Parabolic–Hyperbolic PDE Loops

Abstract

The last chapter of the book is devoted to the study of parabolic–hyperbolic PDE loops by means of the small-gain methodology. Since there are many possible interconnections that can be considered, we focus on two particular cases, which are analyzed in detail. The first case considered in the chapter is the feedback interconnection of a parabolic PDE with a special first-order hyperbolic PDE: a zero-speed hyperbolic PDE. The study of this particular loop is of special interest because it arises in an important application: the movement of chemicals underground. Moreover, the study of this loop can be used for the analysis of wave equations with Kelvin–Voigt damping. The second case considered in the chapter is the feedback interconnection of a parabolic PDE with a first-order hyperbolic PDE by means of a combination of boundary and in-domain terms. The interconnection is effected by linear, non-local terms. For both cases, results for existence/uniqueness of solutions as well as sufficient conditions for ISS or exponential stability in the spatial sup-norm are provided.