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

*is paper is concerned with an optimal control problem governed by a Kirchhoff-type variational inequality. *e 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 Kirchhofftype variational inequality are derived.


Introduction
Let Ω be a bounded domain of R N with a smooth boundary zΩ, U be a nonempty bounded closed and convex subset of the space L q (Ω)(1 < q < min 4, 2 * { }, 2 * � (2N/N − 2)), ψ: Ω ⟶ [0, +∞) be a proper and convex function, and Let H 1 0 (Ω) be endowed with the norm ‖·‖, and let g: Ω × R ⟶ R and l: Ω × R ⟶ R. e objective functional J: U × K ψ ⟶ R is defined by (2) In this paper, we will be discussing the following optimal control problem governed by a state variational inequality: where S(w) is the solution set of the following Kirchhofftype variational inequality: for each w ∈ U, find u � u(w) ∈ K ψ (the state function of the system), such that where λ > 0, μ > 0, f(x, t): Ω × R ⟶ R, τ: U ⟶ L q′ (Ω)(q ′ � (q/q − 1)) and h: R ⟶ R satisfies, (h 1 ) h is continuous on R; (h 2 ) there exists a > 0 such that h(t) > a for all t ∈ R; (h 3 ) there exists b > 0 such that lim t⟶+∞ h(t)/t 2 � b.
A typical example of h is h(t) � a + bt 2 . en, h satisfies (h 1 ) − (h 3 ), and the variational inequality (4) will become to be the usual variational inequality of the Kirchhoff type: for each w ∈ U, find u � u(w) ∈ K ψ , such that e 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][2][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][2][3][4][5][6][7] and the references therein.
e study of variational inequalities like (4) with b � 0 and related optimal control problems was proposed by Lions [8][9][10], and this topic has been widely studied by many authors in different aspects (cf. [11][12][13][14][15][16][17][18][19][20][21][22][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 singlevalue mapping with Moreau-Yosida approximation techniques. is 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 monotonetype 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. e purpose of the present paper is to investigate the optimal control problem governed by the state Kirchhofftype variational inequality (4), i.e., the problem (P) in the case of b > 0. is case is more complicated since the socalled nonlocal term b Ω |∇u| 2 dx Δu 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.
en, as an application, we obtain the existence of solutions for the optimal control problem (P). e 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 f, h, 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 (P).

Preliminary
Let X be a Banach space and X * its dual. e following definitions and theorems can be found in, e.g., [24,25]. Definition 1. Let X be a Banach space, φ: X ⟶ R be a continuously differentiable functional, and let ψ: X ⟶ R ∪ +∞ { } be a proper (i.e., ≠ +∞), convex, and lower semicontinuous functional. e functional φ + ψ: X ⟶ R ∪ +∞ { } is called Szulkin-type functional. φ + ψ is said to satisfy the Palais-Smale condition (the (PS). condition for short), if every sequence u n ⊂ X with φ(u n ) + ψ(u n ) bounded and for which there exists a sequence ε n ⊂ R + , ε n ↘0, such that contains a (strongly) convergent subsequence in X.
e value c of I at a critical point u ∈ X is said to be a critical value of I, that is, I(u) � c.
Journal of Applied Mathematics } be a Szulkintype functional and assume that If I satisfies the (PS) condition, then I has a critical point where Let f(x, t): Ω × R ⟶ R be a function. We denote by e hypotheses on the f are the following: (f 3 ) ere exists a constant c > max 2, q and a function uniformly for almost all x ∈ Ω; (f 4 ) ere exist x 0 ∈ Ω, u 0 ∈ R + and δ > 0 such that where η 0 is a positive constant and U( Let λ > 0, μ > 0, and let us introduce the Euler functional φ λ,μ : H 1 0 (Ω) ⟶ R corresponding to the Kirchhoff-type variational inequality problem (4) as Let K ψ be defined by (1). We define the indicator functional of the set K ψ by Obviously, I λ,μ is a Szulkin-type functional.
e variational inequality (4) will become to be the classic Kirchhoff-type variational inequality: find u � u(w) ∈ K ψ , such that

Existence of Multiple Solutions for a Kirchhoff-Type Variational Inequality
As usual, we denote " ⟶ " and "⇀" by the strong and weak convergence in the space H 1 0 (Ω).

