The core endeavor of this team is to develop methods in control theory for finite-dimensional nonlinear systems, as well as in optimal transport, and to be involved in applications of these techniques. Some mathematical fields like dynamical systems and optimal transport may benefit from control theory techniques. Our primary domain of industrial applications will be space engineering, namely designing trajectories in space mechanics using optimal control and stabilization techniques: transfer of a satellite between two Keplerian orbits, rendez-vous problem, transfer of a satellite from the Earth to the Moon or more complicated space missions. A second field of applications is quantum control with applications to Nuclear Magnetic Resonance and medical image processing.
Our effort is directed toward efficient methods for the control of real (physical) systems, based on a model of the system to be controlled. System refers to the physical plant or device, whereas model refers to a mathematical representation of it.
We mostly investigate nonlinear systems whose nonlinearities admit a strong structure derived from physics; the equations governing their behavior are then well known, and the modeling part consists in choosing what phenomena are to be retained in the model used for control design, the other phenomena being treated as perturbations; a more complete model may be used for simulations, for instance. We focus on systems that admit a reliable finite-dimensional model, in continuous time; this means that models are controlled ordinary differential equations, often nonlinear.
Choosing accurate models yet simple enough to allow control design is in itself a key issue; however, modeling or identification as a theory is not per se in the scope of our project.
The extreme generality and versatility of linear control do not contradict
the often heard sentence “most real life systems are nonlinear”.
Indeed, for many control problems, a linear model is sufficient to capture the important features for control.
The reason is that most
control objectives are local, first order variations around an operating point
or a trajectory are governed by a linear control model, and except in
degenerate situations (non-controllability of this linear
model), the local behavior of a nonlinear dynamic phenomenon is dictated by
the behavior of first order variations.
Linear control is the hard core of control theory and practice; it has been pushed to a high degree of achievement
–see for instance some classics: , – that leads to big successes
in industrial applications (PID, Kalman filtering, frequency domain design,
Linear control by itself however reaches its limits in some important situations:
Non local control objectives. For instance, steering the system from a region to a reasonably remote other one (path planning and optimal control); in this case, local linear approximation cannot be sufficient.
It is also the case when some domain of validity (e.g. stability) is prescribed and is larger than the region where the linear approximation is dominant.
Local control at degenerate equilibria. Linear control yields local stabilization of an equilibrium point based on the tangent linear approximation if the latter is controllable. When it is not, and this occurs in some physical systems at interesting operating points, linear control is irrelevant and specific nonlinear techniques have to be designed.
This is in a sense an extreme case of the second paragraph in point : the region where the linear approximation is dominant vanishes.
Small controls. In some situations, actuators only allow a very small magnitude of the effect of control compared to the effect of other phenomena. Then the behavior of the system without control plays a major role and we are again outside the scope of linear control methods.
Local control around a trajectory. Sometimes a trajectory has been selected (this appeals to point ), and local regulation around this reference is to be performed. Linearization in general yields, when the trajectory is not a single equilibrium point, a time-varying linear system. Even if it is controllable, time-varying linear systems are not in the scope of most classical linear control methods, and it is better to incorporate this local regulation in the nonlinear design, all the more so as the linear approximation along optimal trajectories is, by nature, often non controllable.
Let us discuss in more details some specific problems that we are studying or plan to study: classification and structure of control systems in section , optimal control, and its links with feedback, in section , the problem of optimal transport in section , and finally problems relevent to a specific class of systems where the control is “small” in section .
In most problems, choosing the proper coordinates, or the right quantities that describe a phenomenon, sheds light on a path to the solution. In control systems, it is often crucial to analyze the structure of the model, deduced from physical principles, of the plant to be controlled; this may lead to putting it via some transformations in a simpler form, or a form that is most suitable for control design. For instance, equivalence to a linear system may allow to use linear control; also, the so-called “flatness” property drastically simplifies path planning , .
