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.

Modelling.   Our effort is directed toward efficient methods for the control of real (physical) systems, based on a model of the system to be controlled. Choosing accurate models yet simple enough to allow control design is in itself a key issue. The typical continuous-time model is of the form $\mathrm{d}x/\mathrm{d}t=f(x,u)$ 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. Modelling amounts to deciding the nature and dimension of $x$, as well as the dynamics (roughly speaking the function $f$). Connected to modeling is identification of parameters when a finite number of parameters are left free in “$f$”.

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

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  76, 47, 34, but with a lot of hard open questions. In the end, in order to actually compute these controls, ad-hoc numerical schemes have to be derived for effective computations of the optimal solutions. See more about our research program in optimal control in section 3.2.

Feedback control.   In the above two paragraphs, the control is an explicit function of time. To address in particular the stability issues (sensitivity to errors in the model or the initial conditions for example), the control has to be taken as a function of the (measured) state, or part of it. This is known as closed-loop control; it must be combined with optimal control in many real problems. On the problem of stabilization, there is longstanding research record from members of the team, in particular on the construction of “Control Lyapunov Functions”, see 62, 78. 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 54, 81.

Classification of control systems.   One may perform various classes of transformations acting on systems, or rather on models. The simpler ones come from point-to-point transformations (changes of variables) on the state and control. More intricate ones consist in embedding an extraneous dynamical system into the model. These are dynamic feedback transformations that change the dimension of the state. In most problems, choosing the proper coordinates, or the right quantities that describe a phenomenon, sheds light on a path to the solution; these proper choices may sometimes be found from an understanding of the modelled phenomenons, or it can come from the study of the geometry of the equations and the transformation acting on them. This justifies the investigations of these transformations on models for themselves. These topics are central in control theory; they are present in the team, see for instance the classification aspect in 50 or —although this research has not been active very recently— the study  75 of dynamic feedback and the so-called “flatness” property  65. Likewise, classification tools such as feedback invariants 48 are still currently in use in the team (see, for instance, 20).

3.2 Optimal Control and its Geometry

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

Cut and conjugate loci.    The cut locus —made of the points where the extremals lose optimality— is obviously crucial in optimal control, but usually out of reach (even in low dimensions), and anyway does not have an analytic characterization because it is a non-local object. Fortunately, conjugate points —where the extremals lose local optimality— can be effectively computed with high accuracy for many control systems. Elaborating on the seminal work of the Russian and French schools (see 80, 36, 35 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 45, followed by the Hampath  55 code. Over the years, these codes have allowed for the computation of conjugate loci in a wealth of situations including applications to space mechanics, quantum control, and more recently swimming at low Reynolds number. With in mind the two-dimensional analytic Riemannian framework, a heuristic approach to the global issue of determining cut points is to search for singularities of the conjugate loci; this line is however very delicate to follow on problems stemming from applications in three or more dimensions (see e.g.56 and 41). In all these situations, the fundamental object underlying the analysis is the curvature tensor. In Hamiltonian terms, one considers the dynamics of subspaces (spanned by Jacobi fields) in the Lagrangian Grassmannian 33. 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 60). The cut locus is a central object in Riemannian geometry, control and optimal transport. This is the motivation for a series of conferences on “The cut locus: A bridge over differential geometry, optimal control, and transport”, co-organized by team members and Japanese colleagues.

Riemann and Finsler geometry.    Studying the distance and minimising geodesics in Riemannian Geometry or Finsler Geometry is a particular case of optimal control, simpler because there are no differential constraints; it is studied in the team for the following two reasons. On the one hand, after some tranformations, like averaging or reduction, some more difficult optimal control problems lead to a Riemann or Finsler geometry problem. On the other hand, optimal control, mostly the Hamiltonian setting, brings a fresh viewpoint on problems in Riemann and Finsler geometry. On Riemannian ellipsoids of revolution, the optimal control approach allowed to decide on the convexity of the injectivity domain, which, associated with non-negativity of the Ma-Trudinger-Wang curvature tensor, ensures continuity of the optimal transport on the ambient Riemannian manifold 64, 63. The analysis in the oblate geometry 43 was completed in 59 in the prolate one, including a preliminary analysis of non-focal domains associated with conjugate loci. Averaging in systems coming from space mechanics control with ${\mathrm{L}}^2$-minimization yields a Riemannian metric, thoroughly computed in 42 together with its geodesic flow; in reduced dimension, its conjugate and cut loci were computed in 44 with Japanese Riemannian geometers. Averaging the same systems for minimum time yields a Finsler Metric, as noted in 40. 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 73 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 71 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 39). This link even allowed to successfully introduce concepts of sub-elliptic diffusions in computer vision algorithms thanks to sub-Riemannian geometric structures identified in mammal visual mechanisms 52.

