RESEARCH CENTRE

Inria Centre
at Université Côte d’Azur

IN PARTNERSHIP WITH:

CNRS, Université Côte d’Azur

2023
ACTIVITY REPORT

Project-Team

MCTAO

Mathematics for Control, Transport and
Applications

IN COLLABORATION WITH: Laboratoire Jean-Alexandre Dieudonné (JAD)

DOMAIN

Applied Mathematics, Computation and
Simulation

THEME

Optimization and control of dynamic
systems

Contents

Project-Team MCTAO

1 Team members, visitors, external collaborators

2 Overall objectives

3 Research program

3.1 Control Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Optimal Control and its Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Application domains

4.1 Aerospace Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Optimal control of microbial cells, and other biological applications . . . . . . . . . . . . . .
4.3 Neural dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Highlights of the year

6 New software, platforms, open data

6.1 New software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.1.1 ct

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7 New results

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7.1 Output feedback stabilization of non-uniformly observable systems . . . . . . . . . . . . . .
7.2 Geometry and optimal control for navigation problem . . . . . . . . . . . . . . . . . . . . . .
9
7.3 Small time local controllability for some degenerate two-input systems . . . . . . . . . . . . 10
7.4 Stability of linear time-varying time-delay systems . . . . . . . . . . . . . . . . . . . . . . . . 10
7.5 Second-order averaging of fast-oscillating control systems
. . . . . . . . . . . . . . . . . . . 10
7.6 Control of solar sails: controllability and optimal control . . . . . . . . . . . . . . . . . . . . . 11
7.7 Efficient solution of the low-thrust Lambert’s problem . . . . . . . . . . . . . . . . . . . . . . 12
7.8 Low-thrust satellite collision avoidance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
7.9 Optimal allocation of resources in bacteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
7.10 Biomechanics: control of muscular force response using FES with application to the con-

ception of a smart electrostimulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

7.11 Feedback invariants and optimal control with biological applications and numerical calcu-

lations

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
7.12 State estimation in alcoholic fermentation models . . . . . . . . . . . . . . . . . . . . . . . . 14
7.13 Observer design for Blumenfeld’s model of V1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
7.14 ct: control toolbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

8 Bilateral contracts and grants with industry

15
8.1 Intelligent muscle electrostimulator, Segula Technologies . . . . . . . . . . . . . . . . . . . . 15
8.2 Optimal Control of Solar Sails, ESA (European Space Agency) . . . . . . . . . . . . . . . . . . 15
8.3 Méthodes de contrôle pour l’évitement de collisions entre satellites, Thales Alenia Space . 16

9 Partnerships and cooperations

16
9.1 National initiatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
9.1.1 ANR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
9.1.2 Other . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
9.2 Regional initiatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

10 Dissemination

17
10.1 Promoting scientific activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
10.1.1 Scientific events: organisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
10.1.2 Scientific events: selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
10.1.3 Journal
10.1.4 Invited talks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
10.1.5 Research administration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
10.2 Teaching - Supervision - Juries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
10.2.1 Teaching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
10.2.2 Supervision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
10.2.3 Juries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

11 Scientific production

19
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
11.1 Publications of the year
11.2 Other . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
11.3 Cited publications

Project MCTAO

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Project-Team MCTAO

Creation of the Project-Team: 2013 January 01

Keywords

Computer sciences and digital sciences

A5.10.3. – Planning

A5.10.4. – Robot control

A6.1.1. – Continuous Modeling (PDE, ODE)

A6.1.5. – Multiphysics modeling

A6.2.1. – Numerical analysis of PDE and ODE

A6.2.6. – Optimization

A6.4. – Automatic control

A6.4.1. – Deterministic control

A6.4.3. – Observability and Controlability

A6.4.4. – Stability and Stabilization

A6.4.6. – Optimal control

A6.5. – Mathematical modeling for physical sciences

A8.2.3. – Calculus of variations

A8.12. – Optimal transport

Other research topics and application domains

B1.1.8. – Mathematical biology

B1.1.9. – Biomechanics and anatomy

B1.2.1. – Understanding and simulation of the brain and the nervous system

B2.5.1. – Sensorimotor disabilities

B2.6. – Biological and medical imaging

B2.7.2. – Health monitoring systems

B5.2.3. – Aviation

B5.2.4. – Aerospace

B5.6. – Robotic systems

B5.11. – Quantum systems

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Inria Annual Report 2023

1 Team members, visitors, external collaborators

Research Scientists

• Jean-Baptiste Pomet [Team leader, INRIA, Senior Researcher, HDR]

• Ivan Beschastnyi [INRIA, Researcher, from Dec 2023]

• Lamberto Dell’Elce [INRIA, Researcher]

• Ludovic Sacchelli [INRIA, Researcher]

Faculty Member

• Jean-Baptiste Caillau [UNIV COTE D’AZUR, Professor, HDR]

PhD Students

• Adel Malik Annabi [UNIV COTE D’AZUR]

• Antonin Bavoil [CNRS, from Oct 2023]

• Frank De Veld [INRIA]

• Sandrine Gayrard [SEGULA, until Sep 2023]

• Alesia Herasimenka [UNIV COTE D’AZUR, until Nov 2023]

Interns and Apprentices

• Antonin Bavoil [CNRS, Intern, from Mar 2023 until Sep 2023]

• Martin Fleurial [INRIA, Intern, from Mar 2023 until Aug 2023]

Administrative Assistant

• Claire Senica [INRIA]

External Collaborators

• Bernard Bonnard [Univ. de Bourgogne, founding member of the team, HDR]

• Olivier Cots [TOULOUSE INP, HDR]

• Thierry Dargent [Thales Alenia Space, Cannes]

• Joseph Gergaud [TOULOUSE INP, HDR]

• Jérémy Rouot [Université de Bretagne Occidentale, Brest]

2 Overall objectives

Our goal is to develop methods in geometric control theory for nonlinear systems, mostly finite dimen-
sional, and to transfer our expertise through real applications of these methods. The methodological
developments range from feedback control and observers to optimal control, extending to fields like
sub-Riemannian geometry. Optimal control leads to developments in Hamiltonian dynamics, and also
requires sophisticated numerics, to which the team contributes too. Dynamical systems and modeling
are also part of the background of the team.

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Our primary domain of industrial applications in the past years has been space engineering, in
particular using optimal control and stabilization techniques for mission design: orbit transfer or rendez-
vous problems in the gravity field of a single body (typically satellites around the earth), interplanetary
missions and multi body problems, or control design of solar sails, where propulsion is drastically
constrained.

The team also has continued involvement with applications regarding human bio-mechanics (muscle
stimulation), and various modeling and control questions in biology (Lotka-Volterra models, bacterial
growth, microbiome models, networks of chemical reaction...) The list is not exhaustive; past domains
of application include swimming at low Reynolds number (micro-swimmers) and control of quantum
systems for Magnetic Resonance Imaging.

3 Research program

3.1 Control Problems

McTAO’s major field of expertise is control theory in the broad sense. Let us give an overview of this field.

Modeling. Our effort is directed toward efficient methods for the control of real (physical) systems, based
on a model of the system to be controlled. Choosing accurate models yet simple enough to allow control
design is in itself a key issue. The typical continuous-time model is of the form dx/dt = f (x, u) where x
is the state, ideally finite dimensional, and u the control; the control is left free to be a function of time,
or a function of the state, or obtained as the solution of another dynamical system that takes x as an
input. Modeling amounts to deciding the nature and dimension of x, as well as the dynamics (roughly
speaking the function f ). Connected to modeling is identification of parameters when a finite number of
parameters are left free in “ f ”.

Controllability, path planning. Controllability is a property of a control system (in fact of a model) that
two states in the state space can be connected by a trajectory generated by some control, here taken as an
explicit function of time. Deciding on local or global controllability is still a difficult open question in
general. In most cases, controllability can be decided by linear approximation, or non-controllability
by “physical” first integrals that the control does not affect. For some critically actuated systems, it is
still difficult to decide local or global controllability, and the general problem is anyway still open. Path
planning is the problem of constructing the control that actually steers one state to another.

In optimal control, one wants to find, among the controls that satisfy some constraints
Optimal control.
at initial and final time (for instance given initial and final state as in path planning), the ones that
minimize some criterion. This is important in many control engineering problems, because minimizing a
cost is often very relevant. Mathematically speaking, optimal control is the modern branch of the calculus
of variations, rather well established and mature [78, 45, 31], but still displaying important and hard open
questions. In the end, in order to actually compute these controls, ad-hoc numerical schemes have to
be derived for effective computations of the optimal solutions. See more about our research program in
optimal control in section 3.2.

In the above two paragraphs, the control is an explicit function of time. To address in
Feedback control.
particular the stability issues (sensitivity to errors in the model or the initial conditions for example), the
control has to be taken as a function of the (measured) state, or part of it. This is known as closed-loop
control; it must be combined with optimal control in many real problems. On the problem of stabilization,
there is longstanding research record from members of the team, in particular on the construction of
“Control Lyapunov Functions”, see [65, 79]. It may happen that only part of the state is accessible at
any one time, because of physical or engineering constraints. In that case, a popular strategy is to
pair feedback methods with dynamic estimation of the state, creating so-called output feedback loops.
Simultaneous feedback control and estimation can become a major hurdle for nonlinear systems, see [55,
82].

Classification of control systems. One may perform various classes of transformations acting on sys-
tems, or rather on models. The simpler ones come from point-to-point transformations (changes of

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Inria Annual Report 2023

variables) on the state and control. More intricate ones consist in embedding an extraneous dynamical
system into the model. These are dynamic feedback transformations that change the dimension of
the state. In most problems, choosing the proper coordinates, or the right quantities that describe a
phenomenon, sheds light on a path to the solution; these proper choices may sometimes be found from
an understanding of the modeled phenomenons, or it can come from the study of the geometry of the
equations and the transformation acting on them. This justifies the investigations of these transform-
ations on models for themselves. These topics are central in control theory; they are present in the
team, see for instance the classification aspect in [49] or —although this research has not been active
very recently— the study [77] of dynamic feedback and the so-called “flatness” property [68]. Likewise,
classification tools such as feedback invariants [39] are still currently in use in the team (see, for instance,
[22]).

