Algorithms for exact and approximate linear abstractions of polynomial continuous systems

A polynomial continuous system S = (F,X0) is specified by a polynomial vector field F and a set of initial conditions X0. We study polynomial changes of bases that transform S into a linear system, called linear abstractions. We first give a complete algorithm to find all such abstractions that fit a user-specified template. This requires taking into account the algebraic structure of the set X0, which we do by working modulo an appropriate invariant ideal. Next, we give necessary and sufficient syntactic conditions under which a full linear abstraction exists, that is one capable of representing the behaviour of the individual variables in the original system. We then propose an approximate linearization and dimension-reduction technique, that is amenable to be implemented "on the fly". We finally illustrate the encouraging results of a preliminary experimentation with the linear abstraction algorithm, conducted on challenging systems drawn from the literature.