CAGE's activities take place in the field of mathematical control theory, with applications in three main directions: geometric models for vision, control of quantum mechanical systems, and control of systems with uncertain dynamics.
The relations between control theory and geometry of vision rely on the notion of subRiemannian structure, a geometric framework which is used to measure distances in nonholonomic contexts and which has a natural and powerful control theoretical interpretation. We recall that nonholonomicity refers to the property of a velocity constraint that cannot be recast as a state constraint. In the language of differential geometry, a subRiemannian structure is a (possibly rankvarying) Lie bracket generating distribution endowed with a smoothly varying norm.
SubRiemannian geometry, and in particular the theory of associated (hypoelliptic) diffusive processes, plays a crucial role in the neurogeometrical model of the primary visual cortex due to Petitot, Citti and Sarti, based on the functional architecture first described by Hubel and Wiesel. Such a model can be used as a powerful paradigm for bioinspired image processing, as already illustrated in the recent literature (including by members of our team). Our contributions to this field are based not only on this approach, but also on another geometric and subRiemannian framework for vision, based on pattern matching in the group of diffeomorphisms. In this case admissible diffeomorphisms correspond to deformations which are generated by vector fields satisfying a set of nonholonomic constraints. A subRiemannian metric on the infinitedimensional group of diffeomorphisms is induced by a length on the tangent distribution of admissible velocities. Nonholonomic constraints can be especially useful to describe distortions of sets of interconnected objects (e.g., motions of organs in medical imaging).
Simultaneous control of a continuum of systems with slightly different dynamics is a typical problem in quantum mechanics and also a special case of the third applicative axis to which CAGE is contributing: control of systems with uncertain dynamics. The slightly different dynamics can indeed be seen as uncertainties in the system to be controlled, and simultaneous control rephrased in terms of a robustness task. Robustification, i.e., offsetting uncertainties by suitably designing the control strategy, is a widespread task in automatic control theory, showing up in many applicative domains such as electric circuits or aerospace motion planning. If dynamics are not only subject to static uncertainty, but may also change as time goes, the problem of controlling the system can be recast within the theory of switched and hybrid systems, both in a deterministic and in a probabilistic setting. Our contributions to this research field concern both stabilization (either asymptotic or in finite time) and optimal control, where redundancies and probabilistic tools can be introduced to offset uncertainties.
The activities of CAGE are part of the research in the wide area of control theory. This nowadays mature discipline is still the subject of intensive research because of its crucial role in a vast array of applications.
More specifically, our contributions are in the area of mathematical control theory, which is to say that we are interested in the analytical and geometrical aspects of control applications.
In this approach, a control system is modeled by a system of equations (of many possible types: ordinary differential equations, partial differential equations, stochastic differential equations, difference equations,...), possibly not explicitly known in all its components, which are studied in order to establish qualitative and quantitative properties concerning the actuation of the system through the control.
Motion planning is, in this respect, a cornerstone property: it denotes the design and validation of algorithms for identifying a control law steering the system from a given initial state to (or close to) a target one.
Initial and target positions can be replaced by sets of admissible initial and final states as, for instance, in the motion planning task towards a desired periodic solution.
Many specifications can be added to the pure motion planning task, such as robustness to external or endogenous disturbances, obstacle avoidance or penalization criteria.
A more abstract notion is that of controllability, which
denotes the property of a system for which any two states can be connected by a trajectory corresponding to an admissible control law.
In mathematical terms, this translates into the surjectivity of the socalled endpoint map, which associates with a control and an initial state the final point of the
corresponding trajectory. The analytical and topological properties of endpoint maps are therefore crucial in analyzing the properties of control systems.
One of the most important additional objective which can be associated with a motion planning task is optimal control, which corresponds to the minimization of a cost (or, equivalently, the maximization of a gain) .
Optimal control theory is clearly deeply interconnected with calculus of variations, even if the noninterchangeable nature of the timevariable results in some important specific features, such as the occurrence of abnormal extremals. Research in optimal control encompasses different aspects, from numerical methods to dynamic programming and nonsmooth analysis, from regularity of minimizers to high order optimality conditions and curvaturelike invariants.
