Section: New Results
New results: geometric control
Let us list some new results in subRiemannian geometry and hypoelliptic diffusion obtained by GECO's members.

In [12] and [20] we study the subFinsler geometry as a timeoptimal control problem. In particular, we consider nonsmooth and nonstrictly convex subFinsler structures associated with the Heisenberg, Grushin, and Martinet distributions. Motivated by problems in geometric group theory, we characterize extremal curves, discuss their optimality, and calculate the metric spheres, proving their Euclidean rectifiability.

In [18] we compare different notions of curvature on contact subRiemannian manifolds. In particular we introduce canonical curvatures as the coefficients of the subRiemannian Jacobi equation. The main result is that all these coefficients are encoded in the asymptotic expansion of the horizontal derivatives of the subRiemannian distance. We explicitly compute their expressions in terms of the standard tensors of contact geometry. As an application of these results, we obtain a subRiemannian version of the BonnetMyers theorem that applies to any contact manifold.

In subRiemannian geometry the coefficients of the Jacobi equation define curvaturelike invariants. We show in [21] that these coefficients can be interpreted as the curvature of a canonical Ehresmann connection associated to the metric, first introduced by Zelenko and Li. We show why this connection is naturally nonlinear, and we discuss some of its properties.

On a subRiemannian manifold we define in [22] two type of Laplacians. The macroscopic Laplacian, as the divergence of the horizontal gradient, once a volume is fixed, and the microscopic Laplacian, as the operator associated with a geodesic random walk. We consider a general class of random walks, where all subRiemannian geodesics are taken in account. This operator depends only on the choice of a complement to the subRiemannian distribution. We address the problem of equivalence of the two operators. This problem is interesting since, on equiregular subRiemannian manifolds, there is always an intrinsic volume (e.g. Popp's one) but not a canonical choice of complement. The result depends heavily on the type of structure under investigation: we describe the relationship between the two approaches in the case of contact structures, Carnot groups, quasicontact structures.

In [2] we study 3D almostRiemannian manifolds, that is, generalized Riemannian manifolds defined locally by 3 vector fields that play the role of an orthonormal frame, but could become collinear on some singular set. AlmostRiemannian manifolds were deeply studied in dimension 2. In this paper we start the study of the 3D case which appear to be reacher with respect to the 2D case, due to the presence of abnormal extremals which define a field of directions on the singular set. We study the type of singularities of the metric that could appear generically, we construct local normal forms and we study abnormal extremals. We then study the nilpotent approximation and the structure of the corresponding small spheres. We finally give some preliminary results about heat diffusion on such manifolds.
New results on motion planning are the following.

In [7] (written while D. Prandi was PhD student in the team) we study the complexity of the motion planning problem for controlaffine systems. Such complexities are already defined and rather wellunderstood in the particular case of nonholonomic (or subRiemannian) systems. Our aim is to generalize these notions and results to systems with a drift. Accordingly, we present various definitions of complexity, as functions of the curve that is approximated, and of the precision of the approximation. Due to the lack of timerescaling invariance of these systems, we consider geometric and parametrized curves separately. Then, we give some asymptotic estimates for these quantities. As a byproduct, we are able to treat the longtime local controllability problem, giving quantitative estimates on the cost of stabilizing the system near a nonequilibrium point of the drift.

In [11] and [1] we propose new conditions guaranteeing that the trajectories of a mechanical control system can track any curve on the configuration manifold. We focus on systems that can be represented as forced affine connection control systems and we generalize the sufficient conditions for tracking known in the literature. The new results are proved by a combination of averaging procedures by highly oscillating controls and the notion of kinematic reduction.

[17] we introduce the concept of Developmental Partial Differential Equation (DPDE), which consists of a Partial Differential Equation (PDE) on a timevarying manifold with complete coupling between the PDE and the manifold's evolution. In other words, the manifold's evolution depends on the solution to the PDE, and vice versa the differential operator of the PDE depends on the manifold's geometry. DPDE is used to study a diffusion equation with source on a growing surface whose growth depends on the intensity of the diffused quantity. The surface may, for instance, represent the membrane of an egg chamber and the diffused quantity a protein activating a signaling pathway leading to growth. Our main objective is to show controllability of the surface shape using a fixed source with variable intensity for the diffusion. More specifically, we look for a control driving a symmetric manifold shape to any other symmetric shape in a given time interval. For the diffusion we take directly the LaplaceBeltrami operator of the surface, while the surface growth is assumed to be equal to the value of the diffused quantity. We introduce a theoretical framework, provide approximate controllability and show numerical results. Future applications include a specific model for the oogenesis of Drosophila melanogaster.