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

Authors: Chengfu Wang; Pengcheng Wu; Yuying Zhou
Journal: Journal of Applied Mathematics
 (2020)

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## 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.


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## Body

## 1. Introduction

LetΩ be a bounded domain of ℝN with a smooth boundary ∂Ω, U be a nonempty bounded closed and convex subset of the space LqΩ1<q<min4,2∗,2∗=2N/N−2, ψ:Ω⟶0,+∞ be a proper and convex function, and (1)Kψ=u∈H01Ω:ux≤ψx,a.e.x∈Ω.LetH01Ω be endowed with the norm ⋅, and let g:Ω×ℝ⟶ℝ and l:Ω×ℝ⟶ℝ. The objective functional J:U×Kψ⟶ℝ is defined by(2)Jw,u=∫Ωgx,udx+∫Ωlx,wdx,∀w,u∈U×Kψ.In this paper, we will be discussing the following optimal control problem governed by a state variational inequality:(3)Pinfw,u∈U×Kψ,u∈SwJw,u,where Sw is the solution set of the following Kirchhoff-type variational inequality: for each w∈U, find u=uw∈Kψ (the state function of the system), such that(4)hu∫Ω∇u∇v−∇udx≥λ∫Ωfx,uv−udx−μ∫Ωτwv−udx,∀v∈Kψ,where λ>0, μ>0, fx,t:Ω×ℝ⟶ℝ, τ:U⟶Lq′Ωq′=q/q−1 and h:ℝ⟶ℝ satisfies, h1 h is continuous on ℝ; h2 there exists a>0 such that ht>a for all t∈ℝ; h3 there exists b>0 such that limt⟶+∞ht/t2=b.A typical example ofh is ht=a+bt2. Then, h satisfies h1−h3,(5)hu=a+b∫Ω∇u2,and the variational inequality (4) will become to be the usual variational inequality of the Kirchhoff type: for each w∈U, find u=uw∈Kψ, such that(6)a+b∫Ω∇u2dx∫Ω∇u∇v−∇udx≥λ∫Ωfx,uv−udx−μ∫Ωτwv−udx,∀v∈Kψ.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 b=0 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 P in the case of b>0. This case is more complicated since the so-called nonlocal term b∫Ω∇u2dxΔu 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 P.The paper is structured as follows. Section2 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.


## 2. Preliminary

LetX be a Banach space and X∗ its dual. The following definitions and theorems can be found in, e.g., [24, 25].Definition 1.
LetX 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. The 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 un⊂X with φun+ψun bounded and for which there exists a sequence εn⊂ℝ+,εn↘0, such that(7)φ′un,v−un+ψv−ψun≥−εnv−un,∀v∈X,contains a (strongly) convergent subsequence in X.Definition 2.
LetI=φ+ψ:X⟶X∪+∞ be a Szulkin-type functional. A point u∈X is said to be a critical point of I=φ+ψ if(8)φ′u,v−u+ψv−ψu≥0,∀v∈X.
The valuec of I at a critical point u∈X is said to be a critical value of I, that is, Iu=c.Theorem 1.
LetI=φ+ψ:X⟶X∪+∞ be a Szulkin-type functional which is bounded below. If I satisfies the PS condition for c=infu∈XIu, then c is a critical value.Theorem 2.