3.2 Optimal Control and its Geometry

Let us detail our research program concerning optimal control. Relying on Hamiltonian dynamics is now
prevalent, instead of the Lagrangian formalism in classical calculus of variations. The two points of view
run parallel when computing geodesics and shortest path in Riemannian Geometry for instance, in that
there is a clear one-to-one correspondance between the solutions of the geodesic equation in the tangent
bundle and the solution of the Pontryagin Maximum Principle in the cotangent bundle. In most optimal
control problems, on the contrary, due to the differential constraints (velocities of feasible trajectories do
not cover all directions in the state space), the Lagrangian formalism becomes more involved, while the
Pontryagin Maximum Principle keeps the same form, its solutions still live in the cotangent bundle, their
projections are the extremals, and a minimizing curve must be the projection of such a solution.

Cut and conjugate loci. The cut locus —made of the points where the extremals lose optimality— is
obviously crucial in optimal control, but usually out of reach (even in low dimensions), and anyway
does not have an analytic characterization because it is a non-local object. Fortunately, conjugate points
—where the extremals lose local optimality— can be effectively computed with high accuracy for many
control systems. Elaborating on the seminal work of the Russian and French schools (see [81, 30, 32] and
[50] among others), efficient algorithms were designed to treat the smooth case. This was the starting
point of a series of papers of members of the team culminating in the outcome of the cotcot software
[44], followed by the HamPath [56] code. Over the years, these codes have allowed for the computation
of conjugate loci in a wealth of situations including applications to space mechanics, quantum control,
and more recently swimming at low Reynolds number. With in mind the two-dimensional analytic
Riemannian framework, a heuristic approach to the global issue of determining cut points is to search for
singularities of the conjugate loci; this line is however very delicate to follow on problems stemming from
applications in three or more dimensions (see e.g. [57] and [41]). In all these situations, the fundamental
object underlying the analysis is the curvature tensor. In Hamiltonian terms, one considers the dynamics
of subspaces (spanned by Jacobi fields) in the Lagrangian Grassmannian [29]. This point of view with-
stands generalizations far beyond the smooth case: In L1-minimization, for instance, discontinuous
curves in the Grassmannian have to be considered (instantaneous rotations of Lagrangian subspaces
still obeying symplectic rules [62]). The cut locus is a central object in Riemannian geometry, control
and optimal transport. This is the motivation for a series of conferences on “The cut locus: A bridge over
differential geometry, optimal control, and transport”, co-organized by team members and Japanese
colleagues.

Riemann and Finsler geometry. Studying the distance and minimizing geodesics in Riemannian
Geometry or Finsler Geometry is a particular case of optimal control, simpler because there are no
differential constraints; it is studied in the team for the following two reasons. On the one hand, after
some tranformations, like averaging or reduction, some more difficult optimal control problems lead to
a Riemann or Finsler geometry problem. On the other hand, optimal control, mostly the Hamiltonian
setting, brings a fresh viewpoint on problems in Riemann and Finsler geometry. On Riemannian ellipsoids
of revolution, the optimal control approach allowed to decide on the convexity of the injectivity domain,
which, associated with non-negativity of the Ma-Trudinger-Wang curvature tensor, ensures continuity of
the optimal transport on the ambient Riemannian manifold [66, 67]. The analysis in the oblate geometry

Project MCTAO

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[42] was completed in [60] in the prolate one, including a preliminary analysis of non-focal domains
associated with conjugate loci. Averaging in systems coming from space mechanics control with L2-
minimization yields a Riemannian metric, thoroughly computed in [40] together with its geodesic flow;
in reduced dimension, its conjugate and cut loci were computed in [43] with Japanese Riemannian
geometers. Averaging the same systems for minimum time yields a Finsler Metric, as noted in [38]. In
[48], the geodesic convexity properties of these two types of metrics were compared. When perturbations
(other than the control) are considered, they introduce a “drift”, i.e. the Finsler metric is no longer
symmetric.

Sub-Riemannian Geometry. Optimal control problems that pertain to sub-Riemannian Geometry
bear all the difficulties of optimal control, like the role of singular/abnormal trajectories, while having
some useful structure. They lead to many open problems, see the monograph [75] for an introduction.
The sub-Riemannian problem can be encoded by a non-linear control system with no drift, subjected
to a quadratic energy minimization objective. This allows the sub-Riemannian problem to serve as
rich model spaces for optimal control. The interest of sub-Riemannian geometry can go beyond these
aspects however. It was proved by Hormander in 1967 [73] that local controllability of the system (given
in terms of Lie-brackets of vector fields) is equivalent to sub-ellipticity of a second order differential
operator associated with the vector fields. In this way, sub-Riemannian geometry acts as a bridge between
elements of analysis of PDEs and geometric control theory. For instance, many recent works focus
on framing properties of sub-elliptic operators in terms of minimizers of the optimal control problem
(such as the influence of cut and conjugate points on diffusion asymptotics [36]). This link even allowed
to successfully introduce concepts of sub-elliptic diffusions in computer vision algorithms thanks to
sub-Riemannian geometric structures identified in mammal visual mechanisms [52].

Small controls and conservative systems, averaging. Using averaging techniques to study small per-
turbations of integrable Hamiltonian systems is as old an idea as celestial mechanics. It is very subtle in
the case of multiple periods but more elementary in the single period case, here it boils down to taking
the average of the perturbation along each periodic orbit [33, 80]. This line of research stemmed out
of applications to space engineering (see Section 4.1): the control of the super-integrable Keplerian
motion of a spacecraft orbiting around the Earth is an example of a slow-fast controlled system. Since
weak propulsion is used, the control itself acts as a perturbation, among other perturbations of similar
magnitudes: higher order terms of the Earth potential (including J2 effect, first), potential of more distant
celestial bodies (such as the Sun and the Moon), atmospheric drag, or even radiation pressure. Properly
qualifying the convergence properties (when the small parameter goes to zero) is important and is made
difficult by the presence of control. In [38], convergence is seen as convergence to a differential inclusion;
this applies to minimum time; a contribution of this work is to put forward the metric character of the
averaged system by yielding a Finsler metric (see Section 3.2). Proving convergence of the extremals
(solutions of the Pontryagin Maximum Principle) is more intricate. In [59], standard averaging ([33,
80]) is performed on the minimum time extremal flow after carefully identifying slow variables of the
system thanks to a symplectic reduction. This alternative approach allows to retrieve the previous metric
approximation, and to partly address the question of convergence. Under suitable assumptions on a given
geodesic of the averaged system (disconjugacy conditions, namely), one proves existence of a family of
quasi-extremals for the original system that converge towards the geodesic when the small perturbation
parameter goes to zero. This needs to be improved, but convergence of all extremals to extremals of an
“averaged Pontryagin Maximum Principle” certainly fails. In particular, one cannot hope for C 1-regularity
on the value function when the small parameter goes to zero as swallowtail-like singularities due to the
structure of local minima in the problem are expected. (A preliminary analysis has been made in [58].)

Optimality of periodic solutions/periodic controls. When seeking to minimize a cost with the con-
straint that the controls and/or part of the states are periodic (and with other initial and final conditions),
the notion of conjugate points is more difficult than with straightforward fixed initial point. In [46], for
the problem of optimizing the efficiency of the displacement of some micro-swimmers with periodic
deformations, we used the sufficient optimality conditions established by R. Vinter’s group [84, 70] for
systems with non unique minimizers due to the existence of a group of symmetry (always present with
a periodic minimizer-candidate control). This takes place in a long term collaboration with P. Bettiol

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Inria Annual Report 2023

(Univ. Bretagne Ouest) on second order sufficient optimality conditions for periodic solutions, or in
the presence of higher dimensional symmetry groups, following [84, 70]. Another question relevant to
locomotion is the following. Observing animals (or humans), or numerically solving the optimal control
problem associated with driftless micro-swimmers for various initial and final conditions, we remark
that the optimal strategies of deformation seem to be periodic, at least asymptotically for large distances.
This observation is the starting point for characterizing dynamics for which some optimal solutions are
periodic, and asymptotically attract other solutions as the final time grows large; this is reminiscent of the
“turnpike theorem” (classical, recently applied to nonlinear situations in [83]).

3.3 Software

Optimal control applications (but also the development of theory where numerical experiments can be
very enlightening) require many algorithmic and numerical developments that are an important side
of the team activity. We develop on-demand algorithms and pieces of software, for instance we have
to interact with a production software developed by Thales Alenia Space. A strong asset of the team
is the interplay of its expertise in geometric control theory with applications and algorithms, and the
team has a long-lasting commitment to the development of numerical codes for the efficient resolution
of optimal control problems. Methods for solving optimal control problems with ordinary differential
equations more or less fall into three main categories. Dynamic Programming (or Hamilton Jacobi
Bellman method) computes the global optimum but suffers from high computational costs, the so-called
curse of dimensionality. Indirect methods based on Pontryagin Maximum Principle are extremely fast
and accurate but often require more work to be applied, in terms of mathematical analysis and a priori
knowledge of the solution; this kind of fine geometrical analysis is one of the strong know-how of McTAO.
Direct transcription methods offer a good tradeoff between robustness and accuracy and are widely used
for industrial applications. For challenging problems, an effective strategy is to start with a direct method
to find a first rough solution, then refine it through an indirect method. We develop this further in a
book chapter [17] published this year. Such a combined approach has been for instance used between
McTAO, the former COMMANDS team (Inria Saclay), and CNRS team APO (Université Toulouse, CNRS,
ENSEEIHT) for the optimization of contrast in medical imaging (MRI), and fuel-effective trajectories for
airplanes. This combination of direct and indirect methods has a lot of interest to solve optimal control
problems that contain state or control constraints. In the collaborations mentioned above, the interfacing
between the two solvers BOCOP and HamPath were done manually by ad hoc python or matlab layers.
In collaboration with COMMANDS and colleagues from ENSEEIHT, McTAO leads the ct: control toolbox
project whose goal is to interoperate these solvers using a high level common interface. The project is
an Inria Sophia ADT1 (2019-) in AMDT1 mode supported by Inria Sophia SED. The last sprint session,
closing the project, is planned for February 2023.

4 Application domains

4.1 Aerospace Engineering

Participants:

Jean-Baptiste Caillau, Thierry Dargent, Lamberto Dell’Elce, Frank de
Veld, Alesia Herasimenka, Jean-Baptiste Pomet.

