A hybrid system is a system whose evolution is subject to a continuous evolution (governed by an ordinary differential equation) as well as to a jumping evolution (governed by some logic rules). The way for passing from the continuous mode to the jumping one is a characteristics of the particular model under consideration. A typical example is a bouncing ball on the floor (the scalar velocity is continuous untill the ball hits the floor and then suddenly changes sign), or a system of computers working in parallel that exchange informations to each other only during some suitable interval of time (the state of computers is subject to jumps: exchanging/not exchanging).
Optimal control of hybrid systems (using some parameters in the equations and in the jumping rules, in order to get a desired evolution) is an interesting argument of research nowadays both from a theoretical and applicative point of view. The application of the Dynamic Programming Principle and the study of the corresponding Hamilton-Jacobi equation is an intriguing subject (see the page on Controllo Ottimo).