Necessary optimality conditions for Lagrange problems involving ordinary control systems described by fractional Laplace operators

where k = 1, 2, β > 1/4, f : (0, π)× R ×M → R and f0 : (0, π)× R ×M → R. The first of them, denoted by (OCP1), contains the control system (E1) involving the one-dimensional Dirichlet Laplace operator of order β (−∆1) . The second one (OCP2) includes the system (E2), which is described by the Dirichlet–Neumann Laplace operator (−∆2) . Operators (−∆1) and (−∆2) are defined through the spectral decomposition of the Laplace operator −∆ in (0, π) with zero Dirichlet and Dirichlet–Neumann boundary conditions, respectively (cf. Section 2.2). In the last years, fractional Laplacians are a topic of research of many scientists. There exist many definitions of such operators (e.g. based on Fourier transform [19, 25],


Introduction
In this paper, we consider the following two optimal control problems: x(t), u(t) , t ∈ (0, π) a.e., (E k ) u(t) ∈ M ⊂ R m , t ∈ (0, π), J(x, u) = π 0 f 0 t, x(t), u(t) dt → min, where k = 1, 2, β > 1/4, f : (0, π) × R n × M → R n and f 0 : (0, π) × R n × M → R. The first of them, denoted by (OCP 1 ), contains the control system (E 1 ) involving the one-dimensional Dirichlet Laplace operator of order β (−∆ 1 ) β . The second one (OCP 2 ) includes the system (E 2 ), which is described by the Dirichlet-Neumann Laplace operator (−∆ 2 ) β . Operators (−∆ 1 ) β and (−∆ 2 ) β are defined through the spectral decomposition of the Laplace operator −∆ in (0, π) with zero Dirichlet and Dirichlet-Neumann boundary conditions, respectively (cf. Section 2.2). In the last years, fractional Laplacians are a topic of research of many scientists. There exist many definitions of such operators (e.g. based on Fourier transform [19,25], c 2020 Author. Published by Vilnius University Press This is an Open Access article distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. hypersingular integral [25], Riesz potential operator [24], Bochner's definition [26], spectral decomposition (cf. [6,18])). These different definitions typically lead to different operators (cf. [1,Sect. 2.3.]). Recent intensive investigations show that Laplacians can be applied in various areas; for example, in economics (cf. [5,18]), probability (cf. [5,9,10,17]), mechanics [8,10], material science (cf. [7]), fluid mechanics and hydrodynamics (cf. [11,[14][15][16][29][30][31]). Over the last years, they have been attracted interest of many mathematicians also in the field of optimal control theory. In [12,13], some optimal control problems with a fractional Dirichlet-Laplace operator are investigated. The results concerning the existence, stability, continuous dependence of solutions on controls and existence of optimal solutions minimizing a some integral cost functional have been obtained there. In [28], a some optimal control problem (inspired by considerations in mathematical biology) with a general positive defined fractional operator (so-called diffusion operator) is studied. To be more specific, an evolution equation of a diffusion type with a some integral cost functional is considered. The necessary and sufficient optimality conditions for such a problem have been derived. Results of such a type have been also obtained in [3,4], where the linear-quadratic optimal control problems involving fractional powers of elliptic operators are investigated. Furthermore, a numerical scheme to solve the fractional optimal problems has been proposed there.
The aim of this paper is to derive the necessary optimality conditions for problems (OCP k ). Below, we formulate a result of such a type for the problem (OCP 1 ) obtained in [22]  (i) M ⊂ R m is a closed convex set with nonempty interior, (ii) f, f 0 are measurable on (0, π), continuously differentiable on R n × R m and is a local minimum point for problem (OCP 1 ), one of the following conditions: • β > 1/2 and f x (·, x * (·), u * (·)) L 1 < π/(2ξ(2β)); • β > 1/2 and f x (t, x * (t), u * (t)) 0 for a.e. t ∈ (0, π); • β > 1/4, f x (·, x * (·), u * (·)) ∈ L ∞ and f x (·, x * (·), u * (·)) L ∞ < 1; is fulfilled and (f 0 ) x (·, x * (·), u * (·)), (f 0 ) u (·, x * (·), u * (·)) are not all identically zero, then there exists a function λ ∈ D((−∆ 1 ) β ) such that for any u ∈ M , whereby H : (0, π) × R n × R n × R m → R: However, there exist optimal control problems of type (OCP k ) where the (D-M) method cannot be applied. For example, if M is a finite set containing at least two elements (such a set is often used in optimal control theory), then M does not satisfy assumption (i) of Theorem 1 (the set M is not convex and its interior is empty). Moreover, in view of a specific structure of the set M , the assumption that f and f 0 are differentiable with respect to u does not make sense. So, it is necessary to use an alternative method to obtain optimality conditions. In our study, we use a smooth-convex extremum principle (cf. [23]). In this approach, the assumption of differentiability of f , f 0 with respect to u is replaced with some "convexity assumption" (consequently, maximum conditions obtained in both methods are different). On the other hand, in our approach, compactness of the set M is required (cf. Lemma 4). Consequently, in contrast to the (D-M) approach, it cannot be applied to optimal control problems where M is unbounded (in particular, M = R m ). To sum up, from the above discussion it follows that both methods are useful in practical applications. One can also show that if f , f 0 are smooth and convex in u and M is convex, then both methods are equivalent (more precisely, the minimum conditions in both approaches are equivalent).
The paper is organized as follows. In Section 2, we give necessary notions and facts concerning ordinary Dirichlet and Dirichlet-Neumann Laplace operators of fractional order, as well as the extremum principle for a smooth-convex problem is formulated. In Section 3, we derive the main result of this paper, namely the necessary optimality conditions for problems (OCP k ), k = 1, 2 (Theorem 3). Two illustrative examples are presented in Section 4. We finish with Appendix A containing some basics from the spectral theory of self-adjoint operators in a real Hilbert space.

