Optimal reachability sets using Generalized Independent Parameters

The problem of free-time optimal reachability set computation with alternate integral constraints is motivated and examined. Specific examples of such systems are optimal spacecraft, aircraft and automobile free-time, fuel-limited range computation. An alternate Hamilton Jacobi Bellman PDE formulation is derived using a Generalized Independent Parameter (GIP) associated with the integration constraint and GIP mapping function with respect to time is defined. Necessary conditions on the GIP mapping function are identified and discussed. Singular independent parameter mapping functions, often found in astrodynamics optimal control problems, are shown to be challenging to solve using a simple change of integration variable, motivating an approach to transform such problems before solving. Several short illustrations are used to emphasize theoretical cases of interest, and two simple fully-worked examples are given to demonstrate the potential utility of this approach.