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Hybrid automata are an eminently suitable modelling framework for biological protein regulatory networks, as the protein concentration dynamics inside each biological cell are modelled using linear differential equations; inputs activate or deactivate these continuous dynamics through discrete switches, which themselves are controlled by protein concentrations reaching given thresholds. This paper proposes an iterative refinement algorithm for computing discrete abstractions of a class of hybrid automata with piecewise affine continuous dynamics and forced discrete transitions, defined completely in terms of symbolic variables and parameters. Furthermore, these discrete abstractions are utilised to compute symbolic parametric backward reachable sets from the equilibria of the hybrid automata, that are guaranteed to be exact or conservative under-approximations. The algorithm is then implemented using MATLAB and QEPCAD, to compute reachable sets for the biologically observed equilibria of the multiple cell Delta-Notch protein signalling automaton with symbolic parameters. The results are analysed to show that novel, non-intuitive, and biologically interesting properties can be deduced from the reachability computation, thus demonstrating the utility of the algorithm.

We address the problem of generating provably-safe conflict resolution maneuvers for aircraft in uncertain environments. We assume that a maneuver is composed of a sequence of flight modes, which are segments of constant heading, of constant bank angle, or of constant airspeed. Each of these flight modes has associated to it the kinematics of the aircraft, and hence the maneuver is a hybrid system. While the flight modes are defined ahead of time, their sequencing and parameter values do not necessarily have to be. We present an algorithm for generating provably safe maneuvers, which is based on a general procedure for designing controllers for hybrid systems. The result is a maneuver, proven to be safe within the limits of the models used, which is a familiar sequence of commands easily executable by the flight management systems. The maneuvers may be viewed as protocols, or "rules of the road", and are well-defined for each conflict scenario. We present results for two example maneuvers.

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