Spectral Analysis of Koopman Operator and Nonlinear Optimal Control
Umesh Vaidya
Abstract
In this paper, we present an approach based on the spectral analysis of the Koopman operator for the approximate solution of the Hamilton Jacobi equation that arises while solving the optimal control problem. It is well-known that one can associate a Hamiltonian dynamical system with the Hamilton Jacobi equation. Furthermore, the Lagrangian submanifold of the Hamiltonian dynamical system play a fundamental role in solving the Hamilton Jacobi equation. We show that the principal eigenfunctions of the Koopman operator associated with the Hamiltonian dynamical system can be used in constructing the Lagrangian submanifold, thereby approximating the solution of the Hamilton Jacobi equation. We present simulation results to verify the main findings of the paper.