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I have asked this question in the math SE but no one there knew an answer; there are similar questions here but they differ a bit from what I am trying to understand.

I was wondering if there is an (analytical) solution to the problem (in 2D) where the position $\vec{r}_i = \vec{r}(t=t_i)$, $\vec{r}_f = \vec{r}(t=t_f)$ and the velocities $\vec{v}_i = \vec{v}(t=t_i)$, $\vec{v}_f = \vec{v}(t=t_f)$ as well as the constant magnitude of acceleration $|a|$ are given and the goal is to find the vector $\vec{a}(t)/|a| = \vec{e}_a(t)$ that minimizes the time need to transition from the initial state to the final state assuming there are no other forces present.

There is this paper I found that explains the solution based on it's geometric properties. On page 5 there is a graph that shows that the acceleration is a semicircle rotated and offset from the center with the radius $|a|$ but I don't understand exactly how to compute the actual values for a given initial and final state.

Source is the paper mentioned above.

I know about Lagrangian multipliers and after reading the Wikipedia page I understand the shooting method but I don't know much about optimal control boundary value problems in general so and extended answer would be appreciated. Thanks in advance for your answers!

Ganymed_
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  • If you like this question you may also enjoy reading this Phys.SE post. – Qmechanic Nov 11 '16 at 15:33
  • I do not have the proof that it is optimal, but I suspect it is, but I asked a similar question assuming the direction and magnitude of acceleration would remain constant. This results into solving a quartic equation (forth order polynomial). – fibonatic Nov 11 '16 at 15:57
  • That seems like an interesting post but as far as I understand it the goal seems to be to find the acceleration direction to just pass through the origin, I am interested in matching the final velocity as well! – Ganymed_ Nov 11 '16 at 16:06
  • Are you asking for an analytical solution (ie a formula)? Or a numerical method of solution? – sammy gerbil Nov 11 '16 at 20:13
  • A numerical solution is fine too. In a way a numerical solution is described in the paper I linked, I just don't understand how to actually solve it. – Ganymed_ Nov 11 '16 at 20:26

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