The study of dynamic Systems usually splits into two groups in the literature. The first approach uses continuous or smooth maps, differential equations and flows, whereas the second one deals with discrete maps and discrete evolution equations. Although they usually model different problems, there are certain interesting situations in the overlapping domains of both. Furthermore, there are systems that naturally behave in a continuous (differentiable) way in specific space-time regions and a discrete way in others. Typical examples are electric circuits, switching systems, control systems, bouncing balls, etc. This hybrid situation is the motivation and leitmotiv of this book.

The book presents a clean and self-contained exposition of hybrid systems, starting from the elementary definitions, continuing with the basic tools and finishing with more recent contributions in the literature. Along its nine chapters, the reader is introduced into the hybrid notion of asymptotic behavior, Lyapunov functions, stability, well-posedness, etc. Some examples are fully analyzed.

The authors are researchers in dynamical systems. In particular, the PhD. Thesis of R. Sanfelice (2007, under the guidance of his advisor, A.R. Teel) was titled “Robust Hybrid Control Systems”. The third author (R. Goebel) also has recent contributions in the field. The work is a comprehensive reference and accessible to a broad public interested in dynamical systems from both an applied or a pure point of view.