LetI=φ+ψ:X⟶X∪+∞ be a Szulkin-type functional and assume that(1)
Iu≥d0 for all u=ρ for some d0,ρ>0, and I0=0(2)
There ise∈X with e>ρ and Ie≤0
If I satisfies thePS condition, then I has a critical point u∗∈X with critical value c0=Iu* such that(9)c0≥d0,c0=infg∈Γsupt∈0,1Iγt,where(10)Γ=g∈C0,1,X:g0=0,g1=e.Letfx,t:Ω×ℝ⟶ℝ be a function. We denote by(11)Fx,t=∫0tfx,sds.The hypotheses on thef are the following:f
1
f
x
,
t is measurable in x for every t∈ℝ and continuous in t for a.e. x∈Ω;f
2
f
x
,
t
≤
α
x
+
β
x
t
q
−
1 uniformly for almost all x∈Ω, where(12)α∈L∞Ω,β∈L∞Ω,1<q<min4,2∗,2∗=2NN−2.f
3 There exists a constant γ>max2,q and a function χ∈L∞Ω such that(13)limsupt⟶0Fx,ttγ≤χx,uniformly for almost all x∈Ω;f
4 There exist x0∈Ω, u0∈ℝ+ and δ>0 such that(14)Fx,u0>η0,a.e.x∈Ux0,δ,where η0 is a positive constant and Ux0,δ=x∈Ω:x−x0<δ⊂Ω.Leth:ℝ⟶ℝ+ satisfy the conditions h1−h3. We denote by(15)Ht=∫0thsds,H^t=∫0tHsds.Letλ>0, μ>0, and let us introduce the Euler functional φλ,μ:H01Ω⟶ℝ corresponding to the Kirchhoff-type variational inequality problem (4) as(16)φλ,μu=uHu−H^u−λ∫ΩFx,udx+μ∫Ωτωudx.LetKψ be defined by (1). We define the indicator functional of the set Kψ by(17)ϕu=0,ifu∈Kψ,+∞,ifu∉Kψ.Denote(18)Iλ,μ=φλ,μ+ϕ.Obviously,Iλ,μ is a Szulkin-type functional.Remark 1.
For anys∈ℝ+, if hs=a+bs2, then h satisfies h1−h3, and(19)tHt−H^t=a2t2+b4t4.
The variational inequality (4) will become to be the classic Kirchhoff-type variational inequality: find u=uw∈Kψ, such that(20)a+b∫Ω∇u2dx∫Ω∇u∇v−∇udx≥λ∫Ωfx,uv−udx−μ∫Ωτwv−udx,∀v∈Kψ.


## 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 H01Ω.Proposition 1.
LetU be a nonempty bounded closed and convex subset of the space LqΩ1<q<min4,2∗. If f:Ω×ℝ⟶ℝ satisfies f1,f2, h:ℝ⟶ℝ satisfies conditions h1−h3, and τ:U⟶Lq′Ωq′=q/q−1 is a weakly continuous mapping. Then, every critical point u∈H01Ω of Iλ,μ=φλ,μ+ϕ is a solution of (4), where φλ,μ and Iλ,μ are defined by (16) and (18), respectively.Proof.
It is easy to see thatφλ,μ∈C1H01Ω,ℝ and for each u,v∈H01Ω,(21)φλ,μ′u,v=hu∫Ω∇u∇vdx−λ∫Ωfx,uvdx+μ∫Ωτωvdx.
Ifu∈H01Ω is a critical point of Iλ,μ, from Definition 2, we obtain(22)φλ,μ′u,v−u+ϕv−ϕu≥0,∀v∈H01Ω.
It follows from (17), (21), and (22) that u∈Kψ and (4) holds.Proposition 2.
LetU be a nonempty bounded, closed, and convex subset of the space LqΩ1<q<min4,2∗. If f:Ω×ℝ⟶ℝ satisfies f1,f2, h:ℝ⟶ℝ satisfies conditions h1−h3, and τ:U⟶Lq′Ωq′=q/q−1 is a weakly continuous mapping such that τU is a bounded set. Then, Iλ,μ is coercive in the sense of Iλ,μu⟶+∞ as u⟶+∞ and bounded from below in Kψ, where Kψ is defined by (1) and Iλ,μ is defined (18).Proof.
Byf2, Hölder inequality, we have(23)∫ΩFx,uxdx≤∫Ωαxuxdx+∫Ωβxquxqdx≤Ω1/q′αL∞ΩuLqΩ+βL∞ΩquLqΩq≤c1u+c2uq,where c1 and c2 are positive constants.
