Existence of optimal solutions to Lagrange problems for ordinary control systems involving fractional Laplace operators

In this paper, we study optimal control problems containing ordinary control systems, linear with respect to a control variable, described by fractional Dirichlet and Dirichlet–Neumann Laplace operators and a nonlinear integral performance index. The main result is a theorem on the existence of optimal solutions for such problems. In our approach we use a characterization of a weak lower semicontinuity of integral functionals.

Our study is based on the L 1 weak lower-semicontinuity of integral functionals [26]. The existence of optimal solutions is also investigated in [9], where an optimal control problem with a fractional Dirichlet Laplacian, defined in R n is considered. The control system, studied there, has a variational structure and the cost functional depends also on the fractional Laplacian. The paper is organized as follows. In Sect. 2, we give necessary notions and facts concerning ordinary Dirichlet and Dirichlet-Neumann Laplace operators of fractional order. In Sect. 3, based on a some version of a global implicit function theorem [18], we formulate and prove a theorem on the existence of a unique solution of the control sysytems (E k ), k = 1, 2. In Sect. 4, we derive the main result of this paper, namely a theorem on the existence of optimal solutions for problems (OCP k ), k = 1, 2. Section 5 contains an illustrative, theoretical example. We finish with Sect. A containing some basics from the spectral theory of self-adjoint operators in a real Hilbert space.

Preliminaries
This part of the paper concerns fractional ordinary Dirichlet and mixed Dirichlet-Neumann Laplace operators. Definitions of these operators are based on the spectral integral representation theorem for a self-adjoint operator in a Hilbert space (cf. [19] and Appendix A).

Laplace operators of fractional order
Let us consider the one-dimensional Laplace operator −Δ on the interval (0, π) given by We define the following spaces of functions: where H 1 0 = H 1 0 ((0, π), R n ) and H 2 = H 2 ((0, π), R n ) are classical Sobolev spaces. We recall that conditions z(0) = z(π ) = 0 (hidden in the definition of H D ) and z(0) = z (π ) = 0 are called Dirichlet and Dirichlet-Neumann boundary conditions, respectively. Moreover, H D and H DN are dense subspeces of the space L 2 = L 2 ((0, π), R n ). The operator −Δ : H D ⊂ L 2 → L 2 given by (1) under Dirichlet boundary conditions is called the Dirichlet Laplace operator and denoted by −Δ D . Similarly, by the Dirichlet-Neumann Laplace operator −Δ DN : H DN ⊂ L 2 → L 2 we mean the operator −Δ under Dirichlet-Neumann boundary conditions. In an elementary way one can show that operators −Δ D and −Δ DN are self-adjoint. Moreover, their spectrum is given by respectively and the eigenspaces Eig j (−Δ D ) (associated with the eigenvalues λ j = j 2 ), Eig j (−Δ DN ) (associated with the eigenvalues λ j = j − 1 2 2 ) are sets Eig j (−Δ D ) = {c sin jt; c ∈ R n }, It is well known that systems of functions . . , n, j = 1, 2, . . . , are complete orthonormal systems in L 2 . Now, let us assume that β > 0. We define the operator (here E is the spectral measure for the operator −Δ D and a j 2 π sin jt is the projection of x on the n-dimensional eigenspace Eig j (−Δ D )). It is given by Remark 1 To shorten the notation, in the rest of this paper the fractional Dirichlet (Dirichlet-Neumann) Laplace operator of order β is denoted by (−Δ 1 ) β ((−Δ 2 ) β ). Now, we formulate some useful facts concerning mentioned operators and their domains (cf. [19]).

Lemma 1
The spaces D((−Δ k ) β ), k = 1, 2 are complete with the scalar products The above result follows from the fact that operators (−Δ k ) β ), k = 1, 2 are selfadjoint, so also closed. In our paper we shall use a scalar products given by which generate equivalent norms · k β and · k ∼β in D((−Δ k ) β ) due to the following Poincaré inequalities: The proof of (3) can be found in [19, formula (11)]. Analogously, we prove inequality (4): so embeddings Proof For the convenience of the reader, we recall the proof of the inequality (6) which can be found in [21] (the proof of (5) for n = 1 can be found in [19]).
Hence, we obtain inequality (6). The proof is completed.

