Unifying HJB and Riccati equations: A Koopman operator approach to nonlinear optimal control
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866917002675748864 |
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| author | Breiten, Tobias Höveler, Bernhard |
| author_facet | Breiten, Tobias Höveler, Bernhard |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_20122 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Unifying HJB and Riccati equations: A Koopman operator approach to nonlinear optimal control Breiten, Tobias Höveler, Bernhard Optimization and Control 37L45, 47B33, 47D06, 49L99, 49N35, 93B52, 93D15 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. |
| title | Unifying HJB and Riccati equations: A Koopman operator approach to nonlinear optimal control |
| topic | Optimization and Control 37L45, 47B33, 47D06, 49L99, 49N35, 93B52, 93D15 |
| url | https://arxiv.org/abs/2509.20122 |