Controllability of Linear Systems on Solvable Lie Groups

Abstract

Abstract. Linear systems on Lie groups are a natural generalization of linear system on Euclidian spaces. For such systems, this paper studies controllability by taking in consideration the eigenvalues of an associated derivation D. When the state space is a solvable connected Lie group, controllability of the system is guaranteed if the reachable set of the neutral element is open and the derivation D has only pure imaginary eigenvalues. For bounded systems on nilpotent Lie groups such conditions are also necessary.

The dynamic of a Lie group endomorphism

“…i) If dim(Gφ) = the group Gφ is Abelian and the result is certainly true ii) Let us assume that the result holds for any endomorphism φ such that Gφ is solvable with dim(Gφ) < n. iii) Consider a φ-endomorphism of G with Gφ solvable and dim(Gφ) = n. The assumption of Gφ solvable implies that there exists a nontrivial closed normal Lie subgroup Bφ of Gφ which is Abelian and φ-invariant, (see for instance the proof in Proposition 2.9 of [1]). By considering the homogeneous space Hφ = Gφ/Bφ we obtain a connected solvable Lie group Hφ such that dim(Hφ) = dim(Gφ) − dim(Bφ) < n.…”

“…In [1] was shown that associated to a given continuous ow of automorphisms on a connected Lie group G there are dynamical subgroups of G that are intrinsically connected with the behavior of the ow. The author shows there that only by looking at such subgroups one can get information about the controllability of any control system whose drift generates a 1-parameter ow of automorphisms.…”

Controllability of Linear Systems on Lie Groups with Finite Semisimple Center

“…In particular, to extend the results in [4] as much as possible. It turns out that if the Lie group G has finite semisimple center (see Definition 3.1) such extension is possible.…”

“…For example, controllability of invariant control system, i.e., when the drift below is also an invariant vector field, is a local property for connected groups. Recently, in [4] Da Silva shows that controllability property for linear systems on solvable Lie groups is obtained if A is open and if any eigenvalue of the derivation D has zero real part. Moreover, controllability of restricted systems on nilpotent Lie groups are a very exceptional issue since the above conditions are also necessary.…”

“…Then, we prove that the system is controllable if the Lyapunov spectrum of D reduces to zero. The same sufficient algebraic controllability conditions were applied with success when G is a solvable Lie group, [4]. …”