**Introduction of hybrid dynamical systems:**

The term hybrid systems are used in the literature to refer to systems that feature an interaction between diverse forms of dynamics. Most heavily studied in recent years are hybrid systems that involve the interaction between continuous and discrete dynamics. The study of this class of systems has, to a large extent, been motivated by applications to embedded systems and control. Embedded systems by definition involve the interaction between digital devices and a predominantly analog environment. In optimal control problems, it is typically assumed that a cost is assigned to the different runs of the hybrid system. The objective of the controller is then to minimize this cost by selecting the values of the control variables appropriately. Typically, the cost function assigns a cost to both continuous evolution and discrete transitions. Let us take a predictive control model of power electronics as example. Power electronic systems are used to transform electrical power from one, usually unregulated form, to another regulated one. This transformation is achieved by the use of semiconductor devices that operate as power switches, turning on and off with high switching frequency. The whole system is a hybrid one, since the discrete switch positions are associated with different continuous-time dynamics. Other examples include direct torque control of three-phase induction motors; optimal control of fixed frequency switch-mode de-dc converters; command and control in military operations; communication systems; and flexible manufacturing systems, etc.