Approximation Methods in Optimization of Nonlinear Systems 9783110668520, 9783110668438

Table of contents :
Preface
Contents
Introduction
1. Optimal Distributed Control Problem for an Ill-Posed Strongly Nonlinear Elliptic Equation with p-Laplace Operator and L1-Type of Nonlinearity
2. On Approximation of One Class of Optimal Control Problems for Strongly Nonlinear Elliptic Equations with p-Laplace Operator
3. Neumann Boundary Optimal Control Problem for Strongly Nonlinear Elliptic Equation with p-Laplace Operator
4. Asymptotic Analysis of Optimal Neumann Boundary Control Problem in Domain with Boundary Oscillation for Elliptic Equation with Exponential Non-Linearity
5. On Optimal and Quasi-Optimal Controls in Coefficients for Multi-Dimensional Thermistor Problem with Mixed Dirichlet-Neumann Boundary Conditions
6. Approximation of an Optimal Control Problem in Coefficient for Variational Inequality with Anisotropic p-Laplacian
7. On Unbounded Optimal Controls in Coefficients for Ill-Posed Elliptic Dirichlet Boundary Value Problems
8. On Optimal L1-Control in Coefficients for Quasi-Linear Dirichlet Boundary Value Problem with BMO-Anisotropic p-Laplacian
Bibliography

Citation preview

Peter I. Kogut, Olha P. Kupenko Approximation Methods in Optimization of Nonlinear Systems

De Gruyter Series in Nonlinear Analysis and Applications

Editor in Chief Jürgen Appell, Würzburg, Germany Editors Catherine Bandle, Basel, Switzerland Alain Bensoussan, Richardson, Texas, USA Avner Friedman, Columbus, Ohio, USA Mikio Kato, Tokyo, Japan Wojciech Kryszewski, Torun, Poland Umberto Mosco, Worcester, Massachusetts, USA Louis Nirenberg, New York, USA Simeon Reich, Haifa, Israel Alfonso Vignoli, Rome, Italy Vicenţiu D. Rădulescu, Krakow, Poland

Volume 32

Peter I. Kogut, Olha P. Kupenko

Approximation Methods in Optimization of Nonlinear Systems

Mathematics Subject Classification 2010 49-02, 93-02, 49J20, 49K20, 93C10, 93C20, 35D30, 35B40, 35J57, 35J60, 35J70, 35J75 Authors Prof. Peter I. Kogut Honchar Dnipro National University Gagarin av.,72 Dnipro 49010 Ukraine Prof. Olha P. Kupenko Dnipro Polytechnics Dept of System Analysis and Control Yavornitsky av.,19 Dnipro 49010 Ukraine

ISBN 978-3-11-066843-8 e-ISBN (PDF) 978-3-11-066852-0 e-ISBN (EPUB) 978-3-11-066859-9 ISSN 0941-813X Library of Congress Control Number: 2019952434 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2020 Walter de Gruyter GmbH, Berlin/Boston Printing and binding: CPI books GmbH, Leck www.degruyter.com

| To the memory of Professor Valerii Mel’nik

Preface After many years of increasing scientific interest, optimal control of partial differential equations (PDEs) has developed into a well-established discipline in mathematics with myriad applications to science and engineering. As the field has grown, so has the complexity of the systems it describes; the numerical realization of optimal controls has become increasingly difficult, demanding ever more sophisticated mathematical tools. Effective modeling for optimal control problems involving systems of essentially PDEs and models that are simplifying approximations of both the state equation and control constrains have been the focus of many research initiatives in the last decade. A monograph on the subject Optimal Control of Strongly Nonlinear Partial Differential Equations is intended to address some of the obstacles that face researchers today, particularly with regard to ill-posedness of boundary value problems and problems where we can not expect to have uniqueness of their solutions in the standard functional spaces. Bringing original results together with others previously scattered across the literature, it tackles computational challenges by exploiting methods of approximation and asymptotic analysis and harnessing differences between optimal control problems and their underlying PDEs. The book is focused on self-contained development and study of the asymptotic and approximation methods of optimal control problems for strongly nonlinear elliptic systems which are related to many fields of applied mathematics, including combustion theory, thermal inflammation in chemical reactors, material sciences, mechanics of fluids, nonlinear dielectric composites, material and topology optimization. Specific topics covered in the work include: – a mostly self-contained mathematical theory of optimal control problems of elliptic equations with exponential type of nonlinearity and Dirichlet-Neumann boundary conditions; – a wide class of approximation optimal control problems which can be considered as a basis for the construction of suboptimal controls for the original control problems; – optimal control problems dealing with ill-posed objects such as elliptic equations with degenerate and unbounded coefficients in the principle part, variational inequalities with anisotropic p-Laplacian, elliptic equations with BMO-coefficients and L1 -right hand side; – asymptotic analysis of optimal control problems for nonlinear systems in domains with rugous boundary; – optimal and quasi-optimal control in coefficients for multi-dimensional thermistor problem; – the concept of consistent optimal control problems for ill-posed boundary value problems;

https://doi.org/10.1515/9783110668520-202

VIII | Preface –

optimal control problems in coefficients for elliptic systems with variable p(x)Laplacian.

Served as a monograph where specific applications are explored, this book can be considered as a good reference for graduate students in pure and applied mathematics, as well as engineers, researchers, and practitioners requiring a solid mathematical basis for the solution of practical problems. The book focuses on all of these aspects from two perspectives. First, a rigorous and mostly self-contained mathematical theory of elliptic PDSs with exponential nonlinearity in their right hand side together with well-posedness for the governing optimal control problems is described. Since an ill-posed PDE problem – in the sense of Hadamard – due to the freedom to choose proper control inputs may turn out to be well-posed as an optimization problem, while a well-posed PDE problem easily can exhibit non-well-posedness, once integrated into an optimal control problem, it follows that well-posedness of optimal control problems is a new and interesting issue. This fact was motivating for the development of the concept of consistent optimal control problems for nonlinear and degenerate PDEs, followed by appropriate concepts for feasible solutions and their approximation. Even though there are by now a number of textbooks for PDE-constrained optimal control available, this monograph contains a collection of results that are necessary to treat the optimal control problem for strictly nonlinear and degenerate elliptic systems which are not available in a textbook otherwise. The main statements of optical control problems are strongly motivated by engineering applications in mechanics and material sciences as explained above, but they can be understood without any knowledge from those fields. Requiring only a standard knowledge in the calculus of variations, differential equations, and functional analysis, the book can serve as a text for a graduate course in computational methods of optimal design and optimization, it can be of use also in seminars building on the knowledge from a graduate course, as well as a good reference for applied mathematicians addressing optimization of ill-posed problems. For the convenience of the reader, we sometimes reintroduce basic concepts that are dealt with in the previous chapters; however, they are explicitly geared towards the particular application. Admitting some potential redundancy, we thereby keep the chapters in the book self-consistent for researchers in the field. Many people have influenced our thinking on contents and presentation of this book. But first of all, we wish to express our appreciation to the memory of our friend and teacher Valerii Mal’nik who first made us aware of the elegance and power of variational methods. Prof. V. Mel’nik, one of the most distinguished Ukrainian mathematicians, died on August, 2008, generously shared his insights and ideas with us for making our research collaboration, the fruits of which feature prominently in this publication, so rewarding and enjoyable. We also place on record our gratitude to our colleagues, co-authors, and friends Ciro D’Apice, Rosanna Manzo, Tierry Horsin, Umberto De Maio, Gabriella Zecca, Tiziana

Preface | IX

Durante, for their support and with whom it is a real pleasure to cooperate. We must thank the anonymous referees who painstakingly went through the first draft of the manuscript. Above all, we express our profound gratitude to our families for their constant encouragement and support throughout this seemingly endless writing project. We are perfectly aware that, most likely, there are still at places inadequacies, inconsistencies of notations, inadvertently omitted references, or inappropriate attributions of original results. But any mathematical or otherwise adventure must come to an end, even if its main protagonist is not fully satisfied with it. Or in other words that any mathematician, pure or applied, should read and reread: "The last step for most authors is to stop writing. That’s hard." Dnipro, August 2019

Peter I. Kogut Olha P. Kupenko

Contents Preface | VII Introduction | 1 1 1.1 1.2 1.3 1.4 1.5 1.6

2 2.1 2.2 2.3 2.4 3 3.1 3.2 3.3 3.4 3.5 4

4.1 4.2 4.3 4.4 4.5

Optimal Distributed Control Problem for an Ill-Posed Strongly Nonlinear Elliptic Equation with p-Laplace Operator and L1 -Type of Nonlinearity | 11 Auxiliaries and previous analysis of the boundary value problem | 12 On Pohozaev inequality and a priori estimates for a special class of weak solutions | 16 On reformulation of the original optimal control problem | 25 Auxiliary fictitious optimal control problem and its properties | 27 On a priori estimate for the solutions of variational problem (1.59) | 29 On asymptotic behaviour of the sequence of optimal pairs to the problem (1.47)–(1.50) as ε → 0 | 34 On Approximation of One Class of Optimal Control Problems for Strongly Nonlinear Elliptic Equations with p-Laplace Operator | 39 On consistency of optimal control problem | 40 Approximating optimal control problems and their previous analysis | 44 Asymptotic analysis of approximating OCP | 54 Optimality conditions for approximating OCP | 67 Neumann Boundary Optimal Control Problem for Strongly Nonlinear Elliptic Equation with p-Laplace Operator | 73 Setting of the problem | 74 A priori estimates both for energy solutions and feasible solutions | 80 Existence of optimal boundary controls | 96 On approximation of optimal boundary control problem | 98 On existence of bounded feasible solutions | 108 Asymptotic Analysis of Optimal Neumann Boundary Control Problem in Domain with Boundary Oscillation for Elliptic Equation with Exponential Non-Linearity | 116 Previous analysis of optimal control problem | 118 On consistency of optimal control problem | 129 On uniqueness of optimal solution and optimality conditions | 133 Description of the domain perturbations | 141 Asymptotic analysis of OCP (4.1)–(4.5) | 147

XII | Contents 5

5.1 5.1.1 5.1.2 5.1.3 5.2 5.2.1 5.2.2 5.2.3 5.3 5.4 6 6.1 6.2 6.3 6.4 7 7.1 7.2 7.3 7.4 7.5 7.6 7.6.1 7.6.2 7.6.3 7.6.4 7.6.5 8 8.1

On Optimal and Quasi-Optimal Controls in Coefficients for Multi-Dimensional Thermistor Problem with Mixed Dirichlet-Neumann Boundary Conditions | 164 Introduction and main motivation | 164 Relaxation of the original OCP | 170 Motivation | 172 Main results | 173 Preliminaries and some auxiliary results | 174 On Orlicz spaces | 174 Some special properties of Sobolev-Orlicz spaces | 179 On the weak convergence of fluxes to flux | 183 On approximated optimal control problem in coefficients and its properties | 187 Asymptotic analysis of OCP (5.19)–(5.23) | 198 Approximation of an Optimal Control Problem in Coefficient for Variational Inequality with Anisotropic p-Laplacian | 211 Setting of the optimal control problem | 212 Existence of optimal solutions | 216 Regularization of OCP | 218 Asymptotic analysis of approximating OCP | 222 On Unbounded Optimal Controls in Coefficients for Ill-Posed Elliptic Dirichlet Boundary Value Problems | 226 Notation and preliminaries | 227 Weak convergence in variable L2 -spaces associated with SNsym -matrices | 233 Setting of the optimal control problem | 236 On variational solutions to OCP (7.1)–(7.3) and their approximation | 244 On attainability of non-variational optimal solutions | 261 On some properties of unbounded bilinear forms associated with skew-symmetric L2 (Ω)-matrices | 278 Preliminaries | 279 Motivating example | 282 On formula of integration by parts for measurable functions | 287 On substantiation of formula (7.198) for a non-Lipschitz case | 289 Proof of the uniqueness result | 290 On Optimal L1 -Control in Coefficients for Quasi-Linear Dirichlet Boundary Value Problem with BMO-Anisotropic p-Laplacian | 294 Notation and preliminaries | 296

Contents | XIII

8.2 8.3 8.4 8.5 8.6

Setting of the optimal control problem | 301 On consistency of optimal control problem (8.28)–(8.31) | 305 Existence of Optimal Pairs | 314 On higher integrability of the gradient of an approximation solution | 316 1,p On density of smooth compactly supported functions in W0,B (Ω) | 325

Bibliography | 331

Introduction In a huge collection of published literature dealing with diverse scientific and technological problems, one encounters the mathematical problem ⎧ (︀ )︀ ⎨ − div ψ(y)y = λg(x)Φ(y) + p in Ω, (1) ∂y ⎩ ψ(y) + Bh(x)y = u on ∂Ω, ∂ν where y denotes a scalar field such as the temperature or mass concentration, the term λΦ(y) represents an y-dependent source of the transferred quantity, with λ denoting the intensity level of the source and Φ(y) its dependence on y, Ω is a closed region in RN bounded by surface ∂Ω, with ν denoting the unite outward normal to ∂Ω, the functions g : Ω → R and h : ∂ → R are given and usually required to be positive, and the factor ψ denotes an y-dependent diffusion coefficient, for instance, the thermal conductivity or the coefficient of mass diffusion, ψ(y) > 0 in general. Hereinafter, λ is termed the source intensity, Φ is the source distribution function, u is a boundary control, and p is treated as a distributed control. B is a constant whose magnitude is an indication of the ease of transfer from the boundary ∂Ω to the environment outside Ω relative to the ease of transfer within Ω itself. In particular, when B = ∞, boundary relation in (1) can be reduced to the standard Dirichlet condition y = 0 on ∂Ω. Various analytical aspects of the system (1) have been examined in the literature. For certain source distributions Φ(y), there may be multiple solutions when λ lies in some range λ1 ≤ λ ≤ λ2 and unique solutions when λ lies outside this range. The most popular example of system (1), that has been was studied in the classical papers by Gelfand [62] and Joseph-Lundgrem [73] and mainly motivated by the problems of self-ignition and stellar structure [24], reads as the following Dirichlet boundary value problem {︃ −∆y = λe y + p in Ω, (2) y = 0 on ∂Ω, where Ω is an bounded domain in RN with smooth boundary ∂Ω. From a mathematical point of view, one of the main interests is that Dirichlet problem (2) may have both unbounded (singular) solutions and bounded (regular) solutions (see [14, 62, 73]). Moreover, it is well known that there exists a finite positive number λ* , called the extremal value, such that problem (2) has at least a classical positive solution y ∈ C2 (Ω) if 0 < λ < λ* and p ≡ 0, while there exists no solution of (2) (neither regular nor singular, even in the weak sense) for λ > λ* and p ≡ 0. In particular, considerable mathematical interest accrues to the results of Gelfand concerning the change in the number (less than 3) of solutions of (2) when N = 1, 2 and p ≡ 0 to an infinite number of solutions of problem (2) when λ = 2 and N = 3. However, the existence and properties of the solutions of problems (1) and (2) depend strongly not https://doi.org/10.1515/9783110668520-001

2 | 0 Introduction

only on the dimension N, domain Ω, and nonlinearities ψ(y) and Φ(y), but also on a particular choice of the control function p and u. The boundary value problem (1) appears in many practical considerations and can be considered as a relevant model for the wide range of physical processes, where the source distribution function Φ(y) takes the form of a continuous, positive, increasing and convex function such that Φ(0) > 0 and Φ(y)/y → ∞ as y → ∞. Typical examples of such function are Φ(y) = e y and Φ(y) = (1 + y)p , with p > 1. We will mainly focus on the case when the reaction term Φ(y) has exponential form or it is qualitatively similar (in some sense) to exponential function. In particular, as was mentioned in [73], this type of nonlinearity is an important characteristic of combustion phenomena; without it, critical conditions disappear and the very idea of combustion loses its meaning. It follows from here that, in contrast to many other areas of applied mathematics, the complete linearization of the equations like (2) is unacceptable and, as we will see later on, this point is crucial in optimization problems related to systems like (1) or (2). To give more physical motivation to the study of optimal control problems for objects described by systems (1), we note that the heat generation may occur in different physical problems and this fact can give rise to temperature non-uniformities, cause many undesired effects, and even introduce damage to the system until complete failure occurs. Indeed, in material fatigue studies, when a solid is subjected to a cycled load, a finite amount of heat is generated in each cycle [74, 151]. Heat generation also occurs as a result of chemical reactions, for instance, in a porous catalyst particle [117], in a tubular reactor with axial diffusion [118] and in connection with thermal self-ignition of a chemically active mixture of gases in a vessel [62, 57]. Thus we are faced with the necessity to study the problem of effective control of heat generation that maintain or enhance sustainability of the original objects. Obviously, the interaction of controls and heat generation in the corresponding materials heavily depends on the properties of the system (1). Let us consider the problem in which y represents the temperature field and heat generation occurs as a result of chemical reactions. It is well-known that the nonstationary equation of heat transfer in a medium with continuously distributed sources of heat can be represented as follows )︃ (︃ ∂ ∑︁ C i H i = − div q + p in Ω, (3) ∂t i

where H = (H1 , H2 , . . . ) is the enthalpy of the components of the mixture, C = (C1 , C2 , . . . ) is their concentration, q is the heat flux, and p is the density of the heat sources. In combustion theory, the heat source term p describes the evolution of heat in a chemical reaction. Taking into account the fact that variation of rate coefficients for elementary reactions as a function of temperature is usually parametrized in terms of the Arrhenius expression k = Aexp (−B/T),

0 Introduction | 3

where T is the temperature, A is the pre-exponential factor (related to the change in entropy in the transition state), B is often reported in kinetics studies and can be referred to as the activation temperature, the density of the heat sources can be written in the form (︂ )︂ E p = Qz exp − . (4) RT Here, Q is the heat of reaction, E is the activation energy (which is usually fairly high), R is the gas constant, and z is a constant which is determined by the reaction kinetics. The main assumption upon which the whole of combustion theory is based, is that the dependence of the combustion rate on temperature is stronger than that on all other parameters. Since the heat of the reaction must be large, it follows that diffusive thermal conduction can be neglected. As a result, the heat flux can be expressed as follows q ≈ −λ(T)∇T + c p vT, where λ = λ(T) is the thermal conductivity, v is the velocity of the mixture, and c p is the heat capacity at constant pressure which is assumed to be constant. According to the definition of the heat capacity, we have )︃ (︃ ∂T ∂ ∑︁ Ci Hi = cp ρ , ∂t ∂t i

where ρ is the density. As a result, we arrive at the following basic equation of combustion theory (︂ )︂ (︀ )︀ E ∂T = div λ(T)∇T − c ρ vT + Qz exp − , cp ρ ∂t RT where flow velocity v must, generally speaking, be obtained from the gas dynamic equations. However, if we confine our consideration to the inflammation problem, we can suppose that the medium is stationary, and, therefore, the above equation can be reduced to the form (︂ )︂ (︀ )︀ E ∂T = div λ(T)∇T + Qz exp − . (5) cp ρ ∂t RT For the further simplification of (5), let us assume that there is a stationary distribution of the temperature (or the concentration of active product for chain inflammation) with given boundary conditions (Dirichlet, Neumann, or Robin), and neglecting the dependence of the thermal conductivity λ(T) on the temperature, the derivation with respect to time can be omitted and the stationary equation of the theory of thermal inflammation takes the form (︂ )︂ (︀ )︀ E − div λ(T)∇T = Qz exp − RT or, if the thermal conductivity λ(T) has the representation λ = |∇T |p−2 , with some p ≥ 1, we get (︂ )︂ (︁ )︁ E − div |∇T |p−2 ∇T = Qzexp − . (6) RT

4 | 0 Introduction

Let T* be a temperature such that near T* the reaction of inflammation takes place. In the problem of spontaneous inflammation, this is the initial temperature T0 , and in problems of propagating combustion, it is the maximum chain temperature T m . Then using the method of expanding the exponent, the argument of the exponential term in (6) can be expressed as follows E E E E E 1 ∆T. ≈ = − = RT R (T* + ∆T ) RT* 1 + ∆T RT RT*2 * T* Thus, the exponential term in the expression for the reaction rate is written approximately as E − E e− RT ≈ e RT* e y , where y is the dimensionless temperature difference, y=

E E ∆T = (T − T* ) . 2 RT* RT*2

Taking this representation into account, we finally deduce from (6) the following equation of the theory of thermal inflammation (︂ 2 )︂p−1 (︁ )︁ RT* − E − div |∇y|p−2 ∇y = zQ e RT* e y E

in Ω

(7)

with appropriate boundary conditions on ∂Ω. As was mentioned before, the complete linearization of the equations (7) can lead to meaningless relations and it is an unacceptable approach to the study of such problems. Because of this, qualitative analysis of optimal control problems for strongly nonlinear systems like (1) or (7), is not a trivial task. One of main reasons is the fact that in this case we deal with the ill-posed boundary value problems for elliptic equations for which the corresponding strongly nonlinear differential operator − div(ψ(y)∇y) − λg(x)Φ(y) is not monotone and, in principle, has degeneracies as ∇y tends to zero. Moreover, when the term ψ(y) is regarded as the coefficient of the Laplace operator, we also have the case of unbounded coefficients and the exponential growth of the term f (y) can lead to the blow-up of the solutions of the corresponding evolution problems. It means that there is no reason to assert the existence of weak solutions to the system (1) for given controls u and p, or to suppose that such solution, even if it exists, is unique. Moreover, because of this and L1 -boundedness of the source distribution function g(x)Φ(y) there are serious hurdles to deduce an a priori estimate for the weak solutions of the corresponding BVP in the standard Sobolev space. In view of this, we cannot rely that the correspondent optimal control problems for system (1) are consistent from mathematical point of view. At the same time, it is well-known that there is a principle difference between the theory of optimal control problems for systems with distributed parameters and the theory of partial differential equations. It means that well-posedness for optimal

0 Introduction | 5

control problems governed by systems like (1) and well-posedness for PDEs themselves are different notions, and, consequently, the corresponding theories are not mutually included. Nevertheless, as we will see later, in spite of the ill-posedness of the system (1), the corresponding optimization problems are consistent and solvable. Moreover, in some particular cases we can assert that an optimal solution is unique. In particular, an optimal control problem for the system −∆y = λf (y) + v

in Ω,

y=0

on ∂Ω

(8)

with f (y) = e y , and distributed control v, was first discussed in detail by Casas, Kavian, and Puel [22], where the problem of existence and uniqueness of the underlying boundary value problem and the corresponding optimal control problem was treated and an optimality system has been derived and analyzed. However, analogous results for the case of more general nonlinear elliptic equations of the type (︀ )︀ − div a(|∇y|) = f (y) + v

(9)

with a(∇y) = |∇y|p−2 ∇y, p ≥ 2, and different boundary conditions remained open for nowadays. So, it would be a mistake to claim that the classical approach to the study and analysis of such optimization problems is impossible and, in reality, unnecessary with the great advances of modern computer facilities and technology. Because of the above mentioned difficulties and characteristic features related with nonlinear elliptic equations (9), approximate solutions to the corresponding optimization problems by means of numerical computation have thus become the main stream of research and applications, in both academia and industry. In the endeavor of advancing efficient computational theories and methods for these problems, one successful and unified approach is to formulate various apparently different but intrinsically related optimization problems such that their solutions could provide deep insights and feasible proximity to the optimal characteristics of the original problems. This book is designed as an self-contained introduction to some important aspects of the theory of optimal control problems for essentially nonlinear elliptic systems such as the solvability issue, approximation schemes, convergence analysis, and attainability condition of optimal solutions for several types of nonlinear elliptic boundary value problems. In particular, we study a rather wide class of optimal control problems for the systems like (1) with exponential type of nonlinearity Φ(y), develop the methods of approximation of optimal solutions of the considered problems, derive sufficient conditions of attainability of optimal characteristics in appropriate topologies, and supply the original optimization problems by the corresponding regularizing schemes which would be acceptable for the numerical simulations with a prescribed level of accuracy. In this monograph, we mainly focus on optimal control problems for strictly nonlinear and quasi-linear PDEs that are frequently encountered in applied mathematics and engineering applications related with combustion theory. The presentation is organized as follows.

6 | 0 Introduction

In Chapter 1 we deal with an optimal distributed control problem for nonlinear elliptic equation arising in the theory of thermal inflammation. Since the main characteristic feature of the considered BVP is the fact that because of the specificity of non-linearity f (y), we can not obtain any a priori estimate for the weak solutions in the standard Sobolev space W01,p (Ω), we discuss the necessity of reformulation of the original optimization problem and prove the existence of optimal pairs to the reformulated version of optimal control problem. Our main intention is to show that the original BVP possesses a special type of weak solutions satisfying some extra state constraints. To do so, we introduce a special family of fictitious optimization problems and show that, for some admissible control u ∈ L q (Ω), the original BVP admits a weak solution in W01,p (Ω) with extra properties and this solution can be attained with a prescribed level of accuracy using the sequence of optimal pairs for the special fictitious minimization problems. In Chapter 2 we continue to study the optimal control problem from the previous chapter and focus on the scheme of direct two-level approximation of the strongly nonlinear differential operator − div(|∇y|p−2 ∇y) − f (y) with exponential term f (y) which is not monotone and, in principle, has degeneracies as ∇y tends to zero. Moreover, when the term |∇y|p−2 is regarded as the coefficient of the Laplace operator, we have also the case of operator with unbounded coefficients. Because of this and L1 -boundedness of the function f (y), there are serious hurdles to deduce the differentiability of the state with respect to control and derive an optimality system for the corresponding optimal control problem. Using a monotone and bounded approximation Fk (|∇y|2 ) of |∇y|2 , we introduce a special two-parametric family of optimization problems with fictitious controls and show that an optimal pair to the original optimal control problem can be attained by optimal solutions to the approximating ones provided the parameters k ∈ N and ε > 0 possess some special asymptotic properties. With that in mind we consequently provide the well-posedness analysis for the perturbed partial differential equations as well as for the corresponding fictitious optimal control problems. After that we pass to the limits as k → ∞ and ε → 0. Since the fictitious optimization problems are stated for the quasi-linear elliptic equations with coercive and monotone operators without any state and control constraints, we show that the approximation and regularization approach looks rather attractive option from the numerical simulations point of view. In Chapter 3 we examine an optimal control problem for the mixed DirichletNeumann boundary value problem for the strongly non-linear elliptic equation with p-Laplace operator and the exponential type of nonlinearity in its right-hand side. A density of surface traction u acting on a part of boundary of open domain is taken as the boundary control. The optimal control problem is to minimize the discrepancy between a given distribution y d ∈ L2 (Ω) and the current system state. We deal with such case of nonlinearity when we cannot expect to have a solution of the state equation for any admissible control. After defining a suitable functional class in which we look for solutions and assuming that this problem admits at least one feasible solution, we

0 Introduction | 7

prove the existence of optimal pairs. In order to handle the strong non-linearity in the right-hand side of elliptic equation, we involve a special two-parametric fictitious optimization problem. We derive existence of optimal solutions to the regularized optimization problems at each (ε, k)-level of approximation and discuss the asymptotic behaviour of the optimal solutions to regularized problems as the parameters ε and k tend to zero and infinity, respectively. Our main intention In Chapter 4 is to discuss the existence of optimal pairs to an OCP for strongly non-linear elliptic equation with exponential type of nonlinearity in the right-hand side and the mixed Dirichlet-Neumann boundary conditions that is imposed in a domain with rugous boundary. One of the main characteristic feature of the considered OCP is the fact that we have two different types of controls — distributed and boundary, and the Neumann control zone is supported along of a rough part of ∂Ω ε . The second feature is related with the specificity of non-linearity f (y) and the fact that this term belongs to L1 (Ω ε ) and not to L2 (Ω ε ) as usual. The originality of this problem arises from the conjunction of these two features when ε tends to zero. Because of the specificity of non-linearity f (y), we can not obtain any a priori estimate for the weak solutions in the sense of distributions in the standard Sobolev space W01,2 (Ω). Moreover, we cannot assert that the corresponding BVP admits at least one solution for a given boundary control u ∈ Aad (Γ N,ε ) and fictitious distributed control g ∈ Gad . For indicated problem we derive the corresponding optimality conditions, establish its solvability, and show that an optimal solution is unique (in spite of the fact that we deal with ill-posed boundary value problem for strongly nonlinear elliptic equation). Working in the framework of special class of weak solutions, we realize the limit passage in OCP as ε tends to zero, identify the limit optimization problem, show that the limit problem admits a unique optimal solution, and prove the main variational property: optimal solutions for the original problem converge to the optimal solution of the limit problem. As follows from the results of asymptotic analysis that we provide for OCP in a perturbad domain, even if parameter ε > 0 is small enough, the rugosity of the control zones affects not only the limit boundary value problem, but the limit cost functional and boundary control constraints set as well. Thus, the limit optimal control problem has a structure that drastically differs from the original one. It means that the rugosity effect can not be neglected under precise consideration of the corresponding optimal control problems. In Chapter 5 we consider an optimal control problem in coefficients for the system of two coupled elliptic equations also known as thermistor problem which provides a simultaneous description of the electric field u = u(x) and temperature θ(x). The (︀ )︀ coefficients of operator div B(x) ∇ θ(x) are used as the controls in L∞ (Ω). This model is based on rational mechanics of electrorheological fluids, that takes into account the complex interactions between the electromagnetic fields and the moving liquid. In particular, the electrorheological fluids have the interesting property that their viscosity depends on the electric field in the fluid.

