Rigorous mathematical treatments of engineering topics

Control theory is quite rigorous, one of the more mathematical areas of engineering. Eduardo Sontag's "Mathematical Control Theory: Deterministic Finite Dimensional Systems" is a textbook written by a mathematician that covers a lot of the field. It is not too advanced or too specialized. I can't think of a better place to start. http://www.math.rutgers.edu/~sontag/mct.html

I'm not very knowledgeable in some of the fields that you mention, but here's a quick recommendation.

Mechanics: You may benefit from reading Apostol's calculus books since they provide the rigorous background for things like kinematics and ODEs.

Thermodynamics: Here, I would recommend Stein and Shakarchi's Fourier Analysis book since it is motivated by the initial examples of the wave and heat equations. The methods they introduce for the solution of PDEs have applications in statistical mechanics. If you're interested in a more abstract, less applied treatment, try Folland or Papa Rudin.

Control theory:....I'm not sure about this one. On a related note, if you're interested in signal processing or information theory, I recommend reading Shannon's original paper on the subject. Also, I can't recommend highly enough Hernandez's and Weiss's excellent book on wavelets to anyone interested in signal processing.

Dynamical systems: More applied: Strogatz. More abstract: Glendinning