Mathematics Area | Geometric Control Theory Group Geometric Control Theory sub-Riemannian geometry, topology and applications Research Group Faculty: A. A. Agrachev Postdocs: I. Munive PhD students: D. Barilari (PhD ’11), A. Lerario (PhD ’11), D. Prandi, A. Gentile, L. Rizzi, E. Paoli, P. Silveira Diaz, F. Boarotto, C. Biolo Sub-Riemannian geometry Sub-Riemannian geometry is a generalization of Riemannian geometry. Roughly speaking, to measure distances in a sub-Riemannian manifold, you are allowed to go only along curves tangent to so-called horizontal subspaces. The resulting metric spaces have very interesting and peculiar features, sometimes dramatically different from their Riemannian counterpart. • In [1] we define the curvature of affine optimal control problems, a general framework including geometrical structures such as (sub)-Riemannian and (sub)-Finsler manifolds. • In [2] we develop comparison theorems for conjugate points along sub-Riemannian geodesics. In particular, we prove that if the sub-Riemannian curvature is bounded, then the first conjugate time is controlled by models called LQ optimal control problems. We prove average versions of these results, and a sub-Riemannian Bonnet-Myers theorem. • Any Carnot group has the measure contraction property MCP(0, N ) for some N . There are no sharp results for the best possible value of N . We conjecture that the best N is the geodesic dimension: a new intrinsic invariant, different from the topological and Hausdorff dimensions, obtained in [1]. • In [3] we study how many geodesics join two points on a contact manifold. In particular we perform the count of the number of geodesics between two points on corank-one Carnot groups and we “pass” the count to contact manifold via nilpotent approximation. • The definition of a canonical Laplace operator in sub-Riemannian geometry need a con- struction of a canonical volume. In [4] we investigated the regularity of Hausdorff vol- ume, which is intrinsic, by proving that in general, it is not smooth. • In [5] we study Popp’s volume, an intrinsic smooth measure defined for sub-Riemannian spaces. We obtain an explicit formula in terms of the Lie brackets of the structure and we prove that Popp’s volume is the unique volume preserved by sub-Riemannian isometries. • Sub-Riemannian geometry is the natural geometry underlying the theory of hypoelliptic operators. In [6] we consider the small time asymptotics of the heat kernel at the cut locus, i.e. points of non-smoothness of the distance. In [7] we study the geometric mean- ing of the coefficients in the diagonal heat kernel asymptotics for 3D contact structures, expressing them in terms of the invariants χ and κ. In [8] we are generalizing these results for hypoelliptic operators associated with affine optimal control problems, using pertu- bative methods. In particular, for the LQ case, the order of degeneracy in the expansion on the diagonal is equal to the geodesic dimension, introduced in [1]. • We study global distance estimates and uniform local volume estimates in a large class of sub-Riemannian manifolds. Our main device is the generalized curvature dimension inequality introduced in [9]. We use it to obtain sharp inequalities for solutions of the sub-Riemannian heat equation. • We classified 3D sub-Riemannian structures up to local isometries (see [10]) and con- formal transformations (see [11]). The following picture represents both classifications: κ χ h 3 solv + solv - sh(2) se(2) sl e (2) su(2) sl h (2) a(R) ⊕ R [1] A. A. Agrachev, D. Barilari, L. Rizzi, The curvature: a variational approach, preprint (2013) [2] D. Barilari, L. Rizzi, Comparison theorems for conjugate points in sub-Riemannian geometry, preprint (2014) [3] A. Lerario, L. Rizzi, How many geodesics join two points on a contact sub-Riemannian manifold?, preprint (2014) [4] A. A. Agrachev, D. Barilari, U. Boscain, On the Hausdorff volume in sub-Riemannian geometry, CVPDE (2013) [5] D. Barilari, L. Rizzi, A formula for Popp’s volume in sub-Riemannian geometry - AGMS (2012) [6] D. Barilari, U. Boscain, R. Neel, Small time heat kernel asymptotics at the sub-Riemannian cut locus, JDG (2012) [7] D. Barilari, Trace heat kernel asymptotics in 3D contact sub-Riemannian geometry, JMS (2013) [8] E. Paoli, Small time asymptotics of the heat kernel for hypoelliptic operators with drift (PhD thesis, in preparation) [9] I. Munive et al., Volume comparison theorems for negatively curved sub-Riemannian manifolds, preprint (2012) [10] A. A. Agrachev, D. Barilari, Sub-Riemannian structures on 3D Lie groups, JDCS (2012) [11] F. Boarotto, Conformal classification of 3D contact structures (PhD thesis, in preparation) Topology and real algebraic geometry • In [1] a quantitative study of the topology of loop spaces in sub-Riemannian geometry was initiated, using an infinite-dimensional Morse-Bott theory approach. For step-two, corank k Carnot groups, one can consider the sets Ω s z of all admissible paths joining the origin with a vertical point z with energy less