Generating invariants for non-linear hybrid systems

We describe powerful computational techniques, relying on linear algebraic methods,for generating ideals of non-linear invariants of algebraic hybrid systems. We show thatthe preconditions for discrete transitions and the Lie-derivatives for continuous evolu-tion can be viewed as morphisms, and so can be suitably represented by matrices. Wereduce the non-trivial invariant generation problem to the computation of the associatedeigenspaces by encoding the new consecution requirements as specific morphisms rep-resented by such matrices. More specifically, our methods are the first to establish verygeneral sufficient conditions that show the existence and allow the computation of in-variant ideals. Our methods also embody a strategy to estimate certain degree bounds,leading to the discovery of rich classes of inductive, i.e. provable, invariants. By reduc-ing the problem to related linear algebraic manipulations we are able to address variousdeficiencies of other state-of-the-art invariant generation methods, including the efficienttreatment of non-linear hybrid systems. Our approach avoids first-order quantifier elim-ination, Gröbner basis computation or direct system resolution, thereby circumventingdifficulties met by other recent techniques.