Space engineering is very demanding in terms of safe and high-performance control laws. It is
therefore prone to fruitful industrial collaborations. McTAO now has an established expertise in space
and celestial mechanics. Our collaborations with industry are mostly on orbit transfer problems with
low-thrust propulsion. It can be orbit transfer to put a commercial satellite on station, in which case the
dynamics are a Newtonian force field plus perturbations and the small control. There is also, currently, a
renewed interest in low-thrust missions such as Lisa Pathfinder (ESA mission towards a Lagrange point of
the Sun-Earth system) or BepiColombo (joint ESA-JAXA mission towards Mercury). Such missions look

1ADT is the name of software development actions supported by the service “SED”, by devoting some engineers to these projects.

AMDT means that a group of engineers works on the project part time, rather than sending one engineer in the team.

Project MCTAO

7

more like a controlled multibody system. In all cases the problem involves long orbit transfers, typically
with many revolutions around the primary celestial body. When minimizing time, averaging techniques
provide a good approximation. Another important criterion in practice is fuel consumption minimization
(crucial because only a finite amount of fuel is onboard a satellite for all its “life”), which amounts to
L1-minimization. Both topics are studied by the team. We have a steady relationship with CNES and
Thales Alenia Space (Cannes), that have financed or co-financed 4 PhDs and 2 post-docs in the decade
and are a source of inspiration even at the methodological level. Team members also have connections
with Airbus-Safran (Les Mureaux) on launchers.

Some of the authoritative papers in the field were written by team members, with an emphasis on the
geometric analysis and on algorithms (coupling of shooting and continuation methods). There are also
connections with peers more on the applied side, like D. Scheeres (Colorado Center for Astrodynamics
Research at Boulder), the group of F. Bernelli (Politecnico Milano), and colleagues from University of
Barcelona (A. Farrès, A. Jorba).

Two new directions have been taken recently. The first one is about the control of solar sails (see Sec-
tion 7.6), the second one about collision avoidance for spacecrafts (see Section 7.8). Collision avoidance
is becoming very important in nowadays space missions due to the growing number of various bodies
(garbage, micro-satellites. . . ) orbiting around the earth. A PhD (Frank de Veld) started in December,
supported by Thales Alenia Space. Solar sailing has been actively studied for two decades and recent
missions have demonstrated its interest for "zero-fuel" missions; it poses delicate control questions due
to drastic constraints on the control direction. Alesia Herasimenka, whose PhD had been selected by
ESA for a three-year research co-sponsorship, defended her thesis in September and is now a postdoc at
University of Luxembourg.

4.2 Optimal control of microbial cells, and other biological applications

Jean-Baptiste Caillau, Martin Fleurial,
Participants: Bernard Bonnard,
Sandrine Gayrard‘
Jean-Baptiste Pomet, Ludovic Sacchelli,
Toufik Bakir (Université de Bourgogne Franche Comté, Dijon),
Walid Djema (BIOCORE project-team), Jean-Luc Gouzé (BIOCORE
project-team), Sofya Maslovskaya (Paderborn University, Ger-
many), Jérémy Rouot (Université de Bretagne Occidentale, Brest),
Agustín Yabo (INRAE, Montpellier).

,

The growth of microorganisms is fundamentally an optimization problem which consists in dynamic-
ally allocating resources to cellular functions so as to maximize growth rate or another fitness criterion.
Simple ordinary differential equation models, called self-replicators, have been used to formulate this
problem in the framework of optimal and feedback control theory, allowing observations in microbial
physiology to be explained. The resulting control problems are very challenging due to the nonlinearity
of the models, parameter uncertainty, the coexistence of different time-scales, a dynamically changing
environment, and various other physical and chemical constraints. In the framework of the ANR Maximic
(PI Hidde de Jong, Inria Grenoble Rhône-Alpes), we aim at developing novel theoretical approaches for
addressing these challenges in order to (i) study natural resource allocation strategies in microorganisms
and (ii) propose new synthetic control strategies for biotechnological applications. In order to address (i),
we develop extended self-replicator models accounting for the cost of regulation and energy metabolism
in bacterial cells. We study these models by a combination of analytical and numerical approaches to
derive optimal control solutions and a control synthesis, dealing with the bang-bang-singular structure of
the solutions. Moreover, we define quasi-optimal feedback control strategies inspired by known regulat-
ory mechanisms in the cell. To test whether bacteria follow the predicted optimal strategies, we quantify
dynamic resource allocation in the bacterium Escherichia coli by monitoring, by means of time-lapse
fluorescent microscopy, the expression of selected genes in single cells growing in a microfluidics device.
In order to address (ii), we build self-replicator models that include a pathway for the production of a
metabolite of interest. We also add a mechanism to turn off microbial growth by means of an external
input signal, at the profit of the production of the metabolite. We formulate the maximization of the
amount of metabolite produced as an optimal control problem, and derive optimal solutions and a

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Inria Annual Report 2023

control synthesis, as well as quasi-optimal feedback strategies satisfying chemical and physical design
constraints. The proposed synthetic control strategies are being tested experimentally by growing E. coli
strains capable of producing glycerol from glucose in a mini-bioreactor system. We aim at quantifying the
amount of glucose consumed and glycerol produced, in the case of a predefined input signal (open-loop
control) and the adaptive regulation of the input signal based on on-line measurements of the growth
rate and the expression of fluorescent reporters of selected genes (closed-loop control). New results are
presented in Section 7.9.

The team is also involved in other problems related to biological or medical applications, namely
muscular functional electro-stimulation (new results presented in Section 7.10), Lotka-Volterra models
(new results presented in Section 7.11), and alcoholic fermentation (new results presented in Section 7.12).

4.3 Neural dynamics

Participants:

Adel Annabi, Dario Prandi (CNRS, CentraleSupélec), Jean-Baptiste Po-
met, Ludovic Sacchelli.

Neural fields serve as integro-differential dynamical models for the transmission of activity within
cortical areas [54]. Originating in the 1970s, these models prove particularly advantageous when exploring
the mesoscopic scale. At this level, the neuronal clusters under examination are sufficiently large to be
understood as a continuum, yet compact enough to enable a targeted investigation of specific cortical
functions. A significant appeal of these models lies in their efficacy in describing phenomena within the
perceptual mechanisms of vision and audition. Notably, they have paved the way for sub-Riemannian-
inspired geometric models addressing the anisotropic diffusion of information [51, 53].

Given their successes in characterizing cortical areas, their interplay and their scale, these models also
offer valuable insights into experiments involving the measurement and stimulation of neural activity via
electrodes. Consequently, substantial interest has been directed toward these models from the point of
view of control, where the input-output formalism provides strategic avenues for deep-brain stimulation
techniques. This interest has manifested in recent applications, including the treatment of Parkinson’s
disease [61]. The exploration of this perspective is the topic of A. Annabi’s PhD research, which delves into
the visual cortex, specifically concentrating on observability and observer design for low-dimensional
models within the V1 cortical area.

5 Highlights of the year

Awards

Alesia Herasimenka, who defended her PhD in the team in September, was one of the 35 winners
of the “Prix Jeunes Talents France 2023 L’Oréal-UNESCO Pour les Femmes et la Science” for her
doctoral work, see [18], Section 7.6, and the press release from Fondation L’Oréal.

Other

Ivan Beschastnyi was hired by Inria this year and joined the team in December, 2023.

6 New software, platforms, open data

6.1 New software

6.1.1 ct

Name: control toolbox

Keywords: Optimal control, Ordinary differential equations, Mathematical Optimization, Differential

homotopy, Automatic differentiation

Project MCTAO

9

Scientific Description: Numerical resolution of optimal control problems

Functional Description: The project gathers and allows to interoperate tools designed to solve numeric-
ally optimal control problems on ordinary differential equations. The available approaches include
direct methods (based on a transcription of optimal control problems into mathematical programs)
as well as indirect ones (based on Pontrjagin maximum principle, like the shooting method). The
latter can be coupled to differential continuation. Automatic differentiation (aka Differentiable
Programming) plays a crucial a role in all these algorithms. The project strongly leverages on SED
Sophia support.

Release Contributions: - bocop refactoring - nutopy library - project gallery

URL: https://control-toolbox.org

Contact: Jean-Baptiste Caillau

Participants: Jean-Baptiste Caillau, Pierre Martinon, Olivier Cots, Thibaud Kloczko, Tristan Cabel, Jean-
Luc Szpyrka, Erwan Demairy, Julien Wintz, Carlos Zubiaga Pena, Nicolas Niclausse, Joseph Gergaud

Partners: Université de Toulouse, CNRS, IRIT, ENSEEIHT

7 New results

7.1 Output feedback stabilization of non-uniformly observable systems

Participants:

Ludovic Sacchelli, Lucas Brivadis (CentraleSupélec, Gif-sur-Yvette),
Jean-Paul Gauthier (Université de Toulon).

Stabilization of the state of a system by means of a feedback control is a fundamental problem in
control theory. When only part of the system is known, a usual strategy is to rely on a dynamic algorithm,
known as an observer, in order to provide an estimate of the state that can be fed to the controller. This is
known as output feedback control. Designing a stable closed-loop based on an observer requires that
some necessary information on the state can be accessed through this partial measurement. Critically, for
nonlinear systems, whether or not it is possible to reliably estimate the state can depend on the control.
This fact is known as non-uniform observability and is a root issue for observer design.

Regarding output feedback control, if singular controls exist for observability, there is no clear definit-
ive answer as to how to achieve stabilization. The now published [6] reviews some strategies that showed
to be efficient in tackling the difficulties posed by non-uniform observability, and explores the genericity
side of the matter, including a proof that this critical situation can be generic in some key classes of
systems. In 2023’s IFAC (International Federation of Automatic control) world conference, [11] explores
the issue through the point of view of hybrid systems, a new framing for the group. In the paper, we
propose a method to monitor observability of the system online. We use this method as the backbone of
a dynamically switching Kalman-type observer and feedback to achieve stabilization of bilinear systems.
Allowing Kalman filters in that context was a long term goal but had remained an open problem. This
hybrid approach used to maintain observability of non-uniformly observable systems has been explored
more thoroughly, and will be the topic of future works.

7.2 Geometry and optimal control for navigation problem

Participants: Bernard Bonnard,

Joseph Gergaud, Boris Wembe (ENSEEIHT,
Toulouse), Olivier Cots, Jérémy Rouot (Université de Bretagne Occi-
dentale, Brest).

