 
Summary: FEEDBACK{INVARIANT OPTIMAL CONTROL
THEORY AND DIFFERENTIAL GEOMETRY  I.
REGULAR EXTREMALS
A. A. Agrachev, R. V. Gamkrelidze
Abstract. Feedback{invariant approach to smooth optimal control problems is considered.
A Hamiltonian method of investigating regular extremals is developed, analogous to the
dierential{geometric method of investigation Riemannian geodesics in terms of the Levi{
Civita connection and the curvature tensor.
x0. Introduction
1. Outline of the content. This is the rst in a series of forthcoming papers, devoted
to the unication of the Theory of Smooth Optimal Control Problems and that part of
Dierential Geometry which is dealing with geodesics of dierent kinds. The obtained
results, we believe, not merely suggest a dictionary for translating the known results from
one language into another, but they really extend the scope of applicability of both theories.
The key notions brought into interplay are "Hamiltonian system" in optimal control and
the "curvature tensor" in dierential geometry.
Since the discovery of the Pontryagin maximum principle, cf. [11], nding extremals
in problems of optimal control is reduced to solving Hamiltonian systems of dierential
equations. Even in the classical case of Riemannian geometry, the maximum principle
approach to nding geodesics leads to the nal result much simpler and shorter than the