Small controls and conservative systems, averaging.   Using averaging techniques to study small perturbations of integrable Hamiltonian systems is as old an idea as celestial mechanics. It is very subtle in the case of multiple periods but more elementary in the single period case, here it boils down to taking the average of the perturbation along each periodic orbit  37, 79. 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 40, 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 58, standard averaging (37, 79) 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 57.)

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 46, for the problem of optimizing the efficiency of the displacement of some micro-swimmers with periodic deformations, we used the sufficient optimality conditions established by R. Vinter's group 85, 68 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 85, 68. Another question relevant to locomotion is the following. Observing animals (or humans), or numerically solving the optimal control problem associated with driftless micro-swimmers for various initial and final conditions, we remark that the optimal strategies of deformation seem to be periodic, at least asymptotically for large distances. This observation is the starting point for characterizing dynamics for which some optimal solutions are periodic, and asymptotically attract other solutions as the final time grows large; this is reminiscent of the “turnpike theorem” (classical, recently applied to nonlinear situations in 83).

3.3 Software

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

4.1 Aerospace Engineering

Participants: Jean-Baptiste Caillau, Thierry Dargent, Lamberto Dell'Elce, Frank de Veld, Alesia Herasimenka, Lucas Palazollo, 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 U. Barcelona (A. Farrès, A. Jorba).

Two new directions have been taken recently. The first one is about the control of solar sails (see Section 7.8), the second one about collision avoidance for spacecrafts (see Section 7.10). Collision avoidance is becoming very important in nowadays space missions due to the growing number of various bodies (garbage, micro-satellites...) orbiting around the earth. A PhD (Frank de Veld) started in December, supported by Thales Alenia Space. Solar sailing has been actively studied for two decades and recent missions have demonstrated its interest for "zero-fuel" missions; it poses delicate control questions due to drastic constraints on the control direction. A PhD (Alesia Herasimenka) is in progress, selected by ESA for a three-year research co-sponsorship.

4.2 Optimal control of microbial cells

Participants: Jean-Baptiste Caillau, Walid Djema [BIOCORE project-team], Jean-Luc Gouzé [BIOCORE project-team], Sofya Maslovskaya [Paderborn University, Germany], Jean-Baptiste Pomet, 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). Currently, one PhD (A. Yabo) and one postdoc (S. Maslovskaya) are involved in these tasks and jointly supervised by colleagues from McTAO and Biocore teams at Sophia. Preliminary results concern the definition on extended (higher dimensional) models for the bacteria dynamics, check of second order optimality conditions on the resulting optimal control problem, and study of the turnpike phenomenon for these optimization problems.

4.3 Stability of high frequency active circuits

Participants: Laurent Baratchart [FACTAS project-team], Sébastien Fueyo [CentraleSupélec, Gif-sur-Yvette], Jean-Baptiste Pomet.

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

Some techniques do exist, see 82, based on creating some virtual perturbations and treating them as the input of a (linearized) control system to be “simulated” using the same tools. In an ongoing collaboration between McTAO and the project-team FACTAS, we work on the mathematical ground of these methods and in particular of the relation between stability and the property of the identified time-varying infinite dimensional systems. See recent developments in Section 7.4.

5.1 Awards

• Agustin Yabo (now postdoc at INRAE Montpellier) received the 2nd ATSI prize of the STIC Doctoral School.
• Alesia Herasimenka was awarded the 2022 Pierre Laffitte prize (first prize), rewarding the best second year PhD students working in innovative topics in the Sophia Antipolis region.
• Jean-Baptiste Caillau was awarded the “Défi Défense” prize during the 2022 “Assises des mathématiques” of CNRS.
• The paper 12 won the best paper award of the IFAC conference “Control Applications of Optimisation” (CAO'2022).
• Alesia Herasimenka was awarded the best poster award at the “Journées SMAI-MODE 2022”.

6.1 New software

7.1 Averaging fast-oscillating optimal control systems with two fast variables

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

Research on fast-oscillating optimal control systems is a long-standing topic in McTAO. We investigated in 2021 how trajectories of fast-oscillating control system with a single fast variable converge to their averaged counterpart 61. During 2022, further insight was gained on the averaging of systems with two fast variables. This study was motivated by the need for understanding how to generate “consistent” averaged trajectories of minimum time control systems with two fast variables (i.e., characterized by moderate drift with respect to their original counterpart). For this purpose, we leveraged on an academic example to provide evidence that existing theorems on double averaging of dynamical systems (namely, the Neishtadt theorem) cannot be directly applied to controlled systems. Hence, we developed a proper near-identity transformation of initial adjoint variables that was sufficient to prevent large drift between averaged and original trajectories in non-resonant regions. Finally, we investigated the impact of resonance crossing on the dynamics of adjoint variables and proposed a way to predict this behavior via resonant averaged forms.

