Access the full text.
Sign up today, get DeepDyve free for 14 days.
R. Kamocki (2020)
Necessary optimality conditions for Lagrange problems involving ordinary control systems described by fractional Laplace operatorsNonlinear Analysis-Modelling and Control, 25
Harbir Antil, Deepanshu Verma, M. Warma (2019)
Optimal Control of Fractional Elliptic PDEs with State Constraints and Characterization of the Dual of Fractional-Order Sobolev SpacesJournal of Optimization Theory and Applications, 186
Zhen-Qing Chen, R. Song (2005)
Two-sided eigenvalue estimates for subordinate processes in domainsJournal of Functional Analysis, 226
R. Cont, P. Tankov (2003)
Financial Modelling with Jump Processes
R. Kamocki (2020)
Existence of optimal solutions to Lagrange problems for ordinary control systems involving fractional Laplace operatorsOptimization Letters, 15
Christian Glusa, E. Otárola (2019)
Optimal control of a parabolic fractional PDE: analysis and discretizationarXiv: Optimization and Control
Harbir Antil, M. Warma (2017)
Optimal control of fractional semilinear PDEsESAIM: Control, Optimisation and Calculus of Variations
J. V'azquez (2014)
Recent progress in the theory of Nonlinear Diffusion with Fractional Laplacian OperatorsarXiv: Analysis of PDEs
H. Attouch, G. Buttazzo, G. Michaille (2014)
Variational Analysis in Sobolev and BV Spaces - Applications to PDEs and Optimization, Second Edition, 17
Harbir Antil, M. Warma (2018)
OPTIMAL CONTROL OF THE COEFFICIENT FOR THE REGIONAL FRACTIONAL P -LAPLACE EQUATION: APPROXIMATION AND CONVERGENCE
Jes'us D'iaz, David G'omez-Castro, T. Shaposhnikova, M. Zubova (2019)
A nonlocal memory strange term arising in the critical scale homogenisation of a diffusion equation with a dynamic boundary conditionarXiv: Analysis of PDEs
L. Caffarelli, A. Vasseur (2006)
Drift diffusion equations with fractional diffusion and the quasi-geostrophic equationAnnals of Mathematics, 171
E. Otárola (2015)
A piecewise linear FEM for an optimal control problem of fractional operators: error analysis on curved domainsarXiv: Numerical Analysis
Harbir Antil, M. Warma (2019)
Optimal control of the coefficient for the regional fractional \begin{document} $p$\end{document}-Laplace equation: Approximation and convergenceMathematical Control & Related Fields
H Attouch, G Buttazzo, G Michaille (2006)
10.1137/1.9780898718782Variational Analysis in Sobolev and BV Spaces. Applications to PDEs and Optimization
J. Vázquez (2012)
Nonlinear Diffusion with Fractional Laplacian Operators
D. Applebaum (2004)
Lévy Processes—From Probability to Finance and Quantum Groups
H Antil (2018)
102RIMS Kôkyûroku, 2090
K. Bogdan, T. Byczkowski, T. Kulczycki, M. Ryznar, R. Song, Z. Vondraček, P. Graczyk, A. Stos (2009)
Potential Analysis of Stable Processes and its Extensions
M. D'Elia, Christian Glusa, E. Otárola (2018)
A Priori Error Estimates for the Optimal Control of the Integral Fractional LaplacianSIAM J. Control. Optim., 57
X. Cabré, Jinggang Tan (2009)
Positive solutions of nonlinear problems involving the square root of the LaplacianAdvances in Mathematics, 224
(2000)
Potential theory of Schrödinger operator based on fractional Laplacian
P. Bates, Xinfu Chen, Adam Chmaj, Jianlong Han, Chunlei Zhang, Guangyu Zhao (2008)
ON SOME NONLOCAL EVOLUTION EQUATIONS ARISING IN MATERIALS SCIENCE
D. Bors (2015)
Optimal control of nonlinear systems governed by Dirichlet fractional Laplacian in the minimax frameworkarXiv: Analysis of PDEs
Igor Kossowski, B. Przeradzki (2021)
Nonlinear equations with a generalized fractional LaplacianRevista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 115
