Click the title to view the abstract.

#### Andrei A. AGRACHEV (SISSA, Trieste, Italy and Steklov Mathematical Institute, Moscow, Russia)

Abnormal geodesics and the shape of sub-Riemannian spheres

We discuss local structure of the sub-Riemannian sphere near the endpoint of a geodesic emphasizing the differnce between normal and abnormal geodesics.

#### Yuliy BARYSHNIKOV (University of Illinois, Urbana-Champaign, USA)

Asymptotic of higher winding numbers of random walks and
hypoelliptic diffusions

We consider a hierarchy of "higher winding numbers" for a random walk
in a compact non-simply connected domain (e.g. disk with holes).
The random invariants converge, upon appropriate rescaling, to the law of hypoelliptic left-invariant diffusion on a nilpotent Lie group.
The case of disk with holes (and corresponding free nilpotent groups) will be addressed in some details.

#### Anthony BLOCH (University of Michigan, Ann Arbor, USA)

Optics, Action Principles and Nonholonomic Mechanics

(slides)
In this talk I will discuss the role of various action principles in mechanics and their relationship to action principles in optics.
I will discuss how an optical analogy may be developed for certain nonholonomic systems with both linear and nonlinear constraints
and how this relates to Hamiltonization and to multiparticle dynamics.
In particular, I will show how the dynamics of the knife edge is related to that of the brachistochrone problem.
This is joint work with A. Rojo.

#### Alexey V. BOLSINOV (Loughborough University, UK)

Topological monodromy as an obstruction to Hamiltonization of nonholonomic systems: pro or contra?

The phenomenon of a topological monodromy in integrable Hamiltonian and nonholonomic systems is discussed.
An efficient method for computing and visualizing the monodromy is developed.
The comparative analysis of the topological monodromy is given for the rolling ellipsoid of revolution problem in two cases,
namely, on a smooth and on a rough plane.
The first of these systems is Hamiltonian, the second is nonholonomic.
We show that, from the viewpoint of monodromy, there is no difference between the two systems,
and thus disprove the conjecture by Cushman and Duistermaat stating that
the topological monodromy gives a topological obstruction for Hamiltonization of the rolling ellipsoid of revolution on a rough plane.

#### Bernard BONNARD (INRIA and Université de Bourgogne, Dijon, France)

Lunar and J2 perturbations of the metric associated to the averaged orbital transfer

(slides)
In a series of previous articles we introduced a Riemannian metric associated to the energy minimization orbital transfer
with low propulsion. The aim of this article is the study the deformation of this metric due to standard perturbations
e.g. oblatness of the Earth and Lunar attraction.
Using Hamiltonian formalism we describe the effects of the perturabation to the orbital transfers and the deformation of
the conjugate and cut loci of the original metric.

#### Jean-Baptiste CAILLAU (CNRS / INRIA and Université de Bourgogne, Dijon, France)

$L^1$ minimization in space mechanics

There is a renewed interest in low thrust space missions (see, e.g. Lisa Pathfinder and BepiColombo programs).
Two or three body controlled models provide a nice framework to study these problems.
The objective is to minimize the overall consumption of the spacecraft,
which amounts to minimizing the $L^1$ norm of the control.
The properties of the extremal flow are encoded by the Poisson structure generated by two Hamiltonians;
one observes not only the sparsity of solutions peculiar to $L^1$ minimization,
but also the existence of singular trajectories known after the work of Robbins and Marchal in the late sixties,
Zelikin and Borisov more recently.
Second order conditions for broken extremals will be addressed.

Joint work with Z. Chen and Y. Chitour (Univ. Paris Sud & CNRS, France)

#### Camillo DE LELLIS (UZH, Zürich, Switzerland)

Energy conservation and Onsager's conjecture on the Euler equations

In his famous 1949 paper on statistical hydrodynamics Lars Onsager advanced the
following "conjecture" on weak solutions of the incompressible Euler equations:

*For any $\alpha<\frac{1}{3}$ there are weak solutions which are $\alpha$-Hölder continuous
in space and for which the total kinetic energy is not conserved.*

The first theorem on the existence of an $L^2$ weak solution which does not preserve the kinetic energy was proved by Scheffer in 1993.
In 2007 we introduced methods from the theory of differential inclusions to explain the existence of such solutions and subsequently we were able
to improve our techniques in order to produce Hölder dissipative solutions.
Thanks to our works and to further contributions of P. Isett and T. Buckmaster, the current gap with the Onsager's conjecture is only at the level of summability.
More precisely there are non-conservative solutions which are $(\frac{1}{3}-\varepsilon)$-Hölder continuous in space for almost
every time slice and the corresponding H\"older constant is integrable as a function of time.

