# On Regularization of an Optimal Control Problem for Ill-Posed Nonlinear Elliptic Equations

Authors: Peter I. Kogut; Olha P. Kupenko; Rosanna Manzo
Journal: Abstract and Applied Analysis
 (2020)

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## Abstract

We discuss the existence issue to an optimal control problem for one class of nonlinear elliptic equations with an exponential type of nonlinearity. We deal with the control object when we cannot expect to have a solution of the corresponding boundary value problem in the standard functional space for all admissible controls. To overcome this difficulty, we make use of a variant of the classical Tikhonov regularization scheme. In particular, we eliminate the PDE constraints between control and state and allow such pairs run freely by introducing an additional variable which plays the role of “compensator” that appears in the original state equation. We show that this fictitious variable can be determined in a unique way. In order to provide an approximation of the original optimal control problem, we define a special family of regularized optimization problems. We show that each of these problems is consistent, well-posed, and their solutions allow to attain an optimal solution of the original problem as the parameter of regularization tends to zero. As a consequence, we prove the existence of optimal solutions to the original problem and propose a way for their approximation.


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## Body

## 1. Introduction

The main object of our study is the following optimal control problem for a nonlinear elliptic equation:(1)MinimizeJu,y=12∫Ωy−yd2dx+αp∫Ωupdx,subject to constrains
(2)−Δy=fy+uinΩ,(3)y=0on∂Ω,(4)u∈U∂⊆LpΩ,y∈H01Ω,where Ω is a bounded open domain in ℝN, N≥1, the boundary ∂Ω is assumed to be Lipschitz, fy=F′y, where F∈C1ℝ is a given nonlinear function, yd∈L2Ω is a given distribution, U∂ is a nonempty closed convex subset of LpΩ, 2≤p<+∞, and α≥0 is a given weight coefficient.Optimal control governed by PDEs has been examined thoroughly since the pioneering work of J.L. Lions (see [1, 2], for instance). Other important references that also deal with the numerical approximation and in addition to those already mentioned above without any attempt to be exhaustive, are [3–9]. However, as for the optimal control problem (OCP) (1)–(4) and the corresponding Dirichlet boundary value problem (BVP) (2)–(3), it is well known that they are ill-posed, in general, and it is unknown whether the set of optimal pairs to the problem (1)–(4) is nonempty. In particular, there is no reason to assert the existence of weak solutions to (2)–(3) for a given u∈LpΩ or to suppose that such solution, even if it exists, is unique (see, for instance, I.M. Gelfand [10], M.G. Crandall and P.H. Rabinowitz [11], F. Mignot and J.P. Puel [12], T. Gallouët, F. Mignot and J.P. Puel [13], H. Fujita [14], R.G. Pinsky [15], R. Ferreira, A. De Pablo, J.L. Vazquez [16], J. Dolbeault and R. Stańczy [17]).The novelty of this paper is that we discuss the existence of optimal pairs to OCP (1)–(4) using an indirect approach based on the classical Tikhonov regularization technique in its special implementation. The idea to involve the Tikhonov regularization is inspired by the following reason: the main characteristic feature of BVP (2)–(3) is the fact that because of the specificity of nonlinearity fy (in many particular implementations of the model (2)–(3), Fu=λeu, [18, 19]), we have no a priori estimate for the weak solutions in the standard Sobolev space H01Ω. As a result, the consistency of OCP (1)–(4) and existence of optimal pairs can be established only if we impose rather strict assumptions on the original data. In particular, it was shown in [20] that the set of optimal solutions of (1)–(4) is nonempty provided N>2, p>2, the domain Ω is star-shaped with respect to some interior point x0, and the set of feasible pairs Ξ contains at least one pair u,y such that fu∈L2Ω.Therefore, our main intention is to show that these assumptions can be essentially weakened or even eliminated. With that in mind, in the framework of Tikhonov regularization technique, we introduce the additional variablez (the so-called “defect” in the state equation) into the regularized problem in order to let the pairs “ control-state” u,y run freely in the feasible space LpΩ×H01Ω so that there is no dependence of y on u. At the same time, there is a principle difference between the standard implementation of the Tikhonov regularization of OCPs (see, for instance, [21–23]) and the proposed scheme. This difference lies in the exploitation of the terms εfyH−1Ω2/2 and εfyL1Ω/2 in the perturbed cost functional Jεu,y,z. We show that the boundedness of these terms on the set Ξ of feasible solutions to the original problem plays a crucial role in the study of asymptotic behaviour of global solutions to regularized OCPs. Having introduced a special family of optimization problems, we also show that there exists an optimal solution to the original OCP that can be attained with a prescribed level of accuracy by the sequence of optimal solutions for the regularized minimization problems (for benefit of this approach and its comparison with other ones, we refer to the recent papers [24–31]).The paper is organized as follows. In Section2 we give some preliminaries and describe in details the characteristic features of OCP (1)–(4). The Tikhonov regularization of the original OCP is discussed in Section 3. The key result of this section is Theorem 8, where we announce the sufficient conditions of the existence of optimal solutions to the regularized problems. In Section 4, we focus on deriving and substantiation of optimality conditions for regularized optimal control problem. The details of the indirect approach to the study of the original optimal control problem are discussed in last section. The key points of such approach are summarized in Theorem 13.


