This book presents and develops mathematical tools needed for a description of the motion of two and three-dimensional domains. Such problems have a number of applications in practice (such as free surface flows, shape optimization and contact problems). The authors concentrate on fluid-structure interaction problems and the presentation is based on the Eulerian approach. The book consists of eight chapters and four appendices. The first, introductory chapter presents several typical problems that arise when designing a fluid-structure interaction system. Chapter 2 deals with the identification of a moving boundary that separates a solid and a liquid phase (inverse Stefan problem). Chapter 3 is focused on the weak evolution of measurable sets described by the convection equation for characteristic functions. The concept of transverse variations makes it possible to differentiate functionals associated to evolution tubes. Chapter 4 recalls the concept of the shape differential equation. Applications to a simple shape control problem are given. In addition, it is shown how to proceed when domains are parameterized via the level set formulation.
Chapter 5 is devoted to the dynamic shape control of the Navier-Stokes equations by using the non-cylindrical Eulerian moving shape analysis. In contrast to the previous chapter, the Lagrangian shape analysis approach is used in chapter 6 for solving fluid-solid interaction problems. Chapter 7 presents a complex analysis of an inverse problem arising in the study of bridge deck aeroelastic stability. Finally in chapter 8, the results of the previous chapter are extended to the case of an elastic solid under large displacements inside an incompressible fluid flow. The book ends with four appendices summarizing basic information on function spaces, regularity of domains and the Fourier transform, which is needed for better understanding of the text. The book is intended for researchers and graduate students who are interested in the control of systems involving moving boundaries. A good preliminary knowledge of the topic is required for some parts of the book.