Is abstract formulation any use? If the PDE has a analytic solution, then this solution must can be obtained with classical method. If the PDE does not have a analytic solution, even you use semigroup, you cannot get a analytic solution. In the end, you have to resort to numerical methos. And in the papers, they are always trying to prove something rather than solve something. I'm wondering whether they can be applied to practice.
Your post is a bit too abstract for me to make out, so I'm going to free-associate a bit, and hopefully chance on an answer that is useful to you.
There can be great value in a mathematical test that gives you part of an answer from part of the problem.
For example, if you can apply a test to some formulation that says that there definitely is a solution, or there definitely isn't, then you have value.
Ditto, there's value in a test that puts upper bounds on the best control you'll ever achieve with an approach, or puts lower bounds on the worst control that you will ever achieve with some specific controller. In the former case, it can tell you whether an approach has any chance of ever working, in the latter case it can tell you if your specific controller has reached the threshold of 'good enough', even if you don't know how much better than good enough it is.
Even just counting control loop designs, and not actual loops, I suspect that 99% of the loop designs out there are (or can be) handled quite well with either 1950's 'modern' control design and analysis or with just plain seat-of-the-pants engineering. Others will shudder when I say this, but for 90% of those loops just tossing a PID algorithm in its general direction and 'twiddling the knobs' will probably result in adequate performance, although it'll never result in certainty that things will always work right.
(Counting actual loops makes the figure even higher -- consider that most houses in the civilized world have flush toilets and home heating systems. Then consider that a home heating system can get by just fine with a bimetallic strip and a tilt switch, and that a grotty old flush toilet has not one, but at least three redundant loops to control the level of water in the tank and bowl. And while many control theorists interact with flush toilets and thermostats, none are necessary to the successful design and implementation of one).
It's the remaining 1% of loops that attract all the attention. Frankly, I think that control theory has wandered a bit far off the beaten path, although it seems to be coming back. There are rich troves of pretty math that one can spend a career on, a steadily diminishing supply of high-level control practitioners (as opposed to theorists), and in the US at least, a seemingly unrealistic attitude on the part of both educators and the populace on the value of teaching (or learning) this stuff.
It's good for me as a control practitioner, but I don't think it's good for my country.
Its role has certainly changed, but that's no different to just about every field. Copperplate writing and calculating square roots using the long division method are no longer the widely practised arts that they used to be. For me, digital simulation has largely replaced rigorous solutions, that's mostly due to the sort of systems that I work with - often ill defined and very driven by human factors. I can develop a computer model a lot quicker and more easily than I can solve anything more than a trivial de, or other mathematical problem.
OTOH, studying formal methods does improve your understanding of how things work. I'd always advocate it being part of preparation for a control career.