A decomposition theorem for singular control systems on Lie groups.
The purpose of the paper under review is twofold. In the first place, the concept of a singular control system \({\mathcal S}_G\) on a connected finite-dimensional Lie group \(G\) with Lie algebra \(\text{ Lie}(G)\) has been introduced. This concept depends upon a pair of derivations \((E,D)\) where \(E\) plays on \(\text{ Lie}(G)\) the analogous role as the singular matrix of order \(n \geq 1\) on the simply connected abelian Lie group which is formed by the real vector space \({\mathbb R}^n\), and \(D\) induces the vector field on \(G\) corresponding to the drift vector field of the control system \({\mathcal S}_{{\mathbb R}^n}\). The second aim is to decompose the system into two subsystems: The linear control system and the differential-algebraic control system. In order to obtain the desired decomposition, the authors outline the construction of a principal fibre bundle and a canonical connection associated to any non-invertible derivation on \(\text{ Lie}(G)\) like \(E\). The authors consider the Jordan decomposition of \(E\), denote by \(\text{ Lie}(V)\) its generalized 0-eigenspace and assume that the connected Lie group \(V\) is closed. They also assume that the drift vector field is projectable onto the homogeneous manifold \(G/V\). In particular they extend to the singular system \({\mathcal S}_G\) the decomposition result for the class of singular control systems \({\mathcal S}_{{\mathbb R}^n}\). Finally, an example of decomposition for a singular control system \({\mathcal S}_N\) on the simply connected three-dimensional Heisenberg nilpotent Lie group \(N\) with real Heisenberg Lie algebra \(\text{ Lie}(N)\) of depth or step 2 is studied [\textit{W. Schempp}, Proc. Am. Math. Soc. 92, 103--110 (1984; Zbl 0525.43007); \textit{W. Schempp}, Harmonic analysis on the Heisenberg nilpotent Lie group, with applications to signal theory (1986; Zbl 0632.43001); \textit{A. Bellaïche}, The tangent space in sub-Riemannian geometry, Prog. Math. 144, 1--78 (1996; Zbl 0862.53031); \textit{W. J. Schempp}, Magnetic resonance imaging. Mathematical foundations and applications (1998; Zbl 0930.92015); \textit{W. Schempp}, Math. Meth. Appl. Sci. 22, 867--922 (1999; Zbl 0964.92026); \textit{R. Montgomery}, A tour of subriemannian geometries, their geodesics and applications (2002; Zbl 1044.53022)]. In this first non-trivial example of passing by a Poincaré heterotopy into the sub-Riemannian geo\-metry of \(N\), the role of non-holonomy appears clearly, and the Heisenberg nilpotent Lie group \(N\) serves as a paradigm for the theory.
scientific article; zbMATH DE number 2017214