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Inria Annual Report 2023

This is a long term research contribution that revisits and generalizes the Navigation Problem set
by Carathéodory and Zermelo of a ship navigating on a river with a linear current and aiming to reach
the opposite shore in minimum time. This work is motivated by the displacement of particles in a two
dimensional fluid, in presence of a vortex (initially, a singularity in the Helhmoltz-Kirchhoff equations) in-
ducing a strong current that hampers local controllability. To define a minimum time Zermelo navigation
problem, we consider the particle as the ship of the navigation problem and the control is defined as the
heading angle of the ship axis. It turns out that the historical problem and our recent vortex study are two
examples of the general case of Zermelo navigation problems on surfaces of revolution on which some
contributions appeared this year. Our main contributions in this setting are multiple. In [5] and [47], we
analyze the role of abnormal geodesics in the problem, in particular in relation with cusp singularities
of the geodesics and non continuity properties of the value function. In [2], we relate the existence, in the
geodesics dynamic, of separatrices interpreted as Reeb components, with the Morse–Reeb classification
of the geodesics. Furthermore, we provide, still in [2], explicit computations of the conjugate and cut loci
in case studies such as the averaged Kepler case in space mechanics .

7.3 Small time local controllability for some degenerate two-input systems

Participants:

Laetitia Giraldi (CALISTO project-team), Pierre Lissy (Université Paris
Dauphine, Paris), Clément Moreau (Kyoto University, Japan), Jean-
Baptiste Pomet.

Here, we investigate small time local controllability (STLC) for affine control systems with two controls
around an equilibrium such that the two control vector fields are co-linear at this point. Such a problem
was motivated by the control of planar articulated magnetically actuated swimmers at low Reynolds
number around the straight configuration with all magnetic moments aligned, see C. Moreau’s PhD for
details [76]. We pursued a more general study of local controllability with two controls with the above
mentionned properties; in the paper [8], accepted for publication this year, we introduce novel necessary
conditions for STLC of these systems, based on Chen-Fliess expansions of solutions, in the spirit of [74] or
the more recent [37]. On top of “generalizing” the case of micro-swimmers, this work is the first attempt
to give obstruction to local controllability in the spirit of these references for multi-input systems.

7.4 Stability of linear time-varying time-delay systems

Participants:

Laurent Baratchart (FACTAS project-team), Sébastien Fueyo (Tel-Aviv
University), Jean-Baptiste Pomet.

A linear time-periodic difference-delay systems (periodic LDDS for short) is a dynamical system of the
form z(t ) = A1(t )z(t −τ1)+· · ·+ AN (t )z(t −τN ), where z is finite dimensional and the matrices A j depend
periodically on time. The state of this dynamical system is infinite dimensional. S. Fueyo’s doctoral
work was about testing the stability of nonlinear amplifiers for high frequency signals by frequency
domain methods; linearizing along an internal periodic solution yields a model based on networks
of 1-D hyperbolic PDEs, and a periodic LDDS appears as its ‘high frequency limit”, whose stability
conditions stability of the PDE dynamical system (see [69], or the long introduction of [35]). Coming back
to hyperbolic stability of time-varying LDDS, a well known necessary and sufficient condition for stability
holds in the time-invariant case, due to Hale and Henry [72, 71]; it gives a final answer to the question,
but it is not so easy to check explicitly this criteria, and there is still a vast literature on more specific
sufficient conditions. We obtained a generalization to the periodic time-varying case, presented in the
manuscript [20], submitted for publication, with the proof of a technical part given in [21], separately
submitted for publication.

7.5 Second-order averaging of fast-oscillating control systems

Project MCTAO

11

Participants:

Jean-Baptiste Caillau, Lamberto Dell’Elce, Jean-Baptiste Pomet.

Research on fast-oscillating optimal control systems is a long-standing topic in McTAO. We investig-
ated in 2021 how trajectories of fast-oscillating control system with a single fast variable converge to their
averaged counterpart [64]. During 2022, further insight was gained on the averaging of systems with two
fast variables [63]. Outcomes of these studies were mostly deduced from numerical experiments, but no
proof of convergence of trajecories of the original system to their averaged counterpart was offered, since
such proof requires the inclusion of second-order terms, which were neglected in these works. In 2023,
we carried out a formal computation of second-order terms, which was exploited in the methodology
described in Section 7.7. These developments pave the way to a proof of convergence and to a more
general study for systems with several fast angles (which entails to address resonance issues).

7.6 Control of solar sails: controllability and optimal control

Participants:

Jean-Baptiste Caillau, Lamberto Dell’Elce, Alesia Herasimenka, Jean-
Baptiste Pomet.

The PhD thesis of A. Herasimenka [18], defended in September, 2023, is devoted to the control of
solar sails, which offer a propellantless solution to perform interplanetary transfers, planet escapes, and
de-orbiting maneuvers by leveraging on solar radiation pressure (SRP).

The thesis offers two main contributions, both of which were finalized in 2023. The first one is
dedicated to the controllability study of solar sails. The primary challenge in assessing their controllability
arises from the specific constraints imposed on the control set. Due to the nature of solar radiation
pressure, a solar sail can only generate force whose directions belong to a convex cone with axis toward
the direction of the Sun. Traditional methods for evaluating controllability are inadequate due to these
specific constraints. To address this challenge, we proposed a novel necessary condition and a novel
sufficient condition for controllability. The former involves identifying forbidden dual directions in
the co-tangent bundle associated with the system state manifold that eventually constrain the state
in some “half space” and the latter asserts global or local controllability under conditions that involve
both the system and the shape of the control constraints. These theoretical results are applicable to any
periodic system with a conical constraint on its control set. We also developed an algorithm aimed at
efficiently verifying this necessary condition, based on convex optimization tools and fine properties of
trigonometric polynomials as well as Nesterov’s technique of sum of squares relaxation. This requirement
is aimed at assessing whether a non-ideal solar sail with given optical parameters is capable of decreasing
or increasing all possible functions of the Keplerian integrals of motion over an orbital period. In other
words, we verify whether a solar sail can perturb its orbit in any arbitrary way given its optical properties.
These developments were, for one part, published in the Journal of Guidance, Control, and Dynamics [9]
(a paper that insists on constructive method to check the necessary conditions) and are, for another
part, under revision for publication in SIAM J. on Control & Optim. [24] (a more mathematical paper
that establishes these controllability results in a fairly general context). An extension of the numerical
algorithm to the assessment of the local controllability of periodic orbits in the circular-restricted, three-
body problem was presented at the Spaceflight Mechanics Meeting [13].

The second contribution of the thesis is an algorithm that computes the optimal control inputs for
steering a sail towards a desired direction of the phase space. The algorithm employs convex optimization
to obtain an admissible yet suboptimal control as an initial input. Subsequently, an optimal control
problem is solved to maximize the displacement in the desired direction. By analyzing the Hamiltonian
dynamics of the system, the relevant switching function that governs the structure of the solution is
identified. Additionally, an upper bound on the number of zeros of this function is established, enabling
the efficient implementation of a multiple shooting code using differential continuation. The publication
of this work is currently under review [26].

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Inria Annual Report 2023

7.7 Efficient solution of the low-thrust Lambert’s problem

Participants:

Lamberto Dell’Elce, Alesia Herasimenka, Nicola Baresi (University of
Surrey), Aaron J. Rosengren (University of California San Diego).

Lambert’s problem, in its classical form, consists in finding a Keplerian orbit joining two position
vectors in a given transfer time. Solutions of this problem are extensively used for preliminary mission
design since they offer the identification of launch opportunities and a rough evaluation of their fuel
cost by assuming impulsive maneuvers at the two boundary points. Specifically, the concept of "launch
windows" and the use of graphical representations to find feasible trajectories for planetary missions were
introduced in the early space age. These graphical representations, which include what are now called
pork-chop plots, help mission planners to visualize and analyze the trade-offs between departure and
arrival dates, taking into account the positions and velocities of the planets involved. Pork-chop plots are
often generated to illustrate total ∆V cost (the “∆V ” of a maneuver is a normalized estimate of the total
impulse that the thrusters need to deliver, clearly related to the mass of propellant to be consumed) as a
function of the departure and arrival dates by recursively solving Lambert’s problems. When considering
the low-thrust counterpart of this transfer problem in fixed time, the entire state vector is imposed at
the two boundaries, i.e., both position and velocity of the satellite have to match the ones of departure
and arrival bodies, because impulsive maneuvers are not allowed. In contrast to the original problem, no
exact closed-form solution exists.

Two problems need to be addressed when tackling low-thrust transfers for a fixed maneuvering
time. First, the minimum thrust magnitude necessary to carry out the maneuver has to be identified.
Solutions to this problem are characterized by the absence of coasting arcs and the exploitation of
the maximum control force throughout the entire trajectory. Second, once a sufficiently-large thrust
is chosen, minimum-energy maneuvers can be found. Our research during the 2023 focused on the
first problem, which is of interest because it offers a lower bound on the thrust that cannot be violated
when optimizing other cost functions and it may serve as initial guess for the minimum fuel problem. A
numerical methodology based on the averaging of the extremal flow of the optimal-control system (see
Section 7.5) was proposed: First, a reduced-order solution of the averaged two-point boundary value
problem (TPBVP) parametrized by the adjoint variable of the fast variable is solved. This step requires the
solution of a single shooting problem followed by a numerical continuation procedure. This problem is
independent of the thrust magnitude. Second, sensitivities of the shooting function are computed. These
sensitivities are then used to evaluate perturbations of the averaged TPBVP associated to short-periodic
variations and second-order terms, which are hereby retained to obtain a first-order approximation of
the fast variable from the averaged solution. This information is finally used to find the minimum thrust
required for the transfer.

The methodology was presented at the AIAA/AAS Spaceflight Mechanics Meeting [16], and published

in the Journal of Guidance, Control, and Dynamics [7].

7.8 Low-thrust satellite collision avoidance

Participants:

Lamberto Dell’Elce, Frank de Veld, Jean-Baptiste Pomet.