A better understanding of the “set of nonlinear models”, partly classifying them, has another motivation than facilitating control design for a given system and its model: it may also be a necessary step towards a theory of “nonlinear identification” and modeling. Linear identification is a mature area of control science; its success is mostly due to a very fine knowledge of the structure of the class of linear models: similarly, any progress in the understanding of the structure of the class of nonlinear models would be a contribution to a possible theory of nonlinear identification.
These topics are central in control theory, but raise very difficult mathematical questions: static feedback
classification is a geometric problem which is feasible in principle, although describing invariants explicitly is
technically very difficult; and conditions for dynamic feedback equivalence and linearization raise unsolved
mathematical problems, that make one wonder about decidability
Mathematically speaking, optimal control is the modern branch of the calculus of variations, rather well established and mature , , , . Relying on Hamiltonian dynamics is now prevalent, instead of the standard Lagrangian formalism of the calculus of variations. Also, coming from control engineering, constraints on the control (for instance the control is a force or a torque, which are naturally bounded) or the state (for example in the shuttle atmospheric re-entry problem there is a constraint on the thermal flux) are imposed; the ones on the state are usual but these on the state yield more complicated necessary optimality conditions and an increased intrinsic complexity of the optimal solutions. Also, in the modern treatment, ad-hoc numerical schemes have to be derived for effective computations of the optimal solutions.
What makes optimal control an applied field is the necessity of computing these optimal trajectories, or rather the controls that produce these trajectories (or, of course, close-by trajectories). Computing a given optimal trajectory and its control as a function of time is a demanding task, with non trivial numerical difficulties: roughly speaking, the Pontryagin Maximum Principle gives candidate optimal trajectories as solutions of a two point boundary value problem (for an ODE) which can be analyzed using mathematical tools from geometric control theory or solved numerically using shooting methods. Obtaining the optimal synthesis –the optimal control as a function of the state– is of course a more intricate problem , .
These questions are not only academic for minimizing a cost is very relevant in many control engineering problems. However, modern engineering textbooks in nonlinear control systems like the “best-seller” hardly mention optimal control, and rather put the emphasis on designing a feedback control, as regular and explicit as possible, satisfying some qualitative (and extremely important!) objectives: disturbance attenuation, decoupling, output regulation or stabilization. Optimal control is sometimes viewed as disconnected from automatic control... we shall come back to this unfortunate point.
A control Lyapunov function (CLF) is a function that can be made a Lyapunov function (roughly speaking, a function that decreases along all trajectories, some call this an “artificial potential”) for the closed-loop system corresponding to some feedback law. This can be translated into a partial differential relation sometimes called “Artstein's (in)equation” . There is a definite parallel between a CLF for stabilization, solution of this differential inequation on the one hand, and the value function of an optimal control problem for the system, solution of a HJB equation on the other hand. Now, optimal control is a quantitative objective while stabilization is a qualitative objective; it is not surprising that Artstein (in)equation is very under-determined and has many more solutions than HJB equation, and that it may (although not always) even have smooth ones.
We have, in the team, a longstanding research record on the topic of construction of CLFs and stabilizing feedback controls. This is all the more interesting as our line of research has been pointing in almost opposite directions. , , insist on the construction of continuous feedback, hence smooth CLFs whereas, on the contrary, , , proceed with a very fine study of non-smooth CLFs, yet good enough (semi-concave) that they can produce a reasonable discontinuous feedback with reasonable properties.
We believe that matching optimal transport with geometric control theory is one originality of our team. We expect interactions in both ways.
The study of optimal mass transport problems in the Euclidean or Riemannian setting has a long history which goes from the pioneer works of Monge and Kantorovitch to the recent revival initiated by fundamental contributions due to Brenier and McCann .
