On Rational State Space Realizations (Conference Version)

It is investigated when a polynomial inputoutput differential equation can be realized in rational, explicit state space form, i.e. so that all components of the right hand side are rational functions of the states. In the case where there are no inputs the problem is showed to be equivalent to a famous problem in algebraic geometry, which is solved only in various special cases. For systems with inputs the problem is more complicated, as is the discrete time case. An interpretation of the Luroth problem in terms of observability is made