Then for every
$\Delta u\in L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})$, and every $\Delta
v\in\operatorname{Sol}_{\tau}{(\bm{0})}$ we have

$B_{\tau}[\Delta u,\Delta v]=0\ .$

###### Proof.

Take any such $\Delta u$ and $\Delta v$. Since $Q_{\tau}[\Delta
v]=\widehat{Q}_{\tau}(\bm{0})=0$, for every $\lambda\in\mathbb{R}$ we have

$\displaystyle Q_{\tau}[\Delta u+\lambda\cdot\Delta v]=Q_{\tau}[\Delta
u]+2\lambda B_{\tau}[\Delta u,\Delta v]+\lambda^{2}Q_{\tau}[\Delta
v]=Q_{\tau}[\Delta u]+2\lambda B_{\tau}[\Delta u,\Delta v]$

Note that $\bm{q}^{(1)}(\tau,\Delta u+\lambda\Delta
v)=\bm{q}^{(1)}(\tau,\Delta u)$, as $\bm{q}^{(1)}(\tau,\Delta v)=0$. If
$B_{\tau}[\Delta u,\Delta v]\neq 0$, then by taking $\lambda$ big or small
enough we would find a control steering ($\Lambda_{1}$) from $\bm{0}$ to
$\bm{q}^{(1)}(\tau,\Delta u)$ with arbitrarily small cost. This contradicts
the finiteness of $\widehat{Q}_{\tau}(\bm{q}^{(1)}(\tau,\Delta u))$. ∎

Prop. 4.14 shows that solutions of the characteristic OCP are related with the
kernel of the linear maps $B_{\tau}[\Delta
u,\cdot]:\mathcal{U}_{0}(t_{0},\tau)\rightarrow\mathbb{R}$. On the other hand,
the previous Prop. 4.16 shows that the space
$\operatorname{Sol}_{\tau}{(\bm{0})}$ is contained in that kernel. It turns
out that if the space $\operatorname{Sol}_{\tau}{(\bm{0})}$ is big enough it
may guarantee the existence of solutions of the characteristic OCP. To
formalize this property let us propose the following definition.

######  Definition 4.17.

Let $q(t)$ with $t\in[t_{0},t_{1}]$ be a sub-Riemannian trajectory and
consider the related characteristic OCP. We say that the trajectory satisfies
_Monti’s condition_ at time $\tau\in[t_{0},t_{1}]$ if the space
$\operatorname{Sol}_{\tau}{(\bm{0})}$ is of finite codimension in
$L^{\infty}([t_{0},\tau],\mathbb{R}^{k})$ (equivalently: of finite codimension
in $\mathcal{U}_{0}(t_{0},\tau)$, as the latter is a space of finite
codimension in $L^{\infty}([t_{0},\tau],\mathbb{R}^{k})$ being the kernel of
the linear map
$\bm{q}^{(1)}:L^{\infty}([t_{0},\tau],\mathbb{R}^{k})\rightarrow\mathbb{R}^{n+1}$).

We propose the name Monti’s condition as this is one of the assumptions used
in [BMP20] to derive the third-order optimality conditions.444Actually, the
assumption of [BMP20, Thm 1.2] translated to the notation of this paper is
that the space of elements  $\operatorname{dom}(\mathcal{D}^{3}F):=\\{\Delta
v\in L^{\infty}([t_{0},\tau],\mathbb{R}^{k})\ |\ \bm{q}^{(1)}(\tau,\Delta
v)=0\ \text{and $\bm{q}^{(11)}(\tau,\Delta u,\Delta
v)\in\operatorname{Im}\bm{q}^{(1)}(\tau,\cdot)$ for every $\Delta u\in
L^{\infty}([t_{0},t],\mathbb{R}^{k})$}\\}$ is of finite co-dimension in
$\mathcal{U}_{0}(t_{0},t)$. Here, $2\bm{q}^{(11)}(\tau,\Delta u,\Delta
v):=\bm{q}^{(2)}(\tau,\Delta u+\Delta v)-\bm{q}^{(2)}(\tau,\Delta
u)-\bm{q}^{(2)}(\tau,\Delta v)$ is the bi-linear form related with the
quadratic map $\bm{q}^{(2)}(\tau,\cdot)$. Note that, in particular, every
element $\Delta v\in\operatorname{dom}(\mathcal{D}^{3}F)$ satisfies
$\bm{q}^{(11)}(\tau,\Delta v,\Delta v)=\bm{q}^{(2)}(t,\Delta
v)\in\operatorname{Im}\bm{q}^{(1)}(\tau,\cdot)\subset\ker\psi_{0}$, where
$\psi_{0}$ is as Prop. 4.2. Consequently, $Q_{\tau}[\Delta
v]=\big{\langle}\bm{q}^{(2)}(\tau,\Delta v),\psi_{0}\big{\rangle}=0$ and thus
$Q_{\tau}[\Delta v]=0=\widehat{Q}_{\tau}(\bm{0})$, i.e. $\Delta
v\in\operatorname{Sol}_{\tau}{(\bm{0})}$ whenever $\widehat{Q}_{\tau}(\bm{0})$
is finite. Summing up, if $\widehat{Q}_{\tau}(\bm{0})$ is finite then
$\operatorname{dom}(\mathcal{D}^{3}F)\subset\operatorname{Sol}_{\tau}{(\bm{0})}\subset
L^{\infty}([t_{0},\tau],\mathbb{R}^{k})\ $ and thus the condition of
Definition 4.17 is weaker then the assumption made in [BMP20]. This assumption
is, however, a strong one. Further it may not be satisfied, as is the case in
the two examples discussed in Ssec. 4.4.