This topic is at the core of Frank de Veld’s research, whose PhD started in December, 2022. The
presence of space debris in Earth orbits became an important factor to consider for day-to-day spacecraft
operations. Future trends regarding the number of satellites, number of space debris objects, and
tracking capabilities of these objects suggest that satellites in low-Earth orbits will continue to require
collision avoidance maneuvres (CAM’s) on a regular basis within their lifetime to resume operational
activities safely. At the same time, low thrust satellites are becoming more popular in space flight
replacing conventional chemical propulsion systems with more efficient electrical ones. While propellant
consumption is reduced using low-thrust propulsion systems, CAM design becomes more challenging.
Smaller thrust levels go hand in hand with less maneuvrability and a longer time required to perform a
certain maneuvre, such as a collision avoidance maneuvre.

Project MCTAO

13

In this context, our research focused on the relationship between the thrust arc duration, time of
initiating thrust and thrust magnitude, when performing a collision avoidance maneuvre. We investigated
the relationship between the time instant of initiating a CAM compared to the time of closest approach
(TCA) and the separation distance at the time of closest approach. The resulting outcome of our study has
consequences for CAM design for low-thrust satellites, which often differs significantly from high-thrust
CAM design, as well as space traffic management in general.

7.9 Optimal allocation of resources in bacteria

Participants:

Agustín Yabo (INRAE, Montpellier), Jean-Baptiste Caillau, Jean-
Luc Gouzé (BIOCORE project-team), Mohab Safey El Din (Sorbonne
Université).

In the framework of the ANR Maximic, these last years, we carried on the study of self-replicator
models. These models describe the allocation of resources inside the bacteria: the substrate is used to
produce precursors that, in turn, can be employed either to produce genetic machinery (and increase
the biomass) or metabolic machinery (that will further catalyze the transformation of substrate into
precursors). To this internal control, the model adds and external action that aims, after some genetic
engineering on the bacteria (to create a strain that reacts to light stimuli), at producing a new metabolite
of interest. Then, while the behavior of the untouched bacteria tends to be very well mimicked by biomass
maximization strategies, maximizing the production of the metabolite of interest induces new biological
strategies. This kind of model (and refinements) were studied in [86, 87]. Key properties of the system are:
(i) the Fuller phenomenon as connection between bang and singular arcs requires an infinite number of
switchings in finite time; (ii) the turnpike phenomenon. Indeed, for large fixed final times, trajectories of
the system are essentially singular and close to the optimal (w.r.t. a constant static control) equilibrium
which is a hyperbolic fixed point of the singular flow. See ct gallery for an example, and the recently
defended PhD thesis of A. Yago’s PhD thesis [85] for a discussion of these results. In collaboration with M.
Safey El Din, stability properties of the system were established thanks to a consistency check of a system
of polynomial inequalities [10].

Colleagues from McTAO and Biocore teams at Sophia, together with former students (Agustin Yabo,
now researcher at INRAE), are involved in this task. New results concern the definition on extended
(higher dimensional) models for the bacteria dynamics, check of second order optimality conditions on
the resulting optimal control problem, and study of the turnpike phenomenon for these optimization
problems. One can also mention results on stability of the equilibria of these systems; the analysis in [10]
resorts to real algebraic geometry and associated algorithmic tools in collaboration with Mohab Safey el
Dinh (Sorbonne Université / CNRS / LIP6).

7.10 Biomechanics: control of muscular force response using FES with application

to the conception of a smart electrostimulator

Participants: Bernard Bonnard, Sandrine Gayrard, Jérémy Rouot (Université de
Bretagne Occidentale, Brest), Toufik Bakir (Université de Bourgogne
Franche Comté, Dijon).

This topic started in McTAO in 2017 with a collaboration between B. Bonnard and T. Bakir (ImVia-
UBFC), and J. Rouot (LMBA, Brest), in a collaboration with Segula Technologies. The problem of control of
muscular force is posed in terms of optimization of the train pulses of a Functional Electro-Stimulation
(FES) signal to produce the muscular contraction. Based on preliminary experimental studies, the
dynamical model that was chosen for muscular control is known as Ding et al. force-fatigue model in the
literature. It is a refinement of the historical Hill model (Medicine Nobel Prize in 1922) that takes into
account the variations of the fatigue variable. From the control methodology point of view, this required
some developments on optimal control for sample control systems. This is by itself already a rich topic.

14

Inria Annual Report 2023

In 2020, this project took the industrial transfer direction with a Cifre PhD funding in partnership
with Segula Technologies (see Section 8) whose goal is the design of a smart electrostimulator for force
reinforcement or rehabilitation in the framework of S. Gayrard’s PhD. This PhD was defended in Septem-
ber, 2023, see Section 10.2.2; the manscript is condifential, a summary will soon be available online.
The contribution is twofold. From the theoretical point of view, we have derived a finite dimensional
approximation of the forced dynamics based on the Ding et al. model to provide fast optimizing schemes
aiming to track a reference force or maximize the force, see [34]. On the other hand, S. Gayrard finalized a
prototype of the smart electrostimulator, which was a major objective of the collaboration with Segula
Technologies; extensive tests have been performed; the principle of these tests and parameter estimation
was presented at 2023 European Control Conference [12].

7.11 Feedback invariants and optimal control with biological applications and nu-

merical calculations

Participants: Bernard Bonnard, Jérémy Rouot (Université de Bretagne Occidentale,

Brest), Cristiana J. Silva (Instituto Universitário de Lisboa, Lisbon).

The starting point of the study was the problem of controlled Lotka Volterra dynamics motivated by
curing microbiote infection by a pathogenic agent. This leads to complicated optimal control problems in
the frame of permanent or sampled-data controls, in relation with medical constraints. The problem can
be set as a time-minimal control problem with terminal manifold of codimension one. Our contributions
are presented in the series of works [22, 3, 23, 4]. In relation with previous work regarding the optimal
control of chemical networks, the time minimal synthesis has been described near the terminal manifold
up to codimension two situations in the jets spaces. We have compared both permanent controls and
sampled-data control in relation with numerical issues. Finally, we have obtained feedback invariants
to classify the geodesic dynamics associated to rays of abnormal geodesics which are related to shifted
equilibria of the free Lotka-dynamics and which can be calculated using only linear computations.

7.12 State estimation in alcoholic fermentation models

Participants:

Agustín Yabo (INRAE, Montpellier), Ludovic Sacchelli, Martin Fleurial.

This inquiry folds into the ANR research project STARWINE on real time control of aroma production
in wine fermentation processes, of which A. Yabo is a member. Focused on alcoholic fermentation
in wine-making conditions, the study addresses the challenge of online state estimation during wine
fermentation. This is relevant in industrial scenarios where control laws rely on estimating the full state
from partial measurements of the system, mainly biogas production. This topic has been the subject of
M. Fleurial’s M2 internship and led to the submission of [25] in an international conference on control.
The primary emphasis of [25] lies in investigating the observability properties of an alcoholic ferment-
ation model. Second, a full information estimator algorithm was developed. This type of algorithms are
based on prediction error minimization on expanding time windows. While this method may be costly for
non linear systems (algorithmically speaking), it is well adapted to the context of fermentation processes
that are typically slow (a hundred hours in timeframe and new measurements added at intervals of
30 minutes). To validate the algorithm, comprehensive testing was conducted using both simulated
and experimental data provided by the MISTEA research unit. This work provides the basis for further
application of the estimation algorithm to more modern fermentation models.

7.13 Observer design for Blumenfeld’s model of V1

Participants:

Adel Annabi, Dario Prandi (CNRS, CentraleSupélec), Jean-Baptiste Po-
met, Ludovic Sacchelli.

Project MCTAO

15

The goal of this research is to provide insights into observability and observer synthesis of neural fields
equations, the general topic of A. Annabi PhD. In the case of the visual cortex, neural fields models can
be used to describe the activity dynamics in the specific case of orientation sensitivity of neurons. This
focus allows to map neural fields in the visual cortex onto neural fields in the orientation domain. This
reframing allows to move to Fourier series, which can be truncated to give a significant enough model in
only 3 dimensions, a model of V1 due to Blumenfeld. The work of A. Annabi describes the observability of
this model and highlights the symmetries of the system, and introduces hybrid elements to mitigate their
effects. Crucially, this research also gives insights into the persistence conditions necessary for accurate
estimation of the system’s state, laying groundwork for further exploration of neural field models.

7.14 ct: control toolbox

Participants:

Jean-Baptiste Caillau, Olivier Cots, Joseph Gergaud, Pierre Mar-
tinon (CAGE project-team, on leave).

The ADT ct: control toolbox had its final sprint in 2023. The focus was on initiating new
developments in Julia to take advantage of the powerful features of the language. Julia is indeed a perfect
match for our needs in scientific computing for numerical optimal control; the language has a high level
of abstraction well suited for mathematical descriptions, but still makes no compromise when it comes
to performance thanks to efficient just-in-time compilation. Moreover, it currently has several efficient
backends for AD / DP (automatic differentiation / differentiable programming), including ForwarDiff,
Zygote of Enzyme: this is a crucial step for our project, both for direct and indirect methods. (Some
examples of the project gallery require up to five levels of nested automatic differentiation.) The toolbox
is now a full ecosystem available at control-toolbox.org. These achievements and the use of Julia have
been presented in conferences [15, 14].

8 Bilateral contracts and grants with industry

8.1 Intelligent muscle electrostimulator, Segula Technologies

Participants:

Toufik Bakir (Université de Bourgogne Franche Comté, Dijon), Bern-
ard Bonnard, Sandrine Gayrard.

In the framework of a CIFRE grant, a contract (title: “Réalisation d’un prototype d’électrostimulateur
intelligent”) between Segula Technologies and Université de Bourgogne is partially funding (together
with ANRT) Sandrine Gayrard’s PhD.

This is completed by an additionnal collaboration contract between Segula and SAYENS (represent-

ing Université de Bourgogne), aiming at constructing the prototype of the smart electrostimulator.

• PI: Bernard Bonnard and Toufik Bakir (ImViA, Université de Bourgogne Franche Comté, Dijon).

• Period: 2020–2023.

8.2 Optimal Control of Solar Sails, ESA (European Space Agency)

Participants:

Jean-Baptiste Caillau, Lamberto Dell’Elce, Alesia Herasimenka, Jean-
Baptiste Pomet.

Three year contract starting in 2021 between the team and the European Space agency. Its purpose is

to support the environment of Alesia Herasimenka’s PhD on this topic.

