  Feedback control problems involving autonomous polynomial systems are
prevalent, yet there are limited algorithms and software for approximating
their solution. This paper represents a step forward by considering the special
case of the regulator problem where the state equation has polynomial
nonlinearity, control costs are quadratic, and the feedback control is
approximated by low-degree polynomials. As this represents the natural
extension of the linear-quadratic regulator (LQR) and quadratic-quadratic
regulator (QQR) problems, we denote this class as polynomial-quadratic
regulator (PQR) problems. The present approach is amenable to feedback
approximations with low degree polynomials and to problems of modest model
dimension. This setting can be achieved in many problems using modern model
reduction methods. The Al'Brekht algorithm, when applied to polynomial
nonlinearities represented as Kronecker products leads to an elegant
formulation. The terms of the feedback control lead to large linear systems
that can be effectively solved with an N-way generalization of the
Bartels-Stewart algorithm. We demonstrate our algorithm with numerical examples
that include the Lorenz equations, a ring of van der Pol oscillators, and a
discretized version of the Burgers equation. The software described here is
available on Github.