######  Lemma 4.18.

Let $q(t)$ with $t\in[t_{0},t_{1}]$ be a sub-Riemannian trajectory. Consider
the related characteristic OCP and assume that the minimal 2-jet
$\widehat{Q}_{\tau}(\bm{v})$ is finite for every
$\bm{v}\in\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},\tau)$. If the trajectory
satisfies Monti’s condition at time $\tau$, then the set
$\operatorname{Sol}_{t}{(\bm{v})}$ is non-empty for every
$\bm{v}\in\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},\tau)$.

###### Proof.

We will show that Monti’s condition actually reduces the characteristic OCP to
a finite-dimensional optimization problem.

Consider a control $\Delta u\in L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})$
satisfying $\bm{q}^{(1)}(\tau,\Delta u)=\bm{v}$, and let us introduce any
splitting
$\mathcal{U}_{0}{(t,\tau)}=V\oplus\operatorname{Sol}_{\tau}{(\bm{0})}$, where
$V$ is finite-dimensional. The quadratic map $\Delta v\mapsto Q_{\tau}[\Delta
v]=B_{\tau}[\Delta v,\Delta v]$ is positively-defined on $V$, and so we may
find a $B_{\tau}[\cdot,\cdot]$-orthonormal basis $\\{\Delta
v_{\alpha}\\}_{\alpha\in A}$ of $V$. Denote $\lambda_{\alpha}:=B_{\tau}[\Delta
u,\Delta v_{\alpha}]$ and consider $\Delta u^{\prime}=\Delta u-\sum_{\alpha\in
A}\lambda_{\alpha}\cdot\Delta v_{\alpha}$. By construction
$\bm{q}^{(1)}(\tau,\Delta u^{\prime})=\bm{q}^{(1)}(\tau,\Delta u)=\bm{v}$, and
it is straightforward to check that $B_{\tau}[\Delta u^{\prime},\Delta
v_{\alpha}]=0$ for every $\alpha\in A$. Since, by Prop. 4.16, also
$B_{\tau}[\Delta u^{\prime},\Delta v]=0$ for every $\Delta
v\in\operatorname{Sol}_{\tau}{(\bm{0})}$, the linear map $B_{\tau}[\Delta
u^{\prime},\cdot]$ vanishes on the whole
$\mathcal{U}_{0}{(t,\tau)}=V\oplus\operatorname{Sol}_{\tau}{(\bm{0})}$, and
thus $Q_{\tau}[\Delta u^{\prime}]=\widehat{Q}_{\tau}(\bm{v})$ by Prop. 4.14. ∎

#### Differential conditions

Once the existence of solutions of the characteristic OCP is established, we
may use the Pontryagin Maximum Principle (PMP, in short) [PMBG62] to derive
necessary conditions for optimality. They will give us a lot of information on
the trajectory. When discussing solutions of the characteristic OCP we will
always speak about normal or abnormal _extremals of the characteristic OCP_ ,
not to be confused with the _sub-Riemannian extremal_ that constitutes the
characteristic OCP itself.