Th same transportation problems in the presence of differential constraints on the set of paths —like being an admissible trajectory for a control system— is quite new. The first contributors were Ambrosio and Rigot who proved the existence and uniqueness of an optimal transport map for the Monge problem associated with the squared canonical sub-Riemannian distance on the Heisenberg groups. This result was extended later by Agrachev and Lee , then by Figalli and Rifford who showed that the Ambrosio-Rigot theorem holds indeed true on many sub-Riemannian manifolds satisfying reasonable assumptions. The problem of existence and uniqueness of an optimal transport map for the squared sub-Riemannian distance on a general complete sub-Riemannian manifold remains open; it is strictly related to the regularity of the sub-Riemannian distance in the product space, and remains a formidable challenge. Generalized notions of Ricci curvatures (bounded from below) in metric spaces have been developed recently by Lott and Villani and Sturm , . A pioneer work by Juillet captured the right notion of curvature for subriemannian metric in the Heisenberg group; Agrachev and Lee have elaborated on this work to define new notions of curvatures in three dimensional sub-Riemannian structures. The optimal transport approach happened to be very fruitful in this context. Many things remain to do in a more general context.
Using averaging techniques to study small perturbations of integrable Hamiltonian systems dates back to H. Poincaré or earlier; it gives an approximation of the (slow) evolution of quantities that are preserved in the non-perturbed system. 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; see for instance , .
When the “perturbation” is a control, these techniques may be used after deciding how the control will depend on time and state and other quantities, for instance it may be used after applying the Pontryagin Maximum Principle as in , , , . Without deciding the control a priori, an “average control system” may be defined as in .
The focus is then on studying into details this simpler “averaged” problem, that can often be described by a Riemannian metric for quadratic costs or by a Finsler metric for costs like minimum time.
This line of research stemmed out of applications to space engineering, see section . For orbit transfer in the two-body problem, an important contribution was made by B. Bonnard, J.-B. Caillau and J. Gergaud in explicitly computing the solutions of the average system obtained after applying Pontryagin Maximum Principle to minimizing a quadratic integral cost; this yields an explicit calculation of the optimal control law itself. Studying the Finsler metric issued form the time-minimal case is in progress.
Space engineering is very demanding in terms of safe and high-performance control laws (for instance optimal in terms of fuel consumption, because only a finite amount of fuel is onborad a sattelite for all its “life”). It is therefore prone to real industrial collaborations.
We are especially interested in trajectory control of space vehicles using their own propulsion devices, outside the atmosphere. Here we discuss “non-local” control problems (in the sense of section point ): orbit transfer rather than station keeping; also we do not discuss attitude control.
In the geocentric case, a space vehicle is subject to
- gravitational forces, from one or more central bodies (the corresponding
acceleration is denoted by
- a thrust, the control, produced by a propelling
device; it is the
- other “perturbating” forces (the corresponding acceleration is denoted by
In position-velocity coordinates, its dynamics can be written as
In the case of a single attracting central body (the earth) and in a geocentric frame,
Low thrust means that
Why low thrust ? The common principle to all propulsion devices is to eject particles, with
some relative speed with respect to the vehicle; conservation of momentum
then induces, from the point of view of the vehicle alone, an external force, the “thrust” (and a mass decrease).
Ejecting the same mass of particles with a higher relative speed
results in a proportionally higher thrust; this relative speed (specific impulse,
Electric engines theoretically allow many more maneuvers with the same amount of particles, with the drawback that the instant force is very small; sophisticated control design is necessary to circumvent this drawback. High thrust engines allow simpler control procedures because they almost allow instant maneuvers (strategies consist in a few burns at precise instants).
Let us mention two.
Orbit transfer or rendez-vous.
It is the classical problem of bringing a satellite to its operating position from the orbit where it is delivered by the launcher; for instance from a GTO orbit to the
geostationary orbit at a prescribed longitude (one says rendez-vous when the longitude, or the position on the orbit, is prescribed, and transfer if it is free). In equation
() for the dynamics,
Three body problem.
This is about missions in the solar system leaving the region where the attraction of the earth, or another single body, is preponderant. We are then no longer in the
situation of a single central body,
An issue for future experimental missions in the solar system is interplanetary flight planning with gravitational assistance. Tackling this global problem, that even contains some combinatorial problems (itinerary), goes beyond the methodology developed here, but the above considerations are a brick in this puzzle.
