What is the list of variational techniques for proof of the the Pontryagin's maximum principle that include the needle variation technique?

Question

I have noticed that the rigorous proof of the Pontryagin's maximum principle (in optimal control) is usually based on a technique denoted as the "needle variation". This method is different from the simple technique of making variations in both state and input which is applied in common text-books.

Now, I would like to know the name of the simple method of the text books (e.g. the book by D.E. Kirk) as opposed to the "needle variation" technique (and names of other techniques if there are any).

It would be great if someone knows that there exist a comparative discussion between the techniques in some book/paper/etc.

Thank you

Answer 1

Try Liberzon's book on Optimal Control which presents a detailed comparison between the classical approach based on calculus of variations and the modern approach using needle variations.

See also Sussman's article on the topic, and Andrew Lewis' notes.