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Eikonal Approaches to Free Flight Trajectory Optimization
Martin Weiser, Arturas Jocas, Ralf Borndörfer
Zuse Institute Berlin, Germany
The aviation sector continues to grow rapidly, with projected revenues approaching 1 trillion USD by 2025. This growth raises significant concerns about CO2 emissions and the corresponding fuel consumption. In Europe alone, approximately 1.3 million people travel by air each day, contributing to nearly 240,000 tonnes of CO₂ emissions daily. Addressing these environmental challenges, as well as reducing the airlines' costs, necessitates innovative methods for reducing fuel usage. Here we focus on the implicit reduction by optimizing the routes that aircraft take, such that headwind can be avoided and tailwinds exploited. This is routinely done in flight planning within discrete airway networks. For exploiting the wind conditions more fully, airspaces turn towards free flight, allowing arbitrary continuous trajectories. Numerical studies suggest a potential of 2-3% fuel savings from moving to free flight. This leads to the classic Zermelo navigation problem. The challenge here is the demand for global optima. We present an algorithm capable of finding a continuous globally optimal trajectory for an aircraft in a stationary wind field. The algorithm solves a Hamilton-Jacobi-Bellman (HJB) equation associated with the flight trajectory optimization problem, employing linear finite elements and effective parallelization. Furthermore, we demonstrate a linear order of convergence of the discretization by giving an explicit bound for the error estimate of arrival time. In addition, this holds in the presence of singularities for HJB PDE, i.e., cut loci of the system. Finally, we discuss global optimality guarantees, and by combination with pre-existing optimal control approaches, we can find a globally optimal path within the desired accuracy. The single-trajectory optimization also forms a building block of collaborative routing, a multi-player problem of airlines competing for capacity-bound airspace sectors in planning their flight trajectories. We report first results on a pricing strategy for solving this problem. References R. Borndörfer, F. Danecker, M. Weiser. A Discrete-Continuous Algorithm for Globally Optimal Free Flight Trajectory Optimization. In 22nd Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2022). 2022. R. Borndörfer, A. Jocas, M. Weiser. An Eikonal Approach for Globally Optimal Free Flight Trajectories. arXiv:2603.11830, 2026.
