Safety verification of nonlinear systems based on rational invariants

where x ∈ R is the state variable, and f(x) is a vector of rational functions in x over Q. We consider the dynamics of (1) in a bounded domain of the state space R, given by Ψ , {x ∈ R|ψ1(x) ≥ 0 ∧ · · · ∧ ψr(x) ≥ 0}, with ψi(x) ∈ Q[x] for 1 ≤ i ≤ r. We say that the system (1) is safe if all trajectories of (1) starting from any state in the initial set, can not evolve to the unsafe states. We are interested in the problem of safety verification of nonlinear system (1), described as follows.