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Global Structures and Numerical Methods in Mechanics, Dynamical Systems and Control

Principal Researcher: David Martín de Diego

 

Other Members:  Manuel de León

                        Álvaro Pelayo

                        Daniel Peralta Salas

                        Anthony M. Bloch

                        Gilbert Hector

 

PhD Students:  Fernando Jiménez Alburquerque

                      Leonardo Colombo 

 

Summary

This project seeks to extend and deepen the geometrical and numerical analysis of dynamical systems in classical and quantum mechanics and control theory, already initiated by the proponent group.

Because of the recent research conducted by the group, it has become relevant the use of global geometric structures such as Dirac structures, Lie algebroids, groupoids and foliation theory. These structures find natural applications both in the qualitative analysis of properties of mechanical systems with nonholomonic constraints, Hamilton-Jacobi theory or in the reduction of (classical and quantum) optimal control problems among others. In addition, a natural class of geometrical numerical integrators can be designed using these structures as a guiding framework. To develop these geometrically inspired numerical methods will constitute an important objective of this project that will find applications in the integration of such important problems as algebraic-differential equations and closely related problems such as singular and constrained mechanical and control problems, etc. An ambitious objective of this project will be the analysis of the backwards stability of the proposed numerical integrators. The previous general objectives will be coordinated by subproject at CSIC.

Recent developments in quantum information theory and quantum control have also motivated a number of important mathematical  questions. Solving these equations are instrumental for the actual applicability of these developments. We propose the extension of some of the ideas and methods analyzed in the recent years by our research group to this setting. Among the variety of problems emerging from quantum technologies we are focusing our attention on a set of them that again involve the use of global geometrical structures and original numerical techniques. Thus and objective of the project will be the study of problems in quantum control theory as optimal control problems on Lie groups (both finite and infinite dimensional), the topology of boundary problems and the quantization of classical optimal control problems. New ideas that relate the geometry and the topology of the underlying manifolds with the spectral properties of the systems will be discussed. A long-term idea that will be addresed in this project consists in the use of abstract functional analysis structures to analyze the evolution of dynamical systems. These topics will be coordinated by subproject at UC3M.