标题： 当库普曼遇到哈密顿和雅可比时 显示英文标题

标题： When Koopman Meets Hamilton and Jacobi

摘要： 本文建立了Koopman算子谱理论与Hamilton-Jacobi（HJ）方程解之间的联系。HJ方程在系统理论中占据核心地位，其解在各种控制问题中具有重要意义，包括最优控制、鲁棒控制和输入-输出分析。 一个Hamilton动力学系统可以与HJ方程相关联，并且可以通过Lagrangian子流形的形式从Hamilton系统中提取HJ方程的解。 本文的主要贡献之一是展示可以通过Koopman算子的谱分析获得Lagrangian子流形。 我们提出了两种不同的HJ解的逼近方法。 我们利用与未受控动力学系统和Hamilton系统相关的Koopman算子的谱性质来近似HJ解。 我们提出了一种基于凸优化的计算框架以及收敛性分析，用于近似Koopman特征函数和Lagrangian子流形。 我们使用Koopman理论解决HJ方程的方法为从线性系统到非线性系统的结论提供了自然的扩展。 我们展示了这项工作在求解最优控制问题中的应用。 最后，我们呈现了仿真结果以验证本文的主要发现，并将其与线性二次调节器和基于泰勒级数的逼近控制器进行比较。 显示英文摘要

摘要： In this paper, 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 paper 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 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 the paper's main findings and compare them against linear quadratic regulator and Taylor series based approximation controllers.