[Retracted] Local Null-Controllability for Some Quasi-Linear Phase-Field Systems with Neumann Boundary Conditions by one Control Force

Shi, Shengzhu; Zhang, Dazhi

doi:https://doi.org/10.1155/2022/7645304

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Volume 2022 | Article ID 7645304 | https://doi.org/10.1155/2022/7645304

[Retracted] Local Null-Controllability for Some Quasi-Linear Phase-Field Systems with Neumann Boundary Conditions by one Control Force

Shengzhu Shi¹and Dazhi Zhang¹

Academic Editor: Wei Zhang

Received11 Nov 2021

Revised15 Dec 2021

Accepted07 Feb 2022

Published07 Jul 2022

Abstract

In this article, we study the local null-controllability for some quasi-linear phase-field systems with homogeneous Neumann boundary conditions and an arbitrary located internal controller in the frame of classical solutions. In order to minimize the number of control forces, we prove the Carleman inequality for the associated linear system. By constructing a sequence of optimal control problems and an iteration method based on the parabolic regularity, we find a qualified control in Hölder space for the linear system. Based on the theory of Kakutani’s fixed point theorem, we prove that the quasi-linear system is local null-controllable when the initial datum is small and smooth enough.

1. Introduction

In this paper, we are concerned with the local null-controllability for some quasi-linear phase-field systems with Neumann boundary conditions by a control force acting on an arbitrary small open set . Here is a connected bounded domain with boundary for some . For a given , we consider the cylindrical domain with lateral boundary , and by , we denote the outward unit normal vector to at a point . Consider the following quasi-linear phase-field system:where , is a constant, and there exists a constant such that in , is the temperature function while is the phase-field function, the initial datum is given in . And is a control function in the space

Besides, we assume that is a function with , is a function with , are functions satisfying and the uniform parabolic condition, i.e., there exist constants such that

In addition, we put

Phase-field systems model a large amount of physical phase transition phenomena. Usually, the phase-field function describes the level of liquid solidification or solid crystallization . In particular, the quasi-linear system (1) describes a kind of phenomenon where heat conduction coefficients depend on the temperature in a manner as in the phase transition process. Different from Dirichlet boundary conditions which mean the phase on the boundary should be maintained, Neumann boundary conditions in system (1) describe that there is no flux of phases at the boundary. We refer to [1, 2] and the references therein for more detailed descriptions about the model.

In recent years, lots of researchers have focused on addressing the controllability problems of linear and semilinear equations. For example, Barbu [3] investigated the local controllability of the phase-field system with two control forces and gave the observability estimate for the adjoint system with the global Carleman inequality established in [4]. The work of [3] illuminates us to expand the results of partial differential equations (a variety of results have been established, see for instance [5–10]) to coupled systems [11, 12] which model real physical or biological phenomena. In addition, controlling a system with a minimum number of forces is a common problem in the control theory [13, 14]. From this point of view, Ammar Khodja etc. gave the Carleman inequality for the adjoint system by constructing a suitable functional with suitable weights (see Lemma 3.4 in [13]). It is worth noticing that their results are based on the Carleman inequality given in [15] which makes it possible to improve the regularity of the control function. This method has also been applied to solve the null-controllability for some general reaction-diffusion systems which arise in mathematical biology in [16]. Observing that [13] considered the system on a sub-domain of with , González-Burgos and Pérez-García [17] considered a more general case with an arbitrary and the nonlinearity . They used a method called strategy of fictitious control functions to construct a control function , . Furthermore, [18] studied the case of unbounded domains. In addition, [11] studied a phase-field system with Neumann boundary conditions and showed the relationship between the optimal control and the controllability.

However, few work has been done to address the controllability of the quasi-linear form such as (1). The main problem is that the generalized solution of the quasi-linear system is not good enough, which makes the fixed point theorem not applicable. To overcome this difficulty, we would like to recall some work on the quasi-linear parabolic equation. To our knowledge, [19] proved the local null-controllability of a quasi-linear diffusion equation in one spacial dimension with the Sobolev embedding relation which is only valid for one spacial dimension. And for the multidimensional case, [20] analyzed the controllability in the frame of classical solutions and gave a control in the Hölder space with given initial datum of high regularity and small enough. Enlightened by [19, 20], we also investigate the problem in the frame of classical solutions, while the coupled systems make the case more complex.

