Topology of hybrid systems

We discuss topological issues that arise when differential equations and finite automata interact (hybrid systems). In particular, we examine topologies for achieving continuity of maps from a set of measurements of continuous dynamics to a finite set of input symbols and from a finite set of output symbols into the control space for those continuous dynamics. Finding some anomalies in completing this loop, we discuss a new view of hybrid systems that may broach them and is more in line with traditional control systems. In fact, the most widely used fuzzy control system is related to this new view and does not possess these anomalies. Indeed, we show that fuzzy control leads to continuous maps (from measurements to controls) and that all such continuous maps may be implemented via fuzzy control