From Hamilton-Jacobi-Bellman to Riccati via Carleman: The Scalar Case
Abstract
The Hamilton-Jacobi-Bellman (HJB) equation of a nonlinear system is a nonlinear partial differential equation, and in that form it gives little away. This paper shows that it hides a Riccati equation. Carleman linearization reveals this structure. For scalar input-affine systems with analytic data, the lifted HJB equation is a single infinite-dimensional Riccati equation in a lower-triangular Toeplitz matrix. Its first entry is the Riccati equation of the standard linearization at equilibrium. Its remaining entries hold the nonlinear terms. Nothing about this is an approximation. Under controllability of the standard linearization and detectability of the cost, the equation has exactly one solution with positive diagonal entries, and it can be computed one entry at a time. This solution is the derivative of the value function. We prove that finite-sections of the solution are exact and converge exponentially to the derivative of the value function near the origin. The scalar case has a closed-form solution. This closed form is our benchmark: every result on the lifted equation is checked against it. Nothing in this structure depends on the state dimension. Our methodology points to a general theory of optimal control in which the HJB equation of analytic nonlinear systems becomes an infinite-dimensional Riccati equation.
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BibTeX
@misc{elias2026from,
title = {{From Hamilton-Jacobi-Bellman to Riccati via Carleman: The Scalar Case}},
author = {Philip J. Elias and Qiyu Sun and Nader Motee},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04873},
eprint = {2610.04873},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04873},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - From Hamilton-Jacobi-Bellman to Riccati via Carleman: The Scalar Case
AU - Philip J. Elias
AU - Qiyu Sun
AU - Nader Motee
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.04873
UR - https://arxiv.org/abs/2610.04873
AB - The Hamilton-Jacobi-Bellman (HJB) equation of a nonlinear system is a nonlinear partial differential equation, and in that form it gives little away. This paper shows that it hides a Riccati equation. Carleman linearization reveals this structure. For scalar input-affine systems with analytic data, the lifted HJB equation is a single infinite-dimensional Riccati equation in a lower-triangular Toeplitz matrix. Its first entry is the Riccati equation of the standard linearization at equilibrium. Its remaining entries hold the nonlinear terms. Nothing about this is an approximation. Under controllability of the standard linearization and detectability of the cost, the equation has exactly one solution with positive diagonal entries, and it can be computed one entry at a time. This solution is the derivative of the value function. We prove that finite-sections of the solution are exact and converge exponentially to the derivative of the value function near the origin. The scalar case has a closed-form solution. This closed form is our benchmark: every result on the lifted equation is checked against it. Nothing in this structure depends on the state dimension. Our methodology points to a general theory of optimal control in which the HJB equation of analytic nonlinear systems becomes an infinite-dimensional Riccati equation.
ER -