Algebraic reachability of rational systems

The concept of algebraic reachability refers to the property of a system that every state from a sufficiently big subset of a state-space can be reached from a given inital state by applying an admissible input to the system. In this paper we study algebraic reachability of rational systems. We provide necessary and sufficient conditions for a rational system to be algebraically reachable (from an initial state). These conditions are formulated in terms of ideals of polynomials which makes checking algebraic reachability of a system computationally feasible. We relate the notion of algebraic reachability to the notion of controllability of linear systems, (local and local strong) accessibility and controllability of nonlinear systems.