Stochastic Passivity in Stochastic Differential Equations: A Port-Hamiltonian Perspective
Julia Ackermann, Thomas Kruse, Stefan Tappe
TL;DR
This work develops a stochastic port-Hamiltonian framework by formulating finite-dimensional PHS as stochastic differential equations with a storage function H and multiple passivity notions (passive, stochastic passive, passive SMP, and passive local SMP). It provides necessary and sufficient drift-diffusion conditions via the generator and Stratonovich corrections, and establishes when passivity implies local and global supermartingale properties. For linear systems, the authors derive a central LMI, $\mathfrak{M}_Q\le0$, that characterizes stochastic passivity, local SMP, and stochastic SMP, with observability enabling quadratic storage functions and available storage concepts. They extend these ideas to stochastic linear port-Hamiltonian systems (SLTI-PHS), show interconnection-preservation, and illustrate with stochastic mass-spring-damper and RLC circuit models. Overall, the paper extends deterministic PHS theory to stochastic settings, enabling energy-based analysis, controller design, and interconnection results under noise.
Abstract
We extend deterministic port-Hamiltonian systems (PHS) to a stochastic framework by means of stochastic differential equations. As the dissipation inequality plays a crucial role for deterministic PHS, we develop several passivity concepts for stochastic input-state-output systems and characterize these in terms of the parameters of the system. Afterwards, we examine properties of a certain class of linear stochastic systems that can be regarded as an extension of linear deterministic PHS to a stochastic passivity framework.