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In this course, we will be concerned with the control and stabilization of partial differential equations.
The problem of control is the following. In some dynamical system where it is possible to act by a control, given an initial state and a final state fixed in advance, can we drive the system from the initial one to the second one? In the mathematical point of view, the system can often be modelized by an evolution equation (ODE or a PDE) where the modification takes the form of a source term (or a boundary term) that have some geometrical constraints, for instance to be supported in some specific part of the domain. We will try to give some introduction to these questions. We will begin by introducing the problem in the simpler context of linear ordinary diJerential operators. We will introduce the Kalman rank condition and the general link between observability and controlability. We will then study the wave equation with a specific focus on multiplier methods. In the following part, we will study the Schrödinger equation, and it will be the occasion to present some results of propagation in the simpler case of the dimension 1. If time permits, we will study the heat equation and the use of Carleman estimates.