If there are no restrictions on the thrust direction, i.e., in equation (), if the control
The geographic proximity of Thales Alenia Space, in conjunction with the “Pole de compétitivité” PEGASE in PACA region is an asset for a long term collaboration between Inria - Sophia Antipolis and Thales Alenia Space (Thales Alenia Space site located in Cannes hosts one of the very few European facilities for assembly, integration and tests of satellites).
B. Bonnard and J.-B. Caillau in Dijon have had a strong activity in optimal control for space, in collaboration with the APO Team from IRIT at ENSEEIHT (Toulouse), and sometimes with EADS, for development of geometric methods in numerical algorithms.
These applications started by a collaboration between B. Bonnard and D. Sugny (a physicist from ICB) in the ANR project Comoc, localized mainly at the University of Dijon. The problem was the control of the orientation of a molecule using a laser field, with a model that does take into account the dissipation due to the interaction with the environment, molecular collisions for instance. The model is a dissipative generalization of the finite dimensional Schrödinger equation, known as Lindblad equation. It is a 3-dimensional system depending upon 3 parameters, yielding a very complicated optimal control problem that we have solved for prescribed boundary conditions. In particular we have computed the minimum time control and the minimum energy control for the orientation or a two-level system, using geometric optimal control and appropriate numerical methods (shooting and numerical continuation) , .
More recently, based on this project, we have reoriented our control activity towards Nuclear Magnetic Resonance (MNR).
In MNR medical imaging, the contrast problem is the one of designing a variation of the magnetic field with respect to
time that maximizes the difference,
on the resulting image, between two different chemical species; this is the “contrast”. This research is conducted
with Prof. S. Glaser (TU-München), whose group is performing both in vivo and in vitro experiments;
experiments using our techniques have successfully measured the improvement in contrast between materials chemical
species that have an importance in medicine, like oxygenated and de-oxygenated blood,
see ; this is however still to be investigated and improved.
The model is the Bloch equation for spin
Note that a reference book by B. Bonnard and D. Sugny has been published on the topic .
Optimal Transportation in general has many applications. Image processing, biology, fluid mechanics, mathematical physics, game theory, traffic planning, financial mathematics, economics are among the most popular fields of application of the general theory of optimal transport. Many developments have been made in all these fields recently. Two more specific fields:
- In image processing, since a grey-scale image may be viewed as a measure, optimal transportation has been used because it gives a distance between measures corresponding to the optimal cost of moving densities from one to the other, see e.g. the work of J.-M. Morel and co-workers .
- In representation and approximation of geometric shapes, say by point-cloud sampling, it is also interesting to associate a measure, rather than just a geometric locus, to a distribution of points (this gives a small importance to exceptional “outlier” mistaken points); this was developed in Q. Mérigot’s PhD in the GEOMETRICA project-team. The relevant distance between measures is again the one coming from optimal transportation.
- A collaboration between Ludovic Rifford and Robert McCann from the University of Toronto aims at applications of optimal transportation to the modeling of markets in economy; it was to subject of Alice Erlinger's PhD, unfortunately interrupted.
Applications specific to the type of costs that we consider, i.e. these coming from optimal control, are concerned with evolutions of densities under state or velocity constraints. A fluid motion or a crowd movement can be seen as the evolution of a density in a given space. If constraints are given on the directions in which these densities can evolve, we are in the framework of non-holonomic transport problems.
Control theory (in particular thinking in terms of inputs and reachable set) has brought novel ideas and progresses to mathematics. For instance, some problems from classical calculus of variations have been revisited in terms of optimal control and Pontryagin's Maximum Principle ; also, closed geodesics for perturbed Riemannian metrics where constructed in , using control techniques.
The work in progress is definitely in this line, applying techniques from control to construct some perturbations under constraints of Hamiltonian systems to solve longstanding open questions in the field of dynamical systems. Also, in , L. Rifford and R. Ruggiero applied successfully geometric control techniques to obtain genericity properties for Hamiltonian systems.
