Abstract. Impulse differential inclusions are introduced as a framework for modelling hybrid phenomena. Connections to standard problems in area of hybrid systems are discussed. Conditions are derived that allow one to determine whether a set of states is viable or invariant under the action of an impulse differential inclusion. For sets that violate these conditions, methods are developed for approximating their viability and invariance kernels, that is the largest subset that is viable or invariant under the action of the impulse differential inclusion. The results are demonstrated on examples. 1.

The concept of dynamical consistency is introduced to define a high level transition function for an arbitrary partition of the state space of a finite input-state machine M. The existence of a dynamically consistent transition from a partition member X i to a member X j corresponds to the existence, for every state x in X i , of some (x dependent) control sequence which takes that state directly into X j . This paper analyses what are termed in-block controllable partition machines and between-block controllable partition machines, and establishes the existence of the associated in-block controllable hierarchical lattice IBCP (M). It is shown that any partition machine in IBCP (M) is between-block controllable if and only if the base machine M is controllable. This setting allows the specification of state to state control trajectories of M to be decomposed into high level and low level components. Keywords: discrete event systems, controllability, system decomposition, hierarchical ...

Hybrid systems are models for complex physical systems and are defined as dynamical systems with interacting discrete transitions and continuous evolutions along differential equations. With the goal of developing a theoretical and practical foundation for deductive verification of hybrid systems, we introduce a dynamic logic for hybrid programs, which is a program notation for hybrid systems. As a verification technique that is suitable for automation, we introduce a free variable proof calculus with a novel combination of real-valued free variables and Skolemisation for lifting quantifier elimination for real arithmetic to dynamic logic. The calculus is compositional, i.e., it reduces properties of hybrid programs to properties of their parts. Our main result proves that this calculus axiomatises the transition behaviour of hybrid systems completely relative to differential equations. In a case study with cooperating traffic agents of the European Train Control System, we further show that our calculus is well-suited for verifying realistic hybrid systems with parametric system dynamics.

In this paper, we review rapidly-exploring random trees (RRTs) for motion planning, experiment with them on standard control problems, and extend them to the case of hybrid systems.

. In this paper, we formulate and robustly solve a quite general class of hybrid controller synthesis problems. The type of controller we investigate is the switching control mechanism of a hybrid automaton (via guard and mode invariant sets), and the robustness result is with respect to variations in the right hand sides of the dierential equations that depend continuously on a parameter. We present a novel methodology for controller design and synthesis which uses modal logic as a formalism for reasoning about sets of plant states, and various operators on sets arising from the dierential equations and from metric tolerance relations on the state space. 1 Introduction In general terms, a hybrid system H can be said to satisfy a performance speci cation robustly if every system H 0 in some nominated variation class around H also satis es that speci cation. Likewise, a synthesis procedure for a class of control problems can be called robust if the nominal closed-loop hyb...

KeYmaera is a hybrid verification tool for hybrid systems that combines deductive, real algebraic, and computer algebraic prover technologies. It is an automated and interactive theorem prover for a natural specification and verification logic for hybrid systems. KeYmaera supports differential dynamic logic, which is a real-valued first-order dynamic logic for hybrid programs, a program notation for hybrid automata. For automating the verification process, KeYmaera implements a generalized free-variable sequent calculus and automatic proof strategies that decompose the hybrid system specification symbolically. To overcome the complexity of real arithmetic, we integrate real quantifier elimination following an iterative background closure strategy. Our tool is particularly suitable for verifying parametric hybrid systems and has been used successfully for verifying collision avoidance in case studies from train control and air traffic management.

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Abstract. We introduce a fixedpoint algorithm for verifying safety properties of hybrid systems with differential equations whose right-hand sides are polynomials in the state variables. In order to verify nontrivial systems without solving their differential equations and without numerical errors, we use a continuous generalization of induction, for which our algorithm computes the required differential invariants. As a means for combining local differential invariants into global system invariants in a sound way, our fixedpoint algorithm works with a compositional verification logic for hybrid systems. To improve the verification power, we further introduce a saturation procedure that refines the system dynamics successively with differential invariants until safety becomes provable. By complementing our symbolic verification algorithm with a robust version of numerical falsification, we obtain a fast and sound verification procedure. We verify roundabout maneuvers in air traffic management and collision avoidance in train control.

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Abstract. We generalise dynamic logic to a logic for differential-algebraic programs, i.e., discrete programs augmented with first-order differentialalgebraic formulas as continuous evolution constraints in addition to first-order discrete jump formulas. These programs characterise interacting discrete and continuous dynamics of hybrid systems elegantly and uniformly. For our logic, we introduce a calculus over real arithmetic with discrete induction and a new differential induction with which differential-algebraic programs can be verified by exploiting their differential constraints algebraically without having to solve them. We develop the theory of differential induction and differential refinement and analyse their deductive power. As a case study, we present parametric tangential roundabout maneuvers in air traffic control and prove collision avoidance in our calculus.