Geometric and numerical methods for optimal control of mechanical systems

Abstract

The applications of techniques from the modern Differential Geometry and Topology have helped to a new way of understanding the problems which comes from the theory of Dynamical Systems. These applications have reformulated the analytic mechanics and classical mechanics in a geometric language which attracted new analytic, topologic and numerical methods given rise to a new research line in mathematics and theoretical physics, called Geometric Mechanics. Geometric Mechanics is a meeting point between different areas such as, Analysis, Algebra, Numerical Analysis, Partial Differential Equations... Actually, Geometric Mechanics is a research area with a strong relationship with Nonlinear Control Theory and Numerical Analysis. The applications of Geometric Mechanics in control theory have given great progress in this area. For example, the geometric formulation of mechanical systems subject to nonholonomic constraints has helped to the understanding of problems in locomotion, controllability and trajectory planning, control problems with obstacles and interpolation problems. One of the main goals of the numerical analysis and computational mathematics has been rendering physical phenomena into algorithms that produce sufficiently accurate, affordable, and robust numerical approximations. In the last years, the field of Geometric Integration arose to design and to analyze numerical methods for ordinary differential equations and, more recently, for partial differential equations, that preserve exactly, as much as possible, the underlying geometrical structures. The Discrete Mechanics, understood as the confluence of Geometric Mechanics and Geometric Integration, is both a well-founded research area and a powerful tool in the understanding of dynamical and physical systems, more concretely of those related to mechanics. A key tool of Discrete Mechanics, which has been strongly used in this work, are the variational integrators, i.e., geometric integrators for mechanical problems based on the discretization of variational principles. The work developed in this thesis include new valuable developments in Geometric Mechanics which permits the understanding about mechanical systems, its applications in control theory and the construction of geometric integrators which preserves underlying geometrical structures of great interest to the numerical analysis of control systems. More precisely, we give a new geometric formulation for the dynamics of higher-order mechanical systems subject also to higher-order constraints since an optimal control problem for mechanical systems can be seen as higher-order variational problem with higher-order constraints. We have studied the relation between higher-order Lagrangian systems with constraints (nonholonomics and vakonomics) and higher-order Hamiltonian systems, the reduction by symmetries of this kind of mechanical systems and the geometric integration of control problems. The work developed in this thesis also is in line with new developments in Discrete Mechanics and its relation with control theory, Lie groupoids and Lie algebroids.