01122nas a2200145   4500008004100000245005400041210005100095260001300146520066000159653006000819100002500879700001600904700002000920856003600940       2014                        en  d00aOn conjugate times of LQ optimal control problems0 aconjugate times of LQ optimal control problems  bSpringer3 aMotivated 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 $\vec{H}$.10aOptimal control, Lagrange Grassmannian, Conjugate point1 aAgrachev, Andrei, A.1 aRizzi, Luca1 aSilveira, Pavel  uhttp://hdl.handle.net/1963/722701220nas a2200121   4500008004100000245009100041210006900132260001000201520080700211653002801018100001601046856003601062       2014                        en  d00aThe curvature of optimal control problems with applications to sub-Riemannian geometry0 acurvature of optimal control problems with applications to subRi  bSISSA3 aOptimal 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.10aSub-Riemannian geometry1 aRizzi, Luca  uhttp://hdl.handle.net/1963/732101043nas a2200145   4500008004100000245004200041210003700083260001000120520060700130653006200737100002500799700002100824700001600845856003600861       2013                        en  d00aThe curvature: a variational approach0 acurvature a variational approach  bSISSA3 aThe 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.10aCrurvature, subriemannian metric, optimal control problem1 aAgrachev, Andrei, A.1 aBarilari, Davide1 aRizzi, Luca  uhttp://hdl.handle.net/1963/722601179nas a2200133   4500008004100000245006000041210005500101260001000156520076500166653004100931100001600972700002100988856003601009       2012                        en  d00aA formula for Popp\'s volume in sub-Riemannian geometry0 aformula for Popps volume in subRiemannian geometry  bSISSA3 aFor 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.10asubriemannian, volume, Popp, control1 aRizzi, Luca1 aBarilari, Davide  uhttp://hdl.handle.net/1963/6501