2025‌​‌Activity reportTeamMCTAO​​

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

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 $\mathrm{x‌}$, as well as‌​‌ the dynamics (roughly speaking​​ the function $f$).​​​‌ We are in general‌ not involved in the‌​‌ modeling phase, but sometimes​​ in identification, where the​​​‌ structure of “$\mathrm{f‌}$” is known but‌​‌ it may contain a​​ certain number of parameters​​​‌ that are not a‌ priori known, and are‌​‌ to be identified from​​ measurements.

Controllability, path planning.​​​‌   Controllability is a property‌ of a control system‌​‌ (in fact of a​​ model) that two states,​​​‌ say $x_{\text{init.}}$ and‌ $x_{\text{final}}$, in‌​‌ the state space can​​ be connected by a​​​‌ trajectory generated by some‌ control $t\mapsto \mathrm{u‌‌}\left(t\right)$ on​​ a time-interval. 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 83, 40​‌, 27, 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 66, 84​‌. It may happen​​ that only part of​​​‌ the state is measured​ at any one time,​‌ because of physical or​​ engineering constraints, so that​​​‌ the control cannot be​ assigned as a function​‌ of the whole state.​​ 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​​​‌ 51, 87.​

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 (new state and​‌ control $(z,v)$ given as​​​‌ a smooth function of​ $(x,\mathrm{u‌})$ in a diffeomorphic​​ manner yields a system​​​‌ $\mathrm{d}z/\mathrm{d}t=g(‌z,v)$ from the above control​​​‌ system). 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 43 or​​ —although this research has​​​‌ not been active very‌ recently— the study  82‌​‌ of dynamic feedback and​​ the so-called “flatness” property​​​‌  69. Likewise, classification‌ tools such as feedback‌​‌ invariants 41 are still​​ currently in use in​​​‌ the team (see, for‌ instance, 45).

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 86‌​‌, 29, 28​​ and 44 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​​​‌ 38, followed by‌ the HamPath 54 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.57 and 34​​​‌). 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​​​‌ 26. This point‌ of view withstands generalizations‌​‌ far beyond the smooth​​​‌ case. In ${\mathrm{L}}^{\mathrm{1}}$-minimization, for instance, discontinuous​‌ curves in the Grassmannian​​ have to be considered​​​‌ (instantaneous rotations of Lagrangian​ subspaces still obeying symplectic​‌ rules 64). 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 68​, 67. The​‌ analysis in the oblate​​ geometry 36 was completed​​​‌ in 61 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 ${\mathrm{L}}^{\mathrm{2}}$-minimization yields a Riemannian​​​‌ metric, thoroughly computed in​ 35 together with its​‌ geodesic flow. In reduced​​ dimension, its conjugate and​​​‌ cut loci were computed​ in 37 with Japanese​‌ Riemannian geometers. Averaging the​​ same systems for minimum​​​‌ time yields a Finsler​ Metric, as noted in​‌ 33. In 42​​, 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 80 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 75​​ 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 32). 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 47.

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​​​‌  30, 85.‌ 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 ${\mathrm{J‌}}_2$ 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 33,‌ 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 60,​​ standard averaging (30​​​‌, 85) 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 ${\mathrm{C}}^1$-regularity on the​​​‌ value function when the‌ small parameter goes to‌​‌ zero as swallowtail-like singularities​​ due to the structure​​​‌ of local minima in‌ the problem are expected‌​‌ (a preliminary analysis has​​ been made in 58​​​‌).

Optimality of periodic‌ solutions/periodic controls.    When seeking‌​‌ to minimize a cost​​ with the 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 39​​​‌, 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​​​‌ 90, 71 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 90,​​ 71. 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 88).​‌

In a completely different​​ setting, namely the spectral​​​‌ analysis of Sturm-Liouville operators,​ periodic control has also​‌ been considered in the​​ recent work 53.​​​‌ 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​​ 59. Such a​​ combined approach has been​​​‌ for instance used between‌ McTAO, the former COMMANDS‌​‌ team (Inria Saclay), and​​ CNRS team APO (Université​​​‌ Toulouse, CNRS, ENSEEIHT) for‌ the optimization of contrast‌​‌ in medical imaging (MRI),​​ and fuel-effective trajectories for​​​‌ airplanes. This combination of‌ direct and indirect methods‌​‌ has a lot of​​ interest to solve optimal​​​‌ control problems that contain‌ state or control constraints.‌​‌ In the collaborations mentioned​​ above, the interfacing between​​​‌ the two solvers BOCOP‌ and HamPath were done‌​‌ manually by ad hoc​​python or matlab layers.​​​‌ In collaboration with COMMANDS‌ and colleagues from ENSEEIHT,‌​‌ McTAO leads the ct:​​ control toolbox project whose​​​‌ goal is to interoperate‌ these solvers using a‌​‌ high level common interface.​​ The project is an​​​‌ Inria Sophia ADT1‌ (2019-2023) 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.

3.4‌ Algorithmic differentiation (AD)

Although‌​‌ relevant to the previous​​ section, this topic is​​​‌ only in the research‌ program of the team‌​‌ per se since the​​ end of February, 2025,​​​‌ when Jean-Luc Bouchot and‌ Laurent Hascoët joined the‌​‌ team, due to the​​ ending of the ECUADOR​​​‌ team (in fact, Jean-Luc‌ Bouchot will be in‌​‌ the GAMMA-O team as​​ of 2026).

Algorithmic differentiation​​​‌ (AD), also known as‌ Automatic Differentiation or Differentiable‌​‌ Programming, and the related​​ method of backpropagation, have​​​‌ received growing interest for‌ their many uses in‌​‌ optimization, uncertainty quantification, and​​ machine learning. Given a​​​‌ program that implements some‌ mathematical function, AD creates‌​‌ a new program that​​ computes derivatives of that​​​‌ function. While many approaches‌ and tools have been‌​‌ developed to this end,​​ we focus on reverse​​​‌ AD by source-transformation (ST-AD)‌ which, compared to other‌​‌ approaches, often results in​​ the highest efficiency to​​​‌ obtain gradients. ST-AD transforms‌ the source code that‌​‌ computes the function into​​ a new source code​​​‌ in the same language,‌ that computes the derivatives.‌​‌ Reverse AD, specifically, computes​​ the gradients in the​​​‌ reverse of the original‌ computation order. As a‌​‌ consequence, the complexity of​​ derivative computation becomes independent​​​‌ of the number of‌ inputs. This is a‌​‌ key advantage when computing​​ the gradient of simulations​​​‌ that go from millions‌ of inputs (or more)‌​‌ to a handful of​​ outputs of interest.

Providing​​​‌ a ST-AD tool is‌ a demanding development, comparable‌​‌ to developing a compiler.​​ The required techniques involve​​​‌ code analysis, mostly static‌ data-flow analysis and semantic‌​‌ analysis. The AD tool​​ Tapenade that we distribute​​​‌ results from a development‌ of more than 20‌​‌ person-year.

Application of AD​​ to large codes is​​​‌ always a challenge, and‌ often requires interaction between‌​‌ the developers of the​​ application code and those​​​‌ of the AD tool.‌ Even if the AD‌​‌ tools progressively become more​​ reliable, interaction and new​​​‌ development are all the‌ more necessary when performance‌​‌ is sought. All this​​​‌ is a source of​ new research questions for​‌ AD.

