Extending taylor approximation to hybrid automata with integrals

In [10, 11] we proposed a technique to approximate Hybrid Automata (HA) with Polynomial HA. The idea was to replace functions appearing in formulae with their Taylor Polynomials. Here we extend this technique to HA with formulae admitting integral functions. We prove that we get over-approximations of the original HA. We study the conditions ensuring that: 1. The "distance" between the formulae of the original HA and its approximation get close to 0 when increasing the degree of the Taylor polynomial (syntactical approximation), 2. The "distance" between the configurations reached in n steps by the two HA get close to 0 when increasing the degree of the Taylor polynomial (semantic approximation).