CAGE's activities take place in the field of mathematical control theory, with applications in several directions: control of quantum mechanical systems, stability and stabilization, in particular in presence of uncertain dynamics, optimal control, and geometric models for vision. Although control theory is nowadays a mature discipline, it is still the subject of intensive research because of its crucial role in a vast array of applications. Our focus is on the analytical and geometrical aspects of control applications.

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, motion planning, stability, and
optimal control. The emphasis of such a geometric approach
is in
intrinsic properties, and it is particularly well adapted to study
nonlinear and nonholonomic phenomena 77, 53.
The geometric control approach has historically been
associated with the development of finite-dimensional control theory.
However, its impact in the study 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 by providing dynamical, qualitative, and intrinsic insight 69.
CAGE has the ambition to be at the core of this development.

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 (e.g., Lagrangian or Hamiltonian structures) can be used to characterize minimizing trajectories, prove regularity properties, and describe invariants.
The geometric theory of quantum control, in particular, exploits the rich geometric structure encoded in the Schrödinger equation to
design adapted control schemes and to characterize their qualitative properties.

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) 105.
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 extremals81. 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 92, 82, 98. 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 stability96, 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 90 and models for neural activity 74.
Therapy analysis from the point of view of optimal control has also attracted a great attention 94.

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 102.

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 75.

At the core of the scientific activity of the team
is the geometric control approach.
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 65.
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 in 66
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 100, 93. 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
86, 83, those based on the differentiation of nonlinear flows
such as the return method70, 71, and
those exploiting the differential flatness of the system 73.

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 52 or shape optimization 56. Examples of the second type are inactivation principles in human motricity 58 or neurogeometrical models for image representation of the primary visual cortex in mammals 63.

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 85,
geometric measure theory 54
and hypoelliptic operator theory 59.

Quantum control is one of the bricks of quantum engineering, 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).

Quantum control 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. 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.
CAGE works for the improvement of
quantum control paradigms, especially for what concerns quantum systems evolving in infinite-dimensional Hilbert spaces.
The controllability of quantum system
is
a well-established topic when the state space
is finite-dimensional 72,
thanks to general controllability methods for left-invariant control systems on compact Lie groups 64, 78.
When the state space
is
infinite-dimensional, it is known that in general
the bilinear Schrödinger equation is not exactly controllable 103.
The Lie–Galerkin technique 66
combines finite-dimensional geometric control techniques and the distributed parameter framework
in order to provide the most powerful available tests for the approximate controllability of quantum systems defined on infinite-dimensional Hilbert spaces.
Another important technique to the development of which we
contribute
is adiabatic quantum control.
Adiabatic approximation theory and, in particular, adiabatic evolution 87, 99, 106
is well-known to improve the robustness of the control strategy and is strongly related to time scales analysis.
The advantage of the adiabatic control is that it is constructive and produces control laws which are both smooth and
robust to parameter uncertainty 107, 80, 62.

A control application with a long history and still very challenging open problems is stabilization.
For infinite-dimensional systems, in particular nonlinear ones, the richness of the possible functional analytical frameworks makes feedback stabilization a challenging and active domain of research.
Of particular interest are the different types of stabilization that may be obtained: exponential, polynomial, finite-time, ... Another important aspect of stabilization concerns control of systems with uncertain dynamics, i.e., with dynamics including possibly non-autonomous parameters whose value and
dependence on time cannot be anticipated.
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.
Switched and hybrid
systems constitute a broad framework
for the description
of the heterogeneous 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 95, energy
management 88 and congestion control 84.
Even if both controllability 97 and observability 79 of switched and hybrid systems
raise several important research issues, the central role in their study is played by uniform stability and stabilizabilization
82, 98. Uniformity is considered
with respect to all signals in a given class, and it is well-known
that stability of switched systems depends not only on the dynamics of each subsystem but also on the properties of the considered class of switching signals.
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.
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.
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 57, 68, 104.

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. Geometric optimal control methods are in particular important to handle subtle situations such as rigid optimal paths and, more generally, optimal syntheses exhibiting abnormal minimizers.

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 100, 93.
Applications of optimal control theory considered by CAGE concern, in particular,
motion planning problems for aerospace (atmospheric re-entry, orbit transfer, low cost interplanetary
space missions, ...) 60, 101.

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.
In particular, we use control theory to investigate the properties of sub-Riemannian structures, both
for the sake of mathematical understanding and as a modeling tool for
image and sound perception and processing . We recall that sub-Riemannian geometry is a geometric framework which is used to measure distances in nonholonomic contexts and which has a natural and powerful optimal control interpretation in terms control-linear systems with quadratic cost.
Sub-Riemannian geometry, and in particular the theory of their 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 76, 89, 67, 91. Such a model can be used as a powerful paradigm for bio-inspired image processing, as already illustrated in the literature
63, 61.
Our contributions to geometry of vision 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 55. Nonholonomic constraints can be especially useful to describe distortions of sets of interconnected objects (e.g., motions of organs in medical imaging).

Rémi Robin was awarded with the Prix solennel de thèse de la chancellerie de Paris, the Prix de thèse PGMO 2023, and the Prix de thèse SMAI-GAMNI for his PhD thesis defended in 2022 and obtained while being member of the team CAGE.

Daniele Cannarsa and Mario Sigalotti have been awarded the Brockett-Willems Outstanding Paper Award 2023 for the best paper published in Systems & Control Letters during the two-year period from January 2021 through December 2022 (paper written while Daniele Cannarsa was member of the team CAGE).

Let us list here our new results in quantum control theory.

Let us list here our new results about stability and stabilization of control and hybrid systems.

Let us list here our new results on controllability and motion planning algorithms, including optimal control, optimization beyond the quantum control framework.

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

Contract with MBDA (Palaiseau), 2021–2023. Subject: “Contrôle 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.

Andrei Agrachev (SISSA, Trieste, Italy) made two visits to CAGE (16/1–15/3 and 19/9–18/11) in the framework of his Inria International Chair 2020-2024.

Riccardo Adami (Politecnico di Torino, Italy), March.

The Bourse Emergence(s) de la Ville de Paris “Morphométrie sous contrainte pour l’analyse de données biologiques : un nouvel outil pour la communauté scientifique”, whose principal investigator is Barbara Gris, runs from 2022 to 2025.