Our goal is to develop methods in geometric control theory for finite-dimensional nonlinear systems, and to transfer our expertise through real applications of these techniques.

Our primary domain of industrial applications in the past years has been space engineering, namely using optimal control and stabilization techniques for mission design with low thrust propulsion: orbit transfer or rendez-vous problems in the gravity field of a single body (typically sattelites 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 control of micro-swimmers, i.e. swimming robots where the fluid-structure coupling has a very low Reynolds number; and quantum control, with applications to Nuclear Magnetic Resonance and medical image processing. Recent focus has been put into transfer to other domains that can benefit from a control theory point of view, such as biology, with problems of optimal control of microbial cells, or muscular electro-stimulation.

Finally, part of the team’s core goals is transfer of its expertise to other mathematical fields, where problems in dynamical systems rely on a control theory approach.

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

Modelling. 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
state, ideally finite dimensional, and 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

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.

Optimal control. In optimal control, one wants to find, among the controls that satisfy some contraints 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 74, 44, 29, but with a
lot of 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.

Feedback control. In the above two paragraphs, the control is an explicit function of time.
To address in 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 61, 76.
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 53, 79.

Classification of control systems. One may perform various classes of transformations acting on systems,
or rather on models. The simpler ones come from point-to-point
transformations (changes of 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 modelled 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 transformations on models for themselves.
These topics are central in control theory; they are present in the
team, see for instance the classification aspect in
48 or
—although this research has not been active very recently— the study 73 of dynamic feedback and the so-called “flatness” property 64.

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 78, 31, 30 and
49 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
42, followed by the Hampath 54 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.55 and 38).
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 28.
This point of view withstands generalizations far beyond the smooth case: In

Riemann and Finsler geometry.
Studying the distance and minimising 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 63, 62.
The analysis in the oblate geometry 40 was completed in 59 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 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 71 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 70 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 35). 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 perturbations 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 32, 77.
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

Optimality of periodic solutions/periodic controls.
When seeking to minimize a cost with the constraint 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 43, for the problem of optimizing the efficiency of the
displacement of some micro-swimmers (see Section 4.5) with
periodic deformations, we used the sufficient optimality conditions established by
R. Vinter's group 83, 67 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 (Univ. Bretagne Ouest) on
second order sufficient optimality conditions for periodic solutions, or in the presence
of higher dimensional symmetry groups, following 83, 67.
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 82).

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. 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 hocpython 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.

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 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

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 U. Barcelona (A. Farrès, A. Jorba).

A new action has started in Sep. 2020 with the Phd thesis of Alesia Herasimenka (2020-2023) on the control of solar sails. Solar sailing has been actively studied for two decades and recent missions have demonstrated its interest for "zero-fuel" missions. A lot has to be done to understand even the basic properties of such systems, for instance regarding controllability. Depending on the model used for the sail, not all directions of control are available. Some preliminary studies obtain controllability results by analysing the flow around equilibria of the system, these equilibria being changed when the orientation of the sail is updated. We want to provide a comprehensive understanding of controllability thanks to a systematic use of geometric control theory (orbit theorem, Lie algebraic approach) combined with effective numerical computations to check local controllability properties. The PhD thesis has been selected by ESA for a three-year research co-sponsorship.

The growth of microorganisms is fundamentally an optimization problem which consists in dynamically 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 regulatory 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 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). Currently, one PhD (A. Yabo) and one postdoc (S. Maslovskaya) are involved in these tasks and jointly supervised by colleagues from McTAO and Biocore teams at Sophia. Preliminary 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.

Let us focus on amplifiers, as active circuits. Nonlinear hyper-frequency amplifiers are ubiquitous in cell phone relays and many other devices. They must be as compact as possible, yielding a more complicated design. Computer Assisted Design tools are extensively used; for a given amplifier design, they provide frequency responses but fail to provide information of the stability of the response for each frequency. This stability is crucial for an unstable response will not be observed in practice; the actual device should not be built before stability is asserted. Predicting stability/instability from “simulations” in the Computer Assisted Design tool is of utmost importance (simulation between quotation marks because these simulations are in fact computations in the frequency domain). Potential transfer to industry is important.

