What is the mathematical foundation of Control Theory?
Linear Algebra Underlies Everything.
The power of the Laplace transform derives from the power of concepts like a linear operator and an eigenfunction. The exponential is the eigenfunction of the derrivative operator, which is the main operator in control theory. By projecting the system onto bases which are the eigenfunctions of the operators in your system, you simplify the problem by exposing the symmetries. This is what the Laplace transform does ($\int f(x) e^{-sx}dx$ is like an inner product between co-ordinates ($f(x)$) and the new bases you want to represent your function/vector in (exponentials). The result are new co-ordinates in the exponential space)
The complex exponential is the eigenfunction of the second derrivative operator. So projections into this space expose a different set of symmetries, in this case, the 'frequencies'. So the Fourier Transform is also just linear algebra.
I'd recommend a healthy dose of linear algebra, to satisfy all your inquisitive needs!
Control theory studies systems that have an internal state and can be connected with other systems via inputs and outputs. The dynamics are often but not always described by differential or difference equations.
Because of its engineering motivations, control theory is often interested in detailed analysis results and design techniques. Therefore it tends to spend more time on the easier cases: linear-time invariant systems. Linear algebra, and Laplace and Fourier transforms are extremely useful techniques, and about the most central ones.
Other relevant areas of math include differential geometry, dynamical systems, probability, algebra, analysis, automata, complexity of algorithms, you name it. Control theorists and engineers are always looking to apply other mathematical techniques to their problems, so no one of them is control theory.
As for references,
Feedback Systems: An Introduction for Scientists and Engineers Karl Johan Åström & Richard M. Murray http://press.princeton.edu/titles/8701.html
is a very good book whose Chapter 1 begins with sections "1.1 What Is Feedback? 1.2 What Is Control?" Do not expect precise mathematical definitions but do consider consigning the text that states "nobody knows what feedback is" to the recycling bin.
A more mathematical and more advanced but equally excellent text is Eduardo D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition, Springer, New York, 1998 http://www.math.rutgers.edu/~sontag/mct.html
Arbib and Manes once made an attempt to apply the Automata Theory to Control Systems, see:
Manes, E.G.(ed.) Category theory applied to computation and control. Proc. $1$st Internat. Symp., San Francisco, 1974. Lecture Notes in Comp. Sci., 25.
I am not an expert in the Control Systems and can not judge how successful this attempt is, but from the point of view of the Automata Theory, it looks pretty well.