An Optimal Control Problem Governed by a Kirchhoff-Type Variational Inequality

Wang, Chengfu; Wu, Pengcheng; Zhou, Yuying

doi:https://doi.org/10.1155/2020/2831242

Journal of Applied Mathematics

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Abstract Introduction Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 2831242 | https://doi.org/10.1155/2020/2831242

An Optimal Control Problem Governed by a Kirchhoff-Type Variational Inequality

Chengfu Wang,¹Pengcheng Wu,¹and Yuying Zhou¹

Academic Editor: Sheng Zhang

Received15 Sept 2019

Accepted13 Jan 2020

Published18 Feb 2020

Abstract

This paper is concerned with an optimal control problem governed by a Kirchhoff-type variational inequality. The existence of multiplicity solutions for the Kirchhoff-type variational inequality is established by using some nonlinear analysis techniques and the variational method, and the existence results of an optimal control for the optimal control problem governed by a Kirchhoff-type variational inequality are derived.

1. Introduction

Let be a bounded domain of with a smooth boundary , be a nonempty bounded closed and convex subset of the space , be a proper and convex function, and

Let be endowed with the norm , and let and . The objective functional is defined by

In this paper, we will be discussing the following optimal control problem governed by a state variational inequality:where is the solution set of the following Kirchhoff-type variational inequality: for each , find (the state function of the system), such thatwhere , , , and satisfies, h is continuous on ; there exists such that for all ; there exists such that .

A typical example of h is . Then, h satisfies ,and the variational inequality (4) will become to be the usual variational inequality of the Kirchhoff type: for each , find such that

The Kirchhoff Dirichlet problem was first proposed by Kirchhoff by taking into account a differential equation describing the changes in length of the string produced by transverse vibrations for free vibrations of elastic strings. For more details on the physical and mathematical background of Kirchhoff-type problems, we refer the readers to the papers [1–3] and the references therein. By variational methods, many interesting results about the existence multiplicity of solutions for Kirchhoff-type problems have been established in the last ten years, see, e.g., [1–7] and the references therein.

The study of variational inequalities like (4) with and related optimal control problems was proposed by Lions [8–10], and this topic has been widely studied by many authors in different aspects (cf. [11–23]). One of the most important methods is the approximation of the variational inequality by an equation where the maximal monotone operator (in this case, the subdifferential of a Lipschitz function) is approached by a differentiable single-value mapping with Moreau–Yosida approximation techniques. This method, mainly due to Barbu [11], leads to several existence results and to first-order optimality systems. Lou [12] discussed the regularity of an obstacle control problem, wherein the variational inequality is associated to the Laplace operator. Lou [13] considered the existence and regularity of the control problem governed by the quasilinear elliptic variational inequality. Bergounioux and Lenhart [15] studied obstacle optimal control for semilinear and bilateral obstacle problems. Chen et al. [21] studied an optimal control problem for quasilinear elliptic variational inequality. Ye and Chen [16] studied the existence and necessary condition of an optimal control problem for a quasilinear elliptic obstacle variational inequality in which the obstacle was taken as the control and the cost functional were specific. Zhou et al. [17] established the existence of the optimal control for an optimal control problem governed by an abstract variational inequality and obtained the existence of the optimal control for the optimal control problem governed by a quasilinear elliptic variational inequality with an obstacle. By using nonlinear Lagrangian methods, Zhou et al. [18] studied an optimal control problem where the state of the system is defined by a variational inequality problem for monotone-type mappings. Khan and Sama [19] obtained the existence of an optimal control for a quasivariational inequality with multivalued pseudomonotone maps. Chen et al. [20] studied an optimal control problem for a quasilinear elliptic variational inequality with source term, established the existence results, and derived the optimality system for this optimal control problem. In [22], Migorski et al. investigated an inverse problem of identifying the material parameter in an implicit obstacle problem given by an operator of p-Laplacian type. In [23], Khan et al. studied inverse problems of identifying a variable parameter in variational and quasivariational inequalities.

The purpose of the present paper is to investigate the optimal control problem governed by the state Kirchhoff-type variational inequality (4), i.e., the problem in the case of . This case is more complicated since the so-called nonlocal term is involving in the variational inequality. To the best of our knowledge, the study on optimal control problems controlled by the state Kirchhoff-type variational inequality is still lacking in mathematics literatures. Our first intention is to establish the existence and multiplicity of solutions for the inequality (4) by using some nonlinear analysis techniques and the variational method. Then, as an application, we obtain the existence of solutions for the optimal control problem .

