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 $\mathrm{d}x/\mathrm{d}t=f(x,u)$ 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.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 quite difficult to decide local or global controllability, and the general problem is 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 constraints 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 87, 46, 29, 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.

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 71, 88. 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 59, 91.

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 modeled phenomena, or it can come from the study of the geometry of the equations and the transformation acting on them. This justifies the investigation of these transformations. These topics are central in control theory and they are present in the team, see for instance the classification aspect in 50 or —although this research has not been active very recently— the study  86 of dynamic feedback and the so-called “flatness” property  74. Likewise, classification tools such as feedback invariants 48 are still currently in use in the team (see, for instance, 52).

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 correspondence 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 90, 31, 30 and 51 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 43, followed by the HamPath 61 code and control toolbox 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.64 and 39). 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 ${\mathrm{L}}^1$-minimization, for instance, discontinuous curves in the Grassmannian have to be considered (instantaneous rotations of Lagrangian subspaces still obeying symplectic rules 70). The cut locus is a central object in Riemannian geometry, control and optimal transport. This was 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, simply because there are no differential constraints. It is studied in the team for the following two reasons. On one hand, after some transformations, 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 73, 72. The analysis in the oblate geometry 41 was completed in 68 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 ${\mathrm{L}}^2$-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 42 with Japanese Riemannian geometers. Averaging the same systems for minimum time yields a Finsler Metric, as noted in 38. In 49, 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 83 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 80 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 56.

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, 89. 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 $J_2$ 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 67, standard averaging (32, 89) 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 65).

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 45, 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 94, 76 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 94, 76. 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 92).

In a completely different setting, namely the spectral analysis of Sturm-Liouville operators, periodic control has also been considered in the recent work 16. For periodic conditions, the boundary problem treated in this paper can be considered as problem on the circle, so that the control is by design periodic. The problem is moreover well posed only in this category (translations of a non-constant solution would otherwise provide infinitely many solutions).

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 recent book chapter 66. 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 (2019-2023) in AMDT1 mode supported by Inria Sophia SED. While the ADT ended in 2023, regular interaction between the project members and the SED team still take place. For instance, SED colleagues recently helped us to set up the CI infrastructure of control-toolbox.org on github.

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 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 ${\mathrm{L}}^1$-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 Section 7.6), the second one about collision avoidance for spacecrafts (see Section 7.5). 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, 2022, 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. It was the topic of Alesia Herasimenka's PhD, selected by ESA for a three-year research co-sponsorship, and defended in September, 2023.

4.2 Optimal control of microbial cells, and other biological applications

Participants: Bernard Bonnard, Jean-Baptiste Caillau, Martin Fleurial, 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], Jérémy Rouot [Université de Bretagne Occidentale, Brest], Agustín Yabo [INRAE, Montpellier].

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). New results are presented in Section 7.14.

The team is also involved in other problems related to biological or medical applications, namely muscular functional electro-stimulation (new results presented in Section 7.13), Lotka-Volterra models (new results presented in Section 7.12), and alcoholic fermentation (new results presented in Section 7.15).

4.3 Neural dynamics

Participants: Adel Annabi, Dario Prandi [CNRS, CentraleSupélec], Jean-Baptiste Pomet, Ludovic Sacchelli.

Neural fields serve as integro-differential dynamical models for the transmission of activity within cortical areas 58. 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 55, 57.

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 69. 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.

5.1 Awards

A former PhD student, Alesia Herasimenka, has received the 2nd prize of 2024 "Prix de thèse Maths, entreprises et société" attributed by AMIES for her thesis defended in 2023.

5.2 End of cycle

This year 2024, was the twelfth and last year of the McTAO project-team, for all Inria teams have a maximum duration of 12 years. A new team will succeed, with renewed topics and motivations.

This is the opportunity to thank Prof. Bernard Bonnard, from Université de Bourgogne, who was a founding member of the team —with Ludovic Rifford (Univ. Côte d'Azur) and Jean-Baptiste Pomet— in 2012-13, and has been an active and major member since. This was technically possible thanks to an agreement with Université de Bourgogne, that could not be extended to the full 12 year period; this is why he appears in this report, as an external collaborator although we consider that his role was the one of a member.

6.1 New software

7.1 Control templates for output feedback stabilization of non-uniformly observable systems

Participants: Ludovic Sacchelli, Lucas Brivadis [CentraleSupélec, Gif-sur-Yvette], Ulysse Serres [Université Claude Bernard Lyon 1], Vincent Andrieu [Université Claude Bernard Lyon 1], Jean-Paul Gauthier [Université de Toulon], Itaï Ben Yacoov [Université Claude Bernard Lyon 1].