## 2. Preliminaries

LetΩ be a bounded open subset of ℝN (N≥1). Let F:ℝ⟶0,+∞ be a mapping such that F∈C1ℝ. We specify this mapping as follows: there exists a constant CF>0 such that
(5)CFF′z≥Fz,∀z∈ℝ,and∫−∞0zF′zdz<+∞.Then, it is easy to deduce that(6)Fz≥FaexpCF−1z−a,∀a,z:−∞<a≤z<+∞.Following the standard notation, byH01Ω, we denote the Sobolev space as the closure of C0∞Ω with respect to the norm ∫Ω∇y2dx1/2. Let H−1Ω be the dual space to H01Ω.In order to make a precise meaning of the weak solution to BVP (2)–(3) in the sense of distributions (or shortly, distributional solution), we begin with the following concept.Definition 1.
Letu∈U∂ be a given control function. We say that y=yu is a weak solution to the boundary value problem (2)–(3) in the sense of distributions, if it belongs to the class of functions
(7)Hf=y∈H01Ω:fy∈Lloc1Ω,and the integral identity
(8)∫Ω∇y,∇φdx=∫Ωfyφdx+∫Ωuφdx,holds for every test function φ∈C0∞Ω.
Since for each test functionφ∈C0∞Ω, there exists a compact set K⊂Ω such that
(9)suppφ⊆K⊂Ω,it follows that the second term in (8) is well defined, namely,
(10)∫Ωfyφdx≤∫Kfyφdx≤φCK¯fyL1K.At the same time, it is unknown whether the original BVP admits at least one weak solution in the sense of Definition1 for each admissible control u∈U∂⊆LpΩ. Moreover, as follows from (8), the continuity of form φ↦y,φf≔∫Ωfyφdx on the set H01Ω is not evident. For the details related with this issue, we refer to the classical paper Casas, Kavian, and Puel [20].Before proceeding further, we make use of the following observation. Assume that for a givenu∈U∂, we have y∈Hf, and the pair u,y is related by integral identity (8). Then, for each test function φ∈C0∞Ω, the following estimate
(11)∫Ωfyφdx≤∫Ω∇y,∇φdx+∫Ωuφdx≤yH01ΩφH01Ω+uL2ΩφL2Ω≤byPoincaré′sineq.yH01Ω+CΩΩp−22puLpΩφH01Ω,holds true. Hence, the mapping φ↦y,φf can be extended by continuity onto the set of all φ∈H01Ω using (11) and the standard rule
(12)y,φf=limε→0y,φεf,where φεε>0⊂C0∞ℝN and φε⟶φ strongly in H01Ω as ε⟶0. In particular, if y∈Hf, then we can define the value y,yf, and this one is finite for every y∈Hf. As a consequence, we deduce: if y∈Hf is a weak solution to boundary value problem (2)–(3), then y satisfies the energy equality
(13)∫Ω∇y2dx=y,yf+∫Ωuydx.However, it is unknown whether the valuey,yf preserves a constant sign for all y∈Hf. Therefore, we cannot make use of the energy equality (13) in order to derive a priori estimate in ∥·∥H1Ω-norm for the weak solutions.In particular, to specify the termy,yf, we have the following result (we refer to [20], Lemma 2.1) where this result was proven for a particular nonlinearity fy=ey (see also [27, 28, 32] for the more general cases).Lemma 2.
Lety=yu be a weak solution to BVP (2)–(3) for a given u∈U∂. Then, fy∈H−1Ω,
(14)fyH−1Ω≤yH01Ω+CΩΩp−2/2puLpΩ,(15)y,zf=<fy,z>H−1Ω;H01Ω=∫Ωzfydx,∀z∈H01Ω,and, therefore, zfy∈L1Ω for every z∈H01Ω.Proof.
Taking into account the Friedrich’s inequality(16)yL2Ω≤CΩ∇yL2ΩN,∀y∈H01Ω,and following the definition of the weak solution, we have (see (8))
(17)∫Ωfyφdx≤∫Ω∇y,∇φdx+∫Ωuφdx≤∇yL2Ω∇φL2Ω+uL2ΩφL2Ω≤by13yH01ΩφH01Ω+CΩΩp−2/2puLpΩφH01Ω,∀φ∈C0∞ℝN.
Hence,y∈Hf by Definition 1.
Letz∈H01Ω∩L∞Ω be an arbitrary element. Since fy∈L1Ω, it follows that the term ∫Ωzfydx is well defined. Let φεε>0⊂C∞Ω¯ be a sequence such that φε⟶z in H01Ω. Moreover, in this case, we can suppose that
(18)supε>0∥φε∥L∞Ω<+∞andφε⇀∗zinL∞Ω.
Hence, due to the fact thaty∈Hf, we get
(19)∫Ωzfydx=limε→0∫Ωφεfydx=limε→0y,φεf=by18y,zf.
Thus, we arrive at relation (15) for each z∈H01Ω∩L∞Ω.
Let us take nowz∈H01Ω such that z≥0 almost everywhere in Ω. For every ε>0, let Tε:ℝ⟶ℝ be the truncation operator defined by
(20)Tεs=maxmins,ε−1,−ε−1.
The following property ofTε is well known (see [33]): if z∈H01Ω, then
(21)Tεz∈L∞Ω∩H01Ω∀ε>0andTεz⟶zinH01Ωasε⟶0.
Hence,Tεz⟶z almost everywhere in Ω. Since
(22)Tεzfy≥by51CFTεzFy≥0inΩ,it follows that Tεzfyε>0 is a pointwise nondecreasing sequence, and also, Tεzfy⟶zfy for almost all x∈Ω. Therefore, by monotone convergence theorem, zfy is a measurable function on Ω, and
(23)limε→0∫ΩTεzfydx=∫Ωzfydx.
Thus, (15) holds true for each z∈H01Ω such that z≥0.
As for a general case, i.e.,z∈H01Ω, it is enough to note that z=z+−z− with z+,z−∈H01Ω and z+,z−≥0 in Ω, where z+≔maxz,0, z−≔max−z,0.
To complete the proof, it remains to observe that(24)∫Ωzfydx=by17limε→0∫Ωφεfydx≤by1714limε→0yH01Ωp−1+Ωp−2/2pCΩuLpΩφεH01Ωbythestrongconvergenceofφε⟶zinH01Ω=yH01Ωp−1+Ωp−2/2pCΩuLpΩzH01Ω,holds true for an arbitrary element z∈H01Ω. As a result, we have fy∈H−1Ω and
(25)fy,zH−1Ω;H01Ω=∫Ωzfydx,∀z∈H01Ω,andfyH−1Ω≤yH01Ωp−1+Ωp−2/2pCΩuLpΩ.Remark 3.
As follows from Lemma2, whenever y,u is related by integral identity (8) and y∈Hf, then fy∈H−1Ω, but for a general φ∈H01Ω, it is not necessarily true that the duality action <fy,φ>H−1Ω;H01Ω is given by an integral ∫Ωφfydx, hence the need for a rigorous definition of y,φf.As a direct consequence of Lemma2 and relation (13), we can specify the energy equality (13) as follows.Corollary 4.
Letu∈U∂ be a given control and let y=yu∈H01Ω be a weak solution to BVP (2)–(3) in the sense of Definition 1. Then, the energy equality for y takes the form
(26)∫Ω∇y2dx=∫Ωyfydx+∫Ωuydx.Since it is unknown whether there exists a weak solution to BVP (2)–(3) for a given u∈U∂, or to suppose that such solution, even if it exists, is unique, it motivates us to introduce the following set.Definition 5.
We say that a pairu,y is a feasible solution for optimal control problem (1)–(4) if u∈U∂, y∈Hf, fy∈L1Ω, and the pair u,y is related by integral identity (8). By Ξ⊂LpΩ×H01Ω, we denote the set of all feasible solutions.
As for the optimal control problem (1)–(4), it was mentioned in [20] that its study is a nontrivial matter because of the specific of nonlinearity fy (in [20], the authors consider the case fy=ey). The main troubles in this case are strongly related with the following circumstances:
(i)
The set of feasible solutions can be empty, in general(ii)
Even ifΞ≠∅, we have no a priori estimate for the weak solutions of (2)–(3) with arbitrary u∈U∂(iii)
Some a priori estimates can be established if onlyN>2, the domain Ω has a sufficiently smooth boundary, and it is star-shaped with respect to some interior point x0, i.e,
(27)σ−x0,νσ≥0,fora.a.σ∈∂Ω,where νσ denotes the outward unit normal vector to ∂Ω at the point σ, and the considered weak solutions yu of (2)–(3) satisfies the extra property fy∈L2Ω(iv)
Since we have no estimates for the states (especially without the above mentioned extra propertyfy∈L2Ω), it follows that we cannot deduce the boundedness in LpΩ×H01Ω of a minimizing sequence to the problem (1)–(4)(v)
Even if a minimizing sequenceuk,yk∈Ξn∈ℕ is weakly compact in LpΩ×H01Ω with p≥2, it does not allow to pass to the limit in integral identity (8) as k⟶∞, and, therefore, we are not able to prove the existence of an optimal pair to the problem (1)–(4).Although this list can be extended by many other options, we can summarize this issue by the following existence result (in order to prove this assertion, it is enough to closely follow the arguments of the proofs of Theorems 3.5 and 3.6 in [20]).Theorem 6.
Let us assume that the following conditions hold true:N>2, p>2, the domain Ω has a C0,1 boundary, this domain is star-shaped with respect to some interior point x0, and the set of feasible pairs Ξ contains at least one pair u,y such that fu∈L2Ω. Then, there exists a unique pair u0,y0∈Ξ such that
(28)u0,y0∈Ξ0⊂Ξ,Ju0,y0=infu,y∈Ξ0Ju,y,where
(29)Ξ0=u,y∈ΞN2−1∫Ω∇y2dx≤N∫ΩFydx−∫Ωuxx−x0,∇yxdx.
In spite of the fact that not every pairu,y of Ξ0⊆Ξ needs to be a feasible pair to (1)–(4) with the extra property fy∈L2Ω, and constrained minimization problems
(30)infu,y∈ΞJu,yandinfu,y∈Ξ0Ju,y,are distinguished from a formal point of view, we can deduce the following result (for the proof we refer to [20], Proposition 3.2).Proposition 7.
Assume thatN>2 and Ω is star-shaped with respect to some interior point x0. Assume also that boundary value problem (2)–(3) has a weak solution for some control u∈LpΩ with p≥2. Then, there exists a solution z of (2)–(3) corresponding to the same control u and such that
(31)N2−1∫Ω∇z2dx≤N∫ΩFzdx−∫Ωuxx−x0,∇zxdx.In the next section, we will show that the main restrictions coming from Theorem6 and Proposition 7 can be eliminated by introducing a new additional variable z into the problem which lets pairs u,y run freely in the feasible space LpΩ×H01Ω so that there is no dependence of y on u.


