Controllability of partial differential equations and its semi-discrete approximations

In these notes we analyze some problems related to the controllability and observability of partial differential equations and its space semidiscretizations. First we present the problems under consideration in the classical examples of the wave and heat equations and recall some well known results. Then we analyze the $1-d$ wave equation with rapidly oscillating coefficients, a classical problem in the theory of homogenization. Then we discuss in detail the null and approximate controllability of the constant coefficient heat equation using Carleman inequalities. We also show how a fixed point technique may be employed to obtain approximate controllability results for heat equations with globally Lipschitz nonlinearities. Finally we analyze the controllability of the space semi-discretizations of some classical PDE models: the Navier-Stokes equations and the $1-d$ wave and heat equations. We also present some open problems.