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

Manifold Lution Tate Reso Exter 

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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

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