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We seek to classify the full-rank left-invariant control affine systems evolving on solvable three-dimensional Lie groups. In this paper we consider only the cases corresponding to the solvable Lie algebras of types II, IV , and V in the Bianchi-Behr classification.
We construct the concrete category LiCS of left-invariant control systems (on Lie groups) and point out some very basic properties. Morphisms in this category are examined briefly. Also, covering control systems are introduced and organized into a (comma) category associated with LiCS.
We consider left-invariant control affine systems evolving on Lie groups. In this context, feedback equivalence specializes to detached feedback equivalence. We characterize (local) detached feedback equivalence in a simple algebraic manner. We then classify all (full-rank) systems evolving on three-dimensional Lie groups. A representative is identified for… (More)
We consider left-invariant control affine systems on the matrix Lie group SO (2, 1)0. A classification, under state space equivalence, of all such full-rank control systems is obtained. First, we identify certain subsets on which the group of Lie algebra automorphisms act transitively. We then systematically identify equivalence class representatives (for… (More)
Let T x be the full transformation semigroup on the set X and let S be a subsemigroup of Tx. We may associate with S a digraph g(5) with X as set of vertices as follows: / —► / e g(<S) iff there exists a 6 S such that a(i) = /. Conversely, for a digraph C having certain properties we may assign a semigroup structure, S(G), to the underlying set of G. We are… (More)