8 | 0 Introduction

The optimal control problem, we consider in this chapter, is to minimize the discrepancy between a given distribution θ d ∈ L r (Ω) and the temperature of thermistor θ ∈ W01,γ (Ω) by choosing an appropriate anisotropic heat conductivity matrix B, to the best of the author knowledge, the existence of optimal solutions for the above thermistor problem remains an open question. Only very few articles deal with optimal control for the thermistor problem (see [64, 69] in two dimensional case, [120] for three spatial dimensions, and the recent papers [34, 35, 70] in multi-dimensional case). In spite of the fact that a great deal of attention has been paid by many authors in the study of the thermistor problem during the last two decades, the existence theorems for the corresponding boundary value problem were proved only under Dirichlet boundary conditions for the potential u and some smallness conditions on other parameters. There are several reasons for this: – it is unknown whether the set of feasible points to the corresponding optimization problem is nonempty and weakly closed in appropriate functional spaces; – we have no a priori estimates for the weak solutions to the boundary value problem; – the asymptotic behaviour of a minimizing sequence to the cost functional is unclear in general; – the optimal control problem is ill-posed and requires some relaxation. To circumvent the problems listed above, we propose the so-called indirect approach to the solvability of the optimal control thermistor problem in coefficients. Basing on the perturbation theory of extremal problems and the concept of fictitious controls, we prove the existence of so-called quasi-optimal and optimal solutions to the considered optimization problem and show that they can be attained by the optimal solutions of some appropriate approximations for the original optimal control problem. The main idea of our approach is based on the fact that weak solutions to the Dirichlet-Neumann boundary value thermistor problem can be attained through a special regularization of nonlinear p(x)-Laplacian and introducing fictitious controls with more regular properties. Another important aspect concerns the study of optimal control problems in coefficients for variational inequalities. The interest to variational inequalities whose principle part is an anisotropic p-Laplace-like operator arises from various applied contexts related to composite materials such as nonlinear dielectric composites, whose nonlinear behavior is modeled by the so-called power low. It is sufficient to say that anisotropic )︀ (︀ p−2 p-Laplacian ∆ p (A, y) = div |(A∇y, ∇y)| 2 A∇y has profound background both in the theory of anisotropic and nonhomogeneous media and in Finsler or Minkowski geometry [153]. As a rule, the effect of anisotropy appears naturally in a wide class of geometry — Finsler geometry. A typical and important example of Finsler geometry is Minkowski geometry. In this case, anisotropic Laplacian is closely related to a convex hypersurface in RN , which is called the Wulff shape [153]. Since the topology of the Wulff shape essentially depends on the matrix of anisotropy A(x), it is reasonable to take such matrix as a control. From mathematical point of view, the interest of

0 Introduction | 9

anisotropic p-Laplacian lies on its nonlinearity and an effect of degeneracy, which turns out to be the major difference from the standard Laplacian on RN . Using the direct method in the Calculus of Variations, we show in Chapter 6 that an optimal control problem for variational inequality has a nonempty set of solutions provided the admissible controls A(x) are uniformly bounded in BV-norm, in spite of the fact (︀ )︀ p−2 that the corresponding quasilinear differential operator − div |(A∇y, ∇y)| 2 A∇y , in 1

principle, has degeneracies as |A 2 ∇y| tends to zero. A number of regularizations have been suggested in the literature. See [139] for a discussion for what has come to be known as (ε, p)-Laplace problem, such as − div((ε + p−2 |∇y|2 ) 2 ∇y). While the (ε, p)-Laplacian regularizes the degeneracy as the gradients tend to zero, the term |∇y|p−2 , viewed again as a coefficient, may grow large [20]. Therefore, following ideas of [23] (see also Chapter 2), we introduce yet another regularization 1 that leads to a sequence of monotone and bounded approximations Fk (|A 2 ∇y|2 ) of 1 |A 2 ∇y|2 . As a result, for fixed parameter p ∈ [2, ∞) and control A(x), we arrive at a twop−2 1 parameter variational problem governed by operator − div((ε + Fk (|A 2 ∇y|2 )) 2 A∇y). Finally, we deal with a two-parameter family of optimal control problems in the coefficients for monotone nonlinear variational inequalities. We consequently provide the well-posedness analysis for the perturbed elliptic variational inequalities as well as for the optimal control problem. In particular, we show that the solutions of twoparametric family of perturbed optimal control problems can be considered as appropriate approximations to optimal pairs for the original problem. To the end, we note that the approximation and regularization are not only considered to be useful for the mathematical analysis, but also for the purpose of numerical simulations. The main goal of Chapter 7 is to study an optimal control problem for a linear elliptic equation with unbounded coefficients in the principle part of the elliptic op(︀ )︀ erator. Namely, we deal with the following elliptic operator − div A(x)∇y + a0 (x)y, where matrix A = A sym + A skew ∈ L1 (Ω; SNsym ) ⊕ L2p (Ω; SNskew ) is adopted as a control with p ≥ 1 and its symmetric part A sym has a degenerate spectrum. In spite of the fact that the corresponding Dirichlet boundary value problem can exhibit the so-called Lavrentieff phenomenon, non-uniqueness of the weak solutions as well as other surprising consequences, using the Steklov smoothing operator, we prove that some weak solutions to this problem can be obtained as the limit of solutions of coercive problems with bounded coefficients. As a result, this allows us to prove that the corresponding optimal control problem has a nonempty set of solutions. However, even if the considered OCP has a unique solution (A0 , y0 ), it does not ensure that this pair can be attained in such way. We give in this chapter an example of OCP for which there exists a unique solution that can not be attained through the limit of optimal solutions to the regularized problems with bounded coefficients. This fact motivates us to consider the question about attainability of non-variational optimal solutions. With that in mind we bring into consideration some specification of the original optimal control problem and show that an optimal solution to such problem can inherit a singular character of the

10 | 0 Introduction

original stream matrix A. In order to approximate such solution, we propose a special regularization of the original optimal control problem by optimization problems in perforated domains with special type of fictitious boundary controls. The last topic, considered in Chapter 8 is a Dirichlet optimal control problem for quasi-linear elliptic equation −∆ p (A, y) + |y|p−2 yu = − div f

in Ω

)︀ (︀ p−2 with anisotropic p-Laplacian −∆ p (A, y) = − div |(∇y, A∇y)| 2 A∇y in its principle part and L1 -control u in coefficient of the low-order term. As characteristic feature of such optimal control problem is a specification of the non-symmetric matrix of anisotropy in BMO-space. Since we cannot expect to have a solution of the state equation in the classical Sobolev space, we specify a suitable functional class in which we look for solutions and prove existence of optimal pairs using an approximation procedure and compactness arguments in variable spaces. In particular, we introduce a special functional space Xu,B related to a given control u and symmetric part B of matrix A, and prove that the set of feasible solutions to the original optimal control problem is always nonempty. Moreover, we show that for every admissible control u ∈ L1 (Ω), a weak solutions (in the sense of Minty) to the corresponding BVP can be obtained as the limit of solutions of coercive problems with bounded coefficients, using any L∞ -approximation of BMO-matrix A. Such solutions are called approximation solutions. The characteristic feature of such solutions is that they lay in variable space Xu,B and, in general, do not satisfy the energy equality but rather some energy inequality. We also derive a priori estimates for such solutions that do not depend on the skew-symmetric part D of matrix A. Since we can not exclude the situation when approximation solutions depend on the choice of approximation {A k }k∈N of matrix A, this fact stimulates us to consider some specification of the original optimal control problem. In spite of the fact that the new setting of optimal control problem imposes some additional restriction on the class of feasible solutions, we show that such reformulation of the original optimal control problem is not restrictive from control point of view. Moreover, using the concept of convergence in variable spaces, we show that the new optimal control problem is consistent and has a nonempty set of solutions. At the end of this chapter, using the Gehring’s lemma, we give sufficient conditions for the uniqueness of an approximation solution.

1 Optimal Distributed Control Problem for an Ill-Posed Strongly Nonlinear Elliptic Equation with p-Laplace Operator and L1 -Type of Nonlinearity In this chapter we are concerned with the following optimal control problem for a nonlinear elliptic equation with p-Laplacian: ∫︁ ∫︁ 1 1 Minimize J(u, y) = |y − y d |2 dx + |u|q dx, (1.1) 2 q Ω

Ω

subject to constrains −∆ p y = f (y) + u y=0 q

in Ω,

on ∂Ω,

u ∈ L (Ω), y ∈

W01,p (Ω),

(1.2) (1.3) (1.4)

p where 2 ≤ p < N, q > p−1 , Ω is a bounded domain in RN with sufficiently smooth ′ boundary ∂Ω, f (y) = F (y), F ∈ C1 (K) for any compact set K ⊂ R, F(z) ≥ exp (C−1 F z) for (︀ )︀ all z ∈ R and some constant C F > 0, ∆ p y = div |∇y|p−2 ∇y is the p-Laplacian, and y d ∈ L2 (Ω) is a given distribution. This type of boundary value problems (BVPs) has been studied extensively for p = 2 in various context. In particular, when F(u) = λe u , we have the well-known model of Frank-Kamenetstkii for solid ignition [55]. Other applications of the BVPs like (1.2)–(1.3) appear in the study of stellar structures [24] and in combustion theory for the chemical reactors and etc. In general, the problem (1.2)–(1.3) can be seen as the stationary counterpart of evolution equations with nonlinear diffusion. It is well known that the indicated BVP is ill-posed in general and the exponential growth of the term f (y) can lead to the blow-up of the solutions of the corresponding evolution problems. It means that there is no reason to assert the existence of weak solutions to (1.2)–(1.3) for a given u ∈ L q (Ω), or to suppose that such solution, even if it exists, is unique (see, for instance, I.M. Gelfand [62], M.G. Crandall and P.H. Rabinowitz [29], F. Mignot and J.P. Puel [122], T. Gallouët, F. Mignot and J.P. Puel [59], H. Fujita [58], R.G. Pinsky [132], R. Ferreira, A. De Pablo, J.L. Vazquez [52], P.I. Kogut and G. Leugering [88], J. Dolbeault and R. Stańczy [45]). In this chapter we discuss the existence of optimal pairs to the reformulated version of optimal control (1.1)–(1.4). The necessity of reformulation of the original problem is inspired by the following reason: the main characteristic feature of the indicated BVP (1.2)–(1.3) is the fact that because of the specificity of non-linearity f (y), we can not obtain any a priori estimate for the weak solutions in the standard Sobolev space W01,p (Ω). Moreover, since we cannot assert that the BVP (1.2)–(1.3) admits at least one solution for a given distributed control u ∈ L q (Ω), our main intention is to show that the original BVP possesses a particular type of weak solutions satisfying some extra https://doi.org/10.1515/9783110668520-002

12 | 1 Optimal Distributed Control Problem state constraints. To do so, we introduce a special family of fictitious optimization problems and show that, for some admissible control u ∈ L q (Ω), the BVP (1.2)–(1.3) admits a weak solution in W01,p (Ω) with extra properties and this solution can be attained with a prescribed level of accuracy using the sequence of optimal pairs for the specific fictitious minimization problems (the similar approach can be found in [80, 82, 109]). In fact, this circumstance allows us to prove not only the consistency of the reformulated optimal control problem, but also establish its solvability.

1.1 Auxiliaries and previous analysis of the boundary value problem Let Ω be a bounded open convex subset of RN (N > 2). We assume that the boundary ∂Ω is of the class C1,1 . Through this chapter we assume also that there exists a point (︀ )︀ x0 ∈ int Ω such that Ω is star-shaped with respect to x0 , i.e. σ − x0 , ν(σ) ≥ 0 for H N−1 -a.a. σ ∈ ∂Ω. Let F : R → [0, +∞) be a mapping such that F ∈ C1loc (R), F is a convex function, and there exists a constant C F > 0 satisfying C F F ′ (z) ≥ F(z) ≥ exp (C−1 ∀ z ∈ R, F z), ⃒ ⃒ 0 ⃒ ⃒ ∫︁ ⃒ ⃒ and ⃒⃒ zF ′ (z) dz⃒⃒ < +∞. ⃒ ⃒

(1.5)

−∞

We define the function f ∈ C loc (R) as follows: f (z) = F ′ (z). Typical example of f (z) is f (z) = Ce C F z . p Let p (2 ≤ p < N) and q ( p−1 < q < ∞) be given real values. We assume that each q element of the space L (Ω) can be considered as an admissible control to the problem (1.1)–(1.4). By W01,p (Ω) we denote the Sobolev space as the closure of C∞ 0 (Ω) with respect to the norm ⎛ ⎞1/p ∫︁ (︀ p )︀ ||y||W 1,p (Ω) = ⎝ |y| + |∇y|p dx⎠ . Ω ′

Let W −1,p (Ω) :=

(︁

)︁* W01,p (Ω) be the dual space to W 1,p (Ω), where p′ stands for the

p . conjugate exponent to p ≥ 2, i.e. p′ = p−1 In order to make a precise meaning of a solution to BVP (1.2)–(1.3) and indicate its characteristic properties, we begin with the following concept.

Definition 1.1. We say that a function y = y(u) is a weak solution to the boundary value problem (1.2)–(1.3) for a given control u ∈ L q (Ω) if y belongs to the class of functions ⃒ }︁ {︁ ⃒ (1.6) Y = y ∈ W01,p (Ω) ⃒ f (y) ∈ L1 (Ω) ,

1.1 Auxiliaries and previous analysis |

and the integral identity ∫︁

|∇y|p−2 (∇y, ∇φ) dx =

∫︁

∫︁ f (y)φ dx +

Ω

Ω

13

uφ dx

(1.7)

Ω

holds for every test function φ ∈ C∞ 0 (Ω). Let us associate with the function f ∈ C loc (R) the following form ∫︁ ∞ N [y, φ]f := f (y)φ dx, ∀ y ∈ Y , ∀ φ ∈ C0 (R ).

(1.8)

Ω

In spite of the fact that the integral in the right hand side of (1.8) is well defined for all y ∈ Y and φ ∈ W01,p (Ω), the continuity of the mapping W01,p (Ω) ∋ φ ↦→ [y, φ]f is not evident. This motivates us to introduce the following set. Definition 1.2. We say that an element y ∈ W01,p (Ω) belongs to the set H f if ⃒ ⃒ ⎛ ⎞1/p ⃒ ⃒∫︁ ∫︁ ⃒ ⃒ ⃒ f (y)φ dx⃒ ≤ c(y) ⎝ |∇φ|p dx⎠ , ⃒ ⃒ ⃒ ⃒

∀ φ ∈ C∞ 0 (Ω)

(1.9)

Ω

Ω

with some constant depending on y. As a result, we have: if y ∈ H f then y ∈ Y. Moreover, in this case the mapping φ ↦→ 1,p [y, φ]f can be defined for all φ ∈ W0 (Ω) using (1.9) and the standard rule [y, φ]f = lim [y, φ ε ]f ,

(1.10)

ε→0

1,p N where {φ ε }ε>0 ⊂ C∞ 0 (R ) and φ ε → φ strongly in W 0 (Ω). In particular, if y ∈ H f , then we can define the value [y, y]f and this one is finite for every y ∈ H f , although the “integrand” y f (y) needs not to be integrable on Ω, in general. Taking this fact into account, we deduce:

Proposition 1.1. If u ∈ L q (Ω) is a given distribution and y ∈ H f is a weak solution to BVP (1.2)–(1.3) in the sense of Definition 1.1, then y satisfies the energy equality ∫︁ ∫︁ p (1.11) |∇y| dx = [y, y]f + yu dx. Ω

Ω

We note that by the initial assumptions and Hölder’s and Friedrich’s inequalities, this relation makes sense because ⃒ ⃒ ⃒ ⃒∫︁ ⃒ ⃒ ⃒ uy dx⃒ ≤ ||u|| p′ ||y||L p (Ω) ⃒ ⃒ L (Ω) ⃒ ⃒ Ω

q−p′

≤ |Ω| qp′ (diam Ω) ||u||L q (Ω) ||y||W 1,p (Ω) < +∞. 0

(1.12)

14 | 1 Optimal Distributed Control Problem

However, since it is unknown whether the operator −∆ p y − f (y) is monotone onto the set y ∈ H f , it follows that we cannot leverage the energy equality (1.11) in order to derive a priori estimate in || · ||W 1,p Ω) -norm for the weak solutions. In particular, to specify the 0 term [y, y]f , we give the following result which is mainly inspired by [22, Lemma 2.1]. Lemma 1.1. Let y = y(u) ∈ Y be a weak solution to BVP (1.2)–(1.3) for a given u ∈ L q (Ω). ′ Then y ∈ H f , f (y) ∈ W −1,p (Ω), and ∫︁ ⟨︀ ⟩︀ 1,p (1.13) [y, z]f = f (y), z W −1,p′ (Ω);W 1,p (Ω) = z f (y) dx, ∀ z ∈ W0 (Ω), Ω

i.e. z f (y) ∈ L1 (Ω) for every z ∈ W01,p (Ω). Proof. Taking into account the Friedrich’s inequality ∀y ∈ W01,p (Ω),

||y||L p (Ω) ≤ diam Ω ||∇y||L p (Ω)N ,

(1.14)

N and following the definition of the weak solution, for all φ ∈ C∞ 0 (R ), we have (see (1.7)) ⃒ ⃒ ⃒ ⃒ ⃒ ⃒ ⃒∫︁ ⃒∫︁ ∫︁ ⃒ ⃒ ⃒ ⃒ p−2 ⃒ ⃒ ⃒ f (y)φ dx ≤ ⃒ |∇y| (∇y, ∇φ) dx⃒ + ⃒ uφ dx⃒⃒ ⃒ ⃒ ⃒ ⃒ Ω

Ω

Ω

p−1

≤ ||∇y||L p (Ω) ||∇φ||L p (Ω) + ||u||L p′ (Ω) ||φ||L p (Ω) by (1.14)

p−1

||y|| 1,p ||φ||W 1,p (Ω) + diam Ω ||u||L p′ (Ω) ||φ||W 1,p (Ω) W0 (Ω) 0 0 )︂ (︂ q−p′ by (1.12) p−1 ≤ ||y|| 1,p + |Ω| qp′ diam Ω ||u||L q (Ω) ||φ||W 1,p (Ω) . ≤

W0 (Ω)

(1.15)

0

Hence, y ∈ H f by Definition 1.2. Let z ∈ W01,p (Ω) ∩ L∞ (Ω) be an arbitrary element. Since f (y) ∈ L1 (Ω), it follows ∫︀ that the term Ω z f (y) dx is well defined. Let {φ ε }ε>0 ⊂ C∞ (Ω) be a sequence such that φ ε → z in W01,p (Ω). Moreover, in this case we can suppose that sup ||φ ε ||L∞ (Ω) < +∞

and

*

φ ε ⇀ z in L∞ (Ω).

ε>0

Hence, due to the fact that y ∈ H f , we get ∫︁ ∫︁ z f (y) dx = lim φ ε f (y) dx = lim [y, φ ε ]f ε→0

Ω

ε→0

by (1.10)

=

[y, z]f .

(1.16)

Ω

Thus, we arrive at the relation (1.13) for each z ∈ W01,p (Ω) ∩ L∞ (Ω). Let us take now z ∈ W01,p (Ω) such that z ≥ 0 almost everywhere in Ω. For every ε > 0, let T ε : R → R be the truncation operator defined by {︁ {︁ }︁ }︁ T ε (s) = max min s, ε−1 , −ε−1 . (1.17)

1.1 Auxiliaries and previous analysis |

15

Taking into account that T ε (z)(x) → z(x) for almost all x ∈ Ω and the sequence {︀ }︀ T ε (z) ε↘0 is monotonically increasing, we deduce from the monotone convergence theorem the following well-known property: If z ∈ W01,p (Ω) then T ε (z) → z in W01,p (Ω) as ε → 0

T ε (z) ∈ L∞ (Ω) ∩ W01,p (Ω) ∀ ε > 0 and Moreover, since T ε (z)f (y)

by (1.5)

≥

1 T ε (z)F(y) > 0 in Ω, CF

(1.18)

{︀ }︀ it follows that T ε (z)f (y) ε>0 is a pointwise non-decreasing sequence, and also T ε (z)f (y) → z f (y) for almost all x ∈ Ω. Therefore, by monotone convergence theorem, z f (y) is a measurable function on Ω, and ∫︁ ∫︁ lim T ε (z)f (y) dx = z f (y) dx. ε→0

Ω

Ω

Thus, (1.13) holds true for each z ∈ W01,p (Ω) such that z ≥ 0. As for a general case, i.e. z ∈ W01,p (Ω), it is enough to note that z = z+ − z− with z+ , z− ∈ W01,p (Ω) and z+ , z− ≥ 0 in Ω, where z+ := max {z, 0} ,

z− := max {−z, 0} .

To complete the proof, it remains to observe that ∫︁ ∫︁ by (1.18) z f (y) dx = lim φ ε f (y) dx ε→0

Ω

Ω by (1.15)

≤

(︂ lim ||y|| ε→0

p−1 W01,p (Ω)

)︂ q−p′ + |Ω| qp′ diam Ω ||u||L q (Ω) ||φ ε ||W 1,p (Ω) 0

W01,p (Ω))

(by the strong convergence of φ ε → z in (︂ )︂ q−p′ p−1 = ||y|| 1,p + |Ω| qp′ diam Ω ||u||L q (Ω) ||z||W 1,p (Ω) W0 (Ω)

0

holds true for an arbitrary element z ∈ W01,p (Ω). As a result, we have ′

f (y) ∈ W −1,p (Ω), ∫︁ ⟩︀ ⟨︀ f (y), z W −1,p′ (Ω);W 1,p (Ω) = z f (y) dx, 0

∀ z ∈ W01,p (Ω),

Ω

and (︂ ||f (y)||W −1,p′ (Ω) ≤

p−1 ||y|| 1,p W0 (Ω)

+ |Ω|

q−p′ qp′

)︂ diam Ω ||u||L q (Ω) .

16 | 1 Optimal Distributed Control Problem

As a direct consequence of Lemma 1.1 and Proposition 1.1, we have the following result. Corollary 1.1. Let u ∈ L q (Ω) be a given control and let y = y(u) ∈ W01,p (Ω) be a weak solution to BVP (1.2)–(1.3) in the sense of Definition (1.1). Then the energy equality for y takes the form ∫︁ ∫︁ ∫︁ |∇y|p dx =

Ω

y f (y) dx + Ω

uy dx.

(1.19)

Ω

1.2 On Pohozaev inequality and a priori estimates for a special class of weak solutions In this section we deal with some extra properties of the weak solutions to the boundary value problem (1.2)–(1.3). In some aspects we follow the ideas of the paper E. Casas, O. Kavian, and J.P. Puel [22] where the Dirichlet boundary value problem with linear Laplace operator in the principle part of elliptic equation and exponential nonlinearity has been considered. We begin with the following result, where we establish the Pohozaev-type inequality for some weak solutions of Dirichlet boundary value problem (1.2)–(1.3). Proposition 1.2. Let u ∈ L q (Ω) and let y = y(u) ∈ W01,p (Ω) be a weak solution to BVP ′

(1.2)–(1.3). Assume that f (y) ∈ L p (Ω). Then )︂ ∫︁ (︂ ∫︁ ∫︁ N |∇y|p dx ≤ N F(y) dx − u (x − x0 , ∇y) dx, −1 p Ω

Ω

(1.20)

Ω

(︀

)︀ where x0 ∈ int Ω is a point such that σ − x0 , ν(σ) ≥ 0 for almost all σ ∈ ∂Ω, and ν(σ) denotes the outward unit normal vector to ∂Ω at the point σ. Proof. For the reader’s convenience, we divide this proof onto several steps. Step 1. In view of the initial assumptions, it is easy to see that (︁ )︁ ′ div |∇y|p−2 ∇y ∈ L p (Ω). ′

Indeed, since L q (Ω) ⊆ L p (Ω) for any q > (︁

p p−1 ,

)︁

(1.21)

it follows that ′

− div |∇y|p−2 ∇y = f (y) + u ∈ L p (Ω). Hence, we can multiply the equation (1.2) by any function of φ ∈ L p (Ω) and make the integration over Ω. Let us consider φ := (x − x0 , ∇y) ∈ L p (Ω) such test function (this

1.2 Pohozaev inequality and a priori estimates | 17

is the so-called Pohozaev trick). Then (1.7) implies the relation ∫︁ |∇y|p−2 (∇y, ∇ (x − x0 , ∇y)) dx Ω

∫︁

|∇y(σ)|p−2 ∂ ν y(σ) σ − x0 , ∇y(σ) dσ

(︀

−

)︀

∂Ω

∫︁

∫︁ f (y) (x − x0 , ∇y) dx +

= Ω

u (x − x0 , ∇y) dx,

(1.22)

Ω

where ∇y(σ) = ±|∇y(σ)|ν(σ). Then (︀ )︀ (︀ )︀ σ − x0 , ∇y(σ) = ±|∇y(σ)| σ − x0 , ν(σ) and (︀ )︀ ∂ ν y(σ) = ∇y(σ), ν(σ) = ±|∇y(σ)| (for details, we refer to [22, p. 364]). Taking this fact into account, we can rewrite the equality (1.22) as follows ∫︁ ∫︁ (︀ )︀ |∇y|p−2 (∇y, ∇ (x − x0 , ∇y)) dx − σ − x0 , ν(σ) |∇y(σ)|p dσ Ω

∂Ω

∫︁

∫︁ f (y) (x − x0 , ∇y) dx +

= Ω

u (x − x0 , ∇y) dx,

(1.23)

Ω

Step 2. We apply the integration by parts to the left hand side of (1.22). This yields ∫︁ |∇y|p−2 (∇y, ∇ (x − x0 , ∇y)) dx Ω

=

N ∫︁ ∑︁ i=1 Ω

∫︁ =

⎡ ⎤ N ∑︁ ∂y ∂ ⎣ ∂y ⎦ |∇y|p−2 (x j − x0j ) dx ∂x i ∂x i ∂x j

|∇y|p−2

i=1

Ω

∫︁ =

j=1

Ω

Ω

∫︁

|∇y|p dx +

Ω

(︂

i,j=1

Ω

∫︁ N 1 ∑︁ ∂ p |∇y| dx + (x j − x0j ) |∇y|p dx p ∂x j

∫︁ =

=

⃒ ∫︁ N ∑︁ ⃒ ∂y ⃒2 ∂y ∂2 y ⃒ dx + |∇y|p−2 ⃒ (x j − x0j ) dx ⃒ ∂x i ⃒ ∂x i ∂x i ∂x j

N ⃒ ∑︁

1 p

j=1

|∇y(σ)|p

N ∑︁ j=1

∂Ω

N 1− p

)︂ ∫︁ Ω

1 |∇y| dx + p p

∫︁ ∂Ω

(︀

(σ j − x0j )ν j (σ) dσ −

N p

∫︁ Ω

)︀ σ − x0 , ν(σ) |∇y(σ)|p dσ.

|∇y|p dx

18 | 1 Optimal Distributed Control Problem (︀ )︀ Since, by the star-shaped property of Ω, we have the inequality σ − x0 , ν(σ) ≥ 0 for H N−1 -a.a. σ ∈ ∂Ω, it follows that ∫︁ ∫︁ (︀ )︀ |∇y|p−2 (∇y, ∇ (x − x0 , ∇y)) dx − σ − x0 , ν(σ) |∇y(σ)|p dσ Ω

∂Ω

(︂ =

N 1− p

)︂ ∫︁

N p

)︂ ∫︁

1 |∇y| dx − ′ p p

Ω

(︂ ≤

1−

∫︁

(︀

)︀ σ − x0 , ν(σ) |∇y(σ)|p dσ

∂Ω

|∇y|p dx.

(1.24)

Ω

Step 3. Let us show that the relation ∫︁ ∫︁ ∫︁ (︀ )︀ f (y) (∇y, ψ) dx = F(y(σ)) ν(σ), ψ(σ) dσ − F(y) div ψ dx Ω

(1.25)

Ω

∂Ω

holds true for any vector-valued test function ψ ∈ C1 (Ω)N provided y ∈ W01,p (Ω) is a weak solution to (1.2)–(1.3). To do so, it is enough to prove that f (y)∇y = ∇F(y) as elements of L1 (Ω)N .

(1.26)

Then the equality (1.25) is a direct consequence of the formula of integration by parts. Let T ε : R → R be the truncation operator defined in (1.17). By definition of T ε , we can suppose that T ε (y) → y strongly in W01,p (Ω) and a.e. in Ω as ε → 0.

(1.27)

′

Since ∇y ∈ L p (Ω)N , f (y) ∈ L p (Ω), and f ∈ C loc (R), it follows from (1.27) that (︀ )︀ f T ε (y) → f (y)

′

in L p (Ω) and a.e. in Ω.

Then Lebesgue Dominated Theorem implies: (︀ )︀ f T ε (y) ∇T ε (y) → f (y)∇y in L1 (Ω)N , (︀ )︀ ∇F T ε (y) → ∇F (y) in L1 (Ω)N . Taking into account the fact that (︀ )︀ (︀ )︀ f T ε (y) ∇T ε (y) = ∇F T ε (y) ,

∀ ε > 0,

and passing to the limit in both parts of this relation as ε → 0, we arrive at the desired equality (1.26).

1.2 Pohozaev inequality and a priori estimates |

19

Step 4. Now we are in a position to transform the right hand side in (1.22). Indeed, due to relation (1.25), we have ∫︁ ∫︁ (︀ )︀ by (1.26) f (y) (x − x0 , ∇y) dx = x − x0 , ∇F(y) dx Ω

Ω by (1.25)

=

∫︁

∫︁ F(y) div(x − x0 ) dx +

− Ω

)︀ (︀ F(y(σ)) ν(σ), σ − x0 dσ

∂Ω

∫︁ F(y) dx

≥ −N Ω

because of the star-shaped property of Ω and the fact that F(z) ≥ exp (C−1 F z) over R. Combining this inequality with (1.22) and (1.24), we arrive at the desired relation (1.20). The proof is complete. Remark 1.1. In what follows, we will use the following version of inequality (1.20) ∫︁ ∫︁ ∫︁ Np p p |∇y| dx ≤ F(y) dx − u (x − x0 , ∇y) dx. (1.28) N−p N−p Ω

Ω

Ω

Remark 1.2. In 1965, Pohozaev [134] considered the following nonlinear elliptic problem {︃ −∆u = g(u) in Ω, (1.29) u=0 on ∂Ω, where Ω is a bounded smooth domaain in RN and g is a continuous function on R, and he proved that if u ∈ C2 (Ω) ∩ C1 (Ω) is a solution of (1.29), then ∫︁ ∫︁ ∫︁ (2 − N) ug(u) dx + 2N G(u) dx = |∇u|2 (x, ν) dσ, (1.30) Ω

Ω

∂Ω

∫︁u g(t) dt, ν = ν(x) is the outward unit normal vector at the point x ∈ ∂Ω.

where G(u) = 0

Based on relation (1.30), Pohozzaev established the following well-known non-existence result: If N ≥ 3, Ω is a star-shaped domain with respect to the origin in RN , g(u) ≥ 0 when u ≥ 0, and ∫︁u g(t) dt ≤ 0 provided u ≥ 0,

(2 − N)ug(u) + 2N 0

then the problem (1.29) has no solution.

20 | 1 Optimal Distributed Control Problem

Pohozaev’s identity has many other important applications to the solutions of nonlinear differential equations, and its appearance can be truly associated with a milestone in the development of the modern PDEs theory. For more details we refer to the recent paper [154]. The next result is crucial in this section. Namely, we show that inequality (1.20) implies some a priori estimate for the weak solutions y ∈ Y to the original BVP. Theorem 1.1. Let u ∈ L q (Ω) and let y = y(u) ∈ Y be a weak solution to BVP (1.2)–(1.3) such that y satisfies the inequality (1.20). Then ∫︁ p′ p′ −1 y f (y) dx ≤ C1 ||u||L q (Ω) + C2 ||u||L q (Ω) + C3 , (1.31) Ω p′ −1

||y||W 1,p (Ω) ≤ C4 ||u||L q (Ω) + C5 ,

(1.32)

0

for some positive constants C i , 1 ≤ i ≤ 5, independent of u and y. Proof. Combining the energy equality (1.11) with inequality (1.20), we get (︂ (︂ )︂ ∫︁ )︂ ∫︁ N N y f (y) dx + uy dx −1 −1 p p Ω Ω ∫︁ ∫︁ ≤ N F(y) dx − u (x − x0 , ∇y) dx. Ω

Ω

Then, in view of estimate (1.12), we can rewrite the last relation as follows ∫︁ ∫︁ q−p′ Np Ndiam Ω y f (y) dx ≤ ||u||L q (Ω) ||y||W 1,p (Ω) . F(y) dx + |Ω| qp′ 0 N−p N−p Ω

(1.33)

Ω

For our further analysis, we set {︂ Ω N :=

2NC F p x ∈ Ω : y(x) > N−p

}︂ ,

where the constant C F is defined in (1.5). Since F : R → [0, +∞) is a non-decreasing function, it follows from (1.5) that ∫︁ ∫︁ Np NC F p F(y) dx ≤ f (y) dx N−p N−p Ω Ω (︂ )︂ ∫︁ ∫︁ 2NC F p NC F p 1 f yf (y) dx + dx ≤ 2 N−p N−p ΩN

≤

1 2

Ω\Ω N

∫︁ yf (y) dx + ΩN

Np C |Ω|f N−p F

(︂

2NC F p N−p

)︂ (1.34)

1.2 Pohozaev inequality and a priori estimates | 21

and ∫︁ yf (y) dx ≤ Ω\Ω N

(︂

∫︁

2NC F p N−p

f

2NC F p N−p

)︂ dx

Ω\Ω N

2Np ≤ C |Ω|f N−p F

(︂

2NC F p N−p

)︂ .

