LIMIT BEHAVIOR AND GENERICITY FOR NONLINEAR CONTROL-SYSTEMS

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 to regions of the state space where the system is controllable. In particular, under a rank condition on the linear span of the derived vectorfields 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 sets of a differential equation as intersections of topologically transitive sets of control systems. Topological genericity theorems classify the possible limit behavior of a control system for open and dense sets 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 theoretic arguments and chain transitivity for the control flow on U x M. The results are applied to the Lorenz equations, showing that its strange attractor is contained in a region of controllability, in which its dynamics can be altered, e.g., to yield periodic motions. (C) 1994 Academic Press Inc.