Preliminaries
In the first part of this section, we formulate the so-called smooth-convex optimal control problem and recall the extremum principle for it (cf. [23]). This tool will be used in the proof of the main result of this paper (Theorem 3). The second part concerns fractional ordinary Dirichlet and mixed Dirichlet-Neumann Laplace operators. The definitions of these operators come from the Stone-von Neumann operator calculus and are based on the spectral integral representation theorem for a self-adjoint operator in a Hilbert space (cf. [21,22] and Appendix A).

Smooth-convex extremum principle
Let X, Y be the Banach spaces, and U denotes an arbitrary nonempty set. Let us consider the following problem: http://www.journals.vu.lt/nonlinear-analysis where f 0 , . . . , f n : X × U → R and F : X × U → Y .
If the functions f 0 , . . . , f n and the mapping F satisfy certain conditions of smoothness in x and "convexity" in u (cf. assumptions (a), (b) in Theorem 2), then the above problem is called the smooth-convex problem.

Theorem 2 [Smooth-convex extremum principle].
Let the pair (x * , u * ) satisfies conditions (2)-(4), and assume that there exists a neighborhood V ⊂ X of x * such that (a) for every u ∈ U, the mapping x → F (x, u) and the functions x → f i (x, u), i = 0, . . . , n, are continuously differentiable at the point x * ; (b) for every x ∈ V , the mapping u → F (x, u) and the functions u → f i (x, u), i = 0, . . . , n, satisfy the following convexity condition: for every u 1 , u 2 ∈ U and β ∈ [0, 1], there exists an element u ∈ U such that (c) the range Im F x (x * , u * ) of the linear operator F x (x * , u * ) : X → Y is closed and has a finite codimension in Y (i.e. a complementary subspace to Im F x (x * , u * ) has a finite dimension in Y ).
If (x * , u * ) is a local minimum point of problem (1)-(4), then there exist the Lagrange multipliers λ 0 0, . . . , λ n 0, y * ∈ Y * (not all zero) such that If, additionally, (iv) the image of the set X × U under the mapping contains a neighborhood of the origin of Y and if there exists a point (x, u) such that for all i = 1, . . . , n for which f i (x * , u * ) = 0, then λ 0 = 0 and it can be assumed that λ 0 = 1.

