@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}
}
@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}
}
@article {2010,
	title = {Continuity of optimal control costs and its application to weak KAM theory},
	journal = {Calculus of Variations and Partial Differential Equations. Volume 39, Issue 1, 2010, Pages 213-232},
	number = {arXiv:0909.3826;},
	year = {2010},
	note = {23 pages, 1 figures},
	publisher = {SISSA},
	abstract = {We prove continuity of certain cost functions arising from optimal control of\\r\\naffine control systems. We give sharp sufficient conditions for this\\r\\ncontinuity. As an application, we prove a version of weak KAM theorem and\\r\\nconsider the Aubry-Mather problems corresponding to these systems.},
	doi = {10.1007/s00526-010-0308-4},
	url = {http://hdl.handle.net/1963/6459},
	author = {Andrei A. Agrachev and Paul Lee}
}
@article {2009,
	title = {Controllability on the group of diffeomorphisms},
	journal = {Ann. Inst. H. Poincar{\'e} Anal. Non Lin{\'e}aire 26 (2009) 2503-2509},
	number = {SISSA;79/2008/M},
	year = {2009},
	abstract = {Given a compact manifold M, we prove that any bracket generating family of vector fields on M, which is invariant under multiplication by smooth functions, generates the connected component of identity of the group of diffeomorphisms of M.},
	doi = {10.1016/j.anihpc.2009.07.003},
	url = {http://hdl.handle.net/1963/3396},
	author = {Andrei A. Agrachev and Marco Caponigro}
}
@article {2005,
	title = {On curvatures and focal points of distributions of dynamical Lagrangian distributions and their reductions by first integrals},
	journal = {J. Dyn. Control Syst. 11 (2005)  297-327},
	number = {SISSA;58/2004/M},
	year = {2005},
	abstract = {Pairs (Hamiltonian system, Lagrangian distribution), called dynamical Lagrangian distributions, appear naturally in Differential Geometry, Calculus of Variations and Rational Mechanics. The basic differential invariants of a dynamical Lagrangian distribution w.r.t. the action of the group of symplectomorphisms of the ambient symplectic manifold are the curvature operator and the curvature form. These invariants can be seen as generalizations of the classical curvature tensor in Riemannian Geometry. In particular, in terms of these invariants one can localize the focal points along extremals of the corresponding variational problems. In the present paper we study the behavior of the curvature operator, the curvature form and the focal points of a dynamical Lagrangian distribution after its reduction by arbitrary first integrals in involution. The interesting phenomenon is that the curvature form of so-called monotone increasing Lagrangian dynamical distributions, which appear naturally in mechanical systems, does not decrease after reduction. It also turns out that the set of focal points to the given point w.r.t. the monotone increasing dynamical Lagrangian distribution and the corresponding set of focal points w.r.t. its reduction by one integral are alternating sets on the corresponding integral curve of the Hamiltonian system of the considered dynamical distributions. Moreover, the first focal point corresponding to the reduced Lagrangian distribution comes before any focal point related to the original dynamical distribution. We illustrate our results on the classical $N$-body problem.},
	doi = {10.1007/s10883-005-6581-4},
	url = {http://hdl.handle.net/1963/2254},
	author = {Andrei A. Agrachev and Natalia N. Chtcherbakova and Igor Zelenko}
}
