ABSTRACT

This chapter deals with the analysis of an inverse dynamical shape problem involving a fluid inside a moving domain. This type of inverse problem happens frequently in the design and the control of many industrial devices such as aircraft wings, cable-stayed bridges, automobile shapes, satellite reservoir tanks and more generally of systems involving fluid-solid interactions. The control variable is the shape of the moving domain, and the objective is to minimize a given cost functional that may be chosen by the designer. On the theoretical level, early works concerning optimal control problems for general parabolic equations written in non-cylindrical domains have been considered in [43], [29], [30], [142], [2]. In [140], [151], [152], the stabilization of structures using the variation of the domain has been addressed. The basic principle is to define a map sending the non-cylindrical domain into a cylindrical one. This process leads to the mathematical analysis of non-autonomous PDE’s systems. Recently, a new methodology to obtain Eulerian derivatives for non-cylindrical functionals has been introduced in [157], [156], [58]. This methodology was applied in [59] to perform dynamical shape control of the non-cylindrical NavierStokes equations where the evolution of the domain is the control variable. Hence the classical optimal shape optimization theory has been extended to deal with non-cylindrical domains. The aim of this chapter is to review several results on the dynamical shape control of the Navier-Stokes system and suggest an alternative treatment using the Min-Max principle [45, 46]. Despite its lack of rigorous mathematical justification in the case where the Lagrangian functional is not convex, we shall show how this principle allows, at least formally, to bypass the tedious computation of the state differentiability with respect to the shape of the moving domain.