 
Summary: On the local structure of optimal
trajectories in R 3
Andrei A. Agrachev 1 Mario Sigalotti 2
41/2002/M
1 SISSAISAS, via Beirut 24, 34014  Trieste, Italy and Steklov Mathematical In
stitute, ul. Gubkina 8, Moscow, Russia; Email: agrachev@sissa.it.
2
SISSAISAS, via Beirut 24, 34014  Trieste, Italy; Email: sigalott@sissa.it
Abstract
We analyze the structure of a control function u(t) corresponding to an optimal
trajectory for the system _
q = f(q) + u g(q) in a threedimensional manifold,
nearby a point where some nondegeneracy conditions are satised. The kind
of optimality which is studied includes timeoptimality. The control turns out
to be the concatenation of some bang and some singular arcs. Studying the
index of the second variation of the switching times, the number of such arcs is
bounded by four.
Key Words: optimal control, Lie brackets.
