Mathematical Control Theory is a branch of Mathematics having as one of its main aims the establishment of a sound mathematical foundation for the c- trol techniques employed in several di?erent ?elds of applications, including engineering, economy, biologyandsoforth. Thesystemsarisingfromthese- plied Sciences are modeled using di?erent types of mathematical formalism, primarily involving Ordinary Di?erential Equations, or Partial Di?erential Equations or Functional Di?erential Equations. These equations depend on oneormoreparameters thatcanbevaried, andthusconstitute thecontrol - pect of the problem. The parameters are to be chosen soas to obtain a desired behavior for the system. From the many di?erent problems arising in Control Theory, the C. I. M. E. school focused on some aspects of the control and op- mization ofnonlinear, notnecessarilysmooth, dynamical systems. Two points of view were presented: Geometric Control Theory and Nonlinear Control Theory. The C. I. M. E. session was arranged in ?ve six-hours courses delivered by Professors A. A. Agrachev (SISSA-ISAS, Trieste and Steklov Mathematical Institute, Moscow), A. S. Morse (Yale University, USA), E. D. Sontag (Rutgers University, NJ, USA), H. J. Sussmann (Rutgers University, NJ, USA) and V. I. Utkin (Ohio State University Columbus, OH, USA). We now brie?y describe the presentations. Agrachev's contribution began with the investigation of second order - formation in smooth optimal control problems as a means of explaining the variational and dynamical nature of powerful concepts and results such as Jacobi ?elds, Morse's index formula, Levi-Civita connection, Riemannian c- vature.

Are there universal principles of coordinated group motion and if so what might they be? This carefully edited book presents how natural groupings such as fish schools, bird flocks, deer herds etc. coordinate themselves and move so flawlessly, often without an apparent leader or any form of centralized control. It shows how the underlying principles of cooperative control may be used for groups of mobile autonomous agents to help enable a large group of autonomous robotic vehicles in the air, on land or sea or underwater, to collectively accomplish useful tasks such as distributed, adaptive scientific data gathering, search and rescue, or reconnaissance.

A logic-based switching controller is one whose subsystems include not only familiar dynamical components such as integrators, summers, gains etc. but event-driven logic and associated switches as well. In such a system the predominantly logical component is the supervisor, mode changer, etc. There has been growing interest in recent years in determining what could be gained from utilising "hybrid" controllers of this type. To this end a workshop was held on Block Island with the aim of bringing together individuals to discuss the research and common interest in the field. This volume not only includes contributions from those who were present at Block Island but also additional material from those who were not. Topics covered include: hybrid dynamical systems, control of hard-bound constrained and nonlinear systems, automotive problems involving switching control and system control in the face of large-scale modeling errors.