Christian Glusa, E. Otárola (2021)
Error Estimates for the Optimal Control of a Parabolic Fractional PDESIAM J. Numer. Anal., 59
R. Rockafellar (1976)
Integral functionals, normal integrands and measurable selections
D Applebaum (2004)
1336Notices Am. Math. Soc., 51
L. Cesari (1983)
Optimization-Theory And Applications
D. Bors (2019)
Optimal control of systems governed by fractional Laplacian in the minimax frameworkInternational Journal of Control, 94
R. Kamocki (2020)
On a Differential Inclusion Involving Dirichlet–Laplace Operators of Fractional OrdersBulletin of the Malaysian Mathematical Sciences Society, 43
H Antil, M Warma (2018)
Optimal control of the coefficient for fractional p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p$$\end{document}-Laplace equation: approximation and convergenceRIMS Kôkyûroku, 2090
D. Idczak (2018)
A bipolynomial fractional Dirichlet-Laplace problem.arXiv: Analysis of PDEs
M. Bonforte, J. Vázquez (2013)
A Priori Estimates for Fractional Nonlinear Degenerate Diffusion Equations on Bounded DomainsArchive for Rational Mechanics and Analysis, 218
L. Caffarelli, S. Salsa, L. Silvestre (2007)
Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional LaplacianInventiones mathematicae, 171
A. Bermúdez, C. Saguez (1987)
Optimal control of a Signorini problemSiam Journal on Control and Optimization, 25
S. Dohr, Christian Kahle, Sergejs Rogovs, P. Swierczynski (2018)
A FEM for an optimal control problem subject to the fractional Laplace equationCalcolo, 56
I. Gavrilyuk (2007)
Variational analysis in Sobolev and BV spacesMath. Comput., 76
D. Idczak, S. Walczak (2020)
Lagrange problem for fractional ordinary elliptic system via Dubovitskii–Milyutin methodNonlinear Analysis: Modelling and Control
Harbir Antil, E. Otárola (2014)
A FEM for an Optimal Control Problem of Fractional Powers of Elliptic OperatorsSIAM J. Control. Optim., 53
Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations
Jinggang Tan (2011)
The Brezis–Nirenberg type problem involving the square root of the LaplacianCalculus of Variations and Partial Differential Equations, 42
We consider an optimal control problem containing a control system described by a partial nonlinear differential equation with the fractional Dirichlet–Laplacian, asso- ciated 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. Keywords Lower closure theorem · Implicit measurable function theorem · Fractional Dirichlet–Laplace operator · Existence of optimal solutions Mathematics Subject Classification 49J20 · 49K20 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. [12,14–16,31–33]). Recently, optimal control problems containing control systems described by fractional Laplacians have received a lot of attention. We refer [1,20–22,29], where linear– quadratic 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 Faculty of Mathematics and Computer Science, University of Łódz, ´ Banacha 22, 90-238 Łódz, ´ Poland 123 S1506 Applied Mathematics & Optimization (2021) 84 (Suppl 2):S1505–S1519 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: minimize J (z, u) = f (x , z(x ), u(x ))dx , (1) subject to [(−Δ) ] z (x ) = f (x , z(x ), u(x )), x ∈ Ω a.e., (2) u(x ) ∈ M , x ∈ Ω a.e., N m where Ω ⊂ R , N ≥ 1, is an open and bounded set,β> 0, f , f : Ω ×R×R → R, m β M ⊂ R is a nonempty set and [(−Δ) ] 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 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 , as well as, a general convexity assumption ( H ) 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 f (x , z, u) = g(x , z) + B(x )u, 123 Applied Mathematics & Optimization (2021) 84 (Suppl 2):S1505–S1519 S1507 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 multifunc- tions. 