Computation of Polytopic Invariants for Polynomial Dynamical Systems using Linear Programming

TL;DR

This paper introduces a linear programming-based method for computing and verifying polytopic invariant sets in polynomial dynamical systems, enabling effective analysis of system invariants with practical biological applications.

Contribution

It presents a novel approach combining blossoming, multi-affine properties, and Lagrangian duality to compute certified lower bounds for invariants using linear programs.

Findings

Method effectively verifies polytopic invariants in polynomial systems.

Approach applicable to biological system models.

Iterative computation of invariants via sensitivity analysis.

Abstract

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.