Stabilizing the state of a system by feedback is a fundamental problem in control theory. When only partial information about the system is available, a common approach is to use a dynamic algorithm, known as an observer, to estimate the state and provide it to the controller. This approach is referred to as output feedback control. For a stable closed-loop design, it is essential that sufficient information about the state can be extracted from the partial measurements. In nonlinear systems, however, the ability to estimate the state reliably may depend on the control itself, a phenomenon known as non-uniform observability, which poses a significant challenge for observer design. In the context of output feedback control, the presence of singular controls for observability complicates the stabilization problem, with no definitive solutions available. It has long been recognized that feedback controls based purely on the state may face obstructions, necessitating some form of time-varying component. One approach involves starting with a state-feedback control design and introducing perturbations to achieve output feedback control, even in the presence of observability singularities. The group has been exploring hybrid control techniques to address this challenge.

In 1 (recently accepted for publication in SCL), we introduced a novel technique based on control templates. This approach replaces the feedback with a piecewise approximation, where each segment is modeled on a specific control (the template) known to ensure observability. This generalizes the concept of sample and hold, where the value of the feedback is maintained and periodically updated. Recent work by Lin et al. demonstrated that such approximations still allow stabilization via hybrid output feedback under observability for all inputs. 1 extends this result to analytic systems observable under a null input. The paper also includes a proof that control templates are generic among analytic inputs.

In 8, we applied a variation of this principle to state-affine, control-polynomial systems in order to go beyond this existence results. By leveraging a specialized observer for state-affine systems, we extended the output feedback stabilization result to discontinuous templates. This relaxation allowed to propose an algorithmic procedure for the construction of piecewise constant control templates based on principles from algebraic geometry. This was the topic of a joint journal publication and conference talk at CDC 2024.

7.2 $L^1$-optimal control problems

Participants: Ivan Beschastnyi, Andrei Agrachev [SISSA Trieste, Italy], Michele Motta [SISSA Trieste, Italy].

$L^1$-optimal control problems play an important role in engineering sciences. They appear frequently when we want to minimize work or total cost. For this reason they appear frequently in applications, for example, in space mechanics 60 or in economic epidemiology 82. A notable feature of such minimal solutions to such problems is the presence of long intervals of zero control (sparse control strategy).

We were able to characterize extremal controls for affine systems with any number of controls and obtain no-gap necessary and sufficient conditions for minimality of singular arcs. An interesting feature of the problem is that along singular curves the second variation degenerates only in certain, and not in all, directions. After a suitable relaxation of the problem, no-gap conditions are formulated using a special generalized Legendre-Clebsch condition and absence of non-trivial solutions to the corresponding Jacobi equation. An example relevant to quantum control was studied in detail.

7.3 Random walks in sub-Riemannian manifolds

Participants: Ludovic Sacchelli, Robert W. Neel [Lehigh University, USA], Ivan Beschastnyi, Jules Duhamel.

Laplacian-like operators on sub-Riemannian manifolds can be constructed by incorporating the non-holonomic constraints inherent to their control structure. The study of heat diffusion in this sub-Riemannian framework offers a connection between the geometric analysis of sub-Riemannian geodesics and the behavior of random walks on the manifold. Since Varadhan’s 1984 work on Riemannian manifolds 93, it has been understood that the short-time asymptotics of the heat kernel reveal the distance between two points. This distance reflects the exponentially rare occurrence of continuous-time random walks (e.g., Brownian motion) bridging two points in arbitrarily short time intervals.

In 24 (recently revised and accepted for publication), we extended existing techniques to study bounds on the logarithmic derivatives of the heat kernel. Such bounds, well-established for compact Riemannian manifolds, were recently generalized to complete Riemannian manifolds. Our work further extends these results to incomplete Riemannian manifolds under the least restrictive conditions to date. Additionally, we discuss the unique challenges that arise when transitioning to the sub-Riemannian setting.

During J. Duhamel’s internship, we explored the discrete-time version of these diffusions. Specifically, we studied approximations of Brownian motion by particles following geodesics, randomly drawn at periodic intervals from a finite set. In the context of some well-understood Lie groups equipped with invariant sub-Riemannian structures, this approach generalizes discrete random walks in Euclidean spaces. The resulting approximations can be utilized to recover high-order differential operators that were historically associated with curvature in both discrete and continuous settings. In particular, we investigated the emergence of curvature-dimension inequalities within a few specific model spaces.

7.4 Study of singular geometric structure

Participants: Ivan Beschastnyi, Simão Lucas [Politecnico di Milano], Paula Cerejeiras [CIDMA, Universidade de Aveiro], Victor Nistor [Université de Lorraine], Catarina Carvalho [IST Lisboa], Yu Qiao [Shaanxi Normal University].

Singular geometric structures can arise in many ways. An example could be a domain with corners, a wedge or just a punctured hole. Dynamical systems and PDEs associated to such structures are of great interest. For example, in quantum mechanics singular potentials appear all the time describing the situation when two particles repel each other. These systems behave very differently when compared to their regular counterparts and present a considerable challenge.

