Control of chaos in the presence of uncertainty

Concerns the control of systems whose dynamics would otherwise be chaotic, by the exploitation of the inherent characteristics of the unstable solutions. Specifically, this idea of control of chaos seeks to stabilize one of these solutions so that the controlled system is periodic, but the periodic solution must be a natural solution of the uncontrolled system and cannot be arbitrarily chosen. The fact that this control objective can naturally be thought of as a type of stabilization even in the time-dependent context makes the control problem remarkably well behaved. The methods we propose are along these lines, and have several desirable characteristics not shared with existing methods available for these purposes. Principally, our controllers are designed so as to require very little information about the structure of the system, they are able to achieve global (or semiglobal) results, and the control is robust to bounded external disturbances. At this juncture we acknowledge that the methods outlined have the disadvantage of requiring explicit knowledge of the (natural solution) trajectory. This shortcoming is not unique to our approach but is shared by virtually all available controllers that seek to stabilize unstable solutions of the natural dynamics. While we do not offer a complete resolution of this problem, we do make some contributions in this direction