For the characteristic OCP the _Pontryagin Hamiltonian_
$H:\mathbb{R}\times\mathbb{R}^{n+1}\times(\mathbb{R}^{n+1})^{\ast}\times\mathbb{R}^{k}\longrightarrow\mathbb{R}$
is defined as

$\displaystyle H(t,(q^{(1)},c^{(1)}),$ $\displaystyle(\psi,a),\Delta
v)=H(t,q^{(1)},(\psi,a),\Delta v):=$ $\displaystyle\big{\langle}\sum_{i}\Delta
v_{i}\cdot Y_{i}^{(1)}(t),\psi\big{\rangle}+a\sum_{i}\Delta v_{i}\cdot
u_{i}(t)+p_{0}\big{\langle}\sum_{i}\Delta v_{i}\cdot
Y_{i}^{(2)}(t,q^{(1)}),\psi_{0}\big{\rangle}\ .$

Here, $(\psi,a)\in(\mathbb{R}^{n+1})^{\ast}$ is a covector dual to the pair
$(q^{(1)},c^{(1)})\in\mathbb{R}^{n+1}$ and $p_{0}$ is either -1 (in the normal
case) or $0$ (in the abnormal case). Actually, we may assume that the pair
$(\psi,a)$ belongs to the dual of the reachable set
$\bm{\mathcal{R}}_{(\Lambda_{1})}(t_{0},t_{1})$. Note that since $H$ does not
depends on $c^{(1)}$, the dual object $a$ will always be constant. The
following result is immediate.

######  Proposition 4.19.

Let the control $\Delta u\in L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})$
together with a covector curve $(\psi(t),a(t))$ be an extremal of the the
characteristic OCP related with the SR trajectory $q(t)$, corresponding to the
control $u\in L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})$. Then in the abnormal
case ($p_{0}=0$) both $\psi(t)$ and $a(t)$ are constant. In the normal case
($p_{0}=-1$) $a(t)$ is constant, while $\psi(t)$ is subject to the evolution
equation

(4.13) $\big{\langle}b,\dot{\psi}(t)\big{\rangle}=\big{\langle}\sum_{i}\Delta
u_{i}(t)\cdot Y_{i}^{(2)}(t,b),\psi_{0}\big{\rangle}$

where $b\in\mathbb{R}^{n}$ is arbitrary.

Further, as the Hamiltonian $H(t,q^{(1)},(\psi,a),\Delta v)$ is a linear
function of $\Delta v$, and we have no bounds on the control, the Maximum
Principle gives us the following conditions

(4.14) $\big{\langle}Y_{i}^{(1)}(t),\psi(t)\big{\rangle}+a\cdot
u_{i}(t)+p_{0}\big{\langle}Y_{i}^{(2)}(t,q^{(1)}),\psi_{0}\big{\rangle}=0$

for every $i=1,2,\ldots,k$.

The existence of certain extremals of the PMP for the characteristic OCP
allows to deduce regularity of the underlying SR trajectory $q(t)$.

######  Lemma 4.20.

Under the assumptions of Prop. 4.19

  * •

if $\Delta u$ is an abnormal extremal of the characteristic OCP with $a\neq
0$, the SR trajectory $q(t)$ has to be normal and thus $C^{\infty}$-smooth;

  * •

if $\Delta u$ is a normal extremal of the characteristic OCP with $a\neq 0$,
then the SR trajectory $q(t)$ is $C^{1}$,

  * •

if $\Delta u$ is a normal extremal of the characteristic OCP with $a\neq 0$,
and the SR trajectory $q(t)$ additionally satisfies Goh conditions, then
$q(t)$ is $C^{2}$.

###### Proof.

To prove the first part, let $(\psi,a)\in(\mathbb{R}^{n+1})^{\ast}$ be the
abnormal covector curve of the characteristic OCP (it is constant by Prop.
4.19). The Maximum Principle (4.14) gives us in this case

$\big{\langle}Y_{i}^{(1)}(t),\psi\big{\rangle}+a\cdot u_{i}(t)=0\quad\text{for
$i=1,2,\ldots,k$}.$

These are precisely the equations of the normal SR trajectory. In particular
we get $u_{i}(t)=-\frac{1}{a}\big{\langle}Y_{i}^{(1)}(t),\psi\big{\rangle}$.
Now note that on the left-hand side we have something of class $C^{r}$ (with
$r=0_{-},1_{-},1,2,\ldots$) while, by the results of Lemma 3.1, on the right-
hand side something of class $C^{r+1}$. Hence $u_{i}(t)$ must be
$C^{\infty}$-smooth by the bootstrap argument.

