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On a plane, given initial position (x1,y1), initial velocity (u1,v1), final position (x2,y2), and final velocity (u2,v2), compute the solution to x''= cos(z), y''=sin(z) that has these endpoint conditions and minimizes time.

The control function z can be taken as piecewise continuous.

Pontryagin's optimality criteria implies the fractional linear tangent law: z must be of the form z=arctan((at+b)/(ct+d)).

Starting from this, how do you actually (numerically) compute the optimal path (x(t),y(t))?

This stems from a research paper “Baserunner’s Optimal Path,” published online in The Mathematical Intelligencer, November, 2009, by Frank Morgan and myself.

We did find one old reference: Acceleration-Constrained Time-Optimal Control in N-dimensions, by Feng and Krogh, IEEE Transactions on automatic control, Vol 31, Issue 10, Pages 955-958, Published Oct 1986.

Anyone know any newer work or numerical solutions?

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