Another domain of control theory with countless applications is stabilization. The goal in this case is to make the system
converge towards an equilibrium or some more general safety region. The main difference with respect to motion planning is that here the control law is constructed in feedback form. One of the most important properties in this context is that of robustness, i.e., the performance of the stabilization protocol in presence of disturbances or modeling uncertainties.
A powerful framework which has been developed to take into account uncertainties and exogenous nonautonomous disturbances is that of hybrid and switched systems , , . The central tool in the stability analysis of control systems is that of control Lyapunov function. Other relevant techniques are based on algebraic criteria or dynamical systems. One of the most important stability property which
is studied in the context of control system is inputtostate stability, which measures how sensitive the system is to an external excitation.
One of the areas where control applications have nowadays the most impressive developments is in the field of biomedicine and neurosciences.
Improvements both in modeling and in the capability of finely actuating biological systems
have concurred in increasing the popularity of these subjects.
Notable advances concern, in particular, identification and control for biochemical networks and models for neural activity .
Therapy analysis from the point of view of optimal control has also attracted a great attention .
Biological models are not the only one in which stochastic processes play an important role. Stockmarkets and energy grids are two major examples where optimal control techniques are applied in the nondeterministic setting. Sophisticated mathematical tools have been developed since several decades to
allow for such extensions. Many theoretical advances have also been required for dealing with complex systems whose description is based on distributed parameters representation and partial differential equations. Functional analysis, in particular, is a crucial tool to tackle the control of such systems .
Let us conclude this section by mentioning another challenging application domain for control theory: the decision by the European Union to fund a flagship devoted to the development of quantum technologies is a symptom of the role that quantum applications are going to play in tomorrow's society. Quantum control is one of the bricks of quantum engineering, and presents many peculiarities with respect to standard control theory, as a consequence of the specific properties of the systems described by the laws of quantum physics. Particularly important for technological applications is the
capability of inducing and reproducing coherent state superpositions and entanglement in a fast, reliable, and efficient way .
At the core of the scientific activity of the team
is the geometric control approach, that is,
a distinctive viewpoint issued in particular from
(elementary) differential geometry, to tackle
questions of
controllability, observability,
optimal control... , .
The emphasis of such a geometric approach to control theory is put on
intrinsic properties of the systems and it is particularly well adapted to study
nonlinear and nonholonomic phenomena.
One of the features of the geometric control approach is its capability of exploiting symmetries and intrinsic structures of control systems.
Symmetries and intrinsic structures can be used to characterize minimizing trajectories, prove regularity properties and describe invariants.
An egregious example
is given by mechanical systems, which inherently exhibit Lagrangian/Hamiltonian structures which are naturally expressed using the language of symplectic geometry .
The geometric theory of quantum control, in particular, exploits the rich geometric structure encoded in the Schrödinger equation to
engineer adapted control schemes and to characterize their qualitative properties.
The Lie–Galerkin technique that we proposed starting from 2009
builds on this premises in order to provide powerful tests for the controllability of quantum systems defined on infinitedimensional Hilbert spaces.
Although the focus of geometric control theory is on qualitative properties, its impact can also be disruptive when it is used in combination with quantitative analytical tools, in which case it can dramatically improve the computational efficiency. This is the case in particular in optimal control. Classical optimal control techniques (in particular, Pontryagin Maximum Principle, conjugate point theory, associated numerical methods) can be significantly improved by combining them with powerful modern techniques of geometric optimal control, of the theory of numerical continuation, or of dynamical system theory , . Geometric optimal control allows the development of general techniques, applying to wide classes of nonlinear optimal control problems, that can be used to characterize the behavior of optimal trajectories and in particular to establish regularity properties for them and for the cost function. Hence, geometric optimal control can be used to obtain powerful optimal syntheses results and to provide deep geometric insights into many applied problems. Numerical optimal control methods with geometric insight are in particular important to handle subtle situations such as rigid optimal paths and, more generally, optimal syntheses exhibiting abnormal minimizers.
Optimal control is not the only area where the geometric approach has a great impact.