A similar argument works also for a normal extremal of the characteristic OCP.
In that case, by (4.14) we have

$u_{i}(t)=-\frac{1}{a}\left(\big{\langle}Y_{i}^{(1)}(t),\psi(t)\big{\rangle}-\big{\langle}Y_{i}^{(2)}(t,q^{(1)}(t)),\psi_{0}\big{\rangle}\right)\
.$

Now note that, if $u(t)$ was of class $L^{\infty}$, then, by Lemma 3.1,
$Y_{i}^{(1)}(t)$ and $Y_{i}^{(2)}(t,\cdot)$ are absolutely continuous with
respect to $t$. Further the covector curve $\psi(t)$, as well as the
trajectory $q^{(1)}(t,\Delta u)$ are Lipschitz. Since $Y_{i}^{(2)}(t,q^{(1)})$
depends linearly on $q^{(1)}$, we conclude that $u_{i}(t)$ is absolutely
continuous and, consequently, $q(t)$ is $C^{1}$. This ends the proof of the
second part.

For the last part we differnetiate (4.14) (which we now know is possible
almost everywhere) to get

$\displaystyle a\dot{u_{i}}(t)=$
$\displaystyle-\big{\langle}\partial_{t}Y_{i}^{(1)}(t),\psi(t)\big{\rangle}-\big{\langle}Y_{i}^{(1)}(t),\dot{\psi}(t)\big{\rangle}+\big{\langle}\partial_{t}Y_{i}^{(2)}(t,q^{(1)}(t)),\psi_{0}\big{\rangle}+\big{\langle}Y_{i}^{(2)}(t,\dot{q^{(1)}}(t)),\psi_{0}\big{\rangle}\overset{\eqref{eqn:ev_PMP},\eqref{eqn:CS_deg_1}}{=}$
$\displaystyle-\big{\langle}\partial_{t}Y_{i}^{(1)}(t),\psi(t)\big{\rangle}+\big{\langle}\partial_{t}Y_{i}^{(2)}(t,q^{(1)}(t)),\psi_{0}\big{\rangle}$
$\displaystyle-\big{\langle}\sum_{j}\Delta u_{j}(t)\cdot
Y_{j}^{(2)}(t,Y_{i}^{(1)}(t)),\psi_{0}\big{\rangle}+\big{\langle}Y_{i}^{(2)}(t,\sum_{j}\Delta
u_{j}(t)\cdot Y_{j}^{(1)}(t)),\psi_{0}\big{\rangle}=$
$\displaystyle-\big{\langle}\partial_{t}Y_{i}^{(1)}(t),\psi(t)\big{\rangle}+\big{\langle}\partial_{t}Y_{i}^{(2)}(t,q^{(1)}(t)),\psi_{0}\big{\rangle}$
$\displaystyle\quad\sum_{j}\Delta
u_{j}\big{\langle}Y_{i}^{(2)}(t,Y_{j}^{(1)}(t))-Y_{j}^{(2)}(t,Y_{i}^{(1)}(t)),\psi_{0}\big{\rangle}\overset{\text{Lem.~{}\ref{lem:adapted}}}{=}$
$\displaystyle-\big{\langle}\partial_{t}Y_{i}^{(1)}(t),\psi(t)\big{\rangle}+\big{\langle}\partial_{t}Y_{i}^{(2)}(t,q^{(1)}(t)),\psi_{0}\big{\rangle}+\sum_{j}\Delta
u_{j}\big{\langle}\Phi^{(1)}(t)[X_{i},X_{j}],\psi_{0}\big{\rangle}\ ,$

where the last passage is explained in detail in a remark following the proof
of Thm 4.10. By Lemma 3.1 the $t$-dependence of $Y_{i}^{(1)}(t)$ and
$Y_{i}^{(2)}(t,\cdot)$ is $C^{1}$ (as $u_{i}(t)$’s are now absolutely
continuous), and the term involving the Lie bracket vanishes because of the
Goh condition. By repeating our reasoning from the previous point,
$\dot{u_{i}}(t)$ is absolutely continuous and hence $q(t)$ is actually
$C^{2}$. ∎

With a little more effort, combining the information provided by the PMP with
our previous results about optimality conditions of degree two, we can prove
that actually solutions of the characteristic OCP are precisely the normal
extremals of the PMP. Further, the existence of particular solutions of the
characteristic OCP guarantees the $C^{2}$-regularity of the underlining SR
trajectory.

