Controllability of Nonlocal Partial Differential Equations
This chapter provides a short survey of recent developments in the controllability of nonlocal partial differential equations. We first review controllability results for evolution equations involving nonlocal spatial operators. Particular attention is devoted to unique continuation properties, observability inequalities, and Carleman estimates in establishing controllability properties. We then discuss controllability issues for partial differential equations with memory terms, highlighting how memory effects influence reachable states, null controllability, and exact controllability. We review and compare various analytical techniques developed to overcome the difficulties arising from the nonlocal structure. We also provide a critical assessment of some of these methods. Throughout the chapter, we emphasize the interplay between controllability and observability, the adaptation of classical control methods to nonlocal settings, and the distinctive phenomena that arise in comparison with local equations. We then discuss geometric inequalities and their role in the controllability of diffusion models. The chapter concludes with a discussion of current challenges and several open problems, aiming to provide directions for future research.