Controllability of partial differential equations and its semi-discrete approximations

Abstract

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