This talk aims to introduce, as elementary as possible, some classical or more recent results in infinite dimensional systems theory, with emphasis on controllability and time optimal control questions. After recalling the main controllability types and their characterization by duality, we provide an infinite dimensional version of the Hautus test, which we apply to Schrodinger type systems. The second part of the talk is devoted to time optimal control problems and more precisely to recently developed methods allowing to establish the ban-bag property of optimal controls.