A semigroup approach to stochastic systems with input delay at the boundary

TL;DR

This paper develops a semigroup-based framework to analyze the well-posedness of stochastic systems with boundary input delay, enabling the use of admissible operators and Yosida extensions for solution existence and uniqueness.

Contribution

It introduces a novel semigroup approach to handle boundary input delays in stochastic systems, extending the analysis to unbounded operators and providing solution estimates.

Findings

Established well-posedness of stochastic boundary delay systems

Applied the framework to a stochastic Schrödinger system

Provided conditions for existence and uniqueness of solutions

Abstract

This work focuses on the well-posedness of abstract stochastic linear systems with boundary input delay and unbounded observation operators. We use product spaces and a semigroup approach to reformulate such delay systems into free-delay distributed stochastic systems with unbounded control and observation operators. This gives us the opportunity to use the concept of admissible control and observation operators as well as the concept of Yosida extensions to prove the existence and uniqueness of the solution process and provide an estimation of the observation process in relation to initial conditions and control process. As an example, we consider a stochastic Schr\"odinger system with input delay.