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Inria Annual Report 2023

• Partners: McTAO and ESA.

• Total amount: 24k€.

• Period: 2021–2024.

• Inria reference: 16016.

8.3 Méthodes de contrôle pour l’évitement de collisions entre satellites, Thales

Alenia Space

Participants:

Thierry Dargent, Lamberto Dell’Elce, Frank de Veld, Jean-Baptiste Po-
met.

Thales Alenia Space is co-funding the thesis of Frank de Veld named “Méthodes de Contrôle pour

l’évitement de collisions entre satellites”.

• Partners: McTAO and Thales Alenia Space.

• Period: 2022–2025

• Total amount: 75k€

• Inria reference: 022-0674

9 Partnerships and cooperations

Participants:

Adel Malik Annabi, Antonin Bavoil,
Ivan Beschastnyi, Bern-
ard Bonnard, Jean-Baptiste Caillau, Sandrine Gayrard, Alesia Herasi-
menka, Jean-Baptiste Pomet, Ludovic Sacchelli.

9.1 National initiatives

9.1.1 ANR

MAXIMIC: optimal control of microbial cells by natural and synthetic strategies. Started in 2017, ended

in March, 2023. J.-B. Caillau and J.-B. Pomet were participants. More information and news on the
site of this project.

PDE-AI: partial differential equations for AI. This project on "Numerical analysis, optimal control and
optimal transport for AI", funded by PEPR IA from 2023 to 2027, is led by A. Chambolle (CNRS /
Dauphine) and involves 10 French nodes, including a node in Nice / Sophia supervised by J.-B.
Caillau. Total amount for the node 390 K€.

9.1.2 Other

• Labex CIMI grant on ”Singular control and numerical optimisation in Julia”. Participants are J.-B.

Caillau, J. Gergaud (PI) and O. Cots (both Université de Toulouse). Total amount 8k€.

• "Recherche en réseaux" Université de Bourgogne grant on "Design of an asymmetric copepod-
microswimmer for 2d-motion". Participants are T. Bakir (P.I., Université de Bourgogne), B. Bonnard.
Total amount 5k€.

• McTAO project-team participates in the GdR MOA, a CNRS network on Mathematics of Optimiza-

tion and Applications.

Project MCTAO

17

9.2 Regional initiatives

"Emplois Jeunes Doctorants" Grant from Région SUD – Provence Alpes Côte d’Azur “Emplois
jeunes Chercheurs”, 2022-2025, that co-funds Frank de Veld’s PhD. See also the contract with Thales
Alenia Space in Section 8. Total amount: 54k€.

10 Dissemination

Participants:

Ivan Beschastnyi, Bern-
Adel Malik Annabi, Antonin Bavoil,
ard Bonnard, Jean-Baptiste Caillau, Sandrine Gayrard, Alesia Herasi-
menka, Jean-Baptiste Pomet, Ludovic Sacchelli.

10.1 Promoting scientific activities

10.1.1 Scientific events: organisation

The McTAO project team maintains a recurring seminar on topics of control theory, optimization and
applications (organizer: L. Sacchelli). The seminar has a monthly periodicity and has hosted 9 sessions in
2023:

16/01/23 Ivan Beschastnyi (Université d’Aveiro): optimal control.

24/02/23 Daniel J. Scheeres (University of Colorado Boulder): Quasi-periodic orbits.

08/03/23 Giovanni Federico Gronchi (Università di Pisa): Averaging techniques in space mechanics.

07/04/23 Bruno Cessac (Inria, Biovision): chaos in retinal neuron dynamics.

08/06/23 Agustin Yabo (Inrae, Montpellier): Wine fermentation process and control.

28/06/23 Aaron Rosengren (UC San Diego): xGEO space mechanics.

08/09/23 Bernd Dachwald (Aachen University): neural networks for solar sails trajectory planning.

19/10/23 Jingrui Niu (Inria CAGE, LJLL): controllability of nonlinear Schrödinger equations.

10/11/23 Frédéric Jean (ENSTA Paris): constrained optimal control for reusable lauchers.

10.1.2 Scientific events: selection

Member of conference program committees

J.-B. Caillau is a member of the PGMO scientific council of Programme Gaspard Monge (PGMO),
and was a member of the scientific board of PGMO days’ 2023.

Reviewing activities

All team members take part in a continued effort to offer reviews in various conferences of import-
ance to the community.

10.1.3 Journal

Member of the editorial boards

• B. Bonnard is member of the editorial board of Pacific Journal of Mathematics for Industry.

• J.-B. Caillau is member of the editorial board of ESAIM: Mathematical Modelling and Numerical

Analysis (ESAIM: M2AN).

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Inria Annual Report 2023

Reviewing activities

All team members take part in a continued effort to offer reviews in various journals of importance
to the community.

10.1.4 Invited talks

L. Sacchelli gave invited talks at: Journées Contrôle Optimal, FRUMAM, Marseille, 09/06/23; Lehigh
mathematics department colloquium, Lehigh University, USA, 11/10/2023.

10.1.5 Research administration

• J.-B. Caillau is member of 3IA Côte d’Azur scientific council

• J.-B. Caillau is member of Inria Sophia CDT

• J.-B. Caillau co-organizes LJAD colloquium

• J.-B. Pomet is a member of the steering comitee of Academy of excellence on complex systems, Uni-
versité Côte d’Azur (IDEX) and a member of the executive bureau of “EUR SPECTRUM” (graduate
school, Université Côte d’Azur).

• J.-B. Pomet was an elected member of Inria permanent Evaluation Committee until August 31 (first

elected in 2014).

10.2 Teaching - Supervision - Juries

10.2.1 Teaching

• Engineering school and University: J.-B. Caillau has a full teaching duty of Professor at L (BSc) and

M (Master) level at Polytech Nice Sophia and Université Côte d’Azur.

• Engineering school: L. Dell’Elce, A. Herasimenka and L. Sacchelli each took part as teaching

assistants at L3 and M1 levels at Polytech Nice Sophia, Université Côte d’Azur.

10.2.2 Supervision

PhD students.

• Antonin Bavoil, "Génération optimale d’énergie par un cerf-volant" (Optimal energy genera-
tion from a kite), Université Côte d’Azur, co-supervised by J.-B. Caillau and Alain Nême (ENSTA
Bretagne). Funded by CNRS. Started October, 2023.

• Adel Malik Annabi, "Observability and observer synthesis for neural fields equations", Université
Côte d’Azur, co-supervised by J.-B. Pomet, L. Sacchelli and D. Prandi (CentraleSupélec). Started
October, 2022.

• Frank de Veld, "Méthodes de Contrôle pour l’évitement de collisions entre satellites", Inria, co-

supervised by J.-B. Pomet, L. Dell’Elce and T. Dargent. Started December 2022.

• Sandrine Gayrard, "Réalisation d’un prototype d’électrostimulateur intelligent", Université de
Bourgogne Franche Comté, co-supervised by B. Bonnard and T. Bakir (Université de Bourgogne
Franche Comté), defended her PhD on September 21st, 2023. [condifential manuscript, summary
online soon.]

• Alesia Herasimenka, “Optimal control of solar sails”, Université Côte d’Azur, co-supervised by L.

Dell’Elce and J.-B. Caillau, defended her PhD on September 7th, 2023. See [18].

Project MCTAO

Interns.

19

• Antonin Bavoil, 5th year Polytech Nice. Supervised by J.-B. Caillau. March to September, 2023,

funded by CNRS (Prix défi Défense 2022 des Assises des maths du CNRS).

• Martin Fleurial, INSA Rouen. Co-supervised by L. Sacchelli and A. Yabo (INRAE, Montpellier) on
the topic of "State and parameter estimation of wine fermentation models for real-time aroma
control". March to August, 2023.

10.2.3 Juries

• J.-B. Caillau was reviewer and sat at the jury of Clara Leparoux PhD thesis (ENSTA Paris).

• J.-B. Pomet was reviewer and sat at the jury of Bruno Hérissé HDR thesis (Université Paris saclay).

11 Scientific production

11.1 Publications of the year

International journals

[1] N. Baresi and L. Dell’Elce. ‘Periodic and Quasi-Periodic Orbits near Close Planetary Moons’. In:
Journal of Guidance, Control, and Dynamics (2023), pp. 1–15. DOI: 10.2514/1.G007221. URL:
https://inria.hal.science/hal-03937917.

[2] B. Bonnard, O. Cots and B. Wembe. ‘Zermelo Navigation Problems on Surfaces of Revolution and
Geometric Optimal Control’. In: ESAIM: Control, Optimisation and Calculus of Variations 29.60
(31st July 2023), p. 34. DOI: 10.1051/cocv/2023052. URL: https://hal.science/hal-032094
91.

[3] B. Bonnard and J. Rouot. ‘Optimal Control of the Controlled Lotka-Volterra Equations with Applica-
tions - The Permanent Case’. In: SIAM Journal on Applied Dynamical Systems 22.4 (2023), pp. 2761–
2791. DOI: 10.1137/22M151978X. URL: https://inria.hal.science/hal-03757060.

[4] B. Bonnard, J. Rouot and C. Silva. ‘Geometric Optimal Control of the Generalized Lotka-Volterra
Model of the Intestinal Microbiome’. In: Optimal Control Applications and Methods (2024). DOI:
10.1002/oca.3089. URL: https://hal.science/hal-03861565.

[5] B. Bonnard, J. Rouot and B. Wembe. ‘Accessibility Properties of Abnormal Geodesics in Optimal
Control Illustrated by Two Case Studies.’ In: Mathematical Control and Related Fields 13.4 (Dec.
2023), pp. 1618–1638. DOI: 10.3934/mcrf.2022052. URL: https://inria.hal.science/hal-
03310348.

[6] L. Brivadis, J.-P. Gauthier and L. Sacchelli. ‘Output feedback stabilization of non-uniformly observ-
able systems’. In: Proceedings of the Steklov Institute of Mathematics 321.1 (June 2023), pp. 69–83.
DOI: 10.1134/S0081543823020050. URL: https://hal.science/hal-03589760.

[7] L. Dell’Elce, A. Herasimenka, A. J. Rosengren and N. Baresi. ‘Second-order Averaging of Low-Thrust
Transfers in Fixed Time *’. In: Journal of Guidance, Control, and Dynamics (2024). URL: https://i
nria.hal.science/hal-04387944.

