Are control theorists considered mathematicians?
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?