than s . This set contains approximatively O (s k ) many critical manifolds (families of geodesics), but its topology grows much slower: b (Ω s z ) ≤ O (s k -1 ). This inequality exhibits sharp contrast with Morse-Bott prediction. • The problem of computing the topology of a semialgebraic set X described by quadratic inequalities requires the use of a spectral sequence which encodes the way the number of positive eigenvalues varies on all possible linear combinations of the defining polynomials [2]. This approach is quite effective and leads to many surprising applications [3,4,5]. • How many ovals does a random real algebraic plane curve of degree d has? This is related to a random approach to Hilbert’s Sixteenth Problem. The answer E d depends on the choice of the probability distribution but reduces to classical random univariate polynomials using Random Matrix Theory: slicing the curve with a line produces a random polynomial with N d many zeroes and the random curve has [6,7]: E d ∼ Θ(N 2 d ) many ovals when d →∞. [1] A. A. Agrachev, A. Gentile, A. Lerario, Geodesics and horizontal path-spaces in Carnot Groups, preprint (2013) [2] A. A. Agrachev, A. Lerario, Systems of quadratic inequalities, Proc. London Math. Soc. (2012) [3] A. Lerario, Random matrices and the expected topology of quadric hypersurfaces, to appear in Proc. AMS [4] A. Lerario, Convex pencils of real quadratic forms, Discrete and Computational Geometry 48 (2012) [5] A. Lerario, Complexity of intersection of real quadrics and determinantal varieties, preprint (2012) [6] A. Lerario, E. Lundberg, Statistics on Hilbert’s Sixteenth problem, to appear in IMRN [7] A. Lerario, E. Lundberg, Gap probabilities and applications to geometry and random topology, preprint (2013) Applications • The motion planning problem is a natural problem in control theory, consisting in finding an admissible trajectory connecting a given pair of points, usually under some constraints, e.g., avoiding obstacles. This can be achieved by first finding a non-admissible curve solving the problem, and then tracking it by an admissible trajectory. Our work focused on estimating the complexity of this second step in the case of control-affine systems. [1,2,3] • Another application of sub-Riemannian geometry and hypoelliptic diffusion, image recon- struction, comes from a model of geometry of human vision due to Petitot, Citti, and Sarti. We are developing a more sophisticated algorithm which is fast enough to be usable on real non-academic images. Moreover we are applying the ideas contained in the Petitot- Citti-Sarti model to image recognition, by analysing the associated bispectrum. [4,5] x Γ y [1] D. Prandi, Hölder equivalence of the value function for control-affine systems, to appear on ESAIM: COCV. [2] F. Jean, D. Prandi, Complexity of control-affine motion planning, preprint (2013). [3] F. Jean, D. Prandi, Various notion of complexity for control-affine motion planning, preprint (2013) [4] U. Boscain, R. Chertovskih, J.-P. Gauthier and A. Remizov, Hypoelliptic diffusion and human vision: a semi-discrete new twist, to appear on SIAM J. Imag. Sc. [5] U. Boscain, J.-P. Gauthier, D. Prandi, A. Remizov, Image reconstruction via non-isotropic diffusion in Dubins/Reed- Shepp-like control systems, preprint (2014). Related topics in Control Theory • We studied a LQ optimal control problem with Hamiltonian vector field H (real 2n × 2n Hamiltonian matrix). The solutions are optimal up to some t (the first conjugate time); if H has at least one odd-dimensional Jordan block corresponding to a pure imaginary eigenvalue, the number of conjugate times in the interval [0, T ] grows to infinity for T → ±∞, otherwise, there are no conjugate times. • Time-optimal trajectories do not always have continuous controls. For instance, the system x = f (x )+ ug (x ) in R 2 , with bounded scalar controls |u | 6 1 and suitable properties on f , g , presents time-optimal trajectories with piecewise constant ±1 controls. Our work is focusing on the behaviour of time-optimal control’s switching for control-affine systems with drift, when the space of control parameters is a closed unitary ball. [1] A. A. Agrachev, L. Rizzi, P. Silveira, On conjugate times of LQ optimal control problems, preprint (2014) [2] C. Biolo, Switching in time-optimal problem (PhD thesis, in preparation) Conferences, Workshops and other activities • Geometry, Analysis and Dynamics on Sub-Riemannian Manifolds, IHP Trimester, Paris, Sep.-Dec. 2014 • Geometric control theory and analysis on metric structures - Lake Bajkal - August 2014 • Geometry and Control - Steklov Mathematical Institute, Moscow, April 2014 • Mathematical Control in Trieste, Trieste, December 2013 • ICTP-SISSA-Moscow School on Geometry and Dynamics - June 2013 • Non Linear Control: Geometric Methods and Applications - Florence, April 2013 • INDAM meeting on Geometric Control and sub-Riemannian Geometry, Cortona, May 2012 • Geometric Control Theory blog: notes, seminars and more http://geomcontrol.wordpress.com/