In Section 6, we formulated the problems of minimizing the characteristic velocity for the outgoing trip (Problem P1) and return trip (Problem P2). In these problems, the vectors a,b,c appearing in Eqs. (25) and (28) have dimensions 7, 2, 5 respectively. The vector a includes the major parameters governing the transfer; the vector b includes the components of a that are fixed, namely, the radii of the terminal orbits; the vector c includes the components of a that must be optimized, namely, the terminal velocity impulses, transfer time, spacecraft/planet relative phase angle at departure, and Mars/Earth inertial phase angle difference at departure.
In this section, we modify the previous problems by assuming that the departure date ? is fixed, hence by assuming that ? ?(0) is given. In the new problems, the vectors a, b, c have dimensions 7, 3, 4 respectively as can be seen by transferring ? ?(0) from (26c) to (26b) for the outgoing trip and from (28c) to (28b) for the return trip. Thus, one can formulate the following new problems:
Problem P3. For the outgoing trip, given the triplet ( r LEO, r LMO, ? ?(0)), minimize the performance index (25) w.r.t the parameters ?V LEO, ?V LMO, ?, ? PE(0), subject to the constraints (14).
Problem P4. For the return trip, given the triplet ( r
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