Unifying HJB and Riccati equations: A Koopman operator approach to nonlinear optimal control

Tobias Breiten; Bernhard Höveler

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arXiv · 2509.20122

Unifying HJB and Riccati equations: A Koopman operator approach to nonlinear optimal control

Tobias Breiten·Bernhard Höveler

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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.

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BibTeX RIS

Tobias Breiten, Bernhard Höveler. 2025-09-24. Unifying HJB and Riccati equations: A Koopman operator approach to nonlinear optimal control. https://arxiv.org/abs/2509.20122

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