As for boundary conditions, it is necessary to notice that different boundary conditions describe different physical phenomena, however, there is little difference on mathematic controllability discussion. Thanks to the study in [21], we can find that the Carlemen inequality with Neumann boundary conditions has an additional weight function compared to the one with Dirichlet boundary conditions. Since there are abundant discussions on the case with homogeneous Dirichlet boundary conditions, we pay attention to the system with homogeneous Neumann boundary conditions. Moreover, all of our results established in this paper are valid for homogeneous Dirichlt boundary conditions.

As introduced in [22], in the general case, the observability for the adjoint system implies the controllability for the original system. To prove this, we pay attention to the variational method introduced in [4, 21]. We extend the method in [4] from the case of the parabolic equation to the case of coupled systems. Moreover, our adjoint system is different from the usual one because of the different optimal control problem.

The paper is organized as follows. Section 2 introduces a list of notations and presents preliminaries. In Section 3, we establish a Carleman inequality for the phase-field system which plays a key role in minimizing the number of control forces. In Section 4, we give an observability estimate for the adjoint system. In Section 5, we obtain a qualified control in by constructing a sequence of optimal control problems. Then we prove the control function belongs to by an iteration method based on the parabolic regularity and the embedding theorem. Finally, by utilizing the Kakutani’s fixed point theorem, we prove that the quasi-linear system is local null-controllable when the initial datum is small and smooth enough in Section 6.

2. Preliminaries

In this section, we present some preliminaries. We first introduce some results on the existence, uniqueness and regularity of the solution for the associated linear phase-field systemwhere is a control function to be determined, in , satisfy and the uniform parabolic condition, i.e., there exist constants such thatand . In what follows, we denote

Since is a constant, without loss of generality, we assume in the following demonstration.

We use standard notations , , , in Sobolev space for the estimate and in Hölder space for the Schauder estimate (for more information, one can see [23]). Moreover, we also need the following Hilbert spaces in [11].and the compatibility condition in [24, 25].

In the sequel, the symbols , stand for various positive constants depending on different parameters and the values may change.

By analogy with the proof of Theorem 3.1.2 and Theorem 3.4.2 in [23], we have the following lemma.

Lemma 1. If,,satisfy and the uniform parabolic condition (6),, there exists a unique weak solutionof the system (5) satisfying the estimatewhereandis given in (7).

By recalling the theory of linear parabolic equations of second order (see, for instance, Theorem 6.4 in [26], Theorem 9.2.2 and Theorem 9.2.4 in [23]), we give the result on the estimate.

Lemma 2. For any,,,, satisfy and the uniform parabolic condition (6),, there exists a unique strong solution of the system (5) satisfying the estimatewhereandis given in (7).

If we improve the regularity of and the initial datum in Lemma 2, we can obtain the following Schauder estimate.

Lemma 3. Letbe an arbitrary fixed constant,satisfyand the uniform parabolic condition (6),. If, satisfy the compatibility condition (9), there exists a unique classical solutionof the system (5) satisfying the estimatewhereandis given in (7).

The proof of this Lemma is a standard consequence of the classical Schauder estimate theory. For more information, one can see Theorem 4.2, Theorem 5.1 in [26], Theorem 7.2.24 in [23] and Theorem 3.4 in [24]. In addition, the usual Schauder estimate always includes a maximum norm of the solution on the right-hand side. We use the Moser iteration ([23]) and the result in Lemma 1 to estimate the maximum norm.

Finally, we introduce the embedding theorem of the space introduced in [20, 27].

Lemma 4. Let N be a positive integer and, then the following continuous embedding holds:(1)If , then , where .(2)If , then for any .(3)If is not an integer, then .

3. Carleman Inequality

In this section, we give a Carleman inequality for the adjoint system associated to the linear phase-filed system (5) based on the strategy developed in [21]. We need to notice that, unlike the usual form, the adjoint system we construct is not homogeneous, which is necessary because of the different optimal control problem we give in Section 5.

Let us consider the following adjoint system associated to (5)

Following [21], let us introduce some weight functions with parameter as follows:where and , the function satisfieswith and = .

For simplicity, we define a family of weight functions with parameter and defined in (14). With above weight functions, we have a modified Carleman inequality for Neumann boundary value problems such aswhere , and the coefficients fulfill conditions (6) and (7).

Lemma 5. There exists a constantsuch that for anythere exists a constantsuch that for every, the solution of (16) satisfies the inequalitywhere , , the constant depends continuously on , the constant depends continuously on and the positive constant is independent of .

Remark 1. In [21], Lemma 1.2 gives a detailed proof of the Carleman inequality (17) with . However, it can not be applied to our case directly because of the term (see the proof of Lemma 6). Replacing the auxiliary functionsintroduced in [21] withand following the proof procedure of Lemma 1.2 in [21], one can prove Lemma 5. The method of constructing an auxiliary function is a usual technique in proving Carleman inequality (see, for instance, [15, Theorem 7.1, p.288]).
Applying Lemma 5 to the adjoint system (13), we get the following lemma.