Some techniques do exist, see 80, based on creating some virtual perturbations and treating them as the input of a (linearized) control system to be “simulated” using the same tools. In an ongoing collaboration between McTAO and the project-team FACTAS, we work on the mathematical ground of these methods and in particular of the relation between stability and the property of the identified time-varying infinite dimensional systems. See recent developments in Section 6.5.

The starting point of our interest in optimal control for quantum systems was a collaboration with physicist from ICB, University of Burgundy (Dominique Sugny), motivated by an ANR project where we worked on the control of molecular orientation in a dissipative environment using a laser field, and developed optimal control tools, combined with numerical simulations, to analyze the problem for Qubits. This was related to quantum computing rather than MRI. Using this expertise and under the impulse of Prof. S. Glaser and his group (Chemistry, TU München), we investigated Nuclear Magnetic resonance (NMR) for medical imaging (MRI), where the model is the Bloch equation describing the evolution of the Magnetization vector controlled by a magnetic field, but in fine is a specific Qubit model without decoherence. We worked on, and brought strong contributions to, the contrast problem: typically, given two chemical substances that have an importance in medicine, like oxygenated and de-oxygenated blood, find the (time-dependent) magnetic field that will produce the highest difference in brightness between these two species on the image resulting from Nuclear Magnetic Resonance. This has immediate and important industrial applications in medical imaging. Our contacts are with the above mentioned physics academic labs, who are themselves in contact with major companies. The team has produced and is producing important work on this problem. One may find a good overview in 46, a reference book has been published on the topic 51, a very complete numerical study comparing different optimization techniques was performed in 45. We conduct this project in parallel with S. Glaser team, which validated experimentally the pertinence of the methods, the main achievement being the in vivo experiments realized at the Creatis team of Insa Lyon showing the interest to use optimal control methods implemented in modern softwares in MRI in order to produce a better image in a shorter time. A goal is to arrive to a cartography of the optimal contrast with respect to the relaxation parameters using LMI techniques and numerical simulations with the Hamapth and Bocop code; note that the theoretical study is connected to the problem of understanding the behavior of the extremal solutions of a controlled pair of Bloch equations, and this is an ambitious task. Also, one of the difficulties to go from the obtained results, checkable on experiments, to practical control laws for production is to deal with magnetic field space inhomogeneities.

Following the historical reference for low Reynolds number locomotion 75,
the study of the swimming strategies of micro-organisms is attracting increasing attention in the recent literature. This is both because of
the intrinsic biological interest, and for the possible implications these studies may have on the design of bio-inspired artificial
replicas reproducing the functionalities of biological systems.
In the case of micro-swimmers, the surrounding fluid is dominated by the viscosity effects of the water and becomes reversible.
In this regime, it turns out that the infinite dimensional dynamics of the fluid do not have to be retained as state variables, so that the dynamics of a micro-swimmer can be expressed by ordinary differential equations if its shape has a finite number of degrees of freedom.
Assuming this finite dimension, and if the control is the rate of deformation, one obtains a control system that is linear (affine without drift) with respect to the controls, i.e. the optimal control problem with a quadratic cost defines a sub-Riemannian structure (see section 3.2).
This is the case where the shape is “fully actuated”, i.e. if all the variables describing the shape are angles, there is an actuator on each of these angles.
For artificial micro-swimmers, this is usually unrealistic. For example, (artificial) magneto-elastic micro-swimmers are deformed by an external magnetic field. In this case, the control functions are the external magnetic field.
In both cases, questions are controllability (straightforward in the fully actuated case), optimal control, possibly path planning.
We collaborate with teams that have physical experiments for both.

