# Computing Solutions to the Polynomial-Polynomial Regulator Problem

arXiv: https://arxiv.org/abs/2410.22291 · HTML: https://mlsanity.com/paper/2410.22291/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Oct 2024.

## Built on by (most cited first, 1 of 1)

- [Hessian-augmented Supervised Learning for Hamilton-Jacobi-Bellman PDEs](https://mlsanity.com/paper/2606.23827/) (2026)

## Abstract (arXiv)

We consider the optimal regulation problem for nonlinear control-affine dynamical systems. Whereas the linear-quadratic regulator (LQR) considers optimal control of a linear system with quadratic cost function, we study polynomial systems with polynomial cost functions; we call this problem the polynomial-polynomial regulator (PPR). The resulting polynomial feedback laws provide two potential improvements over linear feedback laws: 1) they more accurately approximate the optimal control law, resulting in lower control costs, and 2) for some problems they can provide a larger region of stabilization. We derive explicit formulas -- and a scalable, general purpose software implementation -- for computing the polynomial approximation to the value function that solves the optimal control problem. The method is illustrated first on a low-dimensional aircraft stall stabilization example, for which PPR control recovers the aircraft from more severe stall conditions than LQR control. Then we demonstrate the scalability of the approach on a semidiscretization of dimension n=129 of a partial differential equation, for which the PPR control reduces the control cost by approximately 75% compared to LQR for the initial condition of interest.