Control issues when observability does not hold for the adjoint system

In this talk, I will present several control problems for which the adjoint system is not observable. Although the theory states that controllability is lost in such cases, we may still have weaker notions of controllability. We will present such property in two different contexts, one related to the wave equation based on a joint work with Belhassen Dehman and Enrique Zuazua in which we managed to prove regional controllability, and another one on the heat equation, which is well-known to be approximately controllable. In this latter case, the question is thus to estimate the cost of approximate controllability, a question linked to the density property of the reachable space within the underlying state space.