Hampath is a software developped to solve optimal control problems but also to study Hamiltonian flow.
It has been developped since 2009 by members of the APO team from Institut de Recherche en Informatique de Toulouse, jointly with colleagues from the Université de Bourgogne. It is now updated with McTAO team members.
See more on http://
These studies aim at optimizing the contrast in Nuclear Magnetic Resonance imaging using advanced optimal control. As said in section , our work on this problem is based on experiments conducted in Prof. S. Glaser in Munich, see .
This is done in collaboration with University of Hawaii, and deals with many theoretical aspects of the contrast problem in NMR: analysis of the optimal flow , feedback classification in relation with the relaxation times of the species , . John Marriott defended his PhD thesis on this topic, on August 28, 2013.
We performed, in a collaboration with Pierre Martinon (COMMANDS team, Inria) and Mathieu Claeys (LAAS CNRS, Toulouse), a thorough comparison of the various available numerical methods in optimal control on this important physical problem. Direct and indirect methods (implemented in the Bocop and Hampath sofwares) were tested in the contrast problem, and LMI techniques were used to obtain global bounds on the extremum (in the contrast problem there are many local optima and the global optimality is a complicated issue). This successful collaboration is accounted for in and was presented at the CDC conference
One of the most important results obtained by B. Bonnard and his collaborators concern the explicit computations of conjugate and cut loci on surfaces. This has obvious applications in optimal control to compute the global optimum; it is also relevent in optimal transport where regularity properties of the transport map in the Monge problem is related to convexity properties of the tangent injectivity domains.
In , we complete the previous results obtained in (we bring them from ellipsoids to general revolutions surfaces).
The conjugate and cut loci in Serret-Andoyer metrics and dynamics of spin particles with Ising coupling are analized in , this is a first step towards the computation of conjugate and cut loci on left invariant Riemannian and sub Riemannian metrics in S0(3) with applications for instance to the attitude control problem of a spacecraft.
An analysis of singular metrics on revolution surfaces, motivated by the average orbital transfer problem when the thrust direction is restricted, is proposed in .
Finally, determines cut and conjugate loci in an enegy minimizing problem that is related to the quantum systems mentionned in the first paragraph of section .
A paper on the construction and properties of an “average control system” appeared this year, it is based on Alex Bombrun's doctoral work (2007). It connects solutions of highly oscillating control systems to those of an average control system, when the frequency of oscillation goes high. It also gives a better ground to averaging for minimum time.
This average system in the case of minimum time for low thrust orbit transfer in the two body problem is currently being explored, in particular the study of its inherent singularities. In we give some properties of this system, like geodesic convexity, and compare it with the one obtained for minimum energy, and Helen Henninger's PhD aims at going further in this direction and then apply this local study to real missions, possibly in a three-body environment.
The very general condition for continuity of the transport map given in motivated exploration of conditions for convexity of the tangent injectivity domain , . Lionel Jassionnesse's PhD is in part devoted to Ma-Trudinger-Wang tensor that also plays an important role in this matter. Ludovic Rifford has an ongoing collaboration with Alession Figalli and Thomas Gallouet on the link between this MTW tansor and the convexity of injectivity domains; They already improved a result by Loeper and Villani (the preprint “Ma-Trudinger-Wang condition vs. convexity of injectivity domains” is available from the authors) and aim at proving a conjecture due to Villani, that would hold in the case of anlaytic surfaces.
The goal of Alice Erlinger's PhD, joint with University of Toronto, is to explore Optimal Transport's application to modeling in economics. She unfortunately stopped her PhD, but some results have already been obtained.
Ludovic Rifford and collaborators have been applying with success, techniques from geometric control theory to open problems in dynamical systems, mostly on genericity properties and using controllability methods to build suitable perturbations.
This has been applied to closing geodesics .
