Exact controllability for a refined Stochastic Wave Equation

TL;DR

This paper establishes the exact controllability of a refined stochastic wave equation with three controls, introducing a novel Carleman estimate and considering practical effects in drift and diffusion terms, with explicit waiting time characterization.

Contribution

It introduces a new Carleman estimate for backward hyperbolic operators and models practical effects in drift and diffusion terms for stochastic wave equations.

Findings

Exact controllability achieved with three controls

Explicit, sharp waiting time in one dimension

Model includes practical effects in drift and diffusion

Abstract

In this paper, we obtain the exact controllability for a refined stochastic wave equation with three controls by establishing a novel Carleman estimate for a backward hyperbolic-like operator. Compared with the known result, the novelty of this paper is twofold: (1) Our model contains the effects in the drift terms when we put controls directly in the diffusion terms, which is more sensible for practical applications; (2) We provide an explicit description of the waiting time which is sharp in the case of dimension one and is independent of the coefficients of lower terms.