The paper is structured as follows. Section 2 contains some basic definitions and preliminary facts needed in the sequel. In Section 3, we shall show that there exist at least two solutions of the Kirchhoff-type variational inequality (4) when some suitable conditions on , and τ are satisfied. In Section 4, we apply the obtained results to study the optimal control problem governed by the state variational inequality (4) and obtain the existence of solutions for the problem .

2. Preliminary

Let X be a Banach space and its dual. The following definitions and theorems can be found in, e.g., [24, 25].

Definition 1. Let X be a Banach space, be a continuously differentiable functional, and let be a proper (i.e., ), convex, and lower semicontinuous functional. The functional is called Szulkin-type functional. is said to satisfy the Palais–Smale condition (the . condition for short), if every sequence with bounded and for which there exists a sequence , such thatcontains a (strongly) convergent subsequence in X.

Definition 2. Let be a Szulkin-type functional. A point is said to be a critical point of ifThe value c of I at a critical point is said to be a critical value of I, that is, .

Theorem 1. Let be a Szulkin-type functional which is bounded below. If I satisfies the condition for , then c is a critical value.

Theorem 2. Let be a Szulkin-type functional and assume that(1) for all for some , and (2)There is with and If I satisfies the condition, then I has a critical point with critical value such thatwhere

Let be a function. We denote by

The hypotheses on the f are the following:

is measurable in x for every and continuous in t for a.e. ;

uniformly for almost all , where

There exists a constant and a function such thatuniformly for almost all ;

There exist , and such thatwhere is a positive constant and .

Let satisfy the conditions . We denote by

Let , , and let us introduce the Euler functional corresponding to the Kirchhoff-type variational inequality problem (4) as

Let be defined by (1). We define the indicator functional of the set by

Denote

Obviously, is a Szulkin-type functional.

Remark 1. For any , if , then h satisfies , andThe variational inequality (4) will become to be the classic Kirchhoff-type variational inequality: find such that

3. Existence of Multiple Solutions for a Kirchhoff-Type Variational Inequality

As usual, we denote “” and “” by the strong and weak convergence in the space .

Proposition 1. Let be a nonempty bounded closed and convex subset of the space . If satisfies , satisfies conditions , and is a weakly continuous mapping. Then, every critical point of is a solution of (4), where and are defined by (16) and (18), respectively.

Proof. It is easy to see that and for each ,If is a critical point of , from Definition 2, we obtainIt follows from (17), (21), and (22) that and (4) holds.

Proposition 2. Let be a nonempty bounded, closed, and convex subset of the space . If satisfies , satisfies conditions , and is a weakly continuous mapping such that is a bounded set. Then, is coercive in the sense of as and bounded from below in , where is defined by (1) and is defined (18).

Proof. By , Hölder inequality, we havewhere and are positive constants.
Since is bounded, there exists , such thatThus,where is a positive constant.
By , we haveLet . By (16), (18), (23), and (25), we haveBy noting that and from (26), we obtain that is coercive and is bounded from below in . The proof is complete.

Proposition 3. Let all conditions in Proposition 2 be satisfied. Then, satisfies the condition in the sense of Definition 1.

Proof. Let such that is bounded, and there exists with andFrom (17), it can be noted that it is easy to obtain . By Proposition 2, is coercive, and then is bounded in . Hence, by the Sobolev embedding theorem, we may assume that going to a subsequence,since .
As is weakly closed, . Setting in (28) and combining with (21), we haveThus,By the Hölder inequality and the condition , for ,Note that (as ), is bounded in and h satisfies ; by (29)–(32), we haveTherefore in .

Theorem 3. Let be a nonempty bounded closed and convex subset of the space . If satisfies , satisfies the conditions , and is a weakly continuous mapping such that is a bounded set. Then, there exists , such that for each , the Kirchhoff-type variational inequality (4) has two solutions , satisfying .