Lemma 3 If β > 1 2 then the operators
Proof The proof of this fact for k = 1 (in the case of n = 1) is given in [19, proof of Lemma 5.1]. It is analogous for vector valuable functions, so we present only the sketch of it in the case of k = 2. Let F ∈ L 2 ((0, π), R n ) be any bounded (by a constant D) set in L 2 ((0, π), R n ) and consider a function In the same way as in [19,Section 5.3] we can show that there exists a unique function Consequently, Let us consider the set of functions Then, for any fixed h ∈ (0, π) we have Using the Hölder inequality (for series) we obtain Similarly we estimate the term I 3 . Now, we estimate the term I 2 .
Analogously, we estimate terms I 1 , Consequently, The proof is completed.

Remark 2
The relatively compactness of F follows from the following Kolmogorov-Fréchet-Riesz theorem (cf. [10,Theorem 4.26]): (Here F |Ω denotes the restrictions to of the functions in F).
Using the above lemma and analogous arguments as in the proof of [19, Lemma 5.2] we obtain

Existence and uniqueness of a solution to the control systems (E 1 ) and (E 2 )
The main result of this section is a theorem on the existence of a unique solution to the control systems (E k ), k = 1, 2. In the proof of this fact we use the following result.
satisfies the Palais-Smale (PS) condition 1 , for any (x, y) ∈ X × Y such that F(x, y) = 0 then, for any y ∈ Y , there exists a unique x y ∈ X such that F(x y , y) = 0.
In the rest of this paper we assume that β > 1 2 . Let us define the following set of controls: We have where (A2) B ∈ L ∞ ((0, π), R n×m ) (A3) for any pair (x, u) ∈ D((−Δ k ) β )×U M one of the following three conditions are satisfied -|I (x l )| ≤ M for all l ∈ N and some M > 0, -I (x l ) → 0, admits a convergent subsequence (I (x l ) denotes the Fréchet differential of I at x l ).
then for any fixed control u ∈ U M there exists a unique solution x u ∈ D((−Δ k ) β ) of the control system (E k ).
Proof Let us fix k = 1, 2 and define the operator It is sufficient to show that F k satisfies all assumptions of Theorem 2.
-Using assumptions (A1), (A2) and analogous arguments as in [19, Proposition 5.1], we check that the mapping F k is continuously differentiable with respect to x ∈ D((−Δ k ) β ) and the differential ( satisfies the Palais-Smale condition. First, let us observe that the growth condition (9) and conditions (11), (12) guarantee coercivity of φ k u for any u ∈ U M (it is sufficient to use the same arguments as in the proof of [19,Lemma 5.3]). Moreover, it is continuously differentiable with respect to x and its differential (φ k u ) : D((−Δ k ) β ) → R is given by Using analogous arguments as in the proof of [19, Proposition 5.3] (including coercivity of φ k u , Corollary 1 and the Lebesque dominated convergence theorem) we conclude that there exists a subsequence (x l j ) j∈N such that The proof is completed.
We also have the following two results