These considerations are of use to generate reliable initial guesses for indirect techniques to solve two-point boundary value problems. This work was presented at the IFAC conference “Control Applications of Optimisation” (CAO'2022) 12.

7.2 Controllability of fast-oscillating system with control constraints

Participants: Jean-Baptiste Caillau, Lamberto Dell'Elce, Alesia Herasimenka, Jean-Baptiste Pomet.

Control of solar sails requires a comprehensive understanding of controllability, that is not a given because forces (control) are drastically constrained. The paper 7 (see Section 7.8) describes effective methods to test controllability, depending on the current orbit and the optical properties of the sail. It is based on a controllability study for general affine control systems where the drift vector fields yields periodic solutions and the controls are constrained in a set that is not a neighborhood of zero. We obtained sufficient conditions for controllability, both local and global, presented in 26, 15; they are formulated in terms of pushforwards along the flow of the drift, rather than in terms of Lie brackets, and take into account not only the vector fields associated to the system but also the definition of the control set; it turns out that they amount to local controllability of a time-varying linear approximation with constrained controls. These conditions are harder to check in practice than these formulated in terms of the rank of a family of vector fields, and the purpose of 7 is, in part, to give a method, based on convex optimization to assess these conditions in the case of a solar sail. (The approach leverages fine properties of trigonometric polynomials as well as Nesterov's technique of sum of squares relaxation.)

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

Participants: Laetitia Giraldi [CALISTO project-team], Pierre Lissy [Université Paris Dauphine, Paris], Clément Moreau [Kyoto University, Japan], Jean-Baptiste Pomet.

Motivated by the study of planar articulated magnetically actuated swimmers at low Reynolds number (see C. Moreau's PhD  74), around the straight configuration, where all magnetic moments are aligned, we studied local controllability (STLC) for a more general, but still particular, class of systems with two controls, around an equilibrium where the two control vector fields are co-linear. The manuscript 29, now provisionally accepted for publication in Control, Optimization and Calculus of Variations, gives novel necessary conditions for STLC of these systems, based on Chen-Fliess expansions of solutions, in the spirit of 72.

7.4 Stability of linear time-varying time-delay systems

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

Sébastien Fueyo's doctoral work (66, PhD defended) on testing the stability of amplifiers in a CAD (computer assisted design) process, described also in Section 4.3, led us to revisiting stability of time-varying systems/networks of 1-D hyperbolic PDEs, or linear "delay difference systems" (i.e. of the type $z\left(t\right)=A_1\left(t\right)z(t-{\tau }_1)+\cdots +A_N\left(t\right)z(t-{\tau }_N)$). In the time-invariant case, a well known necessary and sufficient condition for stability is due to Hale and Henry  70, 69; it gives, in a sense, a final answer to the question, but it is not so easy to check explicitly this criteria, and there is still a vast literature on more specific sufficient conditions. The article 18, submitted to Integral Equations and Operator Theory, proposes a generalization to the periodic time-varying case. In 10 we detail how this and a former sufficient condition, more restrictive but more explicit, stated in 38, apply to various situations.

7.5 State estimation and stabilisation of non-uniformly observable systems

Participants: Ludovic Sacchelli, Lucas Brivadis [CentraleSupélec, Gif-sur-Yvette], Jean-Paul Gauthier [Université de Toulon], Jean-Luc Gouzé [BIOCORE project-team], Théo Rolin.

As was discussed in the research program, stabilization of the state of a system by means of a feedback control is a classical and important problem in control theory. When only part of the system is known, a usual strategy is to rely on a dynamic algorithm, known as an observer, in order to provide an estimate of the state that can be fed to the controller. Designing a stable closed-loop based on an observer requires that some necessary information on the state can be accessed through this partial measurement. Critically, for nonlinear systems, whether or not it is possible to reliably estimate the state can depend on the control. This fact is known as non-uniform observability and is a root issue for observer design. In a collaboration between L. Sacchelli and J.-L. Gouzé, we supervised T. Rolin's M1 internship on the topic of robust state-estimation methods for non-uniformly observable systems. The goal of the internship was to analyze and reproduce the method proposed in 77 for monitoring batch bioreactors that asymptotically lose observability.

