EQUIVALENCE OF CONTROL SYSTEMS WITH LINEAR SYSTEMS ON LIE GROUPS AND HOMOGENEOUS SPACES

作者：Jouan Philippe^*

来源：ESAIM: Control, Optimisation and Calculus of Variations , 2010, 16(4): 956-973.

DOI：10.1051/cocv/2009027

• Lie groups
• homogeneous spaces
• linear systems
• complete vector field
• finite dimensional Lie algebra

The aim of this paper is to prove that a control affine system on a manifold is equivalent by diffeomorphism to a linear system on a Lie group or a homogeneous space if and only if the vector fields of the system are complete and generate a finite dimensional Lie algebra.
A vector field on a connected Lie group is linear if its flow is a one parameter group of automorphisms.
An affine vector field is obtained by adding a left invariant one. Its projection on a homogeneous space, whenever it exists, is still called affine.
Affine vector fields on homogeneous spaces can be characterized by their Lie brackets with the projections of right invariant vector fields.
A linear system on a homogeneous space is a system whose drift part is affine and whose controlled part is invariant.
The main result is based on a general theorem on finite dimensional algebras generated by complete vector fields, closely related to a theorem of Palais, and which has its own interest. The present proof makes use of geometric control theory arguments.

• 出版日期2010-12