## 3. On the Tikhonov Regularization of the Original OCP

Let us introduce the Tikhonov regularized optimal control problem associated to the original OCP (1)–(4). Let ε>0 be a given small parameter. Then, the regularized problem reads as follows (for comparison, we refer to [22, 23]).
(32)MinimizeJεu,y,z=12∫Ωy−yd2dx+αp∫Ωupdx+12ε∫Ω∇z2dx+ε2fyH−1Ω2+fyL1Ω,subject to constraints
(33)u∈U∂⊆LpΩ,y∈H01Ω,fy∈L1Ω∩H−1Ω,(34)−Δz=Δy+fy+uinΩ,(35)z=0on∂Ω.To begin with, let us stress again that the main reason to introduce the additional variablez into the regularized problem is to let pairs u,y run freely in the feasible space LpΩ×H01Ω so that there is no dependence of y on u. On the other hand, there is a principle difference between the standard scheme of the Tikhonov regularization of OCPs (see, for instance, [22, 23]) and the proposed regularization in the form (32)–(35). This difference lies in the exploitation of the terms εfyH−1Ω2/2 and εfyL1Ω/2 in the perturbed cost functional Jεu,y,z. As it will be shown later on, the boundedness of these terms on the set Ξ of feasible solutions to the original problem (see Lemma 2) plays a crucial role in the study of asymptotic behaviour of global solutions uε0,yε0,zε00<ε≤ε0 as ε tends to zero.Our main assumptions are:(a)
Ω is a bounded open domain in ℝN, N≥1(b)
U∂ is a nonempty closed convex subset of LpΩ, 2≤p<+∞(c)
f:ℝ⟶0,+∞ is a given monotonically increasing mapping such that f∈Cℝ.We say that a tupleu,y,z∈LpΩ×H01Ω2 is a feasible solution to regularized problem (32)–(35) (in symbols, u,y,z∈Λε), if u∈U∂, Jεu,y,z<+∞, and the following variational equality
(36)az,φ=Fφ,holds true for all φ∈H01Ω, where
(37)Fφ≔−∇y,∇φL2ΩN+u,φL2Ω+∫Ωfyφdx,and a:H01Ω×H01Ω⟶ℝ denotes the bilinear form
(38)az,φ=∫Ω∇z,∇φdx.Let us show that, for eachε>0, the set of feasible solutions Λε to regularized problem (32)–(35) is nonempty. Indeed, let u,y be an arbitrary pair in LpΩ×H01Ω such that
(39)u∈U∂andfy∈L1Ω∩H−1Ω.Then, the right hand side of (36) is well defined for each test function φ∈H01Ω and satisfies the following estimate (see (12)).
(40)∣Fφ∣≤∇y,∇φL2ΩN+u,φL2Ω+<fy,φ>H−1Ω;H01Ω≤yH01Ω+fyH−1Ω+CΩΩp−2/2puLpΩφH01Ω.Since, the bilinear forma·,· is continuous and uniformly coercive on H01Ω×H01Ω, it follows from Lax-Milgram theorem that the variational problem (36) has a unique solution z=zu,y∈H01Ω. Hence, Jεu,y,z<+∞ and, therefore, u,y,z∈Λε for a given ε>0. Thus, Λε≠∅ and this implies that regularized optimal control problem (32)–(35) is consistent for all ε>0.Our next intention is to discuss the issue related to the existence of optimal solutions of the regularized problems (32)–(35).Theorem 8.
Assume that conditions (a)–(c) indicated before are valid. Then, for eachε>0, there is a triplet uε0,yε0,zε0∈Λε such that
(41)Jεuε0,yε0,zε0=infu,y,z∈ΛεJεu,y,z.Proof.
Letε>0 be a given value. Since the cost functional Jε:Λε⟶ℝ is nonnegative on Λε, it follows that there exist a με≥0 and a sequence uε,k,yε,k,zε,kk∈ℕ⊂Λε such that
(42)με=infu,y,z∈ΛεJεu,y,z=limk→∞Jεuε,k,yε,k,zε,k,(43)με≤Jεuε,k+1,yε,k+1,zε,k+1≤Jεuε,k,yε,k,zε,k≤με+1,∀k∈ℕ.
Then, we can immediately deduce from (43) and definition of the set Λε that yε,k∈H01Ω for each k∈ℕ and the sequences
(44)yε,kk∈ℕ,uε,kk∈ℕ,zε,kk∈ℕ,andfyε,kk∈ℕ,are uniformly bounded in L2Ω, LpΩ, H01Ω, and H−1Ω∩L1Ω, respectively. In particular,
(45)supk∈ℕuε,kLpΩp≤με+1pα,supk∈ℕzε,kH01Ω2≤2εμε+1,(46)supk∈ℕyε,kL2Ω2≤4με+1+2ydL2Ω2,supk∈ℕfyε,kH−1Ω2≤2με+1ε,(47)supk∈ℕfyε,kL1Ω≤2με+1ε.
Hence, without loss of generality, we can suppose that there exist elementsyε∈L2Ω, uε∈LpΩ, ξε∈H−1Ω, and zε∈H01Ω such that
(48)yε,k⇀yεinL2Ω,(49)uε,k⇀uεinLpΩ,(50)zε,k⇀zεinH01Ω,(51)fyε,k⇀ξεinH−1Ω,as k⟶∞.
Let us show that, in fact,yε,k⇀yε in H01Ω. Indeed, from (36), using yε,k∈H01Ω as a test function, we find that
