Section: New Results

New results

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

• On a sub-Riemannian manifold we define two type of Laplacians. The macroscopic Laplacian ${\Delta }_{\omega }$, as the divergence of the horizontal gradient, once a volume $\omega$ is fixed, and the microscopic Laplacian, as the operator associated with a geodesic random walk. In [1] we consider a general class of random walks, where all sub-Riemannian geodesics are taken in account. This operator depends only on the choice of a complement $c$ to the sub-Riemannian distribution, and is denoted $L_c$. We address the problem of equivalence of the two operators. This problem is interesting since, on equiregular sub-Riemannian manifolds, there is always an intrinsic volume (e.g. Popp's one $P$) but not a canonical choice of complement. The result depends heavily on the type of structure under investigation:

  • On contact structures, for every volume $\omega$, there exists a unique complement $c$ such that ${\Delta }_{\omega }=L_c$.

  • On Carnot groups, if $H$ is the Haar volume, then there always exists a complement $c$ such that ${\Delta }_H=L_c$. However this complement is not unique in general.

  • For quasi-contact structures, in general, ${\Delta }_P=L_c$ for any choice of c. In particular, $L_c$ is not symmetric w.r.t. Popp's measure. This is surprising especially in dimension 4 where, in a suitable sense, ${\Delta }_P$ is the unique intrinsic macroscopic Laplacian.

  A crucial notion that we introduce here is the $N$-intrinsic volume, i.e. a volume that depends only on the set of parameters of the nilpotent approximation. When the nilpotent approximation does not depend on the point, a $N$-intrinsic volume is unique up to a scaling by a constant and the corresponding $N$-intrinsic sub-Laplacian is unique. This is what happens for dimension smaller or equal than 4, and in particular in the 4-dimensional quasi-contact structure mentioned above.

• In sub-Riemannian geometry the coefficients of the Jacobi equation define curvature-like invariants. We show in [4] 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.

• In [6] we study the cut locus of the free, step two Carnot groups ${𝔾}_k$ with $k$ generators, equipped with their left-invariant Carnot-Carathéodory metric. In particular, we disprove the conjectures on the shape of the cut loci proposed by O. Myasnichenko, by exhibiting sets of cut points $C_k\subset {𝔾}_k$ which, for $k\ge 4$, are strictly larger than conjectured ones. While the latter were, respectively, smooth semi-algebraic sets of codimension $\Theta \left(k^2\right)$ and semi-algebraic sets of codimension $\Theta \left(k\right)$, the sets $C_k$ are semi-algebraic and have codimension 2, yielding the best possible lower bound valid for all $k$ on the size of the cut locus of ${𝔾}_k$. Furthermore, we study the relation of the cut locus with the so-called abnormal set. Finally, and as a straightforward consequence of our results, we derive an explicit lower bound for the small time heat kernel asymptotics at the points of $C_k$. The question whether $C_k$ coincides with the cut locus for $k\ge 4$ remains open.

New results on complex systems with hybrid or switched components are the following.

• In [2] we address the exponential stability of a system of transport equations with intermittent damping on a network of $N\ge 2$ circles intersecting at a single point $O$. The $N$ equations are coupled through a linear mixing of their values at $O$, described by a matrix $M$. The activity of the intermittent damping is determined by persistently exciting signals, all belonging to a fixed class. The main result is that, under suitable hypotheses on $M$ and on the rationality of the ratios between the lengths of the circles, such a system is exponentially stable, uniformly with respect to the persistently exciting signals. The proof relies on an explicit formula for the solutions of this system, which allows one to track down the effects of the intermittent damping.

• In [3] we study the relative controllability of linear difference equations with multiple delays in the state by using a suitable formula for the solutions of such systems in terms of their initial conditions, their control inputs, and some matrix-valued coefficients obtained recursively from the matrices defining the system. Thanks to such formula, we characterize relative controllability in time $T$ in terms of an algebraic property of the matrix-valued coefficients, which reduces to the usual Kalman controllability criterion in the case of a single delay. Relative controllability is studied for solutions in the set of all functions and in the function spaces $L^p$ and $C^k$. We also compare the relative controllability of the system for different delays in terms of their rational dependence structure, proving that relative controllability for some delays implies relative controllability for all delays that are “less rationally dependent” than the original ones, in a sense that we make precise. Finally, we provide an upper bound on the minimal controllability time for a system depending only on its dimension and on its largest delay.

Finally, a new contribution has been proposed in the domain of the control of quantum systems. More precisely, in [5] we consider the bilinear Schrödinger equation with discrete-spectrum drift. We show, for $n\in \mathbb{N}$ arbitrary, exact controllability in projections on the first $n$ given eigenstates. The controllability result relies on a generic controllability hypothesis on some associated finite-dimensional approximations. The method is based on Lie-algebraic control techniques applied to the finite-dimensional approximations coupled with classical topological arguments issuing from degree theory.