On Dirac structure of infinite-dimensional stochastic port-Hamiltonian systems

TL;DR

This paper extends the concept of Dirac structures to stochastic infinite-dimensional port-Hamiltonian systems with multiplicative noise, establishing a framework for boundary control and stabilization.

Contribution

It introduces a stochastic Dirac structure framework for SPHSs using Stratonovich calculus, enabling new interconnection and control strategies.

Findings

Extended Dirac structures to stochastic systems.

Proved interconnection preserves Dirac structure.

Applied framework to a boundary-controlled stochastic string.

Abstract

Stochastic infinite-dimensional port-Hamiltonian systems (SPHSs) with multiplicative Gaussian white noise are considered. In this article we extend the notion of Dirac structure for deterministic distributed parameter port-Hamiltonian systems to a stochastic ones by adding some additional stochastic ports. Using the Stratonovich formalism of the stochastic integral, the proposed extended interconnection of ports for SPHSs is proved to still form a Dirac structure. This constitutes our main contribution. We then deduce that the interconnection between (stochastic) Dirac structures is again a (stochastic) Dirac structure under some assumptions. These interconnection results are applied on a system composed of a stochastic vibrating string actuated at the boundary by a mass-spring system with external input and output. This work is motivated by the problem of boundary control of SPHSs and will serve as a foundation to the development of stabilizing methods.