[8] L. Giraldi, P. Lissy, C. Moreau and J.-B. Pomet. ‘Necessary conditions for local controllability of a
particular class of systems with two scalar controls’. In: ESAIM: Control, Optimisation and Calculus
of Variations 30.4 (2024). DOI: 10.1051/cocv/2023073. URL: https://hal.science/hal-0217
8973.

[9] A. Herasimenka, L. Dell’Elce, J.-B. Caillau and J.-B. Pomet. ‘Controllability Properties of Solar
Sails’. In: Journal of Guidance, Control, and Dynamics (2023). DOI: 10.2514/1.G007250. URL:
https://hal.science/hal-03331300.

[10] A. G. Yabo, M. Safey El Din, J.-B. Caillau and J.-L. Gouzé. ‘Stability analysis of a bacterial growth
model through computer algebra’. In: MathematicS In Action 12.1 (26th Sept. 2023), pp. 175–189.
DOI: 10.5802/msia.37. URL: https://hal.science/hal-03671432.

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Inria Annual Report 2023

International peer-reviewed conferences

[11] L. Brivadis and L. Sacchelli. ‘Output feedback stabilization of non-uniformly observable systems by
means of a switched Kalman-like observer’. In: IFAC-PapersOnLine. 22nd World Congress of the
International Federation of Automatic Control (IFAC 2023). Yokohama, Japan, 9th July 2023. DOI:
10.1016/j.ifacol.2023.10.250. URL: https://hal.science/hal-03845706.

[12]

S. Gayrard. ‘Fast Optimization Scheme for the Muscular Response to FES Stimulation to Design a
Smart Electrostimulator’. In: 2023 European Control Conference (ECC). Bucharest, France: IEEE,
13th June 2023, pp. 1–6. DOI: 10.23919/ECC57647.2023.10178228. URL: https://inria.hal
.science/hal-03891925.

[13] A. Herasimenka, A. Farrés and L. Dell’Elce. ‘Station-keeping under conical constraint on the control
force’. In: 9th International Conference on Astrodynamics Tools and Techniques (ICATT). Sopot,
Poland, 12th June 2023. URL: https://inria.hal.science/hal-04156856.

Conferences without proceedings

[14]

[15]

J.-B. Caillau, O. Cots, J. Gergaud and P. Martinon. ‘Solving optimal control problems with Julia’. In:
Julia and Optimization Days 2023. Paris, France, 4th Oct. 2023. URL: https://hal.science/hal-
04277441.

J.-B. Caillau, O. Cots, J. Gergaud and P. Martinon. ‘Solving optimal control problems with Julia
(talk)’. In: JuliaCon 2023. Cambridge, Boston, United States, 26th July 2023. URL: https://hal.sc
ience/hal-04277383.

[16] L. Dell’Elce, A. Herasimenka, A. J. Rosengren and N. Baresi. ‘Efficient Numerical Solution of the
Low-Thrust Lambert Problem’. In: SFMM 2023 / 33rd AAS/AIAA Space Flight Mechanics Meeting.
Austin, United States, 15th Jan. 2023. URL: https://inria.hal.science/hal-04050277.

Scientific book chapters

[17]

J.-B. Caillau, R. Ferretti, E. Trélat and H. Zidani. ‘An algorithmic guide for finite-dimensional
optimal control problems’. In: Handbook of numerical analysis: Numerical control, Part B. Vol. 24.
Handbook of Numerical Analysis. North-Holland; Elsevier, 2023, pp. 559–626. DOI: 10.1016/bs.h
na.2022.11.006. URL: https://hal.science/hal-03883822.

Doctoral dissertations and habilitation theses

[18] A. Herasimenka. ‘Optimal control of solar sails’. Université Côte d’Azur, 7th Sept. 2023. URL: https

://theses.hal.science/tel-04304035.

Reports & preprints

[19] V. Andrieu, L. Brivadis, J.-P. Gauthier, L. Sacchelli and U. Serres. Exponential stabilizability and
observability at the target imply semiglobal exponential stabilizability by templated output feedback.
11th Jan. 2024. URL: https://hal.science/hal-04387614.

[20] L. Baratchart, S. Fueyo and J.-B. Pomet. Exponential stability of linear periodic difference-delay

equations. 2023. URL: https://inria.hal.science/hal-03500720.

[21] L. Baratchart, S. Fueyo and J.-B. Pomet. Integral representation formula for linear non-autonomous

difference-delay equations. 2023. URL: https://hal.science/hal-04204500.

[22] B. Bonnard and J. Rouot. Feedback Classification and Optimal Control with Applications to the
Controlled Lotka-Volterra Model. 1st Jan. 2023. URL: https://inria.hal.science/hal-039173
63.

[23] B. Bonnard and J. Rouot. Optimal Control of the Lotka-Volterra Equations with Applications. 2024.

URL: https://hal.science/hal-03829465.

[24]

J.-B. Caillau, L. Dell’Elce, A. Herasimenka and J.-B. Pomet. On the controllability of nonlinear
systems with a periodic drift. Mar. 2024. URL: https://inria.hal.science/hal-03779482.

Project MCTAO

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[25] M. Fleurial, L. Sacchelli and A. G. Yabo. State estimation in alcoholic fermentation models: a case-

study in wine-making conditions. 6th Nov. 2023. URL: https://hal.science/hal-04269299.

[26] A. Herasimenka, L. Dell’Elce, J.-B. Caillau and J.-B. Pomet. Optimal control of a solar sail. 2023. URL:

https://inria.hal.science/hal-04158202.

[27] A. G. Yabo, J.-B. Caillau and J.-L. Gouzé. Optimal bacterial resource allocation strategies in batch

processing. 25th Sept. 2023. URL: https://inria.hal.science/hal-03710681.

11.2 Other

Scientific popularization

[28] A. Herasimenka, A. Farrés and L. Dell’Elce. ‘Controllability of satellites on periodic orbits with
cone-constraints on the thrust direction’. In: Complex Days 2023. Nice, France, 7th Feb. 2023. URL:
https://hal.science/hal-04043023.

11.3 Cited publications

[29] A. A. Agrachev and R. V. Gamkrelidze. ‘Symplectic methods for optimization and control’. In:
Geometry of feedback and optimal control. Vol. 207. Textbooks Pure Appl. Math. Marcel Dekker,
1998, pp. 19–77.

[30] A. A. Agrachev and A. V. Sarychev. ‘Strong minimality of abnormal geodesics for 2-distributions’. In:

J. Dynam. Control Systems 1.2 (1995), pp. 139–176. DOI: 10.1137/S036301290138866X.

[31] A. Agrachev and Y. L. Sachkov. Control theory from the geometric viewpoint. Vol. 87. Encyclopaedia
of Mathematical Sciences. Control Theory and Optimization, II. Berlin: Springer-Verlag, 2004. DOI:
10.1007/978-3-662-06404-7.

[32] A. A. Agrachev and A. V. Sarychev. ‘Abnormal sub-Riemannian geodesics: Morse index and rigidity’.
In: Ann. Inst. H. Poincarré, Anal. non-linéaire 13.6 (1996), pp. 635–690. URL: http://www.numdam
.org/item/AIHPC_1996__13_6_635_0/.

[33] V. I. Arnold. Mathematical methods of classical mechanics. Vol. 60. Graduate Texts in Mathematics.

New York: Springer-Verlag, 1989.

[34] T. Bakir, B. Bonnard, S. Gayrard and J. Rouot. ‘Finite Dimensional Approximation to Muscular
Response in Force-Fatigue Dynamics using Functional Electrical Stimulation’. In: Automatica (Oct.
2022). DOI: 10.1016/j.automatica.2022.110464. URL: https://inria.hal.science/hal-
03154450.

[35] L. Baratchart, S. Fueyo, G. Lebeau and J.-B. Pomet. ‘Sufficient Stability Conditions for Time-varying
Networks of Telegrapher’s Equations or Difference-Delay Equations’. In: SIAM J. Math. Anal. 53.2
(2021), pp. 1831–1856. DOI: 10.1137/19M1301795. URL: http://hal.inria.fr/hal-02385548
/.

[36] D. Barilari, U. Boscain and R. W. Neel. ‘Small-time heat kernel asymptotics at the sub-Riemannian
cut locus’. In: Journal of Differential Geometry 92.3 (Nov. 2012). DOI: 10.4310/jdg/1354110195.
URL: https://doi.org/10.4310/jdg/1354110195.

[37] K. Beauchard and F. Marbach. ‘A unified approach of obstructions to small-time local controllability
for scalar-input systems’. working paper or preprint. May 2022. URL: https://hal.science/hal
-03683867.

[38] A. Bombrun and J.-B. Pomet. ‘The averaged control system of fast oscillating control systems’. In:
SIAM J. Control Optim. 51.3 (2013), pp. 2280–2305. DOI: 10.1137/11085791X. URL: http://hal
.inria.fr/hal-00648330/.

[39] B. Bonnard. ‘Feedback equivalence for nonlinear systems and the time optimal control problem’.
In: SIAM J. Control Optim. 29.6 (1991), pp. 1300–1321. DOI: 10.1137/0329067. URL: https://doi
.org/10.1137/0329067.

[40] B. Bonnard and J.-B. Caillau. ‘Geodesic flow of the averaged controlled Kepler equation’. In: Forum

Mathematicum 21.5 (Sept. 2009), pp. 797–814. DOI: 10.1515/FORUM.2009.038.

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[41] B. Bonnard, J.-B. Caillau and O. Cots. ‘Energy minimization in two-level dissipative quantum
control: The integrable case’. In: Proceedings of the 8th AIMS Conference on Dynamical Systems,
Differential Equations and Applications. Vol. suppl. Discrete Contin. Dyn. Syst. AIMS, 2011, pp. 198–
208. DOI: 10.3934/proc.2011.2011.198.

[42] B. Bonnard, J.-B. Caillau and L. Rifford. ‘Convexity of injectivity domains on the ellipsoid of re-
volution: the oblate case’. In: C. R. Math. Acad. Sci. Paris 348.23-24 (2010), pp. 1315–1318. DOI:
10.1016/j.crma.2010.10.036. URL: https://hal.archives-ouvertes.fr/hal-00545768.