Joint work with László Székelyhidi (Hausdorff Center for Mathematics, Bonn, Germany).

#### Manuel DE LEON (ICMAT, Madrid, Spain)

Reduction of the Hamilton-Jacobi equation

(slides)
We shall discuss a reduction procedure of the Hamilton-Jacobi equation for Hamiltonian systems with symmetries.
A reconstruction process is also developed. Finally we will study the case of complete solutions in this framework.

#### Yuri FEDOROV (Universitat Politècnica de Catalunya, Barcelona, Spain)

Straight line rolling of an ellipsoid on a plane and the Chasles theorem

(slides)
Let a triaxial ellipsoid roll without slipping on a plane in such a way that the contact point on the plane
traces a straight line.
How to describe the trajectory of the contact point on the ellipsoid
and the motion of the ellipsoid in space ?
These kinematical problems, as well as their multi-dimensional generalizations,
have exact solutions based on applications of the classical Chasles theorem on common tangent lines of
confocal quadrics and the Jacobi geodesics on the ellipsoid.

#### Hélène FRANKOWSKA (Université Pierre et Marie Curie, Paris, France)

First and second-order sensitivity relations in optimal control

(slides)
Sensitivity relations in optimal control refer to the interpretation of the gradients of the value function in terms of the costate arc and the Hamiltonian evaluated along an extremal.
Second order sensitivity relations interpret the jets of the value function evaluated along an extremal in terms of the costate arc and the solution of a Riccati equation.

The value function being not differentiable whenever there are multiple optimal solutions,
its gradients and jets have to be understood in a generalized sense.

In this talk I will discuss sensitivity relations involving Fréchet super/subdifferentials
and super/subjets.

When the Hamiltonian is sufficiently smooth, these relations lead to new results on $C^2$ regularity of value function in neighborhoods of optimal trajectories starting from points at which it is proximally
subdifferentiable, that is for almost every initial condition.

#### Janusz GRABOWSKI (IMPAN, Warsaw, Poland)

Dirac Algebroids in Formalisms of Constrained Mechanics

(slides)
We present a unified approach to constrained Lagrangian and Hamiltonian systems,
based on the concepts of the *Tulczyjew triple* and *Dirac algebroid*.
The latter is a certain almost Dirac structure associated with the Courant algebroid $TE^∗ \bigoplus_M T^∗E^∗$
on the dual $E^*$ to a vector bundle $\tau:E\to M$.
If this almost Dirac structure is integrable (Dirac), we speak about *Dirac-Lie algebroid*.

The bundle $E$ plays the role of the bundle of kinematic configurations (quasi-velocities),
while the bundle $E^*$ plays the role of the phase space.
This setting is totally covariant and works well also for singular Lagrangians.
The constraints are part of the framework, so the general approach remains unchanged when nonholonomic constraints are imposed,
producing the (implicit) Euler-Lagrange and Hamilton equations in an elegant geometric way.

The scheme includes all important cases of constrained Lagrangian and Hamiltonian systems,
autonomous and non-autonomous, as well as their reductions, thus constrained systems on Lie algebroids.

#### Bronislaw JAKUBCZYK (IMPAN, Warsaw, Poland)

Connection and curvature of regular control systems and Lagrangian geometry

(slides)
It is well known that many notions of Riemannian geometry on a manifold $M$ can be extended
to Finsler and Lagrange geometries when the basic manifold $M$ is replaced by its tangent bundle $TM$.
This is done by looking at the geodesic equations or the equations of motion as defining
a vector field on $TM$, called semispray.
In particular, any physical system of finite dimension described by the Lagrange formalism
can be assigned a canonical connection and curvature.

