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Abstract. A linear system on a connected Lie group G with Lie algebra g is determined by the family of differential equationṡwhere the drift vector field X is a linear vector field induced by a g-derivation D, the vector fields X j are right invariant and u ∈ U ⊂ L ∞ (R, Ω ⊂ R m ) with 0 ∈ int Ω. Assume that any semisimple Lie subgroup of G has finite center and e ∈ int A τ0 , for some τ 0 > 0. 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].

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Let G be a Lie group. In order to study optimal control problems on a homogeneous space G/H, we identify its cotangent bundle T * G/H as a subbundle of the cotangent bundle of G. Next, this identification is used to describe the Hamiltonian lifting of vector fields on G/H induced by elements in the Lie algebra g of G. As an application, we consider a bilinear control system Σ in R 2 whose matrices generate sl(2). Through the Pontryagin maximum principle, we analyze the time-optimal problem for the angle system PΣ defined by the projection of Σ onto the projective line P 1. We compute some examples, and in particular we show that the bang-bang principle does not need to be true.

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This paper classifies continuous linear flows using concepts and techniques from topological dynamics. Specifically, the concepts of equivalence and conjugacy are adapted to flows on vector bundles, and the Lyapunov decomposition is characterized using the induced flows on the Grassmann and the flag bundles. These results are then applied to bilinear control systems, for which their behavior in R d , on the projective space P d−1 , and on the Grassmannians is characterized.

Let g be an arbitrary finite dimensional Lie algebra over the field R. We give as an additional alternative a detailed overview of an algorithm for finding derivations of g since such procedures are often of interest.

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