Journal ArticleDOI
Left-invariant optimal control problems on Lie groups: classification and problems integrable by elementary functions
TLDR
A survey of left-invariant optimal control problems on Lie groups can be found in this article , where extremal trajectories and their optimality, the cut time and cut locus, and optimal synthesis are discussed.Abstract:Β
Abstract Left-invariant optimal control problems on Lie groups are an important class of problems with a large symmetry group. They are theoretically interesting because they can often be investigated in full and general laws can be studied by using these model problems. In particular, problems on nilpotent Lie groups provide a fundamental nilpotent approximation to general problems. Also, left-invariant problems often arise in applications such as classical and quantum mechanics, geometry, robotics, visual perception models, and image processing. The aim of this paper is to present a survey of the main concepts, methods, and results pertaining to left-invariant optimal control problems on Lie groups that can be integrated by elementary functions. The focus is on describing extremal trajectories and their optimality, the cut time and cut locus, and optimal synthesis. Questions concerning the classification of left-invariant sub-Riemannian problems on Lie groups of dimension three and four are also addressed. Bibliography: 91 titles. read more
Citations
More filters
Journal ArticleDOI
Geodesics and Shortest Arcs of Some Sub-Riemannian Metrics on the Lie Groups $ SU(2)\times{π} $ and $ SO(3)\times{π} $ with Three-Dimensional Generating Distributions
Journal ArticleDOI
On Controllability of a Highly Degenerate Four-Level Quantum System with a ββChainedββ Coupling Hamiltonian
Journal ArticleDOI
ΠΠ΅Π²ΠΎΠΈΠ½Π²Π°ΡΠΈΠ°Π½ΡΠ½ΡΠ΅ Π·Π°Π΄Π°ΡΠΈ ΠΎΠΏΡΠΈΠΌΠ°Π»ΡΠ½ΠΎΠ³ΠΎ ΡΠΏΡΠ°Π²Π»Π΅Π½ΠΈΡ Π½Π° Π³ΡΡΠΏΠΏΠ°Ρ ΠΠΈ, ΠΈΠ½ΡΠ΅Π³ΡΠΈΡΡΠ΅ΠΌΡΠ΅ Π² ΡΠ»Π»ΠΈΠΏΡΠΈΡΠ΅ΡΠΊΠΈΡ ΡΡΠ½ΠΊΡΠΈΡΡ
TL;DR: In this paper , the authors proposed a method to solve the problem of the lack of a suitable solution to the problem by using the knowledge of the user's interaction with the system.
Journal ArticleDOI
Sub-Lorentzian distance and spheres on the Heisenberg group
Yuri L. Sachkov,Elena Sachkova +1 more
Book ChapterDOI
On the Geometry of the Orbits of Killing Vector Fields
TL;DR: In this article , the authors study the classification of the geometry of the orbits of the so-called killing vector fields and find that the most important conservation laws are associated with these transformations. But they do not consider the relationship between the geometrical properties of these fields and the conservation laws.
References
More filters
Journal ArticleDOI
On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents
Book
Nonlinear Dynamical Control Systems
TL;DR: The controlled Invariant Submanifolds and Nonlinear Zero Dynamics and the Disturbance Decoupling problem are studied.
Book
Nonlinear Control Systems: An Introduction
TL;DR: This chapter discusses the development of Geometric Theory of State Feedback for Multi-Input Multi-Output Systems and its applications in control systems.
Journal ArticleDOI
Optimal paths for a car that goes both forwards and backwards.
James A. Reeds,L. A. Shepp +1 more
TL;DR: In this paper, the shortest path a car can travel between two points if its starting and ending directions are specified, and only paths with at most 2 cusps or reversals are considered.
Book
Control Theory from the Geometric Viewpoint
TL;DR: Control Theory from the Geometric Viewpoint as mentioned in this paper is a recent addition to the geometric control theory monograph/textbook literature having Jurdjevic (1997) as its closest neighbor and Nijmeijer and van der Schaft (1995), Isidori (1996) and Bloch (2003) as more distant ones.
Related Papers (5)
The action of a compact Lie group on nilpotent Lie algebras of type {n,2}
Giovanni Falcone,Γgota Figula +1 more