Proposition 2. Let U be a nonempty bounded, closed, and convex subset of the space
is a weakly continuous mapping such that τ(U) is a bounded set. en, I λ,μ is coercive in the sense of I λ,μ (u) ⟶ +∞ as ‖u‖ ⟶ +∞ and bounded from below in K ψ , where K ψ is defined by (1) and I λ,μ is defined (18).
Proof. By (f 2 ), Hölder inequality, we have where c 1 and c 2 are positive constants.
Since τ(U) is bounded, there exists ρ > 0, such that us, where c 3 is a positive constant.
Proof. Noting the conditions (f 3 ) and (f 4 ), we have Denote by where η 0 is given in (f 4 ). Let us define 4 Journal of Applied Mathematics where x 0 is given in (f 4 ). en, where ω N is the volume of unit sphere in R N , and δ, u 0 are defined in (f 4 ). us, ‖u t 0 ‖ is a positive constant which follows from (37). In the following, assume that 0 < μ < 1.
us, I λ,μ attains its global minimum at some u λ,μ ∈ H 1 0 (Ω) by eorem 4. Obviously, Next, we will prove the existence of the second critical point of I λ,μ via the Mountain Pass theorem (see eorem 2).
By the condition (f 3 ) for any ε > 0, there exists t * > 0, such that It follows from (f 2 ) that for all |t| > t * , From (45) and (46), we obtain that for all t ∈ R and a.e. x ∈ Ω, By the conditions (h 1 ) and (h 2 ), we obtain and then there exists a positive constant t such that for all t ∈ (0, t), erefore, by (47) and (49), the Hölder inequality and the Sobolev embedding theorem, for all λ > λ * and where c 4 and c 5 are positive constants. Denote Let λ > λ * and μ < μ * . Note that c > max 2, q . It follows from (50) that Journal of Applied Mathematics Obviously, I λ,μ (0) � 0 and By Proposition 3 and eorem 2, there exists a critical point v λ,μ of I λ,μ . We notice that the v λ,μ cannot be trivial because By Proposition 1, we conclude that u λ,μ and v λ,μ are two solutions of the Kirchhoff-type variational inequality (4). It follows from (44) and (54) that I λ (u λ ) < 0 < I λ (v λ ). e proof is complete.

Existence of an Optimal Control Governed by a Kirchhoff-Type Variational Inequality
is section is concerned with the existence results of an optimal control for the optimal control problem (P). g(x, t): Ω × R ⟶ R satisfies (g 1 )g(x, t) is measurable in x for every t ∈ R and continuous in t for a.e. x ∈ Ω;
en, the functional G defined by is continuous.
Proof. Let u n be a sequence in L r (Ω), such that u n ⟶ u 0 as n ⟶ +∞. Since g satisfies the conditions (g 1 ) and (g 2 ), the operator u ↦ g(x, u) from L r (Ω) to L 1 (Ω) is continuous. erefore, we have as n ⟶ ∞. e proof is complete. □ Lemma 2. Suppose l(x, t): Ω × R ⟶ R is C 1 and for almost all x ∈ Ω, l(x, t) is convex with respect to t ∈ R, and there exist positive constants C i , i � 3, . . . , 6, such that where 1 < q < 2 * . en, the functional L defined by is weakly lower semicontinuous.
Proof. Let w n be a sequence in L q (Ω), such that w n ⇀ w 0 as n ⟶ +∞. Since l(x, t) is C 1 and satisfies the conditions (l 1 ) and (l 2 ), L is Gâteaux differentiable in the space L q (Ω) and L ′ (w 0 ) ∈ L q′ (Ω) (the dual space of L q (Ω)). Note that for each x ∈ Ω, l(x, .) is convex, and then L w n ≥ L w 0 +〈L ′ w 0 , w n − w 0 〉.
As w n ⇀ w 0 , from the inequality mentioned above by taking limits, we obtain lim inf n⟶+∞ L w n ≥ L w 0 .
By the weakly lower semicontinuity of the norm, J: L q (Ω) × L r (Ω) ⟶ R is weakly lower semicontinuous.
Let (w n , u n )⇀(w 0 , u 0 ) ∈ U × K ψ . en, w n ⇀ w 0 in U and u n ⇀ u 0 in K ψ . Since 1 < r < 2 * , it follows from the Sobolev imbedding theorem that the imbedding H 1 0 (Ω) ⟶ L r (Ω) is a compact imbedding. Hence, u n ⟶ u 0 in L r (Ω) and (w n , u n )⇀ (w 0 , u 0 ) in L q (Ω) × L r (Ω). Since J: L q (Ω) × L r (Ω) ⟶ R is weakly lower semicontinuous, J: U × K ψ ⟶ R is weakly lower semicontinuous. en, similar to the rest of the proof given in eorem 5, we obtain our conclusion.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.