Optimal Control of a Nonlinear PDE Governed by Fractional Laplacian

We consider an optimal control problem containing a control system described by a partial nonlinear differential equation with the fractional Dirichlet–Laplacian, associated to an integral cost. We investigate the existence of optimal solutions for such a problem. In our study we use Filippov’s approach combined with a lower closure theorem for orientor fields.


Introduction
In the last years fractional Laplace operators has been attracted the intersts of many scientists. This is mainly due to the fact that such operators better describe nonlocal models of many phenomena. In particular, they appear in many fields of science such as economics (cf. [6,19]), probability (cf. [6,10,11,18]), mechanics (cf. [9,11]), material science (cf. [8]), fluid mechanics and hydrodynamics (cf. [12,[14][15][16][31][32][33]). Recently, optimal control problems containing control systems described by fractional Laplacians have received a lot of attention. We refer [1,[20][21][22]29], where linearquadratic optimal control problems involving fractional partial differential equations are studied. In [21] the numerical aproximation of such a type of problem, where the linear state equation involves a fractional Laplace operator with its spectral definition, is investigated. In [20,22] first order necessary and sufficient optimality conditions as well as a priori error estimates are derived. PDE constraints contain the integral fractional Laplacian. Some numerical schemes are also proposed there. In [1,29], the B Rafał Kamocki rafal.kamocki@wmii.uni.lodz.pl 1 Faculty of Mathematics and Computer Science, University of Łódź, Banacha 22, 90-238 Łódź, Poland state equation is described by a fractional power of the second order a symmetric and uniformly elliptic operator. In this work, some regularity results, numerical schemes to aproximate the optimal solution and a priori error analysis are presented. We mention also [5], where the optimal control of fractional semilinear PDEs with both spectral and integral fractional Laplacians with distributed control is considered and [2]here linear PDEs and integral fractional Laplacian are studied. In these works, the necessary and sufficient optimality conditions for such problems are obtained. The existence results is also investigated in [13], where an optimal control problem with a spectral fractional Dirichlet Laplacian is considered. A nonlinear and nonlocal state equation, studied there, has a variational structure and a cost depends also on the fractional Laplacian. We also refer to [3,4], where a some optimal control problem with a fractional p-Lalacian is studied. In our paper we consider the following optimal control problem: subject to where Ω ⊂ R N , N ≥ 1, is an open and bounded set, β > 0, f , f 0 : Ω ×R×R m → R, M ⊂ R m is a nonempty set and [(−Δ) ω ] β denotes a weak fractional Laplace operator of order β with zero Dirichlet boundary values on ∂Ω (the term "weak" is explained in Sect. 2). This operator is defined through the spectral decomposition of the Laplace operator −Δ in Ω with zero Dirichlet boundary conditions (cf. Sect. 2). The necessary optimality conditions for one-dimensional problem (1)-(2) have been derived in [23] and [27] by using Dubovitskii-Milyutin approach [23] and a smooth-convex extremum principle [27]. In order to obtain results of such a type in the case of Ω ∈ R N more advanced investigations are required. This issue will be addressed in a forthcoming paper.
The main goal of this paper is study the existence of optimal solutions of problem (1)- (2). In view of a nonlinearity of f and f 0 , as well as, a general convexity assumption (H 3 ) a method of the proof of the main result differs from the method presented in [5]. Our study is based on the lower closure theorem for orientor fields ([17, Theorem 10.7.i]) and a measurable selection theorem of Filippov type ([30, Theorem 2J]). To the best of my knowledge, the existence result for such a nonlinear problem was not investigated by other authors. Also, a combination of mentioned Theorems 10.7.i and 2J, used in the proof of the main result, is new. The existence result of such a type for the one-dimensional problem (1)-(2), where has been obtained in [26]. In order to solve such a problem, a characterization of a weak lower semicontinuity of integral functionals was applied there.
The outline of this paper is as follows. In Sect. 2, we recall some necessary notions and facts concerning Dirichlet-Laplace operator of fractional order and multifunctions. In Sect. 3, we formulate and prove the main result of this paper, namely a theorem on the existence of optimal solutions for problem (1)- (2). We finish with an illustrative, theoretical example.

Preliminaries
In this section we provide some necessary notions and results concerning the fractional Dirichlet-Laplace operator in a weak sense (see [25], more details can be found in [24]), as well as, some necessary facts regarding multifunctions are given [17,30].

