Controllability on classical Lie groups
This paper investigates conditions for the global controllability of bilinear systems evolving on a connected Lie group \(G\subset Gl(n,{\mathbb{R}})\) whose Lie algebra L is the normal real form of one of the classical complex Lie algebras g of type An, Bn, Cn or Dn. The systems are described by equations of the form \[ \dot x(t)=(Au+Bv)x(t),\quad u,v\in {\mathbb{R}},\quad x\in G;\quad or\quad \dot x(t)=(A+uB)x(t),\quad u\in {\mathbb{R}},\quad x\in G \] where A, \(B\in L\) and B is supposed to be diagonal. Let \(\Phi\) be the set of nonzero roots of g with respect to a Cartan subalgebra h. B is said to be A-strongly regular if the elements \(\alpha\) (B), \(\alpha \in \Phi_ A\) are nonzero and distinct where \(\Phi_ A\) is the set of roots occuring in the decomposition of A. Under the assumption that B is A-strongly regular it is shown that A, B generate the Lie algebra L if and only if A is P-irreducible. This yields necessary and sufficient conditions for global controllability in situations where this property is equivalent to the controllability rank conditions. The paper extends previous results of Gauthier, Kupka and Sallet obtained under the assumption that B is strongly regular.
scientific article; zbMATH DE number 4077042