Abstract-Hybrid automata provide a language for modeling and analyzing digital and analogue computations in real-time systems. Hybrid automata are studied here from a dynamical systems perspective. Necessary and sufficient conditions for existence and uniqueness of solutions are derived and a class of hybrid automata whose solutions depend continuously on the initial state is characterized. The results on existence, uniqueness, and continuity serve as a starting point for stability analysis. Lyapunov's theorem on stability via linearization and LaSalle's invariance principle are generalized to hybrid automata.Index Terms-Continuity of solutions, dynamical systems; existence, LaSalle's principle, Lyapunov's indirect method, hybrid systems, uniqueness.

Abstract. We propose a framework for a geometric theory of hybrid systems. Given a deterministic, non-blocking hybrid system, we introduce the notion of its hybrifold with the associated hybrid flow on it. This enables us to study hybrid systems from a global geometric perspective as (generally non-smooth) dynamical systems. This point of view is adopted in studying the Zeno phenomenon. We show that it is due to nonsmoothness of the hybrid flow. We introduce the notion of topological equivalence of hybrid systems and locally classify isolated Zeno states in dimension two.

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Abstract-In this paper, we consider the problem of constructing abstractions of affine control systems that preserve reachability properties, and, in particular, local accessibility. In this framework, showing local accessibility of the higher level, abstracted model is equivalent to showing local accessibility of the, more detailed, lower level model. Given an affine control system and a smooth surjective map, we present a canonical construction for extracting an affine control system describing the trajectories of the abstracted variables. We then obtain conditions on the abstraction maps that render the original and abstracted system equivalent from a local accessibility point of view. Such consistent hierarchies of accessibility preserving abstractions of nonlinear control systems are then considered for various classes of affine control systems including linear, bilinear, drift free, and strict feedback systems.

We apply a dynamical systems approach to concatenation of quantum error correcting codes, extending and generalizing the results of Rahn et al.[1] to both diagonal and non-diagonal channels. Our point of view is global: instead of focusing on particular types of noise channels, we study the geometry of the coding map as a discrete-time dynamical system on the entire space of noise channels.In the case of diagonal channels, we show that any code with distance at least three corrects (in the infinite concatenation limit) an open set of errors. For Calderbank-Shor-Steane (CSS) codes, we give a more precise characterization of that set. We show how to incorporate noise in the gates, thus completing the framework. We derive some general bounds for noise channels, which allows us to analyze several codes in detail. Index Terms-Quantum error correction, quantum channels, quantum fault toleranceJ. Fern is with the