We study the well-posedness of a 3D nonlinear stochastic wave equation which derives from the Maxwell system by the Galerkin method. Then we study the approximate controllability of this system by the Hilbert uniqueness method.

Many physical models are described by wave equations, such as the scalar wave equations, the elastic wave equations, and Maxwell’s equations. Therefore, the well-posedness and controllability of wave equations are both important issues in theories and applications.

The well-posedness of partial differential equations (PDEs) has been studied by many authors; there are some useful methods: the semigroup theory [

The controllability of PDEs has been a very active research field since 1960s. From [

For a detailed discussion of HUM, we can refer to [

In a number of applications, the systems in question are subject to stochastic fluctuations arising as a result of either uncertain forcing (stochastic external forcing) or uncertainty of the governing laws of the system. Noise due to experimental uncertainties, intrinsic randomness of the media, and so forth plays an important role in wave propagation, thus calling for the inclusion of stochastic terms in the wave equations. When a physical state is modeled, external random noise can be a serious disturbance. Apparently this affects controllability of the model. In this paper we discuss controllability of a vector wave equation with random noise.

The wave equation with the dissipative term has been extensively studied by many authors (see [

In this paper, we consider the nonlinear stochastic wave equation

System (

Noting the identical equation,

Theorem

For deterministic wave equations, the exact controllability has been extensively investigated for the past few decades (see [

This paper is organized as follows. In Section

Throughout this paper, let

The definitions and properties of these spaces can be found in [

A stochastic process

First, we consider the following system:

By using the Galerkin method in [

Let

Next, we consider the system (

Assume the following conditions:

Then there is a solution to problem (

We formally write

Let

Since

If

Therefore, we apply Theorem

By virtue of (

We have

Then by (

Let

Then we can derive that, for some positive constant

By (

By induction, it follows that

Consequently,

Then

From (

By the diagonal process and taking limits, we get that

In the end, we prove the uniqueness of the solution.

Let

Let

By virtue of (

In the meantime, by the uniqueness of solutions to the linear problem, we can obtain

From Gronwall’s inequality, we can obtain that

Now, we can answer the question in Section

Suppose that assumptions (i)–(iv) of Theorem

In the first part we derive fundamental estimates. The proof of Theorem

By using the method similar to that in [

Let

Let

The estimates (

If

Consider the following initial-boundary value problem:

Let

Let

Assume that

Consider the map

The following property of

For every

In fact, Lemma

The major technical result, essential in our work, is a generalization of the Martingale representation theorem (see [

Given an

The proof of Lemma

Now, we can prove our main result.

The following procedure shows the construction of the approximate control. We use the Hilbert uniqueness method.

Choose an

Solve the forward uncontrolled system

and find the solution

Find the distance of the uncontrolled solution

According to Lemma

We can take

where

From Lemma

Find the solution

According to Lemma

Extend

Then

where

Define

Now we verify that

This completes the proof of Theorem

The author declares that there is no conflict of interests regarding the publication of this paper.

The author would like to sincerely thank Professor Yong Li for many useful suggestions and help.