Stability of Nonlinear Dirichlet BVPs Governed by Fractional Laplacian

Abstract

We consider a class of partial differential equations with the fractional Laplacian and the homogeneous Dirichlet boundary data. Some sufficient condition under which the solutions of the equations considered depend continuously on parameters is stated. The application of the results to some optimal control problem is presented. The methods applied in the paper make use of the variational structure of the problem.

1. Introduction

Consider the following fractional partial differential equation with some variable distributed parameters of the form (−Δ)^𝛼/2𝑢(𝑥)+𝜑(𝑥,𝑢(𝑥),𝜔(𝑥))=0for𝑥∈Ω⊂ℝ^𝑛,𝑢(𝑥)=0for𝑥∈ℝ^𝑛⧵Ω, (1)(2) where 𝑛≥2, Ω is a bounded domain with a Lipschitzian boundary 𝜕Ω, and 𝑢∈𝐻₀^𝛼/2(Ω,ℝ) with 𝛼∈(0,2). We shall assume that the distributed parameter 𝜔 belongs to the space 𝐿^𝑝(Ω,ℝ^𝑚) for some suitably chosen 𝑝>1 and 𝑚≥1.

The equation under consideration is the generalization of the nonlinear Poisson equation involving the Brownian diffusion expressed by the local Laplace operator fully analyzed in [1–3]. We extend our considerations to cover also the case of the nonlocal, fractional Laplace operator being the infinitesimal generator of Lévy processes; see, for instance, [4–7], allowing, contrary to the continuous Brownian motion, for jumps. We prove the analogous stability results as for the Brownian motion with the Laplace operator involved obtained in [1–3].

The problems with the fractional Laplacian attracted in recent years a lot of attention as they naturally arise in various areas of applications to mention only [5–11] and references therein. They appear in probabilistic framework as well as in mathematical finance as infinitesimal generators of stable Lévy processes [4–7]. Moreover one can find the problems involving fractional Laplacian in mechanics and in elastostatics, for example, in Signorini obstacle problem originating from linear elasticity [12–14] as well as in fluid mechanics and in hydrodynamics—appearing in quasi-geostrophic fractional Navier-Stokes equation [15] and describing some porous media flows in the hydrodynamic model like in [11]. The author considered also global solvability of Hammerstein equations derived from BVPs involving fractional Laplacian in recent paper [16].

In the theory of boundary value problems (BVPs) and its applications one considers, first of all, the problem of the existence of a solution, next the question of its stability, uniqueness, and smoothness, and finally the issue of asymptotic analysis. One can say that a given problem is well posed if the problem possesses at least one solution or, more generally, one obtains the set of solutions, which continuously changes along with the change of variable parameters of the system which we call stability. Otherwise we refer to the problem as to ill-posed one. The requirement of stability is necessary if the mathematical formulation is to describe observable natural phenomena, which by its very nature cannot possibly be conceived as rigidly fixed: even the mere process of measuring them involves small errors as was noted by Courant and Hilbert in [17]. The theory of ill-posed problems pays most attention to the requirement of the stability of the boundary value problems.

In this paper we formulate some sufficient condition under which the boundary value problem considered here possesses at least one solution which continuously depends on distributed parameters. The problem of controllability of the related evolution equations driven by the anomalous diffusion governed by the fractional Laplacian was considered, for example, in [18].

The paper is organized as follows. In Section 2 we formulate the problem and list the assumptions appearing throughout the paper. In Section 3, using some variational methods we prove that boundary value problem (1)-(2) is stable with respect to the norm topology in the space of distributed parameters 𝐿^𝑝(Ω,ℝ^𝑚) and the norm topology in the fractional Sobolev space of solutions 𝐻^𝛼/2(Ω,ℝ). We can formulate the main result of Section 3 as follows: if 𝜔_𝑘→𝜔₀ in 𝐿^𝑝(Ω,ℝ^𝑚), then 𝑢_𝑘→𝑢₀ in 𝐻^𝛼/2(Ω,ℝ) where 𝑢_𝑘 is the solution of the boundary value problem (1)-(2) with fixed 𝜔=𝜔_𝑘, 𝑘∈ℕ₀ under suitable conditions imposed on 𝜑. In the case when (1) is linear with respect to 𝜔, we can relax the topology in the space 𝐿^𝑝(Ω,ℝ^𝑚). In short, in Section 4, we prove that 𝑢_𝑘→𝑢₀ in 𝐻^𝛼/2(Ω,ℝ) provided that 𝜔_𝑘⇀𝜔₀ weakly in 𝐿^𝑝(Ω,ℝ^𝑚). In the next section, we present a theorem on the existence of an optimal solution to some control problem with the integral cost functional. The proof of this theorem relies in essential way on the continuous dependence results. In the final part of the paper we give a short survey of the results related to the stability of the initial and boundary value problems for the second-order partial differential systems with parameters.

2. Formulation of the Problem, Introduction of the Fractional Laplacian, and Basic Assumptions

For the definition of the fractional Laplacian one can see [19–25]. In particular, we denote by (𝑢_𝑗,𝜌_𝑗) for 𝑗∈ℕ the system of the eigenfunctions and eigenvalues for the Laplace operator −Δ on Ω with the homogeneous Dirichlet condition on 𝜕Ω. Moreover, by 𝐻₀^𝛼/2(Ω,ℝ), let us denote the Sobolev space of functions 𝑢=𝑢(𝑥) defined on a bounded, smooth domain Ω⊂ℝ^𝑛, 𝑛≥2, such that 𝑢=∑^∞_𝑗=1𝑎_𝑗𝑢_𝑗 and ∑^∞_𝑗=1𝑎²_𝑗𝜌_𝑗^𝛼/2<∞, with the norm in 𝐻₀^𝛼/2(Ω,ℝ) with 𝛼∈(0,2) defined by the equivalent formulas ‖𝑢‖²_𝐻0𝛼/2=^∞_𝑗=1𝑎²_𝑗𝜌_𝑗^𝛼/2=(−Δ)^𝛼/4𝑢²_𝐿2=(−Δ)^𝛼/2𝑢²_{𝐻−𝛼/2}, (3) see, for example, [20, 23] and for the last equality, see, for example, [19]. The fractional Laplacian acts on 𝑢=∑^∞_𝑗=1𝑎_𝑗𝑢_𝑗 as (−Δ)^𝛼/2𝑢=^∞_𝑗=1𝑎_𝑗𝜌_𝑗^𝛼/2𝑢_𝑗. (4) The fractional Sobolev spaces are also referred to as Gagliardo or Slobodeckij spaces. One can give yet another definition of 𝐻^𝛼/2(Ω,ℝ) as follows: 𝐻^𝛼/2()=Ω,ℝ𝑢∈𝐿²()∶𝑢(𝑥)𝑦Ω,ℝ−𝑢𝑥−𝑦^{(𝑛+𝛼)/2}∈𝐿²(Ω×Ω,ℝ) (5) with the norm ‖𝑢‖²_𝐻𝛼/2=_Ω|𝑢(𝑥)|²𝑑𝑥+_Ω×Ω𝑢(𝑥)−𝑢𝑦²𝑥−𝑦^𝑛+𝛼𝑑𝑥𝑑𝑦. (6) For the definition of the fractional Laplacian operator involving singular integrals consistent with ours when 𝑢 is extended by 0 outside Ω, we refer the readers to [23], where one can find the following lemma.

