Air traffic management ensures the safety of flight by optimizing flows and maintaining separation between aircraft. After giving some definitions, some typical feature of aircraft trajectories are presented. Trajectories are objects belonging to spaces with infinite dimensions. The naive way to address such problem is to sample trajectories at some regular points and to create a big vector of positions (and or speeds). In order to manipulate such objects with algorithms, one must reduce the dimension of the search space by using more efficient representations. Some dimension reduction tricks are then presented for which advantages and drawbacks are presented. Then, front propagation approaches are introduced with a focus on Fast Marching Algorithms and Ordered upwind algorithms. An example of application of such algorithm to a real instance of air traffic control problem is also given. When aircraft dynamics have to be included in the model, optimal control approaches are really efficient. We present also some application to aircraft trajectory design. Finally, we introduce some path planning techniques via natural language processing and mathematical programming.

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This work presents a new air traffic complexity metric based on non-linear dynamical systems. Previous work has shown that the structure and organization of traffic are important factors in the perception of the complexity of an air traffic situation. The new metric captures these important aspects of complexity by identifying the organization of trajectories in a traffic pattern. This paper investigates only the features of this new metric without quantifying directly the connection between complexity and the metric. Authors of previous work in this area have proposed metrics that generally have not explicitly addressed the effects of organization in the traffic flow on complexity. In order to capture the effect of organization, the metric is based on a dynamical system which fits as closely as possible the observations given by the aircraft positions and speeds. This approach, uses a non-linear dynamical system model that fits the observations without error. Based on this non linear dynamical system the method consist in the computation of the map of the associated Lyapunov exponents. This metric can be used to identify high (or low) complexity areas on a map, and, by capturing the organization properties of the traffic, captures some of the key factors involved in ATC complexity. Such maps are an example of the usefulness of these methods for comparing the relative complexity of different regions of airspace 1 .

Field observations and focused interviews of Air Traffic Controllers have been used to generate a list of key complexity factors in Air Traffic Control. The underlying structure of the airspace was identified as relevant in many of the factors. A preliminary investigation has revealed that the structure appears to form the basis for abstractions that reduce the difficulty of maintaining Situational Awareness, particularly the projection of future traffic situations. Three examples of such abstractions were identified: standard flows, groupings, and critical points. Preliminary approaches to developing metrics including these structural considerations are discussed.

The ability of air traffic controllers to deal with complex situations is a limiting factor in airspace capacity. The underlying airspace structure and other procedural elements are thought to be important factors in reducing a controller's Cognitive Complexity through the use of structure based abstractions. Because Cognitive Complexity cannot be directly observed it must be investigated indirectly. This paper discusses and presents examples of how directly observable states and controllers subjective responses can be used to indirectly probe and gain insight into how structure based abstractions are used to manage Cognitive Complexity.