Stability and stabilization of polynomial dynamical systems using Bernstein polynomials

In this work, we examine relaxations for the stability analysis and synthesis of stabilizing controllers for polynomial dynamical systems. It is well-known that such problems can be naturally solved using a reduction to polynomial optimization problems. The Sum of Squares (SOS) programming relaxation further relaxes these polynomial optimization problems to convex Semi-Definite Programming (SDP) problems. However, their application, in practice, to formal verification and correct-by-construction synthesis has been made harder due to numerical stability issues encountered while solving SDPs. Our work proposes a new approach to relaxations based on Bernstein polynomials to yield linear programming (LP) relaxations for polynomial optimization problems. This allows us to find polynomial Lyapunov functions that certify the (asymptotic) stability of a system. The approach is also extended to synthesizing a stabilizing feedback control law for a controlled system that will stabilize the closed-loop dynamics to a specified equilibrium.