 
Summary: Wellposed Infinite Horizon Variational
Problems
A. Agrachev
March 20, 2010
Abstract
We give an effective sufficient condition for a variational problem
with infinite horizon on a Riemannian manifold M to admit a smooth
optimal synthesis, i. e. a smooth dynamical system on M whose pos
itive semitrajectories are solutions to the problem. To realize the
synthesis we construct a wellprojected to M invariant Lagrange sub
manifold of the extremals' flow in the cotangent bundle T M. The
construction uses the curvature of the flow in the cotangent bundle
and some ideas of hyperbolic dynamics.
1 Introduction
This paper is dedicated to the 100th anniversary of Lev Semenovich Pon
tryagin.
Let M be a smooth compact ndimensional Riemannian manifold. Given
0, we denote by
the set of all absolutely continuous curves
: [0, +) M such that the integral
