Is there a theorem that garuantees that a variational problem $I[y] = \int_a^bF(x,y,y')dx$ has local/ global maxima and minima?
Perhaps similar to the extreme value theorem for continuous functions on compact sets?!
Specifically once I obtain a solution to the Euler- Lagrange equations how can I show that such a solution is an extemum/ minimum / maximum?
Are there similar theorems for constrained problems? Many thanks!
If $F$ is a continuous function and your constraint set is compact, the Weierstrass theorem still applies. The problem is that compactness in function spaces is much more complicated than in finite-dimensional vector spaces.
What it sounds like you really want is second-order sufficient conditions: "Specifically once I obtain a solution to the Euler- Lagrange equations how can I show that such a solution is an extemum/ minimum / maximum?"
The corresponding versions of the SOSC's for optimal control come in two flavors. Arrow's sufficient conditions are on the fundamentals of the problem, while Mangasarian's are on the optimized Hamiltonian. This is a pretty good reference, if you are at a university:
https://www.jstor.org/stable/2525753?seq=1
Otherwise, googling around for "Arrow-Mangasarian Sufficient Conditions, Optimal Control" will turn up thousands of references.