Exact Controllability for a Refined Stochastic Wave Equation

Abstract

A widely used stochastic wave equation is the classical wave equation perturbed by a term of Itô'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.

A Concise Introduction to Control Theory for Stochastic Partial Differential Equations

“…This, in turn, justifies our modification (6.23). Some further result related to the exact controllability of the above refined stochastic wave equations (even in several space dimensions) can be found in [55].…”

“…As we shall see in this section, the usual stochastic hyperbolic equation, i.e., the classic hyperbolic equation perturbed by a term of Itô's integral, is not exactly controllable even if the controls are effective everywhere in both drift and diffusion terms, which differs dramatically from its deterministic counterpart. This section is based on [55].…”

“…Then, as we did in [55], motivated by Theorem 6.1, we propose below a refined model to describe the DNA molecule considered in Example 2.2. For this purpose, we partially employ a dynamical theory of Brownian motions, developed in [60], to describe the motion of a particle perturbed by random forces.…”

“…A possible way to solve these problems is to establish suitable observability estimate/unique continuation property for some backward stochastic hyperbolic equation. For example, to get the null controllability of (6.4), one should establish the following observability estimate for the equation ( 6 However, until now, we can only prove that (see [55]) • Controllability problem for one dimensional stochastic hyperbolic systems…”

“…• In this notes, we only give a very brief introduction to controllability problems for three kinds of SPDEs. Recently, there are also some works for controllability problems for other types of SPDEs, such as [16] for stochastic complex parabolic equations, [21] for stochastic Kuramoto-Sivashinsky equations, [33,72] for degenerate stochastic parabolic equations, [32] for coupled stochastic parabolic equations, [38] for stochastic Schrödinger equations and [55] for a refined stochastic wave equation.…”