(52)∫Ω∇yε,k2+∇zε,k,∇yε,k−uε,kyε,kdx=fyε,k,yε,kH−1Ω;H01Ω,for all k∈ℕ. Then, utilizing the Poincaré’s inequality, we obtain
(53)∇yε,kL2ΩN2≤zε,kH01Ω+fyε,kH−1Ω+CΩΩp−2/2puε,kLpΩ×∇yε,kL2ΩN.
From this and estimates (45)–(46), we deduce that
(54)supk∈ℕ∇yε,kL2ΩN≤2με+1ε1+ε+CΩΩp−2/2ppε2α≕Cε∗.
Thus, without loss of generality, we can suppose that (up to a subsequence)(55)yε,k⇀yεinH01Ω,yε,k⟶yεinL2Ω,yε,kx⟶yεxa.e.inΩ.
Utilizing the pointwise convergence (55)3 and assumption (c), we see that
(56)fyε,k⟶fyεalmosteverywhereinΩask⟶∞.
Let us show that this fact together with (47) implies the strong convergence
(57)fyε,k⟶fyεinL1Ωask⟶∞.
To begin with, let us show that the sequencefyε,kyε,kk∈ℕ is bounded in L1Ω. With that in mind, for each k∈ℕ, we make use of the decomposition yε,k=yε,k+−yε,k− with
(58)yε,k+=max0,yε,k∈H01Ω,yε,k−=max0,−yε,k∈H01Ω,and set
(59)Tηyx=η,ifyx>η,yx,ifyx≤η.
Then,Tηy∈L∞Ω∩H01Ω for all y∈H01Ω such that y≥0. Using yε,k+∈H01Ω as a test function in (36), we find that
(60)∫Ωfyε,kTηyε,k+dx=∫Ω∇zε,k,∇Tηyε,k+dx+∫Ω∇yε,k,∇Tηyε,k+dx−∫Ωuε,kTηyε,k+dx,∀η∈ℕ.
SinceTηyε,k+⟶yε,k+ in H01Ω as η⟶∞, the limit passage in (60) as η⟶∞ leads to the relation
(61)∫Ωfyε,kyε,k+dx=∫Ω∇zε,k,∇yε,k+dx+∫Ω∇yε,k,∇yε,k+dx−∫Ωuε,kyε,k+dx.
From this and the fact thatyε,k+H01Ω≤yε,kH01Ω, we deduce the estimate
(62)∫Ωfyε,kyε,k+dx≤zε,kH01Ω+yε,kH01Ω+CΩΩp−2/2puε,kLpΩyε,kH01Ω≤by29,2937Cε∗2εμε+1+Cε∗+CΩΩp−2/2pμε+1pα1/p.
Arguing in a similar manner, it can be shown that(63)∫Ωfyε,kyε,k−dx≤by29,2937Cε∗2εμε+1+Cε∗+CΩΩp−2/2pμε+1pα1/p.
Sinceyε,k++yε,k−=∣yε,k∣, it follows from (62)–(63) that there exist positive constants Ci,ε, i=1,2, independent of k and yε,k such that
(64)supk∈ℕfyε,kyε,kL1Ω≤C1,εμε+1+C2,εμε+12/p.
In order to prove the strong convergence (57), we make use of Vitali’s theorem. To do so, we fix an arbitrary δ>0 and take m>0 and τ>0 such that m>2Cε∗∗/δ, τ=δ/2fm, where
(65)Cε∗∗≔C1,εμε+1+C2,εμε+12/p.
Then, for every measurable setS⊂Ω with Lebesgue measure ∣S∣<δ, we have
(66)∫Sfyε,kdx=∫x∈S:yε,kx>mfyε,kdx+∫x∈S:yε,kx≤mfyε,kdx≤1m∫x∈S:yε,kx>myε,kfyε,kdx+∫x∈S:yε,kx≤mfmdx≤1m∫Ω∣yε,k∣fyε,kdx+fm∣S∣≤by45Cε∗∗m+fm∣S∣≤δ2+δ2.
As a result, we see that the sequencefyε,kk∈ℕ is equi-integrable and, hence, the desired convergence (57) is a direct consequence of the pointwise convergence (56) and Vitali’s convergence theorem. From this and (51), we obtain
(67)ξε=fyε,fyε∈H−1Ω∩L1Ω.
Since the set of admissible controlsU∂ is convex and closed in LpΩ, it follows from Mazur’s theorem that it is sequentially closed with respect to the weak topology of LpΩ. Hence, uε∈U∂. Thus, in order to decide that uε,yε,zε is a feasible solution to the regularized problems (32)–(35), it remains to show that this tuple is related by the variational equality (36). To do so, we utilize the following integral identity
(68)∫Ω∇yε,k,∇φ+∇zε,k,∇φ−uε,kφdx=∫Ωfyε,kφdx=fyε,k,φH−1Ω;H01Ω,which holds true for each φ∈H01Ω, k∈ℕ, and ε>0. Taking into account properties (48)–(51), (67), the limit passage in (68) as k⟶∞ becomes trivial. As a result, we arrive at the following relation
(69)∫Ω∇yε,∇φ+∇zε,∇φ−uεφdx=<fyε,φ>H−1Ω;H01Ω=by46∫Ωfyεφdx,∀φ∈H01Ω.
Thus,uε,yε,zε is a feasible solution to the regularized problems (32)–(35).
To conclude the proof, let us show that, in fact, the tripletuε,yε,zε is optimal to the problem (32)–(35). Indeed, in view of the strong convergence (57) and lower semicontinuity of norms in reflexive Banach spaces H01Ω, H−1Ω, and LpΩ with respect to the weak convergence, passing to the limit in (42), we obtain
(70)με=infu,y,z∈ΛεJεu,y,z=limk→∞Jεuε,k,yε,k,zε,k≥liminfk→∞Jεuε,k,yε,k,zε,k≥Jεuε,yε,zε≥infu,y,z∈ΛεJεu,y,z=με.
Thus, the equality (41) holds true with uε0,yε0,zε0=uε,yε,zε and, therefore, the tuple uε,yε,zε is optimal for regularized problems (21)–(24).


