Sum-of-squares-based policy iteration for suboptimal control of polynomial time-varying systems

This paper investigates the suboptimal control design for polynomial time-varying systems. It is known that the solution to this problem relies on the solution of the Hamilton-Jacobi-Bellman (HJB) equation, which is a nonlinear partial differential equation (PDE). A policy iteration (PI) algorithm is developed to solve the HJB equation. The policy evaluation step of this algorithm consists of a sum-of-squares (SOS) program, which is computationally tractable. This algorithm distinguishes from previously known SOS-based adaptive dynamic programming (ADP) algorithms in that it is developed for time-varying systems. The convergence of the iterative algorithm and the global stability of the closed-loop system are proved. At the end, the effectiveness of the proposed algorithm is illustrated through two simulation examples.