Lemma 1. Let 𝛼∈(0,2) and let (−Δ)^𝛼/2 be the fractional Laplacian operator of the form (−Δ)^𝛼/2𝑢(𝑥)=𝐶(𝑛,𝛼)lim_{𝜀→0+ℝ𝑛()⧵𝐵𝑥,𝜀}𝑢(𝑥)−𝑢𝑦𝑥−𝑦^𝑛+𝛼𝑑𝑦, (7) where 𝐶(𝑛,𝛼)=𝜋^{−(𝛼+𝑛/2)}(Γ((𝑛+𝛼)/2)/Γ(−𝛼/2)). Then for any 𝑢 from the Schwartz space of rapidly decaying 𝐶^∞ functions in ℝ^𝑛 we have (−Δ)^𝛼/2𝑢(𝑥)1=−2_ℝ^𝑛𝑢𝑥+𝑦+𝑢𝑥−𝑦−2𝑢(𝑥)𝑥−𝑦^𝑛+𝛼𝑑𝑦 (8) for all 𝑥∈ℝ^𝑛 (cf. [23, Lemma 3.5]).

Throughout the paper, we shall assume that Ω satisfies any condition which guarantees a compact embedding of 𝐻₀^𝛼/2(Ω,ℝ) into 𝐿^𝑠(Ω,ℝ) with 𝑠∈(1,2^∗_𝛼) where 2^∗_𝛼=2𝑛/(𝑛−𝛼) if 𝑛≥2, for example, 𝜕Ω may be Lipschitzian; that is, 𝜕Ω∈𝐶^0,1 (cf. [26] for the definition of 𝐶^0,1). For 𝜕Ω∈𝐶^0,1 it is possible to extend 𝑢 by 0 outside Ω and stay in the same space; see [23, Theorem 5.4].

Further, in this paper we shall use the primitive 𝜙 of the mapping 𝜑∶Ω×ℝ×ℝ^𝑚→ℝ, implying 𝜑 to be defined as the derivative with respect to 𝑢 variable of a function 𝜙∶Ω×ℝ×ℝ^𝑚→ℝ; that is 𝜑(𝑥,𝑢,𝜔)=𝜙_𝑢(𝑥,𝑢,𝜔), (9) where 𝑥∈Ω a.e., 𝑢∈ℝ, and 𝜔∈ℝ^𝑚.

In this case boundary value problem (1)-(2) may be written in the form suitable for variational analysis (−Δ)^𝛼/2𝑢(𝑥)+𝜙_𝑢(𝑥,𝑢(𝑥),𝜔(𝑥))=0for𝑥∈Ω⊂ℝ^𝑛𝑢(𝑥)=0for𝑥∈ℝ^𝑛⧵Ω, (10)(11) where 𝜔∈𝐿^𝑝(Ω,ℝ^𝑚), 𝑝>1 and 𝑚≥1. It is easily seen that (10)-(11) represent the Euler-Lagrange equation for the following functional of action: 𝐹_𝜔(𝑢)=_Ω12(−Δ)^𝛼/4𝑢(𝑥)²+𝜙(𝑥,𝑢(𝑥),𝜔(𝑥))𝑑𝑥, (12) where 𝑢∈𝐻₀^𝛼/2(Ω,ℝ) and 𝜔∈𝐿^𝑝(Ω,ℝ^𝑚). It should be underlined that the solutions of Euler-Lagrange equation (10)-(11) are meant in the weak sense; that is, for any 𝑣∈𝐻₀^𝛼/2(Ω,ℝ) the following equality holds: _Ω(−Δ)^𝛼/4𝑢(𝑥)(−Δ)^𝛼/4𝑣(𝑥)+𝑑𝑥_Ω𝜑(𝑥,𝑢(𝑥),𝜔(𝑥))𝑣(𝑥)𝑑𝑥=0. (13)

To obtain the existence of the weak solutions of the boundary value problem with fractional Laplacian (10)-(11) in the fractional Sobolev space 𝐻₀^𝛼/2(Ω,ℝ) and the continuous dependence of solutions on distributed parameters we shall impose on 𝜙 the following conditions.

(A1) regularity: the functions 𝜙 and 𝜙_𝑢 are measurable with respect to 𝑥 for any (𝑢,𝜔)∈ℝ×ℝ^𝑚 and continuous with respect to (𝑢,𝜔)∈ℝ×ℝ^𝑚 for a.e. 𝑥∈Ω.

(A2) growth: for 𝑝∈(1,∞), there exists a constant 𝑐>0 such that 𝜙(𝑥,𝑢,𝜔)≤𝑐1+|𝑢|^𝑠+|𝜔|^𝑝,𝜙_𝑢(𝑥,𝑢,𝜔)≤𝑐1+|𝑢|^𝑠−1+|𝜔|^{𝑝−𝑝/𝑠}, (14) for a.e. 𝑥∈Ω, 𝜔∈ℝ^𝑚 and 𝑢∈ℝ, where 𝑠∈(1,2^∗_𝛼) where 2^∗_𝛼=2𝑛/(𝑛−𝛼); for 𝑝=∞ and any bounded set 𝑊⊂ℝ^𝑚 there exists a constant 𝑐>0 such that 𝜙(𝑥,𝑢,𝜔)≤𝑐1+|𝑢|^𝑠,𝜙_𝑢(𝑥,𝑢,𝜔)≤𝑐1+|𝑢|^𝑠−1, (15) for a.e. 𝑥∈Ω, 𝜔∈𝑊, 𝑢∈ℝ, and some 𝑠∈(1,2^∗_𝛼), where 2^∗_𝛼=2𝑛/(𝑛−𝛼).

(A3) lower bound: there exist 𝑏∈ℝ and functions 𝛾∈𝐿²(Ω,ℝ), 𝛽∈𝐿¹(Ω,ℝ), such that 𝜙(𝑥,𝑢,𝜔)≥−𝑏|𝑢|²−𝛾(𝑥)𝑢−𝛽(𝑥), (16) for a.e. 𝑥∈Ω, 𝜔∈ℝ^𝑚, and 𝑢∈ℝ, where 𝜌₁^𝛼/2>2𝑏 and 𝜌₁ is the principal eigenvalue of the Laplace operator −Δ defined on the space 𝐻¹₀(Ω,ℝ).

(A4) convexity: the function 𝜙 is convex in 𝑢.

Remark 2. The principal eigenvalue 𝜌₁ of Laplacian appears in the inequality 𝜌₁^𝛼/2∫≤inf^Ω(−Δ)^𝛼/4𝑢(𝑥)²𝑑𝑥∫_Ω|𝑢(𝑥)|²𝑑𝑥;𝑢∈𝐻₀^𝛼/2(Ω,ℝ),𝑢≠0. (17) Indeed, (−Δ)^𝛼/4𝑢₁=𝜌₁^𝛼/4𝑢₁, so infimum on the right hand side of the above inequality is greater or equal to 𝜌₁^𝛼/2. Moreover, the infimum is attained since ‖𝑢‖²_𝐻0𝛼/2=∫_Ω|(−Δ)^𝛼/4𝑢(𝑥)|²𝑑𝑥 is weakly lower semicontinuous, convex, and coercive as the norm in the reflexive space; for details, see [8, 27].
To derive the fractional Poincaré inequality of the form 𝜌₁^𝛼/2_Ω|𝑢(𝑥)|²𝑑𝑥≤_Ω(−Δ)^𝛼/4𝑢(𝑥)²𝑑𝑥 (18) we apply the following theorem with 𝐹(𝑡)=𝑡^𝛼/2.

Theorem 3. Let 𝐹 be a continuous, increasing, and polynomially bounded real-valued functional on [0,∞), in particular, 𝐹(𝑡)>0 for 𝑡>0. Then we have the following fractional order Poincaré inequality: 𝐹√𝜌₁‖𝑢‖_𝐿²≤𝐹√−Δ𝑢_𝐿2, (19) compare [28, Theorem 2.8].

