A vector field on a connected Lie group is said to be linear if its flow is a one parameter group of automorphisms. A controlaffine system is linear if the drift is linear and the controlled vector fields right invariant. The controllability properties of such systems are studied, mainly in the case where the derivation of the group Lie algebra that can be associated to the linear vector field is inner. After some general considerations controllability properties on semi simple, nilpotent and compact Lie groups are stated. The paper ends by many examples.

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The paper is concerned with asymptotic stability properties of linear switched systems. Under the hypothesis that all the subsystems share a non strict quadratic Lyapunov function, we provide a large class of switching signals for which a large class of switched systems are asymptotically stable. For this purpose we define what we call non chaotic inputs, which generalize the different notions of inputs with dwell time.Next we turn our attention to the behaviour for possibly chaotic inputs. To finish we give a sufficient condition for a system composed of a pair of Hurwitz matrices to be asymptotically stable for all inputs.

Abstmcl-This paper deals with the problem of the immersion of a SISO sjstem into a linear up to an output injection one. For this class of sjstems Luenberger-like observes can be designed the djnamics of the error between the states of the obsener and the immersed controlled system is linear. Necessary and sufficient Conditions are firs1 stated within a r e v general framework. Efleetive computations and examples are then provided for uncontrolled and control-affine sjstems In particular the control-affine case is completel) (with a very slight restriction) sohed.