27 research outputs found
New high order sufficient conditions for configuration tracking
In this paper, we propose new conditions guaranteeing that the trajectories
of a mechanical control system can track any curve on the configuration
manifold. We focus on systems that can be represented as forced affine
connection control systems and we generalize the sufficient conditions for
tracking known in the literature. The new results are proved by a combination
of averaging procedures by highly oscillating controls with the notion of
kinematic reduction.Comment: arXiv admin note: text overlap with arXiv:0911.536
Global controllability tests for geometric hybrid control systems
Hybrid systems are characterized by having an interaction between continuous
dynamics and discrete events. The contribution of this paper is to provide
hybrid systems with a novel geometric formulation so that controls can be
added. Using this framework we describe some new global controllability tests
for hybrid control systems exploiting the geometry and the topology of the set
of jump points, where the instantaneous change of dynamics take place.
Controllability is understood as the existence of a feasible trajectory for the
system joining any two given points. As a result we describe examples where
none of the continuous control systems are controllable, but the associated
hybrid system is controllable because of the characteristics of the jump set.Comment: 27 pages, 5 figure
Higher-order Mechanics: Variational Principles and other topics
After reviewing the Lagrangian-Hamiltonian unified formalism (i.e, the
Skinner-Rusk formalism) for higher-order (non-autonomous) dynamical systems, we
state a unified geometrical version of the Variational Principles which allows
us to derive the Lagrangian and Hamiltonian equations for these kinds of
systems. Then, the standard Lagrangian and Hamiltonian formulations of these
principles and the corresponding dynamical equations are recovered from this
unified framework.Comment: New version of the paper "Variational principles for higher-order
dynamical systems", which was presented in the "III Iberoamerican Meeting on
Geometry, Mechanics and Control" (Salamanca, 2012). The title is changed. A
detailed review is added. Sections containing results about variational
principles are enlarged with additional comments, diagrams and summarizing
results. Bibliography is update
Geometric Approach to Pontryagin's Maximum Principle
Since the second half of the 20th century, Pontryagin's Maximum Principle has
been widely discussed and used as a method to solve optimal control problems in
medicine, robotics, finance, engineering, astronomy. Here, we focus on the
proof and on the understanding of this Principle, using as much geometric ideas
and geometric tools as possible. This approach provides a better and clearer
understanding of the Principle and, in particular, of the role of the abnormal
extremals. These extremals are interesting because they do not depend on the
cost function, but only on the control system. Moreover, they were discarded as
solutions until the nineties, when examples of strict abnormal optimal curves
were found. In order to give a detailed exposition of the proof, the paper is
mostly self\textendash{}contained, which forces us to consider different areas
in mathematics such as algebra, analysis, geometry.Comment: Final version. Minors changes have been made. 56 page