Geometric Control Theory€¦ · Geometric Control Theory sub-Riemannian geometry, topology and...
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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/
Transcript of Geometric Control Theory€¦ · Geometric Control Theory sub-Riemannian geometry, topology and...
Mathematics Area | Geometric Control Theory Group
Geometric Control Theorysub-Riemannian geometry, topology and applications
Research GroupFaculty: 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 geometrySub-Riemannian geometry is a generalization of Riemannian geometry. Roughly speaking, tomeasure distances in a sub-Riemannian manifold, you are allowed to go only along curvestangent to so-called horizontal subspaces. The resulting metric spaces have very interestingand peculiar features, sometimes dramatically different from their Riemannian counterpart.
• In [1] we define the curvature of affine optimal control problems, a general frameworkincluding geometrical structures such as (sub)-Riemannian and (sub)-Finsler manifolds.
• In [2] we develop comparison theorems for conjugate points along sub-Riemanniangeodesics. In particular, we prove that if the sub-Riemannian curvature is bounded, thenthe first conjugate time is controlled by models called LQ optimal control problems. Weprove 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. Thereare no sharp results for the best possible value of N. We conjecture that the best Nis the geodesic dimension: a new intrinsic invariant, different from the topological andHausdorff dimensions, obtained in [1].
• In [3] we study how many geodesics join two points on a contact manifold. In particularwe perform the count of the number of geodesics between two points on corank-oneCarnot 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-Riemannianspaces. We obtain an explicit formula in terms of the Lie brackets of the structure and weprove that Popp’s volume is the unique volume preserved by sub-Riemannian isometries.
• Sub-Riemannian geometry is the natural geometry underlying the theory of hypoellipticoperators. In [6] we consider the small time asymptotics of the heat kernel at the cutlocus, 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 resultsfor hypoelliptic operators associated with affine optimal control problems, using pertu-bative methods. In particular, for the LQ case, the order of degeneracy in the expansionon the diagonal is equal to the geodesic dimension, introduced in [1].
• We study global distance estimates and uniform local volume estimates in a large classof sub-Riemannian manifolds. Our main device is the generalized curvature dimensioninequality introduced in [9]. We use it to obtain sharp inequalities for solutions of thesub-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
h3
solv+
solv−
sh(2) se(2)
sle(2) su(2)
slh(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 Ωs
z of all admissible paths joining theorigin with a vertical point z with energy less than s. This set contains approximativelyO(sk) many critical manifolds (families of geodesics), but its topology grows much slower:
b(Ωsz) ≤ O(sk−1).
This inequality exhibits sharp contrast with Morse-Bott prediction.• The problem of computing the topology of a semialgebraic set X described by quadraticinequalities requires the use of a spectral sequence which encodes the way the numberof 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 relatedto a random approach to Hilbert’s Sixteenth Problem. The answer Ed depends on thechoice of the probability distribution but reduces to classical random univariate polynomialsusing Random Matrix Theory: slicing the curve with a line produces a random polynomialwith Nd many zeroes and the random curve has [6,7]:
Ed ∼ Θ(N2d ) 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 curvesolving the problem, and then tracking it by an admissible trajectory. Our work focused onestimating 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, andSarti. We are developing a more sophisticated algorithm which is fast enough to be usableon 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 Γy
[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 imaginaryeigenvalue, 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, thesystem x ′ = f (x) + ug(x) in R2, with bounded scalar controls |u| 6 1 and suitableproperties 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-affinesystems 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/
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