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Abstract. Results from classical dynamical systems are generalized to hybrid dynamical systems. The concept of ω limit set is introduced for hybrid systems and is used to prove new results on invariant sets and stability, where Zeno and non-Zeno hybrid systems can be treated within the same framework. As an example, LaSalle’s Invariance Principle is extended to hybrid systems. Zeno hybrid systems are discussed in detail. The ω limit set of a Zeno execution is characterized for classes of hybrid systems. 1

Polygonal hybrid systems are a subclass of planar hybrid automata which can be represented by piecewise constant differential inclusions. Here, we study the problem of defining and constructing the phase portrait of such systems. We identify various important elements of it, such as viability and controllability kernels, and propose an algorithm for computing them all. The algorithm is based on a geometric analysis of trajectories.

Abstract. By transferring the theory of hybrid systems to a categorical framework, it is possible to develop a homology theory for hybrid systems: hybrid homology. This is achieved by considering the underlying “space ” of a hybrid system—its hybrid space or H-space. The homotopy colimit can be applied to this H-space to obtain a single topological space; the hybrid homology of an H-space is the homology of this space. The result is a spectral sequence converging to the hybrid homology of an H-space, providing a concrete way to compute this homology. Moreover, the hybrid homology of the H-space underlying a hybrid system gives useful information about the behavior of this system: the vanishing of the first hybrid homology of this H-space—when it is contractible and finite—implies that this hybrid system is not Zeno. 1

Polygonal hybrid systems are a subclass of planar hybrid automata which can be represented by piecewise constant differential inclusions. Here, we identify and compute an important object of such systems' phase portrait, namely invariance kernels. An invariant set is a set of initial points of trajectories which keep rotating in a cycle forever and the invariance kernel is the largest of such sets. We show that this kernel is a non-convex polygon and we give a non-iterative algorithm for computing the coordinates of its vertices and edges. Moreover, we present a breadth-first search algorithm for solving the reachability problem for such systems. Invariance kernels play an important role in the algorithm.

In this paper, we investigate conditions for existence of Zeno behaviors in hybrid systems. These are behaviors that occur in a hybrid system when the system undergoes an unbounded number of discrete transitions in a finite and bounded length of time. Zeno behavior occurs, for example, when a controller unsuccessfully attempts to satisfy an invariance specification by switching the system among different configurations faster and faster. Two types of Zeno systems will be investigated: (1) strongly Zeno systems where all runs of the system are Zeno; and (2) (weakly) Zeno systems where only some runs of the system are Zeno. We derive necessary and sufficient conditions for both strong Zenoness and Zenoness, under certain assumptions. Our analysis is based on studying the trajectory set of a certain “equivalent ” continuous-time system that is associated with the dynamic equations of the hybrid system. We also study the relation between the possibility of existence of Zeno behaviors in a system and the problem of existence of non-Zeno safety controllers that prevent the system from entering a suitably defines illegal region of its operating space. In particular, we show

In this work we are concerned with the formal verification of two-dimensional non-deterministic hybrid systems, namely polygonal differential inclusion systems (SPDIs). SPDIs are a class of nondeterministic systems that correspond to piecewise constant differential inclusions on the plane, for which we study the reachability problem. Our contribution is the development of an algorithm for solving exactly the reachability problem of SPDIs. We extend the geometric approach due to Maler and Pnueli [MP93] to non-deterministic systems, based on the combination of three techniques: the representation of the two-dimensional continuous-time dynamics as a one-dimensional discrete-time system (using Poincaré maps), the characterization of the set of qualitative behaviors of the latter as a finite set of types of signatures, and acceleration used to explore reachability according to each of these types.

Abstract. In this paper we address the problem of designing energy minimizing collision-free maneuvers for multiple agents moving on a plane. We show that the problem is equivalent to that of finding the shortest geodesic in a certain manifold with nonsmooth boundary. This allows us to prove that the optimal maneuvers are C 1 by introducing the concept of u-convex manifolds. Moreover, due to the nature of the optimal maneuvers, the problem can be formulated as an optimal control problem for a certain hybrid system whose discrete states consist of different “contact graphs”. We determine the analytic expression for the optimal maneuvers in the two agents case. For the three agents case, we derive the dynamics of the optimal maneuvers within each discrete state. This together with the fact that an optimal maneuver is a C 1 concatenation of segments associated with different discrete states gives a characterization of the optimal solutions in the three agents case. 1 Introduction and

The aim of this course is to introduce some fundamental concepts from the area of hybrid systems, that is dynamical systems that involve the interaction of continuous (real valued) states and discrete (finite valued) states. Applications where these types of dynamics play a prominent role will be highlighted. We will introduce general methods for investigating properties such as existence of solutions, reachability and decidability of hybrid systems. The methods will be demonstrated on the motivating applications. Students who successfully complete the course should be able to appreciate the diversity of phenomena that arise in hybrid systems and how discrete “discrete ” entities and concepts such as automata, decidability and bisimulation can coexist with continuous entities and

Abstract. Polygonal hybrid systems (SPDI) are a subclass of planar hybrid automata which can be represented by piecewise constant differential inclusions. The reachability problem as well as the computation of certain objects of the phase portrait, namely the viability, controllability and invariance kernels, for such systems is decidable. In this paper we show how to compute another object of an SPDI phase portrait, namely semi-separatrix curves and show how the phase portrait can be used for reducing the state-space for optimizing the reachability analysis. 1