Regarding output feedback control, if singular controls exist for observability, there is no clear definitive answer as to how to achieve stabilization. 24 (now in press) reviews some strategies that showed to be efficient in tackling the difficulties posed by non-uniform observability, and explores the genericity side of the matter, including a proof that this critical situation can be generic in some key classes of systems. 25 explores the issue through the point of view of hybrid systems, a new framing for the group. In the paper, we propose a method to monitor observability of the system online. We use this method as the backbone of a dynamically switching Kalman-type observer and feedback to achieve stabilization of bilinear systems. Allowing Kalman filters in that context was a long term goal but had remained an open problem. This new approach will be explored more thoroughly in future works.

7.6 Geometry and optimal control

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

This work was started in the frame of Wembe PhD thesis on time-minimum problems in navigation in presence of a current and a vortex, defended at ENSEEIHT Toulouse, November 2021.

Our contribution was to revisit and generalize the Navigation Problem set by Carathéodory and Zermelo of a ship navigating on a river with a linear current and aiming to reach the opposite shore in minimum time. This work is motivated by the displacement of particles in a two dimensional fluid, in presence of a vortex (initially, a singularity in the Helhmoltz-Kirchhoff equations) inducing a strong current hampering local controllability. To define a minimum time Zermelo navigation problem, we consider the particle as the ship of the navigation problem and the control is defined as the heading angle of the ship axis. It turns out that the historical problem and our recent vortex study are two examples of the general case of Zermelo navigation problems on surfaces of revolution that we tackled this year. Our main contributions in this setting are multiple. In 3, 4, we analyze the role of abnormal geodesics in the problem, in particular in relation with cusp singularities of  the geodesics and non continuity properties of the value function. In 19 (currently in revision), we relate the existence, in the geodesics dynamic, of separatrices interpreted as Reeb components, with the Morse–Reeb classification of the geodesics. Furthermore, we provide explicit computations of the conjugate and cut loci in case studies such as the averaged  Kepler case in space mechanics 19.

7.7 Sub-Riemannian diffusions

Participants: Ludovic Sacchelli, Robert Neel [Lehigh University, USA].

A version of the heat equation on sub-Riemannian manifolds can be obtained by considering a generalized Laplace-Beltrami operator taking into account the non-holonomic constraints of the control structure. It has been known since Varadhan’s formula for Riemannian manifolds 84 that the distance between two points can be recovered by taking the log of the heat kernel and pushing time to 0. Combining tools from the study of Brownian motions on manifolds and geometric analysis of sub-Riemannian length minimizing curves, these type of asymptotics were successively extended to more and more general sub-Riemannian structures. In 30, we expanded on previously developed techniques to study bounds on logarithmic derivatives of the heat kernel. They have been long known on compact Riemannian manifolds, and were recently extended to general complete Riemannian manifolds. We further extend these bounds to incomplete Riemannan manifolds under the least restrictive condition available. We also discuss the difficulties that arise when we consider the sub-Riemannian case.

7.8 Control of solar sails

Participants: Jean-Baptiste Caillau, Tommy Calendini, Paul Chambroux, Lamberto Dell'Elce, Alesia Herasimenka, Jean-Baptiste Pomet.

Solar sails offer a propellantless solution to perform interplanetary transfers, planet escapes, and de-orbiting maneuvers by leveraging on solar radiation pressure (SRP). Although very few solar-sail missions have been launched to this date, the possibility to use SRP as an inexhaustible source of propulsion has attracted the interest of researchers in the last decades. Various contributions on the control of solar sails were obtained in the framework of A. Herasimenka's thesis.

We proposed an efficient test to conclude controllability of non-ideal solar sails in planet-centered orbits. This requirement is aimed at assessing whether a non-ideal solar sail with given optical parameters is capable of decreasing or increasing all possible functions of the Keplerian integrals of motion over an orbital period. In other words, we verify if a solar sail can change the orbit in any desirable way given its optical properties. This requirement is formulated as a worst-case convex optimization problem characterized by a finite number of design variables and a two-parameter family of inequality constraints. By leveraging on the formalism of squared functional systems and on the peculiar form of Gauss variational equations, we recast the semi-infinite problem into a finite-dimensional convex programming problem with a finite number of linear matrix inequalities (LMI) and an unique well-defined solution 7.

The internship of P. Chambroux was focused on a feasibility study on the use of solar sail to carry out observations of the solar corona by occulting the Sun's disk with the Earth. This can be achieved by placing the sail on an orbit in proximity of the Lagrangian point $L_2$ of the Earth-Sun system. The circular restricted three-body problem (Earth-Sun system, where Earth serves as occulting body) was used to model the motion of the satellite. Trajectory design was formulated as a periodic optimal control problem aimed at maximizing the duty cycle of the observations, defined as the fraction of orbital period devoted to observations. Charge of the batteries was part of the state variables, and its rate of change is function of the position of the satellite and increases from zero to its maximum value across the penumbra region. Pontryagin maximum principle (PMP) was then applied to deduce necessary conditions of optimality for the aforementioned cost function (duty cycle) and dynamical model. Numerical solution was achieved by using indirect techniques.

