Calculus of variations is, yes, a part of optimization, now regarded as an old part. But subsequent to Bellman's book was the Pontryagin maximum principle. There is an entry at Wikipedia:

https://en.wikipedia.org/wiki/Pontryagin%27s_maximum_princip...

There is some treatment in

David G. Luenberger,
Optimization by Vector Space Methods.

which I call fun and profit from the Hahn-Banach theorem.

There may be more in

L. S. Pontryagin,
V. G. Boltyanskii,
R. V. Gamkrelidze,
and E. F. Mischenko,
The Mathematical Theory of Optimal Processes.

and in

E. B. Lee and
L. Markus,
Foundations of Optimal Control Theory.

and in

Michael Athans and
Peter L. Falb,
Optimal Control:
An Introduction to the Theory and Its Applications.

At one time I was interested in such math as a way to say how best to climb, cruise, and descend an airplane. So, I chatted with Athans in his MIT office and got a copy of his class notes. And I got accepted to grad school at the Brown University Division of Applied Mathematics where Falb was. Then I applied a little skeptical judgment and didn't go back to Athans and went elsewhere for my Ph.D.

I also visited Brown and Cornell. At Cornell I met with Nemhauser, and he gave me three words of advice that was the real direction for my Ph.D. dissertation. Flying back on the airplane, I figured out what Nemhauser told me.

When I got back, I called a meeting to outline what I'd discovered from Nemhauser, Athans, etc. My manager ordered me not to hold the meeting, but I did anyway. All the C-level people and the founder, COB, CEO showed up. I got a promotion and big office next to the founder. What I presented was the start of two Ph.D. dissertations.

The Fleming reference may be enough. Uh, while I visited Brown, I had lunch with Fleming -- a very bright mathematician.

A good source for prerequisites is the first (real) half of the W. Rudin, Real and Complex Analysis.

Broadly that optimization -- deterministic optimal control -- works with curves where each curve is regarded as one point, vector, in some vector space. The space is short on good assumptions so may be only a Banach space. Actually can still do a lot in Banach space, e.g., Luenberger's book -- one of my favorites. Gee, there can also get a really short treatment of Kalman filtering!

The fraction of mathematicians who know all that math well is tiny. The people who make good money from applications has to be smaller -- I would believe so small there are none. From all I can see, good applications are so rare there almost aren't any.

With the rapidly growing power of computing and the rapidly growing complexity of computer applications,
discrete time stochastic optimal control (that can attack problems that first cut intuitively seem hopelessly difficult yet be comparatively simple mathematically) has a chance of valuable applications, so far not a very good chance, but a chance, probability small but still greater than zero.

That spreadsheet connection I outlined may be a seed for some applications.

Microbiology research gets some motivation from applications, saving the lives of people dying of cancer, heart disease, new viruses, etc. Maybe research in mechanical engineering works on how to put up buildings and bridges that won't fall down. Some years ago I concluded that math is being throttled by too much isolation from important, motivating applications. So, if my startup works, that will be n = 1 cases of evidence that math can do well, at least make money, with some new applications outside of math.