Abstract

Many significant real world challenges arise as optimization problems on different classes of control systems. In particular, ordinary differential equations with symmetries. The purpose of this review article is twofold. First, we give the information we have about the class of Linear Control Systems ΣG on a low dimension matrix Lie group G. Second, we invite the Mathematical community to consider possible applications through the Pontryagin Maximum Principle for ΣG. In addition, we challenge some theoretical open problems. The class ΣG is a perfect generalization of the classical Linear Control System on the Abelian group Rn. Let G be a Lie group of dimension two or three. Related to ΣG, this review describes the actual results about controllability, the time-optimal Hamiltonian equations and, the Pontryagin Maximum Principle. We show how to build ΣG, through several examples on low dimensional matrix groups.

Highlights

  • Many significant real world challenges arise as optimization problems on different classes of control systems

  • We give the information we have about the class of Linear Control Systems ΣG on a low dimension matrix Lie group G

  • The notion of a linear control system on a connected Lie group G depends on two different classes of dynamics: linear and invariant vector fields

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Summary

Introduction

Many significant real world challenges arise as optimization problems on different classes of control systems. It is able to discover the symmetries of classes of differential equations, [13] Examples of these manifolds are the Abelian group Rn , the spheres Sn ⊂ Rn+1 for n = 1, 3 and 7, the set GL(n, R) of the invertible real matrices of order n, and its relevant subgroups SL(n, R) of matrices with determinant 1, the orthogonal group O(n, R), the spinor group Spin(n, R), the unitary group U (n, R) and many others. For a group G of dimension 2 and 3, we describe the main ingredients to build ΣG by showing a basis of the Lie algebra g of G, the face of g-derivations and all possible associated linear vector fields. We include a classical optimal problem on the Euclidean plane, several examples on the 2-dimensional solvable group, and a time-optimal theoretical problems on 3-dimensional groups

Matrix Groups Dynamics and Systems
The Notion of Linear Control System on G
The Classical Linear Control System on Rn
The D -Decomposition of g
Linear Control Systems and Controllability
The Abelian Structure
The Solvable Structure
The Nilpotent Structure
The Finite Semi-Simple Center Structure
The Compact Semi-Simple Structure
The Non-Compact Semi-Simple Structure
The Pontryagin Maximum Principle
We start with the more famous one
Conclusions
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