How to prove the optimal solution exists for a nonlinear optimal control problem?

Question

I am doing a nonlinear optimal control design project, typically a practical engineering problem.

It is easy to solve that with direct method. However, I am wondering is there any way to prove that the optimal solution exists ? Further, since the direct method is employed, is there anyway to prove that the numerical solution will converge to that optimal solution?

Could anyone offer me some reference paper about that ?

thanks

Jie