Lemma 6. Let,be the constants given in Lemma 5. Then, for everyandthe solution of the adjoint system (13) satisfies the estimate

Proof. We apply Lemma 5 to the first equation of the system (13) with , . This implies that there existssuch thatAnd as a consequence, we haveSimilarly, we apply Lemma 5 to the second equation of the system (13) with , . Takingwe getTherefore, we deduce the result by choosing . □
To prove the main result of this section, we need the following estimates of , which will play a crucial role in our following demonstrations.

Lemma 7. For any,, by the definition of, we have, therefore we haveandwhere , .

Proof. For and given in (15), we haveprovided byAgain, in the same way, we getby observing andThus, combined with (28) and (30), we getby virtue ofFinally, noting that , we finish the proof of Lemma 7. □
There are plenty of methods to get the null-controllability by one control force. The method developed in [13] is to estimate the term by the term for any , which is deduced by constructing a weight function . And in [17], the authors used a different weight . Whichever the method is, the computation is complex but necessary. Enlightened by above two methods and the properties of weight functions in Lemma 7, we prove the Carleman inequality for the adjoint system (13) as the following proposition.

Proposition 1. Let,be the constants given in Lemma 5. Then, for every,given in Lemma 6, the solutionof the adjoint system (13) satisfies the estimate

Proof. Let us begin with introducing a truncation function satisfyingwhere with satisfying (35), and is a constant depending on and .
Observe that and integrate in . Integrating by parts, recalling the boundary conditions in (13) and replacing , by their expressions given in (13), we see thatIn order to estimate the first term on the right-estimate the first term on the right-hand side, we need the Young inequality with , i.e.,We will use the Young inequality with different coefficients for many times in the following estimates, and for simplicity, we do not point it out each time. Since the coefficients satisfy the condition (7), we haveby virtue of and . We deduce from the uniform parabolic condition (6) thatSince in , it appears thatIntegrating by parts again, we rewrite asIt follows from the estimates in Lemma 7 and (35) thatThus the above inequality with yieldsUsing similar arguments, we can obtainBy analogy with (42), we getWe deduce from (45) thatHence, by virtue of above estimates, we conclude thatSimilar to the estimate of , we can prove the estimateIt is easy to check thatProceeding as previously, we haveWe bound with the above estimate for in Lemma 7 as followsIt appears thatandBy virtue of above estimates for and the condition in , we have the estimate for (36) as followsOur following goal is to get rid of on the right-hand side of (54). By analogy with above discussions, we integrate in and integrate by parts, then it appears thatTaking into account the estimate on and observing the estimate for in (42) with , we obtainIt is obvious thatAnd it follows thatFurthermore, by virtue of the estimate for and Lemma 7, the following result holdsUtilizing Young inequality again with , we getThus, from the estimates for , we can rewrite (55) asby choosing .
Combining (54) and (61) and Lemma 6, we finish the proof of this part by choosing , , and . □

4. Observability Estimate

This section addresses the study of the observability estimate for (13). Notice that, unlike the general case, the adjoint system (13) is not homogeneous and the observability estimate with can not deduce the null controllability for the linear system (5). To obtain the qualified observability estimate, we need to construct a new sequence of weight functionswherewith the parameters given in Section 3.

Then, we have the following lemma.

Lemma 8. For any, we havewhere and .

Proof. By analogy with the proof of Lemmas 1 and 2 in [6], we denote the auxiliary function with the variable as followswhere , . Then, it can be verified that and is monotone increasing on . Therefore by choosing , we can obtained thatIn addition, we notice that for all , thus for , we haveand the other results can be obtained in the same way. □
Thus, with above estimates, the following result holds.

Proposition 2. Letbe the constant given in Lemma 5andbe the constant given in Lemma 6. Then, for every,, the solutionof the adjoint system (6) satisfies the observability estimate

Proof. Recalling (13) and integrating in , we can easily prove the estimatewhere is a constant to be fixed. By virtue of Young inequality with , we getAnd by virtue ofwe rewrite (69) aswhere and is given in (7). Hence we getfor all . For simplicity, we set . By the second estimate in Lemma 8 and the standard energy estimate (see, for instance, [28]), we haveIt follows from the first inequality in Lemma 8 and Proposition 1 thatThus, by virtue of (74) and (75) and , we obtain the following estimatewhich ends the proof of the proposition with (71).

