An intro?

Nemhauser and Dreyfus and Law are
intended as intros and are quite well written. Bellman, who likely should be named the father of dynamic programming, is the oldest book on my list and is not intended as an intro. But if can get Bellman for less than $10, maybe for the price of a Big Mac, then eat a lighter lunch and get the book.

Bellman's book likely covers the Hamilton–Jacobi–Bellman equation,
not so easy to understand via the other intros.

Not really a biggie but worth knowing, a big part of this subject in computation is the value function, and at least Bertsekas used neural nets as a means to have an efficient approximation. So more recent treatments may include this idea.

Dynamic programing is one of the leading examples of "the curse of dimensionality".

The field has not been still; e.g., multivariate splines have been an idea for handling the value function in computation.

Of course can have software go through the motions of stochastic dynamic programming when the Markov assumption is not satisfied; the software won't object. But when the assumption does old, and the rest of the effort has good quality, can argue that the work provides an optimal solution, the best possible, of course, in expected value, for any means of handling the data. In short, we have what is called "the principal of optimality".

The continuous time case, with all the math details filled in, is more difficult; so for an intro, stay with discrete time.

There has been a lot of attention to this subject in the ORFE (Operations Research and Financial Engineering) department at Princeton. Long the chair was E. Çinlar. The best course I took in school was from a Çinlar student.

Of course, if expect artificial intelligence to be really smart, then as a special case it should be able to be as good as stochastic optimal control. And apparently now some of reinforcement learning is using the core idea of stochastic dynamic programming. But without the Markov assumption, might count that application as a heuristic with no more than weak claims about optimality.

I spent a lot of time in the field. Eventually I concluded that generally in the US economy a lot of knowledge of stochastic optimal control and a dime wouldn't cover a 10 cent cup of coffee. Would have a lot better chance buying a house and supporting a family getting paid to develop Web sites, e.g., to sell, say, used math books, including Bellman's. Maybe there have been and are some niche applications (maybe somewhere in US national security), but that niche is likely much sharper than any razor.

There is something of an organizational problem for a math guy getting hired to apply stochastic optimal control: The person hiring you, your hiring manager, will likely know much less about the math than the math guy, likely know nothing about the math. So this manager will be very reluctant to allocate much of his (her) budget and risk his job to support work he doesn't understand and, that, indeed, might justify the C-level suits promoting the math guy over his manager.

Due to such issues, the math guy with an application potentially valuable in the economy could be better off doing a corresponding startup. Else, to avoid scaring hiring managers, he might be better off omitting such math from his resume. So, for getting hired as an employee, a lot of math background on a resume can be from useless down to, say, a felony conviction.

A recipe for rabbit stew starts out "First catch a rabbit.". A recipe for applied math might start, "First find an application." Can't argue that some topic in math will be useless forever outside of math and in the economy, but forever is a long time. As it is, it can appear that some math papers and books, including what looks like applied math, are written before seeing any rabbits.

Yet, the best of pure/applied math is in some senses super terrific stuff, way up there with the best of Bach and Beethoven, etc., and at times there are good applications. E.g., there is some pure and applied math at the core of my startup; the pure math provides some special support, a version of optimality, for the whole effort. Just what that pure math says is terrific, gives a solid guarantee where intuitively we have only confusion, is so good it's tough to believe, but still it's true. The applied math is original with me, powerful for my startup but no biggie as research.

Good luck.