The evolution of​​ application languages is also​​​‌ a source of new​ questions. While AD of​‌ Fortran or C has​​ been studied for a​​​‌ long time, the novel​ idioms found in new​‌ popular languages (Python, Julia​​ ...), especially their dynamic​​​‌ or interactive aspects, pose​ plenty of challenging questions​‌ to the model of​​ AD, and to the​​​‌ related AD tools.

4.1 Aerospace​‌ Engineering

Participants: Jean-Baptiste Caillau​​, Thierry Dargent,​​​‌ Lamberto Dell'Elce, Frank​ de Veld, 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 ${\mathrm{L}}^{\mathrm{1}}$-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, the​​​‌ second one about collision​ avoidance for spacecrafts (see​‌ Section 7.12). 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),​‌ defended in December 11​​, 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 Quantum control

Participants: Charles‌​‌ Babin [Univ. Bourgogne Europe,​​ CNRS, ICB], Ivan​​​‌ Beschastnyi, Jean-Baptiste Caillau‌, Lamberto Dell'Elce,‌​‌ Jean-Baptiste Pomet, Ludovic​​ Sacchelli, Dominique Sugny​​​‌ [Univ. Bourgogne Europe, CNRS,‌ ICB], David Tinoco‌​‌.

Quantum systems are​​ increasingly used in various​​​‌ applications, from metrology to‌ computation. However, their manipulation‌​‌ is very challenging due​​ to the effects of​​​‌ quantum theory itself as‌ well as interactions with‌​‌ the environment. Therefore, creating​​ efficient and robust control​​​‌ strategies is one of‌ the central tasks of‌​‌ quantum engineering.

At MCTAO​​ we are interested in​​​‌ two research directions. The‌ first is the development‌​‌ of control tools for​​ general quantum control systems.​​​‌ This includes optimal control,‌ adiabatic control, qualitative and‌​‌ quantitative controllability, and the​​ identification of quantum systems.​​​‌ Our goal is to‌ develop a broad range‌​‌ of methods that can​​ be applied to general​​​‌ classes of quantum systems‌ without directly specifying the‌​‌ type of technology used.​​

The second direction involves​​​‌ working with experimental physicists‌ on concrete problems, where‌​‌ the specific nature of​​ a physical system suggests​​​‌ additional tools. Currently, we‌ have established a collaboration‌​‌ with physicists (C. Babin,​​ D. Sugny) from the​​​‌ Laboratoire Interdisciplinaire Carnot de‌ Bourgogne in Dijon on‌​‌ a project related to​​ the control of Nitrogen-Vacancy​​​‌ centers. Our PhD student,‌ David Tinoco, is working‌​‌ on mathematical aspects relevant​​ to the control of​​​‌ such systems. At INPHYNI‌ (Institut de Physique‌​‌ de Nice), we​​ are currently working with​​​‌ J. Etesse on control‌ strategies for quantum memory.‌​‌

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 50. 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 46​​, 48.

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 63. 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​

• Jean-Baptiste Caillau received a​‌ 2025 Prix d'Excellence from​​ Université Côte d'Azur

6.1 Latest​‌ software developments

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 a system​‌ by feedback is a​​ fundamental problem in control​​​‌ theory. When only partial​ measurements are available, output​‌ feedback control relies on​​ an observer to reconstruct​​​‌ the state. In nonlinear​ systems, however, state estimation​‌ may depend on the​​ control itself, a phenomenon​​​‌ known as non-uniform observability,​ which can severely hinder​‌ stabilization. In particular, the​​ presence of controls that​​​‌ are singular for observability​ creates intrinsic obstructions to​‌ output feedback stabilization, for​​ which no general solution​​​‌ is known. It has​ long been recognized that​‌ purely state-based feedback laws​​ may fail in this​​​‌ context, motivating the introduction​ of time-varying or hybrid​‌ mechanisms. One approach consists​​ in perturbing a state-feedback​​​‌ design to restore observability,​ a direction the group​‌ has been exploring through​​ hybrid control techniques.

In​​​‌ 1, published this​ year, we introduced a​‌ new method based on​​ control templates. The feedback​​​‌ is replaced by a​ piecewise-defined approximation, where each​‌ segment follows a control​​ known to ensure observability,​​​‌ generalizing the classical sample-and-hold​ paradigm. Building on recent​‌ results by Lin et​​ al. on hybrid output​​​‌ feedback under observability for​ all inputs (see, e.g.​‌ 78), 1 extends​​ these guarantees to analytic​​​‌ systems observable under a​ null input, and proves​‌ that control templates are​​ generic among analytic controls.​​​‌

7.2 Singular arcs in​ $L^1$-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 52 or​​ in economic epidemiology 79​​​‌. A notable feature‌ of such minimal solutions‌​‌ to such problems is​​ the presence of long​​​‌ intervals of zero control‌ (sparse control strategy).

This‌​‌ year, we have found​​ sufficient conditions for local​​​‌ minimality of singular arcs‌ using the methods of‌​‌ over-maximized Hamiltonians. This solves​​ completely the problem of​​​‌ their minimality. The results‌ are in preprint 12‌​‌.

7.3 Optimal experiment​​ design for parameter identification​​​‌

Participants: Ludovic Sacchelli,‌ Alessandro Scagliotti [Technical University‌​‌ of Munich, Germany].​​

In parameter identification, experiment​​​‌ design aims to plan‌ data acquisition so that‌​‌ the resulting estimation is​​ as informative as possible.​​​‌ A key difficulty is‌ that the outcome of‌​‌ an experiment typically depends​​ on the very parameter​​​‌ being estimated, so that‌ the optimization objective itself‌​‌ depends on the unknown​​ parameter. Standard approaches replace​​​‌ the unknown value with‌ a nominal estimate, such‌​‌ as the prior mean,​​ while more robust strategies​​​‌ account explicitly for parameter‌ uncertainty by averaging the‌​‌ objective over the prior​​ distribution, yielding an infinite-dimensional​​​‌ formulation.

In 22,‌ we considered a linear‌​‌ system with noisy measurements​​ in a Bayesian Gaussian​​​‌ setting and formulated an‌ input-design problem balancing posterior‌​‌ uncertainty reduction with constraints​​ on state magnitude. We​​​‌ first analyzed the classical‌ formulation from the perspective‌​‌ of optimal control, then​​ extended it to the​​​‌ ensemble-control framework. Existing results‌ on the ensemble Pontryagin‌​‌ maximum principle did not​​ cover unbounded parameter ensembles,​​​‌ such as our Gaussian-distributed‌ case, so we introduced‌​‌ a generalization of the​​ Pontryagin Maximum Principle (PMP)​​​‌ to this setting and‌ characterized the resulting optimal‌​‌ flow.

7.4 Controllability of​​ fast-oscillating system with control​​​‌ constraints

Participants: Jean-Baptiste Caillau‌, Lamberto Dell'Elce,‌​‌ Alesia Herasimenka [Univ. of​​ Surrey], Jean-Baptiste Pomet​​​‌.

Control of solar‌ sails requires a comprehensive‌​‌ understanding of controllability, that​​ is a difficult problem​​​‌ because the controls (forces)‌ are drastically constrained. This‌​‌ motivated a controllability study​​ of a class of​​​‌ systems that have periodic‌ behavior without control and‌​‌ where the control is​​ constrained in a convex​​​‌ set that has zero‌ in its boundary. Condition‌​‌ for controllability of these​​ systems are in the​​​‌ paper 5, published‌ this year. They have‌​‌ to be formulated in​​ terms of pushforwards along​​​‌ the flow of the‌ drift, rather than in‌​‌ terms of Lie brackets,​​ and they take into​​​‌ account not only the‌ vector fields associated to‌​‌ the system but also​​ the shape 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; methods to check​​ these conditions in the​​​‌ case of solar sails‌ are given in 74‌​‌ (published earlier): these are​​​‌ based on convex optimization;​ the approach leverages fine​‌ properties of trigonometric polynomials​​ as well as Nesterov's​​​‌ technique of sum of​ squares relaxation.

