Thoughts on Hybrid Systems

There is a growing recognition of the presence of dynamical systems in many fields. In areas ranging from economics to physics and computer science to biology, many open fundamental problems are known to relate to the central problems in dynamical systems theory. Physicists generally work more with continuous dynamical systems, rather than discrete dynamical systems which computer scientists may study. The difference is that a continuous dynamical system is almost always defined as a system with an evolution function that satisfies a certain set of differential equations, while a discrete dynamical system is described by a state space involving discrete states and events which trigger discrete transitions between them. However, as computer science has come into contact with problems such as traffic management, automotive control, and modeling biological cell networks, which involve both continuous and discrete dynamical systems, a new formalism has emerged. This new formalism, the concept of a hybrid system, combines both the continuous and discrete formalisms into one entity that has fueled much research in recent years. Here we will formally introduce the notion of a hybrid system as an extension (or special case) of the concept of a dynamical system that is appropriate for many problems of interest in a wide variety of fields. We will then present summaries of some papers written on hybrid systems in recent years and present ideas for future research in the field.