For example, in 20, we have generalized Gauss-Bonnet-Chern theorem to a class of singular structure called $\alpha$-Grushin manifolds. They can be thought of manifolds, where all the geometric quantities (metric, volume, etc.) degenerate or blow-up uniformly when we approach the singularity. In this case, the integral of the Chern form is not the Euler characteristic and, in most cases, it does not even exist. Nevertheless, we can study Gauss-Bonnet defects, which are asymptotics of the integral of the Gauss-Bonnet form over the manifold excluding a small tubular neighborhood of the singularity. We were able to compute the first terms of this asymptotic expansion using topological and geometric invariants of the manifold and of the singular set.

In 19, we described relevant geometric structures for studying Schrodinger operators with real power potentials. Power potentials are often used to describe non-hard collisions of particles. In our work we desingularized the operator using continuous-family groupoids. That allows us to introduce $C^{*}$-machinery into their study and investigate important properties of such operators, such as self-adjointness or essential spectrum.

7.5 Characterization of the warning time for low-thrust satellite collision avoidance maneuvers

Participants: Lamberto Dell'Elce, Frank de Veld, Jean-Baptiste Pomet.

Owing to the exponential increase in tracked objects in low-Earth orbit (LEO) and geostationary orbit (GEO), the number of alerts for potential collisions is continuously rising. Satellites equipped with low-thrust propulsion need to maneuver for a finite, possibly large time to reduce the risk of collision to a safe threshold. In the context of Frank de Veld's thesis, which began in December 2022 and is carried out in collaboration with Thales Alenia Space, we studied the minimal requirements for such maneuvers during fast encounters between a low-thrust-actuated satellite and a non-cooperative object.

We first introduced a state vector consisting of the time of closest approach and the orbital elements of the primary at this time, and we developed its equations of motion. After deriving necessary conditions for optimality, we proposed an approximate solution that does not require solving a shooting problem but is instead obtained through backward integration from the safe set for any initial state near the advertised collision. A thorough analysis of the optimal synthesis of this solution was performed, with particular focus on computing the cut and conjugate loci. This work was presented at the Astrodynamics Specialist Conference 12.

7.6 Optimal control of solar sails

Participants: Jean-Baptiste Caillau, Vincent Callegari, Lamberto Dell'Elce, Alesia Herasimenka [University of Luxembourg], Jean-Baptiste Pomet.

Our research on the control of solar sails started with the PhD thesis of A. Herasimenka 79, defended in September 2023. Solar sails offer a propellantless solution for performing interplanetary transfers, planetary escapes, and de-orbiting maneuvers by using solar radiation pressure (SRP). The second part of the thesis was devoted to the optimization of trajectories, and it was recently published 7. In 2024, we applied these results to the analysis of two challenging missions detailed below.

First, we investigated how solar sails can enable extra-solar transfers (i.e., trajectories capable of leaving the Solar System). This is achieved by flying the sail on an orbit with a very low perihelion, allowing the sail to significantly increase the orbital energy after a single solar flyby (since SRP is inversely proportional to the squared distance from the Sun). To this end, we formulated an optimal control problem aimed at maximizing the orbital energy at the end of the maneuver. A temperature constraint, crucial to prevent material degradation near the Sun, was also included. From a control perspective, this involved incorporating constraints on both the state variable (distance from the Sun) and the control variable (orientation of the sail) to manage surface temperature fluctuations. This work was presented at the Astrodynamics Specialist Conference 15.

Second, we explored the possibility of using solar sails to perform a solar occultation mission. The idea of the mission is to place a satellite at the tip of the umbra cone created by the Moon to observe the solar corona. Using a solar sail to control the satellite’s trajectory between two observations may enhance the mission’s duration, as solar sails operate without requiring propellant. A preliminary analysis of this problem was undertaken during the internship of V. Callegari. Specifically, after identifying suitable observation windows, we investigated the feasibility of an SRP-actuated transfer.

7.7 Hardware-in-the-loop attitude simulator for the NiceCube mission

Participants: Lamberto Dell'Elce, Salma Janati, Jean-Baptiste Pomet.

The Centre Spatial Universitaire (CSU) (https://nanosat.univ-cotedazur.fr) of the Université Côte d'Azur (UniCA) is planning to build its first nanosatellite, named NiceCube, which has the technological objective of demonstrating data transmission from the satellite to the ground via an optical link. Specifically, NiceCube will be illuminated with a laser beam from the ground. Retro-reflectors will be used to return the light toward the ground station. By modulating the returned signal with occluders, it is possible to transmit data with minimal power requirements onboard. In order to receive the reflected signal on the ground, the satellite needs to be oriented with a few degrees of precision toward the ground station. Hence, a critical aspect of this mission concerns the control of the attitude (i.e., orientation in space) of the satellite.