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. 1 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]. Let Ω ∈ R be an open and bounded set. We shall denote: 2 2 –by L := L (Ω, R) the space of square Lebesque integrable functions endowed with the norm · 2, 2 2 1 1 –by H := H (Ω, R), H := H (Ω, R) the classical Sobolev spaces endowed 0 0 with norms · 2 and · 1, respectively. 1.1 Weak Dirichlet–Laplace Operator of Fractional Order Definition 1 [24] We say that u : Ω → R has a weak (minus) Dirichlet–Laplacian if 1 2 u ∈ H and there exists a function g ∈ L such that ∇ u(x )∇v(x )dx = g(x )v(x )dx Ω Ω for any v ∈ H . The function g, denoted by (−Δ) u, is called the weak Dirichlet– Laplacian and (−Δ) - the weak Dirichlet–Laplace operator. 1 2 2 2 It is well known (cf. [24, Theorem 3.1]) that if −Δ : H ∩ H ⊂ L → L is the 2 2 strong Dirichlet–Laplace operator, (−Δ) : dom((−) ) ⊂ L → L is a weak ω ω 1 2 Dirichlet–Laplace operator then H ∩ H ⊂ dom((−) ) and −Δu = (−Δ) u for ω ω 1 2 all u ∈ H ∩ H . Remark 1 In [7, Sect. 8.2] (−Δ) is called the Laplace–Dirichlet operator (without "weak") and denoted by −Δ. It is known [24] that the spectrum σ((−Δ) ) of (−Δ) contains only the eigenvalues ω ω of (−Δ) that can be written in a non-decreasing sequence, repeating each eigenvalue according to its multiplicity 0 <λ ≤ λ ≤ ··· ≤ λ →∞. Furthemore, a system 1 2 j {e } of eigenfunctions of the operator (−Δ) , corresponding to λ , is a hilbertian j ω j basis in L . 123 S1508 Applied Mathematics & Optimization (2021) 84 (Suppl 2):S1505–S1519 Now, for β> 0 we define the weak fractional Dirichlet–Laplace operator of order β β 2 2 β [(−Δ) ] : dom([(−) ] ) ⊂ L → L as follows (cf. [24, Sect. 3]): ω ω β β [(−Δ) ] u (x ) = (λ ) a e (x ), ω j j j j =1 where β 2 β 2 2 dom [(−) ] = u ∈ L ; ((λ ) ) a < ∞, ω j j =1 wher e a -s is such that u(x ) = a e (x ) . j j j j =1 It is well known [24] that the operator [(−Δ) ] is self-adjoint, bijective and its spec- β β trum σ([(−Δ) ] ) contains only proper values (λ ) , j ∈ N. Moreover, eigenspaces, ω j corresponding to (λ ) ’s and eigenspaces for [(−Δ) ], corresponding to λ ’s are the j ω j same. Let us consider in the space dom([(−) ] ) the following scalar product: β β u,v := u,v 2 + [(−Δ) ] u, [(−Δ) ] v β ω ω 2 which generates the norm: 2 β 2 u = u +[(−Δ) ] u . (3) β ω 2 2 L L β β The operator [(−Δ) ] is closed (as a self-adjoint operator), so the space dom([(−) ] ) ω ω with the scalar product ·, · is a Hilbert space. In the rest of this paper we shall use the space dom([(−) ] ) with the another scalar product ·, · given by ∼β β β u,v := [(−Δ) ] u, [(−Δ) ] v ∼β ω ω 2 which determines the norm u =[(−Δ) ] u 2 . (4) ∼β ω Norms (3) and (4) are equivalent due to the following Poincaré inequality in dom([(−) ] ) ([24, inequality (3.2)]): 2 2 u ≤ N u , (5) 2 β ∼β 123 Applied Mathematics & Optimization (2021) 84 (Suppl 2):S1505–S1519 S1509 where 1 if λ ≥ 1 N = (6) if λ < 1 β 2 ((λ ) ) (here λ > 0 is the first (the smallest) eigenvalue of the operator (−Δ) ). 