This volume constitutes the proceedings of the summer school MIS 2015, “Mathematics In Savoie 2015,” whose theme was: “Evolution Equations: long time behavior and control.”

This summer school was held at the University Savoie Mont Blanc, Chambéry in the period June 15–18, 2015 (see http://lama.univ-savoie.fr/ MIS2015 for details). It was organized by Kaïs Ammari, UR Analysis and Control of PDE, University of Monastir, Tunisia, and Stéphane Gerbi, Laboratoire de Mathématiques, University Savoie Mont Blanc, France.

The summer school consisted of two mini-courses in the morning while the afternoons were devoted to various contributions on the theme.

The first mini-course was held by Farid Ammar-Khodja, University of Franche-Comté, France. The topic was: “Controllability of parabolic systems: the moment method.” This recent point of view on the controllability of parabolic systems permits to overview the moment method for parabolic equations. This course constitutes the first part of this volume.

The second part of this volume is devoted to the second mini-course which was held by Emmanuel Trélat, UPMC, Paris. The topic was “Stabilization of semilinear PDEs, and uniform decay under discretization.” This course was devoted to the numerical stabilization and control of partial differential equations and more specifically it addresses the problem of the construction of numerical feedback control that will preserve the theoretical rate of decay.

Several of the speakers agreed to write review papers related to their contributions to the summer school, while others have written more traditional research papers, which constitute the last part of this volume.

We believe that this volume therefore provides an accessible summary of a wide range of active research topics, along with some exciting new results, and we hope that it will prove a useful resource for both graduate students new to the area and to more established researchers.

The summer school brought together internationally leading researchers from the community of control theory and young researchers who came from all around the world. The organizers’ intention was to provide a wide angle snapshot of this exciting and fast moving area and facilitate the exchange of ideas on recent advances in its various aspects. The numerous formal, informal, and sometimes lively discussions that resulted from this interaction were for us a sign that we achieved something in the direction of fulfilling this aim.

Hybrid systems describe the dynamical interaction between continuous and discrete signals in one common framework (see Figure 16.1). In this chapter we focus our attention on mathematical models of hybrid systems that are particularly suitable for solving finite time-constrained optimal control problems.

Models of Hybrid Systems

The mathematical model of a dynamical system is traditionally expressed through differential or difference equations, typically derived from physical laws governing the dynamics of the system under consideration. Consequently, most of the control theory and tools address models describing the evolution of real-valued signals according to smooth linear or nonlinear state transition functions, typically differential or difference equations. In many applications, however, the system to be controlled also contains discrete-valued signals satisfying Boolean relations, if-then-else conditions, on/off conditions, etc., that also involve the real-valued signals. An example would be an on/off alarm signal triggered by an analog variable exceeding a given threshold. Hybrid systems describe in a common framework the dynamics of real-valued variables, the dynamics of discrete variables, and their interaction.

In this chapter we will focus on discrete-time hybrid systems, which we will call discrete hybrid automata (DHA), whose continuous dynamics is described by linear difference equations and whose discrete dynamics is described by finite state machines, both synchronized by the same clock [276]. A particular case of DHA is the class of piecewise affine (PWA) systems [266]. Essentially, PWA systems are switched affine systems whose mode depends on the current location of the state vector, as depicted in Figure 16.2. PWA and DHA systems can be translated into a form, denoted as mixed logical dynamical (MLD) form, that is more suitable for solving optimization problems. In particular, complex finite time hybrid dynamical optimization problems can be recast into mixed-integer linear or quadratic programs as will be shown in Chapter 17.

In Section 16.7 we will introduce the tool HYSDEL (HYbrid Systems DEscription Language), a high level language for modeling and simulating DHA. Therefore, DHA will represent for us the starting point for modeling hybrid systems. We will show that DHA, PWA, and MLD systems are equivalent model classes, and in particular that DHA systems can be converted to an equivalent PWA or MLD form for solving optimal control problems.