Optimal trajectory for the terminal bunt problem: An analysis by the indirect method

This paper presents an analysis of the optimal trajectory of a generic cruise missile attacking a fixed target. The timeintegrated flight altitude must be minimised and the target struck from above, subject to missile dynamics and path constraints. The generic shape of the optimal trajectory is: level flight, climbing, dive; this combination of the three flight phases is called the bunt manoeuvre. The starting point of the analysis is an existing, approximate solution of the bunt problem obtained via a direct method (using SQP). This initial trajectory enables discerning the structure of the optimal solution which is composed of several arcs, each of which can be identified by the corresponding manoeuvre executed and constraints active. This qualitative analysis is the first main contribution of the paper. It also simplifies considerably the analytical task of deriving the mathematical description of each arc (based on Pontryagin's Minimum Principle) which is the second main contribution. The thus obtained arcs have to be connected into the full trajectory by establishing their duration times. This requires formulating and solving appropriate boundary value problems. The advantage of using the indirect method in this context is twofold: (i) the approximations of the existing initial solution are corrected, (ii) additional insights into the solution structure are obtained, especially the role of constraint activation.