1 ω From [24, Proposition 3.10] follows the following useful result: β 2 Proposition 1 If u u weakly in dom([(−) ] ) then u → u strongly in L and n 0 ω n 0 β β 2 [(−Δ) ] u [(−Δ) ] u weakly in L . ω n ω 0 1.2 Multifunctions Let S be an arbitrary nonempty set equipped with a σ - algebra B and : S s −→ (s) ⊂ R be a closed-valued multifunction. r −1 We shall say that is measurable if for each closed set C ⊂ R the set (C ) given by −1 (C ) := {s ∈ S : (s) ∩ C =∅} −1 is measurable (i.e. (C ) ∈ B). Let us define the set: dom := {s ∈ S : (s) =∅}. A function λ : dom → 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 → R ∪{±∞} is a normal integrand on n n S × R if f is lower semicontinuous on R for all s ∈ S and the epigraph n+1 E (s) := epi f (s, ·) ={(w, ν) ∈ R : ν ≥ f (s,w)} is a measurable multifunction. In the proof of the main result of this paper we apply the following version of Filippov’s lemma (cf. [30, Theorem 2J]): Theorem 1 (Measurable selection theorem) Let : S s −→ (s) ⊂ R be a multifunction of the form (s) := {w ∈ C (s) : F (s,w) = a(s) and f (s,w) ≤ κ , i ∈ J }, i i where C : S s −→ (s) ⊂ R is a measurable (closed-valued) multifunction, r k F : S × R → R is a Carathéodory mapping, ( f : i ∈ J ) is a countable collection r k of normal integrands on S × R and a : S → R , κ : S → R∪{±∞} are measurable. Then is the measurable (closed-valued) multifunction and hence has a measurable selection λ : dom → R . 123 S1510 Applied Mathematics & Optimization (2021) 84 (Suppl 2):S1505–S1519 Now, let us assume that (S,ρ) is a metric space and : S s −→ (s) ⊂ R is an arbitrary multifunction. We say that : S s −→ (s) ⊂ R has property (K) at the point s ∈ S iff (s ) = cl {(s) : ρ(s, s )<δ} , 0 0 δ>0 where cl Z 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. We have (cf. [17, Theorem 8.5.iii]) Theorem 2 Let : S s −→ (s) ⊂ R be a multifunction. Then has property (K) if and only if the graph of given by Gr := {(s,w) : s ∈ S,w ∈ (s)}, is closed in the product space S × 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. ν 1 ν Let G ⊂ R be a measurable set of finite measure, for every x = (x ,..., x ) ∈ G let A(x ) be a given nonempty subset of R and let A ={(x , z) : x ∈ G, z ∈ A(x )}, 1 n whereby z = (z ,..., z ). For every (x , z) ∈ A let Q(x , z) be a given subset of the r +1 space R . Theorem 3 Let us assume that for almost all x ∈ G, the set A(x ) is closed, the sets Q(x , z) are closed, convex and have property (K) with respect to z ∈ A(x ). Let r n ξ, ξ : G → R ,z, z : G → R , λ, η, λ ,η : G → R,k = 1, 2,... , be measurable k k k k 1 r 1 functions, ξ, ξ ∈ (L (G)) , η ∈ L (G), with z → z in measure on G , ξ ξ weakly k k k k 1 r in (L (G)) as k →∞, z (x ) ∈ A(x ), (η (x ), ξ (x )) ∈ Q(x , z (x )), x ∈ G, k = 1, 2,..., k k k k −∞ < i = lim inf η (x )dx < +∞,η (x ) ≥ λ (x ), k k k k→∞ 1 1 λ, λ ∈ L (G), λ → λ weakly in L (G). k k Then there exists a function η ∈ L (G) such that x (x ) ∈ A(x ), (η(x ), ξ(x )) ∈ Q(x , z(x )), x ∈ G, η(x )dx ≤ i . 123 Applied Mathematics & Optimization (2021) 84 (Suppl 2):S1505–S1519 S1511 2 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 U := {u : Ω → R − measur able on Ω; u(x ) ∈ M , t ∈ Ω a.e.} be a set of controls. A pair (z, u) ∈ dom([(−) ] ) × U is called admissible if it satisfies constraints ω M (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 the following conditions: ( H ) the function f is measurable on Ω, continuous on R × R and satisfies the following growth condition: there exist A ≥ 0, a ∈ L (Ω, R ) such that | f (x , z, u)|≤ A|z|+ a(x ), (7) for a.e. x ∈ Ω and all z ∈ R, u ∈ R , ( H ) the function f is measurable on Ω and continuous on R × R , 2 0 ( H ) the sets { } Q(x , z) := (μ ,μ) ∈ R × R;∃ μ ≥ f (x , z, u), μ = f (x , z, u) 0 u∈ M 0 0 (8) for a.e. x ∈ Ω and all z ∈ R, are convex. Remark 3 From [28, Theorem 1] it follows that if A <λ then for any fixed u ∈ U there exists a solution of the control system (2) (here λ is the first eigenvalue of th operator (−Δ) ). Later on we use the following results Proposition 2 If