For nonlinear control systems of the form x = X0(x)+SIGMA(i=1)m(t)X(i)
(x) with constrained control range U subset-of R(m) the limit behavior
of the trajectories is analyzed. The limit sets are closely related t
o regions of the state space where the system is controllable. In part
icular, under a rank condition on the linear span of the derived vecto
rfields on a limit set, this limit set is contained in the interior of
a control set. This result is a mathematical basis for the control of
complicated behavior. It also allows the characterization of Morse se
ts of a differential equation as intersections of topologically transi
tive sets of control systems. Topological genericity theorems classify
the possible limit behavior of a control system for open and dense se
ts in U x M, where U is the space of admissible control functions, and
M is the state space. The methods are a combination of control theore
tic arguments and chain transitivity for the control flow on U x M. Th
e results are applied to the Lorenz equations, showing that its strang
e attractor is contained in a region of controllability, in which its
dynamics can be altered, e.g., to yield periodic motions. (C) 1994 Aca
demic Press Inc.