## 4. Optimality Conditions for Regularized Problem

In this section, we focus on deriving of optimality conditions for regularized optimal control problem (32)–(35) corresponding to the case U∂=LpΩ.We begin with the following observations.Remark 9.
In spite of the natural expectations, the mappingu,y↦z, where z=zu,y is the solution of (34)–(35) associated to u and y, is not of class C1 from LpΩ×H01Ω⟶H01Ω. Indeed, if we assume this property, then the mapping y↦fy should be of class C1 from H01Ω into H−1Ω on the subset Y⊂H01Ω defined by
(71)Y=y∈H01Ω:fy∈L1Ω∩H−1Ω.
Apart from the fact that, under the assumptions on the functionf, it is not clear whether the set Y has a nonempty interior intY. Even if it is so, then the assumption that the mapping y↦fy is of class C1 at some point y0∈intY would imply f′y0h∈H−1Ω for all h∈H01Ω. However, when N≥3, even for C∞ function such as fy=expy, this result does not hold.
Indeed, let us consider the caseN=3, fy=expy, and Ω is the unit ball B10⊂ℝ3. Then, the function yx=−λlog∣x∣, for 0<λ<5/2, satisfies y∈Y. Now, if we set hx=x−αφx for 0<α<1/2 and a function φ∈C0∞Ω such that 0≤φ≤1 and φx=1 for ∣x∣<1/2, we have h∈H01Ω and f′yxhx=x−λ+αφx. However, f′yh∈H−1Ω when α+λ>5/2.Remark 10.
In practice, the numerical simulation of∥·∥H−1Ω-term in the cost functional is quite specific and a delicate matter. Usually, it is associated with providing a very precise numerical analysis. In order to avoid these difficulties, it makes sense to substitute the ∥·∥H−1Ω term in the cost functional by some equivalent norm. For instance, let u∗ be an element of H−1Ω such that
(72)u∗,uH−1Ω;H01Ω=∫ΩF,∇uℝNdx=∫ΩF1∂u∂x1+⋯+FN∂u∂xNdx,∀u∈H01Ω,where F=F1,⋯,FN in L2Ω;ℝN is a given a vector-function.
It is clear that(73)u∗H−1Ω≤∫ΩF12x+⋯+FN2xdx.
On the other hand, due to the Lax-Milgram theorem, the Dirichlet boundary value problem(74)−Δy=u∗inΩ,y=0on∂Ω,has a unique solution y=−Δ−1u∗∈H01Ω for each u∗∈H−1Ω. Moreover, in view of the energy equality
(75)∫Ω∇y,∇yℝNdx=∇yL2Ω;ℝN2=yH01Ω2=u∗,yH−1Ω;H01Ω,which holds true for the weak solution of Dirichlet problems (74), we can deduce the following a priori estimate
(76)yH01Ω≡−Δ−1u∗H01Ω≡∇−Δ−1u∗L2Ω;ℝN≤u∗H−1Ω.
Combining this result with (73), we obtain the following chain of inequalities for the dual norm ∥·∥H−1Ω in H−1Ω:
(77)∇−Δ−1u∗L2Ω;ℝN≤u∗H−1Ω≤∫ΩF12x+⋯+FN2xdx=by50∫Ω∇yℝN2dx=∇yL2Ω;ℝN=∇−Δ−1u∗L2Ω;ℝN.
Hence, in this case, the standard norm inH−1Ω is equivalent to the following one
(78)u∗H−1Ω=∇−Δ−1u∗L2Ω;ℝN,∀u∗∈H−1Ω.
Taking this fact into account, in this section, we specify the cost functional (32) as follows
(79)Jεu,y,z=12∫Ωy−yd2dx+αp∫Ωupdx+12ε∫Ω∇z2dx+ε2∇−Δ−1fyL2Ω;ℝN2+fyL1Ω.As a result, to derive optimality conditions for regularized optimal control problem (79), (33)–(35), we apply the following reasoning. Let ε>0 be a fixed value. In addition to (5), we assume that f0≠0 and fy=F′y are a convex function for which there exists a constant CM>0 such that
(80)fx1+x2−fx1≤CMfx1fx2f0−1,∀x1,x2∈ℝ.Note that this property does not come into conflict with relation (6), and as a particular case of fx satisfying (80) is fx=Cexpkx.LetK be the following subset of H01Ω(81)K=y∈H01Ω:Δy∈LpΩ,fy∈LpΩ.In view of the properties of functionf, it is unknown whether this set has a nonempty interior. However, taking into account that p≥2 and 2>2N/N+2, it follows from Sobolev embedding theorem that fy∈H−1Ω for all y∈K.We know that boundary value problem (34)–(35) has a unique solution z∈H01Ω for every u∈LpΩ and y∈K. Let zε0,uε0,yε0∈H01Ω×U∂×K be an optimal solution to the problem (79), (33)–(35) with fyε0∈LpΩ and Δyε0∈LpΩ.Letw∈C0∞Ω and g∈C0∞Ω be arbitrary chosen functions. Then, property (80) immediately implies that
(82)fyε0+λw∈LpΩ∩H−1Ωforλsmallenough.By convexity off, we have
(83)fyε0+λw≥fyε0+λf′yε0w,fyε0−λw≥fyε0−λf′yε0w.Then,(84)fyε0−fyε0−λw≤λf′yε0w≤fyε0+λw−fyε0,where fyε0−fyε0−λw and fyε0+λw−fyε0 belong to LpΩ∩H−1Ω (see (82)). From this, we deduce that
(85)f′yε0w∈LpΩ∩H−1Ω.For everyλ∈ℝ, λ≠0, we set
(86)uλ=uε0−λΔg−λΔw−fyε0+λw+fyε0,yλ=yε0+λw,zλ=zε0+λg.Then, property (80) implies that yλ∈K and −Δzλ=Δyλ+fyλ+uλ in Ω. So, uλ,yλ,zλ is a feasible point for the problem (32)–(35). As follows from (84)–(85), there exists an element
