Optimal control on Schrödinger Lie group and the behaviour of the dynamics

2016 ◽  
pp. dnw045
Author(s):  
Soumya Ranjan Sahoo ◽  
Soumya Prakash Sahoo ◽  
Amit Jena ◽  
K. C. Pati
Keyword(s):  
Optimal Control ◽  
Lie Group

scholarly journals COMMUTATORS OF SKEW-SYMMETRIC MATRICES

2005 ◽  
Vol 15 (03) ◽  
pp. 793-801 ◽  
Author(s):  
ANTHONY M. BLOCH ◽  
ARIEH ISERLES
Keyword(s):  
Optimal Control ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Lie Group ◽  
Geometric Integration ◽  
Symmetric Matrices ◽  
Frobenius Norm

In this paper we develop a theory for analysing the "radius" of the Lie algebra of a matrix Lie group, which is a measure of the size of its commutators. Complete details are given for the Lie algebra 𝔰𝔬(n) of skew symmetric matrices where we prove [Formula: see text], X, Y ∈ 𝔰𝔬(n), for the Frobenius norm. We indicate how these ideas might be extended to other matrix Lie algebras. We discuss why these ideas are of interest in applications such as geometric integration and optimal control.

scholarly journals Intrinsic Optimal Control for Mechanical Systems on Lie Group

2017 ◽  
Vol 2017 ◽  
pp. 1-8
Author(s):  
Chao Liu ◽  
Shengjing Tang ◽  
Jie Guo
Keyword(s):  
Optimal Control ◽  
Analytical Solution ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Lie Group ◽  
Control Method ◽  
Geodesic Distance ◽  
Infinite Horizon ◽  
Mechanical Systems ◽  
Programming Approach

The intrinsic infinite horizon optimal control problem of mechanical systems on Lie group is investigated. The geometric optimal control problem is built on the intrinsic coordinate-free model, which is provided with Levi-Civita connection. In order to obtain an analytical solution of the optimal problem in the geometric viewpoint, a simplified nominal system on Lie group with an extra feedback loop is presented. With geodesic distance and Riemann metric on Lie group integrated into the cost function, a dynamic programming approach is employed and an analytical solution of the optimal problem on Lie group is obtained via the Hamilton-Jacobi-Bellman equation. For a special case on SO(3), the intrinsic optimal control method is used for a quadrotor rotation control problem and simulation results are provided to show the control performance.

scholarly journals An Underactuated Drift-Free Left Invariant Control System on a Specific Lie Group

2012 ◽  
Vol 2012 ◽  
pp. 1-14
Author(s):  
Camelia Pop Arieşanu
Keyword(s):  
Optimal Control ◽  
Control System ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Lie Group ◽  
Point Of View ◽  
Free Left ◽  
Invariant Control

The paper presents a geometrical overview on an optimal control problem on a special Lie group. The Hamilton-Poisson realization of the dynamics offers us the possibility to study the system from mechanical geometry point of view.

AN OPTIMAL CONTROL PROBLEM ON THE LIE GROUP SO(4)

2008 ◽  
Vol 05 (03) ◽  
pp. 319-327 ◽  
Author(s):  
ANANIA ARON ◽  
IONEL MOŞ ◽  
ANIKO CSAKY ◽  
MIRCEA PUTA
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Lie Group ◽  
Controllable System ◽  
Geometrical Properties

An optimal control problem for a drift-free controllable system on the Lie group SO(4) is discussed and some of its dynamical and geometrical properties are pointed out.

scholarly journals An underactuated drift-free left-invariant control system on the Lie group ISO(3,1)

2019 ◽  
Vol 29 ◽  
pp. 01006
Author(s):  
Camelia Pop ◽  
Ioana Iosif
Keyword(s):  
Optimal Control ◽  
Lie Group ◽  
Sufficient Conditions ◽  
Fixed Time ◽  
Equilibrium States ◽  
Control Pair ◽  
The Maximum Principle ◽  
Invariant Control ◽  
Future Work ◽  
The Given

The purpose of our paper is to study a class of left-invariant, drift-free optimal control problem on the Lie group ISO(3,1). The left-invariant, drift-free optimal control problems involves finding a trajectory-control pair on ISO(3,1), which minimize a cost function and satisfies the given dynamical constrains and boundary conditions in a fixed time. The problem is lifted to the cotangent bundle T*G using the optimal Hamiltonian on G*, where the maximum principle yields the optimal control. We use energy methods (Arnold’s method, in this case) to give sufficient conditions fornonlinear stability of the equilibrium states. Around this equilibrium states we might be able to find the periodical orbits using Moser's theorem, as future work. For the some unstable equilibrium states, a quadratic control is considered in order to stabilize the dynamics.

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