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.

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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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In this paper we address the question of robustness of critical bit rates for the stabilization of networked control systems over digital communication channels. For a deterministic nonlinear system, the smallest bit rate above which practical stabilization (in the sense of setinvariance) can be achieved is measured by the invariance entropy of the system. Under the assumptions of chain controllability and a uniformly hyperbolic structure on the set of interest, we prove that the invariance entropy varies continuously with respect to system parameters. Hence, in this case the critical bit rate is robust with respect to small perturbations.