Stabilization of polynomial dynamical systems using linear programming based on Bernstein polynomials

In this paper, we deal with the problem of synthesizing static output feedback controllers for stabilizing polynomial systems. Our approach jointly synthesizes a Lyapunov function and a static output feedback controller that stabilizes the system over a given subset of the state-space. Specifically, our approach is simultaneously targeted towards two goals: (a) asymptotic Lyapunov stability of the system, and (b) invariance of a box containing the equilibrium. Our approach uses Bernstein polynomials to build a linear relaxation of polynomial optimization problems, and the use of a so-called "policy iteration" approach to deal with bilinear optimization problems. Our approach can be naturally extended to synthesizing hybrid feedback control laws through a combination of state-space decomposition and Bernstein polynomials. We demonstrate the effectiveness of our approach on a series of numerical benchmark examples.