assumption ( H ) is satisfied, whereby A < (9) 2 N (N is given by (6)), then the set of admissible trajectories is bounded in dom([(−) ] ), i.e. there exists a constant C > 0 (independent on u) such that for any control u ∈ U the trajectory z ∈ dom([(−) ] ), corresponding to u, M ω satisfies the inequality z ≤ C . (10) ∼β 123 S1512 Applied Mathematics & Optimization (2021) 84 (Suppl 2):S1505–S1519 Proof Let us fix any control u ∈ U . Assume that z ∈ dom([(−) ] ) is a solution of M ω the control system (2), corresponding to u. Then, using (7) and the Poincaré inequality (5), we obtain 2 β 2 2 z =[(−Δ) ] z = | f (x , z(x ), u(x ))| dx ∼β 2 2 2 2 2 2 2 A |z(x )| dx + 2a ≤ 2 A N z + 2a if A > 0 2 2 ∼β L L a if A = 0. Thus, putting a 2 if A > 0 2 L 1−2 A N C = a 2 if A = 0, we get (10). The proof is completed. Corollary 1 The set of admissible trajectories is weakly relatively compact in dom([(−) ] ). Proposition 3 If assumptions ( H ) – ( H ) are satisfied and the set M is compact then 1 3 the multifunction Q(x , ·) : R z −→ Q(x , z) ∈ R × R given by (8) has property (K ). Proof In view of Theorem 2 it is sufficient to show that the graph Gr( Q(x , ·)) is closed l l l l l l in R × (R × R). Indeed, let {(z ,(μ ,μ ))} ∈ Gr( Q(x , ·)) and (z ,(μ ,μ )) −→ l∈N 0 0 l l l (z ˆ,(μ ˆ , μ)) ˆ in R × (R × R). Since (μ ,μ ) ∈ Q(x , z ) for all l ∈ N, therefore there exists u ∈ M such that l l l l l l μ ≥ f (x , z , u ) and μ = f (x , z , u ), l ∈ N. From the fact that M is compact it follows that there exist a subsequence (u ) ⊂ M j ∈N l m and u ˆ ∈ M such that u −→ u ˆ in R . Consequently, continuity of f and f with j →∞ respect to z implies j l l l l l j j j j j μ = f (x , z , u ) μ ≥ f (x , z , u ) 0 ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ and j →∞ j →∞ μ ˆ = f (x , z ˆ, u ˆ). μ ˆ ≥ f (x , z ˆ, u ˆ) 0 0 This means that (μ ˆ , μ) ˆ ∈ Q(x , z ˆ). Since z ˆ ∈ R, therefore (z ˆ,(μ ˆ , μ)) ˆ ∈ 0 0 Gr ( Q(x , ·)), so the mapping Q(x , ·) has property (K). The proof is completed. In what follows, we assume that for any admissible pair (z, u) the integral (1)isfinite. 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: 123 Applied Mathematics & Optimization (2021) 84 (Suppl 2):S1505–S1519 S1513 ( H ) there exists a constant γ ∈ R such that A := {(z, u) ∈ A : J (z, u) ≤ γ } =∅, 1 γ ( H ) there exists a function λ ∈ L (Ω, R) such that for any pair (z, u) ∈ A f (x , z(x ), u(x )) ≥ λ(x ), x ∈ Ω a.e. ∗ ∗ The pair (z , u ) ∈ A is called an optimal solution to problem (1)–(2)if ∗ ∗ J (z , u ) ≤ J (z, u) 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 )–( H ) with con- 1 5 ∗ ∗ dition (9) are satisfied then problem (1)–(2) has an optimal solution (z , u ) ∈ dom([(−) ] )) × U . ω M Proof Let us denote s := inf J (z, u) = inf J (z, u). (z,u)∈A (z,u)∈A l l From assumptions ( H ) and ( H ) it follows that −∞ < s < ∞. Let {(z , u )} ⊂ 4 5 l∈N A be a minimizing sequence of J , i.e. l l lim J (z , u ) = s. l→∞ l β From Corollary 1 it follows that the sequence of trajectories (z ) ⊂ dom([(−) ] ) l∈N ω l β contains a subsequence (still denoted by (z ) ) weakly convergent in dom([(−) ] ) l∈N ω ∗ β to a some function z ∈ dom([(−) ] ). Proposition 1 implies l ∗ 2 z −→ z strongly in L (11) and β l β ∗ 2 [(−Δ) ] z [(−Δ) ] z weakly in L . (12) ω ω Let us denote: G =Ω, A(x ) = R, A = Ω × R, Q(x , z) = Q(x , z), l β l β ∗ ξ (x ) = [(−Δ) ] z (x ), ξ(x ) = [(−Δ) ] z (x ) ω ω l l l l ∗ η (x ) = f (x , z (x ), u (x )), λ (x ) = λ(x ), z(x ) = z (x ) 123 S1514 Applied Mathematics & Optimization (2021) 84 (Suppl 2):S1505–S1519 for a.e. x ∈ Ω and all l ∈ N. It is clear that for