One-dimensional Dirichlet and Dirichlet-Neumann Laplace operators of fractional order
Let −∆ be the one-dimensional Laplace operator on the interval (0, π) given by Let us define the following spaces of functions: By the one-dimensional Dirichlet Laplace operator −∆ D : H D ⊂ L 2 → L 2 we mean the operator −∆ given by (5) under Dirichlet boundary conditions. Similarly, by the onedimensional Dirichlet-Neumann Laplace operator −∆ DN : H DN ⊂ L 2 → L 2 we mean the operator −∆ under Dirichlet-Neumann boundary conditions.
In an elementary way, one can show that operators −∆ D and −∆ DN are self-adjoint. Moreover, their spectra are given by respectively, and the eigenspaces Eig j (−∆ D ) (associated with the eigenvalues λ j = j 2 ), In what follows, we shall use the fact that systems of functions are complete orthonormal systems in L 2 . Now, assume β > 0 and consider the operator 2) for the operator −∆ D , and a j 2/π × sin jt is the projection of x on the n-dimensional eigenspace Eig j (−∆ D )).
The operator (−∆ D ) β is called the fractional Dirichlet Laplace operator of order β, and the function (−∆ D ) β x -the fractional Dirichlet Laplacian of order β of x.
Similarly, we define the fractional Dirichlet-Neumann Laplace operator of order β. This is the operator given by (here F is the spectral measure for the operator −∆ DN , and b j 2/π sin(j − 1/2)t is the projection of x on the n-dimensional eigenspace Eig j (−∆ DN )).
Remark 1. To shorten the notation, in the next sections, the fractional Dirichlet Laplace operator of order β is denoted by (−∆ 1 ) β . Similarly, by (−∆ 2 ) β we mean the fractional Dirichlet-Neumann Laplace operator.
Remark 2. Completeness of the domain D((−∆ D ) β ) follows from the fact that the operator (−∆ D ) β is self-adjoint (cf. Appendix A), so it is closed. Equivalence of norms · D β and · D ∼β guarantees the following Poincaré inequality on D((−∆ D ) β ) (cf. [21]): Using the similar argumentation (cf. Remark 2), we also obtain Moreover, norms generated by these products are equivalent.
In particular, equivalence of norms · DN β and · DN ∼β is provided due to the following Poincaré inequality on D((−∆ DN ) β ): In the proof of the main result of this paper, we shall use the following and therefore embeddings Proof. The proof of (8) can be found in [21]. Now, let x ∈ D((−∆ DN ) β ). Then Hence, we obtain inequality (9). The proof is completed.

Necessary optimality conditions
In this part of the paper, we derive the necessary optimality conditions for optimal control problems (OCP k ), k = 1, 2.
We define the set of controls Let us fix k = 1, 2. We say that a pair (x * , u * ) ∈ D((−∆ k ) β ) × U M is a locally optimal solution of the problem (OCP k ) if x * is a solution of (E k ) corresponding to the control u * and there exists a neighborhood W k of the point x * in D((−∆ k ) β ) such that for all pairs (x, u) ∈ W k × U M satisfying (E k ).
In the proof of the main result, we shall use the following two lemmas.  Now, we derive the maximum principle for problems (OCP 1 ) and (OCP 2 ). We have Theorem 3. Let us fix k = 1, 2. We assume that M is compact, β > 1/4 and
http://www.journals.vu.lt/nonlinear-analysis If the pair (x * , u * ) ∈ D((−∆ k ) β ) × U M is a locally optimal solution of problem (OCP k ), then there exists a function λ ∈ D((−∆ k ) β ) such that and Proof. Let us fix k = 1, 2 and define the operators: Then problem (OCP k ) can be formulated as We shall show that F k and F 0 k satisfy all assumptions of Theorem 2. First, let us note that from Lemma 4 applied to the function h = (f 0 , f ) it follows that for any x ∈ D((−∆ k ) β ), u 1 , u 2 ∈ U M and γ ∈ [0, 1], there exists a function u ∈ U M such that This means that assumption (b) of Theorem 2 is satisfied.
Using assumptions (B), (C) and analogous arguments as in [21,Prop. 5.1], we check that the mapping F k is continuously differentiable with respect to x ∈ D((−∆ k ) β ) and the differential ( Similarly (using assumption (A) and analogous arguments as in [22,Prop. 3.2]), we obtain a differentiability property of the mapping F 0 k 1 , whereby the differential 1 In order to prove a differentiability property of mappings F 2 and F 0 2 (then (−∆ 2 ) β = (−∆ DN ) β denotes the Dirichlet-Neumann Laplace operator of order β), we use the norm generated by the scalar product (7) and the estimation (9) instead of (6) and (8), respectively.
for any fixed u ∈ U M . The fact that the range Im(F k ) x (x * , u * ) of the mapping (F k ) x (x * , u * ) satisfies assumption (c) of Theorem 2 follows from • condition (10) and [21,Prop. 5.2] in the case of k = 1; • condition (11) and the bijectivity of (F 2 ) x (x * , u * ) 2 in the case of k = 2 (more precisely, in both cases, the range Im(F k ) x (x * , u * ) is a whole space L 2 , so it is closed and its codimension is equal to zero).
So, all assumptions of the smooth-convex extremum principle are satisfied. Consequently, there exist (not all equal to zero) λ 0 0 and λ ∈ L 2 such that for any h ∈ D((−∆ k ) β ) and Since Im(F k ) x (x * , u * ) = L 2 , therefore λ 0 = 0 and, without loss of generality, we can put λ 0 = 1. Then equality (14) can be rewritten as From assumption (C) it follows that V ∈ L 2 . Hence and from the fact that the operator (−∆ k ) β is self-adjoint it follows that λ ∈ D((−∆ k ) β ) and t ∈ (0, π) a.e.
Consequently, condition (12) holds. In order to prove condition (13), let us observe that condition (15) is equivalent to the following one: So, (13) follows from Lemma 5.
The proof is completed.

