How could the following problem be tackled?
Find the curve $v$ between two points $v(0)=p \neq q=v(1)$ with minimum velocity and acceleration (in the $L^\infty$ sense), such that $v'(0)=v'(1)=0$
I tried to suppose the solution to be on a straight line, so that $v$ (upon rotations and translations) can be supposed to be a real valued function, and to impose a symmetric behaviour between the endpoints and enough regularity.
I then discretized the problem with linear hat functions and tried to apply KKT conditions to a problem only on $v'$, but I got an absurd.
Any idea on how to tackle this? Thanks!
Is the problem even well-posed? :) My intuition: To minimize velocity in the $L^\infty$ sense you need to accelerate hard near the end points of the curve, this is thus going to cost you in terms of the sup norm of the acceleration. We thus need some method of comparison for which minimization is more important, e.g. if we want to minimize $$||v'||_\infty + a ||v''||_\infty$$ the result is very much going to depend on a, right?
Edit:
The solution is obviously a straight line (thus the dimension of the space is not important). Sketch of a proof: Consider projections $P$ of velocity and acceleration onto the straight line. A nonzero normal part $v - P v$ or $v' - P v'$ will only increase the sup norm.
We consider the problem in one dimension only. Let $f=v'$, then we need to minimize $||f||_\infty$ and $||f'||_\infty$ for $f$ with $$f(0)=f(1)=0,\\ \int_0^1 f \,\text{d}x = C$$ and $C=\text{const.}$ is the distance between the points. The optimal solution will be a linear increase with slope $f'=||f'||_\infty$ up to the maximum value $||f||_\infty$, $f=||f||_\infty$ over some interval and then a decrease with $f'=-||f'||_\infty$. For if you vary the function locally, you will have to increase it at some point and then you increase either $||f||_\infty$ or $||f'||_\infty$. The choice of the concrete values of both will depend on the cost function you left unspecified.
Also note: If you require $||f||_\infty\leq a$ and $||f'||_\infty \leq b$ with $a$ and $b$ being too small (e.g. $a < C$), there may exist no solution.