Exact Null Controllability of String Equations with Neumann Boundaries

Tis article focuses on the exact null controllability of a one-dimensional wave equation in noncylindrical domains. Both the fxed endpoint and the moving endpoint are Neumann-type boundary conditions. Te control is put on the moving endpoint. When the speed of the moving endpoint is less than the characteristic speed, we can obtain the exact null controllability of this equation by using the Hilbert uniqueness method. In addition, we get a sharper estimate on controllability time that depends on the speed of the moving endpoint.

Consider the motion of a string with one endpoint fxed and the other moving.It can be described by the following wave equation in the noncylindrical domain  Q k T : on(0, T), u(x, 0) � u 0 (x), u t (x, 0) � u 1 (x), in(0, 1), where v is the control variable, u is the state variable, and (u 0 , u 1 ) ∈ L 2 (0, 1) × [V(0, 1)] ′ is any given initial value.Te constant k is called the speed of the moving endpoint.Using the similar method in [1,2], system (4) has a unique solution in the sense of a transposition. ( Control problems can be found everywhere in science, technology, and engineering practice.Fixed-time control has been used in areas such as multiagent systems (MASs), path following in autonomous vehicles, nonlinear parameterisation, nonholonomic systems, and robotic systems (for details, see [3]).In the physical sense, the application of exact controllability of wave equations in noncylindrical domains is also very extensive.A classical example is the interface of an ice-water mixture when temperature rises.Terefore, it is very necessary to study exact controllability of such wave equations.
Te main purpose of this article is to consider the exact null controllability of (4).For the controllability problem of wave equations in cylindrical domains, it has already been studied by diferent authors.However, not much work has been done on the wave equations defned in noncylindrical domains.We refer to [1,[4][5][6][7][8][9][10][11] for some known results in this respect.In [1,4], the exact controllability of a wave equation in a certain noncylindrical domain was studied.In [5], a globally distributed control was obtained by stabilization of the wave equation in a noncylindrical domain.In [6,[8][9][10][11], the exact Dirichlet boundary controllability of the following systems was discussed: and Reference [8] improved the exact controllability time of [6].Reference [7] dealt with the exact controllability of a one-dimensional wave equation with mixed boundary conditions, in which a noncylindrical domain is transformed into a cylindrical domain.Te system is as follows: on(0, T), u(y, 0) � u 0 (y), u t (y, 0) � u 1 (y), in(0, 1).
In this article, we consider the exact null controllability of the wave equation with Neumann-type boundary conditions by taking a direct calculation in a noncylindrical domain when k ∈ (0, � 3 √ /2).But it is still the open problem and we need to overcome in the future when k ∈ ( � 3 √ /2, 1).Tis paper is organized as follows.In Section 2, we give some defnitions and main theorems.In Section 3, we obtain two key inequalities by using the multiplier method used in Section 4. In Section 4, using the Hilbert uniqueness method, we give the proof of exact null controllability of (4).

Preliminary Work and Main Results
Te goal of this paper is to study exact null controllability of (4) in the following sense.Defnition 1. Equation ( 4) is called to be null controllable at the time T, if for any given initial value one can always fnd a control v ∈ [H 1 (0, T)] ′ such that the corresponding solution u of (4) in the sense of a transposition satisfes Defnition 2. Equation ( 4) is called to be exactly controllable at the time T, if for any given initial value and any target function one can always fnd a control v 1 ∈ [H 1 (0, T)] ′ such that the corresponding solution u of (4) in the sense of a transposition satisfes Remark 3. Null controllability of ( 4) is equivalent to exact controllability of (4).
Troughout this article, we set for the controllability time.Te specifc proof will be given later in this paper.

Remark 4. It is easy to verify
Te time T 0 � 2 is in accordance with the controllability time obtained in [12].
Te main results of this paper are the following theorems.

Theorem 5. For any given T > T *
k , equation ( 4) is exactly null controllable at time T in the sense of Defnition 1.
Te key to Proof of Teorem 5 is two important inequalities for the following homogeneous wave equation in the noncylindrical domain  Q k T : Journal of Mathematics where (z 0 , z 1 ) ∈ V(0, 1) × L 2 (0, 1) is any given initial value.
By [1], we know that (16) has a unique weak solution.
We have the following two important inequalities.Te proof of the two important inequalities is given in Section 3. Theorem 6.Let T > T * k .For any (z 0 , z 1 ) ∈ V(0, 1) × L 2 (0, 1), there exists a constant C > 0 such that the corresponding solution z of ( 16) satisfes . (18) Remark 7. In fact, for a more general function α k (t), where 0 /2, we can obtain the same results as in this paper.
Remark 8. We denote by C a positive constant depending only on T and k, which may be diferent from one place to another.

Observability: Proof of Theorem 6
In this section, in order to prove Teorem 6, we need the following lemmas.
We defne the following weighted energy function for (16): where z is the solution of ( 16).It follows that

□ Proof of Teorem 6
Step 1.In the following, we give the proof of the frst inequality in (18).

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From (21), we fnd that From this, one concludes that If T > T * k , we have 2[(T − α k (T)) − 1] > 0, and from this inequality and (37), it holds that . ( Tis completes the proof of the frst inequality in (18).
Step 2. In the following, we give the proof of the second inequality in (18).From ( 28), ( 29), (33), and (34), one concludes that Tis implies that one can fnd a positive constant C such that . (40) Tis completes the proof of the second inequality in (18).By (38) and (40), we get the desired results in Teorem 6.

Controllability: Proof of Theorem 5
In this section, we prove the exact null controllability for wave (4) in the noncylindrical domain ) by the Hilbert uniqueness method.
Proof of Teorem 5. We divide the Proof of Teorem 5 into three steps.
It is worth noting that here G z(α k (t),t) is defned as follows: By [1], we know that (41) has a unique weak solution ξ in the sense of a transposition.We set (43)

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Now, we defne the operator Terefore, Step 2. Multiplying the frst equation of (41) by z(x, t) and integrating on Using the following conditions, By Teorem 6, Γ is bounded and coercive.Hence, by the Lax-Milgram theorem, we can conclude that Γ is an isomorphism.
Hence, we obtain exact null controllability of (4).□ 6 Journal of Mathematics

Conclusion
In this paper, we focus on exact null controllability of a onedimensional wave equation in noncylindrical domains.Both the fxed endpoint and the moving endpoint are Neumanntype boundary conditions.When the speed of the moving endpoint is less than the characteristic speed, we obtain exact null controllability of this equation by using the Hilbert uniqueness method.In the future, we will consider the case of more complex wave equations, such as variable coefcient wave equations.