This course will focus on controllability (and some stabilization) properties for the one dimensional Schrödinger equation with a bilinear control obtained in the last ten years. In this model, the evolution is given by a simple one dimensional partial differential equation on a bounded interval but the control that allows one to act on the dynamic enters the problem in a nonlinear way.
The study of these controllability properties will be an excuse to present several general methods developed to study the controllability properties in the case where the control problem is nonlinear such as the linear test, the return method, power series expansion, the use of Lyapunov functionals …
Instead of giving an abstract presentation of these methods I will show how they can be applied to this particular model.
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