谱Koopman-Hopf公式用于对抗性可达性分析
Spectral Koopman-Hopf Formula for Reachability with Adversary
- Clemson University（克莱姆森大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出谱Koopman-Hopf框架，通过Koopman特征函数坐标将HJI方程转化为状态无关优化问题，利用谱投影和上下界近似，计算非线性对抗系统的向后可达集。
AI中文摘要：
本文针对非线性系统的对抗性可达性分析，提出了一种谱Koopman-Hopf框架。通过将非线性漂移动力学提升到Koopman特征函数坐标中，该方法将原始的状态相关的Hamilton-Jacobi-Isaacs (HJI)方程转化为谱坐标下的近似状态无关优化问题。变换后的控制方向和扰动方向通过最小二乘谱投影进行近似，并利用变换后Hamiltonian的上界和下界推导出值函数和向后可达集的相应界。Koopman特征函数采用基于路径积分的Galerkin框架计算，该框架避免了对相关特征函数偏微分方程的空间离散化。数值算例展示了非线性对抗系统向后可达集的可处理近似。
英文摘要：
This paper develops a spectral Koopman-Hopf framework for adversarial reachability analysis of nonlinear systems. By lifting the nonlinear drift dynamics into Koopman eigenfunction coordinates, the proposed approach transforms the original state-dependent Hamilton-Jacobi-Isaacs (HJI) equation into an approximate state-independent optimization problem in spectral coordinates. The transformed control and disturbance directions are approximated using least-squares spectral projections, and upper and lower bounds on the transformed Hamiltonian are used to derive corresponding bounds on the value function and backward reachable sets. Koopman eigenfunctions are computed using a path-integral-based Galerkin framework that avoids spatial discretization of the associated eigenfunction PDEs. Numerical examples demonstrate tractable approximation of backward reachable sets for nonlinear adversarial systems.