almost all x ∈ Ω,the sets Q(x , z) are convex (assumption ( H )), have property (K) with respect to z ∈ R (Proposition 3), so are also closed l l l l 1 (Remark 2). Of course, functions z, z are measurable on Ω and ξ, ξ ,λ,λ ,η ∈ L , l ∈ N. Moreover, convergences (11) and (12)imply z → z in measure on Ω and l 1 ξ weakly in L as l →∞. We also see that l l l l z ∈ A(x ), (η ,ξ ) ∈ Q(x , z ), x ∈ Ω a.e., l ∈ N, l l l lim inf η (x )dx = lim J (z , u ) = s ∈ (−∞, +∞), l→∞ l→∞ l l l l η (x ) = f (x , z (x ), u (x )) ≥ λ(x ) = λ (x ), x ∈ Ω a.e., l ∈ N and l 1 λ = λ λ weakly in L . Consequently, using Theorem 3, we assert that there exists a function η ∈ L such that (η(x ), ξ(x )) ∈ Q(x , z (x )), x ∈ Ω a.e. and η(x )dx ≤ s. (13) Now, let us consider the multifunction : Ω x −→ (x ) ⊂ R given by ∗ ∗ (x ) ={u ∈ M : η(x ) ≥ f (x , z (x ), u), ξ(x ) = f (x , z (x ), u)}. Let us denote: S = Ω, C : S x → C (x ) = M ⊂ R , m ∗ F : S × R (x , u) → F (x , u) = f (x , z (x ), u) ∈ R, m ∗ f : S × R (x , u) → f (x , u) = f (x , z (x ), u) ∈ R, 1 1 0 a : S x → a(x ) = ξ(x ) ∈ R,κ : S x → κ (x ) = η(x ) ∈ R, x ∈ S. 1 1 Since M is closed and for each closed set D ⊂ R ∅ if M ∩ D =∅ −1 C ( D) = Ω if M ∩ D =∅ 123 Applied Mathematics & Optimization (2021) 84 (Suppl 2):S1505–S1519 S1515 is measurable, therefore C is a closed-valued measurable multifunction. Moreover, the function f is a normal integrand as a Carathéodory function (cf. [30, Proposition 2C]). Of course, functions a and κ are measurable on Ω. Consequently, from Theo- rem 1 it follows that is a closed-valued, measurable multifunction and there exists ∗ m ∗ a measurable function u : Ω → R such that u (x ) ∈ (x ) for a.e. x ∈ Ω.This means that ∗ ∗ η(x ) ≥ f (x , z (x ), u (x )), x ∈ Ω a.e. (14) and β ∗ ∗ ∗ [(−Δ) ] z (x ) = f (x , z (x ), u (x )), x ∈ Ω a.e., ∗ m u (x ) ∈ M ⊂ R , x ∈ Ω. ∗ ∗ Consequently, (z , u ) ∈ A, whereby (see (13) and (14)) ∗ ∗ s ≤ f (x , z (x ), u (x ))dx ≤ η(x )dx ≤ s. Ω Ω ∗ ∗ This means that (z , u ) is an optimal solution to problem (1)–(2). The proof is completed. 3 Illustrative Example In this section we present the following theoretical problem: minimize J (z, u) = sin tz(t ) + u (t ) dt , (15) subject to [(−Δ) ] z (t ) = Az(t ) + u (t ), t ∈ (0,π) a.e., (16) u(t ) ∈[−1, 1], t ∈ (0,π) a.e., where 0 ≤ A < . It is easy to check that all assumptions of [23, Theorem 4] are satisfied. Conse- quently, if the pair (z , u ) ∈ dom((− ) ) × U is a locally optimal solution ∗ ∗ ω [−1,1] to problem (15)–(16) then there exists a function λ ∈ dom((− ) ) such that (−Δ ) λ(t ) = Aλ(t ) + sin(t ), t ∈ (0,π) a.e. (17) 123 S1516 Applied Mathematics & Optimization (2021) 84 (Suppl 2):S1505–S1519 and 3u (t )(λ(t ) + 1)(u − u (t )) ≥ 0, u ∈[−1, 1] (18) for a.e. t ∈ (0,π).Let λ(t ) = b sin jt. Then (17) can be written in the j =1 following way: ∞ ∞ ∞ 2 2 2 jb sin jt = A b sin jt + c sin jt , t ∈ (0,π), j j j π π π j =1 j =1 j =1 whereby c = sin(t ) sin jt dt, j ∈ N. Consequently, 1 π c ; j = 1 1− A 2 b = = j − A 0; j > 1. This means that λ(t ) = sin t , t ∈ (0,π) a.e. 1 − A is a solution of (17). Hence and from (18) we conclude that u (t ) =−1, t ∈ (0,π) a.e. (19) and (−Δ ) z (t ) = Az (t ) − 1, t ∈ (0,π) a.e. (20) ω ∗ ∗ The above equation can be solved the same as (17). Then, we obtain 4 sin t sin 3t sin 5t z (t ) =− + + + ... , t ∈ (0,π) a.e. (21) π 1 − A 3(3 − A) 5(5 − A) It means that the pair (z , u ) given by (21) and (19) is the only pair which can be ∗ ∗ a locally optimal solution of problem (15)–(16). Moreover, the minimal value of the 123 Applied Mathematics & Optimization (2021) 84 (Suppl 