Finally, optimal control of solar sails in planet-centered orbits was investigated in the internship of T. Calendini. Maximization of the Hamiltonian requires the solution of a nonlinear equation (that is well defined thanks to the implicit function theorem). This was obtained numerically by augmenting the Hamiltonian system with an auxiliary variable representing the unknown maximizing control and including the associated equation in the constraints (rather that solving numerically for the control at each step).

7.9 Dynamical systems theory applied to space mechanics

Participants: Nicola Baresi [University of Surrey, UK], Lamberto Dell'Elce, Pini Gurfil [Technion, Israel], Gianpaolo Izzo [Technion, Israel], Aaron Rosengren [University of California San Diego, US].

Two applications of dynamical systems theory to space mechanics were tackled.

First, motivated by upcoming missions towards remote planetary moons that will fly towards chaotic dynamical environments which are significantly perturbed by the oblateness of the host planet, we introduced a new time-periodic set of equations of motion that is based on the analytical solution of the zonal equatorial problem. Such a system, which referred to as the Zonal Hill Problem, remains populated by resonant periodic orbits and families of two-dimensional quasi-periodic invariant tori that we calculated by means of homotopy continuation procedures. The resulting periodic and quasi-periodic trajectories were investigated for the trajectory design of future planetary moons explorers 2.

Second, we proposed a new analytical solution for the first-order, short-periodic perturbations due to planetary oblateness 8. Our approach was based on the Milankovitch vectorial elements and is free of all the mathematical singularities. Being a non-canonical set, our derivation followed the scheme used by Kozai in his oblateness solution. We adopted the mean longitude as the fast variable and presented a compact power-series solution in eccentricity for its short-periodic perturbations that relies on Hansen’s coefficients. We also used a numerical averaging algorithm based on the fast-Fourier transform to further validate our new mean-to-osculating and inverse transformations. This technique constitutes a new approach for deriving short-periodic corrections and exhibits performance that are comparable to other existing and well-established theories, with the advantage that it can be potentially extended to modeling non-conservative orbit perturbations.

7.10 Collision avoidance

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

On the one hand, we have been defining a PhD topic with Thales Alenia Space on collision avoidance during low thrust orbit transfer, that is now funded by Thales Alenia Space together with a grant from Région Provence Alpes Côte d'Azur. The PhD candidate is Frank de Veld, his PhD started in December, 2022.

On the other hand, Lucas Palazzolo's internship (June-July plus a part-time project in the winter-spring) was about “Control Barrier functions” (CBF), after 53 (not chosen to be a reference, rather a current conference paper to study); he conducted numerical studies to improve techniques in that paper or apply them to specific test cases. This is also aimed at safe collision avoidance, but rather in low relative velocity; it is not directly linked to the PhD topic mentioned just above.

7.11 Attitude determination and control system of NiceCUBE

Participants: Lamberto Dell'Elce, Azouaou Ouyoucef, Jean-Baptiste Pomet.

Université Côte d'Azur is planning to build its first single-unit CubeSat (i.e., a recent standard for satellites with mass of the order of few kilograms and volume of few dm${}^3$), named NiceCube. The objective of the mission is to establish an optical link between a ground station in Nice and the satellite by enlightening it with a laser and by moduling the reflected signal with an occulter onboard NiceCube. To this end, the face of the satellite equipped with retroreflectors should be oriented towards the ground station so that the reflected signal can be received. McTAO part of UCA and is in charge of the attitude determination and control system of NiceCube.

The mission is currently in phase A (i.e., feasibility study). This year, we completed the development of the attitude determination component of the numerical simulator that we implemented to simulate the mission.

7.12 Optimal allocation of resources in bacteria

Participants: Agustín Yabo, Jean-Baptiste Caillau, Jean-Luc Gouzé [BIOCORE project-team], Mohab Safey El Din [Sorbonne Université].