[43] B. Bonnard, J.-B. Caillau, R. Sinclair and M. Tanaka. ‘Conjugate and cut loci of a two-sphere of
revolution with application to optimal control’. In: Ann. Inst. H. Poincaré Anal. Non Linéaire 26.4
(2009), pp. 1081–1098. DOI: 10.1016/j.anihpc.2008.03.010.

[44] B. Bonnard, J.-B. Caillau and E. Trélat. ‘Second order optimality conditions in the smooth case and
applications in optimal control’. In: ESAIM Control Optim. Calc. Var. 13.2 (2007), pp. 207–236. DOI:
10.1051/cocv:2007012.

[45] B. Bonnard and M. Chyba. Singular trajectories and their role in control theory. Vol. 40. Math-

ématiques & Applications. Berlin: Springer-Verlag, 2003.

[46] B. Bonnard, M. Chyba, J. Rouot, D. Takagi and R. Zou. ‘Optimal Strokes : a Geometric and Numerical
Study of the Copepod Swimmer’. working paper or preprint. Jan. 2016. URL: https://hal.inria
.fr/hal-01162407.

[47] B. Bonnard, O. Cots, J. Gergaud and B. Wembe. ‘Abnormal Geodesics in 2D-Zermelo Navigation
Problems in the Case of Revolution and the Fan Shape of the Small Time Balls’. In: Systems and
Control Letters 161 (Mar. 2022), p. 105140. DOI: 10 . 1016 / j . sysconle . 2022 . 105140. URL:
https://hal.science/hal-02437507.

[48] B. Bonnard, H. Henninger, J. Nemcova and J.-B. Pomet. ‘Time Versus Energy in the Averaged
Optimal Coplanar Kepler Transfer towards Circular Orbits’. In: Acta Applicandae Math. 135.2 (2015),
pp. 47–80. DOI: 10.1007/s10440-014-9948-2. URL: https://hal.inria.fr/hal-00918633.

[49] B. Bonnard, A. Jacquemard, M. Chyba and J. Marriott. ‘Algebraic geometric classification of the
singular flow in the contrast imaging problem in nuclear magnetic resonance’. In: Math. Control
Relat. Fields (MCRF) 3.4 (2013), pp. 397–432. DOI: 10.3934/mcrf.2013.3.397. URL: https://ha
l.inria.fr/hal-00939495.

[50] B. Bonnard and I. Kupka. ‘Théorie des singularités de l’application entrée-sortie et optimalité des
trajectoires singulières dans le problème du temps minimal’. In: Forum Math. 5 (1993), pp. 111–159.

[51] U. Boscain, R. A. Chertovskih, J. P. Gauthier and A. O. Remizov. ‘Hypoelliptic Diffusion and Human
Vision: A Semidiscrete New Twist’. In: SIAM Journal on Imaging Sciences 7.2 (2014), pp. 669–695.
DOI: 10.1137/130924731. eprint: https://doi.org/10.1137/130924731. URL: https://doi
.org/10.1137/130924731.

[52] U. Boscain, J. Duplaix, J.-P. Gauthier and F. Rossi. ‘Anthropomorphic Image Reconstruction via
Hypoelliptic Diffusion’. In: SIAM J. Control Optim. 50.3 (Jan. 2012), pp. 1309–1336. DOI: 10.1137/1
1082405x.

[53] U. Boscain, D. Prandi, L. Sacchelli and G. Turco. ‘A bio-inspired geometric model for sound recon-
struction’. In: The Journal of Mathematical Neuroscience 11.2 (2021). DOI: 10.1186/s13408-020-
00099-4. URL: https://doi.org/10.1186/s13408-020-00099-4.

[54] P. C. Bressloff. ‘Spatiotemporal dynamics of continuum neural fields’. In: Journal of Physics A:

Mathematical and Theoretical 45.3 (2011), p. 033001.

[55] L. Brivadis, J.-P. Gauthier, L. Sacchelli and U. Serres. ‘Avoiding observability singularities in output
feedback bilinear systems’. In: SIAM J. Control Optim. 59.3 (May 2021), pp. 1759–1780. DOI: 10.113
7/19M1272925. URL: https://hal.archives-ouvertes.fr/hal-02172420.

[56]

[57]

J.-B. Caillau, O. Cots and J. Gergaud. ‘Differential pathfollowing for regular optimal control prob-
lems’. In: Optim. Methods Softw. 27.2 (2012), pp. 177–196.

J.-B. Caillau and B. Daoud. ‘Minimum time control of the restricted three-body problem’. In: SIAM
J. Control Optim. 50.6 (2012), pp. 3178–3202.

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[58]

[59]

[60]

J.-B. Caillau and A. Farrés. ‘On local optima in minimum time control of the restricted three-body
problem’. In: Recent Advances in Celestial and Space Mechanics. Vol. Mathematics for Industry. 23.
Springer, Apr. 2016, pp. 209–302. DOI: 10.1007/978-3-319-27464-5. URL: https://hal.archi
ves-ouvertes.fr/hal-01260120.

J.-b. Caillau, J.-B. Pomet and J. Rouot. ‘Metric approximation of minimum time control systems’.
working paper or preprint. Nov. 2017. URL: https://hal.inria.fr/hal-01672001.

J.-B. Caillau and C. Royer. ‘On the injectivity and nonfocal domains of the ellipsoid of revolution’.
In: Geometric Control Theory and Sub-Riemannian Geometry. Ed. by G. Stefani et al. Vol. 5. INdAM
series. Springer, 2014, pp. 73–85. DOI: 10.1007/978-3-319-02132-4. URL: https://hal.archi
ves-ouvertes.fr/hal-01315530.

[61] R. Carron, A. Chaillet, A. Filipchuk, W. Pasillas-Lépine and C. Hammond. ‘Closing the loop of deep

brain stimulation’. In: Frontiers in systems neuroscience 7 (2013), p. 112.

[62] Z. Chen, J.-B. Caillau and Y. Chitour. ‘L1-minimization for mechanical systems’. In: SIAM J. Control
Optim. 54.3 (May 2016), pp. 1245–1265. DOI: 10.1137/15M1013274. URL: https://hal.archive
s-ouvertes.fr/hal-01136676.

[63] L. Dell’Elce, J.-B. Caillau and J.-B. Pomet. ‘Considerations on Two-Phase Averaging of Time-Optimal
Control Systems’. In: IFAC-PapersOnLine 55.16 (2022). 18th IFAC Workshop on Control Applications
of Optimization (CAO 2022), pp. 7–12. DOI: 10.1016/j.ifacol.2022.08.073. URL: https://in
ria.hal.science/hal-01793704.

[64] L. Dell’Elce, J.-B. Caillau and J.-B. Pomet. ‘On the convergence of time-optimal maneuvers of
fast-oscillating control systems’. In: 2021 European Control Conference (ECC). 2021, pp. 2008–2013.
DOI: 10.23919/ECC54610.2021.9655101. URL: https://hal.inria.fr/hal-03391620.

[65] L. Faubourg and J.-B. Pomet. ‘Control Lyapunov functions for homogeneous “Jurdjevic-Quinn”
systems’. In: ESAIM Control Optim. Calc. Var. 5 (2000), pp. 293–311. DOI: 10.1051/cocv:2000112.

[66] A. Figalli, L. Rifford and C. Villani. ‘Necessary and sufficient conditions for continuity of optimal
transport maps on Riemannian manifolds’. In: Tohoku Math. J. 63.4 (2011), pp. 855–876. URL:
http://hal.inria.fr/hal-00923320v1.

[67] A. Figalli, T. Gallouët and L. Rifford. ‘On the convexity of injectivity domains on nonfocal manifolds’.
In: SIAM J. Math. Anal. 47.2 (2015), pp. 969–1000. DOI: 10.1137/140961821. URL: https://hal
.inria.fr/hal-00968354.

[68] M. Fliess, J. Lévine, P. Martin and P. Rouchon. ‘Flatness and Defect of Nonlinear Systems: Introduct-

ory Theory and Examples’. In: Internat. J. Control 61 (1995), pp. 1327–1361.

[69] S. Fueyo. ‘Systèmes à retard instationnaires et EDP hyperboliques 1-D instationnaires, fonctions
de transfert harmoniques et circuits électriques non-linéaires’. Université Cote d’Azur, 30th Oct.
2020. URL: https://hal.archives-ouvertes.fr/tel-03105344.

[70] C. Gavriel and R. Vinter. ‘Second order sufficient conditions for optimal control problems with
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1044.
<scratchpad>
- Main topic: Research Centre at Inria Centre at Université Côte d’Azur
- Key points: 
  - Project-Team MCTAO (Mathematics for Control, Transport and Applications)
  - Domain: Applied Mathematics, Computation and Simulation
  - Theme: Optimization and control of dynamic systems
  - Research program:
    - Control Problems
    - Optimal Control and its Geometry
    - Small controls and conservative systems, averaging
  - Application domains:
    - Aerospace Engineering
    - Optimal control of microbial cells, and other biological applications
    - Neural dynamics
- Structure: 
  - Document appears to be a research report or annual report
  - Includes sections on project-team, activity report, research program, application domains, highlights of the year, new software, and dissemination
- Potential artifacts to ignore: 
  - Web-related elements such as links, HTML tags, and other web-related elements
</scratchpad>

<final_summary>
The document is a research report or annual report from the Inria Centre at Université Côte d’Azur, focusing on the Project-Team MCTAO (Mathematics for Control, Transport and Applications). The team's research program covers Control Problems, Optimal Control and its Geometry, and Small controls and conservative systems, averaging. The application domains include Aerospace Engineering, Optimal control of microbial cells, and Neural dynamics. The report highlights the team's achievements and collaborations with industry partners, as well as its contributions to various conferences and publications. Overall, the report showcases the team's expertise in applied mathematics and its applications to real-world problems.</final_summary>
The document is a research report or annual report from the Inria Centre at Université Côte d’Azur, focusing on the Project-Team MCTAO (Mathematics for Control, Transport and Applications). The team's research program covers Control Problems, Optimal Control and its Geometry, and Small controls and conservative systems, averaging. The application domains include Aerospace Engineering, Optimal control of microbial cells, and Neural dynamics. The report highlights the team's achievements and collaborations with industry partners, as well as its contributions to various conferences and publications. Overall, the report showcases the team's expertise in applied mathematics and its applications to real-world problems.