In our presentation we will discuss a more general geometry defined by a natural class of control-affine
systems called regular systems.
Replacing the semispray by the drift of control system and the vertical distribution on $TM$
by the control distribution we get a dynamic pair $(X,V)$ of vector field and distribution
on a manifold $N$.
For a given regular dynamic pair one can still define a canonical connection and curvature
(Jacobi endomorphism).
Fully actuated control systems form a particular class of regular systems with holonomic (i.e., integrable)
control distribution.
Systems with nonholonomic control distribution define completely new invariants,
not present in Lagrange geometry.
They can be defined and analyzed by choosing normal generators of control distribution,
and by studying their Lie brackets.

The presentation is, to much extend,
based on the following paper with W. Kryński, where a more general class of systems is considered.

B. Jakubczyk and W. Kryński, *Vector fields with distributions and geometry of ODEs*,
J. Geometric Mechanics, vol. 5 (2013), no. 1, pp. 85-129.

#### Frédéric JEAN (ENSTA ParisTech, Paris, France)

Why don't we move slower? On the cost of time in the neural control of movement

(slides)
In this talk, we investigate the cost of time hypothesis for arm movement control.
According to this hypothesis, movement cannot be too slow because duration has a cost per se.
How the brain penalizes motion time during arm motor planning is however largely unknown
even though some authors have hypothesized linear, quadratic, exponential or hyperbolic growths
for such a cost of time.
Here, we present a methodology to automatically recover the genuine cost of time by sampling it
directly from experimental data.

The theory relies upon a free-time optimal control (OC) formulation of the planning problem
and enables finding infinitesimal values of the cost of time from raw motion capture measurements
and the resolution of adequate fixed-time OC problems.
Consistence of the cost of time with respect to tasks and systems is then investigated.
Properties of uniqueness and robustness are derived for the single-input linear-quadratic scenario.
Extendability of such a cost of time theory is further assessed to account for multi-joint dynamics,
slow/fast verbal instructions and Fitts's law.

#### Fernando JIMÉNEZ (Technische Universität München, München, Germany)

On the discretization of nonholonomic dynamics in $\mathbb{R}^n$

(slides)
We explore the nonholonomic Lagrangian setting of mechanical systems in local coordinates on finite-dimensional configuration manifolds.
We prove existence and uniqueness of solutions by reducing the basic equations of motion to a set of ordinary differential equations on the underlying distribution manifold $D$.
Moreover, we show that any $D$-preserving discretization may be understood as being generated by the exact evolution map
of a time-periodic non-autonomous perturbation of the original continuous-time nonholonomic system.
By means of discretizing the corresponding Lagrange-d'Alembert principle, we construct geometric integrators for the original nonholonomic system.
We give precise conditions under which these integrators generate a discrete flow preserving the distribution $D$.

#### Bozidar JOVANOVIC (Serbian Academy of Sciences and Arts, Belgrade, Serbia)

Various models of multidimensional nonholonomic rigid body dynamics

The origin of existence of invariant measure, Hamiltonization, and integrability in classical nonholonomic rigid body problems
can be seen and understand within a framework of more general $n$-dimensional nonholonomic dynamics.
We shall present several possible nonholonomic models, in particular a generalization of the celebrated Chaplygin ball problem,
describing the rolling without slipping of a balanced ball over a horizontal surface.

#### Matthias KAWSKI (Arizona state University, Phoenix, USA)

Approximating sub-Riemannian geodesics by highly oscillatory controls

Highly oscillatory controls have been used for path planning purposes for a long time. They are well-known (see e.g. the work by Liu and Sussmann) to allow an arbitrarily close approximation of desired motions by admissible trajectories. The complexity of such controls or curves is well studied, and based on relating the growth rate of the geodesic balls to the length of the brackets.

We present simple explicit formulas for sinusoidal controls that annihilate all iterated integral functionals (iif) up to a certain order, and that selectively excite specific modes. While these curves do not form dual bases, their actions are in some sense triangular on bases of iif that are coded by specific Hall words. This work utilizes new computer algebra code to help track the combinatorial relations between basis elements that have the same finely homogeneous degrees. We also provide some explicit lower bounds for the frequencies of controls that can simultaneously annihilate all iif up to a certain order. These may be considered as measures of the complexity of geodesics that are transversal to generic regular distributions, i.e., whose derived flag has maximal growth vectors.

