Limit Behavior and Genericity for Nonlinear Control Systems

For nonlinear control systems of the form = X0(x)+Σmi=1ui(t)Xi(x) with constrained control range U 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 × M, where 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 × 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.