Dynamics and Control of Mechanical Systems

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DMS-0103895 PI: Anthony M. BlochDynamics and Control of Mechanical SystemsAbstract:This is a proposal for the continuation and extension of the proposer's research into the dynamics and control of nonlinear mechanical systems in finite- and infinite-dimensions. In particular, research is proposed in the following areas: integrable dynamical systems in finite- and infinite- dimensions including both Hamiltonian and nonholonomic systems, the stabilization and control of nonlinear mechanical systems via energy methods and the extension of these methods to systems with nonholonomic (nonintegrable) constraints, the geometry of the smooth and discrete dynamics of rigid bodies, and the control of quantum mechanical systems. A general area of interest that encompasses several parts of this proposal is the relationship between energy preservation and asymptotic behavior in dynamical systems and between reversible and irreversible behavior. In the integrable systems area the proposer is interested in the nonabelian Toda lattice, infinite-dimensional generalizations of Toda, and generalized smooth and discrete rigid body systems. Finally he intends to analyze the dynamics and control of various coupled mechanical systems in both the classical and quantum regimes.This proposal is aimed at studying the behavior of various mechanical systems that are important for applications in science and engineering. These include systems such as wheeled or articulated robots, aerospace systems, submarine vehicles, and quantum (microscopic) devices. Quantum devices, where the laws of motion of atoms and molecules play a key role, have become increasingly important in such areas as communication and coding theory. In addition to studying the behavior of these systems the proposer intends to analyze their control and optimal control -- that is, to provide methods for using such systems in engineering applications in a practical, stable, and efficient manner. The proposerintends to analyze various key examples of such mechanical systems with mathematical structure which is particularly amenable to detailedqualitative and quantitative analysis. Further, the proposer intends toanalyze the transition between the behavior of systems at the quantum(microscopic) level and the large scale (macroscopic) level. This transition is important for the application of physical devices in the real world as well as interesting from the scientific point of view.