The questions about optimality of periodic controls raised in Section 3.2 are related to these applications for periodic deformations, or strokes, playing important role in locomotion.

With a strong support from Inria Sophia SED, we are currently working on ct: control toolbox InriaHub project.
The objective of this ADT (started in 2019) is to pave the
way towards a collaborative set of reference tools to solve optimal control problems numerically.
On the basis of the two well established codes BOCOP and
HamPath from the optimal control community, we have designed a high-level modular architecture
to interoperate the solvers and now offer a high-level common user interface for the two codes.
An immediate outcome of the ADT is to integrate state of the art processes into the development of the two solvers
(collaborative dev tools, reliable repositories and continuous integration).
Initially, the
BOCOP project of COMMANDS team aimed at developing an open-source toolbox for solving optimal control problems, with collaborations
involving industrial and academic partners. Optimal control (optimization of dynamical systems governed by differential
equations) has numerous applications in the fields of transportation, energy, process optimization, and biology.
BOCOP implements a direct transcription approach, where the continuous optimal control problem is transformed into a nonlinear program.
The reformulation is done by a discretization of the time interval, with an approximation of the dynamics of the system by a generalized
Runge Kutta scheme.
On the other hand, HamPath is a an open-source package developed to solve optimal control problems by shooting (single or multiple)
and pathfollowing methods for Hamiltonian dynamical systems. It was initially developed by members of the APO
(Algorithmes Parallèles et Optimisation) team from Institut de Recherche en Informatique de Toulouse (ENSEEIHT / CNRS),
jointly with colleagues from the Université de Bourgogne and Inria Sophia.
Applying the maximum principle leads to define a set of Hamiltonians and a Boundary Value Problem, itself described by a
set of nonlinear equations. HamPath compiles the Fortran codes of these (maximized) Hamiltonians and of the shooting function
into a collection of Matlab / Fortran functions which allow to solve the problem very efficiently. The code
makes an intensive use of automatic differentiation, currently thanks to
Tapenade. The well known difficulty of this
approach is to find a good initial guess, so HamPath embeds a differential path following method, as well as
advanced computations such as conjugate point estimation to test second order optimality conditions in optimal control.
Thanks to the ongoing ADT, BOCOP has been entirely rewriten in C++ (BOCOP v3)
and now has a python interface. This allows interoperability with
nutopy, a python port of HamPath.
The gallery of the project mirrors interactions of the two solvers and
contains a bunch of examples coming from mechanics, agronomics, biology, geometry, etc. The project is open to external
contributions and aims to be a living repository for algorithms and applications of optimal control.

This study, started in 2020, is devoted to the understanding of convergence properties of time optimal trajectories of fast oscillating control systems. Specifically, for a control system with one fast periodic variable, with a small parameter measuring the ratio between time derivatives of fast and slow variables, we considered the Hamiltonian equation resulting from applying Pontryagin Maximum Principle for the minimum time problem with fixed initial and final slow variables and free fast variable. One may perform averaging at least under normalization of the adjoint vectors and define a “limit” average system.

When the small parameter approaches 0, we showed that using the right transformations between
initial/final conditions in the "real" and average systems led to a
reconstruction of the fast variable on interval of times of order

We studied the local controllability of non-ideal solar sails in planet-centered orbits. Classical approaches fail when considering this control problem because sails cannot generate forces with positive components toward the Sun direction. More precisely, the control set is delimited by a convex cone of revolution with axis toward the Sun and, as such, it is not defined in a neighborhood of the origin, which precludes the use of standard sufficient conditions for controllability. We introduced both a necessary condition certifying the local non-controllability of the system 13 and a sufficient condition guaranteeing controllabilitly under mild assumptions 26 for given optical properties of the sail and orbital elements. These conditions can be verified by solving two semi-infinite programming problems. By evaluating these conditions on the entire space of slow orbital elements, we identified minimal optical properties that provide local controllability of any planet-centered orbit.