McTAO is in charge of developing the attitude determination and control system (ADCS) of NiceCube. This task, which is more about offering hands-on experience to master's students rather than being a research topic, involves the development of a high-fidelity numerical simulator of the coupled orbital-attitude motion of a satellite in low-Earth orbit. The internship of S. Janati was aimed at integrating this simulator into the FlatSat hosted at the Observatoire de la Côte d'Azur (OCA), which will be used to carry out hardware-in-the-loop simulations. This work is still in progress.

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

Participants: Laetitia Giraldi [CALISTO project-team], Pierre Lissy [Université Paris Dauphine, Paris], Clément Moreau [CNRS, Nantes Université, LS2N], Jean-Baptiste Pomet.

In the paper 6, that appeared this year, 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, STLC holds for some values of the parameters, but does not hold for generic values of these parameters, see C. Moreau's PhD for details 84. The main results are novel necessary conditions for STLC of these systems, based on Chen-Fliess expansions of solutions, in the spirit of 81 or 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.

7.9 Stability of linear time-varying time-delay systems

Participants: Laurent Baratchart [FACTAS project-team], Sébastien Fueyo [DANCE project-team], Jean-Baptiste Pomet.

A linear time-periodic difference-delay systems (periodic LDDS for short) is a dynamical system of the form $z\left(t\right)=A_1\left(t\right)z(t-{\tau }_1)+\cdots +A_N\left(t\right)z(t-{\tau }_N)$, 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 75, 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 78, 77; 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 2, still in review, with the proof of a technical part given in 3, published this year.

7.10 Observer design for low-dimensional models of the visual cortex

Participants: Adel Annabi, Dario Prandi [CNRS, CentraleSupélec, LSS], Jean-Baptiste Pomet, Ludovic Sacchelli.

The goal of this research is to provide insights into observability and observer synthesis for neural fields equations, the general topic of A. Annabi's 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 produce a simplified yet effective three-dimensional model of the V1 area, as introduced (among others) by Blumenfeld et al.

The group's first paper on this topic, 17 (currently in review), investigates the observability of this model, emphasizing the system's inherent symmetries and proposing hybrid elements to counteract their effects. It demonstrates the critical role of nonlinearity in achieving observability and explores persistence conditions required for accurate state estimation. These conditions form the basis for designing a high-gain observer. This work was further extended in the conference paper 11, which explores an alternative observer design. Using techniques based on Hölder-continuous embeddings, this approach enables the development of a finite-time converging observer.

7.11 Geometry and optimal control for navigation problem

Participants: Bernard Bonnard, Joseph Gergaud, Olivier Cots, Jérémy Rouot [Université de Bretagne Occidentale, Brest].

This is a long term research program 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 was motivated by the displacement of particles in a two dimensional fluid, in presence of a vortex (initially, a singularity in the Helhmoltz-Kirchhoff equations) inducing 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.

This year, a new application to spin magnetization reversing was considered. The geometric interpretation of this phenomenon in micromagnetism as a Zermelo navigation problem is a novel point of view. It allowed us to fit the problem in a $L^{\infty }$ accessibility framework, which can be solved by combining algebraic and numerical simulations. The problem can be sorted in two cases (Randers vs Finsler) according to the 4 physical parameters. In the Finsler case, the numerical framework developed by Cots et al. permitted us to completely solve the problem on the 2d-sphere and determine the cut locus (simple branch with two extremities) 47, 44 (see also McTAO talk on 23/09/24).

7.12 Feedback invariants and optimal control with biological application­s and numerical 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 54, 53, 52.

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 jet spaces. We have compared both perma­nent controls and sampled-data control in relation with numerical issues. 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.

Finally, by combining NMPC (Nonlinear Model Predictive Control) techniques and geometric methods, the problem of reducing the Clostridioides difficile (C. difficile) infection of the intestinal microbiote was successfully solved (see 53 for a preliminary presentation of the complete results).

7.13 Biomechanics: control of muscular force response using FES with application to the conception of a smart electrostimulator

Participants: Bernard Bonnard, 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, in the past years, some developments on optimal control for sample control systems. This is by itself already a rich topic. In 2020, this project took the industrial transfer direction with a Cifre PhD funding in partnership with Segula Technologies. The PhD was defended in 2023 by Sandrine Gayrard. In particular, a finite dimensional approximation of the forced dynamics based on the Ding et al. model was derived, that provides fast optimizing schemes aiming to track a reference force or maximize the force (see e.g. 34), and a prototype of the smart electrostimulator was finalized with Segula Technologies.

Previous analysis to study the control problem was completed to estimate the fatigue parameters of the model using backwards horizon methods  33.

7.14 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 96, 95. 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 (with respect to 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 97 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 98.

Recent results also 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 in finite and infinite horizon for batch reactors 9.

7.15 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 was 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 publication of 13 in an international conference on control (ECC 24).

The primary emphasis of 13 lies in investigating the observability properties of an alcoholic fermentation 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.

