In practice, they result in boundary-value problems that are impractical to solve for systems of non-trivial size, and are almost never implemented in practice. Occasionally they are used to construct parameterized solutions for extremum control (e.g. NCO tracking) for very small systems, but these tend to be rarer cases. One runs into dimensionality issues very quickly.
In industrial control systems, optimal control models are almost always discretized and the optimization is done on algebraic systems of equations. Linear algebra dominates there.
When you say 'mathematical analysis' here, is that just a broader category that encompasses e.g. real and complex analysis? Or something else?
Once the models are transformed into discrete form for numerical solution, the tools used lie more in the realm of linear algebra (positive definiteness of Hessians, etc.)
Please tell? I have yet to see anyone give a satisfactory approach on how to deal with the abnormal case.
For context there are sometimes optimal solutions which are not given by Pontryagin's Maximum Principle (PMP). An analogous situation can occur with Lagrange multipliers. The necessary conditions given by the Lagrange multipliers are not related to the maximization of the object functional. I surely think the situation is worse with the PMP because you are now in a continuous setting. I think  offers some good discussion for the abnormal case in Lagrange multipliers.
I would be interested if anyone has made any recent progress in dealing with the abnormal case for the PMP.
 Optimality Conditions: Abnormal and Degenerate Problems
By A.V. Arutyunov
His slides, references and FB livestreamed video, are here:
Both fields are attempting to solve the same problem: choose the optimal action to take at the current time for a given process. Control theorists normally start out with a model, or a family of potential models that describe the behavior of the process and work from there to determine the optimal action. This is very much an area of applied mathematics and academics take rigorous approaches, but, in industry, many engineers just use a PID or LQR controller and call it a day regardless how applicable they are to the actual system theoretically.
Meanwhile, the reinforcement learning folk typically work on problems where the models are too complicated to work with computationally or often even write down, so a more tractable approach is to learn a model and control policy from data. There's plenty of people who analyze properties of learning algorithms, etc., within this framework, and others who don't really care beyond whether or not the system works.
This is the main distinction I've been exposed to, between Optimal Control and Reinforcement Learning. I've heard it summarized as "Optimal Control uses models, Reinforcement Learning tries very hard to stay away from using models". That's probably simplifying things a little bit too much, but it seems like a reasonable starting point to see where the two fields diverge.
In contrast, dynamic programming is based on stitching together optimal sub-solutions.