(87)rw,λ∈LpΩ∩H−1Ω,such that
(88)fyε0+λw=fyε0+λf′yε0w+rw,λ,and rw,λLpΩ∩H−1Ω=o∣λ∣ as λ⟶0. Let us show that, for a given w∈C0∞Ω, the following extra properties hold true
(89)f′yε0w2fyε0∈L1Ω,r2w,λfyε0LpΩ=oλasλ⟶0.Indeed, due to property (80), we deduce from (84) and (88) that
(90)CMfyε0fλwf0−1≥λf′yε0w,CMfyε0fλwf0−1≥rw,λ.Sincefλw∈L∞Ω, the above inequalities imply that
(91)f′yε0wfyε0∈L∞Ω,rw,λfyε0∈L∞Ω.Hence,(92)f′yε0w2fyε0L1Ω≤f′yε0wfyε0L∞Ω∥f′yε0w∥L1Ω<by58+∞,r2w,λfyε0LpΩ≤rw,λfyε0L∞Ω∥rw,λ∥LpΩ<by59+∞.In order to deduce the asymptotic propertyrw,λLpΩ∩H−1Ω=o∣λ∣ as λ⟶0, we utilize property of f. Then, for a given w∈C0∞Ω and λ>0 sufficiently small, we have
(93)fyε0+λw−fyε0≤CMfyε0fλwf0−1.As a result, we see that(94)fyε0+λw⟶fyε0inLpΩasλ⟶0.From this and definition of the directional derivative, we finally deduce(95)limλ→0rw,λλLpΩ=by60limλ→0fyε0+λw−fyε0λ−f′yε0wLpΩbytheLebesgueDominatedTheorem=limλ→0fyε0+λw−fyε0λ−f′yε0wLpΩ=0.Hence,rw,λLpΩ=o∣λ∣ as λ⟶0.In the functionalJε, we will distinguish three terms
(96)Jεu,y,z=J1u+J2,εy+J3,εz,where
(97)J1u=αp∫Ωupdx,J2,εy=12∫Ωy−yd2dx+ε2∇−Δ−1fyL2Ω;ℝN2+fyL1Ω,J3,εz=12ε∫Ω∇z2dx.Now, using Lebesgue’s convergence theorem and the fact thatuε0,yε0,zε0 is an optimal triplet, we get
(98)0≤limλ→0Jεuλ,yλ,zλ−Jεuε0,yε0,zε0λ=limλ→0J1uλ−J1uε0λ+limλ→0J2,εyλ−J2,εyε0λ+limλ→0J3,εzλ−J3,εzε0λ=A1+A2+A3,where
(99)A1=αplimλ→0∫Ωuλp−uε0pλ=α∫Ωuε0p−2uε0−Δw−Δg−f′yε0wdx,(100)A3=12εlimλ→0∫Ω∇zλ2−∇zε02λdx=1ε∫Ω∇zε0,∇gdx.As for the termA2, we notice that
(101)limλ→0∫Ωyε0+λw−yd2−yε0−yd22λdx=∫Ωyε0−ydwdx,(102)limλ→0εfyε0+λwL1Ω−fyε0L1Ω2λ=ε2limλ→0∫Ωf2yε0+λw−f2yε0λfyε0+λw+fyε0dx=by60ε2∫Ωf′yε0wdx+ε2limλ→0Rλ,where
(103)Rλ=∫Ω2fyε0rw,λ+2λf′yε0wrw,λ+λ2f′yε0w2+r2w,λλfyε0+λw+fyε0dx.Here, we have utilized the following obvious equality(104)limλ→0∫Ω2λfyε0f′yε0wλ∣fyε0+λw∣+∣fyε0∣dx=∫Ωfyε0fyε0f′yε0wdx=∫Ωf′yε0wdx,and the fact that fyε0/∣fyε0∣=1 for the nonnegative function f.Since(105)2λf′yε0wrw,λ≤λ2f′yε0w2+r2w,λ,it follows from (89) that
(106)∣Rλ∣≤1λ∫Ω2fyε0rw,λ+2λ2f′yε0w2+2r2w,λfyε0+λw+fyε0dx≤1λ∫Ω2fyε0rw,λ+2λ2f′yε0w2+2r2w,λ∣fyε0∣dx≤2∥rw,λ∥L1Ωλ+2λf′yε0w2fyε0L1Ω+1λr2w,λfyε0L1Ω=by611λoλ.Then, (102) implies that
(107)limλ→0εfyε0+λwL1Ω−fyε0L1Ω2λ=ε2∫Ωf′yε0wdx.Since for any elementζ∈H−1Ω, we have −Δ−1ζ∈H01Ω, it follows that
(108)∇−Δ−1η+ζL2Ω;ℝN2−∇−Δ−1ηL2Ω;ℝN2=2∇−Δ−1η,∇−Δ−1ζL2Ω;ℝN+∇−Δ−1ζL2Ω;ℝN2=−2div∇−Δ−1η,−Δ−1ζH−1Ω;H01Ω+ζH−1Ω2=2η,−Δ−1ζH−1Ω;H01Ω+ζH−1Ω2=2ζ,−Δ−1ηH−1Ω;H01Ω+ζH−1Ω2,∀η∈H−1Ω.Hence,(109)limλ→0ε2λ∇−Δ−1fyε0+λwL2Ω;ℝN2−∇−Δ−1fyε0L2Ω;ℝN2=εf′yε0w,−Δ−1fyε0H−1Ω;H01Ω=ε∫Ω−Δ−1fyε0f′yε0wdx.As a result, utilizing relations (101), (107), and (109), we obtain
(110)A2=∫Ωyε0−yd+ε2f′yε0+ε−Δ−1fyε0f′yε0wdx.From the linearity ofA1, A2, and A3 with respect to w and g, we deduce from (98), (99), (100), and (110) that
(111)∫Ωyε0−yd+ε2f′yε0+ε−Δ−1fyε0f′yε0wdx+α∫Ωuε0p−2uε0−Δw−f′yε0wdx+1ε∫Ω∇zε0,∇gdx−α∫Ωuε0p−2uε0Δgdx=0,for every w∈C0∞Ω and g∈C0∞Ω.Let us setαuε0p−2uε0=με. Then, (111) implies that
(112)∫Ωyε0−yd+ε2f′yε0+ε−Δ−1fyε0f′yε0−μεf′yε0wdx+∫Ωμε−Δwdx=0,∀w∈C0∞Ω,(113)1ε∫Ω∇zε0,∇gdx−∫ΩμεΔgdx=0,∀g∈C0∞Ω.From the last equality, we immediately deduce thatμε=−1/εzε0+d, where dε∈C2Ω is a weakly harmonic function (it satisfies Laplace’s equation Δd=0 in the sense of distributions). As a consequence, we have με∈H1Ω.Taking into account thatC0∞Ω is dense in H01Ω, we see that relations (112)–(113) can be rewritten as follows.
(114)∫Ωyε0−yd+ε2f′yε0+ε−Δ−1fyε0f′yε0wdx−∫Ωμεf′yε0wdx+<−Δμε,w>H−1Ω;H01Ωdx=0,με=−1εzε0+da.e.inΩ,for every w∈C0∞Ω and g∈C0∞Ω.Thus, we can summarize the obtained result as follows.Theorem 11.
LetΩ be a bounded open domain in ℝN with N≥1. For a given p≥2, let K be the subset defined as in (81). Assume that f:ℝ⟶0,+∞ is a monotonically increasing function such that f∈C1ℝ and this function is convex and satisfies property (80). If uε0,yε0,zε0∈U∂×K×H01Ω is an optimal solution to the problem (79), (33)–(35), then
(115)−Δzε0=Δyε0+fyε0+uε0inΩ,zε0=0on∂Ωand setting με=αuε0p−2uε0, one has με=−1/εzε0+d a.e. in Ω, and
(116)yε0−yd+ε2f′yε0+ε−Δ−1fyε0f′yε0−μεf′yε0=ΔμεinD′Ω,Δd=0inD′Ω.