7.16 ct: control toolbox

Participants: Jean-Baptiste Caillau, Olivier Cots, Joseph Gergaud, Pierre Martinon [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 recently been presented in conferences 62, 63. In 2024, the effort has been concentrated on the Julia package OptimalControl.jl, at the heart of the control-toolbox.org ecosystem. This development is strongly tied to an efficient use of sparse linear algebra, numerical optimization and automatic differentiation. Fruitful exchanges with colleagues from the Ecuador team and from Argonne National Lab (visit of A. Montoison) are key to this task and should continue in 2025.

8.1 Méthodes de contrôle pour l’évitement de collisions entre satellites, Thales Alenia Space

Participants: Thierry Dargent, Lamberto Dell'Elce, Frank de Veld, Jean-Baptiste Pomet.

This contract with Thales Alenia Space is co-funding the thesis of Frank de Veld entitled “Méthodes de Contrôle pour l’évitement de collisions entre satellites”; the other source of funding is the grant from Région Provence-Alpes-Côte d'Azur mentioned in Section 9.3

• Partners: McTAO and Thales Alenia Space.
• Period: 2022–2025
• Total amount: 75k€
• Inria reference: 022-0674

9.1 International research visitors

9.2 National initiatives

9.3 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, together with the contract with Thales Alenia Space mentioned in Section 8.1. Total amount: 54k€.
• Idex UCA Welcome package of Ivan Beschastnyi, 2023-2026. Total amount: 50k€.