######  Lemma 4.21.

Let $q(t)$ with $t\in[t_{0},t_{1}]$ be a strictly abnormal SR extremal.

  1. (i)

Assume that $\widehat{Q}_{t}(\bm{0})$ is finite. Then the control $\Delta u\in
L^{\infty}([t_{0},t],\mathbb{R}^{k})$ is a solution of the characteristic OCP
if and only if it is a normal extremal of the characteristic OCP.

  2. (ii)

Assume that the assertion of Thm 4.10 holds. In particular, let
$a_{2}(t)\in\mathbb{R}$ and $\zeta(t)\in\mathrm{T}^{\ast}_{q(t)}M$ be objects
defined by this result. Then if the control $\Delta u\in
L^{\infty}([t_{0},t],\mathbb{R}^{k})$ is such a solution of the characteristic
OCP that $\big{\langle}q^{(1)}(t,\Delta
u),\zeta(t)\big{\rangle}+2a_{2}(t)\cdot c^{(1)}(t,\Delta u)\neq 0$, the SR
trajectory $q(t)$ is of class $C^{2}$.

###### Proof.

If $\Delta u$ is a solution of the characteristic OCP it must be either a
normal, or an abnormal extremal of the PMP. The abnormal case can be easily
excluded. Indeed, let $\Delta u$ be an abnormal extremal corresponding to a
covector $(\psi,a)\in(\mathbb{R}^{n+1})^{\ast}$. By the results of previous
Lemma 4.20, if $a\neq 0$, then $q(t)$ must be a normal SR extremal, which is
impossible since it is assumed to be strictly abnormal.

In the case $a=0$, by (4.14) the covector $\psi$ annihilates all control
fields $Y_{i}^{(1)}(t)$ and, in consequence, the whole reachable set of
($\Lambda_{1}$), so $\psi=0$ and $a=0$ which is impossible, as the covector
defined by the PMP has to be nowhere-vanishing.

Now we can concentrate on the case where $\Delta u$ is a normal extremal of
the characteristic OCP. Let $(\psi(t),a)\in(\mathbb{R}^{n+1})^{\ast}$ be the
related covector curve. For any other control $\Delta v\in
L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})$ we would like to calculate the bi-
linear form $B_{t}[\Delta u,\Delta v]$. Note that

$\displaystyle\big{\langle}\sum_{i}\Delta v_{i}(t)\cdot
Y_{i}^{(2)}(t,q^{(1)}(t,\Delta
u)),\psi_{0}\big{\rangle}\overset{\eqref{eqn:max_princip}}{=}$
$\displaystyle\big{\langle}\sum_{i}\Delta v_{i}(t)\cdot
Y_{i}^{(1)}(t),\psi(t)\big{\rangle}+a\cdot\sum_{i}\Delta v_{i}(t)\cdot
u_{i}(t)\overset{\eqref{eqn:CS_deg_1}}{=}$
$\displaystyle\big{\langle}{\dot{q}}^{(1)}(t,\Delta
v),\psi(t)\big{\rangle}+a\cdot\dot{c^{(1)}}(t,\Delta v)\ .$ On the other hand,
$\displaystyle\big{\langle}\sum_{i}\Delta u_{i}(t)\cdot
Y_{i}^{(2)}(t,q^{(1)}(t,\Delta
v)),\psi_{0}\big{\rangle}\overset{\eqref{eqn:ev_PMP}}{=}$
$\displaystyle\big{\langle}q^{(1)}(t,\Delta v),\dot{\psi}(t)\big{\rangle}$

By adding the two above equalities and integrating over $[t_{0},t]$ we get

(4.15) $B_{t}[\Delta u,\Delta v]=\big{\langle}q^{(1)}(t,\Delta
v),\psi(t)\big{\rangle}+a\cdot c^{(1)}(t,\Delta v)\ .$

Our first conclusion is that whenever $\Delta v\in\mathcal{U}_{0}(t_{0},t)$
then $q^{(1)}(t,\Delta v)=0$ and $c^{(1)}(t,\Delta v)=0$, and thus
$B_{t}[\Delta u,\Delta v]=0$. By Prop. 4.14 this (together with the finiteness
of $\widehat{Q}_{t}(\bm{0})$) implies that $\Delta u$ is optimal, proving
point (i).