## 5. Asymptotic Analysis of Regularized Optimal Control Problem

Our main aim in this section is find out whether the original optimal control problem (1)–(4) is solvable under assumptions (a)–(c) and its optimal solutions can be attained (in some sense) by optimal solutions to the regularized problem (32)–(35).The key point of our consideration is that, in contrast to the well-known approaches (see, for instance, [20, 27, 28]), we do not assume here the fulfillment of the “standard” extra properties such that the domain Ω is an open subset of ℝN with N>2, this domain should be star-shaped and exists a weak solution y∈H01Ω of Dirichlet problem (2)–(3) satisfying fy∈L2Ω. Because of this, the existence of at least one optimal pair to the problem (1)–(4) is an open question provided we restrict our consideration only by assumptions (a)–(c).In what follows, in order to guarantee the consistency of the original problem (1)–(4), we accept the following hypothesis.Hypothesis 12.
The set of feasible solutionsΞ to the problem (1)–(4) is nonempty.It is worth to notice that the verification of Hypothesis12 is not too restrictive from practical implementation point of view. Indeed, let y~∈C0∞Ω be an arbitrary function. Then, it is clear that y~∈H01Ω and fy~∈L1Ω, that is, y∈Hf. Let us define the control u~∈LpΩ as follows u~=−Δy~−fy~ in Ω. Then, u~,y~ is a feasible pair to the problem (1)–(4) if only u~∈U∂. So, this hypothesis is obviously true if we do not impose any additional restrictions on the class of admissible controls, i.e., U∂=LpΩ.The following result is crucial in this paper and it shows that solvability of the original OCP (1)–(4) in some sense is equivalent to its consistency, i.e., OCP (1)–(4) admits at least one solution if and only if Hypothesis 12 is fulfilled. However, in order to establish this fact, we apply an indirect approach based on the variant of Tikhonov regularization which is described in Section 4.Theorem 13.
LetΩ be a bounded open domain in ℝN with N≥1, let U∂ be a nonempty closed convex subset of LpΩ, 2≤p<+∞, and let f:ℝ⟶0,+∞ be a monotonically increasing function such that f∈Cℝ. Let uε0,yε0,zε0∈Λεε→0 be a sequence of optimal solutions to regularized problems (32)–(35) when the parameter ε>0 varies in a strictly decreasing sequence of positive numbers converging to 0. Assume that Hypothesis 12 holds true and the sequence fyε0ε→0 is bounded in L1Ω∩H−1Ω. Then, there is a subsequence of uε0,yε0,zε0ε→0, still denoted by the suffix ε, such that
(117)uε0⇀u0inLpΩ,yε0⇀y0inH01Ω,zε0⟶0inH01Ω,−Δy0=fy0+u0inDΩ,u∈U∂,u0,y0∈Ξ,Ju0,y0=infu,y∈ΞJu,y.Proof.
SinceΞ≠∅, it follows that, for a given δ>0, there exists uδ∗,yδ∗ such that uδ∗,yδ∗ is a feasible solution to optimal control problem (1)–(4), and
(118)infu,y∈ΞJu,y=infu,y∈Ξ12∫Ωy−yd2dx+αp∫Ωupdx≥Juδ∗,yδ∗−δ.
Hence, in view of Definition5 and Lemma 2, uδ∗∈U∂, yδ∗∈H01Ω, fyδ∗∈L1Ω∩H−1Ω, and the pair uδ∗,yδ∗ is related by integral identity (8). Therefore, for each ε>0, the triplet uδ∗,yδ∗,0 is a feasible solution for regularized problem (32)–(35), i,e, uδ∗,yδ∗,0∈Λε for all ε>0. Taking this fact into account, we see that
(119)Jεuε0,yε0,zε0=12∫Ωyε0−yd2dx+αp∫Ωuε0pdx+12ε∫Ω∇zε02dx+ε2fyε0H−1Ω2+fyε0L1Ω=infu,y,z∈ΛεJεu,y,z≤Jεuδ∗,yδ∗,0=12∫Ωyδ∗−yd2dx+αp∫Ωuδ∗pdx+ε2fyδ∗H−1Ω2+fyδ∗L1Ω=C1+εC2<+∞,where
(120)C1≔12∫Ωyδ∗−yd∣2dx+αp∫Ωuδ∗pdx,C2≔12fyδ∗H−1Ω2+fyδ∗L1Ω,and (see (118))
(121)12∫Ωyε0−yd2dx+αp∫Ωuε0pdx≥12∫Ωyδ∗−yd2dx+αp∫Ωuδ∗pdx−δ.
Since this relation holds true for eachε>0 varying in a given interval 0,ε0 and each δ>0, it follows that
(122)supε∈0,ε0uε0LpΩp≤C1+δpα,supε∈0,ε01εzε0H01Ω2≤2C1+ε0C2,(123)supε∈0,ε0yε0L2Ω2≤4C1+δ+2ydL2Ω2.
In addition, the sequencefyε0ε∈0,ε0 is assumed to be bounded in L1Ω∩H−1Ω. Taking this fact into account, we deduce that
(124)supε∈0,ε0fyε0H−1Ω2≤2C2+C1,supε∈0,ε0fyε0L1Ω≤2C2+C1.
Hence, the sequencesuε0ε∈0,ε0, yε0ε∈0,ε0, and fyε0ε∈0,ε0 are weakly compact in LpΩ, L2Ω, and H−1Ω, respectively, whereas estimate (122) implies that the sequence zε0ε→0 is strongly convergent to 0 in H01Ω. So, we can suppose that there exist elements u0∈LpΩ, y∈L2Ω, ξ∈H−1Ω, and a sequence εkk∈ℝ monotonically converging to zero as k⟶∞ such that
(125)uεk0⟶u0weaklyinLpΩ,(126)yεk0⟶y0weaklyinL2Ω,(127)fyεk0⟶ξweaklyinH−1Ω,(128)zεk0⟶0stronglyinH01Ωask⟶∞.
Let us show that, in fact, we have the weak convergenceyεk0⇀y0 in H01Ω. Indeed, arguing as in the proof of Theorem 8, we utilize the integral identity