For the fractional Poincaré inequality with general measures involving nonlocal quantities on unbounded domain see paper by Mouhot et al. [29]. In what follows we shall also use the following result.

Remark 4. The fractional Sobolev inequality extending the above Poincaré inequality to 𝐿^𝑠(Ω,ℝ) with, in general, non optimal constant 𝐶>0, has the form (−Δ)_Ω^𝛼/4𝑢(𝑥)²𝑑𝑥≥𝐶_Ω|𝑢(𝑥)|^𝑠𝑑𝑥^2/𝑠 (20) for any 𝑠∈[1,2^∗_𝛼], 𝑛>𝛼, and every 𝑢∈𝐻₀^𝛼/2(Ω,ℝ). When 𝑠=2^∗_𝛼 the best constant in the fractional Sobolev inequality will be denoted by 𝑆(𝛼,𝑛). This constant is explicit and independent of the domain, its exact value is 𝑆(𝛼,𝑛)=2𝜋^𝛼/2Γ((𝑛+𝛼)/2)Γ((2−𝛼)/2)(Γ(𝑛/2))^𝛼/𝑛Γ(𝛼/2)Γ((𝑛−𝛼)/2)(Γ(𝑛))^𝛼/2, (21) where Γ is the standard Euler Gamma function defined by Γ(𝑎)=∫^∞₀𝑡^𝑎−1𝑒^−𝑡𝑑𝑡, compare [19].
When 𝑠=2 we recover the fractional Poincaré inequality without an optimal constant in general.

Remark 5. The fractional Sobolev space 𝐻₀^𝛼/2(Ω,ℝ) is compactly embedded into 𝐿^𝑠(Ω,ℝ) for 𝑠∈[1,2^∗_𝛼) and 𝜕Ω∈𝐶^0,1; see [23, Corollary 7.2].
Under assumptions (A1)-(A2) the functional of action defined in (12) is well defined and Fréchet differentiable and the derivative of 𝐹_𝜔 acting on 𝑣∈𝐻₀^𝛼/2(Ω,ℝ) has the form 𝐷𝐹_𝜔(𝑢)𝑣=_Ω(−Δ)^𝛼/4𝑢(𝑥)(−Δ)^𝛼/4𝑣(𝑥)+𝜙_𝑢(𝑥,𝑢(𝑥),𝜔(𝑥))𝑣(𝑥)𝑑𝑥. (22)

3. Continuous Dependence: Parameters Converging in the Strong Topology

Define {𝜔_𝑘}_𝑘∈ℕ to be some sequences of parameters distributed on Ω. For 𝑘∈ℕ₀∶={0}∪ℕ, we denote by 𝑈_𝑘, the set of all possible minimizers of the functional 𝐹_𝜔𝑘; that is 𝑈_𝑘=𝑢∈𝐻₀^𝛼/2(Ω,ℝ);𝐹_𝜔𝑘(𝑢)=min_{𝑦∈𝐻0𝛼/2()Ω,ℝ}𝐹_𝜔𝑘𝑦. (23) Since each minimizer 𝑢∈𝑈_𝑘 is a critical point of 𝐹_𝜔𝑘, that is, 𝐷𝐹_𝜔𝑘(𝑢)𝑣=0 for any 𝑣∈𝐻₀^𝛼/2(Ω,ℝ), it follows that 𝑢 is a weak solution of problem (10)-(11). Inversely, if 𝑢 is a weak solution of (10) satisfying (11), then 𝑢∈𝑈_𝑘 provided the functional 𝐹_𝜔𝑘 is convex (cf. [30, 31]). It is clear that, in general, the set 𝑈_𝑘 does not have to be a singleton and hence boundary value problem (10)-(11) does not have to possess a unique solution.

In the following theorem we shall use the definition of the upper Painlevé-Kuratowski limit of the sets (cf. [32]). We say that a set 𝑈⊂𝐻₀^𝛼/2(Ω,ℝ) is an upper limit of the sets 𝑈_𝑘, 𝑘∈ℕ if any point 𝑢∈𝑈 is a cluster point of some sequence {𝑢_𝑘}_𝑘∈ℕ in 𝐻₀^𝛼/2(Ω,ℝ) such that 𝑢_𝑘∈𝑈_𝑘 for 𝑘∈ℕ. By limsup𝑈_𝑘=𝑈, we shall denote the upper Painlevé-Kuratowski limit of the sets 𝑈_𝑘, 𝑘∈ℕ.

Now, we can formulate and prove the main result of this section.

Theorem 6. Assume that(1)the integrand 𝜙 satisfies conditions (A1)–(A3),(2)the sequence of distributed parameters {𝜔_𝑘}_𝑘∈ℕ tends to 𝜔₀ in 𝐿^𝑝(Ω,ℝ^𝑚) with 𝑝>1.
Then(a)for any 𝜔_𝑘, the set 𝑈_𝑘 is a nonempty subset of 𝐻₀^𝛼/2(Ω,ℝ), for 𝑘∈ℕ₀,(b)there exists a ball 𝐵(0,𝜌)⊂𝐻₀^𝛼/2(Ω,ℝ) for some 𝜌>0 such that 𝑈_𝑘⊂𝐵(0,𝜌) for 𝑘∈ℕ₀,(c)any sequence {𝑢_𝑘}_𝑘∈ℕ such that 𝑢_𝑘∈𝑈_𝑘 is relatively compact in 𝐻₀^𝛼/2(Ω,ℝ) and ∅≠limsup𝑈_𝑘⊂𝑈₀.
Additionally, if the sets 𝑈_𝑘 are singletons, that is, 𝑈_𝑘={𝑢_𝑘}, 𝑘∈ℕ₀, then {𝑢_𝑘}_𝑘∈ℕ tends to 𝑢₀ in 𝐻₀^𝛼/2(Ω,ℝ).

Before going to the proof, it is worth noting that, if 𝑈_𝜔 denotes the set of all possible minimizers of the functional 𝐹_𝜔 defined by (12), then assertion (c) of Theorem 6 states that the set valued mapping 𝐿^𝑝(Ω,ℝ^𝑚)∋𝜔↦𝑈_𝜔⊂𝐻₀^𝛼/2(Ω,ℝ) is upper semicontinuous with respect to the strong topology of spaces 𝐿^𝑝(Ω,ℝ^𝑚) and 𝐻₀^𝛼/2(Ω,ℝ).

