Hamilton-Jacobi formulation for reach-avoid problems with an application to air traffic management

A new framework for formulating reachability problems with nonlinear dynamics and state constraints as optimal control problems is developed. The work in this paper is motivated by such problems in the area of air traffic management, in particular the problem of collision avoidance in the presence of 4D constraints, called Target Windows, that the aircraft have to respect to meet their schedule. Earlier approaches to reach-avoid computations are either restricted to linear systems, or face numerical difficulties due to possible discontinuities in the Hamiltonian of the optimal control problem. The main advantage of the approach proposed in this paper is that it can be applied to nonlinear dynamics and has very good properties in terms of its numerical solution, since the value function and the Hamiltonian of the system are both continuous. The performance of the proposed method is demonstrated by applying it to conflict detection and resolution under Target Window constraints in a two aircraft scenario.