7.5 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 $\mathrm{z}\left(t\right)=‌A_1\left(\mathrm{t}\right)z(\mathrm{t‌}-{\tau }_1)+\cdots +{\mathrm{A‌}}_N(t)z(t-‌{\tau }_N)$,​​ 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 70​‌ 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”; the stability​ of this LDDS conditions​‌ stability of the PDE​​ dynamical system (see e.g.​​​‌ the long introduction of​ 31). We give​‌ in the paper 3​​, published this year,​​​‌ a necessary and sufficient​ condition for hyperbolic stability​‌ of periodic time-varying LDDS,​​ generalizing the well-kown one​​​‌ for the time-invariant case,​ due to Hale and​‌ Henry 73, 72​​.

7.6 Random walks​​​‌ in sub-Riemannian manifolds

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

Laplacian-type operators​​​‌ on sub-Riemannian manifolds encode​ the non-holonomic constraints of​‌ the underlying control structure.​​ Their associated heat diffusion​​​‌ links the geometry of​ sub-Riemannian geodesics to the​‌ behavior of random paths​​ on the manifold. Since​​​‌ Varadhan’s seminal work in​ the Riemannian setting 89​‌, it has been​​ known that short-time heat​​​‌ kernel asymptotics recover the​ distance between two points,​‌ reflecting the exponentially rare​​ event that a diffusion​​​‌ connects them in very​ small time.

This program​‌ has been actively developed​​ in the sub-Riemannian setting,​​​‌ and 21 brings together​ and completes several strands​‌ of this line of​​ inquiry. The paper provides​​​‌ a unified treatment of​ small-time asymptotics of the​‌ heat kernel, its derivatives,​​ and its logarithmic derivatives,​​​‌ with new results on​ localization that make the​‌ analysis applicable to incomplete​​ manifolds under essentially optimal​​​‌ assumptions. Away from abnormal​ minimizers, the asymptotics are​‌ governed by the geometry​​ of minimizing geodesics, including​​​‌ subtle behavior near the​ cut locus, leading to​‌ uniform bounds on compacts​​ and full asymptotic expansions​​​‌ in many cases. A​ genuinely new component concerns​‌ logarithmic derivatives: using a​​ law of large numbers​​​‌ for diffusion bridges, we​ show that the leading​‌ behavior of the log-Hessian​​ characterizes the non-abnormal cut​​​‌ locus. The length and​ scope of the paper​‌ reflect several years of​​ work, and this final​​​‌ revision consolidates the results​ into a coherent framework.​‌

7.7 Nonregularity of abnormal​​ sub-Riemannian geodesics

Participants: Ludovic​​​‌ Sacchelli, Yacine Chitour​ [CentraleSupélec, Gif-sur-Yvette], Frederic​‌ Jean [ENSTA, Palaiseau],​​ Roberto Monti [University of​​ Padova, Italy], Ludovic​​​‌ Rifford [Université Côte d'Azur,‌ Nice], Mario Sigalotti‌​‌ [Inria Paris], Alessandro​​ Socionovo [Inria Paris].​​​‌

A distinctive feature of‌ sub-Riemannian geometry is the‌​‌ existence of abnormal minimizing​​ geodesics, whose smoothness is​​​‌ not guaranteed. While normal‌ geodesics are smooth and‌​‌ can be characterized via​​ the Pontryagin maximum principle,​​​‌ abnormal minimizers escape this‌ regular framework, making their‌​‌ analysis particularly challenging. Previous​​ results have established partial​​​‌ regularity only in low-dimensional‌ or analytic cases, leaving‌​‌ open the general question​​ of whether abnormal minimizers​​​‌ can fail to be‌ smooth.

In 17,‌​‌ the above consortium constructed​​ an explicit example of​​​‌ a strictly singular minimizing‌ geodesic that is ${\mathrm{C‌‌}}^3$ but not ${\mathrm{C}}^4$. Establishing this​​​‌ result required precise asymptotic‌ analysis to rule out‌​‌ the possibility that nearby​​ normal trajectories could be​​​‌ shorter, and relied on‌ numerical simulations to identify‌​‌ key qualitative features among​​ short normal competitors to​​​‌ the abnormal curve. The‌ construction provides a definitive‌​‌ instance of a non-smooth​​ minimizing geodesic, contrasting with​​​‌ other recent results on‌ the regularity of abnormal‌​‌ minimizers, and answers a​​ long-standing open question in​​​‌ sub-Riemannian geometry.

7.8 Geometry‌ and optimal control for‌​‌ navigation problem

Participants: Bernard​​ Bonnard, 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​​ (see 91, 62​​​‌) 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 ${\mathrm{L‌}}^{\infty }$ 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) 4.​​​‌

7.9 Pulse control of‌ affine systems with applications‌​‌ to quantum control

Participants:​​ Ivan Beschastnyi, Lamberto​​​‌ Dell'Elce, Jean-Baptiste Pomet‌, Ludovic Sacchelli,‌​‌ David Tinoco.

Time-optimal​​ control problems with unbounded​​​‌ controls naturally arise in‌ quantum engineering, where fast‌​‌ operations are required to​​ mitigate decoherence. In this​​​‌ work, we study time-minimization‌ for affine control systems‌​‌ allowing impulsive (pulse-like) controls.​​​‌ Such problems are typically​ ill-posed in classical control​‌ spaces, as arbitrarily large​​ control amplitudes lead to​​​‌ instantaneous state transitions.

In​ 8, we introduce​‌ a relaxation framework based​​ on a time-rescaling technique​​​‌ that provides a rigorous​ treatment of impulsive controls​‌ without resorting to geometric​​ reduction procedures. The resulting​​​‌ extended system is defined​ on the original state​‌ manifold and admits optimal​​ solutions, albeit with a​​​‌ high degree of symmetry​ and degeneracy. Using Pontryagin’s​‌ Maximum Principle, we characterize​​ the structure of extremal​​​‌ trajectories and recover, in​ a simplified way, several​‌ known results on optimal​​ syntheses.

The approach is​​​‌ illustrated on concrete quantum​ control problems, including time-optimal​‌ gate generation for closed​​ and open two-level systems​​​‌ and population transfer in​ dissipative dynamics. In particular,​‌ we analyze the role​​ of pulse and singular​​​‌ arcs and discuss numerical​ issues arising from the​‌ degeneracy of the relaxed​​ problem.

7.10 Open qubit​​​‌ parameter identification with bounded​ pulses

Participants: Ghaieth Aloui​‌, Ivan Beschastnyi,​​ Ludovic Sacchelli.

Open​​​‌ quantum systems are inherently​ affected by environmental interactions,​‌ which manifest as relaxation​​ and dephasing, and fundamentally​​​‌ limit the performance of​ quantum devices. Accurate identification​‌ of the parameters governing​​ these processes is therefore​​​‌ essential for reliable control​ and calibration of qubit-based​‌ technologies. In 13,​​ we study the problem​​​‌ of parameter identification for​ a single open qubit​‌ subject to dissipation. Our​​ approach relies on a​​​‌ small number of carefully​ designed qubit configurations generated​‌ by saturating control pulses​​ and measured through simple​​​‌ projective readout, leading to​ an explicit and interpretable​‌ identification protocol.

