Advertisement

Examples of Optimal Control Problems on Compact Lie Groups

  • Andrei A. Agrachev
  • Yuri L. Sachkov
Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 87)

Abstract

Let M be a compact Lie group. The invariant scalar product (•, •) in the Lie algebra M = T Id M defines a left-invariant Riemannian structure on M:
$${\langle qu,qv\rangle _q}_ = ^{def}\langle u,u\rangle ,u,v \in M,q \in M,qu,qv \in {T_q}M.$$
So in every tangent space T q M there is a scalar product (•,•)q. For any Lipschitzian curve
$$q:[0,1] \to M$$
its Riemannian length is defined as integral of velocity:
$$\iota = {\text{ }}\int_0^1 {\left| {\dot q\left( t \right)} \right|} dt,\left| {\dot q} \right| = \sqrt {\langle \dot q,\dot q\rangle } .$$

Keywords

Quantum System Optimal Control Problem Optimal Trajectory Transversality Condition Switching Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Andrei A. Agrachev
    • 1
  • Yuri L. Sachkov
    • 2
  1. 1.SISSA-ISASTriesteItaly
  2. 2.Program Systems InstitutePereslavl-ZalesskyRussia

Personalised recommendations