Section: New Results

New results: geometric control

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

• In [22] we study the complexity of the motion planning problem for control-affine systems. Such complexities are already defined and rather well-understood in the particular case of nonholonomic (or sub-Riemannian) 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 time-rescaling invariance of these systems, we consider geometric and parametrized curves separately. Then, we give some asymptotic estimates for these quantities.

• In [23] we study the problem of controlling an unmanned aerial vehicle (UAV) to provide a target supervision and to provide convoy protection to ground vehicles. We first present a control strategy based upon a Lyapunov–LaSalle stabilization method to provide supervision of a stationary target. The UAV is expected to join a pre-designed admissible circular trajectory around the target which is itself a fixed point in the space. Our strategy is presented for both HALE (High Altitude Long Endurance) and MALE (Medium Altitude Long Endurance) types UAVs. A UAV flying at a constant altitude (HALE type) is modeled as a Dubins vehicle (i.e. a planar vehicle with constrained turning radius and constant forward velocity). For a UAV that might change its altitude (MALE type), we use the general kinematic model of a rigid body evolving in ${ℝ}^3$. Both control strategies presented are smooth and unlike what is usually proposed in the literature these strategies asymptotically track a circular trajectory of exact minimum turning radius. We then consider the problem of adding to the tracking task an optimality criterion. In particular, we present the time-optimal control synthesis for tracking a circle by a Dubins vehicle. This optimal strategy, although much simpler than the point-to-point time-optimal strategy obtained by P. Souéres and J.-P. Laumond in the 1990s, is very rich. Finally, we propose control strategies to provide supervision of a moving target, that are based upon the previous ones.

• In [26] we prove the continuity and the Hölder equivalence w.r.t. an Euclidean distance of the value function associated with the $L^1$ cost of the control-affine system $\dot{q}=f_0\left(q\right)+{\sum }_{j=1}^m{u}_j{f}_j\left(q\right)$, satisfying the strong Hörmander condition. This is done by proving a result in the same spirit as the Ball-Box theorem for driftless (or sub-Riemannian) systems. The techniques used are based on a reduction of the control-affine system to a linear but time-dependent one, for which we are able to define a generalization of the nilpotent approximation and through which we derive estimates for the shape of the reachable sets. Finally, we also prove the continuity of the value function associated with the $L^1$ cost of time-dependent systems of the form $\dot{q}={\sum }_{j=1}^m{u}_j{f}_j^t\left(q\right)$.

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

• In [1] we provide normal forms for 2D almost-Riemannian structures, which 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. 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 analyze the cut locus from the singular set and we prove that it is not smooth either. A good candidate appears 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.

• The curvature discussed in [14] is a rather far going generalization of the Riemann sectional curvature. We define it for a wide class of optimal control problems: a unified framework including geometric structures such as Riemannian, sub-Riemannian, Finsler and sub-Finsler structures; a special attention is paid to the sub-Riemannian (or Carnot–Caratheodory) metric spaces. Our construction of the curvature is direct and naive, and it is similar to the original approach of Riemann. Surprisingly, it works in a very general setting and, in particular, for all sub-Riemannian spaces.

• In [15] we provide the small-time heat kernel asymptotics at the cut locus in three relevant cases: generic low-dimensional Riemannian manifolds, generic 3D contact sub-Riemannian manifolds (close to the starting point) and generic 4D quasi-contact sub-Riemannian manifolds (close to a generic starting point). As a byproduct, we show that, for generic low-dimensional Riemannian manifolds, the only singularities of the exponential map, as a Lagragian map, that can arise along a minimizing geodesic are $A_3$ and $A_5$ (in the classification of Arnol'd's school). We show that in the non-generic case, a cornucopia of asymptotics can occur, even for Riemannian surfaces.

• In [19] we study the evolution of the heat and of a free quantum particle (described by the Schroedinger equation) on two-dimensional manifolds endowed with the degenerate Riemannian metric $d{s}^2=d{x}^2+{\left|x\right|}^{-2\alpha }d{\theta }^2$, where $x\in ℝ$, $\theta \in 𝕋$ and the parameter $\alpha \in ℝ$. For $\alpha \le -1$ this metric describes cone-like manifolds (for $\alpha =-1$ it is a flat cone). For $\alpha =0$ it is a cylinder. For $\alpha \ge 1$ it is a Grushin-like metric. We show that the Laplace–Beltrami operator $\Delta$ is essentially self-adjoint if and only if $\alpha \notin (-3,1)$. In this case the only self-adjoint extension is the Friedrichs extension ${\Delta }_F$, that does not allow communication through the singular set $\{x=0\}$ both for the heat and for a quantum particle. For $\alpha \in (-3,-1]$ we show that for the Schroedinger equation only the average on $\theta$ of the wave function can cross the singular set, while the solutions of the only Markovian extension of the heat equation (which indeed is ${\Delta }_F$) cannot. For $\alpha \in (-1,1)$ we prove that there exists a canonical self-adjoint extension ${\Delta }_B$, called bridging extension, which is Markovian and allows the complete communication through the singularity (both of the heat and of a quantum particle). Also, we study the stochastic completeness (i.e., conservation of the $L^1$ norm for the heat equation) of the Markovian extensions ${\Delta }_F$ and ${\Delta }_B$, proving that ${\Delta }_F$ is stochastically complete at the singularity if and only if $\alpha \le -1$, while ${\Delta }_B$ is always stochastically complete at the singularity.