Connection among Stochastic Hamilton–Jacobi–Bellman Equation, Path-Integral, and Koopman Operator on Nonlinear Stochastic Optimal Control

The path-integral control, which stems from the stochastic Hamilton–Jacobi–Bellman equation, is one of methods to control nonlinear systems. This paper gives a new insight into optimal problems perspective Koopman operators. When finite-dimensional dynamical system nonlinear, corresponding operator linear. Although infinite-dimensional, adequate approximation makes it tractable and useful in some discussions applications. Employing perspective, clarified that only specific type observable enough be focused on problem. fact becomes easier understand via control. Furthermore, focus leads natural power-series expansion; coupled ordinary differential equations for discrete-state space systems are derived. A demonstration shows derived work well.