Examples
In this section, we present two theoretical examples, which illustrate obtained maximum principle.
It means that the pair (x * , u * ) given by (26) and (24) is the only pair, which can be a locally optimal solution of problem (20)- (21). Moreover, the minimal value of the cost functional J is equal to

Conclusions
In the paper, we investigated the Lagrange problems containing nonlinear control systems with Dirichlet and Dirichlet-Neumann Laplace operators of fractional orders. The main result, obtained in this work, is the Pontryagin maximum principle for such problems (Theorem 3). Obtained optimality conditions consist of the adjoint system (12) with Dirichlet and Dirichlet-Neumann boundary conditions, respectively, and the minimum condition (13). We derived our result using the smooth-convex extremum principle due to Ioffe (17). The aim of a forthcoming work is studying of the sufficient optimality conditions for problems (OCP 1 ) and (OCP 2 ).

Appendix: Basics of self-adjoint operators in a Hilbert space
In this section, we give the necessary notions and facts from the theory of unbounded selfadjoint operators in a real Hilbert space (cf. [21,22]). More details can be found in [2,27], where all results are obtained in the case of a complex Hilbert space. Nevertheless, their proofs can be reproduced (if required, with small changes) in the case of a real Hilbert space.
So, in this section, we shall assume that H is a real Hilbert space with a scalar product ·, · H .
A.1 Self-adjoint operator For x ∈ D(T * ), we denote T * x = z (this element is uniquely determined due to the density of D(T )). The operator T * : D(T * ) ⊂ H → H is called the adjoint operator to T . If T = T * , then T is called self-adjoint. We note that whenever T is self-adjoint operator one has T x, y H = x, T y H , x, y ∈ D(T ).

A.2 Spectral integral and decomposition theorem
Let B be a σ-algebra of Borel subsets of R, and P(H) denotes the set of all orthogonal projection operators onto closed linear subspaces of H. A set function E : B → P(H) is called a spectral measure (or a decomposition of the identity) if (i) for any x ∈ H, the set function B P → E(P )x is σ-additive; (ii) E(R) = I (here I denotes the identity operator on H); Let W be the union of all open sets V ⊂ R such that E(V ) = 0. Then the complement R \ W is called a support of a spectral measure E and denoted by supp(E). Let us assume that u : R → R defined E-a.e. is a bounded Borel measurable function. Then, in the usual way (via a sequence of simple functions), one can show that for any x ∈ H, there exists the integral (with respect to the vector measure E(·)x) +∞ −∞ u(λ) × E(dλ)x. We define the integral with respect to the spectral measure E  where χ ω is a characteristic function of the set ω.
In order to define the spectral integral in the case of a Borel measurable function u : P → R, where P ∈ B contains the support supp(E), it is sufficient to extend u on R to any Borel measurable function. Now, we formulate a spectral decomposition theorem, which plays a crucial role in the spectral theory of self-adjoint operators. In conclusion of this section, we shall define a function of a self-adjoint operator. Let T : D(T ) ⊂ H → H be a self-adjoint operator with ρ(T ) = ∅. From Theorem A1 it follows that T has the integral representation given by (A.3). For a Borel measurable function u : R → R defined E-a.e., we define the operator u(T ) as follows: According to general properties of the spectral integrals presented above, the domain D(u(T )) is given by (A.1), equality (A.2) holds and u(T ) is self-adjoint. Moreover, its spectrum is given by σ u(T ) = u σ(T ) , provided that u is continuous on σ(T ).