The development of formal methods for control design is an important challenge with potential applications in a wide range of safety-critical cyber-physical systems. Focusing on switched dynamical systems, we propose a new abstraction, based on time-varying regions of invariance (control funnels), that models behaviors of systems as timed automata. The main advantage of this method is that it allows for the automated verification and reactive controller synthesis without discretizing the evolution of the state of the system. Efficient and analytic constructions are possible in the case of linear dynamics whereas bounding funnels with conjectured properties based on numerical simulations can be used for general nonlinear dynamics. We demonstrate the potential of our approach with three examples.