In an​​ idealized regime of infinite-amplitude​​​‌ pulses, we showed that​ all parameters can be​‌ reconstructed analytically from a​​ minimal set of observables.​​​‌ We then analyzed the​ effect of realistic, finite-amplitude​‌ pulses as a perturbation​​ of this ideal setting​​​‌ and derive bounds on​ the resulting estimation error.​‌ This allowed us to​​ separate statistical uncertainty due​​​‌ to finite measurements from​ modeling errors induced by​‌ bounded controls, providing quantitative​​ guarantees for practical implementations.​​​‌

7.11 Control of an​ nitrogen-vacancy center as a​‌ two-qubit system

Participants: David​​ Tinoco, Ivan Beschastnyi​​​‌, Jean-Baptiste Caillau,​ Charles Babin [Université Bourgogne​‌ Europe, CNRS, ICB],​​ Dominique Sugny [Université Bourgogne​​​‌ Europe, CNRS, ICB].​

Nitrogen-vacancy centers constitute a​‌ leading platform for solid-state​​ quantum technologies, where high-fidelity​​​‌ control is limited by​ partial accessibility and environmental​‌ constraints. A collaboration with​​ physicists from ICB (Institut​​​‌ Carnot de Bourgogne) has​ started, with CNRS funding​‌ (CONV project, for​​ Control of nitrogen-vacancy centers),​​​‌ that targets at modeling,​ simulating and optimizing such​‌ systems, and also at​​ making the experiments on​​​‌ small instances (nitrogen-vacancy centers​ inside diamonds). We study​‌ a benchmark control problem​​ consisting of an electronic​​​‌ spin coupled to a​ nuclear spin, in which​‌ only the electronic degree​​ of freedom can be​​​‌ directly driven by microwave​ fields.

Starting from physically​‌ motivated approximations, in paper​​ 23, we analyze​​​‌ the controllability of the​ resulting two-qubit model using​‌ standard tools from geometric​​ control theory. We identified​​ the reachable sets for​​​‌ this approximation and computed‌ the quantum speed limit‌​‌ for two-qubit gate generation​​ by exploiting Lie algebraic​​​‌ and group-theoretic techniques. Finally,‌ numerical optimal control simulations‌​‌ demonstrate that this fundamental​​ limit can be closely​​​‌ approached while maintaining bounded‌ control amplitudes, illustrating the‌​‌ practical relevance of the​​ theoretical bounds.

7.12 Low-thrust​​​‌ satellite collision avoidance maneuvers‌

Participants: Lamberto Dell'Elce,‌​‌ Frank de Veld,​​ Jean-Baptiste Pomet, Joel​​​‌ Amalric [Thales Alenia Space]‌, Roberto Armellin [University‌​‌ of Auckland].

Frank​​ de Veld defended his​​​‌ thesis 11 on the‌ design of collision avoidance‌​‌ maneuvres for a satellite​​ equipped with low-thrust propulsion​​​‌ in encounters with non-cooperative‌ objects (e.g., space debris).‌​‌ The thesis addressed the​​ challenge of optimally controlling​​​‌ low-thrust satellites to avoid‌ collisions, taking into account‌​‌ both the satellite dynamics​​ and operational constraints. Below,​​​‌ we briefly summarize the‌ major contributions of the‌​‌ thesis, all of which​​ were essentially completed during​​​‌ 2025.

The first contribution‌ consisted in characterizing the‌​‌ latest safe time to​​ start a maneuvre. To​​​‌ this end, we introduced‌ a novel set of‌​‌ state variables, comprising the​​ time of closest approach​​​‌ and the satellite state‌ at that instant. This‌​‌ formulation allowed the closest​​ approach to be tracked​​​‌ directly. Necessary conditions for‌ optimality were derived using‌​‌ Pontryagin's Maximum Principle, and​​ approximate solutions for fast​​​‌ encounters were obtained by‌ backward integration from the‌​‌ safe set. We also​​ performed a detailed analysis​​​‌ of the optimal solution‌ structure, including the computation‌​‌ of cut and conjugate​​ loci.

The second contribution​​​‌ was aimed at including‌ operational constraints in the‌​‌ metodology. Namely, we considered​​ a prescribed thrusting direction​​​‌ imposed by mission requirement‌ and we included non-thrusting‌​‌ arcs (e.g., during eclipses).​​

The third contribution, which​​​‌ was developed during F.‌ de Veld's stay at‌​‌ the University of Auckland,​​ revisited the optimization of​​​‌ thrust initiation using Differential‌ Algebra, employing Taylor series‌​‌ expansions to efficiently compute​​ safe maneuvres under three​​​‌ danger metrics: Euclidean distance,‌ Mahalanobis distance, and probability‌​‌ of collision.

No peer-reviewed​​ publications were produced during​​​‌ the year, but several‌ articles were prepared. Two‌​‌ papers discussed the minimum​​ warning time and its​​​‌ extension to operational constraints.‌ One paper presented work‌​‌ from Frank's secondment at​​ the University of Auckland.​​​‌

7.13 Levi-Civita regularization of‌ collisions in the controlled‌​‌ case

Participants: Alain Albouy​​ [Observatoire de Paris, CNRS]​​​‌, Jean-Baptiste Caillau,‌ Riccardo Daluiso.

In‌​‌ the $n$-body problem,​​ $n$ point masses move​​​‌ in space under their‌ mutual gravitational interactions as‌​‌ described by Newton's theory​​ of gravity. The forces​​​‌ acting between particles approach‌ infinity when the mutual‌​‌ distances approach zero. Therefore,​​ at collision, the equations​​​‌ of motion have singularities.‌ Since Levi-Civita $\left(\mathrm{1903‌‌}\right)$ and Sundman $(1907)$, the​​​‌ double collision has been‌ "regularized", i.e. the singularity‌​‌ has been made to​​ disappear by means of​​​‌ algebraic transformations. In his‌ 1920 paper 77,‌​‌ Levi-Civita presented a regularization​​ of the three-body problem​​​‌ which was ignored by‌ almost all the authors‌​‌ with only a few​​​‌ exceptions. The idea of​ Levi-Civita was to first​‌ regularize a parabolic collision​​ orbit of the Kepler​​​‌ problem by means of​ the Hamilton-Jacobi method, following​‌ procedures from Poincaré. These​​ techniques surprisingly provide contact​​​‌ transformations which make the​ Hamiltonian of the three-body​‌ problem perfectly regular. More​​ than fifty years later,​​​‌ Moser $(1970)$ compared the stereographic projection​‌ of the great circles​​ of the sphere with​​​‌ Keplerian odograph, following ideas​ from Fock. This construction​‌ gives rise to a​​ compactification of the phase​​​‌ space of the Keplerian​ orbits for a fixed​‌ level of energy together​​ with the same regularization​​​‌ found by Levi-Civita, which​ is thus designated to​‌ be unique and fundamental.​​ The main issue arising​​​‌ from these technique is​ the inability for the​‌ authors in finding regularizazions​​ which work simultaneously for​​​‌ all the energy surfaces.​ Indeed, it turns out​‌ that the Levi-Civita transformation,​​ defined by

if applied​‌ to the two-body problem​​ sends all the collision​​​‌ states to the same​ value of the energy​‌ (i.e. with the​​ same value of the​​​‌ semi-major axis of the​ collision orbit) on the​‌ same state of the​​ regularized system: the transformation​​​‌ obtained is not bijective.​ In a recent work,​‌ A. Albouy proposes a​​ simple trick to avoid​​​‌ this issue by viewing​ the problem as defined​‌ on 4-manifold embedded into​​ the 5-dimensional Euclidean space​​​‌ on which energy is​ just another coordinate.

