 
Summary: ESAIM: COCV 15 (2009) 173188 ESAIM: Control, Optimisation and Calculus of Variations
DOI: 10.1051/cocv:2008029 www.esaimcocv.org
SMOOTH OPTIMAL SYNTHESIS FOR INFINITE HORIZON
VARIATIONAL PROBLEMS
Andrei A. Agrachev1
and Francesca C. Chittaro2
Abstract. We study Hamiltonian systems which generate extremal flows of regular variational prob
lems on smooth manifolds and demonstrate that negativity of the generalized curvature of such a
system implies the existence of a global smooth optimal synthesis for the infinite horizon problem.
We also show that in the Euclidean case negativity of the generalized curvature is a consequence of
the convexity of the Lagrangian with respect to the pair of arguments. Finally, we give a generic
classification for 1dimensional problems.
Mathematics Subject Classification. 93B50, 49K99.
Received July 30, 2007.
Published online April 26, 2008.
1. Introduction
Given a smooth ndimensional manifold M and a smooth function : T M R we study the functional
J((·)) =
0