In the framework of the ANR Maximic, we carry on the study of self-replicator models. These models describe the allocation of resources inside the bacteria: the substrate is used to produce precursors that, in turn, can be employed either to produce genetic machinery (and increase the biomass) or metabolic machinery (that will further catalyse the transformation of substrate into precursors). To this internal control and external action that aims, after some genetic engineering on the bacteria (to create a strain that reacts to light stimuli), at producing a new metabolite of interest. Then, while the behaviour of the untouched bacteria tends to be very well mimicked by biomass maximization strategies, maximizing the production of the metabolite of interest induces new biological strategies. This kind of model (and refinements) are studied in 67 and 9. Key properties of the system are: (i) the Fuller phenomenon as connection between bang and singular arcs requires an infinite number of switchings in finite time; (ii) the turnpike phenomenon. Indeed, for large fixed final times, trajectories of the system are essentially singular and close to the optimal (wrt. a constant static control) equilibrium which is a hyperbolic fixed point of the singular flow. See ct gallery for an example, and the recently defended PhD thesis of A. Yago 86 for a discussion of these results. In collaboration with M. Safey El Din, stability properties of the system have been established thanks to a consistency check of a system of polynomial inequalities. (To appear in MathS In Action.)

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

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

This topic started in McTAO in 2017 with a collaboration between B. Bonnard and T. Bakir (ImVia-UBFC), and J. Rouot in the frame of Cifre PhD thesis of S. Gayrard with Segula Technologies.

The problem of control of muscular force is posed in terms of optimization of the train pulses of a Functional Electro-Stimulation (FES) signal to produce the muscular contraction. Based on preliminary experimental studies, the dynamical model that was chosen for muscular control is known as Ding et al. force-fatigue model in the literature. It is a refinement of the historical Hill model (Medicine Nobel Prize in 1922) that takes into account the variations of the fatigue variable. From the control methodology point of view, this required some developments on optimal control for sample control systems. This is by itself already a rich topic. In 2020, this project took the industrial transfer direction with a Cifre PhD funding in partnership with Segula Technologies (see Section 8) whose goal is the design of a smart electrostimulator for force reinforcement or rehabilitation in the framework of S. Gayrard's PhD.

This year, our contribution to the project was double. From the theoretical point of view, we have derived a finite dimensional approximation of the forced dynamics based on the Ding et al. model to provide fast optimizing schemes aiming to track a reference force or maximize the force (see 1). On the other hand, S. Gayrard is also finalizing a prototype of the smart electrostimulator, which is a major objective of the collaboration with Segula. It is currently being tested on a cohort of human subjects. See S. Gayrard 2022 Preprint for a succint presentation of this work at ECC 2023 Conference 28.

7.14 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 lead 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 20, 23, 21, 22. In relation with previous work regarding the optimal control of chemical networks, the time minimal synthesis has been described near the terminal manifold up to codimension two situations in the jets spaces. We have compared both perma­nent controls and sampled-data control in relation with numerical issues. Finally, we have obtained feedback invariants to classify the  geodesic dynamics associated to rays of abnormal geodesics which are related to shifted equilibria of the free Lotka-dynamics and which can be calculated using only linear computations.

See Section 9.1.1 for an upcomming project on the topic with C. Silva.

7.15 ct: control toolbox

Participants: Jean-Baptiste Caillau, Olivier Cots, Pierre Martinon [CAGE project-team, on leave].

The ADT ct: control toolbox  had one sprint in 2022. The focus was on to initiate 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.)

Intelligent muscle electrostimulator, Segula Technologies

Participants: Toufik Bakir [Université de Bourgogne Franche Comté, Dijon], Bernard Bonnard, Sandrine Gayrard.

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

This is completed by an additionnal collaboration contract between Segula and SAYENS (representing Université de Bourgogne), aiming at constructing the prototype of the smart electrostimulator.

• PI: Bernard Bonnard and Toufik Bakir (ImViA, Université de Bourgogne Franche Comté, Dijon).
• Period: 2020–2023.

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

Participants: Jean-Baptiste Caillau, Lamberto Dell'Elce, Alesia Herasimenka, Jean-Baptiste Pomet.

Three year contract starting in 2021 between the team and the European Space agency. Its purpose is to support the environment of Alesia Herasimenka's PhD on this topic.

• Partners: McTAO and ESA.
• Total amount: 24k€.
• Period: 2021–2024.
• Inria reference: 16016.

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.

Thales Alenia Space co-funded the thesis of Frank de Veld named “Méthodes de Contrôle pour l’évitement de collisions entre satellites”.

