Section: New Results

New results: geometric control

We start by presenting some results on the design of motion planning and tracking algorithms.

• In [10] we present an iterative steering algorithm for nonholonomic systems (also called driftless control-affine systems) and we prove its global convergence under the sole assumption that the Lie Algebraic Rank Condition (LARC) holds true everywhere. That algorithm is an extension of the one introduced in [65] for regular systems. The first novelty here consists in the explicit algebraic construction, starting from the original control system, of a lifted control system which is regular. The second contribution of the paper is an exact motion planning method for nilpotent systems, which makes use of sinusoidal control laws and which is a generalization of the algorithm described in [83] for chained-form systems.

• [6] and [5] are about motion planning for kinematic systems, and more particularly $\epsilon$-approximations of non-admissible trajectories by admissible ones. This is done in a certain optimal sense. The resolution of this motion planing problem is showcased through the thorough treatment of the ball with a trailer kinematic system, which is a non-holonomic system with flag of type $(2,3,5,6)$.

Application-oriented results about motion planning are contained in [15] . The paper proposes in particular a strategy for providing Unmanned Aerial Vehicles with a certain degree of autonomy, via autonomous planification/replanification strategies.

Let us list some new results in sub-Riemannian geometry and hypoellitpic diffusion.

• In [1] we study the Radon-Nikodym derivative of the spherical Hausdorff measure with respect to a smooth volume for a regular sub-Riemannian manifold . We prove that this is the volume of the unit ball in the nilpotent approximation and it is always a continuous function. We then prove that up to dimension 4 it is smooth, while starting from dimension 5, in corank 1 case, it is $C^3$ (and $C^4$ on every smooth curve) but in general not $C^5$. These results answer to a question addressed by Montgomery about the relation between two intrinsic volumes that can be defined in a sub-Riemannian manifold, namely the Popp and the Hausdorff volume. If the nilpotent approximation depends on the point (that may happen starting from dimension 5), then they are not proportional, in general.

• In [9] we study the Laplace–Beltrami operator on generalized Riemannian structures on orientable surfaces for which a local orthonormal frame is given by a pair of vector fields that can become collinear. Under the assumption that the structure is 2-step Lie bracket generating, we prove that the Laplace–Beltrami operator is essentially self-adjoint and has discrete spectrum. As a consequence, a quantum particle cannot cross the singular set (i.e., the set where the vector fields become collinear) and the heat cannot flow through the singularity.

• For an equiregular sub-Riemannian manifold $M$, Popp's volume is a smooth volume which is canonically associated with the sub-Riemannian structure, and it is a natural generalization of the Riemannian one. In [4] we prove a general formula for Popp's volume, written in terms of a frame adapted to the sub-Riemannian distribution. As a first application of this result, we prove an explicit formula for the canonical sub-Laplacian, namely the one associated with Popp's volume. Finally, we discuss sub-Riemannian isometries, and we prove that they preserve Popp's volume. We also show that, under some hypotheses on the action of the isometry group of $M$, Popp's volume is essentially the unique volume with such a property.

• In [21] , for a sub-Riemannian manifold provided with a smooth volume, we relate the small time asymptotics of the heat kernel at a point $y$ of the cut locus from $x$ with roughly “how much” $y$ is conjugate to $x$. This is done under the hypothesis that all minimizers connecting $x$ to $y$ are strongly normal, i.e. all pieces of the trajectory are not abnormal. Our result is a refinement of the one of Leandre $4tlog{p}_t(x,y)\to -d^2(x,y)$ for $t\to 0$, in which only the leading exponential term is detected. Our results are obtained by extending an idea of Molchanov from the Riemannian to the sub-Riemannian case, and some details we get appear to be new even in the Riemannian context. These results permit us to obtain properties of the sub-Riemannian distance starting from those of the heat kernel and vice versa. For the Grushin plane endowed with the Euclidean volume we get the expansion $p_t(x,y)\sim t^{-5/4}exp(-d^2(x,y)/4t)$ where $y$ is reached from a Riemannian point $x$ by a minimizing geodesic which is conjugate at $y$. In [22] we investigate the small time heat kernel asymptotics on the cut locus on the class of two-spheres of revolution, which is the simplest class of 2-dimensional Riemannian manifolds different from the sphere with nontrivial cut-conjugate locus. We determine the degeneracy of the exponential map near a cut-conjugate point and present the consequences of this result to the small time heat kernel asymptotics at this point. These results give a first example where the minimal degeneration of the asymptotic expansion at the cut locus is attained.

• In [24] we studied normal forms for 2-dimensional almost-Riemannian structures. The latter are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. Generically, there are three types of points: Riemannian points where the two vector fields are linearly independent, Grushin points where the two vector fields are collinear but their Lie bracket is not, and tangency points where the two vector fields and their Lie bracket are collinear and the missing direction is obtained with one more bracket. In [24] we consider the problem of finding normal forms and functional invariants at each type of point. We also require that functional invariants are complete, in the sense that they permit to recognize locally isometric structures. The problem happens to be equivalent to the one of finding a smooth canonical parameterized curve passing through the point and being transversal to the distribution. For Riemannian points such that the gradient of the Gaussian curvature $K$ is different from zero, we use the level set of $K$ as support of the parameterized curve. For Riemannian points such that the gradient of the curvature vanishes (and under additional generic conditions), we use a curve which is found by looking for crests and valleys of the curvature. For Grushin points we use the set where the vector fields are parallel. Tangency points are the most complicated to deal with. The cut locus from the tangency point is not a good candidate as canonical parameterized curve since it is known to be non-smooth. Thus, we analyse the cut locus from the singular set and we prove that it is not smooth either. A good candidate happens to be a curve which is found by looking for crests and valleys of the Gaussian curvature. We prove that the support of such a curve is uniquely determined and has a canonical parametrization.