Let $T>0$, let some function $\kappa(t)$ smooth on $[0,T]$, and let $b$ the unique solution to the ODE $\ddot b + \kappa(t) b = 0$ with initial conditions $b(0)=0$ and $\dot b(0) = 1$.
I was wondering if there exist necessary and sufficient conditions on $\kappa(t)$ so that $b(T)=0$ (or, in general, controlling the next zero of $b$). The only statements I find in this direction never take into account the fact that negative values of $\kappa$ may counteract the positive ones, thereby making the next zero of $b$ happen later.
