On the characterization of the controllability property for linear control systems on nonnilpotent, solvable three-dimensional Lie groups

Abstract

In this paper we show that a complete characterization of the controllability property for linear control system on three-dimensional solvable nonnilpotent Lie groups is possible by the LARC and the knowledge of the eigenvalues of the derivation associated with the drift of the system.

Geometric Structures Generated by the Same Dynamics. Recent Results and Challenges

“…Here, ρ t = e tθ , ∆ = Span Y 1 , Y 2 has dimension 2, and θ is the real matrix of order 2 with all the coefficients 0 except 1 in the position 22. Therefore, we obtain [22].…”

Central periodic points of linear systems

“…Let us denote by R the solvable radical of G. Under the assumption that G 0 is compact, we get that (G/R) 0 = π(G 0 ) is also compact, where π : G → G/R is the canonical projection. Since G/R is semi-simple, Proposition 3.3 of [3] implies that G/R = (G/R) 0 and consequently that G = G 0 R. In particular, G + , G − ⊂ R. On the other hand, Lemma 2.1 of [2] assures that the nilradical r of g contains g α for any nonzero eigenvalue α of D. Therefore, g + , g − ⊂ n and G + , G − ⊂ N . 2.…”

Dynamics of LCSs on the group of proper motions

“…This section introduces the basic notions of control-affine systems, control sets and linear control systems on SE (2), and also to show some preliminaries results we will be using ahead. For more on the subjects, the reader should consult references [1,3,7].…”