Smooth optimal synthesis for infinite horizon variational problems

¹ SISSA, via Beirut 2-4, 34014 Trieste, Italy. agrachev@sissa.it
² Dipartimento di Matematica Applicata “G. Sansone”, via S. Marta 3, 50139 Firenze, Italy. chittaro@math.unifi.it

Received: 30 July 2007

Abstract

We study Hamiltonian systems which generate extremal flows of regular variational problems 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.