A symmetric structure of variational and adjoint systems of stochastic Hamiltonian systems

The authors have extended deterministic port-Hamiltonian systems into stochastic dynamical systems which are described by stochastic differential equations written in the sense of Itô, called stochastic port-Hamiltonian systems. This paper introduces variational systems and their adjoint ones for the stochastic port-Hamiltonian systems. We also reveal some of their properties, particularly an extension of a self-adjoint property of deterministic Hamiltonian systems, which plays an important role in learning optimal control for the deterministic Hamiltonian systems.