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.
The 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 the presence of disturbances, where the goal is to avoid certain undesired behaviors without determining a specific trajectory. (For more details, here is a lecture given by Prof. James A. Yorke about Partial Control)
The surprising advantage of this control method 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 method is what we call safe sets. Recently we developed a general algorithm (the code is available here) for finding these sets in an arbitrary dynamical system, if they exist. The code was written during my research stay at the University of Maryland with Prof. James A. Yorke in 2011. The appearance of these safe sets can be rather complex although they do not appear to have fractal boundaries.
In order to understand better the dynamics on these sets, we have also been doing research on a new concept, the asymptotic safe set. Trajectories in the safe set tend asymptotically to the asymptotic safe set. We have proposed recently two algorithms (the code is available here) for finding such sets, one that sculpts the asymptotic safe set and another that grows it. The code of these algorithms were also written during my research stay at the University of Maryland. To test these new algorithms we have applied them to several paradigmatic models in Nonlinear Dynamics like the Tent map, the Hénon map and the Duffing Oscillator.
Recently we have also been working in the Nonlinear Dynamics, Chaos and Complex Systems Group at Universidad Rey Juan Carlos, in how to apply the partial control method to more pragmatic models. In particular, we have used the partial control method to avoid extinctions in an ecological model, in presence of disturbances, with the goal of minimizing the control applied. Another interesting application has been the use of this control method to a model of cancer, with the goal of avoiding the disappearance of healthy cells and the uncontrolled growth of cancer cells. We hope that this contribution might lead to the development of new disruptive treatments.
This line of research is a joint collaboration with Prof. Miguel A.F. Sanjuán (Universidad Rey Juan Carlos), Dr. Samuel Zambrano (San Raffaele Research Institute) and Prof. James A. Yorke (University of Maryland).