The​‌ purpose of this work​​ is to apply the​​​‌ regularization methods to optimal​ control. Indeed, the compactification​‌ of the state space​​ is relevant for the​​​‌ existence of solutions for​ optimal control problems in​‌ space mechanics, and we​​ consider the control of​​​‌ a spacecraft under the​ attraction of one or​‌ more bodies. The main​​ question we face is​​​‌ the extension of the​ theory to the case​‌ of non-constant energy. Our​​ first results include controllability,​​​‌ existence and extremal flow​ analysis at collisions on​‌ the regularized problem.

7.14​​ Powered landing of reusable​​​‌ space launchers

Participants: Jean-Baptiste​ Caillau, Lamberto Dell'Elce​‌, Samuel Hornus [Inria​​ Nancy Grand Est],​​​‌ Jean-Baptiste Pomet.

We​ propose to extend the​‌ study of the Soft​​ Planetary Landing problem as​​​‌ defined in 25 or,​ more recently, by 76​‌. This problem seeks​​ to determine the best​​​‌ trajectory for landing a​ vehicle at a given​‌ point while consuming as​​ little propellant as possible.​​​‌ We wish to provide​ the detailed mathematical structure​‌ of the optimal trajectories​​ for this problem using​​​‌ the necessary conditions for​ optimality (indirect methods​‌ in the sense of​​ optimal control theory). By​​​‌ ensuring an understanding of​ the structure of optimal​‌ solutions for almost any​​ instance of the problem,​​​‌ we seek to pave​ the way for a​‌ quasi-real-time MPC (Model Predictive​​ Control) controller design without​​​‌ time discretization.

Two American​ companies, Blue Origin and​‌ SpaceX, have demonstrated that​​ reusing a space launcher​​ is viable and advantageous.​​​‌ Unlike previous launchers developed‌ by public agencies, these‌​‌ reusable launchers are developed​​ by private companies, meaning​​​‌ that the expertise and‌ theoretical and practical knowledge‌​‌ they have acquired remain​​ largely inaccessible outside these​​​‌ companies. In the long‌ term, we wish to‌​‌ contribute to the establishment​​ and publication of technical​​​‌ knowledge useful for the‌ successful development of reusable‌​‌ European space vehicles. This​​ optimal control problem seeks​​​‌ to minimize fuel consumption‌ for a rocket-propelled vehicle‌​‌ moving from a given​​ position/velocity above the ground​​​‌ to a rest position‌ on the ground. The‌​‌ associated trajectories must satisfy​​ several constraints, including cone​​​‌ constraints both on the‌ control (thrust angle) and‌​‌ on the position.

We​​ want to describe in​​​‌ detail the mathematical structure‌ of the optimal trajectories‌​‌ for this problem, with​​ the help of the​​​‌ necessary optimality conditions given‌ by the Pontryagin Maximum‌​‌ Principle. We compute indirect​​ solutions for any feasible​​​‌ instance of the problem.‌ This is possible thanks‌​‌ to the determination of​​ the dynamics along constrained​​​‌ arcs for each type‌ of constraint, and to‌​‌ the understanding of the​​ possible sequences of arcs​​​‌ that can arise. For‌ instance, it can be‌​‌ proved that the two​​ constraints (on the state​​​‌ and on the control,‌ see above) probably cannot‌​‌ be active at the​​ same time along a​​​‌ positive interval of time.‌ On top of this‌​‌ mathematical analysis, we develop​​ techniques to determine the​​​‌ structure of the optimal‌ trajectory for a large‌​‌ variety of specific instances​​ of the problem. A​​​‌ crucial step for this‌ is the use of‌​‌ a direct discretization coupled​​ with an efficient nonlinear​​​‌ optimization solver; the resulting‌ approximate solution allows to‌​‌ capture in most cases​​ the structure of the​​​‌ true solution, leading to‌ a precise "indirect" resolution‌​‌ via multiple shooting.

7.15​​ Precise orbit determination of​​​‌ non-cooperative objects in medium‌ and geostationary orbits

Participants:‌​‌ Lamberto Dell'Elce, Eliot​​ Stein, Frederic Cassaing​​​‌ [ONERA], Hanae Labriji‌ [ONERA], Florian Thuillet‌​‌ [ONERA].

The tracking​​ of resident space objects​​​‌ (RSO) with high accuracy‌ is essential for safe‌​‌ and sustainable space operations,​​ enabling early conjunction warnings,​​​‌ improved decision-making, reduced fuel‌ consumption, and longer satellite‌​‌ lifetimes. Orbit determination (OD)​​ is limited by observation​​​‌ accuracy: passive optical stations‌ for medium- and high-Earth‌​‌ orbits (MHO) typically provide​​ few hundreds of milliarcseconds​​​‌ angular accuracy, yielding OD‌ errors of several hundred‌​‌ meters between successive nights​​ of observations.

In the​​​‌ framework of E. Stein’s‌ PhD thesis, we plan‌​‌ to exploit measurements from​​ the CICLOPE telescope, which​​​‌ provides 50 milliarcseconds accuracy‌ (5–10 m at MHO‌​‌ distances), sufficient to detect​​ deviations from gravitational dynamics,​​​‌ including solar radiation pressure‌ shifts of 10 m‌​‌ over 4 h for​​ typical satellites and up​​​‌ to 35 m for‌ debris. This precision allows‌​‌ reliable non-gravitational perturbation estimation​​ and advanced orbit determination.​​​‌

In 2025, we developed‌ and algorithm for precise‌​‌ orbit prediction with decametric​​ accuracy over one day​​​‌ for non-cooperative MHO RSOs‌ using sparse angle-only measurements.‌​‌ A high-fidelity force model​​​‌ is built, and OD​ is performed with weighted​‌ least squares and a​​ Jacobian from variational equations.​​​‌ Using simulated 50 milliarcseconds​ observations, standard gravitational models​‌ give hectometric errors, while​​ including SRP estimation reduces​​​‌ errors to about 20​ m.

7.16 Hardware-in-the-loop attitude​‌ simulator for the NiceCube​​ mission

Participants: Lamberto Dell'Elce​​​‌, Matthieu Estines,​ Laurent Moinet.

The​‌ Centre Spatial Universitaire (CSU)​​ of the Université Côte​​​‌ d'Azur 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. McTAO is in​​​‌ charge of developing the​ attitude determination and control​‌ system (ADCS) of NiceCube.​​ In the previous years,​​​‌ we developed a high-fidelity​ numerical simulator of the​‌ coupled orbital-attitude motion of​​ a satellite in low-Earth​​​‌ orbit. This year, two​ internships were devoted to​‌ the integration of such​​ simulator into the so-called​​​‌ "FlatSat" under development at​ the CSU, which will​‌ enable hardware in the​​ loop simulations of NiceCube.​​​‌ Specifically, L. Moinet implemented​ an interface aimed at​‌ transferring data from the​​ simulator to the on-board​​​‌ computer of the satellite.​ This work was then​‌ finalized by M. Estines,​​ who also initiated the​​​‌ implementation of a simple​ attitude determination and control​‌ algorithm onboard the satellite.​​

7.17 Kite Electrical Energy​​​‌ Production (the KEEP project)​

Participants: Antonin Bavoil,​‌ Jean-Baptiste Caillau, Alain​​ Nême [ENSTA], Christian​​​‌ Jochum [ENSTA], Jean-Baptiste​ Leroux [ENSTA].