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

9.1 International initiatives

9.2 National initiatives

9.3 Regional initiatives

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

10.1 Promoting scientific activities

10.2 Teaching - Supervision - Juries

10.3 Popularization

11.1 Publications of the year

11.2 Cited publications

• 33 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--77
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• 34 bookA.Andrei Agrachev and Y. L.Yuri L. Sachkov. Control theory from the geometric viewpoint.87Encyclopaedia of Mathematical SciencesControl Theory and Optimization, IIBerlinSpringer-Verlag2004
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• 35 articleA. A.Andrey A. Agrachev and A. V.Andrey V. Sarychev. Abnormal sub-Riemannian geodesics: Morse index and rigidity.Ann. Inst. H. Poincarré, Anal. non-linéaire1361996, 635-690URL: http://www.numdam.org/item/AIHPC_1996__13_6_635_0/
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• 36 articleA. A.A. A. Agrachev and A. V.A. V. Sarychev. Strong minimality of abnormal geodesics for 2-distributions.J. Dynam. Control Systems121995, 139--176
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• 37 bookV. I.Vladimir I. Arnold. Mathematical methods of classical mechanics.60Graduate Texts in MathematicsNew YorkSpringer-Verlag1989
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• 38 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.SIAM J. Math. Anal.5322021, 1831--1856
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• 39 articleD.Davide Barilari, U.Ugo Boscain and R. W.Robert W. Neel. Small-time heat kernel asymptotics at the sub-Riemannian cut locus.Journal of Differential Geometry923November 2012, URL: https://doi.org/10.4310/jdg/1354110195
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• 40 articleA.Alex Bombrun and J.-B.Jean-Baptiste Pomet. The averaged control system of fast oscillating control systems.SIAM J. Control Optim.5132013, 2280-2305
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• 41 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--208
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• 42 articleB.Bernard Bonnard and J.-B.Jean-Baptiste Caillau. Geodesic flow of the averaged controlled Kepler equation.Forum Mathematicum215September 2009, 797–814
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• 43 articleB.Bernard Bonnard, J.-B.Jean-Baptiste Caillau and L.Ludovic Rifford. Convexity of injectivity domains on the ellipsoid of revolution: the oblate case.C. R. Math. Acad. Sci. Paris34823-242010, 1315--1318URL: https://hal.archives-ouvertes.fr/hal-00545768
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• 44 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.Ann. Inst. H. Poincaré Anal. Non Linéaire2642009, 1081--1098
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• 45 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.ESAIM Control Optim. Calc. Var.1322007, 207--236
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• 46 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 preprint
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• 47 bookB.Bernard Bonnard and M.Monique Chyba. Singular trajectories and their role in control theory.40Mathématiques & ApplicationsBerlinSpringer-Verlag2003
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• 48 articleB.B. Bonnard. Feedback equivalence for nonlinear systems and the time optimal control problem.SIAM J. Control Optim.2961991, 1300--1321URL: https://doi.org/10.1137/0329067
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• 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.Acta Applicandae Math.13522015, 47-80
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• 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.Math. Control Relat. Fields (MCRF)342013, 397-432
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• 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.Forum Math.51993, 111--159
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• 52 articleU.Ugo Boscain, J.Jean Duplaix, J.-P.Jean-Paul Gauthier and F.Francesco Rossi. Anthropomorphic Image Reconstruction via Hypoelliptic Diffusion.SIAM Journal on Control and Optimization503January 2012, 1309--1336
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• 53 inproceedingsJ.Joseph Breeden and D.Dimitra Panagou. High Relative Degree Control Barrier Functions Under Input Constraints.2021 60th IEEE Conference on Decision and Control (CDC)2021, 6119-6124
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• 54 articleL.Lucas Brivadis, J.-P.Jean-Paul Gauthier, L.Ludovic Sacchelli and U.Ulysse Serres. Avoiding observability singularities in output feedback bilinear systems.SIAM Journal on Control and Optimization593May 2021, 1759--1780
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• 55 articleJ.-B.Jean-Baptiste Caillau, O.O. Cots and J.J. Gergaud. Differential pathfollowing for regular optimal control problems.Optim. Methods Softw.2722012, 177--196
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• 56 articleJ.-B.Jean-Baptiste Caillau and B.B. Daoud. Minimum time control of the restricted three-body problem.SIAM J. Control Optim.5062012, 3178--3202
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• 57 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, 209-302
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• 58 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 preprint
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• 59 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, 73-85
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• 60 articleZ.Zheng Chen, J.-B.Jean-Baptiste Caillau and Y.Yacine Chitour. $L^1$-minimization for mechanical systems.SIAM J. Control Optim.543May 2016, 1245-1265
