A class of problems in air traffic management (ATM) asks for a scheduling algorithm that supplies the air traffic services authority not only with a schedule of arrivals and departures but also with speed advisories. Since advisories must be finite, a scheduling algorithm must ultimately produce a finite data set, hence must either start with a purely discrete model or involve a discretization of a continuous one. The former choice, often preferred for intuitive clarity, naturally leads to mixed-integer programs (MIPs), hindering proofs of correctness and computational cost bounds (crucial for real-time operations). In this paper, a hybrid control system is used to model air traffic scheduling, capturing both the discrete and continuous aspects. This framework is applied to a class of problems, called the fully routed nominal problem. We prove a number of geometric results on feasible schedules and use these results to formulate an algorithm that attempts to compute a collective speed advisory, effectively piecewise linear with finitely many vertices, and has computational cost polynomial in the number of aircraft. This work is a first step toward optimization and models refined with more realistic detail.

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