This​‌ project aims at improving​​ the performance of technologies​​​‌ that meet the challenges​ of the law concerning​‌ the energy transition. The​​ innovative KEEP (Kite Electrical​​​‌ Energy Production) technology developed​ at ENSTA enables the​‌ production of electrical energy​​ from a kite capturing​​​‌ wind energy. It is​ a Ground-Gen system with​‌ a moving-ground-station (rocking arm)​​ simpler than the devices​​​‌ described, e.g., in​ 65. The first​‌ numerical models of the​​ system show an increase​​​‌ of about 50% in​ the efficiency of this​‌ technology compared to already​​ existing ones 92.​​​‌ In the short and​ medium term, the onshore​‌ and shipboard applications with​​ a power production ranging​​​‌ from a few tens​ to hundreds of kW​‌ are targeted, with an​​ extension to an offshore​​​‌ production power capacity at​ a MW scale with​‌ 1000 m${}^2$ kites.​​ To this end, the​​​‌ following developments are required:​ the improvement of the​‌ mechanical model, as well​​ as the theoretical analysis​​​‌ and optimization of the​ complete dynamical system.

In​‌ the preprint 14,​​ we present a brief​​​‌ derivation of the model.​ Our approach consists in​‌ an implicit modeling of​​ the control exerted on​​​‌ the kite: we compute​ the force such that​‌ the motion is indeed​​ prescribed to an eight​​​‌ shaped figure, typical of​ the desired motion. While​‌ the resulting system is​​ a differential algebraic equation​​​‌ (DAE), it is possible​ to make it explicit​‌ as a two-dimensional second-order​​ ODE. We are interested​​​‌ in the limit cycles​ of this equation, and​‌ study them numerically in​​ detail. In order to​​ obtain the relevant periodic​​​‌ motion (only some limit‌ cycles are admissible for‌​‌ the kite), it is​​ convenient to start from​​​‌ specific equilibria on the‌ system so we also‌​‌ examine them. The resulting​​ limit cycle, that depends​​​‌ on several design parameters,‌ is then optimized so‌​‌ as to maximize the​​ energy generated by the​​​‌ motion (the kite is‌ attached to an arm‌​‌ in the ground plane​​ whose oscillations feed a​​​‌ generator). This optimization with‌ respect to a set‌​‌ of finite dimensional parameters​​ leads to an approximate​​​‌ 12% increase of the‌ generated power.

7.18 Parameter‌​‌ and state estimation in​​ neural models

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

This research​​​‌ focuses on dynamic estimation‌ and observer synthesis for‌​‌ neural field equations, the​​ general topic of A.​​​‌ Annabi's PhD. In the‌ visual cortex, neural field‌​‌ models can describe activity​​ dynamics related to orientation​​​‌ sensitivity of neurons. This‌ allows mapping neural fields‌​‌ onto the orientation domain,​​ enabling a Fourier representation​​​‌ that can be truncated‌ to a simplified three-dimensional‌​‌ model of the V1​​ area. In 2,​​​‌ we investigated the observability‌ of this model, highlighting‌​‌ the system's symmetries and​​ proposing hybrid elements to​​​‌ counteract their effects. The‌ study shows the role‌​‌ of nonlinearity in achieving​​ observability and identifies persistence​​​‌ conditions required for accurate‌ state estimation.

In the‌​‌ conference paper 7,​​ presented this year at​​​‌ CDC 2025, we addressed‌ parameter identification in networks‌​‌ of interconnected neural populations,​​ with measurements available from​​​‌ only one. We proposed‌ an online approach that‌​‌ exploits the system's nonlinear​​ characteristics, particularly saturation functions,​​​‌ to recover parameters. By‌ designing targeted control inputs‌​‌ that adapt to unknown​​ states and parameters, parameters​​​‌ can be retrieved online.‌

7.19 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 or 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​​ 55, 56.​​​‌ 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​​​‌ former Ecuador team and​ from Argonne National Lab​‌ are key to this​​ task and benefited in​​​‌ 2025 of visits from​ and to Argonne (J.-B.​‌ Caillau visited Argonne for​​ one week with JLESC​​​‌ fundind during JLESC 2025​ conference). OptimalControl.jl is currently​‌ one of the very​​ few packages to offer​​​‌ state-of-the-art and out-of-the-box high​ level and super fast​‌ optimal control problem solving​​ both on CPU and​​​‌ GPU. To this end,​ we leverage several modelers​‌ and optimization solvers (including​​ ExaModels / MadNLP from​​​‌ our colleagues at MIT,​ Argonne and Mines Paris).​‌ Detailed benchmarks are on​​ their way, as displayed​​​‌ in Figure 1 below​ for our first results.​‌ In December 2025, the​​ project also benefited of​​​‌ a sprint session dedicated​ to BifurcationKit.jl by R.​‌ Veltz (Cronos team) that​​ could be a key​​​‌ tool to add path​ following methods for shooting​‌ in OptimalControl.jl.

The image​​ is a performance profile​​​‌ graph showing CPU time​ across different methods. The​‌ horizontal axis represents the​​ performance ratio ($\mathrm{t‌}au$); the​ vertical axis shows the​‌ proportion of solved instances​​ ($\tau$). Six​​​‌ methods are compared: jump​ with ipopt, adnlp with​‌ ipopt, exa with ipopt,​​ jump with madnlp, adnlp​​​‌ with madnlp, and exa​ with madnlp. The orange​‌ line that corresponds to​​ (exa, madnlp) comnbination consistently​​​‌ shows the highest performance,​ reaching near 100% quicker​‌ than the others. The​​ combination (jump, ipopt) follows,​​​‌ with (adnlp, madnlp) showing​ slower but increasing performance.​‌

7.20 Finite differences for​‌ two-level optimization and applications​​ in learning

Participants: Marco​​​‌ Rando, Samuel Vaiter​ [Univ. Côte d'Azur, CNRS,​‌ LJAD].

This work​​ focuses on two-level optimization,​​​‌ where the aim is​ to minimize a function​‌ whose evaluation depends on​​ the solution to an​​​‌ internal optimization problem. Traditional​ methods are based on​‌ calculating the hypergradient (gradient​​ of the value function),​​​‌ but they become inapplicable​ when first-order information is​‌ not available, as in​​ the zero-order (black box)​​​‌ setting. In the framework​ of Marco Rando's postdoc​‌ (started November 2025, PDE-AI​​ project), Marco and Samuel​​​‌ Vaiter propose ZOBA, the​ first single-loop algorithm based​‌ on finite differences for​​ two-level optimization, avoiding costly​​​‌ and difficult-to-parallelize double-loop schemes.​ The method uses an​‌ approximation of the hypergradient​​ that exploits delayed information,​​​‌ eliminating the need for​ nested loops. A theoretical​‌ analysis establishes convergence guarantees​​ in a non-convex context,​​​‌ with complexity in $\mathrm{O}\left(p\right(\mathrm{d‌}+p)\mathrm{2}\epsilon -2)‌$, which is better​ than some previous approaches​‌ based on Hessian approximation.​​ HF-ZOBA, a Hessian-free variant​​​‌ offering additional complexity gains,​ is also introduced. Experiments​‌ on synthetic functions and​​ a real-world application in​​​‌ black-box adversarial learning confirm​ that these methods achieve​‌ state-of-the-art accuracy while reducing​​ computation time.

