In our chaotic lives we usually do not try to specify our plans in great detail,or if we do, we should be prepared to make major modifications. Our plans forwhat we want to achieve are accompanied with situations we must avoid. Disturbances often disrupt our immediate plans, so we adapt to new situations. Weonly have partial control over our futures. Partial control aims at providing toyexamples of chaotic situations where we try to avoid disasters, constantly revisingour trajectories. More mathematically, partial control of chaotic systems isa new kind of control of chaotic dynamical systems in presence of disturbances.The goal of “partial control” is to avoid certain undesired behaviors withoutdetermining a specific trajectory. The surprising advantage of this control techniqueis that it sometimes allows the avoidance of the undesired behaviors evenif the control applied is smaller than the external disturbances of the dynamicalsystem. A key ingredient of this technique is what we call safe sets. Recently wehave found a general algorithm for finding these sets in an arbitrary dynamicalsystem, if they exist. The appearance of these safe sets can be rather complexthough they do not appear to have fractal boundaries. In order to understandbetter the dynamics on these sets, we introduce in this paper a new concept,the asymptotic safe set. Trajectories in the safe set tend asymptotically tothe asymptotic safe set. We present two algorithms for finding such sets. We illustrate all these concepts for a time-2pi map of the Duffing oscillator. This is joint work with James A Yorke (USA), Samuel Zambrano (Italy) and Juan Sabuco (Spain).