5. Null-Controllability for the Linear System

In this section, we consider the null-controllability for the linear system (5). Firstly, we find a control function with such thatby constructing a sequence of optimal control problems. Then, we prove that the control function belongs to based on the parabolic regularity. Finally, by utilizing an iteration method and the embedding theorem, we find that the control function belongs to the space .

Proposition 3. Letbe the constant given inLemma5andbe the constant given inLemma6. For any,,, there exists a controlwithsuch that the corresponding solutionof (5) satisfies

Moreover,

Proof. To pave a way for further numerical approximation, we consider the following optimal control problem, which is similar to that introduced by Fursikov and Imanuvilov [4] and used in [21, 29, 30]subject toand the terminal constraintsin Banach space , where the parameters , are fixed, with , defined in (14) andAccording to the classical theory about optimal control problem (see [31]), since the control function are imposed on the whole domain , it is easy to prove that, for any , the optimal control problem (80)–(82) has a unique solution, which we denote by . Applying the Lagrange principle to the optimal control problem, we get the following necessary optimality condition (see [31, 32]), i.e., for each , there exists co-state function satisfying the homogeneous boundary conditions in (13) such thatNoting thatand taking into account (84), we obtainRecalling Proposition 2 and the definition of , we getby virtue of , which can be proved by analogy with Lemma 8 by notingHence, it follows from (86) and (87) thatBy virtue of (89), we have a subsequence such thatMoreover, by passing to the limit in (84), we obtain that the pair is a solution of (5) satisfyingwith the control function and is the characteristic function of . Finally, passing limit in (89) and taking into account the third inequality in Lemma 8, we obtain the estimate (79). □
Our following goal is to prove by Lemma 3 (embedding theorem) and the parabolic regularity.

Proposition 4. Letbe the constant given inLemma5andbe the constant given in Lemma 6. For anysatisfying the compatibility condition (9),and(is a fixed finite positive integer), there exists a controlsuch that the corresponding solution of (5) satisfies

Moreover,

Proof. Let , , , and define the following weight functionswhere are defined in (14), are defined in (63) and . Then, with given in Proposition 3, we denoteBy analogy with Lemma 8, it can be verified thatIn addition, for each , , given in Proposition 3, by virtue of Proposition 1, we haveThus, for , passing to the limit in (84) and (97), we see thatPassing to the limit in (89) and estimating in the same way, we arrive atfor . Moreover, similar to the result in Lemma 1, we getBy Lemma 4 with , we have for which impliesOn the other hand, we find thatBy analogy with above estimates, we can arrive atFor , It is easy to prove thatAs noted in the proof of Proposition 4.1 in [20], there exists such that and . Thus, processing as previously and taking into account above estimates, we have with and by the embedding theory and the parabolic regularity.

6. Local Null-Controllability for the Quasi-Linear System

In this section, we show the local null-controllability of the quasi-linear phase-field system (1) by above propositions and the fixed point theorem.

As classical discussion shows (we refer to [33] for more detailed information), we can write

Denote ,and set the following nonempty convex

It is easy to verify that is a compact subset of with small initial datum. For any , consider the following linearized system of (1)

Then, by setting , , and , satisfy the condition in Section 2 related to the linear system (5). By Proposition 4, we can prove that (109) is local null-controllable and has the cost estimate

Combining the Kakutani’s fixed point theorem, we have the following theorem.

Theorem 1. Letandbe the constants given inProposition4. Assume that for every given initial datumsatisfying the compatibility condition (9), there exists a positive constantsuch that. Then, for everyand, one can find a controlsuch that the corresponding solutionof (1) satisfies

Moreover, we have the cost estimate

Proof. For any , putThis defines a map with small enough. Further, for any , is a nonempty convex and compact subset of provided by Lemma 3. Also, is upper semi-continuous. Therefore, by the Kakutani’s fixed point theorem, there exists such that and this ends the proof of the theorem.

7. Conclusion

In this paper, we derive the the local null-controllability for some quasi-linear phase-field systems with homogeneous Neumann boundary conditions and an arbitrary located internal controller under the frame of classical solutions. We first develop the corresponding Carleman inequality and obtain the observability estimation. Then, we derive the null-controllability for the linear system and the get the desired control function by constructing a sequence of optimal control problems. Finally, by the Kakutani’s fixed point theorem, we have the local null-controllability for the quasi-linear system.

Data Availability

No data were used to support this study.

Disclosure

No potential conflict of interest was reported by the authors.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This paper is supported by the Guangdong Basic and Applied Basic Research Foundation (2020B1515310006).

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Copyright © 2022 Shengzhu Shi and Dazhi Zhang. 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.

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