#### Valery V. KOZLOV (Russian Academy of Sciences, Moscow, Russia)

Dynamics in systems with constraints

(slides)
In the presentation we discuss fundamental mathematical models used to describe the motion
of mechanical systems under constraints.
We consider the problems of motivation, axiomatic definition, and physical realization of these models.

#### Jean-Paul LAUMOND (LASS, Toulouse, France)

A geometric perspective of anthropomorphic action

Actions take place in the physical space while they originate in the –robot or human– sensory-motor space.
Geometry is the core abstraction that makes the link between these spaces.
The anthropomorphic body is a complex structure that is both redundant for manipulation tasks,
and underactuated for locomotion.
Considering that the structure of actions inherits from that of the anthropomorphic body,
the talk will address recent research developments tending to explore the computational foundations
of anthropomorphic actions and opening challenging mathematical issues as inverse optimal control problems.

#### Antonio LERARIO (ICJ, Lyon, France)

Quantitative topology of nonholonomic loop spaces

Given two points on a sub-Riemannian manifold the nonholonomic path-space consists of all Lispichitz continuous horizontal curves joining them.
The sub-Riemannian structure allows to define the energy of these curves (a function on the nonholomonic path space) and geodesics are critical points of this function.
I will discuss some quantitative aspects of this picture both on the infinitesimal scale (on Carnot groups),
on the local scale (using a blow-up procedure) and the global scale (generalizing a theorem of Serre).

References

[1] A. A. Agrachev, A. Gentile, A. Lerario: *Geodesics and horizontal path-spaces in Carnot groups*

[2] A. Lerario, L. Rizzi: *How many geodesics are there between two points on a contact sub-Riemannian manifold?*

#### Richard MONTGOMERY (University of California, Santa Cruz, USA)

Some open problems in (and around) sub-Riemannian geometry

I will discuss the status of three open questions in sub-Riemannian geometry.

(1) Must a minimizing geodesic be smooth?

(2) Does Sard's theorem hold for the endpoint map?

(3) What is the constant of proportionality relating Lebesgue to Hausdorff measure on the Heisenberg group?

I will describe partial progress on (1) by Monti, Leonardi, Le Donne, and Vittone [MLLV]
on (2) by Gole, Karidi, Rifford, Trelat, Agrachev, and this trimester in a seminar involving MLLV, Otazzi, Pansu and myself,
and on (3) by Leonardi, Vittoni, and Rigot.
Time and spirit permitting, I may state more open problems.

#### Tudor RATIU (Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland)

Geometry of nonholonomic diffusion

Stochastically perturbed nonholonomic systems are studied from a geometric point of view.
In this setting, it turns out that the probabilistic properties of the perturbed system are intimately linked to the geometry
of the constraint distribution. For $G$-Chaplygin systems, this yields a stochastic criterion for the existence of a smooth
preserved measure. As an application of the results, the motion planning problem for the noisy two-wheeled robot and the noisy snake board
will be considered and presented, time permitting.

#### Mario SIGALOTTI (INRIA Saclay and CMAP, Paris, France)

Controllability of the discrete-spectrum Schrödinger equation

We consider a controlled Schrödinger equation (in a finite- or infinite-dimensional space) driven by one or more control
parameters appearing affinely in the dynamics and we assume the spectrum of each constant-control operator to be discrete.
Based on geometric control techniques, we propose sufficient conditions for its controllability through a constructive approach.
Following a complementary approach, which exploits classical ideas from adiabatic perturbation analysis, we give another family of sufficient conditions for controllability (and motion planning algorithms).

We conclude by comparing the two set of conditions.

