%0 Journal Article
%D 2014
%T On conjugate times of LQ optimal control problems
%A Andrei A. Agrachev
%A Luca Rizzi
%A Pavel Silveira
%K Optimal control, Lagrange Grassmannian, Conjugate point
%X Motivated by the study of linear quadratic optimal control problems, we consider a dynamical system with a constant, quadratic Hamiltonian, and we characterize the number of conjugate times in terms of the spectrum of the Hamiltonian vector field $\vec{H}$. 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 $\vec{H}$ 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 $\vec{H}$.
%I Springer
%G en
%U http://hdl.handle.net/1963/7227
%1 7261
%2 Mathematics
%4 1
%# MAT/05 ANALISI MATEMATICA
%$ Submitted by Andrei Agrachev (agrachev@sissa.it) on 2013-12-03T08:50:15Z
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%R 10.1007/s10883-014-9251-6
%0 Thesis
%D 2014
%T The curvature of optimal control problems with applications to sub-Riemannian geometry
%A Luca Rizzi
%K Sub-Riemannian geometry
%X Optimal control theory is an extension of the calculus of variations, and deals with the optimal behaviour of a system under a very general class of constraints. This field has been pioneered by the group of mathematicians led by Lev Pontryagin in the second half of the 50s and nowadays has countless applications to the real worlds (robotics, trains, aerospace, models for human behaviour, human vision, image reconstruction, quantum control, motion of self-propulsed micro-organism). In this thesis we introduce a novel definition of curvature for an optimal control problem. In particular it works for any sub-Riemannian and sub-Finsler structure. Related problems, such as comparison theorems for sub-Riemannian manifolds, LQ optimal control problem and Popp's volume and are also investigated.
%I SISSA
%G en
%U http://hdl.handle.net/1963/7321
%1 7367
%2 Mathematics
%4 1
%# MAT/03 GEOMETRIA
%$ Submitted by Luca Rizzi (lrizzi@sissa.it) on 2014-05-23T12:10:38Z
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%] Introduction
1) The curvature of optimal control problems
2) Comparison theorems for conjugate points in sub-Riemannian geometry
3) On conjugate times of LQ optimal control problems
4) A formula for Popp's volume in sub-Riemannian geometry
%0 Report
%D 2013
%T The curvature: a variational approach
%A Andrei A. Agrachev
%A Davide Barilari
%A Luca Rizzi
%K Crurvature, subriemannian metric, optimal control problem
%X The curvature discussed in this paper is a rather far going generalization of the Riemannian 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.
%I SISSA
%G en
%U http://hdl.handle.net/1963/7226
%1 7260
%2 Mathematics
%4 1
%# MAT/03 GEOMETRIA
%$ Submitted by Andrei Agrachev (agrachev@sissa.it) on 2013-12-03T08:46:32Z
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%0 Journal Article
%J Analysis and Geometry in Metric Spaces, vol. 1 (2012), pages : 42-57
%D 2012
%T A formula for Popp\'s volume in sub-Riemannian geometry
%A Luca Rizzi
%A Davide Barilari
%K subriemannian, volume, Popp, control
%X For an equiregular sub-Riemannian manifold M, Popp\'s volume is a smooth\r\nvolume which is canonically associated with the sub-Riemannian structure, and\r\nit is a natural generalization of the Riemannian one. In this paper we prove a\r\ngeneral formula for Popp\'s volume, written in terms of a frame adapted to the\r\nsub-Riemannian distribution. As a first application of this result, we prove an\r\nexplicit formula for the canonical sub-Laplacian, namely the one associated\r\nwith Popp\'s volume. Finally, we discuss sub-Riemannian isometries, and we prove\r\nthat they preserve Popp\'s volume. We also show that, under some hypotheses on\r\nthe action of the isometry group of M, Popp\'s volume is essentially the unique\r\nvolume with such a property.
%B Analysis and Geometry in Metric Spaces, vol. 1 (2012), pages : 42-57
%I SISSA
%G en
%U http://hdl.handle.net/1963/6501
%1 6446
%2 Mathematics
%4 1
%# MAT/03 GEOMETRIA
%$ Submitted by Luca Rizzi (lrizzi@sissa.it) on 2013-02-27T08:20:17Z\nNo. of bitstreams: 2\n1211.2325v2.pdf: 219748 bytes, checksum: 03a6289f4ee77387ad54e210dbb84b60 (MD5)\nagms-2012-0004.pdf: 1247984 bytes, checksum: 137bd44ed6f06bbbe02ed9dae33ab42a (MD5)
%R 10.2478/agms-2012-0004