  This paper deals with the computation of polytopic invariant sets for
polynomial dynamical systems. An invariant set of a dynamical system is a
subset of the state space such that if the state of the system belongs to the
set at a given instant, it will remain in the set forever in the future.
Polytopic invariants for polynomial systems can be verified by solving a set of
optimization problems involving multivariate polynomials on bounded polytopes.
Using the blossoming principle together with properties of multi-affine
functions on rectangles and Lagrangian duality, we show that certified lower
bounds of the optimal values of such optimization problems can be computed
effectively using linear programs. This allows us to propose a method based on
linear programming for verifying polytopic invariant sets of polynomial
dynamical systems. Additionally, using sensitivity analysis of linear programs,
one can iteratively compute a polytopic invariant set. Finally, we show using a
set of examples borrowed from biological applications, that our approach is
effective in practice.
