When Koopman Meets Hamilton and Jacobi

• 成果类型: Article
• 署名作者: Vaidya, Umesh
• 署名单位: Clemson University
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2025.3577960
• 发表日期: 2025
• 关键词: operator; DESIGN

摘要

In this article, we establish a connection between the spectral theory of the Koopman operator and the solution of the Hamilton-Jacobi (HJ) equation. The HJ equation occupies a central place in systems theory, and its solution is of interest in various control problems, including optimal control, robust control, and input-output analysis. A Hamiltonian dynamical system can be associated with the HJ equation and the solution of the HJ equation can be extracted from the Hamiltonian system in the form of Lagrangian submanifold. One of the main contributions of this article is to show that the Lagrangian submanifolds can be obtained using the spectral analysis of the Koopman operator. We present two different procedures for the approximation of the HJ solution. We utilize the spectral properties of the Koopman operator associated with the uncontrolled dynamical system and Hamiltonian systems to approximate the HJ solution. We present a convex optimization-based computational framework with convergence analysis for approximating the Koopman eigenfunctions and the Lagrangian submanifolds. Our solution approach to the HJ equation using the Koopman theory provides for a natural extension of results from linear systems to nonlinear systems. We demonstrate the application of this work for solving the optimal control problem. Finally, we present simulation results to validate this article's main findings and compare the performance of the proposed controller against different controllers.