Mathematical control theory is the area of application-oriented mathematics that deals with the basic principles underlying the analysis and design of control systems. The fundamental abstract mathematical notion underlying control theory is that of a dynamical system. There is a natural division of dynamical systems into two main types: continuous and discrete. Here we are interested in the former, for which the fundamental underlying mathematical subject is vector field theory. We aim to understand expansions of ``tame'' (that is, well behaved in some desired sense) structures on the real field by collections of images of solutions of definable vector fields. I shall present an introductory account of this program.