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• 61 inproceedingsL.Lamberto Dell'Elce, J.-B.Jean-Baptiste Caillau and J.-B.Jean-Baptiste Pomet. On the convergence of time-optimal maneuvers of fast-oscillating control systems.ECC 2021 - European Control Conference2021, 2008-2013
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• 62 articleL.L. Faubourg and J.-B.Jean-Baptiste Pomet. Control Lyapunov functions for homogeneous ``Jurdjevic-Quinn'' systems.ESAIM Control Optim. Calc. Var.52000, 293-311
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• 63 articleA.Alessio Figalli, T.Thomas Gallouët and L.Ludovic Rifford. On the convexity of injectivity domains on nonfocal manifolds.SIAM J. Math. Anal.4722015, 969--1000
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• 64 articleA.A. Figalli, L.L. Rifford and C.C. Villani. Necessary and sufficient conditions for continuity of optimal transport maps on Riemannian manifolds.Tohoku Math. J.6342011, 855-876URL: http://hal.inria.fr/hal-00923320v1
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• 65 articleM.Michel Fliess, J.Jean Lévine, P.Philippe Martin and P.Pierre Rouchon. Flatness and Defect of Nonlinear Systems: Introductory Theory and Examples.Internat. J. Control611995, 1327--1361
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• 66 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
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• 67 articleA.Agustín Gabriel Yabo, J.-B.Jean-Baptiste Caillau and J.-L.Jean-Luc Gouzé. Optimal bacterial resource allocation: metabolite production in continuous bioreactors.Mathematical Biosciences and Engineering1762020, 7074-7100
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• 68 articleC.C. Gavriel and R.R. Vinter. Second order sufficient conditions for optimal control problems with non-unique minimizers: an abstract framework.Appl. Math. Optim.7032014, 411--442
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• 69 bookJ. K.Jack K. Hale and S. M.Sjoerd M. Verduyn Lunel. Introduction to functional-differential equations.99Applied Mathematical SciencesSpringer-Verlag, New York1993
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• 70 articleD.Daniel Henry. Linear autonomous neutral functional differential equations.J. Differential Equations151974, 106--128
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• 71 articleL.Lars Hörmander. Hypoelliptic second order differential equations.Acta Mathematica11901967, 147--171URL: https://doi.org/10.1007/bf02392081
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• 72 incollectionM.Matthias Kawski. High-order small-time local controllability.Nonlinear controllability and optimal control133Monogr. Textbooks Pure Appl. Math.Dekker, New York1990, 431--467
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• 73 bookR.Richard Montgomery. A tour of subriemannian geometries, their geodesics and applications.91Mathematical Surveys and MonographsAmerican Mathematical Society, Providence, RI2002
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• 74 thesisC.Clément Moreau. Contrôlabilité en dimension finie et infinie et applications à des systèmes non linéaires issus du vivant.Université Côte d'AzurJune 2020
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• 75 articleJ.-B.Jean-Baptiste Pomet. A necessary condition for dynamic equivalence.SIAM J. on Control and Optimization482009, 925-940URL: http://hal.inria.fr/inria-00277531
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• 76 bookL. S.L. S. Pontryagin, V. G.V. G. Boltjanskiĭ, R. V.R. V. Gamkrelidze and E.E. Mitchenko. Théorie mathématique des processus optimaux.MoscouEditions MIR1974
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• 77 articleA.Alain Rapaport and D. D.Denis D. Dochain. A robust asymptotic observer for systems that converge to unobservable states. A batch reactor case study.IEEE Transactions on Automatic Control656June 2020, 2693-2699
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• 78 articleL.Ludovic Rifford. Stratified semiconcave control-Lyapunov functions and the stabilization problem.Ann. Inst. H. Poincaré Anal. Non Linéaire2232005, 343--384
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• 79 bookJ. A.Jan A. Sanders and F.F. Verhulst. Averaging Methods in Nonlinear Dynamical Systems.56Applied Mathematical SciencesSpringer-Verlag1985
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• 80 articleA. V.A. V. Sarychev. The index of second variation of a control system.Mat. Sb.411982, 338--401
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• 81 articleH.H. Shim and A.A.R. Teel. Asymptotic controllability and observability imply semiglobal practical asymptotic stabilizability by sampled-data output feedback.Automatica3932003, 441 - 454
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• 82 bookA.Almudena Suarez. Analysis and Design of Autonomous Microwave Circuits.Wiley-IEEE Press2009
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• 83 articleE.Emmanuel Trélat and E.Enrique Zuazua. The turnpike property in finite-dimensional nonlinear optimal control.J. Differential Equations25812015, 81--114
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• 84 articleS. R.S. R. S. Varadhan. On the behavior of the fundamental solution of the heat equation with variable coefficients.Comm. Pure Appl. Math.201967, 431--455URL: https://doi.org/10.1002/cpa.3160200210
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• 85 bookR.Richard Vinter. Optimal control.Modern Birkhäuser ClassicsBirkhäuser Boston, Inc.2000
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• 86 phdthesisA. G.Agustin Gabriel Yabo. Optimal resource allocation in bacterial growth : theoretical study and applications to metabolite production.Université Côte d'AzurDecember 2021
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