Are control theorists (electrical engineering) considered mathematicians?
- If so, would they receive a Fields medal if they make a breakthrough in control theory?
- If not, is there a prize similar to the Field medal for control theorists?
I don't have references on the demographics but, traditionally, many and probably most control theorists are engineers by formation and it's in the field of engineering (electrical, mechanical, aerospace, etc.) that most of control theory applications reside.
However, control theory is really a broad interdisciplinary subject comprising engineering, mathematics and, to a lesser extent, physics. For mathematics, topics like algebra, graph theory, dynamical systems theory, optimization theory, and functional analysis are fundamental for the development of control theory, and mathematicians have provided outstanding contributions, as Dan Romik well explained.
My interpretation of your question, however, is whether a control theorists who is engineer by formation is considered a mathematician. I don't like barriers between disciplines, and exceptions always exist, but in my experience engineers are seldom considered pure mathematicians, and their contributions to mathematics are usually far from the mathematics of interest to win a Fields medal.
Thus, there's no Fields medal for control theorists, at least in your probably intended sense, but IEEE awards prizes (e.g., the IEEE Control Systems Award) for outstanding contributions in the field.
But you don't choose a field for the prizes, right?
While it seems true (based on my own quite sketchy observations) that many if not most control theory researchers may be employed in engineering departments rather than mathematics departments, I find the suggestion that control theory is not a branch of mathematics, or that control theorists are not mathematicians, both laughable and offensive (although I'll concede that probably some control theory experts have more of an engineering background and care less about proving theorems, and may not care to define themselves as mathematicians). Some important points to keep in mind are:
Control theory has wide applications to many areas of engineering (mechanical, aeronautical, chemical etc), not just electrical engineering, so certainly OP's description "control theory (electrical engineering)" is inaccurate. In fact the field predates electrical engineering: the first detailed study of a control theory system was published by James Clerk Maxwell (of Maxwell equations fame) in connection with a purely mechanical control system used in steam engines.
One of the other godfathers of control theory is Lev Pontryagin, known for the so-called Pontryagin's maximum principle, a fundamental result in the theory. Pontryagin (who incidentally was blind for most of his life) is regarded as one of the great pure mathematicians of the twentieth century. I am sure he would have laughed at the suggestion that his work on control theory was of a different nature than his work on algebraic topology or harmonic analysis because it was "applied" or "not mathematics but engineering".
I know of at least one notable contemporary counterexample to the claim that control theorists are not "considered mathematicians". Arthur Krener, a leading expert in control theory, is a former professor and chair of my department -- a mathematics department -- and now a professor of mathematics at the Naval Postgraduate School. He is a Fellow of the American Mathematical Society. He has won many awards for his work in control theory, most recently the IEEE Control Systems Award in 2016. I would be offended on his behalf at any suggestion that he is not a first-rate mathematician.