#### Gianna STEFANI (Università degli studi di Firenze, Italy)

Strong-local optimality for trajectories containing a singular arc

(slides)
We consider different optimal control problems associated to an affine system on a $n$-dimensional $C^\infty$ manifold $M$:
$$\mbox{Minimize}\quad J(\xi,u,T)\quad\mbox{subject to}$$
$$\left\{
\begin{array}{l}
\dot{\xi} = \bigg( f_0 + \sum_{i=1}^m u_i f_i \bigg) \circ \xi(t) \\
\xi(0) \in N_0, \quad \xi(T) \in N_f\\
u=(u_1,\ldots,u_m) \in U \subset \mathbb{R}^m
\end{array}
\right.$$
where $N_0$ and $N_f$ are $C^\infty$ sub-manifolds of $M$ and $f_0,\dots,f_m$ are $C^\infty$ vector fields.

I'll give necessary conditions and/or sufficient conditions in the case when the reference trajectory contains a singular
(or partially singular) arc.

The results rely on geometric methods, using both the Hamiltonian approach and the so called
"good needle-like control variations" which give rise to higher order maximum principles.

#### Iskander TAIMANOV (Sobolev institute of mathematics, RAS, Novosibirsk, Russia)

On the integrability of geodesic flows on solvable Carnot 3-groups

To be announced

#### Emmanuel TRÉLAT (Université Pierre et Marie Curie, Paris, France)

The turnpike property in optimal control

(slides)
The turnpike property emerged in the 50's, after the works by the Nobel prize Samuelson in econometry.
It stands for the general behavior of an optimal trajectory solution of an optimal control problem in large time.
This trajectory trends to behave as the concatenation of three pieces: the first and the last arc being rapid transition arcs,
and the middle one being in large time, almost stationary, close to the optimal value of an associated static optimal control problem.
In a recent work with Enrique Zuazua, we have established the turnpike property in a very general framework in finite dimensional nonlinear optimal control.
We prove that not only the optimal trajectory is exponentially close so some (optimal) stationary state,
but also the control and the adjoint vector coming from the Pontryagin maximum principle.
Our analysis shows an hyperbolicity phenomenon which is intrinsic to the symplectic feature of the extremal equations.
We infer a very simple and efficient numerical method to compute optimal trajectories in that framework, with an appropriate variant of the shooting method.

#### Dmitry V. TRESCHEV (Steklov Mathematical Institute, Moscow, Russia)

Does the motion of a rigid ball on a plane look like a nonholonomic system?

We try to understand if the nonholonomic model is adequate in the problem of a motion of a rigid ball on a stiff visco-elastic support plane.
More precisely, the support is regarded as a relatively stiff visco-elastic Kelvin-Voigt medium, coinciding with horizontal plane at undeformed state.
We assume also that while deformed the support induces dry friction forces, that locally are governed by the Coulomb law.
We study the impact appearing as a result of falling a ball on the plane.
Another problem of our interest is the motion of a ball "along the plane".
We present a detailed analysis of various stages of the motion.
We also compare this model with classical models of solid bodies' interaction.

#### Richard VINTER (Imperial College, London, UK)

Optimal Control for Differential Inclusions which have Bounded Variation w.r.t. Time

(slides)
It is well known that properties of optimal state trajectories depend on the regularity of the dynamic constrain with respect to the time variable.
The existing literature focuses for the most part on differences that arise
when the time dependence is merely measurable, Lipschitz continuous or higher order differentiable.
Yet there is another class of problems with distinctive features with have received very little attention,
namely problems for which the dynamic constraint has bounded variation w.r.t. time.
This is a neglected but fruitful area.
In this talk we discuss where these 'bounded variation' optimal control problems originate in control engineering, and give an overview of recent work, in which numerous special properties
of solutions to optimal control problems are established, under the 'bounded variation' hypothesis.
These include regularity properties of value functions,
conditions for boundedness of the optimal controls,
conditions for non-degeneracy of known necessary conditions of optimality, and new sensitivity formulae describing the variation of cost user data perturbations.

#### Dmitry ZENKOV (North Carolina Stare University, Raleigh, USA)

Hamel’s Formalism and Variational Integrators

(slides)
The talk will give a contemporary exposition of Hamel’s formalism
with an emphasis on integral variational principles. Using the technique of variational discretization,
the discrete Hamel equations will then be introduced.
Structure-preserving properties of the resulting discrete dynamics will be discussed.
This development is motivated by earlier results on the discrete Euler–Lagrange and
Euler–Poincaré equations and discrete controlled mechanical systems.