Proof. Consider the following.
Step 1. In the first step we prove assertions (a) and (b) of our theorem.
For 𝑘∈ℕ₀, consider the functional 𝐹_𝜔𝑘(𝑢)=_Ω12(−Δ)^𝛼/4𝑢(𝑥)²+𝜙𝑥,𝑢(𝑥),𝜔_𝑘(𝑥)𝑑𝑥. (24) By assumption (2) of our theorem, ‖𝜔_𝑘‖_𝐿𝑝≤𝐶₀ for some 𝐶₀>0. By (A3), we have 𝐹_𝜔𝑘(𝑢)≥_Ω12(−Δ)^𝛼/4𝑢(𝑥)²−𝑏|𝑢(𝑥)|²−𝛾(𝑥)𝑢(𝑥)−𝛽(𝑥)𝑑𝑥 (25) and therefore the application of the fractional Poincaré inequality (18) gives 𝐹_𝜔𝑘(𝑢)≥12−𝑏𝜌₁^−𝛼/2‖𝑢‖²_𝐻0𝛼/2−𝐶₁‖𝑢‖_𝐻0^𝛼/2−𝐶₂=𝑝‖𝑢‖_𝐻0^𝛼/2 (26) with 𝜌₁^𝛼/2−2𝑏>0 from (A3), where 𝐶₁, 𝐶₂ are some constants independent of 𝜔_𝑘; however, depending on ‖𝛾‖_𝐿2 and ‖𝛽‖_𝐿1. The functional 𝐹_𝜔𝑘 is weakly lower semicontinuous on 𝐻₀^𝛼/2(Ω,ℝ) as a sum involving the norm in 𝐻₀^𝛼/2(Ω,ℝ), compare [8], and the integral term with 𝜙 satisfying the standard regularity and growth conditions (A1) and (A2), compare [33–36], as 𝐻₀^𝛼/2(Ω,ℝ)⊂𝐿^𝑠(Ω,ℝ) for suitably chosen 𝑠 in the embedding. Since, by (26), the functionals 𝐹_𝜔𝑘 are coercive, we infer that the sets 𝑈_𝑘 are nonempty and weakly closed. Moreover, by condition (A2), putting 𝑢=0, we get the following estimates due to the boundedness of 𝜔_𝑘 in 𝐿^𝑝(Ω,ℝ^𝑚)𝐹_𝜔𝑘(0)≤_Ω𝑐1+𝜔_𝑘(𝑥)^𝑝𝑑𝑥≤𝐷₁𝐹if1<𝑝<∞,_𝜔𝑘(0)≤_Ω𝑐𝑑𝑥≤𝐷₂if𝑝=∞, (27)(28) where the constants 𝐷₁ and 𝐷₂ are independent of 𝜔_𝑘. Directly from inequalities (26), (27), and (28) we obtain that for some 𝜌>0𝑈_𝑘⊂𝐵0,𝜌=𝑢∈𝐻₀^𝛼/2(Ω,ℝ)∶‖𝑢‖_𝐻0^𝛼/2≤𝜌. (29) We have thus proved assertions (a) and (b) of our theorem.
Step 2. For 𝑘∈ℕ₀, denote by 𝜇_𝑘 the minimal value of the functional 𝐹_𝜔𝑘; that is 𝜇_𝑘=min_{𝑢∈𝐻0𝛼/2}𝐹_𝜔𝑘(𝑢)=𝐹_𝜔𝑘(𝑢), (30) where 𝑢∈𝑈_𝑘. We shall observe that lim_𝑘→∞𝜇_𝑘=𝜇₀ (31) provided that 𝜔_𝑘→𝜔₀ in 𝐿^𝑝(Ω,ℝ^𝑚) as 𝑘→∞.
We begin by proving that the sequence {𝐹_𝜔𝑘(𝑢)}_𝑘∈ℕ tends to 𝐹_𝜔0(𝑢) uniformly on any ball 𝐵(0,𝜌)⊂𝐻₀^𝛼/2(Ω,ℝ) of radius 𝜌>0. By definition (24), we have 𝐼_𝑘(𝑢)=𝐹_𝜔𝑘(𝑢)−𝐹_𝜔0(𝑢=_Ω)𝜙𝑥,𝑢(𝑥),𝜔_𝑘(𝑥)−𝜙𝑥,𝑢(𝑥),𝜔₀(𝑥)𝑑𝑥. (32) Suppose that, on the contrary, the above integral does not tend to zero uniformly on 𝐵(0,𝜌). It means that there exists 𝜀₀>0 and a sequence {𝑢_𝑘}_𝑘∈ℕ⊂𝐵(0,𝜌) such that |𝐼_𝑘(𝑢_𝑘)|>𝜀₀. Passing to a subsequence if necessary, we can assume that {𝑢_𝑘}_𝑘∈ℕ tends to some 𝑢 weakly in 𝐻₀^𝛼/2(Ω,ℝ). From the fractional Sobolev compact embedding theorem, see Remark 5, we deduce that, up to subsequence, {𝑢_𝑘}_𝑘∈ℕ tends to 𝑢 in 𝐿^𝑠(Ω,ℝ). By assumption (2), we know that {𝜔_𝑘}_𝑘∈ℕ tends to 𝜔₀ in 𝐿^𝑝(Ω,ℝ^𝑚). Applying the Krasnoselskii theorem (cf. [37, 38]) the continuity of the operator 𝐿^𝑠×𝐿^𝑝∋(𝑢,𝑤)↦𝜙(⋅,𝑢(⋅),𝑤(⋅))∈𝐿¹ follows and which together with condition (A2) implies 𝐼_𝑘(𝑢_𝑘)→0 as 𝑘→∞. Thus we have got a contradiction with the inequality |𝐼_𝑘(𝑢_𝑘)|>𝜀₀. It means that 𝐼_𝑘(𝑢) tends to zero uniformly on 𝐵(0,𝜌) and consequently {𝐹_𝜔𝑘(𝑢)}_𝑘∈ℕ converges to 𝐹_𝜔0(𝑢) uniformly on 𝐵(0,𝜌) provided that 𝜔_𝑘→𝜔₀ in 𝐿^𝑝(Ω,ℝ^𝑚).
Consequently, for any 𝜀>0 and 𝑘 chosen to be sufficiently large, we have 𝜇_𝑘=min_{𝑢∈𝐻0𝛼/2()Ω,ℝ}𝐹_𝜔𝑘(𝑢)=min_{𝑢∈𝐵0,𝜌}𝐹_𝜔𝑘(𝑢)≤min_{𝑢∈𝐵0,𝜌}𝐹_𝜔0(𝑢)+𝜀=min_{𝑢∈𝐻0𝛼/2()Ω,ℝ}𝐹_𝜔0(𝑢)+𝜀=𝜇₀+𝜀. (33) Similarly, 𝜇₀≤𝜇_𝑘+𝜀. We have thus proved equality (31).
Step 3. Finally, we shall prove assertion (c). Let {𝑢_𝑘}_𝑘∈ℕ be a sequence of minimizers; that is, 𝑢_𝑘∈𝑈_𝑘. Since 𝑈_𝑘⊂𝐵(0,𝜌) for 𝑘∈ℕ₀, the sequence {𝑢_𝑘}_𝑘∈ℕ is weakly relatively compact in 𝐻₀^𝛼/2(Ω,ℝ). We may assume after passing to a subsequence (still denoted by 𝑢_𝑘) that {𝑢_𝑘}_𝑘∈ℕ tends to some 𝑢∈𝐵(0,𝜌) in the weak topology of 𝐻₀^𝛼/2(Ω,ℝ). Let us prove now that 𝑢∈𝑈₀; that is, 𝑢 is a minimizer of 𝐹_𝜔0. Indeed, suppose that 𝑢∉𝑈₀. The set 𝑈₀ is nonempty and therefore there exists some 𝑢₀∈𝑈₀ such that 𝑢₀≠𝑢. Clearly, since 𝑢₀ is a minimizer of 𝐹_𝜔0, 𝐹_𝜔0(𝑢)−𝐹_𝜔0(𝑢₀)=𝛽>0 and moreover we have 𝜇_𝑘−𝜇₀=𝐹_𝜔𝑘𝑢_𝑘−𝐹_𝜔0𝑢_𝑘+𝐹_𝜔0𝑢_𝑘−𝐹_𝜔0(𝑢)+𝛽. (34) Uniform convergence of {𝐹_𝜔𝑘(𝑢)}_𝑘∈ℕ to 𝐹_𝜔0(𝑢) on 𝐵(0,𝜌) leads to 𝐹_𝜔𝑘(𝑢_𝑘)−𝐹_𝜔0(𝑢_𝑘)→0 as 𝑢_𝑘∈𝐵(0,𝜌) by (b). Furthermore, the weak lower semicontinuity of 𝐹_𝜔0 and the weak convergence of 𝑢_𝑘 to 𝑢 in 𝐻₀^𝛼/2(Ω,ℝ) lead to liminf_𝑘→∞𝐹_𝜔0𝑢_𝑘−𝐹_𝜔0(𝑢)≥0. (35) Thus we have got a contradiction with (31). Consequently, 𝑢∈𝑈₀.
What we need to do now is to demonstrate that any sequence {𝑢_𝑘}_𝑘∈ℕ such that 𝑢_𝑘∈𝑈_𝑘 converges strongly to 𝑢 in 𝐻₀^𝛼/2(Ω,ℝ). By (22), for 𝑘∈ℕ, we have 0=𝐷𝐹_𝜔𝑘𝑢_𝑘−𝐷𝐹_𝜔0(𝑢)𝑢_𝑘−𝑢=𝑢_𝑘−𝑢²_𝐻0𝛼/2+𝐼_𝑘, (36) where 𝐼_𝑘=_Ω𝜙_𝑢𝑥,𝑢_𝑘(𝑥),𝜔_𝑘(𝑥)−𝜙_𝑢𝑥,𝑢(𝑥),𝜔₀(𝑥)×𝑢_𝑘(𝑥)−𝑢(𝑥)𝑑𝑥. (37) The Hölder inequality and the growth condition (A2) allow us to write the following estimates: 𝐼_𝑘≤_Ω𝜙_𝑢𝑥,𝑢_𝑘(𝑥),𝜔_𝑘(𝑥)−𝜙_𝑢𝑥,𝑢(𝑥),𝜔₀(𝑥)^{𝑠/𝑠−1}𝑑𝑥^{(𝑠−1)/𝑠}×_Ω𝑢_𝑘(𝑥)−𝑢(𝑥)^𝑠𝑑𝑥^1/𝑠≤𝐶_4Ω1+𝑢_𝑘(𝑥)^𝑠+|𝑢(𝑥)|^𝑠+𝜔_𝑘(𝑥)^𝑝+𝜔₀(𝑥)^𝑝𝑑𝑥^{(𝑠−1)/𝑠}×𝑢_𝑘−𝑢_𝐿𝑠𝐼if𝑝<∞,_𝑘≤𝐶_5Ω1+𝑢_𝑘(𝑥)^𝑠+|𝑢(𝑥)|^𝑠𝑑𝑥^{(𝑠−1)/𝑠}×𝑢_𝑘−𝑢_𝐿𝑠if𝑝=∞, (38) where 𝐶₄ and 𝐶₅ are some positive constants. Since {𝑢_𝑘}_𝑘∈ℕ converges to 𝑢 in 𝐿^𝑠(Ω,ℝ) and {𝜔_𝑘}_𝑘∈ℕ is bounded in 𝐿^𝑝(Ω,ℝ^𝑚) we see that 𝐼_𝑘→0 as 𝑘→∞ and therefore the first integral ‖𝑢_𝑘−𝑢‖²_𝐻0𝛼/2 tends to zero. Thus the weak convergence of the minimizers 𝑢_𝑘∈𝑈_𝑘 to 𝑢∈𝑈₀ implies the strong convergence of minimizers in 𝐻₀^𝛼/2(Ω,ℝ), which completes the proof.