2):S1505–S1519 S1517 cost functional J is equal to J (z , u ) = sin tz (t ) + u (t ) dt ∗ ∗ ∗ 4 sin t sin 3t sin 5t =− sin t + + + ... dt − π π 1 − A 3(3 − A) 5(5 − A) =− − π. 1 − A Now, we show that (z , u ) is an optimal solution to problem (15)–(16). Indeed, it is ∗ ∗ clear that assumptions ( H ), ( H ), condition (9)( N = 1) are satisfied and the sets 1 2 β 2 3 3 Q(t , z) ={(μ ,μ) ∈ R ; μ ≥ sin tz + u ,μ = Az + u , u ∈[−1, 1]}, 0 0 for a.e. t ∈ (0,π) and all u ∈[−1, 1], are convex. Furthemore, from [23, Lemma 2] and Proposition 2 it follows that | f (t , z(t ), u(t ))|=| sin tz(t ) + u (t )|≤|z(t )|+ 1 2 π ≤ ζ(2)z + 1 ≤ C + 1 π 3 for all admissible pairs (z, u) ∈ dom((− ) ) × U , where C = π and ω [−1,1] 1−2 A ζ(γ ) = denotes the Riemann zeta function. Consequently, for all admissible j =1 pairs (z, u) ∈ dom((− ) ) × U the integral J is finite and hypothesis ( H ) ω [−1,1] 4 and ( H ) hold, whereby in ( H ) 5 5 2π λ(t ) = π + 1, t ∈ (0,π) a.e. 3(1 − 2 A ) This means that all assumptions of Theorem 4 are satisfied, so the pair (z , u ) given ∗ ∗ by (21) and (19) is the optimal solution to problem (15)–(16). 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/. 123 S1518 Applied Mathematics & Optimization (2021) 84 (Suppl 2):S1505–S1519 References 1. Antil, H., Otárola, E.: A FEM for an optimal control problem of fractional powers of elliptic operators. SIAM J. Control Optim. 53(6), 3432–3456 (2015) 2. Antil, H., Verma, D., Warma, M.: Optimal control of fractional elliptic PDEs with state constraints and characterization of the dual of fractional-order Sobolev spaces. J. Optim. Theory Appl. 186, 1–23 (2020) 3. Antil, H., Warma, M.: Optimal control of the coefficient for the regional fractional p-Laplace equation: approximation and convergence. Math. Control Relat. Fields 9, 1–38 (2019) 4. Antil, H., Warma, M.: Optimal control of the coefficient for fractional p-Laplace equation: approxi- mation and convergence. RIMS Kôkyûroku 2090, 102–116 (2018) 5. Antil, H., Warma, M.: Optimal control of fractional semilinear PDEs. ESAIM Control Optim. Calc. Var. 26(1), 5 (2020). https://doi.org/10.1051/cocv/2019003 6. Applebaum, D.: Lévy processes—from probability to finance and quantum groups. Notices Am. Math. Soc. 51, 1336–1347 (2004) 7. Attouch, H., Buttazzo, G., Michaille, G.: Variational Analysis in Sobolev and BV Spaces. Applications to PDEs and Optimization. SIAM-MPS, Philadelphia (2006) 8. Bates, P.W.: On some nonlocal evolution equations arising in materials science. In: Nonlinear Dynamics and Evolution Equations, vol. 48 of Fields Inst. Commun. Amer. Math. Soc., Providence, RI, pp. 13–52 (2006) 9. Bermudez, A., Saguez, C.: Optimal control of a Signorini problem. SIAM J. Control Optim. 25, 576–582 (1987) 10. Bogdan, K., Byczkowski, T.: Potential theory of Schrödinger operator based on fractional Laplacian. Probab. Math. Stat. 20(2), 293–335 (2000) 11. Bogdan, K., Byczkowski, T., Kulczycki, T., Ryznar, M., Song, R., Vondracek, Z.: Potential Theory of Stable Processes and its Extensions. Lecture Notes in Mathematics 1980. Springer, Berlin (2009) 12. Bonforte, M., Vázquez, J.L.: A priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains. The Royal Swedish Academy of Sciences, Mittag-Leffler Institute, Report No. 21, (2013/2014) arXiv:1311.6997 13. Bors, D.: Optimal control of nonlinear systems governed by Dirichlet fractional Laplacian in the minimax framework. J. Control. Int. (2019). https://doi.org/10.1080/00207179.2019.1662091 14. Cabré, X., Tan, J.: Positive solutions of nonlinear problems involving the square root of the Laplacian. Adv. Math. 224, 2052–2093 (2010) 15. Caffarelli, L.A., Salsa, S., Silvestre, L.: Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian. Invent. Math. 171, 425–461 (2008) 16. Caffarelli, L.A., Vasseur, A.: Drift diffusion equations with fractional diffusion and the quasi- geostrophic equation. Ann. Math. 171, 1903–1930 (2010) 17. Cesari, L.: Optimization-Theory and Applications. Springer, New York (1983) 18. Chen, Z.