  Feedback control problems involving autonomous quadratic systems are
prevalent, yet there are only a limited number of software tools available for
approximating their solution due to the complexity of the problem. This paper
represents a step forward in the special case where both the state equation and
the control costs are quadratic. As it represents the natural extension of the
linear-quadratic regulator (LQR) problem, we describe this setting as the
quadratic-quadratic regulator (QQR) problem. This is significantly more
challenging and holds the LQR as special case that must be solved along the
way. We describe an algorithm that exploits the structure of the QQR problem
that arises when implementing Al'Brekht's method. This approach is amenable to
feedback laws with low degree polynomials but have a relatively modest model
dimension that could be achieved by modern model reduction methods. This
problem has an elegant formulation and a solution that introduces several
linear systems where the structure suggests modern tensor-based linear solvers.
We demonstrate this algorithm on a suite of random test problems then apply it
to a distributed parameter control problem that fits the QQR framework.
Comparisons to linear feedback control laws show a modest benefit using the QQR
formulation.
