Disconnected optimal trajectories

Abstract

The Bolza problem of the calculus of variations in modern control notation is extended in scope to include situations in which a number of subarcs occur in a variety of ways. The subarcs are allowed to be overlapping and/or separated. This allows for several subarcs to occur in the same interval of the independent variable and also admits subarcs which are separated by jumps in the independent and state variables. In addition, the differential constraining equations and the integral quantity to be extremized are permitted to have different form from subarc to subarc. The necessary conditions for the extended Bolza problem are obtained by examining a related functional. Whereas the optimizing conditions for the state and control variables for each subarc are given by the usual Euler equations, new conditions associated with the end points of the subarcs are derived using ordinary theory of maxima and minima. The results presented here can be applied to a wide range of space trajectory problems. For some special cases, the theory reduces to results previously obtained and recorded elsewhere. A number of sample problems illustrating the theory are presented. The examples include the problem of inserting two payloads into separate orbits with one vehicle having two upper stages ignited simultaneously and a two-vehicle, dual-rendezvous problem.