 
Summary: Feedback{invariant Optimal Control Theory
and Dierential Geometry, II. Jacobi Curves
for Singular Extremals
A. A. Agrachev
Abstract
This is the second article in the series opened by the paper [4].
Jacobi curves were dened, computed, and studied in that paper for
regular extremals of smooth control systems. Here we do the same for
singular extremals. The last section contains a feedback classication
and normal forms of generic single{input aĆne in control systems on
a 3dimensional manifold.
Introduction
This paper is a continuation of [4]. Jacobi curves were dened, computed,
and studied in [4] for regular extremals of smooth control systems. Here we
do the same for singular extremals.
The points of the Jacobi curves are the Lderivatives of endpoint map
pings. The notion of Lderivative was introduced in [4]; in section 1 of the
present paper we prove a general existence theorem for the Lderivatives of
smooth mappings and indicate a way to compute them. In section 2 we ac
tually compute the Lderivatives of the endpoint mappings and the Jacobi
