@article {2013,
	title = {On conjugate times of LQ optimal control problems},
	number = {Journal of Dynamical and Control Systems},
	year = {2014},
	note = {14 pages, 1 figure},
	publisher = {Springer},
	abstract = {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 $\vec{H}$.},
	keywords = {Optimal control, Lagrange Grassmannian, Conjugate point},
	doi = {10.1007/s10883-014-9251-6},
	url = {http://hdl.handle.net/1963/7227},
	author = {Andrei A. Agrachev and Luca Rizzi and Pavel Silveira}
}
@mastersthesis {2014,
	title = {The curvature of optimal control problems with applications to sub-Riemannian geometry},
	year = {2014},
	note = {The PhD thesis is composed of  211 pages and is recorded in PDF format},
	school = {SISSA},
	abstract = {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{\textquoteright}s volume and are also investigated.},
	keywords = {Sub-Riemannian geometry},
	url = {http://hdl.handle.net/1963/7321},
	author = {Luca Rizzi}
}
@article {2013,
	title = {The curvature: a variational approach},
	number = {arXiv:1306.5318;},
	year = {2013},
	note = {88 pages, 10 figures, (v2) minor typos corrected, (v3) added sections
  on Finsler manifolds, slow growth distributions, Heisenberg group},
	institution = {SISSA},
	abstract = {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.},
	keywords = {Crurvature, subriemannian metric, optimal control problem},
	url = {http://hdl.handle.net/1963/7226},
	author = {Andrei A. Agrachev and Davide Barilari and Luca Rizzi}
}