Let us return to boundary value problem (10)-(11) and, for 𝑘∈ℕ₀, let us denote by 𝑆_𝑘 the set of solutions to the problem which corresponds to the parameter 𝜔_𝑘. It is the well-known fact, see, for instance, [30, 31], that for the convex functional of action the set of minimizers 𝑈_𝑘 coincides with the set of solutions 𝑆_𝑘. Hence for boundary value problem (10)-(11) we have the following corollary.

Corollary 7. If (1)the integrand 𝜙 satisfies conditions (A1)–(A4),(2)the sequence of distributed parameters {𝜔_𝑘}_𝑘∈ℕ tends to 𝜔₀ in 𝐿^𝑝(Ω,ℝ^𝑚) with 𝑝>1,
then the sequence {𝑆_𝑘}_𝑘∈ℕ0 satisfies assertions (a)–(c) of Theorem 6 with 𝑈_𝑘=𝑆_𝑘, 𝑘∈ℕ₀.
Moreover, if the functional of action is strictly convex, then for 𝑘∈ℕ₀, problem (10)-(11) possesses a unique solution 𝑢_𝑘, and lim_𝑘→∞𝑢_𝑘=𝑢₀ in 𝐻₀^𝛼/2(Ω,ℝ).

4. Continuous Dependence: The Parameters Converging in the Weak Topology

To achieve stronger results which are useful in optimization theory, it is necessary to narrow down the class of equations under considerations. Namely, in this section, we shall assume that the integrand 𝜙 is linear with respect to the distributed parameter 𝜔; that is 𝜙(𝑥,𝑢,𝜔)=𝜙¹(𝑥,𝑢)+𝜙²(𝑥,𝑢),𝜔, (39) where 𝜙¹∶Ω×ℝ→ℝ, 𝜙²∶Ω×ℝ→ℝ^𝑚, 𝜔∈ℝ^𝑚 and ⟨⋅,⋅⟩ stands for a scalar product in ℝ^𝑚. In this case, the boundary value problem (10)-(11) takes the form (−Δ)^𝛼/2𝑢(𝑥)+𝜙¹_𝑢(𝑥,𝑢(𝑥))+𝜙²_𝑢(𝑥,𝑢(𝑥)),𝜔(𝑥)=0for𝑥∈Ω⊂ℝ^𝑛,𝑢(𝑥)=0for𝑥∈ℝ^𝑛⧵Ω (40)(41) and the functional of action has the form 𝐹_𝜔(𝑢)=_Ω12(−Δ)^𝛼/4𝑢(𝑥)²+𝜙¹((𝑥+𝜙𝑥,𝑢))₂(𝑥,𝑢(𝑥)),𝜔(𝑥)𝑑𝑥, (42) where 𝑢∈𝐻₀^𝛼/2(Ω,ℝ) and 𝜔∈𝐿^𝑝(Ω,ℝ^𝑚) with 1<𝑝<∞.

We impose the following conditions on 𝜙¹, 𝜙²:(A1^′)regularity: the functions 𝜙¹, 𝜙¹_𝑢, 𝜙², and 𝜙²_𝑢 are measurable with respect to 𝑥 for any 𝑢∈ℝ and continuous with respect to 𝑢 for a.e. 𝑥∈Ω;(A2^′)growth: there exists a constant 𝑐>0 such that 𝜙¹_𝑢(𝑥,𝑢)≤𝑐1+|𝑢|^𝑠−1,𝜙²_𝑢(𝑥,𝑢)≤𝑐1+|𝑢|^{𝑠−1−𝑠/𝑝} (43) for a.e. 𝑥∈Ω, 𝑢∈ℝ and 𝑠∈(1+1/(𝑝−1),2^∗_𝛼) where 2^∗_𝛼=2𝑛/(𝑛−𝛼)>2 and 1<𝑝<∞.

Suppose that 𝜙 meets conditions (A3) and (A4). Obviously, assumptions (A1^′) and (A2^′) imply the function 𝜙 to satisfy (A1) and (A2). For this weaker form of the problem, the claim of the theorem on the existence and the continuous dependence can be strengthened. To draw the same conclusion this time, it suffices to assume the weak convergence of parameters.

