Abstract. This paper has been conceived as an overview on the controllability properties of some relevant (linear and nonlinear) parabolic systems. Specifically, we deal with the null controllability and the exact controllability to the trajectories. We try to explain the role played by the observability inequalities in this context and the need of global Carleman estimates. We also recall the main ideas used to overcome the difficulties motivated by nonlinearities. First, we considered the classical heat equation with Dirichlet conditions and distributed controls. Then we analyze recent extensions to other linear and semilinear parabolic systems and/or boundary controls. Finally, we review the controllability properties for the Stokes and Navier-Stokes equations that are known to date. In this context, we have paid special attention to obtaining the necessary Carleman estimates. Some open questions are mentioned throughout the paper. We hope that this unified presentation will be useful for those researchers interested in the field.Key words. Carleman inequalities, parabolic equations, unique continuation, observability, null controllability AMS subject classifications. 35B37, 70Q05, 93B05, 93B07
DOI. 10.1137/S0363012904439696Introduction. The goal of this paper is to provide a panorama of an important subfield of control theory: the null controllability analysis of parabolic equations and systems.The main contributions to this area are due to O. Yu. Imanuvilov, who popularized the use of global Carleman estimates in the context of null controllability. Many arguments from [36], [30], [39], and [37] will be reproduced here. Another relevant contributor has been E. Zuazua, who was able to deduce global controllability results for some nonlinear systems for the first time in [56].The controllability of partial differential equations has been the object of intensive research during the last few decades. In 1978, Russell [53] made a survey of the most relevant results that were available in the literature at that time. In that paper, the author described a number of different tools that were developed to address controllability problems, often inspired and related to other subjects concerning partial differential equations: multipliers, moment problems, nonharmonic Fourier series, etc. More recently, J.-L. Lions introduced the so-called Hilbert uniqueness method (for instance, see [44], [45], [46]). That was the starting point of a fruitful period on the subject.It would be impossible to present here all the relevant results that have been proved in this area. We will thus reduce our scope drastically, considering only null controllability problems for parabolic equations and systems.In order to get an idea, let us consider the simplest case of the linear heat equation
