Geometric aspects of the Maximum Principle and lifts over a bundle map

A coordinate-free proof of the Maximum Principle is provided in the specific case of an optimal control problem with fixed time. Our treatment heavily relies on a special notion of variation of curves that consist of a concatenation of integral curves of time-dependent vector fields with unit time component, and on the use of a concept of lift over a bundle map. We further derive necessary and sufficient conditions for the existence of so-called abnormal and strictly abnormal extremals.Comment: 38p, accepted for publication in Acta Appl. Mat

Generalised connections over a vector bundle map

A generalised notion of connection on a fibre bundle E over a manifold M is presented. These connections are characterised by a smooth distribution on E which projects onto a (not necessarily integrable) distribution on M and which, in addition, is `parametrised' in some specific way by a vector bundle map from a prescribed vector bundle over M into TM. Some basic properties of these generalised connections are investigated. Special attention is paid to the class of linear connections over a vector bundle map. It is pointed out that not only the more familiar types of connections encountered in the literature, but also the recently studied Lie algebroid connections, can be recovered as special cases within this more general framework.Comment: 31 page

The Berwald-type linearisation of generalised connections

We study the existence of a natural `linearisation' process for generalised connections on an affine bundle. It is shown that this leads to an affine generalised connection over a prolonged bundle, which is the analogue of what is called a connection of Berwald type in the standard theory of connections. Various new insights are being obtained in the fine structure of affine bundles over an anchored vector bundle and affineness of generalised connections on such bundles.Comment: 25 page

Routh reduction for singular Lagrangians

This paper concerns the Routh reduction procedure for Lagrangians systems with symmetry. It differs from the existing results on geometric Routh reduction in the fact that no regularity conditions on either the Lagrangian $L$ or the momentum map $J_L$ are required apart from the momentum being a regular value of $J_L$. The main results of this paper are: the description of a general Routh reduction procedure that preserves the Euler-Lagrange nature of the original system and the presentation of a presymplectic framework for Routh reduced systems. In addition, we provide a detailed description and interpretation of the Euler-Lagrange equations for the reduced system. The proposed procedure includes Lagrangian systems with a non-positively definite kinetic energy metric.Comment: 34 pages, 2 figures, accepted for publicaton in International Journal of Geometric Methods in Modern Physics (IJGMMP

Dynamics of the Tippe Top via Routhian Reduction

We consider a tippe top modeled as an eccentric sphere, spinning on a horizontal table and subject to a sliding friction. Ignoring translational effects, we show that the system is reducible using a Routhian reduction technique. The reduced system is a two dimensional system of second order differential equations, that allows an elegant and compact way to retrieve the classification of tippe tops in six groups as proposed in [1] according to the existence and stability type of the steady states.Comment: 16 pages, 7 figures, added reference. Typos corrected and a forgotten term in de linearized system is adde

Comprehensive evaluation of the Copernicus Atmosphere Monitoring Service (CAMS) reanalysis against independent observations: Reactive gases

The Copernicus Atmosphere Monitoring Service (CAMS) is operationally providing forecast and reanalysis products of air quality and atmospheric composition. In this article, we present an extended evaluation of the CAMS global reanalysis data set of four reactive gases, namely, ozone (O-3), carbon monoxide (CO), nitrogen dioxide (NO2), and formaldehyde (HCHO), using multiple independent observations. Our results show that the CAMS model system mostly provides a stable and accurate representation of the global distribution of reactive gases over time. Our findings highlight the crucial impact of satellite data assimilation and emissions, investigated through comparison with a model run without assimilated data. Stratospheric and tropospheric O-3 are mostly well constrained by the data assimilation, except over Antarctica after 2012/2013 due to changes in the assimilated data. Challenges remain for O-3 in the Tropics and high-latitude regions during winter and spring. At the surface and for short-lived species (NO2), data assimilation is less effective. Total column CO in the CAMS reanalysis is well constrained by the assimilated satellite data. The control run, however, shows large overestimations of total column CO in the Southern Hemisphere and larger year-to-year variability in all regions. Concerning the long-term stability of the CAMS model, we note drifts in the time series of biases for surface O-3 and CO in the Northern midlatitudes and Tropics and for NO2 over East Asia, which point to biased emissions. Compared to the previous Monitoring Atmospheric Composition and Climate reanalysis, changes in the CAMS chemistry module and assimilation system helped to reduce biases and enhance the long-term temporal consistency of model results for the CAMS reanalysis

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