SinceτU is bounded, there exists ρ>0, such that(24)τwLq′<ρ,∀w∈U.
Thus,(25)∫Ωτwudx≤∫Ωτwq′1/q′∫Ωuqdx1/q≤ρuLqΩ≤c3u,where c3 is a positive constant.
Byh3, we have(26)limt⟶+∞tHt−H^tt4=limt⟶+∞ht4t2=b4.
Letu∈Kψ. By (16), (18), (23), and (25), we have(27)Iλ,μu=φλ,μu≥uHu−H^u−λc1+μc3u−c2λuq.
By noting that1<q<4 and from (26), we obtain that Iλ,μ is coercive and Iλ,μ is bounded from below in Kψ. The proof is complete.Proposition 3.
Let all conditions in Proposition2 be satisfied. Then, Iλ,μ satisfies the PS condition in the sense of Definition 1.Proof.
Letun⊂H01Ω such that Iλ,μun is bounded, and there exists εn⊂ℝ+ with εn↘0 and(28)φλ,μ′un,v−un+ϕv−ϕun≥−εnv−un,∀v∈H01Ω.
From (17), it can be noted that it is easy to obtain un⊂Kψ. By Proposition 2, Iλ,μ is coercive, and then un is bounded in Kψ⊂H01Ω. Hence, by the Sobolev embedding theorem, we may assume that going to a subsequence,(29)un⇀u,inH01Ω;un⟶u,inLqΩ,
since1<q<min4,2∗.
AsKψ is weakly closed, u∈Kψ. Setting v=u in (28) and combining with (21), we have(30)hun∫Ω∇un∇un−∇udx−λ∫Ωfx,unun−udx≤εnun−u−μ∫Ωτwun−udx.
Thus,(31)hunu−un2=hun∫Ω∇u∇u−∇un+∇un∇un−∇udx≤hun∫Ω∇u∇u−∇undx+λ∫Ωfx,unun−udx+εnu−un−μ∫Ωτwu−undx.
By the Hölder inequality and the conditionf2, for un⊂Kψ,(32)∫Ωfx,unun−u≤∫Ωαx+βxunq−1un−udx≤αL∞un−uL1+βL∞unLq−1un−uLq.
Note thatεn↘0(as n⟶∞), un is bounded in H01Ω and h satisfies h1−h2; by (29)–(32), we have(33)0≤limn⟶∞au−un2≤limn⟶∞hunu−un2≤0.
Thereforeun⟶u in H01Ω.Theorem 3.
LetU be a nonempty bounded closed and convex subset of the space LqΩ1<q<min4,2∗. If f:Ω×ℝ⟶ℝ satisfies f1−f4, h:ℝ⟶ℝ satisfies the conditions h1−h3, and τ:U⟶Lq′Ω is a weakly continuous mapping such that τU is a bounded set. Then, there exists λ∗>0,0<μ∗<1, such that for each λ>λ∗,μ<μ∗, the Kirchhoff-type variational inequality (4) has two solutions uλ,vλ, satisfying Iλuλ<0<Iλvλ.Proof.
Noting the conditionsf3 and f4, we have(34)0≤M1=αL∞u0+1qβL∞u0q<+∞.
Denote by(35)t0=12M1η0+M11/N+1,where η0 is given in f4. Let us define(36)ut0x=0,x∈ΩUx0,δ,u0,x∈Ux0,t0δ,u0δ1−t0δ−x−x0,x∈Ux0,δ\Ux0,t0δ,where x0 is given in f4. Then,(37)ut02=∫Ω∇ut02dx=∫Bx0,δ\Bx0,t0δu0δ1−t02dx,=ωNδN−21−t0Nu021−t02,where ωN is the volume of unit sphere in ℝN, and δ,u0 are defined in f4. Thus, ut0 is a positive constant which follows from (37). In the following, assume that 0<μ<1. Therefore, from (25) and the condition h1, (38)ut0Hut0−H^ut0+μ∫Ωτwut0dx≤A¯,where A¯ is a positive constant.
