Realization theory of Nash systems

This paper deals with realization theory of so-called Nash systems, i.e. nonlinear systems the right-hand side of which is defined by Nash functions. A Nash function is a semi-algebraic analytic function. The class of Nash systems is an extension of the class of polynomial and rational systems and it is a subclass of analytic nonlinear systems. Nash systems occur in many applications, including systems biology. We formulate the realization problem for Nash systems and present a partial solution to it. More precisely, we provide necessary and sufficient conditions for realizability of a response map by a Nash system. The concepts of semi-algebraic observability and reachability are formulated and their relationship with minimality is explained. In addition to their importance for systems theory, the obtained results are expected to contribute to system identification and model reduction of Nash systems.