10.1 Promoting scientific activities

10.2 Teaching - Supervision - Juries

10.3 Popularization

11.1 Publications of the year

11.2 Cited publications

• 28 incollectionA. A.A. A. Agrachev and R. V.R. V. Gamkrelidze. Symplectic methods for optimization and control.Geometry of feedback and optimal control207Textbooks Pure Appl. Math.Marcel Dekker1998, 19--77back to text
• 29 bookA.Andrei Agrachev and Y. L.Yuri L. Sachkov. Control theory from the geometric viewpoint.87Encyclopaedia of Mathematical SciencesControl Theory and Optimization, IIBerlinSpringer-Verlag2004DOIback to text
• 30 articleA. A.Andrey A. Agrachev and A. V.Andrey V. Sarychev. Abnormal sub-Riemannian geodesics: Morse index and rigidity.1361996, 635--690URL: http://www.numdam.org/item/AIHPC_1996__13_6_635_0/back to text
• 31 articleA. A.A. A. Agrachev and A. V.A. V. Sarychev. Strong minimality of abnormal geodesics for 2-distributions.121995, 139--176DOIback to text
• 32 bookV. I.Vladimir I. Arnold. Mathematical methods of classical mechanics.60Graduate Texts in MathematicsNew YorkSpringer-Verlag1989back to textback to text
• 33 inproceedingsT.Toufik Bakir, B.Bernard Bonnard, I.Ilias Boualam and M.Mokrane Abdiche. Control and Estimation for the Design of a Smart Electrostimulator using Ding et al. Model.FGS 2024 - French-German-Spanish Conference on OptimizationGijon, SpainJune 2024HALback to text
• 34 articleT.Toufik Bakir, B.Bernard Bonnard, S.Sandrine Gayrard and J.Jérémy Rouot. Finite Dimensional Approximation to Muscular Response in Force-Fatigue Dynamics using Functional Electrical Stimulation.October 2022HALDOIback to text
• 35 articleL.Laurent Baratchart, S.Sébastien Fueyo, G.Gilles Lebeau and J.-B.Jean-Baptiste Pomet. Sufficient Stability Conditions for Time-varying Networks of Telegrapher's Equations or Difference-Delay Equations.5322021, 1831--1856URL: http://hal.inria.fr/hal-02385548/DOIback to text
• 36 articleD.Davide Barilari, U.Ugo Boscain and R. W.Robert W. Neel. Small-time heat kernel asymptotics at the sub-Riemannian cut locus.923November 2012, URL: https://doi.org/10.4310/jdg/1354110195DOIback to text
• 37 unpublishedK.Karine Beauchard and F.Frédéric Marbach. A unified approach of obstructions to small-time local controllability for scalar-input systems.May 2022, working paper or preprintHALback to text
• 38 articleA.Alex Bombrun and J.-B.Jean-Baptiste Pomet. The averaged control system of fast oscillating control systems.5132013, 2280--2305URL: http://hal.inria.fr/hal-00648330/DOIback to textback to text
• 39 inproceedingsB.Bernard Bonnard, J.-B.Jean-Baptiste Caillau and O.Olivier Cots. Energy minimization in two-level dissipative quantum control: The integrable case.Proceedings of the 8th AIMS Conference on Dynamical Systems, Differential Equations and Applicationssuppl.Discrete Contin. Dyn. Syst.AIMS2011, 198--208DOIback to text
• 40 articleB.Bernard Bonnard and J.-B.Jean-Baptiste Caillau. Geodesic flow of the averaged controlled Kepler equation.215September 2009, 797--814DOIback to text
• 41 articleB.Bernard Bonnard, J.-B.Jean-Baptiste Caillau and L.Ludovic Rifford. Convexity of injectivity domains on the ellipsoid of revolution: the oblate case.34823-242010, 1315--1318URL: https://hal.archives-ouvertes.fr/hal-00545768DOIback to text
• 42 articleB.Bernard Bonnard, J.-B.Jean-Baptiste Caillau, R.Robert Sinclair and M.Minoru Tanaka. Conjugate and cut loci of a two-sphere of revolution with application to optimal control.2642009, 1081--1098DOIback to text
• 43 articleB.Bernard Bonnard, J.-B.Jean-Baptiste Caillau and E.Emmanuel Trélat. Second order optimality conditions in the smooth case and applications in optimal control.1322007, 207--236DOIback to text
• 44 bookB.Bernard Bonnard, M.Monique Chyba, D.David Holcman and E.Emmanuel Trélat. Ivan Kupka Legacy: A Tour Through Controlled Dynamics.12AIMS2024, URL: https://hal.science/hal-04869632back to textback to text
• 45 unpublishedB.Bernard Bonnard, M.Monique Chyba, J.Jérémy Rouot, D.Daisuke Takagi and R.Rong Zou. Optimal Strokes : a Geometric and Numerical Study of the Copepod Swimmer.January 2016, working paper or preprintURL: https://hal.inria.fr/hal-01162407back to text
• 46 bookB.Bernard Bonnard and M.Monique Chyba. Singular trajectories and their role in control theory.40Mathématiques & ApplicationsBerlinSpringer-Verlag2003back to text
• 47 incollectionB.Bernard Bonnard, O.Olivier Cots, Y.Yannick Privat and E.Emmanuel Trélat. Zermelo navigation on the sphere with revolution metrics.IVAN KUPKA LEGACY: A Tour Through Controlled Dynamics12AIMS Applied Math Books - Special issue in honor of I. Kupka2024, 35--66HALback to text
• 48 articleB.B. Bonnard. Feedback equivalence for nonlinear systems and the time optimal control problem.2961991, 1300--1321URL: https://doi.org/10.1137/0329067DOIback to text
• 49 articleB.Bernard Bonnard, H.Helen Henninger, J.Jana Nemcova and J.-B.Jean-Baptiste Pomet. Time Versus Energy in the Averaged Optimal Coplanar Kepler Transfer towards Circular Orbits.13522015, 47--80URL: https://hal.inria.fr/hal-00918633DOIback to text