Byf2,f4 and (32) and (36) we have(39)∫ΩFx,ut0dx,=∫Ux0,δ\Ux0,t0δFx,u0δ1−t0δ−x−x0dx+∫Ux0,t0δFx,ut0dx≥∫Ux0,t0δη0dx−∫Ux0,δ\Ux0,t0δαu0+βqu0qdx≥ωNδNt0Nη0−∫Ux0,δ\Ux0,t0δM1dx=ωNδNt0Nη0−M1+M1t0N,where η0 is defined in f4. Note that 0<μ<1, it follows from (16), (18), (38), and (39) that(40)Iλ,μut0=ut0Hut0−H^ut0−λ∫ΩFx,ut0dx+μ∫Ωτwut0dx≤A¯−λωNδNη0t0N−M1+t0NM1.
By (35), we obtain that(41)M1η0+M1<t0N<1.
Thus,(42)η0t0N−M1+t0NM1=t0N−M1η0+M1η0+M1>0.
Therefore, there existsλ*>0, such that(43)Iλ,μut00,∀λλ∗.
By Proposition2, the function Iλ,μ is bounded from below. By Proposition 3, the function Iλ,μ satisfies the PS condition. Thus, Iλ,μ attains its global minimum at some uλ,μ∈H01Ω by Theorem 4. Obviously,(44)Iλ,μuλ,μ≤Iλ,μut00,∀λλ∗.
Next, we will prove the existence of the second critical point ofIλ,μ via the Mountain Pass theorem (see Theorem 2).
By the conditionf3 for any ε>0, there exists t∗>0, such that(45)Fx,ttγ≤χx+ε,∀t∈−t∗,t∗.
It follows fromf2 that for all t>t∗,(46)Fx,t≤αx+βxtq−1t≤αxt∗γ−1+βxt∗γ−qtγ.
From (45) and (46), we obtain that for all t∈ℝ and a.e. x∈Ω,(47)Fx,t≤χx+ε+αxt∗γ−1+βxt∗γ−qtγ.
By the conditionsh1 and h2, we obtain(48)limt⟶0tHt−H^tt2=limt⟶0ht2=h02>a2,and then there exists a positive constant t¯ such that for all t∈0,t¯,(49)tHt−H^t>a2t2.
Therefore, by (47) and (49), the Hölder inequality and the Sobolev embedding theorem, for all λ>λ* and u∈Kψ∩u≤t¯:u∈H01Ω, we have(50)Iλ,μu=φλ,μu≥a2u2−λc4uγ−μc5u,where c4andc5 are positive constants.
Denote(51)Λ=min12ut0,a4λc41/γ−2,t¯,μ∗=mina8c5Λ,12.
Letλ>λ∗ and μ<μ∗. Note that γ>max2,q. It follows from (50) that(52)Iλ,μu≥a8Λ2>0,∀u∈H01Ω,u=Λ.
Obviously,Iλ,μ0=0 and(53)Iλ,μut0<0,ut0>Λ.
By Proposition3 and Theorem 2, there exists a critical point vλ,μ of Iλ,μ. We notice that the vλ,μ cannot be trivial because(54)Iλ,μvλ,μ≥a8Λ2>0.
By Proposition1, we conclude that uλ,μ and vλ,μ are two solutions of the Kirchhoff-type variational inequality (4). It follows from (44) and (54) that Iλuλ<0<Iλvλ. The proof is complete.By Theorem3, we haveTheorem 4.
LetU be a nonempty bounded closed and convex subset of the space LqΩ1<q<min4,2∗. If f:Ω×ℝ⟶ℝ satisfies f1−f4 and hs=a+bs2, τ:U⟶Lq′Ω is a weakly continuous mapping such that τU is a bounded set. Then, there exist λ∗>0,0<μ∗<1, such that for each λ>λ∗,μ<μ∗, the Kirchhoff-type variational inequality (20) has two solutions uλ,vλ, satisfying Iλuλ<0<Iλvλ.


