TAB 25

OPTIMAL CONTROLS

No matter what coordinate system you use or how the equations of motion are integrated, there is always a jump discontinuity at the sphere of influence (SOI). The only way to get a reasonable solution is to apply a change of coordinates there and to use statistical estimation methods to drive the system to approximate the solutions on either side of SOI - or to assign the discontinuity a penalty function and to drive the penalty function integral to zero.

The problem is that it is not possible to drive this value to zero. Hence there is a new, unknown force at SOI. This implies the existence of a new integral of motion, which can be found by setting up the trajectory optimization through SOI as a convex or concave problem, which restricts the problem without loss of generality and eliminates a variable. This is an unlikely and unexpected result, brought about because the assumption in basic calculations of variations theory is OK for simple trajectories but not for interplanetary trajectories because they traverse a fractal change of level. This can be perceived by characterizing the extremal path of a low eccentricity ellipse as the variation from the unit circle.

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© 2004 WH Clark