(129)∫Ω∇yεk02+∇zεk0,∇yεk0−uεk0yεk0dx=fyεk0,yεk0H−1Ω;H01Ω,which holds true for each ε∈0,ε0 and reflects the fact that the triplets uεk0,yεk0,zεk0k∈ℝ are feasible to the problem (32)–(35) for each k∈ℝ. Then, we deduce from (129) that
(130)∥∇yεk0∥L2ΩN2≤zεk0H01Ω+fyεk0H−1Ω+CΩΩp−2/2puεk0LpΩ∇yεk0L2ΩN.
Hence, estimates (122)–(124) imply that
(131)supk∈ℕ∇yεk0L2ΩN≤2ε0C1+ε0C2+2C2+CΩΩp−2/2pC1+δpα≕C∗<+∞.
Thus, without loss of generality, we can suppose that (up to a subsequence)(132)yεk0⇀y0inH01Ω,yεk0⟶y0inL2Ω,yεk0x⟶y0xa.e.inΩ.
Utilizing the pointwise convergence (132)3 and (c)-property, we see that fyεk0⟶fy0 almost everywhere in Ω as k⟶∞. Let us show that, in fact, we have the strong convergence
(133)fyεk0⟶fy0inL1Ωask⟶∞,and, as a consequence of (127), ξ=fy0 and fy0∈H−1Ω∩L1Ω.
To this end, we make use of some arguments of the proof of Theorem8. With that in mind, for each k∈ℕ, we make use of the decomposition yεk−=yεk+−yεk− with
(134)yεk+=max0,−yεk∈H01Ω,yεk−=max0,−yεk∈H01Ω,and set Tηyx=η,ifyx>η,yx,ifyx≤η. Then, Tηy∈L∞Ω∩H01Ω for all y∈H01Ω such that y≥0. Using yε,k+∈H01Ω as a test function in (25), we find that
(135)∫Ωfyε,kTηyε,k+dx=∫Ω∇zε,k,∇Tηyε,k+dx+∫Ω∇yε,k,∇Tηyε,k+dx−∫Ωuε,kTηyε,k+dx,∀η∈ℕ.
SinceTηyε,k+⟶yε,k+ in H01Ω as η⟶∞, the limit passage in (135) as η⟶∞ leads to the relation
(136)∫Ωfyεkyεk+dx=∫Ω∇zεk,∇yεk+dx+∫Ω∇yεk,∇yεk+dx−∫Ωuεkyεk+dx.
From this and the fact thatyεk+H01Ω≤yεkH01Ω, we deduce the estimate
(137)∫Ωfyεkyεk+dx≤zεkH01Ω+yεkH01Ω+CΩΩp−2/2puεkLpΩyεkH01Ω≤by84,8476C∗2C1+ε0C2+C∗+CΩΩp−2/2pC1+δpα1/p.
Arguing in a similar manner, it can be shown that(138)∫Ωfyεkyεk−dx≤C∗2C1+ε0C2+C∗+CΩΩp−2/2pC1+δpα1/p.
Sinceyεk++yεk−=yεk, it follows from (137)–(138) that there exists a positive constant C^, independent of ε and yεk such that
(139)supk∈ℕfyεkyεkL1Ω≤C^.
In order to prove the strong convergence (133), it remains to make use of Vitali’s theorem. We fix an arbitrary ζ>0 and take m>0 and τ>0 such that m>2C^/ζ, τ=ζ/2fm. Then, for every measurable set S⊂Ω with Lebesgue measure ∣S∣<ζ, we have
(140)∫Sfyεkdx=∫x∈S:yεkx>mfyεkdx+∫x∈S:yεkx≤mfyεkdx≤1m∫x∈S:yεkx>myεkfyεkdx+∫x∈S:yεkx≤mfmdx≤1m∫Ω∣yεk∣fyεkdx+fm∣S∣≤by91C^m+fm∣S∣≤ζ2+ζ2.
As a result, we see that the sequencefyεkk∈ℕ is equi-integrable and, hence, the desired convergence (133) is a direct consequence of the pointwise convergence fyεk0⟶fy0 almost everywhere in Ω and Vitali’s convergence theorem.
We are now in a position to show thatu0,y0 is a feasible solution to the original OCP (1)–(4). Indeed, in view of the initial assumptions, the set U∂ is sequentially closed with respect to the weak topology of LpΩ. Hence, u0∈U∂. It remains to show that the pair u0,y0 is related by the integral identity (8). To this end, we note that uεk0,yεk0,zεk0∈Λεk for all k∈ℝ. Hence, the equality
(141)∫Ω∇yε,k,∇φ+∇zε,k,∇φ−uε,kφdx=fyε,k,φH−1Ω;H01Ω,holds true for each test function φ∈C0∞Ω. As a result, the limit passage in (141) as k⟶∞ becomes trivial, and it immediately leads us to the expected integral identity (8). Thus, combining all properties of the pair u0,y0 established here and before, we finally deduce that u0,y0∈Ξ.
To conclude the proof, we have to show thatu0,y0∈Ξ in an optimal pair to the problem (1)–(4). To do so, we assume the converse, namely, there is a pair u^,y^∈Ξ such that Ju^,y^<Ju0,y0. Then, the triplet u^,y^,0 is feasible to the regularized problem i.e.,u^,y^,0∈Λεk for each k∈ℕ. Hence,
(142)Ju^,y^+εk2fy^H−1Ω2+fy^L1Ω=Jεku^,y^,0≥infu,y,z∈ΛεkJεku,y,z=Jεkuεk0,yεk0,zεk0,∀k∈ℕ.
Therefore, passing in (142) to the limit as k⟶∞ and using the properties
(143)liminfk→∞∫Ωyεk0−yd2dx=by85∫Ωy0−yd2dx,liminfk→∞∫Ωuεk0pdx≥by8579∫Ωu0pdx,limk→∞1εkzεk0H01Ω2=by857976const>0,liminfk→∞εk2fyεk0H−1Ω2=by857976810,limk→∞εk2fyεk0L1Ω=by85797681860,we obtain
(144)Ju^,y^≥liminfk→∞Jεkuεk0,yεk0,zεk0≥Ju0,y0+limk→∞1εkzεk0H01Ω2≥Ju0,y0.
As a result, it leads us to a contradiction. Thus,u0,y0∈Ξ in an optimal pair to the problem (1)–(4).
To the end of proof, we note if the original OCP admits a unique solution, then the asymptotic analysis given before remains valid for each subsequenceuεk0,yεk0,zεk0k∈ℝ of the sequence of optimal solutions uε0,yε0,zε0∈Λεε→0. Therefore, the limits in (125)–(128) do not depend on the choice of a subsequence, and, hence, u0,y0∈Ξ is a unique limit pair for the entire sequence of optimal triplets uε0,yε0ε>o.



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*Source: 7418707-2020-11-17.xml*