Section: Research Program
Scientific foundations
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... [70], [111]. 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 [93]. The geometric theory of quantum control, in particular, exploits the rich geometric structure encoded in the SchrÃ¶dinger equation to engineer adapted control schemes and to characterize their qualitative properties. The Lieâ€“Galerkin technique that we proposed starting from 2009 [96] 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 [152], [139]. 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 [132], [122], those based on the differentiation of nonlinear flows such as the return method [101], [100], and those exploiting the differential flatness of the system [105].
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:

geometric control theory as a tool to investigate properties of mathematical structures;

geometric control theory as a modeling tool for neurophysical phenomena and for synthesizing biomimetic algorithms based on such models.
Examples of the first type, concern, for instance, hypoelliptic heat kernels [68] or shape optimization [76]. Examples of the second type are inactivation principles in human motricity [79] or neurogeometrical models for image representation of the primary visual cortex in mammals [90].
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 [130], geometric measure theory [72] and hypoelliptic operator theory [82].
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 [99]. CAGE's ambition is to be at the core of this development in the years to come.