Let us mention, for instance, motion planning, where different geometric approaches have been developed:
those based on the Lie algebra associated with the control system
, , those based on the differentiation of nonlinear flows
such as the return method, , and
those exploiting the differential flatness of the system .
Geometric control theory is not only a powerful framework to investigate control systems,
but also a useful tool to model and study phenomena that are not a priori controlrelated.
Two occurrences of this property play an important role in the activities of CAGE:
Examples of the first type, concern, for instance, hypoelliptic heat kernels or shape optimization . Examples of the second type are inactivation principles in human motricity or neurogeometrical models for image representation of the primary visual cortex in mammals .
A particularly relevant class of
control systems, both from the point of view of
theory and applications,
is characterized by the linearity of the controlled vector field with respect to the
control parameters.
When the controls are unconstrained in norm, this means that the admissible velocities form a distribution in the tangent bundle to the state manifold. If the distribution is equipped with a pointdependent quadratic form (encoding the cost of the control), the resulting geometrical structure is said to be
subRiemannian.
SubRiemannian geometry appears as the underlying geometry of nonlinear control systems: in a similar way
as the linearization of a control system provides local informations
which are
readable using the Euclidean metric scale, subRiemannian geometry provides an adapted nonisotropic class of lenses which are often much more informative.
As such, its study is fundamental for control design. The importance of subRiemannian geometry goes beyond control theory and it is an active field of research both in differential geometry ,
geometric measure theory
and hypoelliptic operator theory .
The geometric control approach has historically been related to the development of finitedimensional control theory. However, its impact in the analysis of distributed parameter control systems and in particular systems of controlled partial differential equations has been growing in the last decades, complementing analytical and numerical approaches, providing dynamical, qualitative and intrinsic insight . CAGE's ambition is to be at the core of this development in the years to come.
A suggestive application of subRiemannian geometry and in particular of hypoelliptic diffusion comes from a model of geometry of vision describing the functional architecture of the primary visual cortex V1.
In 1958, Hubel and Wiesel (Nobel in 1981)
observed that the visual cortex V1 is endowed with the socalled pinwheel structure, characterized by neurons grouped into orientation columns, that are sensible both to positions and directions .
The mathematical rephrasing of this discovery is that the visual cortex lifts an image from
A simplified version of the model can be described as follows: neurons of V1 are grouped into orientation columns, each of them
being sensitive to visual stimuli at a given point of the retina and for a given direction
on it. The retina is
modeled by the real plane, i.e., each point is represented by a pair
Orientation columns are connected between them in two different ways. The first kind of connections are the vertical (inhibitory) ones, which connect orientation columns belonging to the same hypercolumn and sensible to similar directions. The second kind of connections are the horizontal (excitatory) connections, which connect neurons belonging to different (but not too far) hypercolumns and sensible to the same directions. The resulting metric structure is subRiemannian and the model obtained in this way provides a convincing explanation in terms of subRiemannian geodesics of gestalt phenomena such as Kanizsa illusory contours.
The subRiemannian model for image representation of V1
has a great potential of yielding powerful
bioinspired
image processing
algorithms , .
Image inpainting, for instance, can be implemented by reconstructing an incomplete image by activating orientation columns in the missing regions
in accordance with subRiemannian nonisotropic constraints.
The process intrinsically defines an hypoelliptic heat equation on
We have been working on the model and its software implementation since 2012. This work has been supported by several project, as the ERC starting grant GeCoMethods and the ERC Proof of Concept ARTIV1 of U. Boscain, and the ANR GCM.
A parallel approach that we will pursue and combine with this first one is based on pattern matching in the group of diffeomorphisms. We want to extend this approach, already explored in the Riemannian setting , , to the general subRiemannian framework.
The paradigm of the approach is the following:
consider a distortable object, more or less rigid, discretized into a certain number of points. One may track its distortion by considering the paths drawn by these points. One would however like to know how the object itself (and not its discretized version) has been distorted. The study in , shed light on the importance of Riemannian geometry in this kind of problem. In particular, they study the Riemannian submersion obtained by making the group of diffeomorphisms act transitively on the manifold formed by the points of the discretization, minimizing a certain energy so as to take into account the whole object.