7.21 Algorithmic​​ Differentiation support and collaboration​​​‌ with Tapenade users

Participants:‌ Laurent Hascoët, Jean-Luc‌​‌ Bouchot, Michael Vossbeck​​ [The Inversion Lab, Hamburg,​​​‌ Germany], Sri Hari‌ Krishna Narayanan [Argonne National‌​‌ Lab. (Illinois, USA)],​​ Shreyas Gaikwad [U. of​​​‌ Texas, Austin, USA].‌

We support both academic‌​‌ and industrial users of​​ the AD tool Tapenade​​​‌ for their own applications.‌ This involves AD on‌​‌ codes of all sizes,​​ providing us with suggestions​​​‌ for improvement and possibly‌ new research.

This year's‌​‌ main applications are on:​​

• the global circulation model​​​‌ MIT GCM, in collaboration‌ with Shreyas Gaikwad and‌​‌ Patrick Heimbach (University of​​ Texas in Austin) for​​​‌ end-users in all fields‌ of climatology including glaciology‌​‌ (Dan Goldberg, University of​​ Edinburgh, UK). The goal​​​‌ is to blend MIT‌ GCM with an approved‌​‌ tool for adjoint differentiation​​ that is open-source and​​​‌ free of cost for‌ end-users. This effort has‌​‌ run over several years​​ and now results in​​​‌ Tapenade AD officially adopted‌ by the maintainers of‌​‌ the GCM.
• the Biosphere​​ model NUCAS (The Inversion​​​‌ Lab, Hamburg, Germany) and‌ closely related codes BEPS‌​‌ and BETHY, in collaboration​​ with Michael Vossbeck (The​​​‌ Inversion Lab). NUCAS was‌ already differentiated (tangent and‌​‌ adjoint) with Tapenade, in​​ its sequential version. This​​​‌ year, Tapenade differentiated the‌ MPI-parallel version of NUCAS.‌​‌ This required a few​​ improvements to Tapenade handling​​​‌ of MPI codes, resulting‌ in adMPI: a differentiable‌​‌ wrapper around MPI maintained​​ and shipped with the​​​‌ Tapenade distribution.
• the nuclear‌ physics code HFBTHO (Lawrence‌​‌ Livermore National Lab, USA),​​ in collaboration with Krishna​​​‌ Narayanan (Argonne National Lab).‌ HFBTHO is a quantum‌​‌ physics code that models​​ the orbits of particles​​​‌ inside the atomic nucleus.‌ HFBTHO approximates the underlyling‌​‌ complex physics through a​​ number of hidden parameters,​​​‌ that must be guessed‌ by parameter estimation. This‌​‌ uses AD-computed derivatives. Only​​ tangent derivatives are required​​​‌ at present. This work‌ is discussed in an‌​‌ article 6 published in​​ "Computer Physics Communications". In​​​‌ parallel, Krishna Narayanan differentiates‌ the BLAS library (Linear‌​‌ Algebra) by AD with​​ Tapenade.

7.22 Jacobian Sparsity​​​‌

Participants: Laurent Hascoët,‌ Jean-Luc Bouchot, Alexis‌​‌ Montoison [Argonne National Laboratory,​​ Illinois, USA].

Jacobian​​​‌ Matrices (matrix of the‌ first-order derivatives of each‌​‌ output with respect to​​ each input) are one​​​‌ key derivative object. As‌ they are often very‌​‌ large, it is profitable​​ to exploit their sparsity,​​​‌ in particular with compression‌ models that use coloring.‌​‌ The sparse structure of​​ the Jacobian, a matrix​​​‌ of boolean values describing‌ the structural zeros of‌​‌ the Jacobian, can be​​ computed by source-transformation of​​​‌ the original model with‌ an AD tool. This‌​‌ requires a special mode​​ of AD, “sparsity-AD”, which​​​‌ is already know in‌ the literature, but almost‌​‌ always implemented in Overloading-based​​ AD tools, with the​​​‌ associated performance loss. We‌ instead study sparsity-AD for‌​‌ source-transformation AD models.

Sparsity-AD​​ exhibits interesting properties. In​​​‌ particular, activity analysis for‌ this mode is able‌​‌ to simplify the derivative​​ code further, detecting and​​​‌ removing many more unnecessary‌ instructions of the original‌​‌ model. Performance of the​​​‌ “sparsity-AD” code is thus​ improved. Moreover, propagation of​‌ the sparsity structure in​​ the form of bitvectors​​​‌ is well adapted to​ vectorization on GPUs. We​‌ implemented this special “sparsity-AD”​​ mode in Tapenade, in​​​‌ both tangent and adjoint​ modes. We validated this​‌ extension on large test​​ cases. We presented our​​​‌ first implementation at the​ 27th EuroAD workshop in​‌ Kaiserslautern, Germany April 3-4.​​ An article is in​​​‌ preparation.

In the future,​ we will continue sparsity-AD​‌ for Jacobian matrices, exploring​​ alternative storage strategies of​​​‌ the boolean matrix, for​ instance replacing bit vectors​‌ with sets of non-zero​​ ranks when sparsity exceeds​​​‌ some threshold. We will​ also study extension to​‌ higher-order derivative objects such​​ as Hessians, that deserve​​​‌ different sparsity propagation rules.​

7.23 Algorithmic Differentiation for​‌ Julia

Participants: Laurent Hascoët​​, Jean-Luc Bouchot.​​​‌

We develop an extension​ of the AD tool​‌ Tapenade devoted to the​​ Julia language. Tapenade can​​​‌ so far handle languages​ Fortran (77, 90, 2003)​‌ and C. Our development​​ principle is that Tapenade​​​‌ internal representation is language​ independent, and manipulates programming​‌ constructs (types, variables, control​​ structures, procedures, packages...) independently​​​‌ of the particular source​ language. This principle facilitated​‌ many aspects of the​​ extension to C, and​​​‌ we expect the same​ kind of advantage in​‌ extending to Julia. We​​ developed a Julia parser​​​‌ that targets Tapenade's internal​ imperative language, together with​‌ the reciprocal unparser that​​ generates Julia source from​​​‌ Tapenade's internal representation.

In​ addition to the standard​‌ programming constructs, every language​​ has a few specific​​​‌ ones with original semantics,​ that require some research​‌ or development. In Julia,​​ the novel constructs that​​​‌ required some research for​ their differentiation contain: the​‌ type system and the​​ multiple dispatch semantics of​​​‌ call, the tuple construct,​ or the more interpreter-like​‌ form of programs. These​​ constructs (and a few​​​‌ others) required to extend​ the source-transformation AD model​‌ that we promote. Consequently,​​ we implemented and tested​​​‌ these extensions inside our​ tool Tapenade. A first​‌ Julia example code (a​​ solver for the Burgers​​​‌ equation) has been successfully​ differentiated in tangent mode.​‌ In the future, we​​ will progressively extend the​​​‌ dialect of Julia that​ we can differentiate, also​‌ in adjoint mode, in​​ a call-by-need fashion guided​​​‌ by applications. We presented​ this effort towards AD​‌ of Julia at the​​ 27th EuroAD workshop in​​​‌ Kaiserslautern, Germany April 3-4.​

8.1​​ Méthodes de contrôle pour​​​‌ l’évitement de collisions entre​ satellites (Control Methods for​‌ satellite collision avoidance), 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 9.4​

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

9.1 International​‌ initiatives

9.2‌ European initiatives

Participants: Laurent‌​‌ Hascoët, Jean-Luc Bouchot​​.