(1.35)

Then inequality (1.33) yields the following relation ∫︁ ∫︁ ∫︁ y f (y) dx y f (y) dx = y f (y) dx − ΩN

Ω

Ω\Ω N

by (1.34)

≤

∫︁

1 2

yf (y) dx +

Np C |Ω|f N−p F

(︂

2NC F p N−p

)︂

ΩN q−p′

∫︁

y f (y) dx + |Ω| qp′

−

N diam Ω ||u||L q (Ω) ||y||W 1,p (Ω) . 0 N−p

Ω\Ω N

Therefore, 1 2

∫︁ y f (y) dx ≤

Np C |Ω|f N−p F

(︂

2NC F p N−p

)︂

∫︁ y f (y) dx

−

ΩN

Ω\Ω N

+ |Ω|

q−p′ qp′

N diam Ω ||u||L q (Ω) ||y||W 1,p (Ω) . 0 N−p

Fp Taking into account that the product f (t)|t| remains bounded for any real t ≤ 2 NC N−p (see (1.5)), we can deduce from (1.35) and the previous inequality the existence of a constant C* (N) > 0 such that (︂ )︂ ∫︁ 2NC F p 2Np C |Ω|f ||u||L q (Ω) ||y||W 1,p (Ω) y f (y) dx ≤ 0 N−p F N−p

Ω q−p′

+ C* (N) + 2|Ω| qp′

N diam Ω N−p

̂︀ 1 + 2C ̂︀ 2 ||u|| q ||y|| 1,p . =C L (Ω) W (Ω)

(1.36)

0

Finally, using the energy equality (1.11) and the standard form of Young’s inequality, we obtain p ̂︀ 1 + 3C ̂︀ 2 ||u|| q ||y|| 1,p ||y||W 1,p (Ω) ≤ C L (Ω) W0 (Ω) ⎡ ⎤ )︃p′ /p (︃ p′ ̂︀ 2 ||u||L q (Ω) 1 6 C p ̂︀ 1 + 3C ̂︀ 2 ⎣ ⎦. ≤C ||y|| 1,p + W0 (Ω) ̂︀ 2 p p′ 6C

Hence, p ||y|| 1,p W0 (Ω)

̂︀ 1 + 2 ≤ 2C p′

(︃

̂︀ 2 6C p

)︃p′ /p

p′

||u||L q (Ω)

22 | 1 Optimal Distributed Control Problem

and this implies the estimate (1.32). In order to establish the estimate (1.31), it is enough to make use of (1.32) in (1.36). The proof is complete. Remark 1.3. It is worth to notice that inequality (1.20) makes sense even if we do not ′ assume fulfillment of the inclusion f (y) ∈ L p (Ω) but have only that y ∈ Y and u ∈ L q (Ω). At the same time it is unknown whether this inequality holds for an arbitrary weak solution to BVP (1.2)–(1.3). Since the existence and uniqueness of the weak solutions to the original BVP is an open question for arbitrary given control u ∈ L q (Ω) with q > p p−1 , the following result reflects some interesting properties of weak solutions satisfying inequality (1.20). {︀ }︀ Proposition 1.3. Let (u, y) be an arbitrary pair in L q (Ω) × W01,p (Ω). Let (u k , y k ) k∈N be a sequence in L q (Ω) × Y such that, for each k ∈ N, the pairs (u k , y k ) are related by the integral identity (1.7), satisfy inequality (1.20), and (u k , y k ) ⇀ (u, y) weakly in L q (Ω) × W01,p (Ω) as k → ∞

(1.37)

p for given q > p−1 . Then y is a weak solution to BVP (1.2)–(1.3) for the given u ∈ L q (Ω), the pair (u, y) satisfies the inequality (1.20), and

f (y k ) → f (y) in L1 (Ω) as k → ∞.

(1.38)

Proof. By the Sobolev Embedding Theorem, the injection W01,p (Ω) ˓→ L p (Ω) is compact. Hence, the weak convergence y k ⇀ y in W01,p (Ω) implies the strong convergence in L p (Ω). Therefore, up to a subsequence, we can suppose that y k (x) → y(x) for almost every point x ∈ Ω. As a result, we have the pointwise convergence: f (y k ) → f (y) almost everywhere in Ω. Let us show that this fact implies the strong convergence (1.38). With that in mind we recall that a sequence {f k }k∈N is called equi-integrable on ∫︀ Ω if for any δ > 0, there is a τ = τ(δ) such that S |f k | dx < δ for every measurable {︀ }︀ subset S ⊂ Ω of Lebesgue measure |S| < τ. Let us show that the sequence f (y k ) k∈N is equi-integrable on Ω. To do so, we take m > 0 such that (︂ )︂ p′ p′ −1 m > 2 C1 sup ||u k ||L q (Ω) + C2 sup ||u k ||L q (Ω) + C3 δ−1 , (1.39) k∈N

k∈N

where the constants C i , i = 1, 2, 3, are as in (1.31). We also set τ = δ/(2f (m)). Then for every measurable set S ⊂ Ω with |S| < τ, we have ∫︁ ∫︁ ∫︁ f (y k ) dx f (y k ) dx + f (y k ) dx = S

{x∈S : y k (x)≤m}

{x∈S : y k (x)>m}

≤

∫︁

∫︁

1 m

{x∈S : y k (x)≤m}

{x∈S : y k (x)>m} p′ by (1.31) C 1 ||u k ||L q (Ω)

≤

by (1.39)

≤

f (m) dx

y k f (y k ) dx + +

p′ −1 C2 ||u k ||L q (Ω)

m δ δ + . 2 2

+ C3

+ f (m)|S|

1.2 Pohozaev inequality and a priori estimates | 23

As a result, the assertion (1.38) is a direct consequence of Lebesgue’s Convergence Theorem. Hence, y ∈ Y. Let us show now that the limit pair (u, y) is related by the integral identity (1.7). Indeed, in view of the initial assumptions and property (1.38), the limit passage in the right-hand side of the equality ∫︁ ∫︁ ∫︁ N |∇y k |p−2 (∇y k , ∇φ) dx = f (y k )φ dx + u k φ dx, ∀ φ ∈ C∞ (1.40) 0 (R ) Ω

Ω

Ω

becomes trivial. As for the limit passage as k → ∞ in the left-hand side of (1.40), we make use of the following result (see L. Boccardo and F. Murat [10, Theorem 2.1]): if (i) y k → y weakly in W01,p (Ω), strongly in L p (Ω) and a.e. in Ω; ′

(ii) u k → u strongly in W −1,p (Ω); (iii) the sequence {f k }k∈N is bounded in L1 (Ω); (iv) − div(|∇y k |p−2 ∇y k ) = f k + u k in D′ (Ω) for all k ∈ N, then, within a subsequence, ∇y k → ∇y strongly in L r (Ω)N for any 1 ≤ r < p and a.e. in Ω.

(1.41)

In our case, instead of (ii), we have the weak convergence u k ⇀ u in L q (Ω for some p q > p−1 . Since, by Sobolev embedding theorem, W01,p (Ω) is continuously embedded in (︁ * )︁* * pN L p (Ω) with p* = N−p , we have by duality arguments that L p (Ω) is continuously ′

embedded in W −1,p (Ω). So, if we define p* = (p* )′ =

pN , pN − N + p

then we have ′

L r (Ω) ⊂ L p* (Ω) ⊂ W −1,p (Ω) ∀ r >

pN . pN − N + p

However, it is easy to check that p pN < pN − N + p p − 1

for all p ≥ 2.

Hence, the weak convergence u k ⇀ u in L q (Ω) with q > −1,p

p p−1 ,

implies the strong

′

convergence in W (Ω). As for the rest assumptions of Boccardo–Murat Theorem they are obviously satisfied in our case. Hence, the pointwise convergence property (1.41), continuity of the mapping ξ ↦→ |ξ |p−2 ξ , and Vitali’s theorem imply that |∇y k |p−2 ∇y k → |∇y|p−2 ∇y

strongly in L r (Ω)N for all 1 ≤ r < p′ .

(1.42)

Thus, taking these facts into account and passing to the limit in the integral identity (1.40) as k → ∞, we see that y is a weak solution to BVP (1.2)–(1.3) for the given u ∈ L q (Ω).

24 | 1 Optimal Distributed Control Problem

It remains to prove that the limit pair (u, y) satisfies the inequality (1.20). With that in mind let us show that, in fact, the condition (1.41) implies the strong convergence of ′ gradients ∇y k → ∇y in L q (Ω)N with q′ = q/(q − 1). Indeed, for an arbitrary small set A, by the Hölder inequality for any r, r′ ≥ 1 such that 1/r + 1/r′ = 1, we have ⎛ ⎞1/r ⎛ ⎞1/r′ ∫︁ ∫︁ ∫︁ ′ ′ ′ |∇y k − ∇y|q dx ≤ ⎝ |∇y k − ∇y|q r dx⎠ ⎝ 1r dx⎠ . A

A

Having chosen r >

Np2 N−p

A

′

such that q r = p, we obtain

1 1 q′ p − q′ p(q − 1) − q = 1 − = 1 − = = r p p p(q − 1) r′ Then

∫︁

′

|∇y k − ∇y|q dx ≤ |A|

p(q−1)−q p(q−1)

or

r′ =

p(q − 1) . p(q − 1) − q

q′

sup ||∇y k − ∇y||L p (Ω)N ≤ C|A|

p(q−1)−q p(q−1)

,

k∈N

A

{︁ }︁ ′ that is, the sequence |∇y k − ∇y|q

k∈N

is equi-integrable. Combining this fact with

the pointwise convergence (1.41), by Lebesgue Convergence Theorem, we conclude: ′

|∇y k − ∇y|q → 0 strongly in L1 (Ω),

and, therefore, ′

strongly in L q (Ω)N with q′ = q/(1 − q).

∇y k → ∇y

As a result, we get ∫︁ ∫︁ by (1.43) lim u k (x − x0 , ∇y k ) dx = u (x − x0 , ∇y) dx k→∞

Ω

Ω

(as product of weakly and strongly convergent sequences), ∫︁ ∫︁ by (1.38) F(y) dx, lim F(y k ) dx =

k→∞

Ω

Ω

∫︁

p

|∇y k | dx

lim inf k→∞

by (1.37)

∫︁

≥

Ω

|∇y|p dx.

Ω

Then we can pass to the limit in the inequality (1.20) to finally obtain (︂ (︂ )︂ ∫︁ )︂ ∫︁ N N |∇y|p dx ≤ −1 − 1 lim inf |∇y k |p dx p p k→∞ Ω Ω ⎡ ⎤ ∫︁ ∫︁ ≤ lim inf ⎣N F(y k ) dx − u k (x − x0 , ∇y k ) dx⎦ k→∞

Ω

∫︁ =N

F(y) dx − Ω

The proof is complete.

Ω

∫︁ u (x − x0 , ∇y) dx. Ω

(1.43)

1.3 On reformulation of the original optimal control problem | 25

1.3 On reformulation of the original optimal control problem The main question we are going to discuss in this section is about the reformulation of the optimal control problem (1.1)–(1.4) and existence of optimal pairs to the reformulated problem. To do so, we bring into consideration the following sets ⎫ ⃒ ⎧ ⃒ ⎪ ⎪ u ∈ L q (Ω), y ∈ Y , ⎪ ⃒ ⎪ ⎪ ⎪ ⎪ ⃒ ⎪ ∫︁ ⎪ ⎪ ⎪ ⃒ ⎪ ⎪ ⎪ p−2 ⎬ ⃒ ⎨ ∇ y, ∇ φ dx |∇ y | ( ) ⃒ (1.44) Ξ = (u, y) ⃒ Ω ⃒ ∫︁ ⎪ ⎪ ∫︁ ⎪ ⎪ ⃒ ⎪ ⎪ ⎪ N ⎪ ⃒ = f (y)φ dx + uφ dx, ∀ φ ∈ C∞ ⎪ ⎪ ⎪ 0 (R ) ⎪ ⃒ ⎪ ⎪ ⎭ ⎩ ⃒ Ω

Ω

and ⃒ ⎧ ⃒ ⎪ u ∈ L q (Ω), y ∈ Y , ⃒ ⎪ ⎪ ⃒ ∫︁ ⎪ ∫︁ ⎪ ⃒ ⎪ ⎪ p−2 ⃒ ⎪ |∇ y | ∇ y, ∇ φ dx = f (y)φ dx ( ) ⎪ ⃒ ⎪ ⎪ ⃒ ⎪ ⎪ Ω ⃒ Ω ⎪ ∫︁ ⎪ ⃒ ⎪ ⎪ ∞ N ⃒ ⎨ + uφdx, ∀ φ ∈ C 0 (R ), ⃒ Ξ ∆ = (u, y) ⃒ ⃒ ⎪ ∫︁ Ω ∫︁ ⎪ ⃒ ⎪ Np ⎪ p ⃒ ⎪ |∇ y | dx ≤ F(y) dx ⎪ ⃒ ⎪ N−p ⎪ ⃒ ⎪ ⎪ Ω Ω ⃒ ⎪ ∫︁ ⎪ ⃒ ⎪ p ⎪ ⃒ ⎪ u x − x , − ( ⎪ 0 ∇ y ) dx. ⃒ ⎪ N−p ⎩ ⃒ Ω

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

(1.45)

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

At the first glance it looks rather questionable and provocative to assert that each of these sets can be considered as the set of feasible solutions to the optimal control problem (1.1)–(1.4). On the one hand, it is clear that the set Ξ ∆ imposes some additional restrictions on the class of feasible solutions. Moreover, as it has been mentioned in ′ Remark 1.3, not every pair (u, y) ∈ Ξ ∆ needs to guarantee the inclusion f (y) ∈ L p (Ω) whereas it is unknown whether the inequality (1.20) holds for an arbitrary feasible pair (u, y) ∈ Ξ. So, in general, we have the obvious inclusion Ξ ∆ ⊂ Ξ and, therefore, it is plausible to suppose that inf J(u, y) ≤

(u,y)∈Ξ

inf

(u,y)∈Ξ ∆

J(u, y).

On the other hand, since the solvability of the optimal control problem (1.1)–(1.4) in its original statement, i.e. when we interpret it as the minimization problem ⟨︀ ⟩︀ inf (u,y)∈Ξ J(u, y) , is an open question for nowaday, it is worth to notice that if we reformulate the original optimal control problem (1.1)–(1.4) as follows ∫︁ ∫︁ 1 1 Minimize J(u, y) = |y − y d |2 dx + |u|q dx, 2 q (1.46) Ω Ω subject to the constrains (u, y) ∈ Ξ ∆ ,

26 | 1 Optimal Distributed Control Problem

then the new optimization problem becomes consistent in the following sense: the set of its feasible solutions Ξ ∆ is always nonempty. Indeed, it is enough to take an arbitrary function ̃︀ y ∈ C∞ u := −∆ p ̃︀ y − f (̃︀ y). Since ̃︀ u ∈ L q (Ω), ̃︀ y ∈ Y, and 0 (Ω) and put ̃︀ ′ p ̃︀ f (y) ∈ L (Ω), it follows from Proposition 1.2 and Definition 1.1 that the pair (̃︀ u, ̃︀ y) is related by integral identity (1.7) and satisfies the inequality (1.20). Hence, (̃︀ u, ̃︀ y) ∈ Ξ ∆ . Thus, ∅ ̸= Ξ ∆ ⊂ Ξ. In view of this fact, we have the following result. p Theorem 1.2. Let p ∈ [2, N), q > p−1 , and let Ω be a star-shaped domain with respect to one of its interior points. Then the reformulated optimal control problem (1.46) has at least one solution for each y d ∈ L2 (Ω).

Proof. Since J(u, y) ≥ 0 for all (u, y) ∈ Ξ ∆ , it follows that there exists a non-negative {︀ }︀ value μ ≥ 0 such that μ = inf (u,y)∈Ξ ∆ J(u, y). Let (u k , y k ) k∈N be a minimizing sequence to the problem (1.46), i.e. (u k , y k ) ∈ Ξ ∆ ∀ k ∈ N

and

lim J(u k , y k ) = μ.

k→∞

So, we can suppose that J(u k , y k ) ≤ μ+1 for all k ∈ N. Taking into account the definition of the set Ξ ∆ , it follows from Theorem 1.1 that p′ −1

||y k ||W 1,p (Ω) ≤ C4 ||u k ||L q (Ω) + C5 0

for some constants C4 and C5 independent of k and u k . Then )︀ 1 (︀ )︀ p′ −1 (︀ ||y k ||W 1,p (Ω) + ||u k ||L q (Ω) ≤ C5 + C4 qJ(u k , y k ) q + qJ(u k , y k ) q 0

)︀ 1 (︀ )︀ p′ −1 (︀ ≤ C5 + C4 q(μ + 1) q + q(μ + 1) q ,

∀ k ∈ N.

Thus, without loss of generality, we can suppose that there exists a subsequence of {︀ }︀ (u k , y k ) k∈N (still denoted by the same index) and a pair (u0 , y0 ) ∈ L q (Ω) × W01,p (Ω) such that (u k , y k ) ⇀ (u0 , y0 ) weakly in L q (Ω) × W01,p (Ω) as k → ∞, and (u0 , y0 ) ∈ Ξ ∆ by Proposition 1.3. It remains to make use of the lower semi-continuity property of the cost functional J : L q (Ω) × W01,p (Ω) → R with respect to the weak convergence in L q (Ω) × W01,p (Ω). This yields μ= 0

inf

(u,y)∈Ξ ∆

J(u, y) = lim J(u k , y k ) ≥ J(u0 , y0 ). k→∞

0

Thus, (u , y ) ∈ Ξ ∆ is an optimal pair to the problem (1.46). Our main intension that we are going to realize in the next sections, is to show that, in fact, the passage from the original statement of the optimal control problem (1.1)–(1.4) to its reformulated version (1.46) is not restrictive with respect to the controls. By this assertion we basically mean the following one: if (u, y) ∈ Ξ is an arbitrary feasible pair to the problem (1.1)–(1.4) such that (u, y) ̸∈ Ξ ∆ , then there exists an element z ∈ W01,p (Ω) such that (u, z) ∈ Ξ ∆ .

1.4 Auxiliary fictitious optimal control problem and its properties | 27

1.4 Auxiliary fictitious optimal control problem and its properties Let (u, y) ∈ Ξ be a given pair such that (u, y) ̸∈ Ξ ∆ . Let us consider the following family of constrained minimization problems ∫︁ ′ −1 Minimize J ε (u, z) = ε |v − T ε (f (z))|p dx (1.47) Ω

subject to the constrains −∆ p z = v + u z=0

in Ω,

(1.48)

on ∂Ω,

(1.49)

′

v ∈ L p (Ω), z ∈ W01,p (Ω),

(1.50)

where the cut-off operator T ε : R → R is defined in (1.17). We consider the function ′ v ∈ L p (Ω) as a fictitious control and ε as a small parameter. Hereinafter, we assume that the parameter ε varies within a strictly decreasing sequence of positive real numbers which converge to zero. As was mentioned in Introduction, the idea to involve the fictitious optimization problems to the analysis of solvability of nonlinear boundary value problems was mainly inspired by the book of V.S. Mel’nik and V.I. Ivanenko [71]. Our aim is to show that the minimization problem (1.47)–(1.50) is solvable for each ε > 0. ′

Lemma 1.2. For given ε > 0 and u ∈ L q (Ω), there exists a pair (v0ε , z0ε ) ∈ L p (Ω) × W01,p (Ω) which is optimal to the problem (1.47)–(1.50). v = −∆ p ̃︀z − u. Then Proof. Indeed, let ̃︀z ∈ C∞ 0 (Ω) be an arbitrary function. We set ̃︀ ′ ̃︀ v ∈ L p (Ω) and, therefore, the pair (̃︀ v, ̃︀z) is feasible to the problem (1.47)–(1.50) Let }︀ {︀ (v ε,k , z ε,k ) k∈N be a minimizing sequence. Without loss of generality, we can suppose that ∫︁ ′ J ε (v ε,k , z ε,k ) = ε−1 |v ε,k − T ε (f (z ε,k ))|p dx Ω

< ε−1

∫︁

′

′

̃︀ p , |̃︀ v − T ε (f (̃︀z))|p dx ≤ ε−1 C

∀ k ∈ N and ∀ ε > 0.

(1.51)

Ω

Since T ε (f (z ε,k )) ∈ L∞ (Ω) for all k ∈ N, it follows from (1.51) that the sequence of {︀ }︀ ′ fictitious controls v ε,k k∈N is uniformly bounded in L p (Ω). So, we can admit the ′

existence of an element v0ε ∈ L p (Ω) such that (up to a subsequence) ′

v ε,k ⇀ v0ε in L p (Ω) as k → ∞. ′

(1.52) ′

Since for every admissible fictitious control v ∈ L p (Ω) we have (v + u) ∈ L p (Ω) ′ ′ and the injection L p (Ω) ˓→ W −1,p (Ω) is compact by Sobolev Embedding Theorem, it

28 | 1 Optimal Distributed Control Problem

follows from (1.52) that ′

v ε,k → v0ε strongly in W −1,p (Ω) as k → ∞

(1.53)

and moreover, by the well-known results of the theory of monotone operators, the Dirichlet boundary value problem (1.48)–(1.49) admits a unique weak solution z ∈ W01,p (Ω) for which the a priori estimate 1 (︁ )︁ p−1 ||z||W 1,p (Ω) ≤ diam (Ω) ||v + u||L p′ (Ω) 0

(︂ ≤

1 [︂ ]︂)︂ p−1 q−p′ ′ qp diam (Ω) ||v||L p′ (Ω) + |Ω| ||u||L q (Ω)

1 )︁ p−1 (︁ ≤ C ||v||L p′ (Ω) + ||u||L q (Ω)

(1.54)

and the integral identity ∫︁ ∫︁ ∫︁ vφ dx + uφ dx, |∇z|p−2 (∇z, ∇φ) dx =

∀ φ ∈ C∞ 0 (Ω)

(1.55)

hold true (for the details, we refer to [113, 131]). Hence, ∫︁ ∫︁ ∫︁ )︀ p−2 (︀ |∇z ε,k | ∇z ε,k , ∇φ dx = v ε,k φ dx + uφ dx, ∀ φ ∈ C∞ 0 (Ω),

(1.56)

Ω

Ω

Ω

Ω

Ω

Ω 1 )︂ p−1

(︂ sup ||z ε,k ||W 1,p (Ω) ≤ C sup ||v ε,k ||L p′ (Ω) + ||u||L q (Ω) k∈N

0

< +∞.

(1.57)

k∈N

Then, Boccardo–Murat Theorem (see [10] and the proof of Proposition 1.3) implies the existence of element z0ε ∈ W01,p (Ω) such that, within a subsequence, the following limit properties z ε,k → z0ε weakly in W01,p (Ω), strongly in L p (Ω), and a.e. in Ω , ∇z ε,k → ∇z0ε strongly in L r (Ω)N for any 1 ≤ r < p and a.e. in Ω, |∇z ε,k |p−2 ∇z ε,k → |∇z0ε |p−2 ∇z0ε

strongly in L r (Ω)N for all 1 ≤ r < p′

(1.58)

hold true (see (1.42)). In view of (1.58) and (1.52), the limit passage in the integral identity (1.56) becomes trivial. As a result, we have ∫︁ ∫︁ ∫︁ (︁ )︁ 0 p−2 0 0 |∇z ε | ∇z ε , ∇φ dx = v ε φ dx + uφ dx, ∀ φ ∈ C∞ (1.59) 0 (Ω), Ω

Ω

Ω

that is, z0ε is a unique weak solution to the Dirichlet problem (1.48)–(1.49) with v = v0ε . (︀ )︀ ′ The fact that v0ε , z0ε ∈ L p (Ω) × W01,p (Ω) is an optimal pair to the problem (1.47)–(1.50) immediately follows from the following observations: in accordance to (1.58), we have T ε (f (z ε,k )) → T ε (f (z0ε )) a.e. in Ω and supk∈N ||T ε (f (z ε,k ))||L p′ (Ω) ≤ ε

−

1 p′

1

| Ω | p′ .

1.5 On a priori estimate for the solutions of variational problem (1.59) | 29

′

Hence, T ε (f (z ε,k )) → T ε (f (z0ε )) weakly in L p (Ω). Combining this fact with (1.52) and the lower semi-continuity of the norm || · ||L p′ (Ω) with respect to the weak convergence ′

in L p (Ω), we finally obtain (︁ )︁ inf J ε (v, z) = lim J ε (v ε,k , z ε,k ) ≥ J ε v0ε , z0ε .

(1.60)

k→∞

(︀ )︀ Thus, v0ε , z0ε is an optimal pair to the problem (1.47)–(1.50).

1.5 On a priori estimate for the solutions of variational problem (1.59) The main goal at this stage is to derive an a priori estimate for the weak solutions of Dirichlet boundary value problem (1.48)–(1.49) with v = v0ε . We begin with the following result. (︀ )︀ Lemma 1.3. Let v0ε , z0ε be an optimal pair to the problem (1.47)–(1.50). Then this pair is related by inequality )︂ ∫︁ (︂ ∫︁ ∫︁ (︁ )︁ N |∇z0ε |p dx ≤ N F(z0ε ) dx − u x − x0 , ∇z0ε dx −1 p Ω Ω Ω ∫︁ [︁ ]︁ (︁ )︁ 0 v ε − T ε (f (z0ε )) x − x0 , ∇z0ε dx. − (1.61) Ω

Proof. Let us make use of the following equality [︁ ]︁ − ∆ p z0ε = T ε (f (z0ε )) + v0ε − T ε (f (z0ε )) + u

in Ω.

(1.62)

[︀ ]︀ ′ Since T ε (f (z0ε )) ∈ L∞ (Ω) and v0ε − T ε (f (z0ε )) ∈ L p (Ω), it follows that we can multiply the equality (1.62) by any function of φ ∈ L p (Ω) and make the integration over Ω. Let (︀ )︀ us consider φ := x − x0 , ∇z0ε ∈ L p (Ω) as this function. Arguing as in the proof of Proposition 1.2, after integration in (1.62) over Ω, we get (︂ )︂ ∫︁ ∫︁ (︀ )︀ 1 N 0 p |∇z ε | dx − ′ 1− σ − x0 , ν(σ) |∇z0ε (σ)|p dσ p p Ω ∂Ω ∫︁ (︁ )︁ = T ε (f (z0ε )) x − x0 , ∇z0ε dx Ω

∫︁ +

)︁ (︁ u x − x0 , ∇z0ε dx

Ω

+

∫︁ [︁ Ω

v0ε − T ε (f (z0ε ))

]︁ (︁

x − x0 , ∇z0ε

)︁

dx.

(1.63)

30 | 1 Optimal Distributed Control Problem

Let us define the function F ε : R → R as follows {︃ F(t) F ε (t) = F(f −1 (ε−1 )) + ε−1 (t − f −1 (ε−1 ))

if f (t) ≤ ε−1 , if f (t) > ε−1 .

Then F ε (t) is the primitive of T ε (f (t)) and, moreover, because of the relation F(f −1 (ε−1 )) + ε−1 (t − f −1 (ε−1 )) = F(f −1 (ε−1 )) + F ′ (f −1 (ε−1 ))(t − f −1 (ε−1 )) ≤ F(t) which is obviously valid for all t > f −1 (ε−1 ) by convexity of F, we have F ε (t) ≤ F(t) ∀ t ∈ R. Now we are in a position to transform the right hand side in (1.22). Indeed, due to relation (1.25), we have ∫︁ ∫︁ (︁ (︁ )︁ )︁ T ε (f (z0ε )) x − x0 , ∇z0ε dx= x − x0 , ∇F ε (z0ε ) dx Ω

Ω

∫︁ =−

F ε (z0ε ) div(x

∫︁ − x0 ) dx +

Ω

(︀ )︀ F ε (z0ε (σ)) ν(σ), σ − x0 dσ

∂Ω

∫︁ ≥ −N

F ε (z0ε ) dx

∫︁ ≥ −N

Ω

F(z0ε ) dx

Ω

because of (1.5) and the star-shaped property of Ω. As a result, combining now this relation with (1.63) and using again the star-shaped property of Ω, we arrive at the expected inequality (1.61). Our next step is to obtain an a priori estimate on the state z0ε making use of the inequality (1.61). (︀ )︀ Lemma 1.4. Let v0ε , z0ε be an optimal pair to the problem (1.47)–(1.50). Then there exist ̂︀ * , and C ̂︀ * , independent of v0ε , z0ε , u, and ε such that positive constants C, C1 , C 0 1 ]︁ [︁ ′ ′ p ̂︀ 1 + C ̂︀ ||u||pq + ||v0ε − T ε (f (z0ε ))||p ′ ||z0ε || 1,p ≤ 2C , (1.64) p L (Ω) W0 (Ω) L (Ω) ∫︁ [︁ ]︁ ′ ′ ̂︀ *0 + C ̂︀ *1 ||u||pq + ||v0ε − T ε (f (z0ε ))||p ′ z0ε T ε (f (z0ε )) dx ≤ C . (1.65) p L (Ω) L (Ω)

Ω

Proof. Since the integral identity (1.59) implies the equality ∫︁ Ω

|∇z0ε |p dx =

∫︁ Ω

z0ε T ε (f (z0ε )) dx +

∫︁ [︁ Ω

∫︁ ]︁ v0ε − T ε (f (z0ε )) z0ε dx + uz0ε dx, Ω

(1.66)

1.5 On a priori estimate | 31

it follows from (1.61) that ∫︁ ∫︁ Np z0ε T ε (f (z0ε )) dx ≤ F(z0ε ) dx N−p Ω Ω ∫︁ [︂ )︁]︂ p (︁ 0 0 x − x0 , ∇z ε dx − u zε + N−p Ω ∫︁ [︁ ]︁ [︂ )︁]︂ p (︁ v0ε − T ε (f (z0ε )) z0ε + − x − x0 , ∇z0ε dx. N−p

(1.67)

Ω

Making use of the obvious estimates ⃒ ∫︁ [︂ )︁]︂ ⃒ p (︁ ⃒ ⃒ 0 x − x0 , ∇z0ε dx⃒ ⃒ u zε + N−p Ω by (1.12)

≤

|Ω|

q−p′ qp′

0

(diam Ω) ||u||L q (Ω) ||z ε ||W 1,p (Ω) 0

′

q−p p (diam Ω) + |Ω| qp′ ||u||L q (Ω) ||z0ε ||W 1,p (Ω) 0 N−p ′

=

q−p N (diam Ω) |Ω| qp′ ||u||L q (Ω) ||z0ε ||W 1,p (Ω) 0 N−p

and )︁]︂ ⃒ p (︁ ⃒ x − x0 , ∇z0ε dx⃒ N−p

⃒ ∫︁ [︁ ]︁ [︂ ⃒ 0 0 v − T (f (z )) z0ε + ⃒ ε ε ε Ω

≤ ||v0ε − T ε (f (z0ε ))||L p′ (Ω) ||z0ε ||L p (Ω) p (diam Ω) 0 ||v ε − T ε (f (z0ε ))||L p′ (Ω) ||∇z0ε ||L p (Ω)N N−p N 0 0 0 = (diam Ω) ||v ε − T ε (f (z ε ))||L p′ (Ω) ||z ε ||W 1,p (Ω) , 0 N−p +

we derive from (1.66) and (1.67) {︂ }︂ ∫︁ ∫︁ q−p′ N Np 0 0 0 ′ qp F(z ε ) dx + z ε T ε (f (z ε )) dx ≤ (diam Ω) max 1, |Ω| N−p N−p ⏟ ⏞ Ω Ω C*

× ||z0ε ||W 1,p (Ω)

[︁

0

∫︁ Ω

|∇z0ε |p dx ≤

∫︁

||u||L q (Ω) + ||v0ε

−

T ε (f (z0ε ))||L p′ (Ω)

]︁

,

(1.68)

z0ε T ε (f (z0ε )) dx

Ω

[︁ ]︁ + C* ||z0ε ||W 1,p (Ω) ||u||L q (Ω) + ||v0ε − T ε (f (z0ε ))||L p′ (Ω) . 0

By analogy with the proof of Theorem 1.1, we set {︂ }︂ 2NC F p Ω ε := x ∈ Ω : z0ε (x) > , N−p

(1.69)

32 | 1 Optimal Distributed Control Problem

where the constant C F is defined in (1.5). Since F : R → [0, +∞) is a non-decreasing function, it follows from (1.5) that ∫︁ ∫︁ Np NC F p F(z0ε ) dx ≤ f (z0ε ) dx N−p N−p Ω Ω ⎡ ⎤ ∫︁ ∫︁ (︁ )︁ NC F p ⎣ = T ε (f (z0ε )) dx + f (z0ε ) − T ε (f (z0ε )) dx⎦ N−p Ω Ω (︂ )︂ ∫︁ ∫︁ 1 2NC F p NC F p 0 0 f z ε T ε (f (z ε )) dx + dx ≤ 2 N−p N−p Ωε

Ω\Ω ε

NC F p + N−p

∫︁ (︁

)︁ f (z0ε ) − T ε (f (z0ε )) dx

Ω

1 ≤ 2

∫︁

z0ε T ε (f (z0ε )) dx

+

Np C |Ω|f N−p F

(︂

2NC F p N−p

)︂

Ωε

NC F p + N−p

∫︁ ⃒ ⃒ ⃒ 0 0 ⃒ ⃒f (z ε ) − T ε (f (z ε ))⃒ dx

(1.70)

Ω

and ∫︁

z0ε T ε (f (z0ε )) dx ≤

2NC F p N−p

(︂

∫︁ f

2NC F p N−p

)︂ dx

Ω\Ω ε

Ω\Ω ε

2Np ≤ C |Ω|f N−p F

(︂

2NC F p N−p

)︂ .

