 
Summary: Chapter 1
Is it possible to recognize
Local Controllability in a
nite number of
dierentiations?
Andrei A. Agrachev
Steklov Mathematical Institute
ul. Gubkina 8
Moscow 117966
RUSSIA
agrachev@mi.ras.ru
1.1 Description of the problem
Let f 1 ; : : : ; f k ; k 2, be real{analytic vector elds dened on a neighbor
hood of the origin in R n , and t > 0. The point x 2 R n is called attainable
from the origin for a time less than t and with no more than N switch
ings, if there exists a subdivision 0 = t 0 < t 1 < : : : < t N+1 < t of the
segment [0; t] and solutions j (t), t 2 [t j ; t j+1 ] of the dierential equations
_
x = f i j
(x), for some i j 2 f1; : : : ; kg, such that 0 (0) = 0, j 1 (t j ) = j (t j )
