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 sub-Riemannian 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 sub-Riemannian structure is a (possibly rank-varying) Lie bracket generating distribution endowed with a smoothly varying norm.

Sub-Riemannian 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 bio-inspired 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 sub-Riemannian 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 sub-Riemannian metric on the infinite-dimensional 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).

Control theory is one of the components of the forthcoming quantum revolution 1, since manipulation of quantum mechanical systems is ubiquitous in applications such as quantum computation, quantum cryptography, and quantum sensing (in particular, imaging by nuclear magnetic resonance). The efficiency of the control action has a dramatic impact on the quality of the coherence and the robustness of the required manipulation. Minimal time constraints and interaction of time scales are important factors for characterizing the efficiency of a quantum control strategy. Time scales analysis is important for evaluation approaches based on adiabatic approximation theory, which is well-known to improve the robustness of the control strategy. CAGE works for the improvement of evaluation and design tools for efficient quantum control paradigms, especially for what concerns quantum systems evolving in infinite-dimensional Hilbert spaces.

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 so-called end-point 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) 168.
Optimal control theory is clearly deeply interconnected with calculus of variations, even if the non-interchangeable nature of the time-variable results in some important specific features, such as the occurrence of abnormal extremals130. Research in optimal control encompasses different aspects, from numerical methods to dynamic programming and non-smooth analysis, from regularity of minimizers to high order optimality conditions and curvature-like 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 non-autonomous disturbances is that of hybrid and switched systems 171, 132, 159. 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 input-to-state stability155, 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 149 and models for neural activity 118.
Therapy analysis from the point of view of optimal control has also attracted a great attention 152.

Biological models are not the only one in which stochastic processes play an important role. Stock-markets and energy grids are two major examples where optimal control techniques are applied in the non-deterministic 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 165.

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 119.

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... 78, 123.
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 105.
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 108
builds on this premises in order to provide powerful tests for the controllability of quantum systems defined on infinite-dimensional 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 163, 151. 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
144, 134, those based on the differentiation of nonlinear flows
such as the return method112, 113, and
those exploiting the differential flatness of the system 117.

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 control-related.
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 76 or shape optimization 87. Examples of the second type are inactivation principles in human motricity 90 or neurogeometrical models for image representation of the primary visual cortex in mammals 101.

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 point-dependent quadratic form (encoding the cost of the control), the resulting geometrical structure is said to be
sub-Riemannian.
Sub-Riemannian 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, sub-Riemannian geometry provides an adapted non-isotropic class of lenses which are often much more informative.
As such, its study is fundamental for control design. The importance of sub-Riemannian geometry goes beyond control theory and it is an active field of research both in differential geometry 142,
geometric measure theory 81
and hypoelliptic operator theory 94.

The geometric control approach has historically been related to the development of finite-dimensional 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 111. CAGE's ambition is to be at the core of this development in the years to come.

A suggestive application of sub-Riemannian 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 so-called pinwheel structure, characterized by neurons grouped into orientation columns, that are sensible both to positions and directions 122.
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 sub-Riemannian and the model obtained in this way provides a convincing explanation in terms of sub-Riemannian geodesics of gestalt phenomena such as Kanizsa illusory contours.

The sub-Riemannian model for image representation of V1
has a great potential of yielding powerful
bio-inspired
image processing
algorithms 116, 101.
Image inpainting, for instance, can be implemented by reconstructing an incomplete image by activating orientation columns in the missing regions
in accordance with sub-Riemannian non-isotropic 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 164, 139, to the general sub-Riemannian 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 164, 139 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 sub-Riemannian 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 sub-Riemannian 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 low-dimensional models (e.g., single-spin 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 well-established topic when the state space
is finite-dimensional 114, thanks to general controllability methods for left-invariant control systems on compact Lie groups 104, 124.
When the state space
is
infinite-dimensional, it is known that in general
the bilinear Schrödinger equation is not exactly controllable 166. 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 88. A similar description for higher-dimensional manifolds seems intractable and at the moment only approximate controllability results
are available 140, 146, 125.
The most widely applicable tests for controllability of quantum systems in infinite-dimensional Hilbert spaces are based on the Lie–Galerkin technique108, 96, 97. They allow, in particular, to show that the controllability property is generic among this class of systems 137.