Then inequality (1.68) together with (1.70) yields to the following relation ∫︁ ∫︁ ∫︁ 0 0 0 0 z ε T ε (f (z ε )) dx = z ε T ε (f (z ε )) dx − z0ε T ε (f (z0ε )) dx Ωε

Ω

1 ≤ 2

∫︁

z0ε T ε (f (z0ε )) dx

Ω\Ω ε

Np C |Ω|f + N−p F

(︂

2NC F p N−p

)︂

Ωε

+

NC F p N−p

∫︁ ∫︁ ⃒ ⃒ ⃒ 0 0 ⃒ ⃒f (z ε ) − T ε (f (z ε ))⃒ dx − Ω

z0ε T ε (f (z0ε )) dx

Ω\Ω ε

[︁ ]︁ N C* 0 ||z ε ||W 1,p (Ω) ||u||L q (Ω) + ||v0ε − T ε (f (z0ε ))||L p′ (Ω) . + 0 N−p

(1.71)

1.5 On a priori estimate | 33

Therefore, (︂ )︂ ∫︁ ∫︁ 1 2NC F p Np 1 z0ε T ε (f (z0ε )) dx ≤ C F |Ω|f − 2 N−p N−p 2 Ω

z0ε T ε (f (z0ε )) dx

Ω\Ω ε *

]︁ NC ||z0ε ||W 1,p (Ω) ||u||L q (Ω) + ||v0ε − T ε (f (z0ε ))||L p′ (Ω) 0 N−p ∫︁ ⃒ ⃒ NC F p ⃒ 0 0 ⃒ + ⃒f (z ε ) − T ε (f (z ε ))⃒ dx. N−p [︁

+

Ω Fp Taking into account that the product T ε (f (t))|t| remains bounded for any real t ≤ 2 NC N−p 1 (see (1.5)) and T ε (f ) → f strongly in L loc (R) as ε → 0, we deduce from (1.71) and the previous inequality the existence of a constant C* > 0 such that ∫︁ ⃒ ∫︁ ⃒ 1 NC F p ⃒ 0 0 ⃒ z0ε T ε (f (z0ε )) dx ≤ C* ⃒f (z ε ) − T ε (f (z ε ))⃒ dx − N−p 2

Ω

Ω\Ω ε

and as a consequence (︂ )︂ ∫︁ 2NC F p 2Np C F |Ω|f + C* z0ε T ε (f (z0ε )) dx ≤ N−p N−p Ω

[︁ ]︁ N C* 0 ||z ε ||W 1,p (Ω) ||u||L q (Ω) + ||v0ε − T ε (f (z0ε ))||L p′ (Ω) 0 N−p [︁ ]︁ 0 ̂︀ 1 + C ̂︀ 2 ||z ε || 1,p =C ||u|| q + ||v0ε − T ε (f (z0ε ))|| p′ . +

L (Ω)

W0 (Ω)

(1.72)

L (Ω)

It remains to apply the equality (1.69) and the standard form of Young’s inequality. As a result, we obtain [︁ ]︁ p ̂︀ 1 + (C ̂︀ 2 + C* )||z0ε || 1,p ||z0ε ||W 1,p (Ω) ≤ C ||u||L q (Ω) + ||v0ε − T ε (f (z0ε ))||L p′ (Ω) W (Ω) 0

(︃ ̂︀ 1 + (C ̂︀ 2 + C* ) ≤C [︁ ×

̂︀ 2 + C* ) 2(C p

)︃p′ /p

||u||L q (Ω) + ||v0ε − T ε (f (z0ε ))||L p′ (Ω) p′

]︁p′ +

1 0 p ||z || . 2 ε W01,p (Ω)

Hence,

||z0ε ||

p W01,p (Ω)

(︁ )︁p′ (︂ )︂p′ −1 2(C ̂︀ 2 + C* ) ̂︀ 1 + 2 ≤ 2C p p′ [︁ p′ p′ × ||u||L q (Ω) + ||v0ε − T ε (f (z0ε ))|| p′

L (Ω)

]︁

(1.73)

34 | 1 Optimal Distributed Control Problem

and combining this estimate with inequality (1.72), we finally obtain ∫︁ ̂︀ ̂︀ 1 + C2 ||z0ε ||p 1,p z0ε T ε (f (z0ε )) dx ≤ C W0 (Ω) p Ω

]︁p′ ̂︀ 2 [︁ C 0 0 ||u|| + ||v − T (f (z ))|| ′ q ε ε ε p L (Ω) L (Ω) p′ ]︃ [︃(︂ )︂ ′ ′ p (︁ )︁p′ ̂︀ 2 2p −1 C ̂︀ 1 C ̂︀ 2 2 C 1 * ̂︀ 1 + ̂︀ 2 + C ) + 1 2(C ≤C + p p p′ ⏟ ⏞ ⏟ ⏞

+

̂︀ C*0

[︁

̂︀ C*1

p

′

× ||u||L q (Ω) + ||v0ε − T ε (f (z0ε ))||

′

p L p′ (Ω)

]︁

.

(1.74)

1.6 On asymptotic behaviour of the sequence of optimal pairs to the problem (1.47)–(1.50) as ε → 0 The following result is crucial for the substantiation of the consistency of reformulated optimal control problem (1.46). Lemma 1.5. Let u ∈ L q (Ω) be a fixed control. Assume that the boundary value problem {︀(︀ )︀}︀ (1.2)–(1.3) has a solution for the given u. Let v0ε , z0ε ε>0 be a sequence of optimal pairs to the problem (1.47)–(1.50). Then there exist an element z ∈ W01,p (Ω) such that (u, z) ∈ Ξ ∆ . Proof. Let y be a solution of (1.2)–(1.3) associated to the control u ∈ L q (Ω). Let us define y0 ∈ W01,p (Ω) as a unique solution to the problem −∆ p y0 = u

in Ω

and

y0 = 0

on ∂Ω.

Then we have the following obvious inequalities −∆ p y0 ≤ T ε (f (y0 )) + u

and

− ∆ p y ≥ T ε (f (y)) + u

a.e. in Ω.

From this it follows that y0 is a subsolution and y is a supersolution of the problem −∆ p z = T ε (f (z)) + u

in Ω

and

z=0

on ∂Ω.

Moreover, as follows from definition of the functions y0 and y, we have (︀ )︀ −∆ p y0 + ∆ p y ≤ u − T ε (f (y)) + u = −T ε (f (y)) ≤ 0 a.e. in Ω, that is ∆ p y0 ≥ ∆ p y in Ω. By the strict monotonicity of operator −∆ p , it is easy to deduce from this that y0 (x) ≤ y(x) a.e. in Ω. Combining all these facts, by the classical techniques introduced by F.H. Sattinger [146], we get the existence of a pair

1.6 On Asymptotic Behaviour of Optimal Pairs | 35

′

(v*ε , z*e ) ∈ L p (Ω) × W01,p (Ω) such that, for a given ε > 0, (v*ε , z*e ) is feasible to the problem (1.47)–(1.50) and ∫︁ ′ |v − T ε (f (z))|p dx ≤ εC* Ω

with some positive constant C* independent of ε. Since ∫︁ ′ ε−1 |v0ε − T ε (f (z0ε ))|p dx = J ε (v0ε , z0ε ) ≤ J ε (v*ε , z*e ) Ω

=ε

−1

∫︁

′

|v − T ε (f (z))|p dx ≤ C* ,

Ω

it follows that

1

||v0ε − T ε (f (z0ε ))||L p′ (Ω) ≤ ε p′ C* .

(1.75)

Hence, for optimal pairs we have the following remarkable property v0ε − T ε (f (z0ε )) → 0 ′

′

strongly in L p (Ω) and strongly in W −1,p (Ω) as ε → 0.

(1.76)

On the other hand, the estimates (1.72) and (1.75) imply the compactness of the sequence {︀ 0 }︀ z ε ε>0 with respect to the weak convergence in W01,p (Ω). So, without loss of generality {︀ }︀ we can suppose that there exists an element z ∈ W01,p (Ω) and a subsequence of z0ε ε>0 , still denoted by the same index, such that z0ε → z weakly in W01,p (Ω) and a.e. in Ω as ε → 0.

(1.77)

Our next intension is to pass to the limit in the integral identity (1.59), or what is equivalent ∫︁ ∫︁ (︁ )︁ 0 p−2 0 |∇z ε | ∇z ε , ∇φ dx = φT ε (f (z0ε )) dx Ω

Ω

+

∫︁ [︁

v0ε

−

Ω

T ε (f (z0ε ))

]︁

∫︁ φ dx +

uφ dx

(1.78)

Ω

where φ ∈ C∞ 0 (Ω) is an arbitrary test function. With that in mind, let us show that T ε (f (z0ε )) → f (z) strongly in L1 (Ω) as ε → 0.

(1.79)

Indeed, in view of (1.77), we have T ε (f (z0ε )) → f (z) almost everywhere in Ω. In order to deduce the strong convergence (1.79), it remains to show that the sequence {︀ }︀ T ε (f (z0ε )) ε>0 is equi-integrable on Ω. The argumentation will be similar to the one

36 | 1 Optimal Distributed Control Problem

in the proof of Proposition 1.3. Let δ > 0 be an arbitrary small enough value. We take m > 0 such that (︁ [︁ ]︁)︁ ′ ′ ̂︀ *0 + C ̂︀ *1 ||u||pq + ||v0ε − T ε (f (z0ε ))||p ′ m>2 C δ−1 , (1.80) p L (Ω) L (Ω)

̂︀ * and C ̂︀ * are as in (1.74). We also set τ = δ/(2f (m)). Then for where the constants C 0 1 every measurable set S ⊂ Ω with |S| < τ, we have ∫︁ ∫︁ 0 T ε (f (z ε )) dx = T ε (f (z0ε )) dx S

{x∈S : z0ε (x)>m}

∫︁

T ε (f (z0ε )) dx

+ {x∈S : z0ε (x)≤m}

∫︁

∫︁

1 m

≤

f (m) dx

{x∈S :

′ ̂︀ * ||u||pq C 1 L (Ω)

[︁

̂︀ * + C 0

by (1.79)

∫︁

z0ε f (z0ε ) dx +

z0ε (x)>m}

{x∈S :

f (z0ε ) dx

{x∈S : z0ε (x)≤m}

{x∈S : z0ε (x)>m}

≤

∫︁

f (z0ε ) dx +

≤

+ ||v0ε

z0ε (x)≤m}

− T ε (f (z0ε ))||

p′ L p′ (Ω)

m

]︁ + f (m)|S|

δ δ + . 2 2

by (1.80)

≤

As a result, the property (1.79) is a direct consequence of Lebesgue’s Convergence Theorem. Taking (1.79) and (1.76) into account, we see that all assumptions of Boccardo– Murat Theorem they are obviously satisfied in our case. Hence, we may conclude that {︀ }︀ {︀ }︀ there exists a subsequence of z0ε ε>0 still denoted by z0ε ε>0 such that z0ε → z weakly in W01,p (Ω) and ∇z0ε → ∇z almost everywhere in Ω. Then by continuity of the mapping ξ ↦→ |ξ |p−2 ξ and Vitali’s theorem, we deduce that |∇z0ε |p−2 ∇z0ε → |∇z|p−2 ∇z

strongly in L r (Ω)N for all 1 ≤ r < p′ .

As a result, combining the above properties, for any φ ∈ C∞ 0 (Ω) we get ∫︁ ∫︁ (︁ )︁ by (1.81) lim |∇z0ε |p−2 ∇z0ε , ∇φ dx = |∇z|p−2 (∇z, ∇φ) dx, ε→0

Ω

Ω

∫︁ lim ε→0

by (1.78) φT ε (f (z0ε )) dx =

Ω

lim

∫︁ [︁

ε→0

Ω

v0ε

−

∫︁ φf (z) dx, Ω

T ε (f (z0ε ))

]︁

φ dx

by (1.75)

=

0.

(1.81)

1.6 On Asymptotic Behaviour of Optimal Pairs | 37

Thus, z ∈ W01,p (Ω) is a weak solution the Dirichlet boundary value problem (1.2)–(1.3) for a given control u ∈ L q (Ω). It remains to show that the pair (u, z) is feasible to the optimal control problem (1.46), i.e. (u, z) ∈ Ξ ∆ . With that in mind we have establish that the pair (u, z) satisfies the inequality (1.20). Using precisely the same reasoning as in the proof of Proposi′ tion 1.3, we find that |∇z0ε − ∇z|q → 0 strongly in L1 (Ω) as ε → 0, and, therefore, ′

strongly in L q (Ω)N with q′ = q/(1 − q).

∇z0ε → ∇z

(1.82)

As a result, we get ∫︁ (︁ ∫︁ )︁ by (1.82) lim u x − x0 , ∇z0ε dx = u (x − x0 , ∇z) dx ε→0

Ω

∫︁ lim ε→0

Ω by (1.79) F(z0ε ) dx =

Ω

∫︁ F(z) dx, Ω

∫︁ lim inf ε→0

|∇z0ε |p

dx

by (1.77)

Ω

lim

∫︁ [︁

ε→0

∫︁

≥

|∇z|p dx,

Ω

v0ε

−

T ε (f (z0ε ))

]︁ (︁

x − x0 , ∇z0ε

)︁

dx

by (1.76) and(1.77)

=

0

Ω

(as product of weakly and strongly convergent sequences). Then we can pass to the limit in the inequality (1.61) to finally obtain (︂ (︂ )︂ ∫︁ )︂ ∫︁ N N |∇z|p dx ≤ −1 − 1 lim inf |∇z0ε |p dx p p ε→0 Ω Ω ⎡ ⎤ ∫︁ ∫︁ (︁ )︁ ≤ lim inf ⎣N F(z0ε ) dx − u x − x0 , ∇z0ε dx⎦ ε→0

Ω

− lim

∫︁ [︁

ε→0

v0ε

Ω

−

]︁ (︁

T ε (f (z0ε ))

x − x0 , ∇z0ε

)︁

dx

Ω

∫︁ =N

∫︁ F(z) dx −

Ω

u (x − x0 , ∇z) dx. Ω

Thus, (u, z) ∈ Ξ ∆ and the proof is complete. As immediately follows from Lemmas 1.2–1.5, we have the following result that can be considered as a substantiation of the fact that the passage from the original statement of the optimal control problem (1.1)–(1.4) to its reformulated variant (1.46) is not restrictive from the control point of view. Theorem 1.3. Let Ω be a bounded open convex subset of RN and p ∈ [2, N). Let u ∈ p L q (Ω) with q > p−1 be an arbitrary admissible control such that the boundary value

38 | 1 Optimal Distributed Control Problem

problem (1.2)–(1.3) is solvable. Then the Dirichlet boundary value problem (1.2)–(1.3) admits a weak solution z ∈ Y ⊂ W01,p (Ω) such that (u, z) ∈ Ξ ∆ .

2 On Approximation of One Class of Optimal Control Problems for Strongly Nonlinear Elliptic Equations with p-Laplace Operator Let Ω be a bounded open subset of RN (N ≥ 3). We assume that its boundary ∂Ω is of the class C1,1 and there exists a point x0 ∈ int Ω such that Ω is star-shaped with respect (︀ )︀ to x0 , i.e. σ − x0 , ν(σ) ≥ 0 for H N−1 -a.a. σ ∈ ∂Ω. Let F : R → [0, +∞) be a mapping satisfying conditions F ∈ C1loc (R), F is a convex function, and F(z) ≥ exp (C F z) for all z ∈ R with some constant C F > 0. Let f (z) = F ′ (z) and we assume that ⃒ 0 ⃒ ⃒ ∫︁ ⃒ ⃒ ⃒ ⃒ (2.1) f (z) ≥ F(z), ∀ z ∈ R, and ⃒ zf (z) dz⃒⃒ < +∞. ⃒ ⃒ −∞

We are concerned with the following optimal control problem ∫︁ ∫︁ ∫︁ ′ 1 1 α 2 q J(u, y) = |f (y)|p dx → inf, |y − y d | dx + |u| dx + ′ 2 q p Ω

(2.2)

Ω

Ω

subject to constrains −∆ p y = f (y) + u y=0

in Ω,

(2.3)

on ∂Ω,

q

u ∈ L (Ω), y ∈

(2.4)

W01,p (Ω),

(2.5)

where α > 0 is a given weight which is assumed to be small enough, 2 ≤ p < N, q > p′ , (︀ )︀ p′ = p/(p − 1) ∈ (1, 2] stands for the conjugate exponent, ∆ p y = div |∇y|p−2 ∇y is the p-Laplacian, and y d ∈ L2 (Ω) is a given distribution. By analogy with Chapter 1 (see Definition 1.1) we say that the state y ∈ W01,p (Ω) is a weak solution of (2.3)–(2.4) if y belongs to the set ⃒ }︁ {︁ ⃒ (2.6) Y = y ∈ W01,p (Ω) ⃒ f (y) ∈ L1 (Ω) , and the integral identity ∫︁ ∫︁ ∫︁ |∇y|p−2 (∇y, ∇φ) dx = f (y)φ dx + uφ dx Ω

Ω

(2.7)

Ω

C∞ 0 (Ω).

holds for every test function φ ∈ A physical motivation to the study of optimal control problem (2.2)–(2.5), various applications of this type of boundary value problems (BVPs), and their main characteristic features are described in details in Chapter 1 (see also [109]). We just mention here that, in contrast to OCP (1.1)–(1.4) the cost functional J(u, y) in (2.2) contains the extra term

p′ α ||f (y)|| p′ q′ L (Ω)

and, as it will be shown later on, this term plays a special

https://doi.org/10.1515/9783110668520-003

40 | 2 Approximation of OCPs role, because it is unknown whether OCP (2.2)–(2.5) will be consistent without this stabilizing factor. In Section 2.1 we clarify this point in more details. In this chapter we mainly focus on the scheme of direct two-level approximation of the strongly nonlinear differential operator − div(|∇y|p−2 ∇y) − f (y) which is not monotone and, in principle, has degeneracies as ∇y tends to zero. Moreover, when the term |∇y|p−2 is regarded as the coefficient of the Laplace operator, we have also the case of operator with unbounded coefficients. Because of this and L1 -boundedness of the function f (y), there are serious hurdles to deduce the differentiability of the state with respect to control and derive an optimality system for optimal control problem (2.2)–(2.5). Using a monotone and bounded approximation Fk (|∇y|2 ) of |∇y|2 , we introduce a special two-parametric family of optimization problems with fictitious controls and show that an optimal pair to the original optimal control problem can be attained by optimal solutions to the approximating ones provided the parameters k ∈ N and ε > 0 possess some special asymptotic properties. With that in mind we consequently provide the well-posedness analysis for the perturbed partial differential equations as well as for the corresponding fictitious optimal control problems. After that we pass to the limits as k → ∞ and ε → 0. Since the fictitious optimization problems are stated for the quasi-linear elliptic equations with coercive and monotone operators without any state and control constraints, the approximation and regularization approach looks rather attractive option for the numerical simulations. The plan of the chapter is as follows. In Section 2.1 we study the existence of a solution for the original problem (2.2)–(2.5). A two-parametric family of approximating optimal control problems with a fictitious control is introduced in Section 2.2. We show here that each of these problems is consistent and admits at least one solution at each (ε, k)-level of approximation. Section 2.3 contains the proof of the main results (Theorems 2.6 and 2.7) and deals with the asymptotic analysis of the sequences of optimal solutions to the approximating problems (2.26)–(2.29), provided the parameter ε varies within a strictly decreasing sequence {ε k }k∈N of positive real numbers satisfying some special condition. Finally, in Section 2.4 we deduce the differentiability of the state for approximating problem with respect to the controls u and v and derive an optimality system for each (ε, k)-level of approximation based on the Lagrange principle.

2.1 On consistency of optimal control problem Before proceeding further with qualitative analysis of optimal control problem (2.2)– (2.5), for the reader’s convenience, we make use of the following results (see Lemma 1.1, Proposition 1.2, and Theorem 1.1 in Chapter 1). Lemma 2.1. Let y = y(u) ∈ Y be a weak solution to BVP (2.3)–(2.4) for a given u ∈ ′

′

L q (Ω). Then f (y) ∈ W −1,p (Ω), where W −1,p (Ω) stands for the dual space to W01,p (Ω)

2.1 On consistency of OCP (2.2)–(2.5) |

41

with p′ = p/(p − 1), ⟨︀

f (y), z

∫︁

⟩︀ W

−1,p′

(Ω);W 1,p (Ω)

z f (y) dx,

=

∀ z ∈ W01,p (Ω),

(2.8)

Ω

and y satisfies the energy equality ∫︁ ∫︁ ∫︁ p |∇y| dx = y f (y) dx + yu dx. Ω

Ω

(2.9)

Ω

Proposition 2.1. Let u ∈ L q (Ω) and let y = y(u) ∈ W01,p (Ω) be a weak solution to BVP ′

(2.3)–(2.4). Assume that f (y) ∈ L p (Ω) and f satisfies properties (2.1). Then (︂ )︂ ∫︁ ∫︁ ∫︁ N p |∇y| dx ≤ N F(y) dx − u (x − x0 , ∇y) dx, −1 p Ω

Ω

(2.10)

Ω

(︀ )︀ where x0 ∈ int Ω is a point such that σ − x0 , ν(σ) ≥ 0 for H N−1 -a.a. σ ∈ ∂Ω. Theorem 2.1. Let u ∈ L q (Ω) and let y = y(u) ∈ Y be a weak solution to BVP (2.3)–(2.4) such that y satisfies the inequality (2.10) and properties (2.1) hold true. Then ∫︁ p′ p′ −1 y f (y) dx ≤ C1 ||u||L q (Ω) + C2 ||u||L q (Ω) + C3 , (2.11) Ω p′ −1

||y||W 1,p (Ω) ≤ C4 ||u||L q (Ω) + C5 ,

(2.12)

0

for some positive constants C i , 1 ≤ i ≤ 5, independent of u and y. It spite of the fact that inequality (2.10) makes sense even if we do not assume fulfillment ′ of the inclusion f (y) ∈ L p (Ω) but have only that y ∈ Y and u ∈ L q (Ω), it is unknown whether this inequality holds for an arbitrary weak solution to BVP (2.3)–(2.4). Since the existence and uniqueness of the weak solutions to the original BVP is an open question for arbitrary given control u ∈ L q (Ω) with q > p′ , the following result reflects some interesting properties of the Dirichlet boundary value problem (2.3)–(2.4) (see Theorem 1.2). Theorem 2.2. Let p ∈ [2, N) and let u ∈ L q (Ω) with q > p′ be an arbitrary admissible control such that the boundary value problem (2.3)–(2.4) is solvable. Then the Dirichlet boundary value problem (2.3)–(2.4) admits a weak solution y ∈ Y ⊂ W01,p (Ω) satisfying the inequality (2.10).

42 | 2 Approximation of OCPs

Mainly inspired by this theorem, it has been proposed in Chapter 1 to consider the following optimal control problem: ∫︁ ∫︁ 1 1 Minimize J(u, y) = |y − y d |2 dx + |u|q dx, (2.13) 2 q Ω

Ω

subject to constrains −∆ p y = f (y) + u y=0 ∫︁ Ω

|∇y|p dx ≤

Np N−p

in Ω,

(2.14)

on ∂Ω,

∫︁

(2.15) ∫︁

F(y) dx −

p N−p

q

Ω 1,p W0 (Ω),

Ω

u ∈ L (Ω), y ∈

u (x − x0 , ∇y) dx,

(2.16) (2.17)

where, from the formal point of view, the inequality (2.16) plays the role of an extra control-state constraint. As follows from Theorem 2.2, the reformulated version (2.13)–(2.17) becomes a consistent optimization problem with a nonempty set of feasible solutions. Moreover, this problem has at least one solution for each y d ∈ L2 (Ω) (see Theorem 4.1 in [81]). However, because of the inequality (2.16), there are serious hurdles to derive the corresponding optimality conditions for the problem (2.13)–(2.17) and provide its numerical simulations. On the other hand, the validity of inequality (2.16) is a direct consequence of ′ the condition f (y) ∈ L p (Ω). Hence, it is reasonable to consider instead of the problem (2.13)–(2.17), its regularized version in the form of the optimal control problem (2.2)–(2.5). As a result, its consistency immediately follows from Proposition 2.1, and moreover, the set of feasible solutions to the problem (2.2)–(2.5) ⎫ ⃒ ⎧ ⃒ u ∈ L q (Ω), y ∈ Y , f (y) ∈ L p′ (Ω), ⎪ ⎪ ⎪ ⃒ ⎪ ⎪ ⎪ ⎪ ⃒ ∫︁ ⎪ ∫︁ ⎪ ⎪ ⎪ ⃒ ⎪ ⎪ ⎪ p−2 ⎬ ⃒ ⎨ |∇ y | ∇ y, ∇ φ dx = f (y)φ dx ( ) ⃒ (2.18) Ξ = (u, y) ⃒ Ω ⃒ Ω ⎪ ⎪ ∫︁ ⎪ ⎪ ⃒ ⎪ ⎪ ⎪ ⎪ ⃒ ⎪ ⎪ + uφ dx, ∀ φ ∈ W01,p (Ω) ⎪ ⎪ ⃒ ⎪ ⎪ ⎭ ⎩ ⃒ Ω

y ∈ C∞ u := is always nonempty. Indeed, if we take an arbitrary function ̃︀ 0 (Ω) and put ̃︀ ′ 1,p q p y − f (̃︀ y), then ̃︀ −∆ p ̃︀ u ∈ L (Ω), ̃︀ y ∈ W0 (Ω), and f (̃︀ y ∈ Y ⊂ W01,p (Ω) y) ∈ L (Ω). Hence, ̃︀ u and (̃︀ is a weak solution to the boundary value problem (2.3)–(2.4) for given ̃︀ u, ̃︀ y) ∈ Ξ. Let us show that the optimal control problem (2.2)–(2.5) is solvable. Theorem 2.3. Let p ∈ [2, N) and q > p′ . Then, for a given y d ∈ L2 (Ω), the optimal control problem (2.2)–(2.5) has at least one solution. Proof. Argumentation has much in common with the proof of Proposition 1.3 and Theorem 2.1. Since J(u, y) ≥ 0 for all (u, y) ∈ Ξ, it follows that there exists a non-

2.1 On consistency of OCP (2.2)–(2.5) |

43

}︀ {︀ negative value μ ≥ 0 such that μ = inf (u,y)∈Ξ J(u, y). Let (u k , y k ) k∈N be a minimizing sequence to the problem (2.2)–(2.5), i.e. (u k , y k ) ∈ Ξ ∀ k ∈ N and

lim J(u k , y k ) = μ.

k→∞

So, we can suppose that J(u k , y k ) ≤ μ+1 for all k ∈ N. Taking into account the definition of the set Ξ, it follows from Proposition 2.1 and Theorem 2.1 that p′ −1

||y k ||W 1,p (Ω) ≤ C4 ||u k ||L q (Ω) + C5 0

for some constants C4 and C5 independent of k and u k . Then )︀ 1 (︀ )︀ p′ −1 (︀ ||y k ||W 1,p (Ω) + ||u k ||L q (Ω) ≤ C5 + C4 qJ(u k , y k ) q + qJ(u k , y k ) q 0

||f (y k )||L p′ (Ω)

(︀ )︀ p′ −1 (︀ )︀ 1 ≤ C5 + C4 q(μ + 1) q + q(μ + 1) q , )︂1/p′ (︂ ′ p (μ + 1) . ≤ α

∀ k ∈ N,

Thus, without loss of generality, we can suppose that there exists a subsequence of {︀ }︀ (u k , y k ) k∈N (still denoted by the same index) and a pair (u0 , y0 ) ∈ L q (Ω) × W01,p (Ω) such that (u k , y k ) ⇀ (u0 , y0 ) weakly in L q (Ω) × W01,p (Ω) as k → ∞, Let us show now that the limit pair (u0 , y0 ) is related by the integral identity (2.7). By the Sobolev Embedding Theorem, the injection W01,p (Ω) ˓→ L p (Ω) is compact. Hence, the weak convergence y k ⇀ y in W01,p (Ω) implies the strong convergence in L p (Ω). Therefore, up to a subsequence, we can suppose that y k (x) → y(x) for almost every point x ∈ Ω. As a result, we have the pointwise convergence: f (y k ) → f (y) almost {︀ }︀ ′ everywhere in Ω. Since the sequence f (y k ) k∈N is bounded in L p (Ω), it follows that ′

f (y k ) ⇀ f (y0 ) weakly in L p (Ω). As a result, the limit passage in the right-hand side of the integral identity ∫︁ ∫︁ ∫︁ N f (y k )φ dx + u k φ dx, ∀ φ ∈ C∞ |∇y k |p−2 (∇y k , ∇φ) dx = 0 (R ) Ω

Ω

(2.19)

(2.20)

Ω

becomes trivial. As for the limit passage as k → ∞ in the left-hand side of (2.20), we make use of the following result of L. Boccardo and F. Murat (see [10, Theorem 2.1]): if (i) y k → y0 weakly in W01,p (Ω), strongly in L p (Ω) and a.e. in Ω; ′

(ii) u k → u0 strongly in W −1,p (Ω); (iii) the sequence {f k }k∈N is bounded in L1 (Ω);

44 | 2 Approximation of OCPs (iv) − div(|∇y k |p−2 ∇y k ) = f k + u k in D′ (Ω) for all k ∈ N, then, within a subsequence, ∇y k → ∇y0 strongly in L r (Ω)N for any 1 ≤ r < p and a.e. in Ω.

