The controllability for the semi-discrete wave equation with a finite element method

where the initial value $(y_0,y_1)$ belongs to $H_0^1(0,1)\times L^2(0,1)$ and $u(\cdot )$ is a control function taken from the space $L^2(0,T;L^2(\omega ))$.

System (1.1) is said to be exactly controllable from the initial value $(y_0,y_1)\in H_0^1(0,1)\times L^2(0,1)$ in time T if there exists a control function $u(\cdot )\in L^2(0,T;L^2(\omega ))$ such that the solution of (1.1) matches that $(y(T),{\partial }_{t}y(T))=(0,0)$. We have already known that the controllability property for the above continuous one-dimensional wave equation holds for any given $T>2$ (see [1]).

Throughout this paper, ${\chi }_{\omega }$ will be treated as an operator from $L^2(0,1)$ to $L^2(\omega )$, by setting ${\chi }_{\omega }(f)=f{|}_{\omega }$ for all $f\in L^2(0,1)$. Clearly, ${\tilde{V}}^h$ is a subspace of $H^1(\omega )$.

In this paper, we mainly deal with the controllability for semi-discrete system (1.4). The main result of the paper is presented as follows.

Theorem 1.1 For each $T>0$, controlled system (1.4) is exactly controllable in time T. Namely, there exists a control function $v_h(\cdot )\in L^2(0,+\mathrm{\infty };{\tilde{V}}^h)$ such that the solution of (1.4) satisfies $(y_h(T),y_h^{\prime }(T))=(0,0)$.

Remark 1.2 In this paper, our main purpose is to discuss whether or not the semi-discrete internally controlled systems have the exactly controllable property which the original controlled systems have. This is a very valuable problem in control theory. In [2], the authors established such a controllability result for the semi-discrete one-dimensional boundary controlled wave equation by the numerical approximation method. The authors also got that the semi-discrete systems are not uniformly controllable as the discretization parameter h goes to zero. The main differences between [2] and our paper are as follows. In [2], the authors focused on a one-dimensional boundary controlled wave equation and they obtained the controllability for the semi-discrete wave system, with the controls taken from an infinite dimensional space. In our paper, we discuss an internally controlled one-dimensional wave equation. In this case, we obtain the exact controllability of the semi-discrete wave equation, with the controls taken from a finite dimensional space. Regarding other works related to this problem, we would like to mention [3, 4], and [5].

The rest of the paper is structured as follows. In Section 2, we give a sufficient condition for controllability. By making use of this sufficient condition presented in Section 2, we provide the proof of Theorem 1.1 in Section 3.

where the initial data $(y_0^h,y_1^h)\in V_0^h\times V_0^h$. Clearly, it is a controlled system governed by ordinary differential equations. However, the control functions for this system are taken from the infinite dimensional space $L^2(0,T;L^2(\omega ))$.

In this section, we discuss some controllability result for system (2.1). More concisely, a sufficient condition for the exact controllability property of (2.1) will be presented. The proofs of the following Lemmas 2.1, 2.2 and 2.4 can be found in [6].

for all $({\phi }_0^h,{\phi }_1^h)\in V_0^h\times V_0^h$, where ${\phi }_h$ is the corresponding solution of equation (2.2).

where ${\phi }_h$ is the solution of (2.2) with initial data $({\phi }_0^h,{\phi }_1^h)\in V_0^h\times V_0^h$.

We have the following result.

is a control which drives the initial data $(y_0^h,y_1^h)$ of controlled system (2.1) to zero in time T.

To get the sufficient condition that ensures the existence of a minimizer for $\mathcal{J}$, we need to give the following definition.

for any $({\phi }_0^h,{\phi }_1^h)\in V_0^h\times V_0^h$, where ${\phi }_h$ is the solution of (2.2) with initial data $({\phi }_0^h,{\phi }_1^h)$.

Inequality (2.6) is called observability inequality. The following conclusion shows that observability inequality (2.6) is the sufficient condition for the exact controllability of system (2.1).

Lemma 2.4 Suppose equation (2.2) is observable in time T. Then the functional $\mathcal{J}$ defined by (2.4) has a unique minimizer $({\hat{\phi }}_0^h,{\hat{\phi }}_1^h)\in V_0^h\times V_0^h$.

forms an orthogonal basis of $V_0^h$. The following two results are quoted from [7]. They will be used later.

Lemma 3.1 For any non-zero vector ${\xi }_h$ in the space $V_0^h$, we have ${\xi }_h={\sum }_{s=1}^q{r}_s{f}_s^h$, where $f_s^h$, $s\in \{1,2,\dots ,q\}$, is a normalized eigenfunction in the eigenspace $W_s^h$, and $r_1,r_2,\dots ,r_q$ are real numbers satisfying ${\sum }_{s=1}^q{r}_s^2={\parallel {\xi }_h\parallel }^2$, where $\parallel \cdot \parallel$ denotes the usual norm of $L^2(0,1)$.

Theorem 3.2 Suppose that $X^h$ is an eigenfunction to the operator $-{\mathrm{△}}_h$, and ω is an open and nonempty subset of $(0,1)$. Then ${\chi }_{\omega }{X}^h\ne 0$.

Now, we will prove the controllability for system (2.1).

Remark 3.4 Since $V_0^h\times V_0^h$ is a finite dimensional space, thus all norms of this space are equivalent, and then we can get that inequality (3.1) implies observability of semi-discrete system (2.2).

Now, noting that $V_h^0$ is an $N_h-1$ dimensional space, we can choose $f_{q+1}^h,\dots ,f_{N_h-1}^h$ which are normalized eigenfunctions of $-{\mathrm{△}}_h$ such that $f_1^h,\dots ,f_q^h,f_{q+1}^h,\dots ,f_{N_h-1}^h$ constitute a complete standard orthogonal basis of $V_h^0$. Let ${\lambda }_1^h,\dots ,{\lambda }_{N_h-1}^h$ be $N_h-1$ corresponding eigenvalues to eigenfunctions $f_1^h,\dots ,f_q^h,f_{q+1}^h,\dots ,f_{N_h-1}^h$.

which leads to a contradiction to the assumption that $w_0^h\ne 0$, and then we can complete the proof of (3.2).

where $r=0,1,2,\dots$ .

In the above second identity, corresponding to each $j\in \{\alpha +1,\dots ,q\}$, we can take s as the number $s(j)$ given in (3.8). Then it follows that ${\xi }_j=0$ for all $j\in \{\alpha +1,\dots ,q\}$. Hence, we prove (3.5) and finish the proof for this theorem. □

is a control which drives the initial data $(y_0^h,y_1^h)$ of controlled system (2.1) to zero in time T. It is easy to see that ${\chi }_{\omega }{\hat{\phi }}_h\in L^2(0,+\mathrm{\infty };{\tilde{V}}^h)$. This completes the proof of this theorem. □