A family of algorithms
which are specific to quantum systems are those based on
adiabatic evolution 170, 169, 128.
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 quasi-static 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 95, 145, 162,
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 73, 100, 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 154, energy management 147 and congestion control 138.

Even if both controllability 158 and observability 126 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 time-dependent signal is uniformly stable or can be uniformly stabilized 132, 159. 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 131. 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 141. Moreover, provided that the system has finitely many modes, the Lyapunov function can be taken polyhedral or polynomial 93, 92, 115. 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 133, 153 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 136. More refined tools rely on multiple and non-monotone Lyapunov functions 103. 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 75.

For systems evolving in the plane, more geometrical tests apply, and yield a complete characterization of the stability 102, 82. Such a geometric approach also yields sufficient conditions for uniform stability in the linear planar case 98.

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 dwell-time.
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 (shift-invariance, 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' worst-case 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 89, 110, 167.

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 sub-Riemannian structures.

We recall that sub-Riemannian geometry is a generalization of Riemannian geometry, whose birth dates back to Carathéodory's seminal paper on the foundations of Carnot thermodynamics 106, followed by E. Cartan's address at the International Congress of Mathematicians in Bologna 107. In the last twenty years, sub-Riemannian 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) 83, 84.

One of the main open problems in sub-Riemannian geometry concerns the regularity of length-minimizers 79, 143.
Length-minimizers are solutions to a variational problem with constraints and satisfy a first-order necessary condition resulting from the Pontryagin Maximum Principle (PMP).
Solutions of the PMP are either normal or abnormal.
Normal length-minimizer are well-known to be smooth, i.e.,

An interesting set of recent results in sub-Riemannian 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 77, 129, 74, 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 curvature-dimension inequality 85, 86 or on optimal transport and the generalization of Ricci curvature 157, 156, 135, 80.

The PhD thesis “Stabilisation de systèmes hyperboliques non-linéaires en dimension un d'espace” of our former PhD student Amaury Hayat received two prizes

Let us list here our new results in the geometry of vision axis and, more generally, on hypoelliptic diffusion and sub-Riemannian geometry.

We would also like to mention the monograph 42, which was finally published in 2020.

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 sub-Riemannian framework.

Contract CIFRE with ArianeGroup (les Mureaux), 2019–2021, funding the thesis of A. Nayet. Participants : M. Cerf (ArianeGroup), E. Trélat (coordinator).

A new contract is being signed by E. Trélat with MBDA.

New grant by AFOSR ( Air Force Office of Scientific Research), 2020–2023. Participants : Mohab Safey El Din (LIP6), E. Trélat.

Andrei Agrachev (SISSA, Italy) was expected to start his Inria International Chair in 2020. However, due to Covid-related restrictions, he had to cancel his visit scheduled for the Fall. He will start his Chair spending two months in Paris in Fall 2021.

Hoai-Minh Nguyen spend two months visitng CAGE (October and December), working in particular with Jean-Michel Coron.

Ugo Boscain visited SISSA in January 2020.

Program: H2020-EU.1.3.1. - Fostering new skills by means of excellent initial training of researchers

Call for proposal: MSCA-ITN-2017 - Innovative Training Networks

Project acronym: QUSCO

Project title: Quantum-enhanced 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.

Emmanuel Trélat was the main organizer of the SRGI conference: Sub-Riemannian Geometry and Interactions which was held in Paris in September 2020.
Ugo Boscain was also in the Organizing Committe.

Emmanuel Trélat is Head of the Laboratoire Jacques-Louis Lions (LJLL).