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

?⃗? = (b1, b2, . . . , bn) ∈ W 1,∞ (Ω;R), c ∈ L(Ω), h ∈ L2(Q T ) is the control function, and χ ω is the characteristic function of ω. We note that a(x)may be allowed to vanish at some points on the later boundary ∂Ω, and thus (1) may be degenerate on the set {(x, t) ∈ ∂Ω×(0, T) : a(x) = 0}, 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:

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.

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  ∈  2 (  ).
The weak solution of problem ( 13) is defined as follows.
Definition 1.A function (, ) is called a weak solution of problem ( 13), if  ∈ ([0, ];  2 (Ω))∩B and for any function  ∈  ∞ (  ) ∩ B with / ∈  2 (  ) and (⋅, )| Ω = 0, the following integral equality holds: Here, we use B to denote the closure of the set  ∞ 0 (  ) with respect to the norm As to the set B, we give the following remark whose proof can be found in [19] Corollary 2.1 and Remark 2.1.

Lemma 2. If 𝑢 ∈ B, then
in the trace sense.
Next, we establish the well-posedness for problem (13).
Property (a) follows from Theorem 3, and property (b) can be deduced from the unique continuation of the nondegenerate parabolic equation [20,21].
Proposition 4. (⋅) is a strictly convex and continuous functional defined on  2 (Ω) and satisfies lim inf Furthermore, the functional (⋅) achieves its minimum at a unique point V in  2 (Ω) and Proof.One can easily prove that (⋅) is strictly convex by the linearity of L and the convexity of  2 (Ω) norm.Moreover, the continuity of (⋅) can be derived from Theorem 3 and the continuity of L. Now we prove (41) by contradiction.Otherwise, there exists a sequence For  = 1, 2, . .., we denote Ṽ()  = V ()  /‖V ()  ‖  2 (Ω) .There exist a subsequence of {Ṽ ()   } ∞ =1 , denoted by {Ṽ Then, it follows from Theorem 3 that  (V which contradicts (43) and completes the proof of (41).From (41), we get that lim inf This, together with the strict convexity and the continuity of (⋅), implies that the functional (⋅) achieves its minimum at a unique point in  2 (Ω).