Université Côte d'Azur is planning to build its first single-unit CubeSat (i.e., a recent
standard for satellites with mass of the order of few kilograms and volume
of few dm

The mission is currently in phase A (i.e., feasibility study). This year, we carried out a detailed analysis of the mission focusing on the agility of attitude maneuvers when establishing the optical link with the ground station (these events last few minutes and occur twice per day in general). We quantified the power consumption of such maneuvers and verified its compatibility with the available resources of NiceCube.

In the framework of the ANR Maximic, we carry 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 catalyse the transformation of substrate into precursors). To this internal control 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 behaviour 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) are studied in 66 and 6. 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 (wrt. 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 15 for a discussion of these results.

Sébastien Fueyo's doctoral work (PhD defended in 2020, see 65) on testing the stability of amplifier in a CAD (computer assisted design) process, described also in Section 4.3, led us to revisiting stability of time-varying linear "delay difference systems" (continuous time, but strictly difference equations). In the time-invariant case, a well known necessary and sufficient condition for stability is due to Hale and Henry 69, 68; it gives, in a sense, 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. The time-varying case, on the contrary, had seldom been touched.

The article 2 presents a novel sufficient condition for hyperbolic stability in the time-varying case; it has a passivity interpretation that is exactly fit to classical assumptions on the behavior of electronic devices "at high frequency". A more general result is presented in 17, to be submitted soon; it can be viewed as the proper generalization to the time-varying case of the Henry-Hale criteria mentionned above. The criteria is however less easily checkable than the one in 2.

One part of C. Moreau's PhD 72 was on controllability properties of planar articulated magnetically actuated swimmers. The sinularities in controllability around the straight configuration, where all magnetic moments are aligned, motivated research on necessary conditions for local controllability of a more general, but still particular, class of systems with two controls. Technical results based on the study of the Chen-Fliess series associated to these systems were obtained. The manuscript 25 is under revision and possible generalization are currently being explored.

B. Bonnard took an active part in the research around the doctoral
work of B. Wembe (PhD defended in November 84) on
time-minimum problems in navigation in presence of a current and a vortex.
As explained in 4, 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).
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.
We have two cases, that coexist in different regions: strong versus
weak current. In the weak case, the time minimal problem defines a
2-D Randers metric (a specific Finsler metric) in a portion of the
the plane, while in the other case one cannot consider the problem as
locally metric. Because of the singularity, we have a non trivial
extension of the classical case.
In a subsequent work 18, we analyze
the effect of the same vortex singularity on more general
surfaces of revolutions.
Some salient points are:
integrability of the
extremal flow of the Pontryagin Maximum Principle, due to the
rotational symmetry,
absence of conjugate points (where an extremal curve ceases
to be optimal for the

Parallel efforts have been dedicated to the particular case of ship navigation with trailers. The work is motivated by the optimization of turns and maneuvers of marine vessels towing a set of long and fragile underwater cables for seismic data acquisition. In a variant of the Zermelo-Markov-Dubins problem, we consider a model where the towed cables are represented with articulated trailers. Prior work had been achieved on a simpler trailer model 57. Recently, in 10 and during internships, we theoretically and numerically studied the properties and optimality of trajectories of with more complex trailer models. These results serve as a basis for meetings with the subsurface exploration company Sercel on an expected first collaboration in 2022.

This topic started in McTAO in 2017 with a collaboration between
B. Bonnard and T. Bakir (ImVia-UBFC).
Based on preliminary experimental studies, the chosen model to
muscular control integrates the fatigue variables and is known as
Ding et al force-fatigue model in the literature.
It is a refinement of the historical Hill model (Medecine Nobel Prize
in 1922) that takes into account the variations of the fatigue
variable.
The problem is the one of optimizing the train pulses of the
Functional Electro-Stimulation (FES) signal to produce the muscular
contraction.
From the control methodology point of view, this required some
developments on optimal control for sample control systems. This is
by itself a rich topic, on which a workshop was organised in
September in Brest on this occasion, see Section 9.1.1.
Results on applications of sampled geometric optimal control to the
isometric case are presented in 33 and
1, with preliminary work on extension to the
non isometric case in 50 (non isometric
means that movements and deformations are taken into account).