Let {𝜔_𝑘}_𝑘∈ℕ be some sequence of the distributed parameters. Denote by 𝑈_𝑘 a set of all minimizers of the functional of action (42) with 𝜔=𝜔_𝑘 given in (23). We shall prove the following.

Theorem 8. Suppose that (1)the integrand 𝜙 is of the form (39) and satisfies conditions (𝐴1^′), (𝐴2^′), and (𝐴3),(2)the sequence of distributed parameters {𝜔_𝑘}_𝑘∈ℕ tends to 𝜔₀ in the weak topology of 𝐿^𝑝(Ω,ℝ^𝑚).
Then the sequence {𝑈_𝑘}_𝑘∈ℕ0 satisfies assertions (a)–(c) of Theorem 6.

Proof. As in the proof of Theorem 6, in the similar manner, we obtain assertions (a) and (b) of our theorem taken from Theorem 6. Let {𝑢_𝑘}_𝑘∈ℕ⊂𝐻₀^𝛼/2(Ω,ℝ) be an arbitrary sequence such that 𝑢_𝑘∈𝑈_𝑘⊂𝐵(0,𝜌), for 𝑘∈ℕ, where the sets 𝑈_𝑘 are defined by formula (23). The sequence {𝑢_𝑘}_𝑘∈ℕ is bounded and therefore weakly relatively compact. Passing, if necessary, to a subsequence, we can assume that 𝑢_𝑘⇀𝑢₀ weakly in 𝐻₀^𝛼/2(Ω,ℝ). We shall show that 𝑢₀∈𝑈₀, but now we present different approach than in the proof of Theorem 6. By conditions (A1^′) and (A2^′) and formula (22), for 𝑘∈ℕ and ℎ∈𝐻₀^𝛼/2(Ω,ℝ), we have 0=𝐷𝐹_𝜔𝑘𝑢_𝑘ℎ=_Ω(−Δ)^𝛼/4𝑢_𝑘(𝑥)(−Δ)^𝛼/4ℎ(𝑥)+𝜙¹_𝑢𝑥,𝑢_𝑘(𝑥ℎ(𝑥)+𝜙_2𝑢)𝑥,𝑢_𝑘(𝑥)ℎ(𝑥),𝜔_𝑘(𝑥)𝑑𝑥. (44) It is easy to observe that since 𝑢_𝑘⇀𝑢₀ weakly in 𝐻₀^𝛼/2(Ω,ℝ) for any ℎ∈𝐻₀^𝛼/2(Ω,ℝ)lim_𝑘→∞Ω(−Δ)^𝛼/4𝑢_𝑘(𝑥)(−Δ)^𝛼/4ℎ(𝑥)=𝑑𝑥_Ω(−Δ)^𝛼/4𝑢₀(𝑥)(−Δ)^𝛼/4ℎ(𝑥)𝑑𝑥. (45) By the fractional Sobolev compact embedding theorem, after passing to a subsequence (still denoted by 𝑢_𝑘) if necessary, we can assume that {𝑢_𝑘}_𝑘∈ℕ tends to 𝑢₀ in 𝐿^𝑠(Ω,ℝ) for 𝑠∈(1,2^∗_𝛼). By (A2^′), the superposition operator 𝜙¹_𝑢(⋅,𝑢(⋅))ℎ(⋅) acting on 𝑢∈𝐿^𝑠(Ω,ℝ) to 𝐿¹(Ω,ℝ) is continuous; that is, for any ℎ∈𝐻₀^𝛼/2(Ω,ℝ)lim_𝑘→∞Ω𝜙¹_𝑢𝑥,𝑢_𝑘(𝑥)ℎ(𝑥)𝑑𝑥=_Ω𝜙¹_𝑢𝑥,𝑢₀(𝑥)ℎ(𝑥)𝑑𝑥. (46) Let us consider the integral 𝐼_𝑘=_Ω𝜙²_𝑢𝑥,𝑢_𝑘(𝑥)ℎ(𝑥),𝜔_𝑘(𝑥)𝑑𝑥 (47) which can be represented as 𝐼_𝑘=𝐼¹_𝑘+𝐼²_𝑘, (48) where 𝐼¹_𝑘=_Ω𝜙²_𝑢𝑥,𝑢₀(𝑥)ℎ(𝑥),𝜔_𝑘(𝑥𝐼)𝑑𝑥,_2𝑘=_Ω𝜙²_𝑢𝑥,𝑢_𝑘(𝑥)ℎ(𝑥)−𝜙²_𝑢𝑥,𝑢₀(𝑥)ℎ(𝑥),𝜔_𝑘(𝑥)𝑑𝑥. (49) Since ‖𝜔_𝑘‖_𝐿𝑝‖ℎ‖_𝐿^𝑠≤𝐶, we see that 𝐼²_𝑘≤𝐶_Ω𝜙²_𝑢𝑥,𝑢_𝑘(𝑥)−𝜙²_𝑢𝑥,𝑢₀(𝑥)^𝛾𝑑𝑥^1/𝛾, (50) where 𝛾=𝑝𝑠/((𝑠−1)𝑝−𝑠) and moreover due to growth estimate (A2^′) we get the bound 𝐶(1+‖𝑢_𝑘‖_𝐿𝑠+‖𝑢₀‖_𝐿𝑠), since 𝑝𝑠()()𝑠−1𝑝−𝑠𝑠−1𝑝−𝑠𝑝=𝑠. (51) Hence, up to subsequence, we have that 𝐼²_𝑘→0 as 𝑘→∞. Similarly, by (2) and (A2^′) we get lim_𝑘→∞𝐼¹_𝑘=_Ω𝜙²_𝑢𝑥,𝑢₀(𝑥)ℎ(𝑥),𝜔₀(𝑥)𝑑𝑥. (52) Thus lim_𝑘→∞𝐼_𝑘=_Ω𝜙²_𝑢𝑥,𝑢₀(𝑥)ℎ(𝑥),𝜔₀(𝑥)𝑑𝑥 (53) for all ℎ∈𝐻₀^𝛼/2(Ω,ℝ). Taking into account equalities (45), (46), and (53), we infer that 𝐷𝐹_𝜔0(𝑢₀)=0. It means that 𝑢₀∈𝑈₀. To complete the proof, we shall verify that the sequence {𝑢_𝑘}_𝑘∈ℕ converges to 𝑢₀ in 𝐻₀^𝛼/2(Ω,ℝ). By (22), we have 0=𝐷𝐹_𝜔𝑘𝑢_𝑘−𝐷𝐹_𝜔0𝑢₀𝑢_𝑘−𝑢₀=_Ω(−Δ)^𝛼/4𝑢_𝑘(𝑥)−(−Δ)^𝛼/4𝑢₀(𝑥)²+𝑑𝑥_Ω𝜙¹_𝑢𝑥,𝑢_𝑘(𝑥)−𝜙¹_𝑢𝑥,𝑢₀(𝑥)𝑢_𝑘(𝑥)−𝑢₀(𝑥)+𝑑𝑥_Ω𝜙²_𝑢𝑥,𝑢_𝑘(𝑥)𝑢_𝑘(𝑥)−𝑢₀(𝑥),𝜔_𝑘(𝑥−)𝑑𝑥_Ω𝜙²_𝑢𝑥,𝑢₀(𝑥)𝑢_𝑘(𝑥)−𝑢₀(𝑥),𝜔₀(𝑥=𝑢)𝑑𝑥_𝑘−𝑢₀²_𝐻0𝛼/2+𝐼¹_𝑘+𝐼²_𝑘+𝐼³_𝑘. (54) Since for 𝐼¹_𝑘=_Ω𝜙¹_𝑢𝑥,𝑢_𝑘(𝑥)−𝜙¹_𝑢𝑥,𝑢₀(𝑥)𝑢_𝑘(𝑥)−𝑢₀(𝑥)𝑑𝑥, (55) by the Hölder inequality and the growth condition (A2^′), we get 𝐼¹_𝑘≤_Ω𝜙¹_𝑢𝑥,𝑢_𝑘(𝑥)−𝜙¹_𝑢𝑥,𝑢₀(𝑥)^{𝑠/(𝑠−1)}𝑑𝑥^{(𝑠−1)/𝑠}×_Ω𝑢_𝑘(𝑥)−𝑢₀(𝑥)^𝑠𝑑𝑥^1/𝑠≤𝐶_4Ω1+𝑢_𝑘(𝑥)^𝑠+𝑢₀(𝑥)^𝑠𝑑𝑥^{(𝑠−1)/𝑠}×𝑢_𝑘−𝑢_0𝐿𝑠if𝑝<∞, (56) and in a similar manner using the Hölder inequality, 𝐼²_𝑘 and 𝐼³_𝑘 can be estimated by the terms involving ‖𝑢_𝑘−𝑢₀‖_𝐿𝑠, ‖𝑤_𝑘‖_𝐿𝑝, for 𝑘∈ℕ₀, and finally ‖𝜙²_𝑢‖_{𝐿𝑝𝑠/(𝑝(𝑠−1)−𝑠)}. The latter term due to the growth condition imposed on 𝜙²_𝑢 can be estimated as before from the above by ‖𝑢_𝑘‖_𝐿𝑠 for 𝑘∈ℕ₀. Since {𝑢_𝑘}_𝑘∈ℕ converges to 𝑢₀ in 𝐿^𝑠(Ω,ℝ) and {𝜔_𝑘}_𝑘∈ℕ converges to 𝜔₀ in 𝐿^𝑝(Ω,ℝ^𝑚) we have 𝐼_𝑘=(𝐼¹_𝑘+𝐼²_𝑘+𝐼³_𝑘)→0 as 𝑘→∞. Consequently, ‖𝑢_𝑘−𝑢₀‖_𝐻0^𝛼/2→0 as 𝑘→0. Thus, the weak convergence of minimizers 𝑢_𝑘∈𝑈_𝑘 to 𝑢₀∈𝑈₀ implies the strong convergence of minimizers in 𝐻₀^𝛼/2(Ω,ℝ). Therefore, the proof of our theorem is complete.

