TY - JOUR
T1 - On conjugate times of LQ optimal control problems
Y1 - 2014
A1 - Andrei A. Agrachev
A1 - Luca Rizzi
A1 - Pavel Silveira
KW - Optimal control, Lagrange Grassmannian, Conjugate point
AB - 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 $\vec{H}$.
PB - Springer
UR - http://hdl.handle.net/1963/7227
N1 - 14 pages, 1 figure
U1 - 7261
U2 - Mathematics
U4 - 1
U5 - MAT/05 ANALISI MATEMATICA
ER -
TY - THES
T1 - The curvature of optimal control problems with applications to sub-Riemannian geometry
Y1 - 2014
A1 - Luca Rizzi
KW - Sub-Riemannian geometry
AB - 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.
PB - SISSA
UR - http://hdl.handle.net/1963/7321
N1 - The PhD thesis is composed of 211 pages and is recorded in PDF format
U1 - 7367
U2 - Mathematics
U4 - 1
U5 - MAT/03 GEOMETRIA
ER -
TY - RPRT
T1 - The curvature: a variational approach
Y1 - 2013
A1 - Andrei A. Agrachev
A1 - Davide Barilari
A1 - Luca Rizzi
KW - Crurvature, subriemannian metric, optimal control problem
AB - 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.
PB - SISSA
UR - http://hdl.handle.net/1963/7226
N1 - 88 pages, 10 figures, (v2) minor typos corrected, (v3) added sections
on Finsler manifolds, slow growth distributions, Heisenberg group
U1 - 7260
U2 - Mathematics
U4 - 1
U5 - MAT/03 GEOMETRIA
ER -
TY - JOUR
T1 - A formula for Popp\'s volume in sub-Riemannian geometry
JF - Analysis and Geometry in Metric Spaces, vol. 1 (2012), pages : 42-57
Y1 - 2012
A1 - Luca Rizzi
A1 - Davide Barilari
KW - subriemannian, volume, Popp, control
AB - 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.
PB - SISSA
UR - http://hdl.handle.net/1963/6501
N1 - 16 pages, minor revisions
U1 - 6446
U2 - Mathematics
U4 - 1
U5 - MAT/03 GEOMETRIA
ER -