This problem is arising in robotic motion control.
I am trying to find a function that describes $\theta(t)$ where all derivatives vanish at the boundaries 0 and $\tau$ as well as:
$$ \theta(0) = -\pi/2 $$ $$ \theta(\tau) = \pi/2 $$ In addition, the function $T(t)$ and all of its derivatives must vanish at the boundaries, where $$ T(t) = a\ddot{\theta} + b\cos\theta $$
I would think this is a well-studied problem, since $T(t)$ describes the dynamics of a rod rotating against gravity, but I can't come up with anything on my own or find anything in the literature.