Computing Differential Invariants as Fixed Points

We introduce a fixed-point algorithm for verifying safety properties of hybrid systems with differential equations whose right-hand sides are polynomials in the state variables. In order to verify nontrivial systems without solving their differential equations and without numerical errors, we use differential induction as a continuous generalisation of induction, for which our algorithm computes the required differential invariants. As a means for combining local differential invariants into global system invariants in a sound way, our fixed-point algorithm works with differential dynamic logic as a compositional verification logic for hybrid systems. To improve the verification power, we further introduce a saturation procedure that refines the system dynamics successively by differential cuts with differential invariants until the property becomes provable. By complementing our symbolic verification algorithm with a robust version of numerical falsification, we obtain a fast and sound verification procedure. We verify roundabout manoeuvres in air traffic control and collision avoidance in train control.