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| 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 for what we want to achieve are accompanied with situations we must avoid. Disturbances often disrupt our immediate plans, so we adapt to new situations. We only have partial control over our futures. Partial control aims at providing toy
examples of chaotic situations where we try to avoid disasters, constantly revising
our trajectories. More mathematically, partial control of chaotic systems is
a new kind of control of chaotic dynamical systems in presence of disturbances.
The goal of partial control is to avoid certain undesired behaviors without
determining a specific trajectory. The surprising advantage of this control technique
is that it sometimes allows the avoidance of the undesired behaviors even
if the control applied is smaller than the external disturbances of the dynamical
system. A key ingredient of this technique is what we call safe sets. Recently we
have found a general algorithm for finding these sets in an arbitrary dynamical
system, if they exist. The appearance of these safe sets can be rather complex
though they do not appear to have fractal boundaries. In order to understand
better the dynamics on these sets, we introduce in this paper a new concept,
the asymptotic safe set. Trajectories in the safe set tend asymptotically to the asymptotic safe set. I will present two algorithms for finding such sets. I will illustrate all these concepts for a time-$2\pi$ map of the Duffing oscillator. Furthermore I will show two examples of applications to a cancer model and to a species extinction model in ecology. |
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