This is an example of an emergency landing of a two stage space transport vehicle. After the separation of the first stage the engine of the upper stage cannot be re-ignited. Due to this propulsion system damage the upper stage cannot reach a safe orbit.
The optimal control problem is fully discretized using Euler's method and implemented in AMPL; this leads to a sparse large-scale nonlinear optimization problem. The problem dimensions are determined by the number of points n used for the discretization, specifically it has N = 8n+1 optimization variables subject to M = 6(n-1)+4n+7 constraints.
The picture on the right shows an example of an emergency trajectory. The reentry point is situated over the city of Bremen, Germany and the objective function is to maximize the flight distance without propulsion from this reentry point.
The following table summarizes the results for different number of discrete points. The tests were run on an Intel Core2 Quad CPU Q6600 @ 2.40GHz with tolerances for constraint violations of 10-6 and for KKT conditions of 10-5.