Controllability of Partial Differential Equations and its SemiDiscrete Approximations

Dedicated to the memory of Jacques-Louis Lions. Abstract. In these notes we analyze some problems related to the control-lability and observability of partial differential equations and its space semi-discretizations. First we present the problems under consideration in the clas-sical examples of the wave and heat equations and recall some well known results. Then we analyze the 1−d wave equation with rapidly oscillating coef-ficients, 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. 1. Introduction. I