Solving the maximum-crossrange problem via successive second-order cone programming with a line search

The maximum-crossrange problem is an optimal control problem of computing the maximum crossrange reachable by a hypersonic entry vehicle at a specified downrange, which has long known to be very difficult to solve due to its high nonlinearities and non-convexity. This paper presents how to convexify the problem so that it can be efficiently solved by successive second-order cone programming (SOCP). Particular focus is given on equivalent transformation of the original optimization objective and rigorous establishment of validity of the relaxation process used for convexification. In addition, it is observed that iteratively solving the SOCP problems may not always guarantee convergence to the original problem, a simple line search approach is proposed which is found critical to ensure the convergence of the successive SOCP method. Numerical demonstrations are provided to illustrate the effectiveness and efficiency of the proposed method and its applicability to both orbital and suborbital missions.