Ayadi Lazrag's PhD also deals with such problems; applying techniques close to these in
, one goal is to establish a version of Francks' lemma for geodesic flows and to apply this to
persitence problems. The approach relies on control theory results, with order 2 conditions.
See and another preprint (“Franks' lemma for
In , a non trivial conjecture on generic hyperbolicity of the so-called Aubry set of a
Hamiltonian is solved on compact surfaces and in the
“Transfert orbital dans le problème des deux et trois corps avec la technique de propulsion faible”.
This contract started October, 2012 for 3 years. It partially supports Helen Heninger's PhD.
The goal is to improve transfer strategies for guidance of a spacecraft in the gravitation field of one central body (the two-body problem) or two celestial bodies (three-body problem).
This three year contract will formally start in 2014, but discussion and preliminary work started in 2013.
It involves CNES and McTAO both through Inria and through Université de Bourgogne. It concerns averaging techniques in orbit transfers around the earth while taking into acount many perturbation of the main force (gravity for the earth considered as circular). The objective is to validate numerically and theoretically the approximations made by using averaging, and to propose methods that refine the approximation.
The “région” Provence Alpes Côte d'Azur (PACA) partially supports Helen Heninger's PhD . The other part comes from Thales Alenia space, see section .
The “région” Provence Alpes Côte d'Azur (PACA) partially supports Jérémy Rouot's PhD.
The team is officially a common team with University of Nice, but also has very strong links with Université de
Bourgogne and IMB (Institute of Mathematics in Burgundy).
Bernard Bonnard is currently on leave from Université de Bourgogne; Jean-Baptiste Caillau collaborates actively with us;
there is also an active common seminar http://
A formal convention between Inria and Université de Bourgogne is planned for 2014. It will make the IMB control team a part of McTAO.
This is a four year project ending in 2013, on Geometric Control Methods, Sub-Riemannian Geometry and Applications.
It is organized in four “poles” and gathers people from Université du Sud Toulon-Var, Université de Bourgogne (Dijon), École Polytechnique (Paris),
Nancy-Université, Université Joseph Fourier (Grenoble 1), Université Paris Sud, ParisTech ENSTA and Université Nice
Sophia-Antipolis. Bernard Bonnard, Jean-Baptiste Caillau and Ludovic Rifford (leader of one pole) are members of this project.
More details on the site;
http://
Bernard Bonnard and Ludovic Rifford participate in the GDR MOA, a CNRS network on Mathematics of Optimization and Applications.
http://
Jean-Baptiste Caillau is in the board of governors of the group SMAI-MODE (http://
Jean-Baptiste Caillau is a member of the Centre de Compétences Techniques (CCT) Mécanique orbitale du CNES
Jean-Baptiste Caillau is the corresponding member in Dijon for the Labex AMIES (http://
Jean-Baptiste Caillau is a member of the SADCO network (FP7-PEOPLE-2010-ITN, grant no. 264735-SADCO), cf. http://
Technische Universität München, Department of Chemistry (Germany).
The applications of optimal control to MNR (see sections ) are conducted with the group of Prof. Steffen Glaser in Munich.
University of Hawaii, Department of Mathematics (U. S. A.)
There is a long term collaboration on optimal control and control of quantum systems, see mostly section . Besides, Gautier Picot, a former Phd student from Dijon has a temporary position at the Math Department and collaborates with M. Chyba and G. Patterson (second Phd student from M. Chyba) in relation with the Laboratoire d'Astronomie de Paris, to apply the Hampath code to make rendez-vous with quasi-asteroids entering in the solar system near the L1-Lagrange point, in the continuation of the work developed by G. Picot and B. Daoud. This collaboration is very active and has to be emphasized.
University of Toronto, Department of Mathematics (Canada)
Optimal Transport. Alice Erlinger's PhD is co-supervised by Ludovic Rifford and John Mc Cann from University of Toronto. See section .
Alessio Figalli, from University of Texas at Austin, visited twice, for a total of slightly more than a month.
