Summary: ESAIM: COCV 15 (2009) 173­188 ESAIM: Control, Optimisation and Calculus of Variations
DOI: 10.1051/cocv:2008029 www.esaim-cocv.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 1-dimensional problems.
Mathematics Subject Classification. 93B50, 49K99.
Received July 30, 2007.
Published online April 26, 2008.
1. Introduction
Given a smooth n-dimensional manifold M and a smooth function : T M R we study the functional
J((·)) =

0

Source: Agrachev, Andrei - Functional Analysis Sector, Scuola Internazionale Superiore di Studi Avanzati (SISSA)

Collections: Engineering; Mathematics