(2.21)

In our case, the fulfilment of condition (iii) is guaranteed by the property (2.19). However, instead of (ii), we have the weak convergence u k ⇀ u in L q (Ω for the given q > p′ . Since, * by Sobolev embedding theorem, W01,p (Ω) is continuously embedded in L p (Ω) with (︁ * )︁* pN p* = N−p , we have by duality arguments that L p (Ω) is continuously embedded in ′

W −1,p (Ω). So, if we define p* = (p* )′ =

pN , pN − N + p

then we have ′

L r (Ω) ⊂ L p* (Ω) ⊂ W −1,p (Ω) ∀ r > It is easy to check that q

pN pN−N+p ′

< p′ =

p p−1

pN . pN − N + p

for all p ≥ 2. Hence, the weak convergence ′

u k ⇀ u in L (Ω) with q > p , implies the strong convergence in W −1,p (Ω). As for the rest assumptions of Boccardo–Murat Theorem they are obviously satisfied in our case. Hence, the pointwise convergence property (2.21), continuity of the mapping ξ ↦→ |ξ |p−2 ξ , and Vitali’s theorem imply that |∇y k |p−2 ∇y k → |∇y0 |p−2 ∇y0

strongly in L r (Ω)N ∀ r : 1 ≤ r < p′ .

(2.22)

Thus, taking these facts into account and passing to the limit in the integral identity (2.20) as k → ∞, we see that y0 is a weak solution to BVP (2.3)–(2.4) for the given u0 ∈ L q (Ω). Hence, (u0 , y0 ) is a feasible pair to the problem (2.2)–(2.5). To conclude the proof, it remains to take into account the lower semi-continuity of the cost functional J : L q (Ω) × W01,p (Ω) → R with respect to the weak convergence in L q (Ω) × W01,p (Ω) and property (2.19). This yields μ = inf J(u, y) = lim J(u k , y k ) ≥ J(u0 , y0 ). (u,y)∈Ξ

k→∞

Thus, (u0 , y0 ) ∈ Ξ is an optimal pair to the problem (2.2)–(2.5).

2.2 Approximating optimal control problems and their previous analysis We introduce, as in [23], the following two-parameter family of perturbed operators (︂(︁ )︂ (︁ )︁)︁ p−2 2 ε + Fk |∇y|2 ∆ ε,k,p (y) = div ∇y , (2.23)

2.2 Approximating Optimal Control Problems | 45

where Fk : R+ → R+ is a non-decreasing C1 (R+ )-function such that [︁ ]︁ Fk (t) = t, if t ∈ 0, k2 , Fk (t) = k2 + 1, if t > k2 + 1, and t ≤ Fk (t) ≤ t + δ, if k2 ≤ t < k2 + 1

for some δ ∈ (0, 1),

Fk′ (t) ≤ δ* , if k2 ≤ t < k2 + 1

for some δ* > 1,

and the constants δ and δ* are independent of k ∈ N. In particular, if ⎧ ⎪ if 0 ≤ t ≤ k2 , ⎨ t, 2 3 2 2 Fk (t) = (k − t) + (k − t) + t, if k2 ≤ t ≤ k2 + 1, ⎪ ⎩ k2 + 1, if t ≥ k2 + 1.

(2.24)

(2.25)

then δ = 4/27 and δ* = 4/3 satisfy (2.24). Hereinafter, we assume that the parameter ε varies within a strictly decreasing sequence of positive real numbers which converge to zero. We now introduce the following perturbed optimal control problem (see, for comparison, [20, 21]). ∫︁ ∫︁ {︁ ′ 1 k Minimize I ε,k (u, v, y) = |v − T ε (f (y))|p dx |y − y d |2 dx + ′ 2 p Ω Ω ∫︁ ∫︁ }︁ ′ α 1 |u|q dx + ′ + |v|p dx (2.26) q p Ω

Ω

subject to the constraints −∆ ε,k,p (y) = v + u y=0 ′

v ∈ L p (Ω),

in

Ω,

on ∂Ω,

u ∈ L q (Ω),

(2.27) (2.28)

y ∈ H01 (Ω).

Here, T ε : R → R is the truncation operator defined by {︁ {︁ }︁ }︁ T ε (s) = max min s, ε−1 , −ε−1 .

(2.29)

(2.30)

′

We consider the function v ∈ L p (Ω) as a fictitious control. For our further analysis, we make use of the following notation

||φ||ε,k

⎞1/p ⎛ ∫︁ (︁ )︁ p−2 2 |∇φ|2 dx⎠ ∀φ ∈ H01 (Ω). =⎝ ε + Fk (|∇φ|2 ) Ω

It is clear that the effect of such perturbation of ∆ p (y) is to provide its regularization around points where |∇y(x)| is equal to zero and becomes unbounded. Indeed, for an arbitrary element y* ∈ H01 (Ω) let us consider the level set Ω k (y* ) :=

46 | 2 Approximation of OCPs {︁

x ∈ Ω : |∇y* (x)| >

√

}︁ k2 + 1 . Then

∫︁

|Ω k (y* )| :=

1 dx ≤ √ Ω k (y* )

∫︁

1 k2 + 1

|∇y* (x)| dx

Ω k (y* )

⎞ 12

⎛ ≤

1 ⎜ 1 |Ω (y* )| 2 ⎝ k k

∫︁

|∇y* |2 dx⎠

⎟

Ω k (y* ) *

=

|Ω k (y )|

1 2

(︂

k

1 ε + k2 + 1

)︂ p−2 4

⎞ 21

⎛ ∫︁

(︁

⎜ ⎝

ε + Fk (|∇y* |2 )

)︁ p−2 2

|∇y* |2 dx⎠

⎟

Ω k (y* ) p 1 1 2 . ≤ p |Ω k (y* )| 2 ||y* ||ε,k k2

Hence, the Lebesgue measure of the set Ω k (y* ) satisfies the estimate |Ω k (y* )| ≤

1 * p ||y ||ε,k , kp

∀ y* ∈ H01 (Ω).

(2.31) ′

In what follows, we say that for given ε > 0, k ∈ N, u ∈ L q (Ω), and v ∈ L p (Ω), a distribution y ε,k ∈ H01 (Ω) is the weak solution to boundary value problem (2.27)–(2.28) if ∫︁ ∫︁ (︀ )︀ 2 p−2 2 ∇y ε,k , ∇φ RN dx = (u + v)φ dx, ∀ φ ∈ C∞ (ε + Fk (|∇y ε,k | )) 0 (Ω), (2.32) Ω

Ω

or equivalently ∫︁

(ε + Fk (|∇φ|2 ))

p−2 2

(︀

∇φ, ∇φ − ∇y ε,k

)︀ RN

dx

Ω

∫︁ (u + v)(φ − y ε,k ) dx,

≥

∀ φ ∈ C∞ 0 (Ω).

(2.33)

Ω

The existence of a unique solution to the boundary value problem (2.27)–(2.28) follows from an abstract theorem on monotone operators; see, for instance, [113, Theorem 2.1.2] or [147, Section II.2]. Theorem 2.4. Let V be a reflexive separable Banach space. Let V * be the dual space, and let A : V → V * be a bounded, semicontinuous, coercive and strictly monotone operator. Then the equation Ay = f has a unique solution for each f ∈ V * . Moreover, Ay = f if and only if ⟨Aφ, φ − y⟩ ≥ ⟨f , φ − y⟩ for all φ ∈ V * .

2.2 Approximating Optimal Control Problems |

47

Here, the above mentioned properties of the strict monotonicity, semicontinuity, and coercivity of the operator A have respectively the following meaning: ⟨Ay − Av, y − v⟩V * ;V ≥ 0,

∀ y, v ∈ V;

(2.34)

⟨Ay − Av, y − v⟩V * ;V = 0 =⇒ y = v;

⟨︀

the function t ↦→ A(y + tv), w

(2.35)

⟩︀ V * ;V

is continuous for all y, v, w ∈ V; ⟨Ay, y⟩V * ;V

lim

||y||V

||y||V →∞

(2.36)

= +∞.

(2.37)

In our case, we can define the operator A ε,k,u as a mapping H01 (Ω) → H −1 (Ω) by ∫︁ (︀ )︀ p−2 ⟨A ε,k,u φ, v⟩H −1 (Ω);H 1 (Ω) := ε + Fk (|∇φ|2 ) 2 (∇φ, ∇v)RN u dx. 0 Ω

Then it is easy to show that A ε,k,u y = −∆ ε,k,p (u, y). Now, we establish the following results. Proposition 2.2. For every u ∈ Aad , k ∈ N, and ε > 0, the operator A ε,k,u := −∆ ε,k,p (u, ·) : H01 (Ω) → H −1 (Ω) (︀ )︀ p−2 is bounded and ||A ε,k,u || ≤ ε + k2 + 1 2 . Proof. From the assumptions on Fk and the boundedness of u we obtain ||A ε,k,u || =

sup ||y||H 1 (Ω) ≤1

||A ε,k,u y||H −1 (Ω)

0

=

sup

⟨︀

sup

A ε,k,u y, v

⟩︀

||y||H 1 (Ω) ≤1 ||v||H 1 (Ω) ≤1 0

=

sup

sup

||y||H 1 (Ω) ≤1 ||v||H 1 (Ω) ≤1 0

(︁

H −1 (Ω);H01 (Ω)

0

≤ ε + k2 + 1

⎤ ⎡ ∫︁ )︀ p−2 (︀ 2 2 ⎣ ε + Fk (|∇y| ) (∇y, ∇v)RN u dx⎦

0

)︁ p−2 2

Ω

sup

sup

||y||H 1 (Ω) ≤1 ||v||H 1 (Ω) ≤1 0

||y||H 1 (Ω) ||v||H 1 (Ω) 0

0

0

(︁ )︁ p−2 2 = ε + k2 + 1 , which concludes the proof. Proposition 2.3. For every u ∈ Aad , k ∈ N, and ε > 0, the operator A ε,k,u is strictly monotone.

48 | 2 Approximation of OCPs

Proof. To begin with, we make use of the following algebraic inequality: (︂

(︀

(︀ )︀ p−2 )︀ p−2 ε + Fk (|a|2 ) 2 a − ε + Fk (|b|2 ) 2 b, a − b

)︂ RN

≥ε

p−2 2

∀ a, b ∈ RN .

| a − b |2 ,

(2.38)

In order to prove it, we note that the left hand side of (2.38) can be rewritten as follows )︂ (︂ (︀ )︀ p−2 (︀ )︀ p−2 ε + Fk (|a|2 ) 2 a − ε + Fk (|b|2 ) 2 b, a − b RN

⎛

∫︁1

=⎝

d ds

{︂

(︀

ε + Fk (|sa + (1 − s)b|2 )

)︀ p−2 (︀ 2

sa + (1 − s)b

)︀

⎞

}︂

ds, a − b⎠

0

RN

∫︁1 =

(︀

ε + Fk (|sa + (1 − s)b|2 )

)︀ p−2 2

|a − b|2 dx

0

+(p − 2)

∫︁1 {︁ (︀

)︀ p−4 ε + Fk (|sa + (1 − s)b|2 ) 2 Fk′ (|sa + (1 − s)b|2 )

0

⃒(︀ )︀ ⃒2 }︁ × ⃒ sa + (1 − s)b, a − b RN ⃒ ds = I1 + I2 . Since p ≥ 2 and Fk : R+ → R+ is a non-decreasing C1 (R+ )-function, it follows that I2 ≥ 0 for all a, b ∈ RN . It remains to observe that (︀ Hence, I1 ≥ ε

p−2 2

)︀ ε + Fk (|sa + (1 − s)b|2 ) ≥ ε,

∀ a, b ∈ RN .

|a − b|2 and we arrive at the inequality (2.38). With this we obtain

⟨︀

⟩︀ − ∆ ε,k,p (u, y) + ∆ ε,k,p (u, v), y − v H −1 (Ω);H 1 (Ω) 0 ∫︁ )︁ (︁ p−2 2 p−2 2 = u(x) (ε + Fk (|∇y| )) ∇y − (ε + Fk (|∇v|2 )) 2 ∇v, ∇y − ∇v dx Ω

≥ αε

p−2 2

∫︁

|∇y − ∇v|2 dx = αε

p−2 2

2

||y − v||H01 (Ω) ≥ 0.

Ω

Since the relation ⟨︀

A ε,k,u y − A ε,k,u v, y − v

⟩︀

H −1 (Ω);H01 (Ω)

=0

implies that y = v almost everywhere in Ω, it follows that the strict monotonicity property (2.35) holds in this case. Proposition 2.4. For every u ∈ Aad , k ∈ N, and ε > 0, the operator A ε,k,u is coercive (in the sense of relation (2.37)).

2.2 Approximating Optimal Control Problems | 49

Proof. In order to check this property it is enough to observe that for any y ∈ H01 (Ω), k ∈ N, ε > 0, and u ∈ Aad , we have ⟨︀ ⟩︀ ⟨︀ ⟩︀ A ε,k,u y,y H −1 (Ω);H 1 (Ω) = − ∆ ε,k,p (u, y), y H −1 (Ω);H 1 (Ω) 0 0 ∫︁ (︁ )︁ p−2 p−2 2 2 2 2 ε + Fk (|∇y| ) = |∇y| u dx ≥ αε 2 ||y||H 1 (Ω) . 0 Ω

To prove the well-posedness of the boundary value problem (2.27)–(2.28), it is enough ′ now to show that L p (Ω) is continuously embedded in H −1 (Ω). Indeed, by Sobolev [︀ )︀ embedding theorem, we have: H01 (Ω) ˓→ L r (Ω) compactly for all r ∈ 1, 2N/(N − 2) . ′

Hence, the compactness of injection L r (Ω) ˓→ H −1 (Ω) holds true if only r′ > 2N/(N + 2). [︀ )︀ Since p′ > 2N/(N + 2) for each p ∈ 2, 2N/(N − 1) , it follows by duality arguments ′

that L p (Ω) ˓→ H −1 (Ω) with a compact embedding. So, there exists a constant C p > 0 ′ such that ||v||H −1 (Ω) ≤ C p ||v||L p′ (Ω) for all v ∈ L p (Ω). Moreover, taking into account the estimate ∫︁ vφ dx = ⟨v, φ⟩H −1 (Ω);H 1 (Ω) ≤ ||v||H −1 (Ω) ||φ||H 1 (Ω) 0

0

Ω

≤ C p ||v||L p′ (Ω) ||φ||H 1 (Ω) , 0

∀ φ ∈ C∞ 0 (Ω)

′

and the inclusion u + v ∈ L p (Ω), we see that the right-hand side of (2.32) can be extended to a linear continuous functional on H01 (Ω), ∫︁ L(φ) := ⟨v, φ⟩H −1 (Ω);H 1 (Ω) + uφ dx, ∀ φ ∈ H01 (Ω). 0

Ω

H01 (Ω)

Since the operator −∆ ε,k,p (·) : → H −1 (Ω) is bounded, strictly monotone, semi-continuous, and coercive (see [23]), it follows from Theorem 2.4 (for the details we refer to the general theory of monotone operators [113, 139, 147]) that for each ε > 0, [︀ )︀ ′ k ∈ N, p ∈ 2, 2N/(N − 1) , u ∈ L q (Ω), and v ∈ L p (Ω), the boundary value problem (2.27)–(2.28) admits a unique weak solution y ε,k ∈ H01 (Ω) satisfying the energy equality (see also Theorem 4.5 in [23]) ∫︁ ⟨︀ ⟩︀ p ||y ε,k ||ε,k = v, y ε,k H −1 (Ω);H 1 (Ω) + uy ε,k dx. (2.39) 0

Ω

From this it is easy to deduce that, for every positive value ε > 0 and integer k ∈ N, the set of feasible solutions to the problem (2.26)–(2.29) is nonempty, i.e. ⎧ ⃒ ⎫ ′ ⃒ u ∈ L q (Ω), v ∈ L p (Ω), ⎪ ⎪ ⎪ ⃒ ⎪ ⎨ ⎬ ⃒ 1 Ξ ε,k = (u, v, y) ⃒⃒ ̸= ∅. (2.40) y ∈ H0 (Ω), I ε,k (u, v, y) < +∞, ⎪ ⎪ ⃒ ⎪ ⎪ ⎩ ⎭ ⃒ (u, v, y) are related by identity (2.32)

50 | 2 Approximation of OCPs [︀ )︀ For our further analysis, we assume that p ∈ 2, 2N/(N−1) . We also need to obtain some appropriate a priory estimates for the weak solutions to problem (2.27)–(2.28). With that in mind, we make use of the following auxiliary results. Proposition 2.5. Let k ∈ N and ε > 0 be given. Then, for arbitrary u ∈ L q (Ω), v ∈ ′ L p (Ω), and y ∈ H01 (Ω), we have ⃒ ⃒ [︁ p−2 p ]︁ ⃒ ⃒ 2 , (2.41) ⃒⟨v, y⟩H −1 (Ω);H01 (Ω) ⃒ ≤ C p ||v||L p′ (Ω) |Ω| 2p ||y||ε,k + ||y||ε,k ⃒ ⃒ ⃒ ⃒∫︁ [︁ p−2 q−p′ p ]︁ ⃒ ⃒ ⃒ uy dx⃒ ≤ C p |Ω| qp′ ||u||L q (Ω) |Ω| 2p ||y||ε,k + ||y|| 2 . (2.42) ⃒ ⃒ ε,k ⃒ ⃒ Ω

Proof. Let us fix an arbitrary element y of H01 (Ω). We associate with this element the {︀ }︀ set Ω k (y), where Ω k (y) := x ∈ Ω : |∇y(x)| > k . Then ∫︁ uy dx ≤ C p ||u||L p′ (Ω) ||y||H 1 (Ω) 0

Ω q−p′ (︀ )︀ ≤ C p |Ω| qp′ ||u||L q (Ω) ||∇y||L2 (Ω\Ω k (y))N + ||∇y||L2 (Ω k (y))N ,

(2.43)

⟨v, y⟩H −1 (Ω);H 1 (Ω) ≤ C p ||v||L p′ (Ω) ||y||H 1 (Ω) 0

0

(︀ )︀ = C p ||v||L p′ (Ω) ||∇y||L2 (Ω\Ω k (y))N + ||∇y||L2 (Ω k (y))N . Using the fact that p−2

||∇y||L2 (Ω\Ω k (y))N ≤ |Ω| 2p ||∇y||L p (Ω\Ω k (y))N ⎞ 1p ⎛ ∫︁ p−2 p−2 ⎟ ⎜ ≤ |Ω| 2p ⎝ (ε + |∇y|2 ) 2 |∇y|2 dx⎠ Ω\Ω k (y)

and Fk (|∇y|2 ) = |∇y|2 a.e. in Ω \ Ω k (y), and k2 ≤ Fk (|∇y|2 ) ≤ k2 + 1 a.e. in Ω k (y),

∀ k ∈ N,

(2.44)

2.2 Approximating Optimal Control Problems | 51

we obtain ⎞ 1p

⎛ p−2 ⎜ ||∇y||L2 (Ω\Ω k (y))N ≤ |Ω| 2p ⎝

∫︁

(ε + Fk (|∇y|2 ))

p−2 2

|∇y|2 dx⎠

⎟

Ω\Ω k (y) p−2 2p

||∇y||L2 (Ω k (y))N

= |Ω| ||y||ε,k , ⎛ ⎞ 21 ∫︁ p−2 ⎜ ⎟ ≤⎝ (ε + Fk (|∇y|2 )) 2 |∇y|2 dx⎠

(2.45)

Ω k (y) p 2 = ||y||ε,k .

(2.46)

As a result, inequalities (2.41)–(2.42) immediately follows from (2.43)–(2.46). ′

Definition 2.1. Let {u ε,k , v ε,k } ε>0 ⊂ L q (Ω) × L p (Ω) be an arbitrary sequence of admisk∈N {︀ }︀ sible controls. We say that a two-parametric sequence y ε,k ε>0 ⊂ H01 (Ω) is bounded k∈N

with respect to the || · ||ε,k -quasi-seminorm if sup ||y ε,k ||ε,k < +∞. ε>0 k∈N

′

Let us show that for every (u, v) ∈ L q (Ω) × L p (Ω), the sequence of weak solutions to the {︀ }︀ boundary value problem (2.27)–(2.28) y ε,k = y ε,k (u, v) ε>0 is bounded with respect k∈N

to the || · ||ε,k -quasi-seminorm in the sense of Definition 2.1. Indeed, the energy equality (2.39) together with estimates (2.41)–(2.42) immediately lead us to the relation ∫︁ (︀ )︀ p−2 p ||y ε,k ||ε,k := ε + Fk (|∇y ε,k |2 ) 2 |∇y ε,k |2 dx Ω

⟨︀

= v, y ε,k

⟩︀

∫︁ H −1 (Ω);H01 (Ω)

uy ε,k dx

+ Ω

)︂ (︂ q−p′ ≤ C p ||v||L p′ (Ω) + |Ω| qp′ ||u||L q (Ω) [︁ p−2 p ]︁ 2 × |Ω| 2p ||y ε,k ||ε,k + ||y ε,k ||ε,k .

(2.47)

As a result, it follows from (2.47) that {︂ 2 }︂ 1 p p−1 , , C u,v ||y ε,k ||ε,k ≤ max C u,v

(2.48)

′

∀ ε > 0, ∀ k ∈ N, ∀ u ∈ L q (Ω), ∀ v ∈ L p (Ω),

where

(︂ C u,v := C p ||v||L p′ (Ω) + |Ω|

q−p′ qp′

||u||L q (Ω)

)︂ (︁

p−2

)︁

|Ω| 2p + 1 .

To conclude this section, we give the following result.

(2.49)

52 | 2 Approximation of OCPs Theorem 2.5. Let p ∈ [2, 2N/(N − 1)) and q > p′ . Then, for every positive value ε > 0 and integer k ∈ N, the approximating optimal control problem (2.26)–(2.29) has at least one solution. ′

Proof. Since I ε,k (u, v, y) ≥ 0 for all (u, v, y) ∈ L q (Ω) × L p (Ω) × H01 (Ω), it follows that there exists a non-negative value μ ε,k such that μ ε,k =

inf

(u,v,y)∈Ξ ε,k

I ε,k (u, v, y).

{︀ }︀ Let (u ε,k,m , v ε,k,m , y ε,k,m ) m∈N be a minimizing sequence, i.e. (u ε,k,m , v ε,k,m , y ε,k,m ) ∈ Ξ ε,k ∀ m ∈ N and lim I ε,k (u ε,k,m , v ε,k,m , y ε,k,m ) = μ ε,k .

m→∞

So, without loss of generality, we can suppose that ∫︁ 1 I ε,k (u ε,k,m , v ε,k,m , y ε,k,m ) = |y ε,k,m − y d |2 dx 2 Ω ∫︁ ′ k |v ε,k,m − T ε (f (y ε,k,m ))|p dx + ′ p Ω ∫︁ ∫︁ ′ 1 α q + |v ε,k,m |p dx |u ε,k,m | dx + ′ q p Ω

Ω

≤ μ ε,k + 1

for all m ∈ N.

(2.50)

Since T ε (f (y ε,k,m )) ∈ L∞ (Ω), it follows from (2.50) that the sequence of fictitious {︀ }︀ ′ controls v ε,k,m k∈N is uniformly bounded in L p (Ω). The similar conclusion can be {︀ }︀ made for the sequence u ε,k,m k∈N . So, we can admit the existence of elements v0ε,k ∈ ′

L p (Ω) and u0ε,k ∈ L q (Ω) such that (up to a subsequence) ′

v ε,k,m ⇀ v0ε,k in L p (Ω) and u ε,k,m ⇀ u0ε,k in L q (Ω) as m → ∞.

(2.51)

{︀ }︀ Moreover, in view of estimate (2.48), we see that the sequence y ε,k,m k∈N is bounded in H01 (Ω). Indeed, setting {︀ }︀ Ω k (y ε,k,m ) := x ∈ Ω : |∇y ε,k,m (x)| > k for each k ∈ N, we have ||y ε,k,m ||H 1 (Ω) ≤ ||∇y ε,k,m ||L2 (Ω\Ω k (y ε,k,m ))N + ||∇y ε,k,m ||L2 (Ω k (y ε,k,m ))N 0 p ]︁ by (2.48) p−2 by (2.45)–(2.46) [︁ 2 < +∞. ≤ |Ω| 2p ||y ε,k,m ||ε,k + ||y ε,k,m ||ε,k

(2.52)

2.2 Approximating Optimal Control Problems | 53

}︀ {︀ As a result, we deduce the existence of a subsequence of y ε,k,m k∈N , denoted in the same way, and an element y0ε,k ∈ H01 (Ω) such that y ε,k,m ⇀ y0ε,k in H01 (Ω) as m → ∞. Let us prove that y0ε,k is the solution of (2.27)-(2.28) with v = v0ε,k and u = u0ε,k . Let us fix an arbitrary test function φ ∈ C∞ 0 (Ω) and pass to the limit in the Minty inequality ∫︁ )︀ p−2 (︀ (ε + Fk (|∇φ|2 )) 2 ∇φ, ∇φ − ∇y ε,k,m RN dx Ω

⟨︀ ⟩︀ ≥ v ε,k,m , (φ − y ε,k,m ) H −1 (Ω);H 1 (Ω) +

∫︁ u ε,k,m (φ − y ε,k,m ) dx,

0

(2.53)

Ω

as m → ∞. In view of the convergences ′

v ε,k,m ⇀ v0ε,k in L p (Ω), u ε,k,m ⇀ u0ε,k in L q (Ω), and y ε,k,m → y0ε,k strongly in L2 (Ω), we obtain ∫︁ lim

m→∞

(ε + Fk (|∇φ|2 ))

p−2 2

(︀

∇φ, ∇y ε,k,m

)︀ RN

dx

Ω

∫︁ =

(ε + Fk (|∇φ|2 ))

p−2 2

(︁

∇φ, ∇y0ε,k

)︁ RN

dx,

Ω

⟨ ⟩ ⟩︀ lim v ε,k,m , (φ − y ε,k,m ) H −1 (Ω);H 1 (Ω) = v0ε,k , (φ − y0ε,k ) , 0 m→∞ H −1 (Ω);H01 (Ω) ∫︁ ∫︁ lim u ε,k,m (φ − y ε,k,m ) dx = u0ε,k (φ − y0ε,k ) dx. ⟨︀

m→∞

Ω

Ω

Thus, passing to the limit in relation (2.53) as m → ∞, we arrive at the inequality ∫︁ (︁ )︁ p−2 dx (ε + Fk (|∇φ|2 )) 2 ∇φ, ∇φ − ∇y0ε,k RN

Ω

⟨ ⟩ ≥ v0ε,k , (φ − y0ε,k )

∫︁ H −1 (Ω);H01 (Ω)

+

u0ε,k (φ − y0ε,k ) dx,

∀ φ ∈ C∞ 0 (Ω).

Ω

1 Finally, from the density of C∞ 0 (Ω) in H 0 (Ω), we infer that the integral identity ∫︁ ∫︁ ⟨ ⟩ (︁ )︁ p−2 + u0ε,k φ dx dx = v0ε,k , φ (ε + Fk (|∇y0ε,k |2 )) 2 ∇y0ε,k , ∇φ 1 RN

Ω

H01 (Ω),

H −1 (Ω);H0 (Ω)

Ω

holds for every φ ∈ and hence y ∈ H01 (Ω) is the solution to the boundary value problem (2.27)-(2.28) for v = v0ε,k and u = u0ε,k . Since the solution of (2.27)-(2.28) is unique, the whole sequence {y ε,k,m }m∈N converges weakly to y0ε,k in H01 (Ω). Thus, (︀ 0 )︀ u ε,k , v0ε,k , y0ε,k ∈ Ξ ε,k .

54 | 2 Approximation of OCPs )︀ (︀ To deduce the fact that u0ε,k , v0ε,k , y0ε,k is an optimal solution to the problem (2.26)–(2.29), we make use of the following observations: in accordance to the strong convergence y ε,k,m → y0ε,k in L2 (Ω), we have T ε (f (y ε,k,m )) → T ε (f (y0ε,k )) a.e. in Ω and 1

1

supm∈N ||T ε (f (y ε,k,m ))||L q (Ω) ≤ ε− q |Ω| q . ′

Since q > 2, it follows that the sequence T ε (f (y ε,k,m )) → T ε (f (y0ε,k )) strongly in L p (Ω). ′

Combining this fact with the weak convergence v ε,k,m ⇀ v0ε,k in L p (Ω) and the lower ′

semi-continuity of the norm || · ||L p′ (Ω) with respect to the weak convergence in L p (Ω), we finally obtain (︁ )︁ inf I ε,k (u, v, y) = lim I ε,k (u ε,k,m , v ε,k,m , y ε,k,m ) ≥ I ε,k u0ε,k , v0ε,k , y0ε,k . m→∞

(u,v,y)∈Ξ ε,k

Thus, u0ε,k , v0ε,k , y0ε,k is an optimal solution to the problem (2.26)–(2.29). (︀

)︀

2.3 Asymptotic analysis of approximating OCP Our main intention in this section is to show that optimal solutions to the original OCP (2.2)–(2.5) can be attained (in some sense) by optimal solutions to the approximating problems (2.26)–(2.29). With that in mind, we make use of the concept of variational convergence of constrained minimization problems (see [88]) and study the asymptotic behaviour of a family of OCPs (2.26)–(2.29) as ε → 0 and k → ∞. We begin with some auxiliary results concerning the weak compactness in H01 (Ω) of || · ||ε,k -bounded sequences. ′

Lemma 2.2. Let {u ε,k } ε>0 ⊂ L q (Ω) and {v ε,k } ε>0 ⊂ L p (Ω) be arbitrary bounded k∈N k∈N {︀ }︀ sequences of admissible controls with associated states y ε,k ε>0 ⊂ H01 (Ω), i.e. k∈N {︀ }︀ y ε,k = y ε,k (u ε,k , v ε,k ) is a weak solution of (2.27)–(2.28). Then the sequence y ε,k ε>0 k∈N {︀ }︀ is bounded in H01 (Ω). Moreover, each cluster point y of the sequence y ε,k ε>0 with k∈N

respect to the weak convergence in H01 (Ω), satisfies: y ∈ W01,p (Ω). For the proof of this Lemma we refer to [23, Lemma 5.1]. ′

Lemma 2.3. Let {ε i }i∈N , {k i }i∈N , {u i }i∈N ⊂ L q (Ω), and {v i }i∈N ⊂ L p (Ω) be sequences such that ′ ε i → 0, k i → ∞, u i ⇀ u in L q (Ω), v i ⇀ v in L p (Ω), (2.54) where p′ = p/(p − 1) and 2 ≤ p < 2N/(N − 1). Let y = y(u, v) and y i = y ε i ,k i (u i , v i ) be the solutions of (︁ )︁ − div |∇y|p−2 ∇y = v + u in Ω, (2.55) y=0

on ∂Ω

(2.56)

2.3 Asymptotic analysis of approximating OCP | 55

and − div

(︂(︁

)︂ (︁ )︁)︁ p−2 2 ε i + Fk i |∇y|2 ∇y = v i + u i y=0

in

Ω,

on ∂Ω,

(2.57) (2.58)

respectively. Then y i → y in H01 (Ω) as i → ∞, p

(2.59) N

χ Ω\Ω k (y i ) ∇y i → ∇y strongly in L (Ω) , ∫︁ ∫︁ (︁ )︁ p−2 2 |∇y i |2 dx = |∇y|p dx, lim ε i + Fk i (|∇y i |2 )

i→∞

(2.60) (2.61)

Ω

Ω

where Ω k i (y i ) is defined as {︂ Ω k i (y i ) :=

x ∈ Ω : |∇y i (x)| >

√︁

}︂ k2i + 1 .