To prove (ii), take any $\Delta v\in
L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})$, $\varepsilon\in\mathbb{R}$, and
denote $\bm{q}^{(1)}(t,\Delta
v)=\bm{v}=(v,c)\in\mathbb{R}^{n}\times\mathbb{R}$. We have

$\displaystyle Q_{t}[\Delta u+\varepsilon\cdot\Delta v]=$ $\displaystyle
Q_{t}[\Delta u]+\varepsilon B_{t}[\Delta u,\Delta
v]+\varepsilon^{2}Q_{t}[\Delta v]=$
$\displaystyle\widehat{Q}_{t}(\bm{q}^{(1)}(t,\Delta
u))+\varepsilon\big{\langle}\bm{v},(\psi(t),a)\big{\rangle}+\varepsilon^{2}Q_{t}[\Delta
v]\geq$ $\displaystyle\widehat{Q}_{t}(\bm{q}^{(1)}(t,\Delta
u))+\varepsilon\big{\langle}\bm{v},(\psi(t),a)\big{\rangle}+\varepsilon^{2}\widehat{Q}_{t}(\bm{v})\
,$

and thus

(4.16) $\widehat{Q}_{t}(\bm{q}^{(1)}(t,\Delta
u)+\varepsilon\bm{v})\geq\widehat{Q}_{t}(\bm{q}^{(1)}(t,\Delta
u))+\varepsilon\big{\langle}\bm{v},(\psi(t),a)\big{\rangle}+\varepsilon^{2}\widehat{Q}_{t}(\bm{v})\
.$

On the other hand, by Thm 4.10 we know that the minimal 2-jet
$\widehat{Q}_{t}(\cdot)$ is a quadratic function and hence

$\displaystyle\widehat{Q}_{t}(\bm{q}^{(1)}(t,\Delta u)+\varepsilon\bm{v})=$
$\displaystyle\bm{\Psi}_{2}[\bm{q}^{(1)}(t,\Delta
u)+\varepsilon\bm{v},\bm{q}^{(1)}(t,\Delta u)+\varepsilon\bm{v}]=$
$\displaystyle\bm{\Psi}_{2}[\bm{q}^{(1)}(t,\Delta u),\bm{q}^{(1)}(t,\Delta
u)]+2\varepsilon\bm{\Psi}_{2}[\bm{v},\bm{q}^{(1)}(t,\Delta
u)]+\varepsilon^{2}\bm{\Psi}_{2}[\bm{v},\bm{v}]=$
$\displaystyle\widehat{Q}_{t}(\bm{q}^{(1)}(t,\Delta
u))+2\varepsilon\bm{\Psi}_{2}[\bm{v},\bm{q}^{(1)}(t,\Delta
u)]+\varepsilon^{2}\widehat{Q}_{t}(\bm{v})\overset{\eqref{eqn:ineq_Q}}{\geq}$
$\displaystyle\widehat{Q}_{t}(\bm{q}^{(1)}(t,\Delta
u))+\varepsilon\big{\langle}\bm{v},(\psi(t),a)\big{\rangle}+\varepsilon^{2}\widehat{Q}_{t}(\bm{v}).$

As $\varepsilon$ and $\bm{v}=(v,c)$ were arbitrary, the above inequality may
hold if and only if

$\displaystyle\big{\langle}v,\psi(t)\big{\rangle}+c\cdot a=$
$\displaystyle\big{\langle}\bm{v},(\psi(t),a)\big{\rangle}=2\bm{\Psi}_{2}[\bm{v},\bm{q}^{(1)}(t,\Delta
u)]=$ $\displaystyle 2\Psi_{2}(q^{(1)}(t,\Delta
u),v)+\big{\langle}v,\zeta(t)\big{\rangle}c^{(1)}(t,\Delta
u)+c\cdot\left(\big{\langle}q^{(1)}(t,\Delta
u),\zeta(t)\big{\rangle}+2a_{2}(t)\cdot c^{(1)}(t,\Delta u)\right)\ .$

In particular, $a=\big{\langle}q^{(1)}(t,\Delta
u),\zeta(t)\big{\rangle}+2a_{2}(t)\cdot c^{(1)}(t,\Delta u)$. By Lemma 4.20 if
the latter number is different than zero and the Goh conditions are satisfied
(this is a consequence of Thm 4.10), the trajectory $q(t)$ is of class
$C^{2}$. ∎

We end our consideration by stating the following consequences of Monti’s
condition.

