Journal of Function Spaces
Volume 2015 (2015), Article ID 151484, 7 pages
http://dx.doi.org/10.1155/2015/151484

Research Article

Approximate Controllability of the Degenerate System with the First-Order Term

Runmei Du

School of Basic Science, Changchun University of Technology, Changchun 130012, China

Received 30 May 2015; Revised 15 July 2015; Accepted 22 July 2015

Academic Editor: Hugo Leiva

Copyright © 2015 Runmei Du. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We consider the approximate controllability of the degenerate system with the first-order term. The first-order term in the equation cannot be controlled by the diffusion term. The system is shown to be approximately controllable by constructing a control by means of its conjugate problem.

1. Introduction

In this paper, we investigate the approximate controllability of the equation

where is a bounded domain of , is an open and nonempty subset of , and is positive in , , , is the control function, and is the characteristic function of . We note that may be allowed to vanish at some points on the later boundary , and thus (1) may be degenerate on the set , a portion of the lateral boundary.

In recent years, various controllability problems for linear and nonlinear differential equations have been considered. There are a great number of results on constrained controllability (see [1–3] and the references therein) and unconstrained controllability (see [4–6] and the references therein). Among these, some authors have investigated the null controllability of one-dimensional linear and semilinear equations with boundary degeneracy. In particular, the null controllability of the following degenerate equation is considered:

where , . Equation (2) may be used to describe some physical models (see [7, 8] and the references therein). In [7–14], the degeneracy of (2) is divided into weak one and strong one according to the value of , and different boundary conditions are proposed for the two cases. More precise, the boundary value condition is

in the weakly degenerate case with , while it is

in the strongly degenerate case with . On the other hand, the following initial value condition is proposed for both cases:

Then, system (2), (3) or (4), (5) is null controllable if [7–9, 12, 14], while it is not if [13]. However, the first-order term in (2) is controlled by the diffusion term. Further, the equation

has been investigated. Different from (2), the convection term in (6) cannot be controlled by the diffusion term. In [15], the authors studied the null controllability for system (6), (3), and (5) only in the case .

Since the above one-dimensional degenerate systems may be not null controllable, a natural question is whether the systems are approximately controllable. More generally, the multidimensional degenerate systems have been investigated. In [16], the authors considered the equation

Similar to [16], the lateral boundary is decomposed into three parts: the nondegenerate boundary , the weakly degenerate boundary , and the strongly degenerate boundary . The boundary value condition is prescribed on ; namely,

Then the authors proved the approximate controllability of system (7), (8) and the initial value condition

where .

In the present paper, we assume

which means that the strongly degenerate boundary is empty. For example, if , , , then (10) implies (the nondegenerate case) or (the weakly degenerate case). In this case, we consider (1) subject to

where . In the present paper, our method is similar to [5, 16–18]. The control is constructed via the conjugate problem.

The paper is organized as follows. In Section 2, we establish the well-posedness of system (1), (11), and (12) under condition (10). The approximate controllability of the system is proved in Section 3 subsequently.

2. The Well-Posedness of the Problem

In this section, we establish the well-posedness of problem (1), (11), and (12) in case (10). More generally, let us consider the problem

where .

The weak solution of problem (13) is defined as follows.

Definition 1. A function is called a weak solution of problem (13), if and for any function with and , the following integral equality holds:

Here, we use to denote the closure of the set with respect to the norm

As to the set , we give the following remark whose proof can be found in [19] Corollary and Remark .

Lemma 2. If , then

in the trace sense.

Next, we establish the well-posedness for problem (13).

Theorem 3. For any and , problem (13) admits a unique weak solution satisfying

where is a constant depending only on , , and . Moreover, if and , then and .