(2.62)

Proof. For the reader’s convenience, we divide the proof into five steps. Step 1: y i ⇀ y in H01 (Ω). Taking into account the a priori estimate (2.48), we have [︂ ]︂ [︁ ]︁ q−p′ p p−2 p ′ qp (2.63) ||y i ||ε i ,k i ≤ C p ||v i ||L p′ (Ω) + |Ω| ||u i ||L q (Ω) × |Ω| 2p ||y i ||ε i ,k i + ||y i ||ε2i ,k i . It follows from (2.63) that {︂ 2 }︂ 1 ||y i ||ε i ,k i ≤ max C ip , C ip−1 ,

∀ i ∈ N,

(2.64)

)︂ (︁ (︂ )︁ q−p′ p−2 |Ω| 2p + 1 . where C i := C p ||v i ||L p′ (Ω) + |Ω| qp′ ||u i ||L q (Ω) Then from Lemma 2.2 we deduce the existence of a subsequence, denoted in the same way {y i }i∈N ⊂ H01 (Ω) and an element y ∈ W01,p (Ω) such that y i ⇀ y in H01 (Ω). Let us prove that y is the solution of (2.55)-(2.56). With that in mind we fix an arbitrary test function φ ∈ C∞ 0 (Ω) and pass to the limit in the Minty inequality ∫︁

(ε i + Fk i (|∇φ|2 ))

p−2 2

(∇φ, ∇φ − ∇y i )RN dx

Ω

∫︁ ≥ ⟨v i , φ − y i ⟩H −1 (Ω);H 1 (Ω) +

u i (φ − y i ) dx,

0

Ω

as i → ∞. Taking into account that (ε i + Fk i (|∇φ|2 ))

p−2 2

′

∇φ → |∇φ|p−2 ∇φ strongly in L p (Ω)N ,

with p′ = p/(p − 1),

(2.65)

56 | 2 Approximation of OCPs in view of the convergences ∇y i ⇀ ∇y in L2 (Ω)N , u i ⇀ u in L q (Ω), v i → v in H −1 (Ω) ′ (by compactness of the embedding L p (Ω) ˓→ H −1 (Ω)), and y i → y in L2 (Ω), we obtain ∫︁ ∫︁ p−2 lim (ε i + Fk i (|∇φ|2 )) 2 (∇φ, ∇φ)RN dx = |∇φ|p−2 (∇φ, ∇φ)RN dx, i→∞

Ω

Ω

∫︁ lim

i→∞

2

(ε i + Fk i (|∇φ| ))

p−2 2

∫︁ (∇φ, ∇y i )RN dx =

|∇φ|p−2 (∇φ, ∇y)RN dx,

Ω

Ω

∫︁

∫︁ u i (φ − y i ) dx =

lim

i→∞ Ω

u(φ − y) dx, Ω

lim ⟨v i , φ − y i ⟩H −1 (Ω);H 1 (Ω) = ⟨v, φ − y⟩W −1,p′ (Ω);W 1,p (Ω) . 0

i→∞

0

Thus, passing to the limit in relation (2.65) as i → ∞, we arrive at the inequality ∫︁

|∇φ|p−2 (∇φ, ∇φ − ∇y)RN dx ≥ ⟨v, φ − y i ⟩W −1,p′ (Ω);W 1,p (Ω) +

∫︁ u(φ − y) dx

0

Ω

Ω

1,p ∞ for every φ ∈ C∞ 0 (Ω). Finally, from the density of C 0 (Ω) in W 0 (Ω), we infer that this 1,p 1,p inequality holds for every φ ∈ W0 (Ω), and hence y ∈ W0 (Ω) is the solution to the boundary value problem (2.55)–(2.56) in the sense of distributions. Since the solution of (2.55)–(2.56) is unique, the whole sequence {y i }i∈N converges weakly to y = y(u, v) in H01 (Ω).

Step 2: χ Ω\Ω k (y i ) ∇y i ⇀ ∇y in L p (Ω)N . Following the definition of the sets Ω k i (y i ), we obtain ∫︁ ∫︁ |∇y i |p dx |χ Ω\Ω k (y i ) ∇y i |p dx = i

Ω

Ω\Ω k i (y i )

∫︁

(︁

≤

ε i + Fk i (|∇y i |2 )

)︁ p−2 2

|∇y i |2 dx,

Ω\Ω k i (y i ) p

≤ ||y i ||ε i ,k i

by (2.48)

≤

C < +∞,

∀ i ∈ N.

Hence, taking a new subsequence if necessary, we infer the existence of a vectorvalued function g ∈ L p (Ω)N such that χ Ω\Ω k (y i ) ∇y i ⇀ g in L p (Ω)N as i → ∞, i.e. i

∫︁ lim

i→∞ Ω\Ω k i (y i )

∫︁ (g, ∇φ) dx,

(∇y i , ∇φ) dx = Ω

∀φ ∈ C∞ 0 (Ω).

(2.66)

2.3 Asymptotic analysis of approximating OCP | 57

On the other hand, in view of the weak convergence ∇y i ⇀ ∇y in L2 (Ω)N , ∫︁ ∫︁ ∫︁ (∇y i , ∇φ) dx (∇y, ∇φ) dx = lim (∇y i , ∇φ) dx = lim i→∞

Ω

i→∞ Ω\Ω k i (y i )

Ω

∫︁ + lim

i→∞ Ω k i (y i )

(∇y i , ∇φ) dx.

(2.67)

Since ⃒ ⃒ ⎞1/2 ⎛ ⃒ ⃒ ∫︁ ∫︁ √︁ ⃒ ⃒ ⃒ ⃒ ⎟ ⎜ |∇y i |2 dx⎠ ⃒ (∇y i , ∇φ) dx⃒ ≤ ||φ||C1 (Ω) |Ω k i (y i )| ⎝ ⃒ ⃒ ⃒ ⃒Ω k (y i ) Ω k (y i ) i

i

√︁ p ||φ||C1 (Ω) |Ω k i (y i )|||y i ||ε2 ,k ≤ (︀ p−2 )︀ i i ε i + k2i + 1 4 by (2.31),(2.48)

≤

||φ||C1 (Ω)

C k p−1 i

→ 0 as i → ∞,

it follows from (2.66) and (2.67) that ∫︁ ∫︁ g, ∇ φ dx = (∇y, ∇φ) dx, ( )

∀φ ∈ C∞ 0 (Ω).

Ω

Ω

Hence, g = ∇y almost everywhere in Ω and χ Ω\Ω k (y i ) ∇y i ⇀ ∇y in L p (Ω)N holds. Step 3: Proof of (2.61) and χ Ω\Ω k (y i ) ∇y i → ∇y in L p (Ω)N . For each i ∈ N, we have the energy equalities ∫︁ ∫︁ p−2 (ε i + Fk i (|∇y i |2 )) 2 |∇y i |2 dx = ⟨v i , y i ⟩H −1 (Ω);H 1 (Ω) + u i y i dx, 0

Ω

Ω

∫︁

∫︁

p

|∇y| dx = ⟨v, y⟩W −1,p′ (Ω);W 1,p (Ω) +

uy dx.

(2.68)

0

Ω

Ω

H01 (Ω)

From (2.68) and the fact that y i ⇀ y in and v i → v in H −1 (Ω), we deduce ∫︁ p−2 lim (ε i + Fk i (|∇y i |2 )) 2 |∇y i |2 dx i→∞

Ω

⎡

⎤ ∫︁

= lim ⎣⟨v i , y i ⟩H −1 (Ω);H 1 (Ω) +

u i y i dx⎦

0

i→∞

Ω

∫︁ uy dx

= ⟨v, y⟩W −1,p′ (Ω);W 1,p (Ω) +

by (2.68)2

∫︁

=

0

Ω

Ω

|∇y|p dx.

(2.69)

58 | 2 Approximation of OCPs Thus, property (2.61) holds true. Moreover, we have ∫︁ ∫︁ (︁ )︁ p−2 2 |∇y|p dx = lim ε i + Fk i (|∇y i |2 ) |∇y i |2 dx i→∞

Ω

Ω

∫︁

(︁

≥ lim sup i→∞

ε i + Fk i (|∇y i |2 )

)︁ p−2 2

|∇y i |2 dx

Ω\Ω k i (y i ) by (2.62)

≥

∫︁

(︁

lim sup i→∞

ε i + |∇y i |2

)︁ p−2 2

|∇y i |2 dx

Ω\Ω k i (y i )

∫︁

χ Ω\Ω k (y i ) |∇y i | dx ≥ lim inf

≥ lim sup i→∞

∫︁

p

i→∞

i

Ω

χ Ω\Ω k (y i ) |∇y i |p dx. i

(2.70)

Ω

Since χ Ω\Ω k (y i ) ∇y i ⇀ ∇y in L p (Ω)N , it follows from (2.70) that i

∫︁

|∇y|p dx ≥ lim sup i→∞

Ω

∫︁

χ Ω\Ω k (y i ) |∇y i |p dx ≥ lim inf

∫︁

i→∞

i

Ω

χ Ω\Ω k (y i ) |∇y i |p dx i

Ω p

p

∫︁

= lim inf ||χ Ω\Ω k (y i ) ∇y i ||L p (Ω)N ≥ ||∇y||L p (Ω)N = i→∞

i

|∇y|p dx.

Ω

It remains to note that the weak convergence χ Ω\Ω k (y i ) ∇y i ⇀ ∇y in L p (Ω)N i

and the convergence of their norms ||χ Ω\Ω k (y i ) ∇y i ||L p (Ω)N → ||∇y||L p (Ω)N i

imply the strong convergence χ Ω\Ω k (y i ) ∇y i → ∇y in L p (Ω)N . i

Step 4: y i → y in

H01 (Ω).

From (2.60) and (2.70) we see that

∫︁ lim

i→∞ Ω k i (y i )

(︁

ε i + Fk i (|∇y i |2 )

)︁ p−2 2

|∇y i |2 dx = 0.

Then, utilizing (2.71), we deduce ∫︁ ∫︁ p−2 lim |∇y i |2 dx ≤ lim (ε i + Fk (|∇y i |2 )) 2 |∇y i |2 dx = 0. i→∞ Ω k (y i )

i→∞ Ω k (y i )

Now, combining this estimate and (2.60), we finally conclude that ∇y i = χ Ω k (y i ) ∇y i + χ Ω\Ω k (y i ) ∇y i → ∇y strongly in L2 (Ω)N .

(2.71)

2.3 Asymptotic analysis of approximating OCP | 59

The following noteworthy property is crucial for our further analysis. {︀ )︀ Theorem 2.6. Let (u0ε,k , v0ε,k , y0ε,k ) } ε>0 be an arbitrary sequence of optimal solutions k∈N

to the approximating problems (2.26)–(2.29). Assume that 2 ≤ p < 2N/(N − 1) and the parameter ε varies within a strictly decreasing sequence {ε k }k∈N of positive real numbers such that )︁ (︁ ′ lim k ε pk = 0. (2.72) k→∞

{︀ )︀ ′ Then the sequence (u0ε,k , v0ε,k , y0ε,k ) } ε>0 is bounded in L q (Ω) × L p (Ω) × H01 (Ω) and any k∈N

′

its cluster triple (u0 , v0 , y0 ) with respect to the weak topology of L q (Ω) × L p (Ω) × H01 (Ω) is such that v0 = f (y0 ) and (u0 , y0 ) is a feasible solution of the OCP (2.2)–(2.5). Proof. Let us take an arbitrary function ̃︀ y ∈ C∞ u := −∆ p ̃︀ y − f (̃︀ y). Then 0 (Ω) and put ̃︀ q 1 2 ̃︀ u ∈ L (Ω), ̃︀ y ∈ H0 (Ω), and f (̃︀ y ∈ Y ⊂ H01 (Ω) is a weak solution to y) ∈ L (Ω). Hence, ̃︀ the boundary value problem (2.3)–(2.4) for given ̃︀ u, and, therefore, (̃︀ u, ̃︀ y) ∈ Ξ. Having set ̃︀ v ε,k := −∆ ε,k,p (̃︀ y) + ∆ p ̃︀ y + f (̃︀ y), (2.73) it is easy to check that ̃︀ v ε,k ∈ L2 (Ω) and (̃︀ u, ̃︀ v ε,k , ̃︀ y) ∈ Ξ ε,k . Let us show that, for sufficiently small ε > 0 and k ∈ N large enough, there exists a constant C > 0 such that (︀ )︀ p′ ||̃︀ v ε,k − T ε f (̃︀ y) || p′

L (Ω)

′

≤ Cε p .

(2.74)

Indeed, using the fact that ̃︀ y ∈ C∞ 0 (Ω), we see that (︀ )︀ f (̃︀ y) = T ε f (̃︀ y)

for sufficiently small ε > 0 and

2

Fk (|∇̃︀ y| ) = |∇̃︀ y|2

for k ∈ N large enough.

In what follows, we make use of the following notation (︁ )︁ ∆∞ (̃︀ y) = ∇̃︀ y, D2 (̃︀ y)∇̃︀ y . Then, for indicated ε and k, by smoothness of the function ̃︀ y, we have )︂ (︂ (︁ )︁ p−2 (︀ )︀ 2 ̃︀ ∇̃︀ y v ε,k − T ε f (̃︀ y) = div |∇̃︀ y|p−2 ∇̃︀ y − ε + |∇̃︀ y|2 = |∇̃︀ y|p−2 ∆̃︀ y + (p − 2)|∇̃︀ y|p−4 ∆∞ (̃︀ y) (︁ )︁ p−2 (︁ )︁ p−4 2 2 − ε + |∇̃︀ y|2 ∆̃︀ y − (p − 2) ε + |∇̃︀ y|2 ∆∞ (̃︀ y) ]︂ [︂(︁ )︁ p−2 2 − |∇̃︀ y|p−2 ∆̃︀ y y |2 = − ε + |∇̃︀ − (p − 2)

[︂(︁

2

ε + |∇̃︀ y|

)︁ p−4 2

− |∇̃︀ y|

p−4

]︂ ∆∞ (̃︀ y).

(2.75)

60 | 2 Approximation of OCPs (︀ )︀ p−2 Setting Φ(ε) := ε + |∇̃︀ y|2 2 , we see that Φ ∈ C([0, 1]) and Φ ∈ C1 (0, 1) because p ≥ 2. Hence, by the mean value theorem, we deduce that (︁

ε + |∇̃︀ y |2

)︁ p−2 2

− |∇̃︀ y|p−2 = Φ(ε) − Φ(0) = εΦ′ (ε0 ) =

)︁ p − 2 (︁ ε0 + |∇̃︀ y|2 2

p−4 2

ε,

(2.76)

where ε0 ∈ (0, 1). The similar inference can be done for the function Ψ(ε) := (︀ )︀ p−4 ε + |∇̃︀ y|2 2 provided p ≥ 4. Indeed, in this case we have (︁

ε + |∇̃︀ y |2

)︁ p−4 2

− |∇̃︀ y|p−4 = Ψ(ε) − Ψ(0) = εΨ ′ (ε1 ) =

)︁ p − 4 (︁ ε1 + |∇̃︀ y|2 2

p−6 2

ε,

(2.77)

for some ε1 ∈ (0, 1). Utilizing (2.76)-(2.77), and using the fact that |∇̃︀ y|, ∆∞ (̃︀ y), ∆̃︀ y ∈ L∞ (Ω),

we can deduce from (2.75) existence of a positive constant C1 , depending only on ε0 , ε1 , ̃︀ y, and p, such that the following estimate ∫︁ (︀ )︀ r (︀ (︀ )︀)︀r ̃︀ ||̃︀ v ε,k − T ε f (̃︀ y) ||L r (Ω) = v ε,k − T ε f (̃︀ dx ≤ C1 ε r (2.78) y) Ω

holds true for all r ∈ [1, ∞), ε small enough, and k large enough, provided p ≥ 4. In the case if p ∈ [2, 4), we have ⃒(︁ ⃒ ⃒ ⃒ (︁ )︁ 4−p )︁ p−4 ⃒ ⃒ ⃒ 2 2 p−4 ⃒ 4−p ⃒ 2 2 ⃒ ̃︀ ̃︀ ̃︀ ̃︀ ̃︀ − |∇y| ⃒∆∞ (y) = ⃒ ε + |∇y| − |∇y| ⃒ ⃒ ε + |∇y| )︂ (︂ |∇̃︀ y|p−2 ∇̃︀ y ∇̃︀ y 2 ̃︀ × (︀ , D ( y) )︀ 4−p |∇̃︀ y| |∇̃︀ y| ε + |∇̃︀ y |2 2 ⃒ ⃒(︁ )︁p−2− 4−p )︁ 4−p ⃒ (︁ ⃒ 2 2 y|2 − |∇̃︀ y|4−p ⃒⃒ ε + |∇̃︀ ||D2 (̃︀ y)|| y|2 ≤ ⃒⃒ ε + |∇̃︀ (︁ )︁ 32 (p−2) )︁ 2−p by (2.77) 4 − p (︁ 2 ||D2 (̃︀ y)|| ε 1 + |∇̃︀ y |2 ε2 + |∇̃︀ y |2 ≤ 2 Combining this estimate with (2.76) and using the same arguments as we did it before, we also can deduce the existence of a constant C2 , depending only on ε0 , ε2 , ̃︀ y, and p ∈ [2, 4), such that ∫︁ (︀ )︀ r (︀ (︀ )︀)︀r ̃︀ ̃︀ ̃︀ ||v ε,k − T ε f (y) ||L r (Ω) = v ε,k − T ε f (̃︀ dx ≤ C2 ε r , ∀ r ≥ 1. (2.79) y) Ω

2.3 Asymptotic analysis of approximating OCP |

Hence, the estimate (2.74) is valid. Taking this fact into account, we see that (︁ )︁ )︀ (︀ u, ̃︀ v ε,k , ̃︀ y I ε,k (u, v, y) ≤ I ε,k ̃︀ inf I ε,k u0ε,k , v0ε,k , y0ε,k = (u,v,y)∈Ξ ε,k ∫︁ ∫︁ ∫︁ ′ ′ 1 1 k α |̃︀ v ε,k |p dx, ≤ |̃︀ y − y d |2 dx + ′ Cε p + |̃︀ u|q dx + ′ 2 q p p Ω

61

(2.80)

Ω

Ω

where ∫︁

′

|̃︀ v ε,k |p dx

by (2.73)

≤

⎤ ⎡ ∫︁ ∫︁ ′ ′ ⃒ ⃒ ⃒ ⃒ p p y)⃒ dx⎦ y⃒ dx + ⃒f (̃︀ y) + ∆ p ̃︀ 2p ⎣ ⃒−∆ ε,k,p (̃︀ ′

Ω

Ω

Ω p′

′

p′

p ≤ Cε + 2 ||f (̃︀ y)||L∞ (Ω) |Ω|.

(2.81)

Utilizing the estimates (2.80) and(2.81), and assuming that the sequence {ε} = {ε k }k∈N satisfies the condition (2.72), we obtain (︁ )︁ sup I ε,k u0ε,k , v0ε,k , y0ε,k ε>0 k∈N

= sup ε>0 k∈N

[︁ 1 ∫︁ 2 ∫︁

|y0ε,k − y d |2 dx +

k p′

∫︁

′

|v0ε,k − T ε (f (y0ε,k ))|p dx

Ω

Ω

∫︁ ]︁ ′ α |v0ε,k |p dx |u0ε,k |q dx + ′ p Ω Ω ∫︁ ∫︁ [︁ 1 ∫︁ ]︁ ′ k ′ 1 α 0 2 |y ε,k − y d | dx + ′ ε p + |u0ε,k |q dx + ′ ≤ sup |v0ε,k |p dx 2 q p p ε>0 k∈N Ω Ω Ω ∫︁ ∫︁ 1 α p′ 1 p′ 2 q y)||L∞ (Ω) |Ω| < +∞ |̃︀ y − y d | dx + |̃︀ u| dx + ′ 2 ||f (̃︀ ≤ 2 q p 1 + q

Ω

Ω

and, as a consequence, we can deduce the existence of a constant C* > 0 independent on ε and k such that q

sup ||v0ε,k ||L p′ (Ω) < C* , sup ||u0ε,k ||L q (Ω) < C* , ε>0 k∈N

ε>0 k∈N

and sup ||v0ε,k − T ε (f (y0ε,k ))|| ε>0 k∈N

p′ L p′ (Ω)

< C* k−1 .

(2.82)

Then the estimate (2.64) implies that sup ||y0ε,k ||ε i ,k i ε>0 k∈N

(︂

*

{︂

2 p

1 p−1

}︂

≤ max C* , C*

with C* = C p C + |Ω|

q−p′ qp′

C

*

)︂ (︁

p−2

(2.83) )︁

|Ω| 2p + 1 ,

62 | 2 Approximation of OCPs

i.e., in view of (2.52), we can suppose that the sequence

{︀

y0ε,k

}︀

H01 (Ω) {︀(︀

in lems

ε>0 k∈N

is bounded

and, therefore, the sequence of solutions to the approximating prob)︀}︀ u0ε,k , v0ε,k , y0ε,k ) ε>0 is compact with respect to the weak convergence in k∈N

′

L q (Ω) × L p (Ω) × H01 (Ω). Let (u0 , v0 , y0 ) be its any cluster triplet, i.e. up to a subsequence, we have ′

y0ε,k ⇀ y0 in H01 (Ω), u0ε,k ⇀ u0 in L q (Ω), v0ε,k ⇀ v0 in L p (Ω).

(2.84)

Then Lemma 2.3 implies that y0 = y(u0 , v0 ) is a weak solution to the problem (2.55)– (2.56). Moreover, for any φ ∈ C∞ 0 (Ω) we have ∫︁ ∫︁ ⃒ ⃒ ⃒ ∫︁ ⃒ ∫︁ ⃒ ⃒ ⃒ ⃒ 0 0 0 0 ⃒ v φ dx − f (y )φ dx⃒ ≤ ⃒ v ε,k φ dx − v φ dx⃒ Ω

Ω

Ω

Ω

∫︁ ⃒ ⃒ ∫︁ ⃒ ⃒ 0 0 ⃒ v ε,k φ dx − f (y )φ dx⃒

+

Ω

Ω

⃒ ∫︁ ⃒ ⃒ ∫︁ (︁ (︁ )︁⃒ )︁ ⃒ ⃒ ⃒ ⃒ v0ε,k − v0 φ dx⃒ + ⃒v0ε,k − T ε f (y0ε,k ) ⃒ |φ| dx ≤⃒ Ω

Ω

∫︁ ⃒ (︁ ⃒ )︁ ⃒ ⃒ + ⃒T ε f (y0ε,k ) − f (y0ε,k )⃒ |φ| dx Ω

∫︁ ⃒ ⃒ ⃒ ⃒ + ⃒f (y0ε,k ) − f (y0 )⃒ |φ| dx = J0 + J1 + J2 + J3 ,

(2.85)

Ω

where ⃒ by (2.84)3 ⃒ ∫︁ (︁ )︁ ⃒ ⃒ v0ε,k − v0 φ dx⃒ → 0, J0 = ⃒

(2.86)

Ω

)︁ (︁ J1 ≤ ||v0ε,k − T ε f (y0ε,k ) ||L p′ (Ω) ||φ||L p (Ω)

by (2.82)

(︁ )︁ J2 ≤ ||T ε f (y0ε,k ) − f (y0ε,k )||L1 (Ω) ||φ||L∞ (Ω)

→

0,

(2.87)

by definition of T ε

→

0,

(2.88)

J3 ≤ ||f (y0ε,k ) − f (y0 )||L1 (Ω) ||φ||L∞ (Ω) , as ε → 0 and k → ∞. Let us show that f (y0ε,k ) → f (y0 ) strongly in L1 (Ω) as ε → 0 and k → ∞.

(2.89)

With that in mind, we note that by compactness of the injection H01 (Ω) ˓→ L2 (Ω), we {︀ }︀ can deduce the existence of a subsequence of y0ε,k ε>0 , denoted in the same way, k∈N

such that f (y0ε,k ) → f (y0 ) almost everywhere in Ω. So, in order to conclude (2.89), it {︀ (︀ )︀}︀ remains to establish the equi-integrability on Ω of the sequence f y0ε,k ε>0 . For this k∈N

2.3 Asymptotic analysis of approximating OCP |

63

purpose, we make use of the following relation, coming from energy identity (2.47), ∫︁

(︁ )︁ p y0ε,k f y0ε,k dx = ||y0ε,k ||ε,k

Ω

⟨ (︁ )︁ ⟩ − v0ε,k − f y0ε,k , y0ε,k

∫︁ H −1 (Ω);H01 (Ω)

−

u0ε,k y0ε,k dx.

(2.90)

Ω

Then ||y0ε,k ||ε,k

by (2.83)

{︂ 2 }︂ 1 A1p := max C*p , C*p−1 ,

u0ε,k y0ε,k dx

by (2.42)

[︁ p−2 q−p′ p ]︁ 2 C p |Ω| qp′ ||u0ε,k ||L q (Ω) |Ω| 2p ||y0ε,k ||ε,k + ||y0ε,k ||ε,k

p

∫︁

≤

≤

(2.91)

Ω by (2.91), (2.82)

≤

[︁ p−2 q−p′ p ]︁ A2 := C p |Ω| qp′ C* |Ω| 2p A1 + A12

(2.92)

and ⃒⟨︀ ⃒ (︁ )︁ ⃒ 0 ⃒ 0 0 ⟩︀ ⃒ v ε,k −f y ε,k , y ε,k H −1 (Ω);H 1 (Ω) ⃒ 0 ⃒ ⃒⟨ (︁ (︁ )︁)︁ ⟩ ⃒ ⃒ ≤ ⃒ v0ε,k − T ε f y0ε,k , y0ε,k ⃒ H −1 (Ω);H01 (Ω) ⃒ ⟨ (︁ )︁ (︁ (︁ )︁)︁ ⟩ ⃒ + ⃒ f y0ε,k − T ε f y0ε,k , y0ε,k

H −1 (Ω);H01 (Ω)

⃒ ⃒ ⃒

by (2.41)

[︁ p−2 (︁ (︁ )︁)︁ p ]︁ 2 ≤ C p ||v0ε,k − T ε f y0ε,k ||L p′ (Ω) |Ω| 2p ||y0ε,k ||ε,k + ||y0ε,k ||ε,k [︁ p−2 (︁ )︁ (︁ (︁ )︁)︁ p ]︁ 2 + C p ||f y0ε,k − T ε f y0ε,k ||L p′ (Ω) |Ω| 2p ||y0ε,k ||ε,k + ||y0ε,k ||ε,k )︁ [︁ p−2 p ]︁ (︁ ̃︀ , (2.93) ≤ C p |Ω| 2p A1 + A12 εC* + ε C where (︁ (︁ )︁)︁ ||v0ε,k − T ε f y0ε,k ||L p′ (Ω) ≤ εC* by (2.72) and (2.82), (︁ )︁ (︁ (︁ )︁)︁ ̃︀ by (2.78) and (2.88). ||f y0ε,k − T ε f y0ε,k ||L p′ (Ω) ≤ ε C Utilizing the estimates (2.93), (2.92), (2.91), it follows from (2.90) that there exists a constant M > 0 independent of ε and k such that ⃒ ⃒ ⃒∫︁ (︁ )︁ ⃒⃒ ⃒ (2.94) sup ⃒⃒ y0ε,k f y0ε,k dx⃒⃒ ≤ M. ε>0 ⃒ ⃒ k∈N Ω

We recall that a sequence {f k }k∈N is called equi-integrable on Ω if for any δ > 0, there ∫︀ is a τ = τ(δ) such that S |f k | dx < δ for every measurable subset S ⊂ Ω of Lebesgue

64 | 2 Approximation of OCPs )︀}︀ {︀ (︀ measure |S| < τ. So, in order to show that the sequence f y0ε,k ε>0 is equi-integrable k∈N

on Ω, we take m > 0 such that m > 2Mδ−1 .

(2.95)

We also set τ = δ/(2f (m)). Then for every measurable set S ⊂ Ω with |S| < τ, we have ∫︁ ∫︁ ∫︁ f (y0ε,k ) dx f (y0ε,k ) dx + f (y0ε,k ) dx = S

{︁

≤

x∈S : y0ε,k (x)>m

1 m {︁

}︁

{︁

∫︁ x∈S : y0ε,k (x)>m

}︁

∫︁ f (m) dx x∈S : y0ε,k (x)≤m

}︁

y0ε,k f (y0ε,k ) dx

+ {︁

x∈S : y0ε,k (x)≤m

by (2.94)

≤

by (2.95) δ M δ + f (m)|S| ≤ + . m 2 2

}︁

As a result, the assertion (2.89) is a direct consequence of Lebesgue’s Convergence Theorem. Thus, J3 ≤ ||f (y0ε,k ) − f (y0 )||L1 (Ω) ||φ||L∞ (Ω)

by (2.89)

→

0 as ε → 0 and k → ∞.

Combining this fact with properties (2.86)–(2.88), we deduce from (2.85) that v0 = f (y0 ) almost everywhere on Ω. Hence, by (2.84), we have ′

f (y0 ) ∈ L p (Ω) and

′

v0ε,k ⇀ f (y0 ) in L p (Ω).