• 50 articleB.Bernard Bonnard, A.Alain Jacquemard, M.Monique Chyba and J.John Marriott. Algebraic geometric classification of the singular flow in the contrast imaging problem in nuclear magnetic resonance.342013, 397--432URL: https://hal.inria.fr/hal-00939495DOIback to text
• 51 articleB.Bernard Bonnard and I.I. Kupka. Théorie des singularités de l'application entrée-sortie et optimalité des trajectoires singulières dans le problème du temps minimal.51993, 111--159back to text
• 52 articleB.Bernard Bonnard and J.Jérémy Rouot. Feedback Classification and Optimal Control with Applications to the Controlled Lotka-Volterra Model.Optimization2024, URL: https://inria.hal.science/hal-03917363DOIback to textback to text
• 53 inbookB.Bernard Bonnard and J.Jérémy Rouot. Optimal Control of the Lotka-Volterra Equations with Applications.12IVAN KUPKA LEGACY: A Tour Through Controlled Dynamics2024, 27-45HALback to textback to text
• 54 articleB.Bernard Bonnard, J.Jérémy Rouot and C.Cristiana Silva. Geometric Optimal Control of the Generalized Lotka-Volterra Model of the Intestinal Microbiome.Optimal Control Applications and MethodsFebruary 2024HALDOIback to text
• 55 articleU.U. Boscain, R. A.R. A. Chertovskih, J. P.J. P. Gauthier and A. O.A. O. Remizov. Hypoelliptic Diffusion and Human Vision: A Semidiscrete New Twist.SIAM Journal on Imaging Sciences722014, 669-695URL: https://doi.org/10.1137/130924731DOIback to text
• 56 articleU.Ugo Boscain, J.Jean Duplaix, J.-P.Jean-Paul Gauthier and F.Francesco Rossi. Anthropomorphic Image Reconstruction via Hypoelliptic Diffusion.503January 2012, 1309--1336DOIback to text
• 57 articleU.Ugo Boscain, D.Dario Prandi, L.Ludovic Sacchelli and G.Giuseppina Turco. A bio-inspired geometric model for sound reconstruction.The Journal of Mathematical Neuroscience1122021, URL: https://doi.org/10.1186/s13408-020-00099-4DOIback to text
• 58 articleP. C.Paul C Bressloff. Spatiotemporal dynamics of continuum neural fields.Journal of Physics A: Mathematical and Theoretical4532011, 033001back to text
• 59 articleL.Lucas Brivadis, J.-P.Jean-Paul Gauthier, L.Ludovic Sacchelli and U.Ulysse Serres. Avoiding observability singularities in output feedback bilinear systems.593May 2021, 1759--1780URL: https://hal.archives-ouvertes.fr/hal-02172420DOIback to text
• 60 articleJ.-B.J.-B Caillau, Z.Z Chen and Y.Y Chitour. L 1 -minimization for mechanical systems.2016, URL: https://hal.archives-ouvertes.fr/hal-01136676back to text
• 61 articleJ.-B.Jean-Baptiste Caillau, O.O. Cots and J.J. Gergaud. Differential pathfollowing for regular optimal control problems.2722012, 177--196back to text
• 62 inproceedingsJ.-B.Jean-Baptiste Caillau, O.Olivier Cots, J.Joseph Gergaud and P.Pierre Martinon. Solving optimal control problems with Julia (talk).JuliaCon 2023Cambridge, Boston, United StatesJuly 2023HALback to text
• 63 inproceedingsJ.-B.Jean-Baptiste Caillau, O.Olivier Cots, J.Joseph Gergaud and P.Pierre Martinon. Solving optimal control problems with Julia.Julia and Optimization Days 2023Paris, FranceOctober 2023HALback to text
• 64 articleJ.-B.Jean-Baptiste Caillau and B.B. Daoud. Minimum time control of the restricted three-body problem.5062012, 3178--3202back to text
• 65 incollectionJ.-B.Jean-Baptiste Caillau and A.Ariadna Farrés. On local optima in minimum time control of the restricted three-body problem.Recent Advances in Celestial and Space MechanicsMathematics for Industry23SpringerApril 2016, 209--302URL: https://hal.archives-ouvertes.fr/hal-01260120DOIback to text
• 66 inbookJ.-B.Jean-Baptiste Caillau, R.Roberto Ferretti, E.Emmanuel Trélat and H.Hasnaa Zidani. An algorithmic guide for finite-dimensional optimal control problems.24Handbook of numerical analysis: Numerical control, Part BHandbook of Numerical AnalysisNorth-Holland; Elsevier2023, 559-626HALDOIback to text
• 67 unpublishedJ.-B.Jean-Baptiste Caillau, J.-B.Jean-Baptiste Pomet and J.Jeremy Rouot. Metric approximation of minimum time control systems.November 2017, working paper or preprintURL: https://hal.inria.fr/hal-01672001back to text
• 68 incollectionJ.-B.Jean-Baptiste Caillau and C.Clément Royer. On the injectivity and nonfocal domains of the ellipsoid of revolution.Geometric Control Theory and Sub-Riemannian Geometry5INdAM seriesSpringer2014, 73--85URL: https://hal.archives-ouvertes.fr/hal-01315530DOIback to text
• 69 articleR.Romain Carron, A.Antoine Chaillet, A.Anton Filipchuk, W.William Pasillas-Lépine and C.Constance Hammond. Closing the loop of deep brain stimulation.Frontiers in systems neuroscience72013, 112back to text
• 70 articleZ.Zheng Chen, J.-B.Jean-Baptiste Caillau and Y.Yacine Chitour. $L^1$-minimization for mechanical systems.543May 2016, 1245--1265URL: https://hal.archives-ouvertes.fr/hal-01136676DOIback to text
• 71 articleL.L. Faubourg and J.-B.Jean-Baptiste Pomet. Control Lyapunov functions for homogeneous ``Jurdjevic-Quinn'' systems.52000, 293--311DOIback to text
