This paper introduces a framework for solving time-autonomous nonlinear infinite horizon optimal control problems, under the assumption that all minimizers satisfy Pontryagin’s necessary optimality conditions. In detail, we use methods from the field of symplectic geometry to analyze the eigenvalues of a Koopman operator that lifts Pontryagin’s differential equation into a suitably defined infinite dimensional symplectic space. This has the advantage that methods from the field of spectral analysis can be used to characterize globally optimal control laws. A numerical method for constructing optimal feedback laws for nonlinear systems proceeds by computing the eigenvalues and eigenvectors of a matrix that is obtained by projecting the Pontryagin–Koopman operator onto a finite dimensional space. We illustrate the effectiveness of this approach by computing accurate approximations of the optimal nonlinear feedback law for a Van der Pol control system, which cannot be stabilized by a linear control law.

本文介绍了一个解决时间自治非线性无限视界最优控制问题的框架，假设所有极小值都满足庞特里亚金的必要最优条件。具体而言，我们使用辛几何领域的方法来分析 Koopman 算子的特征值，该算子将 Pontryagin 的微分方程提升到适当定义的无限维辛空间中。这样做的优点是可以使用频谱分析领域的方法来表征全局最优控制律。通过计算矩阵的特征值和特征向量来构建非线性系统最优反馈定律的数值方法，该矩阵是通过将 Pontryagin-Koopman 算子投影到有限维空间上而获得的。