Thus, (u0 , y0 ) a feasible solution of the OCP (2.2)–(2.5). The proof is complete. We are now in a position to show that optimal solutions to the approximating OCP (2.26)–(2.29) lead in the limit to optimal pairs of the original OCP (2.2)–(2.5). {︀ )︀ Theorem 2.7. Let 2 ≤ p < 2N/(N − 1) and let (u0ε,k , v0ε,k , y0ε,k ) } ε>0 be an arbitrary sek∈N

quence of optimal solutions to the approximating problems (2.26)–(2.29), where the parameter ε varies within a strictly decreasing sequence {ε k }k∈N of positive real numbers ′

satisfying condition (2.72). Then, this sequence is bounded in L q (Ω) × L p (Ω) × H01 (Ω) and any its cluster point (u0 , v0 , y0 ) with respect to the weak topology is such that v0 = f (y0 ) and (u0 , y0 ) is solution of the OCP (2.2)–(2.5). Moreover, if for one subsequence we have ′ y0ε,k ⇀ y0 in H01 (Ω), u0ε,k ⇀ u0 in L q (Ω), and v0ε,k ⇀ v0 in L p (Ω), then the following

2.3 Asymptotic analysis of approximating OCP |

65

properties hold u0ε,k → u0 in L q (Ω),

y0ε,k → y0 in H01 (Ω), k p′

′

v0ε,k → f (y0 ) in L p (Ω),

∫︁

(2.96)

′

|v0ε,k − T ε (f (y0ε,k ))|p dx → 0,

(2.97)

Ω

χ Ω\Ω k (y0 ) ∇y0ε,k → ∇y0 strongly in L p (Ω)N , ε,k ∫︁ ∫︁ (︁ )︁ p−2 2 |∇y0ε,k |2 dx = |∇y0 |p dx, lim ε + Fk (|∇y0ε,k |2 ) ε→0 k→∞

(2.98) (2.99)

Ω

Ω

lim I ε,k (u0ε,k , v0ε,k , y0ε,k ) ε→0 k→∞

=

lim J(u0ε,k , y0ε,k ) ε→0

= J(u0 , y0 ).

(2.100)

k→∞

{︀ )︀ Proof. The boundedness of the sequence (u0ε,k , v0ε,k , y0ε,k ) } ε>0 has been proved in k∈N

Theorem 2.6. Let (u0 , v0 , y0 ) be its any cluster point with respect to the weak topology of ′ L q (Ω) × L p (Ω) × H01 (Ω). Let us take a subsequence, denoted in the same way, satisfying the property (2.84). Then y0ε,k ⇀ y0 in H01 (Ω), u0ε,k ⇀ u0 in L q (Ω),

(2.101)

′

v0ε,k ⇀ v0 in L p (Ω) as ε → 0 and k → ∞,

and from Lemma 2.3 we get that y0ε,k → y0 strongly in H01 (Ω). As for the convergences (2.98) and (2.99), they follow from (2.60) and (2.61), respectively. Moreover, Theorem 2.6 ′ implies that v0 = f (y0 ), f (y0 ) ∈ L p (Ω), and y0 is a weak solution of (2.3)-(2.4) corresponding to u = u0 . Let us prove that (u0 , y0 ) is an optimal pair to the problem (2.2)–(2.5). Given an arbitrary feasible (u, y) ∈ Ξ, we define u ε,k = u, v ε,k = T ε (f (y)), and y ε,k as the ′

solution of the boundary value problem (2.27)-(2.28). Since v ε,k ∈ L p (Ω), it follows that (u ε,k , v ε,k , y ε,k ) ∈ Ξ ε,k . By definition of the cut-off operator T ε , we have v ε,k → f (y)

′

strongly in L p (Ω) as ε → 0 and k → ∞.

(2.102)

Then Lemma 2.3 implies the existence of an element y* ∈ W01,p (Ω) such that y ε,k → y* in H01 (Ω) and y* satisfies the equality (in the sense of distributions) (︁ )︁ − div |∇y* |p−2 ∇y* = f (y) + u in Ω. On the other hand, the condition (u, y) ∈ Ξ leads to the relation (︁ )︁ − div |∇y|p−2 ∇y = f (y) + u in Ω. Hence, (︁ )︁ (︁ )︁ − div |∇y* |p−2 ∇y* + div |∇y|p−2 ∇y = 0 and, therefore, (︁ ⟨ )︁ (︁ )︁ ⟩ − div |∇y* |p−2 ∇y* + div |∇y|p−2 ∇y0 , y* − y

W −1,p′ (Ω);W01,p (Ω)

= 0.

66 | 2 Approximation of OCPs Since the p-Laplace operator is strictly monotone, it follows that y* = y as element of W01,p (Ω). Thus, from (2.102) and Lemma 2.3 we get that Ξ ε,k ∋ (u ε,k , v ε,k , y ε,k ) −→ (u, f (y) , y) strongly in L q (Ω) × L p′ (Ω) × H01 (Ω).

(2.103)

Further, we make use of the following observation. Since f ∈ C loc (R), it follows from (2.30) and (2.102) that ′

v ε,k − T ε (f (y ε,k )) −→ 0 strongly in L p (Ω) as ε → 0 and k → ∞. Hence, there exists a mapping ε ↦→ k(ε), increasing to +∞ and arguably depending on y, such that (see Section 1.2.2 in [7]) ⎡ ⎤ ⎡ ⎤ ∫︁ ∫︁ ′ ′ lim ⎣ |v ε,k − T ε (f (y ε,k ))|p dx⎦ = lim ⎣ |v ε,k(ε) − T ε (f (y ε,k(ε) ))|p dx⎦ = 0. (2.104) ε→0 k→∞

ε→0

Ω

Ω

Utilizing (2.103) and (2.104), we have ∫︁ ∫︁ ]︁ [︁ 1 ∫︁ ′ 1 α |v ε,k |p dx |y ε,k − y d |2 dx + |u ε,k |q dx + ′ lim ε→0 2 q p k→∞ Ω Ω Ω ∫︁ ∫︁ ∫︁ ]︁ [︁ 1 ′ 1 α 2 |y ε,k(ε) − y d | dx + |u ε,k(ε) |q dx + ′ = lim |v ε,k(ε) |p dx q ε→0 2 p Ω Ω Ω ∫︁ ∫︁ ∫︁ 1 1 α p′ 2 q = |f (y)| dx. (2.105) |y − y d | dx + + |u| dx + ′ 2 q p Ω

Ω

Ω

Since ⎤ ∫︁ ′ k p |v ε,k(ε) − T ε (f (y ε,k(ε) ))| dx⎦ 0 ≤ lim sup ⎣ ′ p ε→0 k→∞ Ω ⎤ ⎡ ∫︁ ′ k ≤ lim sup ⎣ ′ lim sup |v ε,k(ε) − T ε (f (y ε,k(ε) ))|p dx⎦ = 0, p ε→0 k→∞ ⎡

(2.106)

Ω

it follows from (2.105) and (2.106) that lim I ε,k (u ε,k(ε) , v ε,k(ε) , y ε,k(ε) ) ε→0 k→∞

[︂ ]︂ = lim sup lim sup I ε,k (u ε,k(ε) , v ε,k(ε) , y ε,k(ε) ) = J(u, y). k→∞

ε→0

(2.107)

2.4 Optimality conditions for approximating OCP |

67

Now, using (2.101), (2.82), (2.107), and the fact that the triplet (u0ε,k , v0ε,k , y0ε,k ) is a solution of (2.26)–(2.29), we get J(u0 , y0 ) ≤ lim inf I ε,k (u0ε,k , v0ε,k , y0ε,k ) ≤ lim inf I ε,k (u0ε,k(ε) , v0ε,k(ε) , y0ε,k(ε) ) ε→0 k→∞

ε→0 k→∞

≤ lim inf I ε,k (u ε,k(ε) , v ε,k(ε) , y ε,k(ε) ) ε→0 k→∞

≤ lim sup I ε,k (u ε,k(ε) , v ε,k(ε) , y ε,k(ε) ) = J(u, y). ε→0 k→∞

Since (u, y) is an arbitrary pair in Ξ, this implies that (u0 , y0 ) is a solution of the original optimal control problem (2.2)–(2.5). Moreover, taking (u, y) = (u0 , y0 ) in the above inequalities, the relations (2.100) is proved. Finally, (2.96)2 and (2.97) are the direct consequences of (2.100) and the convergence properties (2.101) established before.

2.4 Optimality conditions for approximating OCP The aim of this section is to derive the optimality system for approximating optimal control problem (2.26)–(2.29). With that in mind, we assume that the mapping F : R → [0, +∞) satisfies condition F ∈ C2loc (R) and begin with investigation of differentiability of the mapping (u, v) ↦→ y ε,k (u, v). It is well known that in the case ε = 0 and k = ∞ this mapping is not necessarily Gâteaux differentiable even if f (y) ≡ 0. Indeed, let us consider the following boundary value problem (︁ )︁ − div |∇y|p−2 ∇y = u in Ω, y=0

on ∂Ω,

where p > 2 and Ω is the unite open ball in R N centered at the origin, Ω = B(0, 1). It is easy to check that the states associated to u0 (x) = 0, u1 (x) = −N, and u t (x) := u0 (x) + tu1 (x)(x) = −tN, for each t > 0, are )︁ p 1 p − 1 (︁ p−1 |x| − 1 , and y t (x) = t p−1 y1 (x), y0 (x) = 0, y1 (x) = p respectively. Then the mapping u ↦→ y(u) is not Gâteaux differentiable at u = u0 , because the sequence }︁ {︁ y − y t 0 , with α = (2 − p)/(p − 1) = t α y1 (x) t t>0 does not converge as t → 0, if p > 2. So, in order to derive an optimality system to the original optimal control problem (2.2)–(2.5) and provide its rigour mathematical substantiation, a direct application of the implicit function theorem or Ioffe–Tikhomirov theorem looks rather questionable. On the other hand, Theorem 2.7 reveals another way to characterize the optimal pairs

68 | 2 Approximation of OCPs

to the problem (2.2)–(2.5). Namely, we can do it deriving an optimality system for the approximating problem (2.26)–(2.29) and studying then its asymptotic behaviour as ε → 0 and k → ∞. We know that the boundary value problem (2.27)-(2.28) has a unique solution ′ ′ y ε,k ∈ H01 (Ω) for every u ∈ L q (Ω) and v ∈ L p (Ω). Let G ε,k : L q (Ω) × L p (Ω) −→ H01 (Ω) be the mapping defined by G ε,k (u, v) = y ε,k (u, v), where y ε,k (u, v) solution of (2.27)-(2.28) associated to u and v. Let SNsym be the set of all N × N symmetric matrices. ′

Theorem 2.8. The mapping G ε,k is of the class C1 and for any u ∈ L q (Ω), v ∈ L p (Ω), ′

h u ∈ L q (Ω), and h v ∈ L p (Ω) the element z(h u , h v ) = D u G ε,k (u, v) [h u ] + D v G ε,k (u, v) [h v ] is the unique solution in H01 (Ω) of the equation (︀ )︀ − div ρ ε,k (y ε,k )Aε,k (y ε,k )∇z = h u + h v ,

(2.108)

where y ε,k = G ε,k (u, v) and ⎡ (︁

)︁

Aε,k (y) = I N + (p − 2)Fk′ |∇y|2 ⎣ √︁

⎤ ∇y

∇y

⎦ (︀ (︀ )︀ ⊗ √︁ )︀ , ε + Fk |∇y|2 ε + Fk |∇y|2

(︁ )︁ p−2 ρ ε,k (y) = (ε + Fk |∇y|2 ) 2 .

(2.109) (2.110)

Proof. We apply the implicit function theorem. To this end we define the function ′ F : H01 (Ω) × L q (Ω) × L p (Ω) −→ H −1 (Ω) by F(y, u, v) = −∆ ε,k,p (y) − u − v. It is immediate that F is of class C1 . Moreover the partial derivative H01 (Ω)

∂F ∂y (y, u, v)

:

−1

−→ H (Ω) is an isomorphism. Indeed, it is easy to see that (︀ )︀ ∂F (y, u, v) [z] = − div ρ ε,k (y ε,k )Aε,k (y ε,k )∇z , ∂y

where the matrix Aε,k (y) and the scalar function ρ ε,k (y) are given by (2.109)–(2.110) and possess the following properties: Aε,k (y) ∈ L∞ (Ω; SNsym ); *

||Aε,k (y)||L∞ (Ω;SNsym ) ≤ 1 + (p − 2)δ , p−2 2

(2.111) ∀ ε > 0 and ∀ k ∈ N; p−2 2

≤ ρ ε,k (y) ≤ (ε + k2 + 1) a.e. in Ω; ε (︁ )︁ (︀ )︀ |η|2 ≤ η, Aε,k (y)η RN ≤ 1 + (p − 2)δ* |η|2 a.e. in Ω, ∀ η ∈ RN .

(2.112) (2.113) (2.114)

2.4 Optimality conditions for approximating OCP |

69

We note that properties (2.111)–(2.113) immediately follow from (2.109)–(2.110) and definition of the C1 (R+ )-function Fk : R+ → R+ . To prove the property (2.114), it is enough to take into account the following chain of estimates ⎛ ⎞2 (︁ )︁ ∇ y ⎠ |η|2 ≤ (η, Iη)RN ≤ (η, Iη)RN + (p − 2)Fk′ |∇y|2 ⎝ √︁ )︀ , η (︀ 2 ε + Fk |∇y| RN ⃒2 ⃒ ⃒ ⃒ (︁ )︁ ⃒ ⃒ (︀ )︀ ∇y 2 ⃒ = η, Aε,k (y)η RN ≤ |η|2 + (p − 2)Fk′ |∇y|2 ⃒⃒ √︁ (︀ )︀ ⃒ |η| ⃒ ε + Fk |∇y|2 ⃒ (︁ )︁ ≤ 1 + (p − 2)δ* |η|2 a.e. in Ω, (︀ )︀ because Fk′ |∇y|2 = 0 a.e. on the set {︁ }︁ √ Ω k (y) := x ∈ Ω : |∇y(x)| > k2 + 1 . As a result, the isomorphism of the mapping ∂F (y, u, v) : H01 (Ω) −→ H −1 (Ω) ∂y is a direct consequence of estimates (2.113)–(2.114) and the Lax-Milgram theorem. ′ In addition, for every u ∈ L q (Ω), v ∈ L p (Ω), and y ε,k = G ε,k (u, v), we have that F(y ε,k , u, v) = 0. Hence, by application of the implicit function theorem, we deduce ′

that for any (u0 , v0 ) ∈ L q (Ω) × L p (Ω) there exists a neighborhood U × V of (u0 , v0 ) in ′ L q (Ω) × L p (Ω) and a mapping g : U × V −→ H01 (Ω) of class C1 such that F(g(u, v), u, v) = 0 ∀ (u, v) ∈ U × V. The mapping g obviously coincides with G ε,k , which proves that G ε,k is of class C1 and the expression of the derivative follows from the equation [︁ ]︁ ∂F (y ε,k , u, v) D u G ε,k (u, v) [h u ] + D v G ε,k (u, v) [h v ] ∂y ∂F ∂F + (y , u, v) [h u ] + (y , u, v) [h v ] = 0. ∂u ε,k ∂v ε,k Now we observe that the problem (2.26)–(2.29) can be written in the form ∫︁ 1 |y ε,k (u, v) − y d |2 dx J ε,k (u, v) = Minimize 2 (u,v)∈L q (Ω)×L p′ (Ω) Ω ∫︁ ′ k + ′ |v − T ε (f (y ε,k (u, v)))|p dx p Ω ∫︁ ∫︁ ′ 1 α + |u|q dx + ′ |v|p dx. q p Ω

Ω

(2.115)

70 | 2 Approximation of OCPs

In the functional J ε,k we distinguish two terms J ε,k (u, v) = F ε,k (u, v) + j(u, v) with ∫︁ ∫︁ ′ 1 k F ε,k (u, v) = |v − T ε (f (y ε,k (u, v)))|p dx |y ε,k (u, v) − y d |2 dx + ′ 2 p Ω

Ω

and j(u, v) =

1 q

∫︁

|u|q dx +

α p′

∫︁

′

|v|p dx.

Ω

Ω

Now we make use of the Stampacchia’s Theorem (see, for instance, Theorem 1.19 in [127]) which says that if Φ : R → R is a Lipschitz continuous function and z ∈ W01,p (Ω), then ∇Φ(z) belongs to L p (Ω)N and ∇Φ(z) = Φ′ (z)∇z almost everywhere in Ω. Setting Φ(t) = T ε (f (t)) and using the fact that f is of class C1 , from the differentiability of G ε,k and the chain rule it immediately follows that ∫︁ (︀ )︀′ (︀ )︀′ F ε,k u (u, v)[h u ] + F ε,k v (u, v)[h v ] = (y ε,k (u, v) − y d )z(h u , h v ) dx Ω

∫︁ +k

′

|v − T ε (f (y ε,k (u, v)))|p −2 v − T ε (f (y ε,k (u, v)))

(︀

)︀

Ω

(︁ )︁ × h v − f ′ (y ε,k (u, v))χ{|f (y ε,k (u,v))|≤ ε−1 } z(h u , h v ) dx ∫︁ ∫︁ = Ψ1 z(h u , h v ) dx + Ψ2 h v dx Ω

Ω

with z(h u , h v ) = D u G ε,k (u, v) [h u ] + D v G ε,k (u, v) [h v ] , ′

′

for any (u, v) ∈ L q (Ω) × L p (Ω) and (h u , h v ) ∈ L q (Ω) × L p (Ω). Here, ′

Ψ1 = y ε,k (u, v) − y d − k|v − T ε (f (y ε,k (u, v)))|p −2 (︀ )︀ × v − T ε (f (y ε,k (u, v))) f ′ (y ε,k (u, v))χ{|f (y ε,k (u,v))|≤ ε−1 } (︀ )︀ ′ Ψ2 = k|v − T ε (f (y ε,k (u, v)))|p −2 v − T ε (f (y ε,k (u, v))) . As for the functional j(u, v), we have ∫︁ j′u (u, v)[h u ] + j′v (u, v)[h v ] = |u|q−2 uh u dx Ω

∫︁ +α

|v|

p′ −2

′

vh v dx, ∀ (h u , h v ) ∈ L q (Ω) × L p (Ω).

Ω

Now we introduce the adjoint state as follows ⎧ )︁ (︁ ⎨ − div ρ (y (u, v))A (y (u, v))∇μ ε,k ε,k ε,k ε,k ε,k = Ψ 1 in Ω, ⎩ μ = 0 on ∂Ω. ε,k

(2.116)

2.4 Optimality conditions for approximating OCP | 71

Using (2.108), we obtain (︀

F ε,k

)︀′

(︀ )︀′ (u, v)[h u ] + F ε,k v (u, v)[h v ] = u

∫︁ [h u + h v ] μ ε,k dx Ω

+k

∫︁ [︁

′

|v − T ε (f (y ε,k (u, v)))|p −2 v − T ε (f (y ε,k (u, v)))

(︀

)︀]︁

h v dx.

(2.117)

Ω

We are now in a position to establish the main result of this section. Theorem 2.9. For given ε > 0, k ∈ N, 2 ≤ p < 2N/(N − 1), q > p′ , and y d ∈ L2 (Ω), let (︀ 0 )︀ u ε,k , v0ε,k be a local solution of (2.115). Assume that F ∈ C2loc (R). Then there exist elements y0ε,k , μ ε,k ∈ H01 (Ω) such that the tuple (u0ε,k , v0ε,k , y0ε,k , μ ε,k ) satisfies the following Euler-Lagrange system to the problem (2.26)–(2.29) (︁ )︁ − div ρ ε,k (y0ε,k )∇y0ε,k = v0ε,k + u0ε,k in Ω, y0ε,k = 0 on ∂Ω,

(2.118)

)︁ (︁ ′ − div ρ ε,k (y0ε,k )Aε,k (y0ε,k )∇μ ε,k = y0ε,k − y d − k|v0ε,k − T ε (f (y0ε,k ))|p −2 (︁ )︁ }︁ in Ω, × v0ε,k − T ε (f (y0ε,k )) f ′ (y0ε,k )χ{︁ 0 −1 |f (y ε,k )|≤ ε

μ ε,k = 0 on ∂Ω, ⃒ 1 ⃒ u0ε,k = − ⃒μ ε,k ⃒ q−1 sign(μ ε,k ) a.e. in Ω,

(2.119)

′

μ ε,k = −α|v0ε,k |p −2 v0ε,k [︁ (︁ )︁]︁ ′ − k |v0ε,k − T ε (f (y0ε,k ))|p −2 v0ε,k − T ε (f (y0ε,k )) ,

a.e. in Ω.

(2.120)

(︀ )︀ Remark 2.1. Here, we say that u0ε,k , v0ε,k is a local solution of (2.115) if there is a closed (︀ )︀ ′ neighborhood U(u0ε,k ) × V(v0ε,k ) of u0ε,k , v0ε,k in the norm topology of L q (Ω) × L p (Ω) satisfying (︁ )︁ J ε,k u0ε,k , v0ε,k < J ε,k (u, v) ∀ u ∈ U(u0ε,k ) and ∀ v ∈ V(v0ε,k ) (︀ )︀ such that (u, v, y ε,k (u, v)) is a feasible triplet for (2.27)–(2.28) and (u, v) ̸= u0ε,k , v0ε,k . (︀ )︀ ′ Proof. Given u ∈ L q (Ω) and v ∈ L p (Ω), since u0ε,k , v0ε,k is a local optimal solution of (2.115), we have that J ε,k (u0ε,k + ρ(u − u0ε,k ), v0ε,k + ρ(v − v0ε,k )) ≥ J ε,k (u0ε,k , v0ε,k ) for all ρ > 0 small enough.

72 | 2 Approximation of OCPs

Hence, 0≤ =

)︁ 1 (︁ J ε,k (u0ε,k + ρ(u − u0ε,k ), v0ε,k + ρ(v − v0ε,k )) − J ε,k (u0ε,k , v0ε,k ) ρ F ε,k (u0ε,k + ρ(u − u0ε,k ), v0ε,k + ρ(v − v0ε,k )) − F ε,k (u0ε,k , v0ε,k + ρ(v − v0ε,k )) ρ +

F ε,k (u0ε,k , v0ε,k + ρ(v − v0ε,k )) − F ε,k (u0ε,k , v0ε,k ) ρ

+

j((u0ε,k + ρ(u − u0ε,k )), v0ε,k + ρ(v − v0ε,k )) − j(u0ε,k , v0ε,k + ρ(v − v0ε,k )) ρ

+

j(u0ε,k , v0ε,k + ρ(v − v0ε,k )) − j(u0ε,k , v0ε,k ) . ρ

Now, taking ρ → 0, we get [︁ ]︁ (︀ [︁ ]︁ (︀ )︀′ )︀′ 0 ≤ F ε,k u (u0ε,k , v0ε,k ) u − u0ε,k + F ε,k v (u0ε,k , v0ε,k ) v − v0ε,k [︁ ]︁ [︁ ]︁ + j′u (u0ε,k , v0ε,k ) u − u0ε,k + j′v (u0ε,k , v0ε,k ) v − v0ε,k . Finally, using the expression of F ′ε,k given by (2.117) we obtain 0≤

∫︁ [︁

u−

u0ε,k

]︁

μ ε,k dx +

Ω

∫︁ [︁

]︁ v − v0ε,k μ ε,k dx

Ω

∫︁ +

[︁

]︁

|u0ε,k |q−2 u0ε,k u − u0ε,k dx + α

Ω

+k

∫︁

′

[︁

]︁

|v0ε,k |p −2 v0ε,k v − v0ε,k dx

Ω

∫︁ [︁

′

(︁

|v0ε,k − T ε (f (y0ε,k ))|p −2 v0ε,k − T ε (f (y0ε,k ))

)︁]︁ [︁

]︁ v − v0ε,k dx.

Ω ′

Since u ∈ L q (Ω) and v ∈ L p (Ω) are independent and arbitrary functions, we deduce from this relation the following equalities μ ε,k = −|u0ε,k |q−2 u0ε,k ,

a. e. in Ω

(2.121)

′

μ ε,k = −α|v0ε,k |p −2 v0ε,k [︁ (︁ )︁]︁ ′ − k |v0ε,k − T ε (f (y0ε,k ))|p −2 v0ε,k − T ε (f (y0ε,k ))

(2.122) a. e. in Ω.

(2.123)

Thus, the optimality system (2.118)-(2.120) immediately follows from (2.116) and (2.121).

3 Neumann Boundary Optimal Control Problem for Strongly Nonlinear Elliptic Equation with p-Laplace Operator This chapter focuses on the study of an optimal control problem for the mixed Dirichlet-Neumann boundary value problem for the strongly non-linear elliptic equation with p-Laplace operator and the exponential type of nonlinearity in their right-hand side. A density of surface traction u acting on a part of boundary of an open domain is taken as the boundary control. The optimal control problem is to minimize the discrepancy between a given distribution y d ∈ L2 (Ω) and the current system state. We deal with such case of nonlinearity when we cannot expect to have a solution of the state equation for any admissible control. After defining a suitable functional class where we look for solutions and assuming that this problem admits at least one feasible solution, we prove the existence of optimal pairs. In order to handle the strong non-linearity in the right-hand side of elliptic equation, we involve a special twoparametric fictitious optimization problem. We derive existence of optimal solutions to the regularized optimization problems at each (ε, k)-level of approximation and discuss the asymptotic behaviour of the optimal solutions to regularized problems as the parameters ε and k tend to zero and infinity, respectively. It is worth to note here that an optimal control problem for the system −∆y = λf (y) + v

in Ω,

y=0

on ∂Ω

(3.1)

with f (y) = e y , and distributed control v, was first discussed in detail by Casas, Kavian, and Puel [22], where the problem of existence and uniqueness of the underlying boundary value problem and the corresponding optimal control problem was treated and an optimality system has been derived and analyzed. However, analogous results for the case of general nonlinear elliptic equations of the type (︀ )︀ − div a(∇y) = f (y) + v remained open. In this chapter we treat the case of the p-Laplacian, where a(∇y) = |∇y|p−2 ∇y and p ≥ 2. The corresponding strongly nonlinear differential operator − div(|∇y|p−2 ∇y) − f (y) is not monotone and, in principle, has degeneracies as ∇y tends to zero. Moreover, when the term |∇y|p−2 is regarded as the coefficient of the Laplace operator, we also have the case of unbounded coefficients. Because of this and L1 -boundedness of the function f (y) there are serious hurdles to deduce an a priori estimate for the weak solutions of the corresponding BVP in the standard Sobolev space W01,p (Ω). As a result, we focus on the case when for some admissible control ′

u ∈ L p (Γ N ), the original BVP possesses a special type of weak solutions satisfying some extra state constraint. However, in this case it is not an easy matter to touch directly on this special set because its structure and the main topological properties are https://doi.org/10.1515/9783110668520-004

74 | 3 Neumann Boundary Control Problem

unknown in general. To lighten this problem and make the corresponding optimization procedure more feasible, and show that some regularization and approximation of the considered optimal control problem are necessary.

3.1 Setting of the problem Let us start with some notations and preliminaries. Let Ω be a bounded open subset of RN (N > 2). We assume that its boundary ∂Ω is of the class C1 . So, the unit outward normal ν = ν(x) is well-defined for H N−1 -a.a. x ∈ ∂Ω, where the abbreviation ’a.a.’ means here for ’almost all’ x ∈ ∂Ω with respect to the (N − 1)-dimensional Hausdorff measure H N−1 . We also assume that the boundary ∂Ω consists of two disjoint parts ∂Ω = Γ D ∪ Γ N , where the sets Γ D and Γ N have positive (N − 1)-dimensional measures. Throughout this chapter we assume that there exists a point x0 ∈ int Ω such that Ω satisfies the star-shaped property with respect to x0 , i.e. (︀

)︀ σ − x0 , ν(σ) ≥ 0 for H N−1 -a.a. σ ∈ ∂Ω.

Let F : R → [0, +∞) be a mapping with properties defined in Section 1.1, i.e., F ∈ C1loc (R), F is a non-decreasing positive function, and there exists a constant C F > 0 satisfying ⃒ ⃒ 0 ⃒ ⃒ ∫︁ ⃒ ⃒ ′ ′ ⃒ (3.2) F (z) ≥ C F F(z), ∀ z ∈ R and ⃒ zF (z) dz⃒⃒ < +∞. ⃒ ⃒ −∞

We define the function f ∈ C loc (R) as follows: f (z) = F ′ (z). Typical example of f (z) is f (z) = Ce C F z . Let {︁ }︁ N ∞ N C∞ 0 (R ; Γ D ) = φ ∈ C 0 (R ) : φ = 0 on Γ D be the class of test functions. For a given exponent p ≥ 2, we define the Banach space N W01,p (Ω; Γ D ) as the closure of C∞ 0 (R ; Γ D ) with respect to the norm

||y||W 1,p (Ω) 0

⎛ ⎞1/p ∫︁ p = ⎝ |∇y| dx⎠ . Ω

So, we can suppose that each element of the space W01,p (Ω; Γ D ) has zero trace at the )︁* (︁ ′ Γ D -part of boundary ∂Ω. Let W −1,p (Ω; Γ D ) := W01,p (Ω; Γ D ) be the dual space to W01,p (Ω; Γ D ). Let p, r, and q be real numbers such that p ≥ 2, q ≥ p p′ = p−1 is the conjugate exponent to p.

pN pN−N+p ,

and r ≥ p′ , where

3.1 Setting of the problem | 75

We are concerned with the following optimal control problem for a nonlinear elliptic equation with p-Laplacian: ∫︁ ∫︁ ∫︁ ′ 1 α 1 J(u, y) = |u|p dx + |y − y d |2 dx + ′ |f (y)|r dx → inf, (3.3) 2 r p ΓN

Ω

Ω

subject to constraints )︁ − div |∇y|p−2 ∇y = f (y) + g in Ω, (︁

y=0

|∇y|p−2 ∂ ν y = u

on Γ D ,

(3.4)

on Γ N ,

(3.5)

′

u ∈ Aad ⊂ L p (Γ N ), y ∈ W01,p (Ω; Γ D ),

(3.6)

where α > 0 is a given weight which is assumed to be small enough, Aad is a closed ′ convex subset of L p (Γ N ), g ∈ L q (Ω) and y d ∈ L2 (Ω) are given distributions, and u is a density of surface traction acting on the Γ N -part of boundary ∂Ω is taken as a boundary control. So, the optimal control problem (OCP) (3.3)–(3.6) is to minimize the discrepancy between a given distribution y d ∈ L2 (Ω) and the current state of system (3.4)–(3.6). Equations like (3.4) with homogeneous Dirichlet boundary conditions appear in a number of applications. In particular, it can be considered as a relevant model for the description of a ball of isothermal gas in gravitational equilibrium, proposed by lord Kelvin in the study of stellar structures [24]. It has been also actively investigated in connection with combustion theory (see, for instance, [62, Section 15], [55, Chapter VI], [73, Theorem 2]). However, the case of mixed boundary conditions (3.5) makes the corresponding optimal control problem much more difficult because there is no reason to suppose that this problem is consistent from mathematical point of view. ′

Definition 3.1. We say that, for a given control u ∈ L p (Γ N ), a function y = y(u) is a weak solution to the boundary value problem (3.4)–(3.5) (in the sense of distributions) if y belongs to the class ⃒ }︁ {︁ ⃒ (3.7) Y = y ∈ W01,p (Ω; Γ D ) ⃒ f (y) ∈ L1 (Ω) and the integral identity ∫︁ ∫︁ ∫︁ ∫︁ |∇y|p−2 (∇y, ∇φ) dx = f (y)φ dx + uφ dH N−1 + gφ dx Ω

ΓN

Ω

(3.8)

Ω

N holds for every test function φ ∈ C∞ 0 (R ; Γ D ). ′

Definition 3.2. We say that (u, y) ∈ L p (Γ N ) × W01,p (Ω; Γ D ) is a feasible solution to OCP (3.3)–(3.6) if – u is an admissible control, i.e. u ∈ Aad ; – J(u, y) < +∞;

76 | 3 Neumann Boundary Control Problem the function y = y(u) ∈ W01,p (Ω; Γ D ) is a weak solution to BVP (3.4)–(3.5) for given control u in the sense of Definition 3.1. We denote by Ξ the set of all feasible solutions to the problem (3.3)–(3.6).

