Well-posedness of Stochastic Port-Hamiltonian Systems on Infinite-dimensional Spaces

TL;DR

This paper introduces stochastic port-Hamiltonian systems in infinite-dimensional spaces, extending finite-dimensional concepts, and proves their well-posedness using an example of a vibrating string with noise.

Contribution

It generalizes the concept of well-posedness for stochastic port-Hamiltonian systems to infinite-dimensional spaces and demonstrates this with a practical example.

Findings

Stochastic port-Hamiltonian systems are well-posed in infinite-dimensional spaces.

The theory extends finite-dimensional concepts to infinite-dimensional settings.

An example with a vibrating string illustrates the application of the theory.

Abstract

Stochastic port-Hamiltonian systems on infinite-dimensional spaces governed by It\^o stochastic differential equations (SDEs) are introduced and some properties of this new class of systems are studied. They are an extension of stochastic port-Hamiltonian systems defined on a finite-dimensional state space. The concept of well-posedness in the sense of Weiss and Salamon is generalized to the stochastic context. Under this extended definition, stochastic port-Hamiltonian systems are shown to be well-posed. The theory is illustrated on an example of a vibrating string subject to a Hilbert space-valued Gaussian white noise process.