Settled as such, the problem is Riemannian, but if one considers objects involving connections, or submitted to nonholonomic constraints,
like in medical imaging where one tracks the motions of organs, then one comes up with a subRiemannian problem. The transitive group is then far bigger, and the aim is to lift curves submitted to these nonholonomic constraints into curves in the set of diffeomorphisms satisfying the corresponding constraints, in a unique way and minimizing an energy (giving rise to a subRiemannian structure).
The goal of quantum control is to design efficient protocols for tuning the occupation probabilities of the energy levels of a system. This task is crucial in atomic and molecular physics, with applications ranging from photochemistry to nuclear magnetic resonance and quantum computing. A quantum system may be controlled by exciting it with one or several external fields, such as magnetic or electric fields. The goal of quantum control theory is to adapt the tools originally developed by control theory and to develop new specific strategies that tackle and exploit the features of quantum dynamics (probabilistic nature of wavefunctions and density operators, measure and wavefunction collapse, decoherence, ...). A rich variety of relevant models for controlled quantum dynamics exist, encompassing lowdimensional models (e.g., singlespin systems) and PDEs alike, with deterministic and stochastic components, making it a rich and exciting area of research in control theory.
The controllability of quantum system
is
a wellestablished topic when the state space
is finitedimensional , thanks to general controllability methods for leftinvariant control systems on compact Lie groups , .
When the state space
is
infinitedimensional, it is known that in general
the bilinear Schrödinger equation is not exactly controllable . Nevertheless, weaker
controllability properties, such as approximate controllability or controllability between eigenstates of the internal Hamiltonian
(which are the most relevant physical states), may hold.
In certain cases, when the state space
is a function space on a 1D manifold, some rather precise description of the set of reachable states has
been provided . A similar description for higherdimensional manifolds seems intractable and at the moment only approximate controllability results
are available , , .
The most widely applicable tests for controllability of quantum systems in infinitedimensional Hilbert spaces are based on the Lie–Galerkin technique, , . They allow, in particular, to show that the controllability property is generic among this class of systems .
A family of algorithms
which are specific to quantum systems are those based on
adiabatic evolution , , .
The basic principle of adiabatic control is
that the flow of a slowly varying Hamiltonian can be approximated (up to a phase factor) by a quasistatic evolution, with a precision proportional to the velocity of variation of the Hamiltonian.
The advantage of the adiabatic approach is that it is constructive and produces control laws which are both smooth and
robust to parameter uncertainty. The paradigm is based on
the adiabatic perturbation theory developed in mathematical physics , , ,
where it plays an important role for understanding molecular dynamics.
Approximation theory by adiabatic perturbation can be used to
describe the evolution of the occupation probabilities of
the energy levels of a slowly varying
Hamiltonian.
Results from the last 15 years, including those by members of our team , , have highlighted the effectiveness of control techniques based on adiabatic path following.
Switched and hybrid systems constitute a broad framework for the description of the heterogeneous aspects of systems in which continuous dynamics (typically pertaining to physical quantities) interact with discrete/logical components. The development of the switched and hybrid paradigm has been motivated by a broad range of applications, including automotive and transportation industry , energy management and congestion control .
Even if both controllability and observability of switched and hybrid systems have attracted much research efforts, the central role in their study is played by the problem of stability and stabilizability. The goal is to determine whether a dynamical or a control system whose evolution is influenced by a timedependent signal is uniformly stable or can be uniformly stabilized , . Uniformity is considered with respect to all signals in a given class. Stability of switched systems lead to several interesting phenomena. For example, even when all the subsystems corresponding to a constant switching law are exponentially stable, the switched systems may have divergent trajectories for certain switching signals . This fact illustrates the fact that stability of switched systems depends not only on the dynamics of each subsystem but also on the properties of the class of switching signals which is considered.
The most common class of switching signals which has been considered in the literature is made of all piecewise constant signals.
