## Section: Application Domains

### Joint theoretical core

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 [92], followed by E. Cartan's address at the International Congress of Mathematicians in Bologna [93]. 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) [60], [61].

One of the main open problems in sub-Riemannian geometry concerns the regularity of length-minimizers [63], [130].
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., ${C}^{\infty}$, as it follows by the Hamiltonian nature of the PMP.
The question of regularity is then reduced to abnormal length-minimizers.
If the sub-Riemannian structure has step 2, then abnormal length-minimizers can be excluded and thus every length-minimizer is smooth. For step 3 structures, the situation is already more complicated and smoothness of length-minimizers is known only for
Carnot groups [114], [149].
The question of regularity of length-minimizers is not restricted to the smoothness in the ${C}^{\infty}$ sense.
A recent result prove that length-minimizers, for sub-Riemannian structures of any step, cannot have corner-like singularities [107].
When the sub-Riemannian structure is analytic, more is known on the size of the set of points where a length-minimizer can lose
analyticity [148], regardless of the rank and of the step of the distribution.

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 [67], [116], [64], 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 [72], [73] or on optimal transport and the generalization of Ricci curvature [145], [144], [122], [69].