Weak Dirichlet-Laplace Operator of Fractional Order
Definition 1 [24] We say that u : Ω → R has a weak (minus) Dirichlet-Laplacian if u ∈ H 1 0 and there exists a function g ∈ L 2 such that for any v ∈ H 1 0 . The function g, denoted by (−Δ) ω u, is called the weak Dirichlet-Laplacian and (−Δ) ω -the weak Dirichlet-Laplace operator.
In the rest of this paper we shall use the space dom([(− ) ω ] β ) with the another scalar product ·, · ∼β given by Norms (3) and (4) where From [24,Proposition 3.10] follows the following useful result:

Multifunctions
Let S be an arbitrary nonempty set equipped with a σ -algebra B and : S s −→ (s) ⊂ R r be a closed-valued multifunction. We shall say that is measurable if for each closed set C ⊂ R r the set −1 (C) given by Let us define the set: dom := {s ∈ S : (s) = ∅}.
A function λ : dom → R r such that λ(s) ∈ (s) for all s ∈ dom , is called a selection of the multifunction . We shall say that a function f : S × R n → R ∪ {±∞} is a normal integrand on S × R n if f is lower semicontinuous on R n for all s ∈ S and the epigraph In the proof of the main result of this paper we apply the following version of Filippov's lemma (cf. [30, Theorem 2J]): is a countable collection of normal integrands on S ×R r and a : S → R k , κ i : S → R∪{±∞} are measurable. Then is the measurable (closed-valued) multifunction and hence has a measurable selection λ : dom → R r . Now, let us assume that (S, ρ) is a metric space and : S s −→ (s) ⊂ R r is an arbitrary multifunction.
We say that : S s −→ (s) ⊂ R r has property (K) at the point s 0 ∈ S iff where clZ denotes the closure of the set Z .
We say that has property (K) in S if it has property (K) at every point s ∈ S.

be a multifunction. Then has property (K) if and only if the graph of given by
is closed in the product space S × R r .

Remark 2
From the above theorem it follows that if has property (K) then its values are closed.
In conclusion, we formulate a key result in our study, namely, a lower closure theorem ([17, Theorem 10.7.i]). First, we give the necessary notation. Let G ⊂ R ν be a measurable set of finite measure, for every x = (x 1 , . . . , x ν ) ∈ G let A(x) be a given nonempty subset of R n and let whereby z = (z 1 , . . . , z n ). For every (x, z) ∈ A letQ(x, z) be a given subset of the space R r +1 .

Existence of Optimal Solutions
In this section we shall prove the main result of this paper, namely a theorem on the existence of optimal solutions for problem (1)- (2). Let (2). Then, z is called an admissible trajectory, while u is an admissible strategy.
In what follows, we assume that the system (2) is controllable in the sense that at least one admissible pair exists. Furthemore, we impose on functions f and f 0 the following conditions: (H 1 ) the function f is measurable on Ω, continuous on R × R m and satisfies the following growth condition: there exist A ≥ 0, a ∈ L 2 (Ω, for a.e. x ∈ Ω and all z ∈ R, u ∈ R m , (H 2 ) the function f 0 is measurable on Ω and continuous on R × R m , for a.e. x ∈ Ω and all z ∈ R, are convex.

Remark 3 From [28, Theorem 1] it follows that if
A < λ β 1 then for any fixed u ∈ U M there exists a solution of the control system (2) (here λ 1 is the first eigenvalue of th operator (−Δ) ω ).

Proposition 2 If assumption (H 1 ) is satisfied, whereby
(N β is given by (6) Proof Let us fix any control u ∈ U M . Assume that z ∈ dom([(− ) ω ] β ) is a solution of the control system (2), corresponding to u. Then, using (7) and the Poincaré inequality (5), we obtain Thus, putting we get (10). The proof is completed.
In what follows, we assume that for any admissible pair (z, u) the integral (1) is finite.
The set of such pairs we will denote by A. Since the control system (2) is controllable, therefore A = ∅. Moreover, the following two additional hypotheses are required: The pair (z * , u * ) ∈ A is called an optimal solution to problem (1)- (2) if for any pair (z, u) ∈ A. Now, we formulate and prove the main result of this paper:

Theorem 4
Assume that M is a compact set. If assumptions (H 1 )-(H 5 ) with condition (9) are satisfied then problem (1)-(2) has an optimal solution (z * , Proof Let us denote and Let us denote: for a.e. x ∈ Ω and all l ∈ N. It is clear that for almost all x ∈ Ω, the setsQ(x, z) are convex (assumption (H 3 )), have property (K) with respect to z ∈ R (Proposition 3), so are also closed (Remark 2). Of course, functions z, z l are measurable on Ω and ξ , ξ l , λ, λ l , η l ∈ L 1 , l ∈ N. Moreover, convergences (11) and (12) imply z l → z in measure on Ω and ξ l ξ weakly in L 1 as l → ∞. We also see that x ∈ Ω a.e., l ∈ N and λ l = λ λ weakly in L 1 .
Consequently, using Theorem 3, we assert that there exists a function η ∈ L 1 such that Now, let us consider the multifunction : Ω x −→ (x) ⊂ R m given by Let us denote: Since M is closed and for each closed set D ⊂ R m is measurable, therefore C is a closed-valued measurable multifunction. Moreover, the function f 1 is a normal integrand as a Carathéodory function (cf. [30,Proposition 2C]). Of course, functions a and κ 1 are measurable on Ω. Consequently, from Theorem 1 it follows that is a closed-valued, measurable multifunction and there exists a measurable function u * : Ω → R m such that u * (x) ∈ (x) for a.e. x ∈ Ω. This means that and Consequently, (z * , u * ) ∈ A, whereby (see (13) and (14)) This means that (z * , u * ) is an optimal solution to problem (1)- (2). The proof is completed.
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