Taylor approximation for hybrid systems

We propose a new approximation technique for Hybrid Automata. Given any Hybrid Automaton H, we call Approx(H,k) the Polynomial Hybrid Automaton obtained by approximating each formula φ in H with the formulae φk obtained by replacing the functions in φ with their Taylor polynomial of degree k. We prove that Approx(H,k) is an over–approximation of H. We study the conditions ensuring that, given any e > 0, some k0 exists such that, for all k>k0, the “distance” between any vector satisfying φk and at least one vector satisfying φ is less than e. We study also conditions ensuring that, given any e > 0, some k0 exists such that, for all k > k0, the “distance” between any configuration reached by Approx(H,k) in n steps and at least one configuration reached by H in n steps is less than e.