Suppose an object has initial velocity $v_0$, and it's going to reach a point at $x = L$ with final velocity $v_f$.

Assume it could take bounded acceleration at any given time. i.e.

$ a_{min} \le a(t) = x''(t) \le a_{max}$

What's the optimal plan for $a(t)$ (or $a(x)$) such that the arrival time $T$ is minimum?


Let $x(t)$ be the trajectory of this object with $x(0) = 0$ and $v(0) = v_0$.

Find $a(t)$ (or $a(x)$) such that $T$ is minimum, subject to $x(T) = L$, $v(T) = v_f$, and

$ a_{min} \le a(t) = x''(t) \le a_{max}, \forall t \in [0, T] $

I've surveyed a while but I could only found solutions on 2-dimensional motion like this, which are far more complicated than I need. I believe there's a much simpler solution such as solving an integral equation here. Could anyone point me out how to do this?


In the one-dimensional case the problem is "bang-bang", meaning that you almost always operate the control at the extremes, a=amax or a=amin, with a single switch between the two. Using this makes it fairly straightforward to compute the solution.

Here are some references for the two dimensional version:


(see the remark after lemma 1)

  • $\begingroup$ Thanks a lot for your explanation $\endgroup$ – hiiwave Nov 22 '17 at 0:01

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