{"ID":2866594,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.20122","arxiv_id":"2509.20122","title":"Unifying HJB and Riccati equations: A Koopman operator approach to nonlinear optimal control","abstract":"This paper proposes an operator-theoretic framework that recasts the minimal value function of a nonlinear optimal control problem as an abstract bilinear form on a suitable function space. The resulting bilinear form is shown to satisfy an operator equation with quadratic nonlinearity obtained by formulating the Lyapunov equation for a Koopman lift of the optimal closed-loop dynamics to an infinite-dimensional state space. It is proven that the minimal value function admits a rapidly convergent sum-of-squares expansion, a direct consequence of the fast spectral decay of the bilinear form. The framework thereby establishes a natural link between the Hamilton-Jacobi-Bellman and a Riccati-like operator equation and further motivates numerical low-rank schemes.","short_abstract":"This paper proposes an operator-theoretic framework that recasts the minimal value function of a nonlinear optimal control problem as an abstract bilinear form on a suitable function space. The resulting bilinear form is shown to satisfy an operator equation with quadratic nonlinearity obtained by formulating the Lyapu...","url_abs":"https://arxiv.org/abs/2509.20122","url_pdf":"https://arxiv.org/pdf/2509.20122v2","authors":"[\"Tobias Breiten\",\"Bernhard Höveler\"]","published":"2025-09-24T13:46:44Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[\"Large Language Model\"]","has_code":false}