Let us suppose we have a very simple particle movement simulation:
$\dot{x}(t) = f(t,F(t))$
where F is applied force. Integrating over t and F we can get the final position of the object.
Now, at some point t' I need the object to be located at x' and move with the speed $\dot{x'}$. I need to find such F(t) that the condition meets. Moreover, I may need to minimize either time or amount of energy spent.
What will be the definition of the problem? What methods can I use to solve it? What branches of mathematics/engineering deal with such problems?
Optimal control or more generally calculus of variation
https://en.wikipedia.org/wiki/Optimal_control