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
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
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
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.
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