In this case uniform stability of the system is equivalent to the existence of a common quadratic Lyapunov function . Moreover, provided that the system has finitely many modes, the Lyapunov function can be taken polyhedral or polynomial , , . A special role in the switched control literature has been played by common quadratic Lyapunov functions, since their existence can be tested rather efficiently (see the surveys , and the references therein). It is known, however, that the existence of a common quadratic Lyapunov function is not necessary for the global uniform exponential stability of a linear switched system with finitely many modes. Moreover, there exists no uniform upper bound on the minimal degree of a common polynomial Lyapunov function . More refined tools rely on multiple and nonmonotone Lyapunov functions . Let us also mention linear switched systems technics based on the analysis of the Lie algebra generated by the matrices corresponding to the modes of the system .
For systems evolving in the plane, more geometrical tests apply, and yield a complete characterization of the stability , . Such a geometric approach also yields sufficient conditions for uniform stability in the linear planar case .
In many
situations,
it is interesting for modeling purposes to
specify the features
of the switched system by introducing
constrained switching rules. A typical constraint is that each mode is activated for at least a fixed minimal amount of time, called the dwelltime.
Switching rules can also be imposed, for instance, by
a timed automata.
When constraints apply, the common Lyapunov function approach becomes conservative and new tools have to be developed to give more detailed characterizations of stable and unstable systems.
Our approach to constrained switching is based on the idea of relating the analytical properties of the classes of constrained switching laws (shiftinvariance, compactness, closure under concatenation, ...) to the stability behavior of the corresponding switched systems.
One can introduce
probabilistic uncertainties by endowing the classes of admissible signals with suitable probability measures.
One then looks at the corresponding Lyapunov exponents, whose existence is established by the multiplicative ergodic theorem.
The interest of this approach is that probabilistic stability analysis filters out highly `exceptional' worstcase trajectories.
Although less explicitly characterized from a dynamical viewpoint than its deterministic counterpart, the probabilistic notion of uniform exponential stability can be studied
using several
reformulations of Lyapunov exponents proposed in the literature , , .
The theoretical questions raised by the different applicative area will be pooled in a research axis on the transversal aspects of geometric control theory and subRiemannian structures.
We recall that subRiemannian geometry is a generalization of Riemannian geometry, whose birth dates back to Carathéodory's seminal paper on the foundations of Carnot thermodynamics , followed by E. Cartan's address at the International Congress of Mathematicians in Bologna . In the last twenty years, subRiemannian geometry has emerged as an independent research domain, with a variety of motivations and ramifications in several parts of pure and applied mathematics. Let us mention geometric analysis, geometric measure theory, stochastic calculus and evolution equations together with applications in mechanics and optimal control (motion planning, robotics, nonholonomic mechanics, quantum control) , .
One of the main open problems in subRiemannian geometry concerns the regularity of lengthminimizers , .
Lengthminimizers are solutions to a variational problem with constraints and satisfy a firstorder necessary condition resulting from the Pontryagin Maximum Principle (PMP).
Solutions of the PMP are either normal or abnormal.
Normal lengthminimizer are wellknown to be smooth, i.e.,
An interesting set of recent results in subRiemannian geometry concerns the extension to such a setting of the Riemannian notion of sectional curvature. The curvature operator can be introduced in terms of the symplectic invariants of the Jacobi curve , , , a curve in the Lagrange Grassmannian related to the linearization of the Hamiltonian flow. Alternative approaches to curvatures in metric spaces are based either on the associated heat equation and the generalization of the curvaturedimension inequality , or on optimal transport and the generalization of Ricci curvature , , , .
Barbara Gris was awarded the GSI Women in Machine Learning & Data Science Prize at the conference GSI2021
Rémi Robin was part of the winning team of the challenge mathématiques et enterprises de l'AMIES “Modélisation du profil longitudinal de la voie à partir d'enregistrement inclinométriques"

Let us list here our new results in the geometry of vision axis and, more generally, on hypoelliptic diffusion and subRiemannian geometry.
Let us list here our new results in quantum control theory.
Let us list here our new results about stability and stabilization of control systems, on the properties of systems with uncertain dynamics.
Let us list here our new results on controllability beyond the quantum control framework.
Let us list here our new results in optimal control theory beyond quantum control and the subRiemannian framework.
