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Let $\mathbf{x}(t) : \mathbb{R}\rightarrow\mathbb{R}^{n}$ be a vector-valued function, and $\{\mathbf{x}_{0}, \mathbf{x}_{t}, \mathbf{v}_{0}, \mathbf{v}_{t}, c\}$ constants. With $\mathbf{x}(0) = \mathbf{x}_{0}$, $\mathbf{x}(T) = \mathbf{x}_{t}$, $\mathbf{x}'(0) = \mathbf{v}_{0}$, $\mathbf{x}'(T) = \mathbf{v}_{t}$, $|\mathbf{x}''(t)| = c$, find $\mathbf{x}(t)$ that minimizes $T$.

I’ve been looking into variational calculus and optimal control theory, but all the methods I’ve seen are for optimizing an objective function over a fixed interval, as opposed to optimizing the interval itself.

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    $\begingroup$ How do we know there is any solution $\mathbf x(t)$ at all that satisfied the various conditions? $\endgroup$ Nov 2, 2023 at 22:47
  • $\begingroup$ It's relatively straightforward to see that a solution exists for any c > 0. You could always construct x to accelerate arbitrarily far away, then cancel any velocity, then do a bang-bang approach to a point x1 (also at 0 velocity) such that you can go from x1 to xt at direct acceleration and satisfy the final constraints. Clearly not optimal, but clearly exists. $\endgroup$
    – D0SBoots
    Nov 2, 2023 at 22:50

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