5. Existence of Optimal Solutions

We now formulate the optimal control problem to which this section is dedicated. It transpires that the continuous dependence results from Section 4 enable us to prove a theorem on the existence of optimal processes to some optimal control problem. Specifically, we shall consider control problem governed by boundary value problem (40)-(41) with the integral cost functional 𝐽(𝑢,𝜔)=_Ω𝜃𝑥,𝑢(𝑥),(−Δ)^𝛼/4𝑢(𝑥),𝜔(𝑥)𝑑𝑥, (57) where 𝜃∶Ω×ℝ×ℝ×ℝ^𝑚→ℝ is a given function. Here 𝑢∈𝐻₀^𝛼/2(Ω,ℝ) is the trajectory and 𝜔∈𝒲 is the distributed control where 𝒲=𝜔∈𝐿^𝑝Ω,ℝ^𝑚∶𝜔(𝑥)∈𝑊fora.e.𝑥∈Ω (58) with 𝑝>1 and 𝑊 being a compact and convex subset of ℝ^𝑚.

Let 𝒟 be the set of all admissible pairs; that is 𝒟=(𝑢,𝜔)∈𝐻₀^𝛼/2()().Ω,ℝ×𝒲∶𝑢satisfies40for𝜔∈𝒲 (59) It should be noted that under assumptions of Theorem 8 the set of all admissible pairs 𝒟 is nonempty. In this section, our aim is to find a pair (𝑢_𝜔∗,𝜔^∗)∈𝐻₀^𝛼/2(Ω,ℝ)×𝒲 such that 𝐽𝑢_𝜔∗,𝜔^∗=min_{()𝑢,𝜔∈𝒟}𝐽(𝑢,𝜔). (60) On the integrand 𝜃 we impose the following conditions.(A5)The function 𝜃=𝜃(𝑥,𝑢,𝑝,𝜔) is measurable with respect to 𝑥 for all 𝑢∈ℝ, 𝑝∈ℝ, 𝜔∈𝑊, continuous with respect to (𝑢,𝑝,𝜔) for a.e. 𝑥∈Ω, and convex with respect to 𝜔 for all 𝑢∈ℝ, 𝑝∈ℝ, and a.e. 𝑥∈Ω. Moreover there exists a constant 𝑐>0 such that 𝜃𝑥,𝑢,𝑝,𝜔≤𝑐1+|𝑢|^𝑠+𝑝² (61) for a.e. 𝑥∈Ω, all 𝑢∈ℝ, 𝑝∈ℝ, 𝜔∈𝑊, and for some 𝑠∈(1,2^∗_𝛼) where 2^∗_𝛼=2𝑛/(𝑛−𝛼).(A6)There exists a function 𝜂∈𝐿¹(Ω,ℝ) and a constant 𝑀>0 such that 𝜃𝑥,𝑢,𝑝,𝜔≥𝜂(𝑥)−𝑀|𝑢|+𝑝+|𝜔| (62) for all 𝑢∈ℝ, 𝑝∈ℝ, 𝜔∈𝑊, and a.e. 𝑥∈Ω.

Now we prove a theorem on the existence of optimal processes to our optimal control problem (60).

Theorem 9. If the functions 𝜙 of the form (39) satisfies (𝐴1^′), (𝐴2^′), (A3), (A4), and the integrand 𝜃 meets assumptions (A5), (A6), then the optimal control problem (60) possesses at least one optimal process (𝑢_𝜔∗,𝜔^∗).

Proof. From (A5), (A6), and classical theorems on semicontinuity of integral functional (cf. [33–36]), we deduce that 𝐽 is lower semicontinuous with respect to the strong topology in the space 𝐻₀^𝛼/2(Ω,ℝ) and the weak topology of 𝐿^𝑝(Ω,ℝ^𝑚), since convergence of any sequence {𝑢_𝑘}_𝑘∈ℕ in 𝐻₀^𝛼/2(Ω,ℝ) implies the strong convergence of {𝑢_𝑘}_𝑘∈ℕ in 𝐿^𝑠(Ω,ℝ) with 𝑠∈(1,2^∗_𝛼) and the strong convergence of {(−Δ)^𝛼/4𝑢_𝑘}_𝑘∈ℕ in 𝐿²(Ω,ℝ).
Let {(𝑢_𝑘,𝜔_𝑘)}_𝑘∈ℕ⊂𝒟 be a minimizing sequence for optimal control problem (60); that is lim_𝑘→∞𝐽𝑢_𝑘,𝜔_𝑘=inf_{()𝑢,𝜔∈𝒟}𝐽(𝑢,𝜔)=𝜗, (63) Since the set 𝑊 is compact and convex, the sequence {𝜔_𝑘}_𝑘∈ℕ is compact in the weak topology of 𝐿^𝑝(Ω,ℝ^𝑚). Passing to subsequence, if necessary, we can assume that 𝜔_𝑘 tends to some 𝜔₀∈𝒲 weakly in 𝐿^𝑝(Ω,ℝ^𝑚). By assumption (A4) the set of the weak solutions of problem (40)-(41) coincides with the set of minimizers of the functional 𝐹_𝜔 on the space 𝐻₀^𝛼/2(Ω,ℝ). By Theorem 8, the sequence {𝑢_𝑘}_𝑘∈ℕ, or at least some of its subsequence, tends to 𝑢₀ in 𝐻₀^𝛼/2(Ω,ℝ) and the pair (𝑢₀,𝜔₀) is an admissible pair for control problem (40)-(41).
Due to the lower semicontinuity of 𝐽, we have 𝐽𝑢₀,𝜔₀≤liminf_𝑘→∞𝐽𝑢_𝑘,𝜔_𝑘 (64) provided 𝑢_𝑘 tends to 𝑢₀ in 𝐻₀^𝛼/2(Ω,ℝ) and 𝜔_𝑘⇀𝜔₀ weakly in 𝐿^𝑝(Ω,ℝ^𝑚). Furthermore, by (63) and (64), we have 𝜗≤𝐽𝑢₀,𝜔₀≤liminf_𝑘→∞𝐽𝑢_𝑘,𝜔_𝑘=inf_{()𝑢,𝜔∈𝒟}𝐽(𝑢,𝜔)=𝜗. (65) Thus, 𝐽(𝑢₀,𝜔₀)=𝜗=inf_{(𝑢,𝜔)∈𝒟}𝐽(𝑢,𝜔). It means that the process (𝑢_𝜔∗,𝜔^∗)=(𝑢₀,𝜔₀) is optimal for the problem (60).

