Mathematics

Regularity properties of subanalytic sets are often used in control theory to establish regularity of various characteristics of real analytic control systems. In some cases, the regularity of the corresponding characteristics of generic smooth control systems, while being of interest, is not known. We suggest an approach to these questions based on older observations due to Yu. Ilyashenko and S. Yakovenko, and S. Yakovenko and the speaker (obtained in a different context). Specifically, we intend to sketch a proof of a partial result in this direction. It is convenient to formulate this result as the logic-theoretic assertion that the extension of the first order theory of the real numbers by (a finite number of) generic smooth functions, restricted to the unit cube, is o-minimal. We will attempt to give a clear and concise explanation of the concepts involved, stressing the usefulness of the logic-theoretic point of view in the present context. In particular, no background in mathematical logic is assumed.