Let us also mention the publication , in honor of Enrique Zuazua, at the occasion of his sixtieth birthday.
Contract CIFRE with ArianeGroup (les Mureaux), 2019–2021, funding the thesis of A. Nayet. Participants : M. Cerf (ArianeGroup), E. Trélat (coordinator).
Contract with MBDA (Palaiseau), 2021–2023. Subject: “Controôle optimal pour la planification de trajectoires et l’estimation des ensembles accessibles". Pariticpants: V. Askovic (MBDA & CAGE), E. Trélat (coordinator).
Grant by AFOSR (Air Force Office of Scientific Research), 2020–2023. Participants : Mohab Safey El Din (LIP6), E. Trélat.
Program: H2020EU.1.3.1.  Fostering new skills by means of excellent initial training of researchers
Call for proposal: MSCAITN2017  Innovative Training Networks
Project acronym: QUSCO
Project title: Quantumenhanced Sensing via Quantum Control
Duration: From November 2017 to October 2021.
Coordinator: Christiane Koch
Coordinator for the participant Inria: Ugo Boscain
Abstract: Quantum technologies aim to exploit quantum coherence and entanglement, the two essential elements of quantum physics. Successful implementation of quantum technologies faces the challenge to preserve the relevant nonclassical features at the level of device operation. It is thus deeply linked to the ability to control open quantum systems. The currently closest to market quantum technologies are quantum communication and quantum sensing. The latter holds the promise of reaching unprecedented sensitivity, with the potential to revolutionize medical imaging or structure determination in biology or the controlled construction of novel quantum materials. Quantum control manipulates dynamical processes at the atomic or molecular scale by means of specially tailored external electromagnetic fields. The purpose of QuSCo is to demonstrate the enabling capability of quantum control for quantum sensing and quantum measurement, advancing this field by systematic use of quantum control methods. QuSCo will establish quantum control as a vital part for progress in quantum technologies. QuSCo will expose its students, at the same time, to fundamental questions of quantum mechanics and practical issues of specific applications. Albeit challenging, this reflects our view of the best possible training that the field of quantum technologies can offer. Training in scientific skills is based on the demonstrated tradition of excellence in research of the consortium. It will be complemented by training in communication and commercialization. The latter builds on strong industry participation whereas the former existing expertise on visualization and gamification and combines it with more traditional means of outreach to realize target audience specific public engagement strategies.
The Inria Exploratory Action “StellaCage” is supporting since Spring 2020 a collaboration between CAGE, Yannick Privat (Inria team TONUS), and the startup Renaissance Fusion, based in Grenoble (Francesco Volpe, CEO & Chris Smiet, CSO).
StellaCage approaches the problem of designing better stellarators (yielding better confinement, with simpler coils, capable of higher fields) by combining geometrical properties of magnetic field lines from the control perspective with shape optimization techniques.
Barbara Gris is the PI of a
Bourse Emergence(s) by the Ville de Paris.
Barbara Gris was the organizer of the Journée thématique MIA “Registering Medical Images’' at Henri Poincaré Institute(IHP), October 21, 2021.
Ugo Boscain organized (with Dario Prandi and Alessandro Sarti) the session “SubRiemannian geometry and neuromathematics” at GSI2021, Paris, July 2021.
JeanMichel Coron organized (with Tatsien Li and Zhiqiang Wang) two LIASFMA (Laboratoire International Associé SinoFrançais de Mathématiques Appliquées) International Graduate School on Applied Mathematics, online, the first one in April 2021 and the second one in NovemberDecember 2021.
Ugo Boscain, Eugenio Pozzoli, and Mario Sigalotti were the organizers of the workshop “Quantum day: analysis and control” at LJLL, September 13, 2021.
Mario Sigalotti was one of the organizers of the “Padua Paris SubRiemannian seminar”, Padua, Italy, September 67, 2021.
Emmanuel Trélat is Head of the Laboratoire JacquesLouis Lions (LJLL).
Emmanuel Trélat is member of the Comité d'Honneur du Comité International des Jeux Mathématiques
Emmanuel Trélat gave a general public lecture at Les forums régionaux du savoir, Rouen, October 2021.