Approximating Nonlinear Feedback Controls for Polynomial Systems

The calculation of optimal feedback controllers for nonlinear systems remains elusive since it requires the solution or approximation of the Hamilton-Jacobi-Bellman equations.  By restricting our attention to quadratic regulator problems and polynomial systems, we are able to calculate polynomial feedback laws for systems with hundreds of states.  We describe our approximation algorithm, which relies on introducing a Kronecker structure and provide examples for discretized PDEs such as Burgers, Chafee-Infante, Kuramoto-Sivashinsky, etc. where nonlinear feedback performs up to 40% better than a frequently used linear feedback strategy.