RUI: Combinatorial Control of Chaos, Symbolic Dynamics, Optimal Control and Inverse Frobenius-Perron Problem

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NSF Award Abstract - DMS-0071314Mathematical Sciences: RUI: Combinatorial Control of Chaos, Symbolic Dynamics, Optimal Control and Inverse Frobenius-Perron ProblemAbstract0071314 BolltThis project addresses global control strategies for chaotic dynamical systems, based on transfer operator representations of dynamics on an invariant subspace. Emphasis is on developing techniques useful to experimentalists. During the last decade, much research has been focused on the realization that sensitive dependence on initial conditions and parameters allows flexible and efficient control of a chaotic dynamical system. Major difficulties have been connected with representing the action of the dynamical system on its phase space. In contrast, a complete global representation of a dynamical system is available in terms of its symbolic dynamics, and a graph approximation of the symbolic grammar is a highly efficient way to completely encode coarse-grained control strategies. This project reduces the difficult problems of developing a global control strategy to much easier problems, concerning linear algebra if targeting invariant density, or combinatorics of path searching if targeting optimal trajectories. In this coarse-grained approximation, paths through graphs model trajectories of the dynamical system. Local feedback control and small parameter variations are used to realize paths through the graph as true trajectories. Questions under study are closely related to Ulam's method in the theory of Frobenius-Perron operators and also to the concept of partition in information theory and in symbolic dynamics.This project develops techniques for targeting, finding optimal trajectories, and highlighting desirable states. Potential engineering applications include Navy ship-to-ship transfer by cargo crane, space mission design, design of electronic circuits that stay within desirable operating regimes, and perhaps someday preemptive control of epileptic seizures. The work involves collaboration with experimentalists to pursue real world applications. In addition, the project investigates applications of controlling symbolic dynamics to encode information in chaotic oscillations.