Polytopic Invariant Verification and Synthesis for Polynomial Dynamical Systems via Linear Programming

This paper deals with the verification and the synthesis 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 can be verified by solving a set of optimization problems involving multivariate polynomials on bounded polytopes. Using the blossoming principle for polynomials together with properties of multi-affine functions 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, we show that using sensitivity analysis of linear programs, we can synthesize a polytopic invariant set. Finally, we show using a set of examples borrowed from engineering or biological applications, that our approach is effective in practice.