Exact Controllability for a Refined Stochastic Wave Equation

TL;DR

This paper introduces a refined stochastic wave equation based on stochastic Newton's law, demonstrating its exact controllability with three controls and highlighting the limitations of the classical model.

Contribution

The paper proposes a new stochastic wave equation model and proves its exact controllability using a novel Carleman estimate, addressing shortcomings of the classical model.

Findings

Classical stochastic wave equation is not exactly controllable.

Refined model achieves exact controllability with three controls.

Three controls are necessary for controllability.

Abstract

A widely used stochastic wave equation is the classical wave equation perturbed by a term of It\^o's integral. We show that this equation is not exactly controllable even if the controls are effective everywhere in both the drift and the diffusion terms and also on the boundary. In some sense this means that some key feature has been ignored in this model. Then, based on a stochastic Newton's law, we propose a refined stochastic wave equation. By means of a new global Carleman estimate, we establish the exact controllability of our stochastic wave equation with three controls. Moreover, we give a result about the lack of exact controllability, which shows that the action of three controls is necessary. Our analysis indicates that, at least from the point of view of control theory, the new stochastic wave equation introduced in this paper is a more reasonable model than that in the existing literatures.