There is a strong collaboration with the control group in the University of Hawaii around M. Chyba. B. Bonnard visited the group twice in 2012-2013 (a total of 3 months). The purpose of the collaboration is to study the aspects of the contrast problem in Nuclear Magnetic Resonance, see section .
Ludovic Rifford was invited to the program “Optimal Transport: Geometry and Dynamics”
(http://
Bernard Bonnard was invited ot the Japanese forum “Math-for-Industry” 2013 on
The Impact of Applications on Mathematics,
November 4 to 8, 2013, Fukuoka. See http://
Ludovic Rifford belongs to the editorial board of Journal of Dynamical and Control Systems and Discrete and Continuous Dynamical Systems-A.
All members of the team are active reviewers in journals of the field.
The team organized two one-day seminars http://
February 13 in Dijon. Speakers: F. Chazal, B. Bonnard, G. Carlier, E. Trélat.
June 4 in Nice. Speakers: S. Rigot, Q. Mérigot, M. Mirrahimi, A. Farres.
A two days conference was organized September 6 to 7 (co-organisers: Monique Chyba and Bernard Bonnard), on “Control and Observation of Nonlinear Control Systems with Application to Medicine” at the University of Hawaii. It was supported by supported by a NSF grant and by the Engineering Department (P.E. Crouch); it followed J. Marriott's PhD defense.
Jean-Baptiste Caillau organized a mini-symposium on Optimization in aeronautics and space machanics in the SMAI annual congress in Seignosse, France.
J.-B. Pomet is the president of the “Comité de Suivi Doctoral”, and in charge of “formation par la recherche”. This includes organising local visits for students, organising PhD candidates selection, managing PhD students working at Inria Sophia (that are from two different “écoles doctorales” from Université de Nice, not counting these in Montpellier).
Jean-Baptiste Caillau, Contrôle optimal : introduction au cas déterministe en dimension
finie, 15 hours, 7ème école d’été de Peyresq en traitement du signal et des images (http://
Jean-Baptiste Caillau, Introduction to optimal control and application to space
mechanics, 15 hours, Gravasco trimester, Institut Henri Poincaré, France (http://
PhD: John Marriott, Geometric Optimal Control with an Application to Imaging in Nuclear Magnetic Resonance, University of Hawaii, defended september 5, 2013, advisors: Monique Chyba and Bernard Bonnard.
PhD in progress: Alice Erlinger, subject: Economics and Optimal Transport, Université de Nice Sophia Antipolis, started october, 2012, advisor: Ludovic Rifford. She unfortunately decided to stop in 2013, within her first year.
PhD in progress: Helen Heninger, subject: Étude des solutions du transfert orbital avec poussée faible dans le probleme des deux ou trois corps, Université de Nice Sophia Antipolis, started october, 2012, advisors: Bernard Bonnard and Jean-Baptiste Pomet.
PhD in progress: Ayadi Lazrag, subject: Control methods in dynamical systems, Université de Nice Sophia Antipolis, started october, 2011, advisor: Ludovic Rifford.
PhD in progress: Lionel Jassionnesse, subject: Lieu conjugés et de coupure pour des métriques de Liouville et applications, Université de Bourgogne, started october, 2010, advisor: Bernard Bonnard.
PhD in progress: Jérémy Rouot, subject: Moyennisation en contrôle et en contrôle optimal, effet des perturbations non périodiques, Université de Nice Sophia Antipolis, started october, 2013, advisors: Bernard Bonnard and Jean-Baptiste Pomet.
MSc: Jasseur Abidi, Optimisation d'une commande appliquée à un satellite, Ensta ParisTech, supervisors: Jean-Baptiste Caillau and Jean-Baptiste Pomet.
Jean-Baptiste Caillau was in the jury of X. Dupuis (Ecole Polytechnique, PhD) as President and in the jury of T. Bayen (university of Montpellier, HDR) as a referee.
Jean-Baptiste Caillau: “Mathematics for planet Earth 2013”:
http://