Remark 10. From the proof of Theorem 9 one can see that it suffices to assume weaker assumption on controls than 𝑊 to be compact and convex, namely only boundedness of 𝜔_𝑘 in 𝐿^𝑝(Ω,ℝ^𝑚).

Remark 11. By a direct calculation, one can check that the quadratic functional ℱ(𝑢)=12_Ω(−Δ)^𝛼/4𝑢(𝑥)²−𝜉|𝑢(𝑥)|²𝑑𝑥 (66) is strictly convex for 𝜉<𝜌₁^𝛼/2 and convex for 𝜉=𝜌₁^𝛼/2 where 𝜌₁ is the principal eigenvalue of the operator −Δ defined on 𝐻¹₀(Ω,ℝ).
Since 𝐹_𝜔(𝑢)(𝑢)+=ℱ_Ω𝜉2|𝑢(𝑥)|²+𝜙¹((𝑥+𝜙𝑥,𝑢))₂(𝑥,𝑢(𝑥)),𝜔(𝑥)𝑑𝑥 (67)𝑢∈𝐻₀^𝛼/2(Ω,ℝ). Theorem 9 implies the following.

Corollary 12. The optimal control system (60) possesses at least one optimal process (𝑢_𝜔∗,𝜔^∗) provided the functions 𝜙 of the form (39) satisfy (𝐴1^′), (𝐴2^′), and (𝐴3), the integrand 𝜃 meets assumptions (A5) and (A6) and the function (𝜉/2)|𝑢|²+𝜙¹(𝑥,𝑢)+⟨𝜙²(𝑥,𝑢),𝜔⟩ is convex in 𝑢 for some 𝜉≤𝜌₁^𝛼/2, all 𝜔∈𝑊 and a.e. 𝑥∈Ω.

Example 13. Let Ω be a cube of the form Ω=𝑃³(0,𝜋)=𝑥∈ℝ³∶0<𝑥^𝑖<𝜋,𝑖=1,2,3. (68) Note that 𝑢₁=sin𝑥₁sin𝑥₂sin𝑥₃ and 𝜌₁=3 are eigenfunction and eigenvalue for −Δ on 𝐻¹₀(Ω,ℝ) since −Δ𝑢₁=3𝑢₁. Similarly, (−Δ)^𝛼/2𝑢₁=3^𝛼/2𝑢₁ hence, by (4), 3^𝛼/2 is the first eigenvalue for (−Δ)^𝛼/2 in this case. The equation is of the form (−Δ)^𝛼/2𝑢(𝑥)−𝑎𝑢(𝑥)+𝑠|𝑥|²𝑢^𝑠−1(𝑥)𝜔¹(𝑥)−|𝑥|𝜔²(𝑥)=0for𝑥∈𝑃³(),𝑢(𝑥)0,𝜋=0onℝ³⧵𝑃³(0,𝜋) (69) for 1<𝑠<6/(3−𝛼), 1<𝑝 sufficiently large and the cost is given by 𝐽(𝑢,𝜔)=_Ω𝑢^𝑠(𝑥)+(−Δ)^𝛼/4𝑢(𝑥)²𝜔¹(𝑥)−|𝑥|(−Δ)^𝛼/4𝑢(𝑥)+|𝜔(𝑥)|²−𝜔²(𝑥)𝑑𝑥, (70) where 𝑎<3^𝛼/2, 0≤𝜔¹(𝑥)≤1, and 0≤𝜔²(𝑥)≤1. Obviously, the functional of action for system (69) has the form 𝐹_𝜔(𝑢)=_Ω12(−Δ)^𝛼/4𝑢(𝑥)²−𝑎2𝑢²(𝑥)+|𝑥|²𝑢^𝑠(𝑥)𝜔¹(𝑥)−𝑢(𝑥)|𝑥|𝜔²(𝑥)𝑑𝑥. (71) It is easy to check that the functionals 𝐹_𝜔 and 𝐽 satisfy all assumptions of Theorems 8 and 9. By Remark 11, 𝐹_𝜔 is strictly convex. Thus, Theorem 9 implies that for any control 𝜔 there exists exactly one solution 𝑢_𝜔 of (69) and the solution continuously depends on control 𝜔. Moreover, by Corollary 12, we infer that there exists optimal control (𝑢_𝜔∗,𝜔^∗) described by (69) with the cost functional given by (70).

6. Summary

In this paper we formulate some sufficient condition under which the boundary value problem considered in the paper possesses at least one solution which continuously depends on distributed parameters. We based our approach on the variational methods and we have investigated the stability problem or continuous dependence problem for the problem involving fractional Laplace operator in the fractional Sobolev space 𝐻₀^𝛼/2(Ω,ℝ) with distributed parameters 𝜔 from the space 𝐿^𝑝(Ω,ℝ^𝑚) thus generalizing the stability results obtained for the boundary value problem with the Laplace operator in [1–3]. The stability results enable us to prove the theorem on the existence of optimal processes to some control problem with the integral cost functional.

The question of the existence of a solution for the boundary value problem of the Dirichlet type, periodic, homoclinic or heteroclinic type, and so forth was investigated in many papers and monographs. One can find a wide survey of results and research methods in monographs [30, 31, 38–41] and the references to be found therein. On the contrary to the initial value problem the literature on the stability problems for the boundary value problems governed by the differential equation of the elliptic type is not very vast. The stability of solutions of scalar second-order ordinary differential equation with two-point boundary conditions based on some direct methods related to the implicit function theorem was considered among others in the papers [42–46].

The question of the continuous dependence of solutions of the linear elliptic equations with the variable Dirichlet boundary data and parameters was investigated in the pioneering paper of Oleĭnik compare [47]. In this work sufficient conditions for stability of the linear partial differential equation defined in the classical spaces of smooth solutions were formulated. Analogous results for the scalar linear partial differential equation with the Dirichlet boundary conditions defined on the Sobolev spaces were proved in the paper [48]. The results on the stability of multidimensional nonlinear boundary value problems with variable parameters appeared in papers [49–51] where ordinary differential equations with two-point boundary conditions and variable functional parameters were investigated, and the stability conditions with respect to the strong and weak topology were proved. Similar results for partial differential equation with distributed parameters are given in papers [1–3, 52, 53].

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.