Control systems on regular time scales and their differential rings

The paper describes an algebraic construction of the inversive differential ring, associated with a nonlinear control system, defined on a nonhomogeneous but regular time scale. The ring of meromorphic functions in system variables is constructed under the assumption that the system is submersive, and equipped with three operators (delta- and nabla-derivatives, and the forward shift operator) whose properties are studied. The formalism developed unifies the existing theories for continuous- and discrete-time nonlinear systems, and accommodates also the case of non-uniformly sampled systems. Compared with the homogeneous case the main difficulties are noncommutativity of delta (nabla) derivative and shift operators and the fact that the additional time variable t appears in the definition of the differential ring. The latter yields that the new variables of the inversive closure, depending on t , have to be chosen to be smooth at each dense point t of the time scale.