Exact Controllability for Stochastic Transport Equations

TL;DR

This paper investigates the exact controllability of stochastic transport equations using boundary and internal controls, employing duality and Carleman estimates, and highlights the necessity of both controls for controllability.

Contribution

It introduces a new approach using global Carleman estimates to establish controllability and demonstrates the necessity of two controls, advancing understanding of stochastic PDE control.

Findings

Controllability is achieved with boundary and internal controls.

A new global Carleman estimate is developed for backward stochastic transport equations.

Lack of controllability results show both controls are essential.

Abstract

This paper is addressed to studying the exact controllability for stochastic transport equations by two controls: one is a boundary control imposed on the drift term and the other is an internal control imposed on the diffusion term. By means of the duality argument, this controllability problem can be reduced to an observability problem for backward stochastic transport equations, and the desired observability estimate is obtained by a new global Carleman estimate. Also, we present some results about the lack of exact controllability, which show that the action of two controls is necessary. To some extent, this indicates that the controllability problems for stochastic PDEs differ from their deterministic counterpart.