A hybrid automaton is a formal model for a mixed discrete-continuous system. We classify hybrid automata acoording to what questions about their behavior can be answered algorithmically. The classification reveals structure on mixed discrete-continuous state spaces that was previously studied on purely discrete state spaces only. In particular, various classes of hybrid automata induce finitary trace equivalence (or similarity, or bisimilarity) relations on an uncountable state space, thus permitting the application of various model-checking techniques that were originally developed for finite-state systems.

. We present algorithms for computing similarity relations of labeled graphs. Similarity relations have applications for the refinement and verification of reactive systems. For finite graphs, we present an O(mn) algorithm for computing the similarity relation of a graph with n vertices and m edges (assuming m n). For effectively presented infinite graphs, we present a symbolic similarity-checking procedure that terminates if a finite similarity relation exists. We show that 2D rectangular automata, which model discrete reactive systems with continuous environments, define effectively presented infinite graphs with finite similarity relations. It follows that the refinement problem and the 8CTL model-checking problem are decidable for 2D rectangular automata. 1 Introduction A labeled graph G = (V; E;A; hh\Deltaii) consist of a (possibly infinite) set V of vertices, a set E ` V 2 of edges, a set A of labels, and a function hh\Deltaii : V ! A that maps each vertex v to a label hh...

Rectangular hybrid automata model digital control programs of analog plant environments. We study rectangular hybrid automata where the plant state evolves continuously in real-numbered time, and the controller samples the plant state and changes the control state discretely, only at the integer points in time. We prove that rectangular hybrid automata have nite bisimilarity quotients when all control transitions happen at integer times, even if the constraints on the derivatives of the variables vary between control states. This is in contrast with the conventional model where control transitions may happen at any real time, and already the reachability problem is undecidable. Based on the nite bisimilarity quotients, we give an exponential algorithm for the symbolic sampling-controller synthesis of rectangular automata. We show our algorithm to be optimal by proving the problem to be EXPTIME-hard. We also show that rectangular automata form a maximal class of systems for which the sampling-controller synthesis problem can be solved algorithmically.

In order to study control problems for hybrid systems, we generalize hybrid automata to hybrid games -- say, controller vs. plant. If we specify the continuous dynamics by constant lower and upper bounds, we obtain rectangular games. We show that for rectangular games with objectives expressed in Ltl (linear temporal logic), the winning states for each player can be computed, and winning strategies can be synthesized. Our result is sharp, as already reachability is undecidable for generalizations of rectangular systems, and optimal -- singly exponential in the size of the game structure and doubly exponential in the size of the Ltl objective. Our proof systematically generalizes the theory of hybrid systems from automata (single-player structures) [9] to games (multi-player structures): we show that the successively more general infinite-state classes of timed, 2d rectangular, and rectangular games induce successively weaker, but still finite, quotient structures called game bisimilarity, game similarity, and game trace equivalence. These quotients can be used, in particular, to solve the Ltl control problem.

Abstract. In this paper we investigate the decidability of the reachability problem for planar non-linear hybrid systems. A planar hybrid system has the property that its state space corresponds to the standard Euclidean plane, which is partitioned into a nite number of (polyhedral) regions. To each of these regions is assigned some vector eld which governs the dynamical behaviour of the system within this region. We prove the decidability of point to point and region to region reachability problems for planar hybrid systems for the case when trajectories within the regions can be described by polynomials of arbitrary degree. 1

Abstract. An important case of hybrid systems are the rectangular automata. First, rectangular dynamics can naturally and arbitrarily closely approximate more general, nonlinear dynamics. Second, rectangular automata are the most general type of hybrid systems for which model checking |in particular, Ltl model checking | is decidable. However, on one hand, the original proofs of decidability did not suggest practical algorithms and, on the other hand, practical symbolic model-checking procedures |such as those implemented in HyTech | were not known to terminate on rectangular automata. We remedy this unsatisfactory situation: we present a symbolic method for Ltl model checking which can be performed by HyTech and is guaranteed to terminate on all rectangular automata. We dosoby proving that our method for symbolic Ltl model checking terminates on an in nite-state transition system if the trace-equivalence relation of the system has nite index, which is the case for all rectangular automata. 1

1 Introduction The design of control techniques for nonholonomic vehicles is a topic of extensive research #see e.g. #1#, #2#, #3#, #4##. For nonholonomic systems, the problem of tracking a path is simpler in principle than stabilizing to a point 1 . By #path" we refer to a curve #with some regularity requirements# in the plane were the vehicle moves. See #6#, #7#, #8#, for path following controllers. Besides the kinematic constraints imposed by the nonholonomy of the model of the vehicle, most often the additional constraint that the radius of curvature of the paths of the vehicle are lower bounded must be considered. This restriction makes the kinematic model more similar to real#world vehicles encountered in most applications. Some fundamental results in this area have a direct bearing to the work reported in this paper. In particular, it was shown that the kinematic model of a car that can drive both forwards and backwards with bounded curvature #but allowing cusps in the path#,...