TITLE: What is the mathematical foundation of Control Theory?
QUESTION [41 upvotes]: There is a question which I'm wondering again and again in recent months. I have taken courses on elementary differential equations, signals and systems, linear control systems, general theory of circuits and networks but I still do not know how can a simple linear integral transformation (Laplace transform) be so much helpful in analyzing and designing linear systems. For example, I think I somewhat know what "time domain" is, but how about the "frequency domain"?
Consider a block diagram in the frequency domain. Conceptually, what do the blocks represent? What is traveling(?) on the connections between blocks? First, I have thought that instead of taking care of the statement "when does some phenomenon happens" we consider "how many times does a phenomenon oscillate", but how can this approach be more helpful? Is it the best (optimal) way of treating such problems?
I think control systems and even systems have a more general meaning in mathematics. I even think that the very problem of "controlling a system" does have multiple "paradigms" — A.I., neural networks, state space approach, fuzzy logic, robust control, adaptive control, etc.
What I'm looking for is a book (preferred) or paper discussing the very beginning notions of system, control system, its history, and more importantly, what is the field or language in mathematics to study systems rigorously? Dynamical systems? Game theory? Differential equations? Linear algebra? A.I?
Any help is appreciated. Thanks.

P.S.: I've already studied these books:

Alan Oppenheim's Signals and Systems (I think it is the best in the field)

Ogata's Modern Control Engineering

Desore's Basic Circuits Theory

Charles A - Kuh and Ernest S's A rather brief book on the history of control systems ($H_\infty$ methods and so on, but I didn't get much; at least, this book induced some information on me that nobody actually knows what "feedback" is, why is it so helpful and why we should take samples of the output! It wasn't a precise book, I admit)



26 May 2013:
As I said, I am looking for a book or article discussing the origins of Control Theory and Systems. Seven days are passed since I started the bounty, yet I didn't find an answer. I'll get to the bottom of this issue as soon as I get some free time. Thanks.

REPLY [3 votes]: To the asker's desire 

"for a book or article discussing the origins of Control Theory and
  Systems"

I can recommend the 2014 article Control: A Perspective by Astrom and Kumar.
Control Theory is a branch of applied mathematics, and as such doesn't really have a predefined "Mathematical Foundation". Techniques from complex variable theory, differential equations, linear algebra, optimization and calculus of variations, functional analysis, differential geometry, graph theory, Lie groups, etc. all make appearances in different places throughout the vast literature on systems and control. 
One of the answers mentions automata theory, which puzzled me, since I've rarely seen this used outside of applications that deal with information systems. It's certainly not central to control theory in the way a control theorist would think of it.