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 semi-discretizations. 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.