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Summary: On the local structure of optimal
trajectories in R 3
Andrei A. Agrachev 1 Mario Sigalotti 2
41/2002/M
1 SISSA-ISAS, via Beirut 2-4, 34014 - Trieste, Italy and Steklov Mathematical In-
stitute, ul. Gubkina 8, Moscow, Russia; E-mail: agrachev@sissa.it.
2
SISSA-ISAS, via Beirut 2-4, 34014 - Trieste, Italy; E-mail: 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 three-dimensional manifold,
nearby a point where some nondegeneracy conditions are satis ed. The kind
of optimality which is studied includes time-optimality. 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.
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