Approximate Controllability of the Degenerate System with the First-Order Term
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.
In this paper, we investigate the approximate controllability of the equationwhere 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 isin the weakly degenerate case with , while it isin 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 . However, the first-order term in (2) is controlled by the diffusion term. Further, the equationhas been investigated. Different from (2), the convection term in (6) cannot be controlled by the diffusion term. In , 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 , the authors considered the equation Similar to , 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 conditionwhere .
In the present paper, we assumewhich 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
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  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 satisfyingwhere 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 problemAccording 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 getwhere is a constant depending only on , , and . From estimate (23), there exist a subsequence of , denoted by itself, and a function , such thatSince 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 yieldswhere is independent of . Denote . Then satisfiesFrom (23), we havewhere 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
The study on the approximate controllability of the control system is related to its conjugate problemDefine 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
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 satisfiesFurthermore, 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 satisfyingFor , 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 ).
Lemma 5. For any with , Here we say , if when , while when .
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 thatwhere 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 haveOn the other hand, since is the weak solution to problem (38) with , we obtainCombining (57), (58), and (55) yields Let to get which implies (54) owing to the arbitrariness of . The proof is complete.
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 satisfieswhere is the weak solution to the problemThen we can get (61) from (63) and the uniqueness result in Theorem 3. The proof is complete.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
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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