Section: New Results

New results: geometric control

Let us list some new results in sub-Riemannian geometry and hypoelliptic diffusion obtained by GECO's members.

• The article [14] presents simple controls that generate motion in the direction of high order Lie brackets. Whereas the naive use of piecewise constant controls requires the number of switchings to grow exponentially with the length of the bracket, we show that such motion is possible with sinusoidal controls whose sum of frequencies equals the length of the bracket. This work is closely related and motivated by the study of the complexity of sub-Riemannian geodesics for generic regular distributions, i.e., whose derived flag has maximal growth vector. Of particular interest is the approximation of curves transversal to the distribution by admissible curves. We also present a surprising example that shows that it is possible to simultaneously kill higher moments without increasing the number of self-intersections of the base curve.

• The curvature discussed in [18] 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 by Riemann. Surprisingly, it works in a very general setting and, in particular, for all sub-Riemannian spaces.

• In [19] we prove sectional and Ricci-type comparison theorems for the existence of conjugate points along sub-Riemannian geodesics. In order to do that, we regard sub-Riemannian structures as a special kind of variational problems. In this setting, we identify a class of models, namely linear quadratic optimal control systems, that play the role of the constant curvature spaces. As an application, we prove a version of sub-Riemannian Bonnet–Myers theorem and we obtain some new results on conjugate points for 3D left-invariant sub-Riemannian structures.

• In the study of conjugate times in sub-Riemannian geometry, linear quadratic optimal control problems show up as model cases. In [1] we consider a dynamical system with a constant, quadratic Hamiltonian $h$, and we characterize the number of conjugate times in terms of the spectrum of the Hamiltonian vector field $H$. We prove the following dichotomy: the number of conjugate times is identically zero or grows to infinity. The latter case occurs if and only if $H$ has at least one Jordan block of odd dimension corresponding to a purely imaginary eigenvalue. As a byproduct, we obtain bounds from below on the number of conjugate times contained in an interval in terms of the spectrum of $H$.

• A 3D almost-Riemannian manifold is a generalized Riemannian manifold defined locally by 3 vector fields that play the role of an orthonormal frame, but could become collinear on some set called the singular set. Under the Hormander condition, a 3D almost-Riemannian structure still has a metric space structure, whose topology is compatible with the original topology of the manifold. Almost-Riemannian manifolds were deeply studied in dimension 2. In [21] 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.

• In [22] we study spectral properties of the Laplace-Beltrami operator on two relevant almost-Riemannian manifolds, namely the Grushin structures on the cylinder and on the sphere. As for general almost-Riemannian structures (under certain technical hypothesis), the singular set acts as a barrier for the evolution of the heat and of a quantum particle, although geodesics can cross it. This is a consequence of the self-adjointness of the Laplace-Beltrami operator on each connected component of the manifolds without the singular set. We get explicit descriptions of the spectrum, of the eigenfunctions and their properties. In particular in both cases we get a Weyl law with dominant term $ElogE$. We then study the effect of an Aharonov-Bohm non-apophantic magnetic potential that has a drastic effect on the spectral properties. Other generalized Riemannian structures including conic and anti-conic type manifolds are also studied. In this case, the Aharonov-Bohm magnetic potential may affect the self-adjointness of the Laplace-Beltrami operator.

• In [28] we investigate the number of geodesics between two points $p$ and $q$ on a contact sub-Riemannian manifold $M$. We show that the count of geodesics on $M$ is controlled by the count on its nilpotent approximation at $p$ (a contact Carnot group). For contact Carnot groups we give sharp bounds for a generic point $q$. Removing the genericity condition for $q$, geodesics might appear in families and we prove a similar statement for their topology. We study these families, and in particular we focus on the unexpected appearance of isometrically non-equivalent geodesics: families on which the action of isometries is not transitive. We apply the previous study to contact sub-Riemannian manifolds: we prove that for any given point $p\in M$ there is a sequence of points $p_n$ such that $p_n\to p$ and that the number of geodesics between $p$ and $p_n$ grows unbounded (moreover these geodesics have the property of being contained in a small neighborhood of $p$).

New results on automatic control and motion planning for various type of applicative domains are the following.

• [8] is devoted to the problem of model-based prognostics for a Waste Water Treatment Plant (WWTP). Our aim is to predict degradation of certain parameters in the process, in order to anticipate malfunctions and to schedule maintenance. It turns out that a WWTP, together with the possible malfunction, has a specific structure: mostly, the malfunction appears in the model as an unknown input function. The process is observable whatever this unknown input is, and the unknown input can itself be identified through the observations. Due to this property, our method does not require any assumption of the type “slow dynamics degradation", as is usually assumed in ordinary prognostic methods. Our system being unknown-input observable, standard observer-based methods are enough to solve prognostic problems. Simulation results are shown for a typical WWTP.

• In [9] we study the problem of controlling an unmanned aerial vehicle (UAV) to provide a target supervision and/or 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 predesigned admissible circular trajectory around the target which is itself a fixed point in the space. Our strategy is presented for both high altitude long endurance (HALE) and medium altitude long endurance (MALE) types of UAVs.

• In [12] we study how a particular spatial structure with a buffer impacts the number of equilibria and their stability in the chemostat model. We show that the occurrence of a buffer can allow a species to setup or on the opposite to go to extinction, depending on the characteristics of the buffer. For non-monotonic response function, we characterize the buffered configurations that make the chemostat dynamics globally asymptotically stable, while this is not possible with single, serial or parallel vessels of the same total volume and input flow. These results are illustrated with the Haldane kinetic function.

• In [15] and [25] we present new results on the path planning problem in the case study of the car with trailers. We formulate the problem in the framework of optimal nonholonomic interpolation and we use standard techniques of nonlinear optimal control theory for deriving hyperelliptic signals as controls for driving the system in an optimal way. The hyperelliptic curves contain as many loops as the number of nonzero Lie brackets generated by the system. We compare the hyperelliptic signals with the ordinary Lissajous-like signals that appear in the literature, we conclude that the former have better performance.

• In [27] we consider affine-control systems, i.e., systems in the form $\dot{q}\left(t\right)=f_0\left(q\left(t\right)\right)+{\sum }_{i=1}^m{u}_i\left(t\right)f_i\left(q\left(t\right)\right)$. Here, the point $q$ belongs to a smooth manifold $M$, the $f_i$'s are smooth vector fields on $M$. This type of system appears in many applications for mechanical systems, quantum control, microswimmers, neuro-geometry of vision...

We conclude the section by mentioning the book [17] that we edited, collecting some papers in honour of Andrei A. Agrachev for his 60th birthday. The book contains new results on sub-Riemannian geometry and more generally on the geometric theory of control.