######  Lemma 4.22.

Let $q(t)$, with $t\in[t_{0},t_{1}]$, be a minimizing strictly abnormal SR
geodesic. Assume additionally that it satisfies Monti’s condition at time
$t_{1}$ (Definition 4.17).

Consider the division of $q(t)$ into a finite number of pieces provided by Thm
4.11. Then on each of these pieces $q(t)$ is either a 2-abnormal extremal in
the sense of Definition 4.13, or of class $C^{2}$.

  1. (i)

The assertion of Thm 4.11 holds.

  2. (ii)

Let $a_{2}(t)\in\mathbb{R}$ and $\zeta(t)\in\mathrm{T}^{\ast}_{q(t)}M$ be
objects defined by Thm 4.10. Then $a_{2}(t_{1})\neq 0$ or $\zeta(t_{1})\neq 0$
then $q(t)$ is of class $C^{2}$. Moreover, there exists $\tau\in[t_{0},t_{1})$
such that $q(t)$ is of class $C^{\infty}$ on $[\tau,t_{1}]$.

###### Proof.

By our previous results we may divide $q(t)$ into pieces
$[\tau_{i},\tau_{i+1}]$, such that inside each of them, i.e on
$[t_{0}^{\prime},t_{1}^{\prime}]\subset(\tau_{i},\tau_{i+1})$, the assertion
of Thm 4.10 holds. In particular, we can assure the finteness of the minimal
2-jet $\widehat{Q}_{t}(\cdot)$, and the existence of the curve $a_{2}(t)$ for
$t\in[t_{0}^{\prime},t_{1}^{\prime}]$. Observe that if the Monti’s condition
holds at $t_{1}$ then also it holds at $t_{1}^{\prime}$ on the shorter
trajectory as all we have to do is to take the finite-codimensional extension
$\operatorname{Sol}_{t_{1}}{(}\bm{0})\subset
L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})$ and interset it with
$L^{\infty}([t_{0}^{\prime},t_{1}^{\prime}],\mathbb{R}^{k})$. Consequenlty, we
may apply the results of Lemma 4.18, which guarantees that the characteristic
OCP has solutions on the sub-interval $[t_{0}^{\prime},t_{1}^{\prime}]$.

Now we have two possibilities, either the piece $q(t)$, with
$t\in[t_{0},t_{1}^{\prime}]$, is a 2-abnormal extremal, or $a_{2}(t)\neq 0$
for every $t$ (the 2-normal case). In the latter situation, we simply take any
solution of the characteristic OCP described by Lemma 4.21 point (ii) which
guarantees that $q(t)$ is of class $C^{2}$ on
$[t_{0}^{\prime},t_{1}^{\prime}]$. This ends the proof. ∎

######  Remark 4.23.

Actually is seems to be possible to further enhance the regularity of the
abnormal geodesic if we assume that Monti’s condition holds. Namely, note that
if $\Delta u\in L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})$ is a solution of the
characteristic OCP then $\big{\langle}\bm{q}^{(2)}(t,\Delta
u),\psi_{0}\big{\rangle}-\widehat{Q}_{t}(\bm{q}^{(1)}(t,\Delta u))=0$ for
every $t$. In particular, the differential of the latter expression is zero,
and, by repeating the argument from the proof of Thm 4.10, the quadratic form

$\dot{\Psi}_{2}(t)[q^{(1)},q^{(1)}]+\big{\langle}q^{(1)},\dot{\zeta}(t)\big{\rangle}c^{(1)}+\dot{a_{2}}(t)(c^{(1)})^{2}$

has zero at $\bm{q}^{(1)}(t,\Delta u)$. Consequently if the Monti’s condition
holds at $t_{1}$, then we may find a solution of the characteristic OCP giving
every possible value of $\bm{q}^{(1)}(t_{1},\Delta u)$, hence the above
quadratic form must be zero at $t_{1}$ (and also in some neighbourhood of
$t_{1}$ by the continuity argument). Hence the bilinear form
$\bm{\Psi}_{2}(t)$ defined in the assertion of Thm 4.10 has a trivial
evolution in a neighbourhood of $t_{1}$, and so in the 2-normal case equation
(4.5) would give us $C^{\infty}$-regularity of the control $u_{i}(t)$ in a
neighbourhood of $t_{1}$.

