Hello everyone!
I have a system of equations, the solution of which defines a trajectory on a plane:
dx/dt = Vx(x,y,t,p),
dy/dt = Vy(x,y,t,p),
where p is some unknown parameter. I know the starting point of the trajectory, i.e. (x[0], y[0]) = (xA, yA). My task is to find the value of the parameter p at which the trajectory will arrive at the point (xB, yB) in the shortest time. Also, I need to find the corresponding trajectory and travel time T.
I’m not sure what type this problem is and whether it can be solved with DifferentialEquations.jl? Note, that one can transform the variable t' = t/T and then the timespan of the problem becomes fixed [0,1], but we have an additional unknown parameter T in the transformed system of equations. So, how can we find the values of the parameters (p, T) of a system of equations if the solution is known at two fixed points?