Exact Controllability for a Refined Stochastic Wave Equation

Abstract

A widely used stochastic plate equation is the classical plate equation perturbed by a term of Itô's integral. However, it is known 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, a one-dimensional refined stochastic plate equation is proposed and its exact controllability is established in [28]. In this paper, by means of a new global Carleman estimate, we establish the exact controllability of the multidimensional refined stochastic plate equation with two interior controls and two boundary controls. Moreover, we give a result about the lack of exact controllability, which shows that the action of two interior controls and at least one boundary control is necessary.

Determination of two unknowns for a stochastic plate equation

“…The weight functions used in this paper are different from those in [41]. And the terms involving the random source g(t, x) appear on the right-hand side of the Carleman estimate in [41], while on the left-hand side of the Carleman estimates in this paper. There are no Carleman estimates for the stochastic plate equations (1.1) and (1.2) with N ≥ 2 and the weight function which is used in this paper.…”

“…However, the plate equations in this paper are not exactly controllable, even if the controls act everywhere on the domain and boundary (see e.g., [41]). The weight functions used in this paper are different from those in [41]. And the terms involving the random source g(t, x) appear on the right-hand side of the Carleman estimate in [41], while on the left-hand side of the Carleman estimates in this paper.…”

“…Although there are numerous results for Carleman estimates for deterministic partial differential equations, there are just a few Carleman estimates for stochastic partial differential equations (see e.g., [8,16,37,38,39,40,41,42,44,53]). In [41], global Carleman estimates are established to proved the exact controllability of a refined stochastic plate equation. The refined stochastic plate equation is quite different from the plate equations in this paper.…”

“…For example, the refined stochastic plate equations are exactly controllable at any time when the controls act on the diffusion terms and boundary (see e.g., [41]). However, the plate equations in this paper are not exactly controllable, even if the controls act everywhere on the domain and boundary (see e.g., [41]). The weight functions used in this paper are different from those in [41].…”