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

The Quadratic-Quadratic Regulator Problem: Approximating feedback controls for quadratic-in-state nonlinear systems