# What's the minimum time plan given bounded accelration and velocity at endpoints? (on one dimensional motion)

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?

## Formulation

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?