• 72 articleA.Alessio Figalli, T.Thomas Gallouët and L.Ludovic Rifford. On the convexity of injectivity domains on nonfocal manifolds.4722015, 969--1000URL: https://hal.inria.fr/hal-00968354DOIback to text
• 73 articleA.A. Figalli, L.L. Rifford and C.C. Villani. Necessary and sufficient conditions for continuity of optimal transport maps on Riemannian manifolds.6342011, 855--876URL: http://hal.inria.fr/hal-00923320v1back to text
• 74 articleM.Michel Fliess, J.Jean Lévine, P.Philippe Martin and P.Pierre Rouchon. Flatness and Defect of Nonlinear Systems: Introductory Theory and Examples.611995, 1327--1361back to text
• 75 thesisS.Sébastien Fueyo. Systèmes à retard instationnaires et EDP hyperboliques 1-D instationnaires, fonctions de transfert harmoniques et circuits électriques non-linéaires.Université Cote d'AzurOctober 2020, URL: https://hal.archives-ouvertes.fr/tel-03105344back to text
• 76 articleC.C. Gavriel and R.R. Vinter. Second order sufficient conditions for optimal control problems with non-unique minimizers: an abstract framework.7032014, 411--442DOIback to textback to text
• 77 bookJ. K.Jack K. Hale and S. M.Sjoerd M. Verduyn Lunel. Introduction to functional-differential equations.99Applied Mathematical SciencesSpringer-Verlag, New York1993DOIback to text
• 78 articleD.Daniel Henry. Linear autonomous neutral functional differential equations.151974, 106--128DOIback to text
• 79 thesisA.Alesia Herasimenka. Optimal control of solar sails.Université Côte d'AzurSeptember 2023HALback to text
• 80 articleL.Lars Hörmander. Hypoelliptic second order differential equations.1191967, 147--171URL: https://doi.org/10.1007/bf02392081DOIback to text
• 81 incollectionM.Matthias Kawski. High-order small-time local controllability.Nonlinear controllability and optimal control133Monogr. Textbooks Pure Appl. Math.Dekker, New York1990, 431--467back to text
• 82 unpublishedH.Helmut Maurer and M. D.Maria Do Rosário de Pinho. Optimal Control of Epidemiological SEIR models with L1-Objectives and Control-State Constraints.2014, Submitted, 21 pagesHALback to text
• 83 bookR.Richard Montgomery. A tour of subriemannian geometries, their geodesics and applications.91Mathematical Surveys and MonographsAmerican Mathematical Society, Providence, RI2002back to text
• 84 thesisC.Clément Moreau. Contrôlabilité en dimension finie et infinie et applications à des systèmes non linéaires issus du vivant.Université Côte d'AzurJune 2020, URL: https://hal.archives-ouvertes.fr/tel-03106682back to text
• 85 unpublishedR. W.Robert W. Neel and L.Ludovic Sacchelli. Localized bounds on log-derivatives of the heat kernel on incomplete Riemannian manifolds.2022, working paper or preprintHALback to text
• 86 articleJ.-B.Jean-Baptiste Pomet. A necessary condition for dynamic equivalence.482009, 925--940URL: http://hal.inria.fr/inria-00277531DOIback to text
• 87 bookL. S.L. S. Pontryagin, V. G.V. G. Boltjanskiı̆, R. V.R. V. Gamkrelidze and E.E. Mitchenko. Théorie mathématique des processus optimaux.MoscouEditions MIR1974back to text
• 88 articleL.Ludovic Rifford. Stratified semiconcave control-Lyapunov functions and the stabilization problem.2232005, 343--384DOIback to text
• 89 bookJ. A.Jan A. Sanders and F.F. Verhulst. Averaging Methods in Nonlinear Dynamical Systems.56Applied Mathematical SciencesSpringer-Verlag1985back to textback to text
• 90 articleA. V.A. V. Sarychev. The index of second variation of a control system.411982, 338--401back to text
• 91 articleH.H. Shim and A.A.R. Teel. Asymptotic controllability and observability imply semiglobal practical asymptotic stabilizability by sampled-data output feedback.3932003, 441--454DOIback to text
• 92 articleE.Emmanuel Trélat and E.Enrique Zuazua. The turnpike property in finite-dimensional nonlinear optimal control.25812015, 81--114DOIback to text
• 93 articleS. R.S. R. S. Varadhan. On the behavior of the fundamental solution of the heat equation with variable coefficients.201967, 431--455URL: https://doi.org/10.1002/cpa.3160200210DOIback to text
• 94 bookR.Richard Vinter. Optimal control.Modern Birkhäuser ClassicsBirkhäuser Boston, Inc.2000DOIback to textback to text
• 95 articleA. G.Agust\'in Gabriel Yabo, J.-B.Jean-Baptiste Caillau, J.-L.Jean-Luc Gouzé, H.Hidde de Jong and F.Francis Mairet. Dynamical analysis and optimization of a generalized resource allocation model of microbial growth.2112022, 137-165HALDOIback to text
• 96 articleA. G.Agustín Gabriel Yabo, J.-B.Jean-Baptiste Caillau and J.-L.Jean-Luc Gouzé. Optimal bacterial resource allocation: metabolite production in continuous bioreactors.1762020, 7074--7100URL: https://hal.inria.fr/hal-02974185DOIback to text
• 97 thesisA. G.Agustin Gabriel Yabo. Optimal resource allocation in bacterial growth : theoretical study and applications to metabolite production.Université Côte d'AzurDecember 2021, URL: https://theses.hal.science/tel-03636842back to text
• 98 articleA. G.Agustín Gabriel Yabo, M.Mohab Safey El Din, J.-B.Jean-Baptiste Caillau and J.-L.Jean-Luc Gouzé. Stability analysis of a bacterial growth model through computer algebra.MathematicS In Action121September 2023, 175-189HALDOIback to text