# How do you minimize t instead of an objective function over t?

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.

• How do we know there is any solution $\mathbf x(t)$ at all that satisfied the various conditions? Nov 2, 2023 at 22:47
• 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. Nov 2, 2023 at 22:50