L. Hascoët and​​​‌ J.-L. Bouchot participate regularly‌ to the EuroAD Workshops‌​‌ (bi-annual) and to the​​ attached community. They went​​​‌ to Kaiserslautern in April‌ (talk: "Source transformation AD‌​‌ for Julia with Tapenade")​​ and to CERN, Geneva​​​‌ in December.

9.3 National‌ initiatives

9.4‌​‌ Regional initiatives

Participants: Lamberto​​ Dell'Elce, Jean-Baptiste Pomet​​​‌, Frank de Veld‌, Ivan Beschastnyi,‌​‌ Ludovic Sacchelli.

• Grant​​​‌ from Région SUD –​ Provence Alpes Côte d'Azur​‌ “Emplois jeunes Doctorants”, 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€.
• CSI Univ.​ Côte d'Azur (Incentive Scientific​‌ Credits of Université Côte​​ d'Azur): 2025 Project COEPLEX​​​‌, 6k€, P.I. L.​ Sacchelli, on optimal control​‌ for experimental design.
• CSI​​ Univ. Côte d'Azur 2025:​​​‌ Project COSQUO, 5k€, P.I.​ I. Beschastnyi, on quantum​‌ control.
• Idex Univ. Côte​​ d'Azur: Welcome package of​​​‌ Ivan Beschastnyi, 2023-2026. Total​ amount: 50k€.

10.1 Promoting scientific activities​​

10.2 Teaching - Supervision​​ - Juries - Educational​​​‌ and pedagogical outreach

10.3‌ Popularization

11.1 Publications of the​ year

11.2 Cited publications

• 25​‌ articleB.Behcet Açıkmeşe​​ and S. R.Scott​​​‌ R. Ploen. Convex​ Programming Approach to Powered​‌ Descent Guidance for Mars​​ Landing.Journal of​​​‌ Guidance, Control, and Dynamics​3052007,​‌ 1353-1366DOIback to​​ text
• 26 incollectionA.​​​‌ A.A. A. Agrachev​ and R. V.R.​‌ V. Gamkrelidze. Symplectic​​ methods for optimization and​​​‌ control.Geometry of​ feedback and optimal control​‌207Textbooks Pure Appl.​​ Math.Marcel Dekker1998​​​‌, 19--77back to​ text
• 27 bookA.​‌Andrei Agrachev and Y.​​ L.Yuri L. Sachkov​​​‌. Control theory from​ the geometric viewpoint.​‌87Encyclopaedia of Mathematical​​ SciencesControl Theory and​​​‌ Optimization, IIBerlinSpringer-Verlag​2004DOIback to​‌ text
• 28 articleA.​​ A.Andrey A. Agrachev​​​‌ and A. V.Andrey​ V. Sarychev. Abnormal​‌ sub-Riemannian geodesics: Morse index​​ and rigidity.13​​​‌61996, 635--690​URL: http://www.numdam.org/item/AIHPC_1996__13_6_635_0/back to​‌ text
• 29 articleA.​​ A.A. A. Agrachev​​​‌ and A. V.A.​ V. Sarychev. Strong​‌ minimality of abnormal geodesics​​ for 2-distributions.1​​​‌21995, 139--176​DOIback to text​‌
• 30 bookV. I.​​Vladimir I. Arnold.​​​‌ Mathematical methods of classical​ mechanics.60Graduate​‌ Texts in MathematicsNew​​ YorkSpringer-Verlag1989back​​​‌ to textback to​ text
• 31 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/DOI​​back to text
• 32​​​‌ 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/1354110195​​​‌DOIback to text​
• 33 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/DOI​​​‌back to textback​ to text
• 34 inproceedings​‌B.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 Applications​​suppl.Discrete Contin. Dyn.​​​‌ Syst.AIMS2011,​ 198--208DOIback to​‌ text
• 35 articleB.​​Bernard Bonnard and J.-B.​​​‌Jean-Baptiste Caillau. Geodesic​ flow of the averaged​‌ controlled Kepler equation.​​215September 2009​​​‌, 797--814DOIback​ to text
• 36 article​‌B.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​‌
• 37 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
• 38 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.132​​​‌2007, 207--236DOI‌back to text
• 39‌​‌ 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
• 40 book‌​‌B.Bernard Bonnard and​​ M.Monique Chyba.​​​‌ Singular trajectories and their‌ role in control theory‌​‌.40Mathématiques &​​ ApplicationsBerlinSpringer-Verlag2003​​​‌back to text
• 41‌ articleB.B. Bonnard‌​‌. Feedback equivalence for​​ nonlinear systems and the​​​‌ time optimal control problem‌.2961991‌​‌, 1300--1321URL: https://doi.org/10.1137/0329067​​DOIback to text​​​‌
• 42 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-00918633DOI‌​‌back to text
• 43​​ 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.34​​​‌2013, 397--432URL:‌ https://hal.inria.fr/hal-00939495DOIback to‌​‌ text
• 44 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​​
• 45 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-03917363‌​‌DOIback to text​​
• 46 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/10.1137/130924731‌​‌DOIback to text​​
• 47 articleU.Ugo​​​‌ Boscain, J.Jean‌ Duplaix, J.-P.Jean-Paul‌​‌ Gauthier and F.Francesco​​ Rossi. Anthropomorphic Image​​​‌ Reconstruction via Hypoelliptic Diffusion‌.503January‌​‌ 2012, 1309--1336DOI​​back to text
• 48​​​‌ 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 Neuroscience112​2021, URL: https://doi.org/10.1186/s13408-020-00099-4​‌DOIback to text​​
• 49 thesisJ.-L.Jean-Luc​​​‌ Bouchot. A scientific​ journey in the realm​‌ of signal and data​​ processing: Compressed sensing, sparse​​​‌ approximation, and applications.​Université Côte D'AzurDecember​‌ 2025, URL: https://inria.hal.science/tel-05464427​​back to text
• 50​​​‌ articleP. C.Paul​ C Bressloff. Spatiotemporal​‌ dynamics of continuum neural​​ fields.Journal of​​​‌ Physics A: Mathematical and​ Theoretical4532011​‌, 033001back to​​ text
• 51 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-02172420​​​‌DOIback to text​
• 52 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​​​‌
• 53 inbookJ.-B.Jean-Baptiste​ Caillau, Y.Yacine​‌ Chitour, P.Pedro​​ Freitas and Y.Yannick​​​‌ Privat. Extremal determinants:​ the periodic one-dimensional essentially​‌ bounded case.12​​IVAN KUPKA LEGACY: A​​​‌ Tour Through Controlled Dynamics​AIMS Applied Math Books​‌ - Special issue in​​ honor of I. Kupka​​​‌February 2024, 155--172​HALback to text​‌
• 54 articleJ.-B.Jean-Baptiste​​ Caillau, O.O.​​​‌ Cots and J.J.​ Gergaud. Differential pathfollowing​‌ for regular optimal control​​ problems.272​​​‌2012, 177--196back​ to text
• 55 inproceedings​‌J.-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 2023​‌HALback to text​​
• 56 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, France​​October 2023HALback​​​‌ to text
• 57 article​J.-B.Jean-Baptiste Caillau and​‌ B.Bilel Daoud.​​ Minimum time control of​​​‌ the restricted three-body problem​.5062012​‌, 3178--3202back to​​ text
• 58 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 Industry23Springer​​​‌April 2016, 209--302​URL: https://hal.archives-ouvertes.fr/hal-01260120DOIback​‌ to text
• 59 inbook​​J.-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; Elsevier​2023, 559-626HAL​‌DOIback to text​​
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