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.