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.