Controllability and Observability of Partial Differential Equations: Some Results and Open Problems

Abstract In this chapter we present some of the recent progresses done on the problem of controllability of partial differential equations (PDE). Control problems for PDE arise in many different contexts and ways. A prototypical problem is that of controllability. Roughly speaking it consists in analyzing whether the solution of the PDE can be driven to a given final target by means of a control applied on the boundary or on a subdomain of the domain in which the equation evolves. In an appropriate functional setting this problem is equivalent to that of observability which concerns the possibility of recovering full estimates on the solutions of the uncontrolled adjoint system in terms of partial measurements done on the control region. Observability/controllability properties depend in a very sensitive way on the class of PDE under consideration. In particular, heat and wave equations behave in a significantly different way, because of their different behavior with respect to time reversal. In this paper we first recall the known basic controllability properties of the wave and heat equations emphasizing how their different nature affects their main controllability properties. We also recall the main tools to analyze these problems: the so-called Hilbert uniqueness method (HUM), multipliers, microlocal analysis and Carleman inequalities. We then discuss some more recent developments concerning equations with low regularity coefficients, equations with potentials, bang-bang controls, etc. We also analyze the way control and observability properties depend on the norm and regularity of these coefficients, a problem which is also relevant when addressing nonlinear models. We then present some recent results on coupled models of wave–heat equations arising in fluid–structure interaction. We also present some open problems and future directions of research.