Symmetry Reduction of Optimal Control Systems and Principal Connections

This paper explores the role of symmetries and reduction in nonlinear control and optimal control systems. The focus of the paper is to give a geometric framework of symmetry reduction of optimal control systems as well as to show how to obtain explicit expressions of the reduced system by exploiting the geometry. In particular, we show how to obtain a principal connection to be used in the reduction for various choices of symmetry groups, as opposed to assuming such a principal connection is given or choosing a particular symmetry group to simplify the setting. Our result synthesizes some previous works on symmetry reduction of nonlinear control and optimal control systems. Affine and kinematic optimal control systems are of particular interest: We explicitly work out the details for such systems and also show a few examples of symmetry reduction of kinematic optimal control problems.Comment: 23 pages, 2 figure

Lagrangian Reduction, the Euler--Poincar\'{e} Equations, and Semidirect Products

There is a well developed and useful theory of Hamiltonian reduction for semidirect products, which applies to examples such as the heavy top, compressible fluids and MHD, which are governed by Lie-Poisson type equations. In this paper we study the Lagrangian analogue of this process and link it with the general theory of Lagrangian reduction; that is the reduction of variational principles. These reduced variational principles are interesting in their own right since they involve constraints on the allowed variations, analogous to what one finds in the theory of nonholonomic systems with the Lagrange d'Alembert principle. In addition, the abstract theorems about circulation, what we call the Kelvin-Noether theorem, are given.Comment: To appear in the AMS Arnold Volume II, LATeX2e 30 pages, no figure

On the global version of Euler-Lagrange equations

The introduction of a covariant derivative on the velocity phase space is needed for a global expression of Euler-Lagrange equations. The aim of this paper is to show how its torsion tensor turns out to be involved in such a version.Comment: 5 pages, 1 figur

Lagrangian reduction, the Euler--Poincaré Equations, and semidirect products

There is a well developed and useful theory of Hamiltonian reduction for semidirect products, which applies to examples such as the heavy top, compressible fluids and MHD, which are governed by Lie-Poisson type equations. In this paper we study the Lagrangian analogue of this process and link it with the general theory of Lagrangian reduction; that is the reduction of variational principles. These reduced variational principles are interesting in their own right since they involve constraints on the allowed variations, analogous to what one finds in the theory of nonholonomic systems with the Lagrange d'Alembert principle. In addition, the abstract theorems about circulation, what we call the Kelvin-Noether theorem, are given

Analisa Karakteristik Karbon Aerosol (Oc dan Ec) dari Emisi Pm2.5 dan Rekomendasi Perlindungan Lingkungan dari Emisi Pm2.5 Kebakaran Lahan Gambut secara Pembaraan (Smouldering) (Studi Kasus : Kabupaten Siak dan Kabupaten Kampar Provinsi Riau)

Penelitian ini memiliki tujuan untuk mengetahui karakteristik karbon organik (OC) dan karbon elemental (EC) dalam PM2.5 yang diketahui sebagai salah satu polutan udara akibat kebakaran lahan gambut dan rekomendasi perlindungan lingkungan. Metode yang digunakan untuk mengetahui konsentrasi PM2.5 adalah gravimetri dengan bantuan alat Sartorius ME5-F dan metode analisa konsentrasi karbon aerosol adalah metode pemantulan cahaya dan thermal (IMPROVE A) dengan bantuan alat Carbon Analyzer Model DRI 2001. Konsentrasi rata-rata dan tertinggi PM2.5 emisi kebakaran lahan gambut terutama pada fase pembaraan adalah 996,71 ± 531,01 µm/g3 dan 2163.49 µg/m3. Nilai tersebut lebih tinggi dari konsentrasi PM2.5 ketika tidak terjadi kebakaran (background) sebesar 48 kali. Rata-rata komposisi OC (sebagai salah satu penyusun utama PM2.5) dan EC dalam karbon total (TC) adalah 98,58 ± 0,91% dan 1,42 ± 0,91%. Fraksi OC (Organic Carbon) yang dominan adalah OC1 dan OC2 dengan rata-rata komposisi dalam karbon total (TC) adalah 40,34 ± 5,43% dan 31,58 ± 5,58%. Rasio OC/EC pada penelitian ini lebih besar dari rasio OC/EC pada kebakaran reruntuhan kayu dan kebakaran pohon pinus pada fase yang sama. Rasio OC/EC menunjukkan pengaruh emisi kebakaran lahan gambut terhadap emisi sumber kebakaran lain. Perlindungan lingkungan dari dampak yang ditimbulkan dari kebakaran lahan gambut dapat dilakukan dengan pencegahan penyebaran kebakaran dan penurunan konsentrasi PM2.5. Pencegahan penyebaran kebakaran dilakukan dengan menciptakan sistem pelindung lahan terhadap kebakaran dengan bantuan parit buatan. Penurunan konsentrasi PM2.5 dilakukan dengan menyediakan zona penyangga/penyerapan (buffer zone) menggunakan vegetasi khusus pada luas dan jarak tertentu

Routh's procedure for non-Abelian symmetry groups

We extend Routh's reduction procedure to an arbitrary Lagrangian system (that is, one whose Lagrangian is not necessarily the difference of kinetic and potential energies) with a symmetry group which is not necessarily Abelian. To do so we analyse the restriction of the Euler-Lagrange field to a level set of momentum in velocity phase space. We present a new method of analysis based on the use of quasi-velocities. We discuss the reconstruction of solutions of the full Euler-Lagrange equations from those of the reduced equations.Comment: 30 pages, to appear in J Math Phy

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

Routhian reduction for quasi-invariant Lagrangians

In this paper we describe Routhian reduction as a special case of standard symplectic reduction, also called Marsden-Weinstein reduction. We use this correspondence to present a generalization of Routhian reduction for quasi-invariant Lagrangians, i.e. Lagrangians that are invariant up to a total time derivative. We show how functional Routhian reduction can be seen as a particular instance of reduction of a quasi-invariant Lagrangian, and we exhibit a Routhian reduction procedure for the special case of Lagrangians with quasi-cyclic coordinates. As an application we consider the dynamics of a charged particle in a magnetic field.Comment: 24 pages, 3 figure