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.