-Q., Song, R.: Two-sided eigenvalue estimates for subordinate Brownian motion in bounded domains. J. Funct. Anal. 226, 90–113 (2005) 19. Cont, R., Tankov, P.: Financial Modelling with Jump Processes. Chapman & Hall/CRC Financial Mathematics Series, Boca Raton (2004) 20. D’Elia, M., Glusa, C., Otárola, E.: A priori error estimates for the optimal control of the integral fractional Laplacian. SIAM J. Control Optim. 57(4), 2775–2798 (2019) 21. Dohr, S., Kahle, C., Rogovs, S., Swierczynski, P.: A FEM for an optimal control problem subject to the fractional Laplace equation. Calcolo 56, 37 (2019). https://doi.org/10.1007/s10092-019-0334-3 22. Glusa, C., Otárola, E.: Error estimates for the optimal control of a parabolic fractional PDE. arXiv e-prints arXiv:1905.10002 (2019) 23. Idczak, D., Walczak, S.: Lagrange problem for fractional ordinary elliptic system via Dubovitskii- Milyutin method. Nonlinear Anal. 25(2), 321–340 (2020) 24. Idczak, D.: A bipolynomial fractional Dirichlet-Laplace problem. Electron. J. Differ. Eq. 2019(59), 1–17 (2019) 25. Kamocki, R.: On a Differential inclusion involving Dirichlet–Laplace operators of fractional orders. Malays. Math. Sci. Soc. Bull. 43, 4089–4106 (2020) 26. Kamocki, R.: Existence of optimal solutions to Lagrange problems for ordinary control systems involv- ing fractional Laplace operators. Lett. Optim. 15(2), 779–801 (2021) 123 Applied Mathematics & Optimization (2021) 84 (Suppl 2):S1505–S1519 S1519 27. Kamocki, R.: Necessary optimality conditions for Lagrange problems involving ordinary control sys- tems described by fractional Laplace operators. Nonlinear Anal. 25(5), 884–901 (2020) 28. Kossowski, I., Przeradzki, B.: Nonlinear equations with a generalized fractional Laplacian. RACSAM 115, 58 (2021). https://doi.org/10.1007/s13398-021-00998-5 29. Otárola, E.: A piecewise linear FEM for an optimal control problem of fractional operators: error estimates on curved domains. ESAIM 51(4), 1473–1500 (2017) 30. Rockafellar, R.T.: Integral functionals, normal integrands and measurable selections. In: Gossez, J.P., Lami Dozo, E.J., Mawhin, J., Waelbroeck, L. (eds.) Nonlinear Operators and the Calculus of Variations. Lecture Notes in Mathematics, vol. 543. Springer, Berlin (1976) 31. Tan, J.: The Brezis-Nirenberg type problem involving the square root of the Laplacian. Calc. Var. 42, 21–41 (2011) 32. Vázquez, J.L.: (2012) Nonlinear diffusion with fractional Laplacian operators. Nonlinear Partial Dif- ferential Equations vol. 7 of Abel Symposia, pp. 271–298 33. Vázquez, J.L.: Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators. Discret. Contin. Dyn. Syst. Ser. 7, 857–885 (2014) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Applied Mathematics and Optimization – Springer Journals
Published: Dec 1, 2021
Keywords: Lower closure theorem; Implicit measurable function theorem; Fractional Dirichlet–Laplace operator; Existence of optimal solutions; 49J20; 49K20
You can share this free article with as many people as you like with the url below! We hope you enjoy this feature!
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get DeepDyve free for 14 days.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.