Proposition 1 If M is a bounded set and assumptions (A1), (A2), (A3) of Theorem 3 are satisfied then there exists constants C
Proof Let us fix k = 1, 2 and any control u ∈ U M . Let C be a constant such that |u(t)| ≤ C for a.e. t ∈ (0, π). Assume that x u ∈ D((−Δ k ) β ) is a solution of the control system (E k ), corresponding to u. Then, using (9), we obtain Thus and from Lemma 2 we have This means that The proof is completed.
Proof Let us fix k = 1, 2 and consider a sequence of controls (u l ) l∈N ∈ U M and a sequence of corresponding solutions (x l ) l∈N ⊂ D((−Δ k ) β ) of the system (E k ). Using the standard arguments we check that compactness and convexity of the set M imply a convexity, boundedness and closure of the set U M in L 2 ((0, π), R m ). This means that U M is sequentially weakly compact, while L 2 ((0, π), R m ) is a reflexive space. Consequently, there exist a subsequence (u l i ) i∈N and u 0 ∈ U M such that u l i i→∞ u 0 weakly in L 2 ((0, π), R m ), so the condition (Z3) of this proposition is satisfied. From Proposition 1 it follows that the sequence of norms x l k ∼β is bounded, so, there exist a subsequence (x l i ) i∈N and a function x 0 ∈ D((−Δ k ) β ) such that Consequently, Corollary 1 implies convergences (Z1) and (Z2). Now, we show that the x 0 is a solution of (E k ), corresponding to u 0 . Indeed, first we note that since the matrix B is essentially bounded on (0, π), therefore Bu l i i→∞ Bu 0 weakly in L 2 .
Moreover, using condition (9) and Lemma 2 we have where C k , k = 1, 2 are constants from Proposition 1. Consequently, from the Lebesque dominated convergence theorem it follows that Then, of course Thus, using (Z2) we get weakly in L 2 ((0, π), R n ). On the other hand, (x l i ) is a solution of (E k ), corresponding to (u l i ), so we have This means that ∈ (0, π) a.e.
The proof is completed.

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 of the problems (OCP k ), k = 1, 2. Let us fix k = 1, 2. We shall say, that a pair (x * , u * ) ∈ D((−Δ k ) β ) × U M is a globally optimal solution of the problem (OCP k ), if x * is the solution of the control system (E k ), corresponding to the control u * and We have Theorem 4 Let us fix k = 1, 2 and assume that 1. M is convex and compact, 2. hypothesis (A1), (A2) and (A3) of Theorem 3 are satisfied, 3. f 0 (·, x, u) is measurable on (0, π) for all x ∈ R n and u ∈ M, 4. f 0 (t, ·, ·) is continuous on R n × M for a.e. t ∈ (0, π), 5. f 0 (t, x, ·) is convex on M for a.e. t ∈ (0, π) and all x ∈ R n , 6. there exist a summable function ψ : (0, π) → R + 0 and a constant c ≥ 0 such that for a.e. t ∈ (0, π) and all x ∈ R n , u ∈ M.

Illustrative example
In this section we present the the following theoretical problems where k = 1, 2, β > 1 2 and a > 0. We see that B : (0, π) → R 2×1 and It is clear that f is measurable with respect to t, continuously differentiable on R 2 and 4 , for a.e. t ∈ (0, π) and all x ∈ R 2 . Consequently, conditions (9) then conditions (13), (14) are satisfied. Of course, the function f 0 satisfies assumptions 3,4,5 of Theorem 4. The assumption 6 also holds because ∈ (0, π) and all x ∈ R 2 , u ∈ [−1, 1]. Consequently, we proved the following Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

A Basics of self-adjoint operators in a Hilbert space
In this section we give the necessary notions and facts concerning a theory of unbounded self-adjoint operators in a real Hilbert space (cf. [19]). More details can be found in [1,25], 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 * and D(T ) = D(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 , f or all x, y ∈ D(T ).

A.2 Spectral integral and decomposition theorem
We define the integral with respect to the spectral measure E One proves that the above operator is linear, continuous and Hermitian. Now, let u : R → R defined a.e. E be an unbounded Borel measurable function. Let us define the sequence of functions u n : u n (λ) = u(λ), i f |u(λ)| ≤ n 0, i f |u(λ)| > n. Let us consider the set One can show that D is a dense linear subspace of H and for x ∈ D there exists the limit So, we can define the operator 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 6 it follows that T has the integral representation given by (22). For a Borel measurable function u : R → R defined a.e. 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 (20), the equality (21) 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 ).