CAREER: Computational Geometric Mechanics: Foundations, Computation, and Applications

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Date: November 21, 2007Proposal: DMS- 0747659PI: Leok, Melvin Institution: Purdue UniversityTitle: CAREER: Computational Geometric Mechanics: Foundations, Computation, and ApplicationsABSTRACTSymmetry, and the study of invariant and equivariant objects, is a deep and unifying principle underlying a variety of mathematical fields. In particular, geometric mechanics is characterized by the application of symmetry and differential geometric techniques to Lagrangian and Hamiltonian mechanics, and geometric integration is concerned with the construction of numerical methods with geometric invariant and equivariant properties. Computational geometric mechanics blends these fields, and uses a self-consistent discretization of geometry and mechanics to systematically construct geometric structure-preserving numerical schemes. The proposed research will combine theoretical and computational tools arising from Dirac mechanics and geometry, noncommutative harmonic analysis, and uncertainty quantification to dramatically extend the applicability of computational geometric mechanics and geometric control to engineering problems that evolve intrinsically on nonlinear spaces, such as Lie groups and homogeneous spaces. This will provide insights into the canonical discretization of Dirac constraints, nonholonomic constraints, and interconnected systems. In addition, the study of uncertainty in the context of geometric control will improve the robustness and reliability of the resulting numerical and computational tools.This research will improve our ability to control interconnected systems of autonomous vehicles in a robust and efficient fashion, by explicitly taking into account the uncertainty inherent in our knowledge of the surrounding environment. Our results will be applicable to the control of distributed sensor networks, consisting of an interconnected set of satellites, unmanned aerial vehicles and underwater vehicles. Such sensor networks are an exciting new development in the field of remote sensing that has the potential to dramatically increase the efficiency, coverage, and reliability of the information we obtain about our oceans, environment, and climate. More broadly, most complex engineering systems can be expressed as an interconnected system of more elementary components, and our mathematical framework will allow us to more readily understand complex systems in terms of the behavior of its component parts and the manner in which they are interconnected.