## 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 problemP.Lemma 1.
Supposegx,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;
g
2 There exist positive constants Ci,i=1,2, such that(55)gx,t≤C1+C2tr,∀x,t∈Ω×ℝ.where 1<r<2∗.Then, the functionalG defined by(56)Gu=∫Ωgx,udx,u∈LrΩ,is continuous.Proof.
Letun be a sequence in LrΩ, such that un⟶u0 as n⟶+∞. Since g satisfies the conditions g1 and g2, the operator u↦gx,u from LrΩ to L1Ω is continuous. Therefore, we have(57)∫Ωgx,undx⟶∫Ωgx,udx,
asn⟶∞. The proof is complete.Lemma 2.
Supposelx,t:Ω×ℝ⟶ℝ is C1 and for almost all x∈Ω, lx,t is convex with respect to t∈ℝ, and there exist positive constants Ci,i=3,…,6, such that(58)l1lx,t≤C3+C4tq,∀x,t∈Ω×ℝ,l2∂lx,t∂t≤C5+C6tq−1,∀x,t∈Ω×ℝ,where 1<q<2∗. Then, the functional L defined by(59)Lw=∫Ωlx,wdx,w∈LqΩ,is weakly lower semicontinuous.Proof.
Letwn be a sequence in LqΩ, such that wn⇀w0 as n⟶+∞. Since lx,t is C1 and satisfies the conditions l1 and l2, L is Gâteaux differentiable in the space LqΩ and L′w0∈Lq′Ω (the dual space of LqΩ). Note that for each x∈Ω, lx,. is convex, and then(60)Lwn≥Lw0+L′w0,wn−w0.
Aswn⇀w0, from the inequality mentioned above by taking limits, we obtain(61)lim infn⟶+∞Lwn≥Lw0.
The proof is complete.Theorem 5.
LetU be a nonempty bounded closed and convex subset of the space LqΩ1<q<min4,2∗. Assume that h:ℝ⟶ℝ satisfies the conditions h1−h3, f:Ω×ℝ⟶ℝ satisfies the conditions f1−f4, g:Ω×ℝ⟶ℝ satisfies g1,g2, τ:U⟶Lq′Ω a weakly continuous mapping such that τU is a bounded set, and gx,t:Ω×ℝ⟶ℝ and l:Ω×ℝ⟶ℝ satisfy all the conditions in Lemma 1 and Lemma 2, respectively. There exist λ∗>0,0<μ∗<1, such that for each λ>λ∗,μ<μ∗, there exists an optimal control w0,u0∈U×Sw0 for the optimal control problem P.Proof.
From Theorem3, there exist λ∗>0,0<μ∗<1, such that for each λ>λ∗,μ<μ∗, Sw≠ϕ, where Sw is the solution set of the variational inequality (4).
Letwn,unn∈N (un∈Swn) be a minimizing sequence for problem P such that(62)limn⟶∞Jwn,un=infw,u∈U×SwJw,u,where the cost function J is defined by (2).
Asun∈Swn, we have(63)hun∫Ω∇un∇v−∇undx≥λ∫Ωfx,unv−undx−μ∫Ωτwnv−undx,∀v∈Kψ.
We claim thatwn,unn∈N is bounded. In fact, as U is bounded, wn is bounded. Let v=0 in (63). We get(64)hunun2≤λ∫Ωfx,unundx−μ∫Ωτwnundx≤λ∫Ωfx,unundx+μ∫Ωτwnundx≤λ∫Ωαun+βunqdx+μ∫Ωτwnundx≤λαL∞unL1+βL∞unLqq+μτwnLq′unLq≤c6λ+μun+c7λunq,where c6andc7 are positive constants. As 1<q<4 and h satisfies h1−h3, un is bounded.