The attempt to an industrial transfer started last fall (2020) with a contract (CIFRE PhD funding) with SEGULA Technologies (see Section 7) whose goal is the construction of an electro-stimulator prototype, in the framework of S. Gayrard's PhD. Preliminary results lead to construct a nonlinear observer and the optimized pulses trains are computed using MPC methods 34. Serious effort is put in identifying the dynamics; in 16, a detailed study of the muscle response is presented and used to build the necessary finite-dimensional dynamic model to perform closed-loop control. We have hope to improve over state-of-the-art isometric stimulators and be able to tackle the non-isometric case, which is richer in applications.

The ADT ct: control toolbox had two sprints in 2021. The focus was on python
as a high level backend to interoperate existing solvers. We also initiated 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 and
Zygote: this is a crucial step for our project, both for direct and indirect
methods. (We added to the project gallery some examples stemming from Riemannian geometry in the gallery that require up to four levels of automatic
differentiation.)

Co-financed by Segula Technologies on the topic “Réalisation d’un prototype d’électrostimulateur intelligent”. The PhD etudent on this topic is Sandrine Gayrard.

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

Period: 2020–2023.

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.

Partners: McTAO and ESA.

Total amount: 24k€.

Period: 2021–2024.

Inria reference: 16016.

Maximic: optimal control of microbial cells by natural and synthetic strategies.
Started 2017, duration: 6 years.
J.-B. Caillau, L. Giraldi, J.-B. Pomet, S. Maslovkaya and A. Yabo are participants.
More information and news on the site of this project.

PGMO grant (2019-2021) on "Sampled-Data Control Systems and Applications" (PI B. Bonnard).

PGMO grant (2020-2022) on "Extremal determinants". Participants are Y. Chitour (Université Paris Saclay), J.-B. Caillau, P. Freitas (University of Lisbon), Y. Privat (Université de Strasbourg)

The McTAO team participates in the GdR MOA, a CNRS network on Mathematics of Optimization and Applications.

J.-B. Caillau is associate researcher of the CNRS team Parallel Algorithms & Optimization at ENSEEIHT, Univ. Toulouse.

UCAJEDI, académie d'excellence “Systèmes complexes” grant (2021) for organization of a workshop on “observability defects in nonlinear systems” (P.I. L. Sacchelli).

B. Bonnard co-organized a workshop on sampled-data-control optimal control, September 8 to 10, 2021, at Université de Bretagne Occidentale, Brest, France.

J.-B. Caillau co-organized the SEME / MSGI AI & companies week in Sophia.

L. Sacchelli was the principal organizer of Mini-workshop : Défauts d’observabilité des systèmes non linéaires. December 1-2 2021, Inria Sophia Antipolis. The event was made possible thanks to a grant from UCA JEDI Académie d’Excellence “Systèmes complexes”.

J.-B. Caillau was member of the Organizing Committee of 2021 Journées de la Statistique (Nice), of the Organizing Committee of the CIMPA school Optimal Control and Applications in Engineering, held at Arba Minch University (Ethiopia), and of the Scientific Committee of PGMO days 2021.

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).

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

J.-B. Caillau was plenary speaker at the Dynamic Control and Optimization conference held in honor of Andrey Sarychev in Aveiro, Portugal (2021).

J.-B. Caillau was

J.-B. Pomet was

J.-B. Caillau reviewed and sat at the jury of Victor Cantu PhD thesis (Université Toulouse)

J.-B. Caillau is member of MASTIC and of Terra Numerica.

L. dell'Elce is member of MASTIC.