  A new framework for formulating reachability problems with competing inputs,
nonlinear dynamics and state constraints as optimal control problems is
developed. Such reach-avoid problems arise in, among others, the study of
safety problems in hybrid systems. 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 a general class of target hitting continuous dynamic
games with 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 a two aircraft collision avoidance scenario under target window
constraints and in the presence of wind disturbance. Target Windows are a novel
concept in air traffic management, and represent spatial and temporal
constraints, that the aircraft have to respect to meet their schedule.