##  5 Further study

The presented material provokes a few interesting questions:

  * •

Is it possible to improve the time-regularity of the minimal 2-jet
$\widehat{Q}_{\tau}(\bm{v})$ in Thm. 4.9? Answering this question would be an
important step in solving the regularity problem of SR geodesics. Namely, if
the $\tau$-regularity of $\widehat{Q}_{\tau}(\bm{v})$ could be improved to,
say being of class $C^{k}$, then in the 2-normal case equation (4.8) would
guarantee that the control $u(t)$ is also of class $C^{k}$, and thus the
trajectory $q(t)$ be of class $C^{k+1}$ on each piece from a finite
collection.

We hypothesize that it is possible to prove that if the control $u(t)$ is of
class $C^{r}$ (where $r=0_{-},1_{-},1,2,\ldots$) then the dependence
$\tau\mapsto\widehat{Q}_{\tau}(\bm{v})$ is of class $C^{r+1}$. This, by the
standard bootstrap argument, will imply that every 2-normal trajectory $q(t)$
is piece-wise smooth (with a finite number of pieces), thus smooth by the
results of [HLD16].

  * •

For 2-abnormal trajectories, as equation (4.8) does not determines the control
$u(t)$, the question of regularity remains open regardless of the regularity
properties of the minimal 2-jet $\widehat{Q}_{\tau}(\bm{v})$. It seems that
for such curves, one needs to understand the geometry of third-order (and
higher) order expansion of the extended end-point map. Some research in this
direction was initiated in [BMP20], yet a fully satisfactory theory is yet to
be developed.

  * •

Sub-Riemannian normal extremals are known to be locally minimizing. It is
natural to ask whether the same is true for 2-normal extremals. We hypothesize
that the answer to this question is positive, and could be provided by
extending the methods from [JR16b].

  * •

Our proof is based on Agrachev-Sarychev Index Lemma (Lem. 4.1). It seems to us
that is is possible to (slightly) modify the proof of this result to guarantee
the following result.

Hypothesis. Let $q(t)$, with $t\in[t_{0},t_{1}]$ be a minimizing strictly
abnormal SR geodesic of co-rank $r$. Then there exists a covector
$\varphi_{0}:\mathrm{T}_{q(t_{1})}M\rightarrow\mathbb{R}$, and a symmetric
2-form
$\bm{\Phi}_{0}:\mathrm{T}_{q(t_{1})}(M\times\mathbb{R})\times\mathrm{T}_{q(t_{1})}(M\times\mathbb{R})\rightarrow\mathbb{R}$
such that:

    * –

$\big{\langle}\bm{b}^{(1)}(t_{1},\Delta u),\varphi_{0}\big{\rangle}=0$ for
every $\Delta u\in L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})$

    * –

the negative index of the 2-form

$L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})\ni\Delta
u\longmapsto\big{\langle}\bm{b}^{(2)}(t_{1},\Delta
u),\varphi_{0}\big{\rangle}-\bm{\Phi}_{0}[\bm{b}^{(1)}(t_{1},\Delta
u),\bm{b}^{(1)}(t_{1},\Delta u)]$

is at most $r$.

In the standard version the bi-linear term $\bm{\Phi}_{0}$ is not present, as
the 2-form is restricted to a subspace
$\ker\operatorname{D}_{u}{\mathbf{End}}^{[t_{0},t_{1}]}_{q_{0}}\subset
L^{\infty}([t_{0},t_{1}],\mathbb{R}^{k})$.

It should be also possible to prove that necessarily
$\bm{\Phi}_{0}[(0,1),(0,1)]\geq 0$, and so $a_{2}(t)$ described by Thm 4.11 is
always non-negative.

Such an improvement on Agrachev-Sarychev Lemma will allow to simplify the
cutting procedure that we used in order to guarantee the finiteness of all
minimal 2-jets $\widehat{Q}_{\tau}(\bm{v})$, and thus reduce the estimate on
the maximal number of divisions in Thm 4.11.

We plan to address these questions in a future series of publications.

## Acknowledgements

I would like to thank Witold Respondek for bringing my attention to the topic
of sub-Riemannian geodesics, our initial work on this topic during my stay in
Rouen in 2016, and his vast knowledge of control theory which he eagerly
shared with me. My special thanks goes to Piotr Mormul for his words of
encouragement and support, often expressed in his haiku-like emails.

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