Proof. Noting the conditions and , we haveDenote bywhere is given in . Let us definewhere is given in . Then,where is the volume of unit sphere in , and are defined in . Thus, is a positive constant which follows from (37). In the following, assume that . Therefore, from (25) and the condition , where is a positive constant.
By and (32) and (36) we havewhere is defined in . Note that , it follows from (16), (18), (38), and (39) thatBy (35), we obtain thatThus,Therefore, there exists , such thatBy Proposition 2, the function is bounded from below. By Proposition 3, the function satisfies the condition. Thus, attains its global minimum at some by Theorem 4. Obviously,Next, we will prove the existence of the second critical point of via the Mountain Pass theorem (see Theorem 2).
By the condition for any , there exists , such thatIt follows from that for all ,From (45) and (46), we obtain that for all and a.e. ,By the conditions and , we obtainand then there exists a positive constant such that for all ,Therefore, by (47) and (49), the Hölder inequality and the Sobolev embedding theorem, for all and , we havewhere are positive constants.
DenoteLet and . Note that . It follows from (50) thatObviously, andBy Proposition 3 and Theorem 2, there exists a critical point of We notice that the cannot be trivial becauseBy Proposition 1, we conclude that and are two solutions of the Kirchhoff-type variational inequality (4). It follows from (44) and (54) that . The proof is complete.

By Theorem 3, we have

Theorem 4. Let be a nonempty bounded closed and convex subset of the space . If satisfies and , is a weakly continuous mapping such that is a bounded set. Then, there exist , such that for each , the Kirchhoff-type variational inequality (20) has two solutions , satisfying .

4. Existence of an Optimal Control Governed by a Kirchhoff-Type Variational Inequality

This section is concerned with the existence results of an optimal control for the optimal control problem .

Lemma 1. Suppose satisfies
is measurable in x for every and continuous in t for a.e. ;
There exist positive constants such thatwhere .

Then, the functional defined byis continuous.

Proof. Let be a sequence in , such that as . Since satisfies the conditions and , the operator from to is continuous. Therefore, we haveas . The proof is complete.

Lemma 2. Suppose is and for almost all , is convex with respect to , and there exist positive constants such thatwhere . Then, the functional defined byis weakly lower semicontinuous.

Proof. Let be a sequence in , such that as . Since is and satisfies the conditions and , is Gâteaux differentiable in the space and (the dual space of ). Note that for each , is convex, and thenAs , from the inequality mentioned above by taking limits, we obtainThe proof is complete.

Theorem 5. Let be a nonempty bounded closed and convex subset of the space . Assume that satisfies the conditions , satisfies the conditions , satisfies , a weakly continuous mapping such that is a bounded set, and and satisfy all the conditions in Lemma 1 and Lemma 2, respectively. There exist , such that for each , there exists an optimal control for the optimal control problem .

Proof. From Theorem 3, there exist , such that for each , , where is the solution set of the variational inequality (4).
Let () be a minimizing sequence for problem such thatwhere the cost function is defined by (2).
As , we haveWe claim that is bounded. In fact, as is bounded, is bounded. Let in (63). We getwhere are positive constants. As and h satisfies , is bounded.
Since and are reflexive spaces, there exists a weakly convergent subsequence of . Without loss of generality, we may assume as . Hence, in and in as . Since and are weakly closed sets, and . Then, let in (63). We getSince f satisfies the conditions and τ is bounded,Then, from (65) and (66), we haveSince is bounded and h satisfies ,Note that in as , and we haveBy (68) and (69), we getThus, in as . Therefore, by (63), we havethat is, . DenoteSince in as and the embedding is continuous compact (), in as . By Lemma 1 and Lemma 2, is continuous and is weakly lower semicontinuous. Therefore,That is, is an optimal control for problem . The proof is complete.

Theorem 6. Let be a nonempty bounded closed and convex subset of the space . Assume that h satisfies the conditions , f satisfies the conditions , and is a weakly continuous mapping such that is a bounded set. Let , , and , for some constants and , where . There exist , such that for each , there exists an optimal control for the optimal control problem .

Proof. Firstly, we claim that is weakly lower semicontinuous. In fact,By the weakly lower semicontinuity of the norm, is weakly lower semicontinuous.
Let . Then, in and in . Since , it follows from the Sobolev imbedding theorem that the imbedding is a compact imbedding. Hence, in and in . Since is weakly lower semicontinuous, is weakly lower semicontinuous. Then, similar to the rest of the proof given in Theorem 5, we obtain our conclusion.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors are grateful to the referees for their valuable comments for improving the paper. This research was supported by National Natural Science Foundation of China (11771319 and 11971339).

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