Stabilization and Optimal Control of Interconnected SDE - Scalar PDE System

TL;DR

This paper develops a control strategy for an interconnected stochastic system involving SDEs and PDEs, using backstepping and stochastic control techniques to stabilize the system and minimize state variance.

Contribution

It introduces a novel control design for coupled SDE-PDE systems by transforming the PDE and applying stochastic control methods, addressing stabilization and optimality.

Findings

Controller successfully stabilizes the mean of the system states.

Variance of the states remains bounded under the designed control.

The approach effectively handles the coupling between SDE and PDE components.

Abstract

In this paper, we design a controller for an interconnected system consisting of a linear Stochastic Differential Equation (SDE) actuated through a linear hyperbolic Partial Differential Equation (PDE). Our approach aims to minimize the variance of the state of the SDE component. We leverage a backstepping technique to transform the original PDE into an uncoupled stochastic PDE. As such, we reformulate our initial problem as the control of a delayed SDE with a non-deterministic drift. Under standard controllability assumptions, we design a controller steering the mean of the states to zero while keeping its covariance bounded. As final step, we address the optimal control of the delayed SDE employing Artstein's transformation and Linear Quadratic stochastic control techniques.