SinceLqΩ and H01Ω are reflexive spaces, there exists a weakly convergent subsequence of wn,un. Without loss of generality, we may assume wn,un⇀w0,u0 as n⟶∞. Hence, wn⇀w0 in LqΩ and un⇀u0 in H01Ω as n⟶∞. Since U and Kψ are weakly closed sets, w0∈U and u0∈Kψ. Then, let v=u0 in (63). We get(65)hun∫Ω∇un∇u0−undx≥λ∫Ωfx,unu0−undx−μ∫Ωτwnu0−undx.
Sincef satisfies the conditions f1−f4 and τ is bounded,(66)limn⟶∞λ∫Ωfx,unu0−undx=0,limn⟶∞∫Ωτwnu0−undx=0.
Then, from (65) and (66), we have(67)limsupn⟶∞hun∫Ω∇un∇un−u0dx≤0.
Sinceun is bounded and h satisfies h1andh2,(68)limsupn⟶∞∫Ω∇un∇un−u0dx≤0.
Note thatun⇀u0 in H01Ω as n⟶∞, and we have(69)limn⟶∞∫Ω∇u0∇un−u0dx=0.
By (68) and (69), we get(70)limsupn⟶∞∫Ω∇un−u02dx≤0.
Thus,un⟶u0 in H01Ω as n⟶+∞. Therefore, by (63), we have(71)hu0∫Ω∇u0∇v−∇u0dx≥λ∫Ωfx,u0v−u0dx−∫Ωτwv−u0dx,∀v∈Kψ,that is, u0∈Sw0. Denote(72)Gu=∫Ωgx,udx,u∈Kψ,Lw=∫Ωlx,wdx,w∈U.
Sinceun⟶u0 in H01Ω as n⟶+∞ and the embedding H01Ω↪LrΩ is continuous compact (1<r<2∗), un⟶u0 in LrΩ as n⟶∞. By Lemma 1 and Lemma 2, G:Kψ⟶ℝ is continuous and L:LqΩ⟶ℝ is weakly lower semicontinuous. Therefore,(73)Jw0,u0=Gu0+Lw0≤lim infn⟶+∞Gun+Lwn,=lim infn⟶+∞Jwn,un,=infw∈U,u∈SwJw,u.That is, w0,u0∈U×Sw0 is an optimal control for problem P. The proof is complete.Theorem 6.
LetU be a nonempty bounded closed and convex subset of the space LqΩ1<q<min4,2∗. Assume that h satisfies the conditions h1−h3, f satisfies the conditions f1−f4, and τ:U⟶Lq′Ω is a weakly continuous mapping such that τU is a bounded set. Let 1<r<p∗, gu=C7u−udr, and lw=C8w−wdq, for some constants C7 and C8, where ud∈LrΩ,wd∈LqΩ. There exist λ∗>0,0<μ∗<1, such that for each λ>λ∗,μ<μ∗, there exists an optimal control w0,u0∈U×Sw0 for the optimal control problem P.Proof.
Firstly, we claim thatJ:U×Kψ⟶ℝ is weakly lower semicontinuous. In fact,(74)Jw,u=∫Ωgu+∫Ωlw,=C7∫Ωu−udr+C8∫Ωw−wdq,=C7u−udLrr+C8w−wdLqq.
By the weakly lower semicontinuity of the norm,J:LqΩ×LrΩ⟶ℝ is weakly lower semicontinuous.
Letwn,un⇀w0,u0∈U×Kψ. Then, wn⇀w0 in U and un⇀u0 in Kψ. Since 1<r<2∗, it follows from the Sobolev imbedding theorem that the imbedding H01Ω⟶LrΩ is a compact imbedding. Hence, un⟶u0 in LrΩ and wn,un⇀w0,u0 in LqΩ×LrΩ. Since J:LqΩ×LrΩ⟶ℝ is weakly lower semicontinuous, J:U×Kψ⟶ℝ is weakly lower semicontinuous. Then, similar to the rest of the proof given in Theorem 5, we obtain our conclusion.



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*Source: 2831242-2020-02-18.xml*