Proof. First, we prove the existence. For any , we take such that

Consider the problem

According to the classical theory on parabolic equations, problem (19)–(21) admits a unique weak solution .
Multiply (19) by and then integrate over to get

Using the Hölder inequality and the Grönwall inequality, we can get

where is a constant depending only on , , and . From estimate (23), there exist a subsequence of , denoted by itself, and a function , such that

Since is the weak solution of problem (19)–(21), the following integral equality holds for any function with and :

Note that due to (10). Let in (25) to get

which means that is the weak solution of problem (13).
Moreover, if and , the maximum principle yields

where is independent of . Denote . Then satisfies

From (23), we have

where is independent of . It follows from (27) and (29) that there exists a subsequence of , denoted by itself, such that

Thus, and .
Finally, we prove the uniqueness by the Holmgren method. Let and be two weak solutions of problem (13) and denote

Then and for any function with and , the following integral equality holds:

For any , the above existence result shows that the problem

admits a weak solution with . Taking in (32), we get

This leads to

owing to the arbitrariness of . Therefore,

namely, the weak solution of problem (13) is unique. The proof is complete.

3. Approximate Controllability of the Control System

In this section, we investigate the approximate controllability of control system (1), (11), and (12).

First, we consider control system (1), (11) with null initial data; namely,

The study on the approximate controllability of the control system is related to its conjugate problem

Define a mapping

where is the weak solution of conjugate problem (38). Then the mapping satisfies the following:

(a) is a continuous linear operator from to ;(b)if , then it holds

Property (a) follows from Theorem 3, and property (b) can be deduced from the unique continuation of the nondegenerate parabolic equation [20, 21].

Fix and . For , we introduce a functional

For this functional, we have the following proposition.

Proposition 4. is a strictly convex and continuous functional defined on and satisfies

Furthermore, the functional achieves its minimum at a unique point in and

Proof. One can easily prove that is strictly convex by the linearity of and the convexity of norm. Moreover, the continuity of can be derived from Theorem 3 and the continuity of .
Now we prove (41) by contradiction. Otherwise, there exists a sequence satisfying

For , we denote There exist a subsequence of , denoted by , and , such that and

Then, it follows from Theorem 3 that

where and are the weak solution of conjugate problem (38) with and , respectively. Additionally, (43) yields

Hence

This and (b) lead to in and thus in . Therefore,

which contradicts (43) and completes the proof of (41).
From (41), we get that

This, together with the strict convexity and the continuity of , implies that the functional achieves its minimum at a unique point in .
Finally, we prove (42). On the one hand, if , it follows from the Hölder inequality that

and thus . On the other hand, if , then

that is,

Letting yields . The proof is complete.

For the functional , we have the following lemma (Proposition [16]).

Lemma 5. For any with ,

Here we say , if when , while when .

Now we can get the approximate controllability of control system (1), (11), and (12) with null initial data by Proposition 4 and Lemma 5.

Theorem 6. Problem (1), (11), and (37) is approximate controllability. That is to say, for any given number and function , there exists a control , such that

Proof. Since we can take to get (54) when , we only consider the case .
From Proposition 4, there exists with realizing the minimum of . Note that is subdifferentiable at . Therefore, . By Lemma 5, for any , there exists such that

where and are the weak solutions of conjugate problem (38) with and , respectively. Here, satisfying

From the definition of the weak solution to problem (1), (11), and (37) with , we have

On the other hand, since is the weak solution to problem (38) with , we obtain

Combining (57), (58), and (55) yields

Let to get

which implies (54) owing to the arbitrariness of . The proof is complete.

Finally, we prove the approximate controllability of control system (1), (11), and (12).

Theorem 7. Control system (1), (11), and (12) is approximately controllable. That is to say, for any given initial datum , the desired datum , and the admissible error value , there exists a control such that the weak solution of problem (1), (11), and (12) satisfies

Proof. It follows from Theorem 6 that there exists a control such that the weak solution to the problem

satisfies

where is the weak solution to the problem

Then we can get (61) from (63) and the uniqueness result in Theorem 3. The proof is complete.

From the proof of Theorems 6 and 7, it is worthy to note the following.

Remark 8. The controls obtained in Theorems 6 and 7 are both quasi bang-bang controls.

Conflict of Interests

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

Acknowledgments

The author sincerely thanks the referees for their comments that improved this paper. This work was supported by the National Natural Science Foundation of China (Grants no. 11401049).

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