Controllability of Linear Systems on Low Dimensional Nilpotent and Solvable Lie Groups

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

“…For restricted linear control systems on nilpotent Lie groups such condition is also necessary for controllability. In the same direction, Dath and Jouan show in [6] that linear control systems (restricted or not) on a two-dimensional solvable Lie group present the same behavior, they are controllable if and only if they satisfy the Lie algebra rank condition and the associated derivation has only zero eigenvalues. Here, the LARC is equivalent to the ad-rank condition which implies, in particular, the openness of the reachable set (see [4]).…”

“…A natural extension of a linear control system on Lie groups appears first in [13] for matrix groups and then in [4] for any Lie group. In the subsequent years, several works addressing the main problems in control theory for such systems, such as controllability, observability and optimization appeared (see [1,2,3,5,6,8,9]). In [9] P. Jouan shows that such generalization is also important for the classification of general affine control systems on abstract connected manifolds.…”

The control set of a linear control system on the two dimensional Lie group

“…Before showing that the control system (3) is controllable it is important to notice that in [8] the authors prove that b = 0 and the LARC are equivalent to the controllability of a linear control system. The difference here is that we show explicitly "the waysuch controllability is obtained.…”

Isometries of almost-Riemannian structures on Lie groups

“…For more details, see [12]. An ARS on G is defined by an orthonormal frame {B 1 , B 2 , X }, where B 1 , B 2 are left-invariant vector fields and X is a linear one with associated derivation D.…”

“…On an n-dimensional connected Lie group the simplest ARSs are defined by a set of n − 1 left-invariant vector fields and one linear vector field, the rank of which is equal to n on a proper open and dense subset and that satisfy the rank condition (a vector field on a Lie group is linear if its flow is a one parameter group of automorphisms, see Section 2, and [5], [6], [12], [15], [16] about linear systems on Lie groups).…”