–

It is well known that (see [14, Section 5]) BVP (3.4)–(3.5) is ill-posed, in general. So ′ we can not assert that, for a given function g ∈ L q (Ω) and a control u ∈ L p (Γ N ), there exists at least one weak solutions y = y(u, g) to (3.4)–(3.5), or suppose that such solution, even if it exists, is unique and satisfies condition J(u, y) < +∞. Moreover, to the best knowledge of authors, the existence and uniqueness of the weak solutions to the original BVP is an open question for now. In view of this it is worth to emphasize the following result (see [13]): there exists a finite positive number λ* , called the extremal value, such that the boundary value problem −∆y = λe y + v

in Ω,

y=0

on ∂Ω

(3.9)

has at least a classical positive solution y ∈ C2 (Ω) provided 0 < λ < λ* and v = 0, while no solution exists, even in the weak sense, for λ > λ* . In the case λ = λ* and v = 0, this problem admits existence of the so-called singular solutions u ∈ H01 (Ω) that do not belong to L∞ (Ω). Thus, in the context of the optimal control problem that we deal with in this chapter, it is not an easy matter to touch directly the set of feasible solutions Ξ to the original optimal control problem because its structure and the main topological properties are unknown in general. Let us associate with the function f ∈ C loc (R) (f (z) = F ′ (z)) the following form ∫︁ ∞ N (3.10) [y, φ]f := f (y)φ dx, ∀ y ∈ Y , ∀ φ ∈ C0 (R ; Γ D ). Ω

By analogy with Chapter 1 (see Definition 1.2), let us introduce the following set. Definition 3.3. We say that an element y ∈ Y belongs to the set H f if ⃒ ⃒ ⎛ ⎞1/p ⃒ ⃒∫︁ ∫︁ ⃒ ⃒ p ⃒ f (y)φ dx⃒ ≤ c(y) ⎝ |∇φ| dx⎠ , ⃒ ⃒ ⃒ ⃒ Ω

N ∀ φ ∈ C∞ 0 (R ; Γ D )

(3.11)

Ω

with some constant depending arguably on y. As a result, we have: if y ∈ H f , then the mapping φ ↦→ [y, φ]f can be defined for all φ ∈ W01,p (Ω; Γ D ) using (3.11) and the rule [y, φ]f = lim [y, φ ε ]f . ε→0

(3.12)

The following result which can be considered as another motivation to introduce the set H f .

3.1 Setting of the problem | 77

′

Proposition 3.1. If u ∈ L p (Γ N ) and g ∈ L q (Ω) are given distributions and y ∈ Y is a weak solution to BVP (3.4)–(3.5) in the sense of Definition 3.1, then y belongs to the set H f , and y satisfies the energy equality ∫︁ ∫︁ (3.13) |∇y|p dx = [y, y]f + γ0 (y)u dH N−1 + ⟨g, y⟩W −1,p′ (Ω;Γ );W 1,p (Ω;Γ ) , D

0

D

ΓN

Ω

where ⟨·, ·⟩W −1,p′ (Ω;Γ

′

1,p D );W 0 (Ω;Γ D )

: W −1,p (Ω; Γ D ) × W01,p (Ω; Γ D ) → R ′

denotes the duality pairing between W −1,p (Ω; Γ D ) and W01,p (Ω; Γ D ), and ′

γ0 : W01,p (Ω; Γ D ) → W 1/p ,p (Γ N ) stands for the trace operator, γ0 (y) = y|Γ D ,

∀ y ∈ W01,p (Ω; Γ D ) ∩ C(Ω).

Proof. To begin with, let us show that the duality pairing ⟨g, y⟩W −1,p′ (Ω;Γ

1,p D );W 0 (Ω;Γ D )

is well defined for all y ∈ W01,p (Ω; Γ D ) and g ∈ L q (Ω) provided q≥

pN . pN − N + p

Indeed, by Sobolev embedding theorem, Banach space W01,p (Ω; Γ D ) is continu(︁ * )︁* * pN ously embedded in L p (Ω) with p* = N−p . Hence, by duality arguments, L p (Ω) is ′

continuously embedded in W −1,p (Ω; Γ D ). So, if we define p* = (p* )′ =

pN , pN − N + p

(3.14)

′

then we have: L q (Ω) ⊂ L p* (Ω) ⊂ W −1,p (Ω; Γ D ) continuously for all q ≥ fore, there exists a constant C > 0 such that ⃒ ⃒ ⃒ ⃒ ⃒ ⟨g, y⟩W −1,p′ (Ω;Γ );W 1,p (Ω;Γ ) ⃒ ≤ ||g||W −1,p′ (Ω;Γ ) ||y||W 1,p (Ω;Γ D ) D

0

D

D

pN pN−N+p .

There-

0

≤ C||g||L q (Ω) ||y||W 1,p (Ω;Γ D ) , 0

∀ y ∈ W01,p (Ω; Γ D ).

(3.15)

′

We also note that the injection W 1/p ,p (Γ N ) ˓→ L p (Γ N ) is compact by Sobolev embedding theorem. Hence, by continuity of the trace operator γ0 : W01,p (Ω; Γ D ) → ′

W 1/p ,p (Γ N ) (see [115, Theorem 8.3]), there is a constant C γ0 > 0 such that ||γ0 (y)||L p (Γ N ) ≤ C γ0 ||y||W 1,p (Ω;Γ D ) , 0

∀ y ∈ W01,p (Ω; Γ D ).

(3.16)

78 | 3 Neumann Boundary Control Problem

As a result, we have the estimate ⃒ ⃒ ⃒ ⃒∫︁ ⃒ ⃒ ⃒ uγ0 (y) dH N−1 ⃒ ≤ ||u|| p′ ||y||L p (Γ N ) ≤ C γ0 ||u||L p′ (Γ ) ||y||W 1,p (Ω;Γ D ) < +∞. ⃒ ⃒ L (Γ N ) N 0 ⃒ ⃒

(3.17)

ΓN

We are now in a position to show that the weak solution y belongs to the set H f . In order to verify the validity of this assertion it is enough to rewrite the integral identity (3.8) in the form ∫︁ ∫︁ ∫︁ (3.18) [y, φ]f = |∇y|p−2 (∇y, ∇φ) dx − uφ dH N−1 − gφ dx ΓN

Ω

Ω

and apply the Hölder’s inequality to the right-hand side of (3.18). As a result, taking into account the Friedrich’s inequality ||y||L p (Ω) ≤ diam Ω ||∇y||L p (Ω)N ,

∀y ∈ W01,p (Ω; Γ D ),

(3.19)

and estimates (3.15)–(3.17), we obtain ⃒ ⃒ ⃒ ⃒ ⃒ ⃒ ⃒∫︁ ⃒∫︁ ∫︁ ⃒ ⃒ ⃒ ⃒ [y, φ]f := f (y)φ dx ≤ ⃒⃒ |∇y|p−2 (∇y, ∇φ) dx⃒⃒ + ⃒⃒ uφ dH N−1 ⃒⃒ ⃒ ⃒ ⃒ ⃒ ΓN Ω Ω ⃒ ⃒ ⃒ ⃒∫︁ ⃒ ⃒ p−1 + ⃒⃒ gφ dx⃒⃒ ≤ ||∇y||L p (Ω)N ||∇φ||L p (Ω)N ⃒ ⃒ Ω

+ C γ0 ||u||L p′ (Γ ) ||φ||W 1,p (Ω;Γ D ) + C||g||L q (Ω) ||φ||W 1,p (Ω;Γ D ) N 0 (︁ )︁ 0 p−1 ≤ ||y|| 1,p + C γ0 ||u||L p′ (Γ ) + C||g||L q (Ω) ||φ||W 1,p (Ω;Γ D ) . W0 (Ω;Γ D )

N

0

N Since this relation is valid for all φ ∈ C∞ 0 (R ; Γ D ), it follows that y ∈ H f by Definition 3.3. To conclude the proof, it remains to establish the energy equality (3.13). With that in mind it is enough to consider the integral identity (3.8) with test functions {φ := φ ε }ε>0 such that φ ε → y strongly in W01,p (Ω; Γ D ), and pass to the limit in it as ε → 0 using the rule (3.12). The existence of such sequence immediately follows from Meyers-Serrin’s theorem. The proof of this result is fairly delicate and we refer to R. Adams [1] for the details.

To specify the term [y, y]f in (3.8), we give the following result. Lemma 3.1. Let y = y(u) ∈ Y be a weak solution to BVP (3.4)–(3.5) for a given u ∈ ′ ′ L p (Γ N ). Then f (y) ∈ W −1,p (Ω; Γ D ), and ∫︁ ⟨︀ ⟩︀ 1,p (3.20) [y, z]f = f (y), z W −1,p′ (Ω;Γ );W 1,p (Ω;Γ ) = z f (y) dx, ∀ z ∈ W0 (Ω; Γ D ), D

0

D

Ω 1

i.e. z f (y) ∈ L (Ω) for every z ∈

W01,p (Ω; Γ D ).

3.1 Setting of the problem | 79

Proof. The similar result has been established in Section 1.1 (see Lemma 1.1). However, for the reader convenience we reproduce it for the case of BVP (3.4)–(3.5). Let z ∈ W01,p (Ω; Γ D ) ∩ L∞ (Ω) be an arbitrary distribution. Since f (y) ∈ L1 (Ω), it follows that ∫︀ the term Ω z f (y) dx is well defined. Let {φ ε }ε>0 ⊂ C∞ (RN ; Γ D ) be a sequence such that φ ε → z in W01,p (Ω; Γ D ). Moreover, in this case we can suppose that sup ||φ ε ||L∞ (Ω) < +∞

and

*

φ ε ⇀ z in L∞ (Ω).

ε>0

Hence, due to the fact that y ∈ H f , we get ∫︁ ∫︁ z f (y) dx = lim φ ε f (y) dx = lim [y, φ ε ]f ε→0

Ω

by (3.12)

ε→0

=

[y, z]f .

(3.21)

Ω

Thus, we arrive at the relation (3.20) for each z ∈ W01,p (Ω; Γ D ) ∩ L∞ (Ω). Let us take now z ∈ W01,p (Ω; Γ D ) such that z ≥ 0 almost everywhere in Ω. For every ε > 0, let T ε : R → R be the truncation operator defined in (1.17). Then by the well-known property of T ε , we have: if z ∈ W01,p (Ω; Γ D ) then T ε (z) ∈ L∞ (Ω) ∩ W01,p (Ω; Γ D ), ∀ ε > 0 and T ε (z) → z in W01,p (Ω; Γ D ) as ε → 0. Generally speaking, this property means that we can cut the functions of the first order Sobolev space at certain level and the truncated function is still in the same Sobolev space, whereas higher order Sobolev spaces do not enjoy this property (for counter-example we refer to [76, Example 1.6]). {︀ }︀ Since T ε (z) → z almost everywhere in Ω and T ε (z)f (y) ε>0 is a pointwise nondecreasing sequence and T ε (z)f (y) → z f (y) for almost all x ∈ Ω, it follows by monotone convergence theorem that z f (y) is a measurable function on Ω and ∫︁ ∫︁ lim T ε (z)f (y) dx = z f (y) dx. ε→0

Ω

Ω

Thus, (3.20) remains valid for each z ∈ W01,p (Ω; Γ D ) such that z ≥ 0. As for the general case, i.e. for z ∈ W01,p (Ω; Γ D ), it is enough to note that (see [75, Proposition II.5.3]) z = z+ − z− with z+ , z− ∈ W01,p (Ω; Γ D ) and z+ , z− ≥ 0 in Ω, where z+ := max {z, 0} ,

z− := max {−z, 0} .

To complete the proof, we observe that ∫︁ ∫︁ z f (y) dx = lim φ ε f (y) dx ε→0

Ω

Ω by (3.18)

≤

(︁ p−1 lim ||y|| 1,p ε→0

W0 (Ω;Γ D )

)︁ + C γ0 ||u||L p′ (Γ ) + C||g||L q (Ω) ||φ ε ||W 1,p (Ω;Γ D ) N

0

W01,p (Ω; Γ D ))

(by the strong convergence of φ ε → z in )︁ (︁ p−1 + C γ0 ||u||L p′ (Γ ) + C||g||L q (Ω) ||z||W 1,p (Ω;Γ D ) = ||y|| 1,p W0 (Ω;Γ D )

N

0

80 | 3 Neumann Boundary Control Problem holds true for an arbitrary element z ∈ W01,p (Ω; Γ D ). Hence, ′

⟨︀

f (y), z

⟩︀

W −1,p′ (Ω;Γ D

f (y) ∈ W −1,p (Ω; Γ D ), ∫︁ = z f (y) dx, 1,p );W (Ω;Γ )

∀ z ∈ W01,p (Ω; Γ D ),

D

0

Ω

and ||f (y)||W −1,p′ (Ω;Γ

D)

(︁ p−1 ≤ ||y|| 1,p

W0 (Ω;Γ D )

)︁ + C γ0 ||u||L p′ (Γ ) + C||g||L q (Ω) . N

The proof is complete. As a direct consequence of Lemma 3.1 and Proposition 3.1, we have the following result. ′

Corollary 3.1. Let u ∈ L p (Γ N ) be a given control and let y = y(u) ∈ W01,p (Ω; Γ D ) be a weak solution to BVP (3.4)–(3.5) in the sense of Definition (3.1). Then the energy equality for y takes the form ∫︁ ∫︁ ∫︁ (3.22) |∇y|p dx = y f (y) dx + γ0 (y)u dH N−1 + ⟨g, y⟩W −1,p′ (Ω;Γ );W 1,p (Ω;Γ ) . D

Ω

Ω

0

D

Ω

Following Brezis (see [14, p. 445]) we call the weak solutions of BVP (3.4)–(3.5) satisfying energy equality (3.22) energy or variational solutions. However, the substantiation of ′ energy equality (3.22) can be simplified if, for a given control u ∈ L p (Γ N ) the pair (u, y) ′ is feasible to OCP (3.3)–(3.6). Indeed, if (u, y) ∈ Ξ, then f (y) ∈ L p (Ω) and, therefore, ∫︀ the form [y, φ]f := Ω f (y)φ dx is continuous on the set Y. In this case, for each test N function φ ∈ C∞ 0 (R ; Γ D ), we have ⃒ ⎛ ⃒ ⎞1/p′ ⎛ ⎞1/p ⃒ ⃒∫︁ ∫︁ ∫︁ ⃒ ⃒ ′ p p ⃒ f (y)φ dx⃒ ≤ ⎝ |f (y)| dx⎠ ⎝ |∇φ| dx⎠ ⃒ ⃒ ⃒ ⃒ Ω Ω Ω ⎛ ⎞1/r ⎛ ⎞1/p ∫︁ ∫︁ 1 1 − ≤ |Ω| p′ r ⎝ |f (y)|r dx⎠ ⎝ |∇φ|p dx⎠ Ω

≤ |Ω|

1 p′

− 1r

(︁ r α

Ω

J(u, y)

)︁1/r

||φ||W 1,p (Ω;Γ D ) .

(3.23)

0

Thus, it is easy now to deduce by continuity that the integral identity (3.8) remains valid for all φ ∈ W 1,p (Ω; Γ D ).

3.2 A priori estimates both for energy solutions and feasible solutions In this section we deal with some extra properties of the weak solutions to the boundary value problem (3.4)–(3.5). In some aspects we follow the ideas of the paper E. Casas,

3.2 A Priori Estimates | 81

O. Kavian, and J.P. Puel [22], where the Dirichlet boundary value problem with linear Laplace operator in the principle part of elliptic equation and exponential nonlinearity has been considered. Since it is unknown whether the operator (︁ )︁ −∆ p y − f (y) := − div |∇y|p−2 ∇y − f (y) is monotone onto the set y ∈ H f , we cannot use the energy equality (3.13) in order to derive a priori estimate in || · ||W 1,p Ω) -norm for the weak solutions of BVP (3.4)–(3.5). 0

However, as we will see later on, in the case of feasible solutions (u, y) ∈ Ξ, we will be able to leverage the condition J(u, y) < +∞ in order to deduce some a priori estimates for them. We begin with the following result, where we establish the Pohozaev-type inequality for the solutions of Dirichlet-Nuemann boundary value problem (3.4)–(3.5). ′

Proposition 3.2. Let u ∈ L p (Γ N ) and let y = y(u) ∈ W01,p (Ω; Γ D ) be a weak solution to ′

BVP (3.4)–(3.5). Assume that 2 ≤ p < N, f (y) ∈ L p (Ω), and g ∈ L q (Ω) with q ≥ p′ . Then (︂

)︂ ∫︁ ∫︁ N |∇y|p dx ≤ N F(y) dx −1 p Ω Ω ∫︁ ∫︁ ′ β − g (x − x0 , ∇y) dx + |u|p dH N−1 , p

(3.24)

ΓN

Ω

(︀

)︀ where β = (1 + p) diam Ω, x0 ∈ int Ω is a point such that σ − x0 , ν(σ) ≥ 0 for a.a. σ ∈ ∂Ω and ν(σ) denotes the outward unit normal vector to ∂Ω at the point σ. Remark 3.1. Since p′ >

pN pN−N+p

and q ≥ p′ , it follows that the term ∫︁ g (x − x0 , ∇y) dx Ω

in (3.24) is well defined. Moreover, utilizing the Hölder inequality, we get ∫︁ ∫︁ g (x − x0 , ∇y) dx ≤ diam Ω |g ||∇y| dx Ω

Ω

≤ diam Ω||g||L p′ (Ω) ||∇y|L p (Ω)N q−p′

≤ diam Ω|Ω| qp′ ||g||L q (Ω) ||y||W 1,p (Ω;Γ D ) < +∞. 0

Proof. For the reader’s convenience, we divide this proof onto several steps. Step 1. In view of the initial assumptions and Remark 3.1, it is easy to see that (︁ )︁ ′ div |∇y|p−2 ∇y ∈ L p (Ω).

(3.25)

Hence, we can multiply the equation (3.4) by any function φ ∈ L p (Ω) and make the integration over Ω. Let us consider φ := (x − x0 , ∇y) ∈ L p (Ω) as the test function. Then

82 | 3 Neumann Boundary Control Problem

(3.8) implies the relation ∫︁ |∇y|p−2 (∇y, ∇ (x − x0 , ∇y)) dx Ω

∫︁ −

|∇y(σ)|p−2 ∂ ν y(σ) σ − x0 , ∇y(σ) dH N−1

(︀

)︀

∂Ω

∫︁

∫︁ f (y) (x − x0 , ∇y) dx +

= Ω

g (x − x0 , ∇y) dx.

(3.26)

Ω

Further, we note that the boundary ∂Ω consists of two disjoint parts ∂Ω = Γ D ∪ Γ N and we have y(σ) = 0 on Γ D . Hence, the tangential component of ∇y(σ) vanish on the (︀ )︀ Γ D -part of boundary. Therefore, ∇y(σ) = ±|∇y(σ)|ν(σ) on Γ D . Then σ − x0 , ∇y(σ) = (︀ )︀ (︀ )︀ ±|∇y(σ)| σ − x0 , ν(σ) and ∂ ν y(σ) = ∇y(σ), ν(σ) = ±|∇y(σ)| on Γ D (for details, we refer to [22, p. 364]). As for the Γ N -part of ∂Ω, we can postulate the existence of a couple of functions ζ1 , ζ2 ∈ L∞ (Γ N ) with |ζ i (σ)| ≤ 1 H N−1 -a.e. on Γ N such that (︀

)︀ σ − x0 , ∇y(σ) = ζ1 (σ) diam Ω |∇y(σ)|, (︀ )︀ ∂ ν y(σ) = ∇y(σ), ν(σ) = ζ2 (σ)|∇y(σ)| H N−1 -a.e. on Γ N . Taking this fact into account, we can rewrite the equality (3.26) as follows ∫︁ |∇y|p−2 (∇y, ∇ (x − x0 , ∇y)) dx Ω

∫︁ −

ζ1 (σ)ζ2 (σ) diam Ω |∇y(σ)|p dH N−1

ΓN

∫︁ −

(︀

)︀ σ − x0 , ν(σ) |∇y(σ)|p dH N−1

ΓD

∫︁

∫︁ f (y) (x − x0 , ∇y) dx +

= Ω

g (x − x0 , ∇y) dx. Ω

(3.27)

3.2 A Priori Estimates | 83

Step 2. We apply the integration by parts to the first term in left hand side of (3.26). This yields ∫︁ |∇y|p−2 (∇y, ∇ (x − x0 , ∇y)) dx Ω

=

N ∫︁ ∑︁ i=1 Ω

∫︁ =

⎡ ⎤ N ∑︁ ∂y ⎦ ∂y ∂ ⎣ (x j − x0j ) dx |∇y|p−2 ∂x i ∂x i ∂x j j=1

|∇y|p−2

N ⃒ ∑︁

⃒ ∫︁ N ∑︁ ⃒ ∂y ⃒2 ∂y ∂2 y ⃒ ⃒ dx + |∇y|p−2 (x j − x0j ) dx ⃒ ∂x i ⃒ ∂x i ∂x i ∂x j

i=1

Ω

1 p

∫︁ ∑︁ N

Ω

Ω

∫︁

1 p

∫︁

∫︁ =

=

|∇y|p dx +

|∇y|p dx +

Ω

−

N p

∂ |∇y|p dx ∂x j

(x j − x0j )

j=1 N ∑︁

|∇y(σ)|p

(σ j − x0j )ν j (σ) dH N−1

j=1

∂Ω

∫︁

i,j=1

Ω

|∇y|p dx

Ω

(︂ =

N 1− p

)︂ ∫︁

∫︁

1 |∇y| dx + p p

Ω

(︀

)︀ σ − x0 , ν(σ) |∇y(σ)|p dH N−1 .

∂Ω

(︀ )︀ Since, by the star-shaped property of Ω, we have σ − x0 , ν(σ) ≥ 0 for H N−1 -a.a. σ ∈ Γ D ′

and |∇y|p = |u|p on Γ N , it follows that ∫︁ )︀ 1 (︀ σ − x0 , ν(σ) |∇y(σ)|p dH N−1 p ∂Ω ∫︁ − ζ1 (σ)ζ2 (σ) diam Ω |∇y(σ)|p dH N−1 ΓN

∫︁ −

(︀

)︀ σ − x0 , ν(σ) |∇y(σ)|p dH N−1

ΓD

1 ≤− ′ p

∫︁

(︀

)︀ σ − x0 , ν(σ) |∇y(σ)|p dH N−1

ΓD

+ diam Ω

1+p p

∫︁ ΓN

′

|u|p dH N−1 ≤

β p

∫︁ ΓN

′

|u|p dH N−1

84 | 3 Neumann Boundary Control Problem

with β = (1 + p) diam Ω. Hence, ∫︁ |∇y|p−2 (∇y, ∇ (x − x0 , ∇y)) dx Ω

∫︁ −

|∇y(σ)|p−2 ∂ ν y(σ) σ − x0 , ∇y(σ) dH N−1

(︀

)︀

∂Ω

(︂ ≤

N 1− p

)︂ ∫︁

β |∇y| dx + p p

∫︁

′

|u|p dH N−1 .

Step 3. Let us show that the relation ∫︁ ∫︁ ∫︁ (︀ )︀ f (y) (∇y, ψ) dx = F(y(σ)) ν(σ), ψ(σ) dσ − F(y) div ψ dx Ω

(3.28)

ΓN

Ω

(3.29)

Ω

∂Ω

holds true for any vector-valued test function ψ ∈ C1 (Ω)N provided y ∈ W01,p (Ω; Γ D ) is a weak solution to (3.4)–(3.5). To do so, it is enough to prove that f (y)∇y = ∇F(y) as elements of L1 (Ω)N .

(3.30)

Then the equality (3.29) is a direct consequence of the formula of integration by parts. Let T ε : R → R be the truncation operator defined in (1.17). By definition of T ε , we can suppose that T ε (y) → y strongly in W01,p (Ω; Γ D ) and a.e. in Ω as ε → 0.

(3.31)

′

Since f (y) ∈ L p (Ω) and f ∈ C loc (R), it follows from (3.31) that (︀ )︀ f T ε (y) → f (y)

′

in L p (Ω) and a.e. in Ω.

Then, utilizing the fact that ∇y ∈ L p (Ω)N and ∇T ε (y) → ∇y a.e. in Ω, the Lebesgue Dominated Theorem implies: (︀ )︀ f T ε (y) ∇T ε (y) → f (y)∇y in L1 (Ω)N , (︀ )︀ ∇F T ε (y) → ∇F (y) in L1 (Ω)N . Taking into account that (︀ )︀ (︀ )︀ f T ε (y) ∇T ε (y) = ∇F T ε (y) ,

∀ ε > 0,

and passing to the limit in both parts of this relation as ε → 0, we arrive at the desired equality (3.30).

3.2 A Priori Estimates | 85

Step 4. Now we are in a position to provide some transformation of the right hand side in (3.26). Due to relation (3.29) and the star-shape properties of Ω, we have ∫︁ ∫︁ (︀ )︀ by (3.30) f (y) (x − x0 , ∇y) dx = x − x0 , ∇F(y) dx Ω

Ω by (3.29)

=

∫︁

∫︁ F(y) div(x − x0 ) dx +

− Ω

(︀ )︀ F(y(σ)) ν(σ), σ − x0 dσ

∂Ω

∫︁ F(y) dx.

≥ −N Ω

Then combining this inequality with (3.26) and (3.28), we finally arrive at the desired relation (3.24). The proof is complete. Remark 3.2. In what follows, we will use the following version of inequality (3.24) ∫︁ ∫︁ ∫︁ Np p |∇y|p dx ≤ F(y) dx − g (x − x0 , ∇y) dx N−p N−p Ω Ω Ω ∫︁ ′ β |u|p dH N−1 . (3.32) + N−p ΓN

Our next intention is to show that Pohozaev-type inequality (3.24) implies some a priori estimate for the weak solutions y ∈ Y to the original BVP. ′

Theorem 3.1. Let u ∈ L p (Γ N ), g ∈ L q (Ω) with q ≥ p′ and 2 ≤ p < N, and let y = y(u) ∈ Y be a weak solution to BVP (3.4)–(3.5) satisfying inequality (3.24). Then ∫︁ p′ p′ −1 + C3 , (3.33) y f (y) dx ≤ C1 ||u|| p′ + C2 ||u|| p′ L (Γ N )

L (Γ N )

Ω

||y||W 1,p (Ω;Γ D ) ≤ C4 ||u|| 0

p′ −1 L p′ (Γ N )

+ C5 ,

(3.34)

for some positive constants C i , 1 ≤ i ≤ 5, independent of u and y. Proof. Combining the energy equality (3.13) with inequality (3.24), we get ⎞ ⎛ )︂ ∫︁ (︂ ∫︁ N N−1 − 1 ⎝ y f (y) dx + γ0 (y)u dH + ⟨g, y⟩W −1,p′ (Ω;Γ );W 1,p (Ω;Γ ) ⎠ D D 0 p Ω Ω ∫︁ ∫︁ ∫︁ ′ β ≤ N F(y) dx − g (x − x0 , ∇y) dx + |u|p dH N−1 . p Ω

Ω

ΓN

86 | 3 Neumann Boundary Control Problem

Then, in view of estimate (3.18), we can rewrite the last relation as follows ∫︁ ∫︁ Np y f (y) dx ≤ F(y) dx N−p Ω Ω )︁ p (︁ + C γ0 ||u||L p′ (Γ ) + C||g||L q (Ω) ||y||W 1,p (Ω;Γ D ) N 0 N−p ∫︁ ′ β p + diam Ω |g ||∇y| dx + ||u|| p′ L (Γ N ) p Ω ∫︁ β Cp Np p′ + ||g||L q (Ω) |y||W 1,p (Ω;Γ D ) F(y) dx + ||u|| p′ ≤ L (Γ N ) 0 N−p p N−p Ω q−p′

(︂

+ diam Ω|Ω| qp′ ||g||L q (Ω) +

)︂ C γ0 p ||u||L p′ (Γ ) ||y||W 1,p (Ω;Γ D ) . N 0 N−p

(3.35)

}︂ 2NC−1 F p , where the constant N−p C F is defined in (3.2). Since F : R → [0, +∞) is a non-decreasing function, it follows from (3.2) that ∫︁ ∫︁ NC−1 Np F p F(y) dx ≤ f (y) dx N−p N−p {︂

For our further analysis, we set Ω N :=

Ω

x ∈ Ω : y(x) >

Ω

1 ≤ 2

NC−1 F p yf (y) dx + N−p

∫︁ ΩN

1 ≤ 2

∫︁

(︂

∫︁ f

2NC−1 F p N−p

)︂

2NC−1 F p N−p

)︂

dx

Ω\Ω N

Np −1 yf (y) dx + C |Ω|f N−p F

(︂

(3.36)

ΩN

and ∫︁ Ω\Ω N

2NC−1 F p yf (y) dx ≤ N−p

(︂

∫︁ f

2NC−1 F p N−p

)︂ dx

Ω\Ω N

2Np −1 C |Ω|f ≤ N−p F

(︂

2NC−1 F p N−p

)︂ .

(3.37)

3.2 A Priori Estimates | 87

Then inequality (3.35) yields the following relation ∫︁ ∫︁ ∫︁ y f (y) dx y f (y) dx = y f (y) dx − Ω

ΩN

1 2

by (3.36)

≤

Ω\Ω N

∫︁ yf (y) dx +

Np −1 C |Ω|f N−p F

(︂

2NC−1 F p N−p

)︂

ΩN

∫︁ y f (y) dx +

−

β Cp p′ ||u|| p′ ||g||L q (Ω) ||y||W 1,p (Ω;Γ D ) + L (Γ N ) 0 p N−p

Ω\Ω N

)︂ (︂ q−p′ C γ0 p ||u||L p′ (Γ ) ||y||W 1,p (Ω;Γ D ) . + diam Ω|Ω| qp′ ||g||L q (Ω) + N 0 N−p Therefore, 1 2

∫︁

Np −1 y f (y) dx ≤ C |Ω|f N−p F

(︂

2NC−1 F p N−p

)︂

∫︁ y f (y) dx

−

ΩN

Ω\Ω N

β Cp p′ ||u|| p′ ||g||L q (Ω) ||y||W 1,p (Ω;Γ D ) + L (Γ N ) 0 p N−p (︂ )︂ q−p′ C p γ 0 ′ qp ||u||L p′ (Γ ) . + diam Ω|Ω| ||g||L q (Ω) + N N−p +

NC−1 p

F Taking into account that the product f (t)|t| remains bounded for any real t ≤ 2 N−p (see (3.2)), we can deduce from (3.37) and the previous inequality the existence of a constant C* = C* (N, p, C F , |Ω|) > 0 such that ∫︁ β Cp p′ y f (y) dx ≤ C* + ||u|| p′ + ||g||L q (Ω) ||y||W 1,p (Ω;Γ D ) L (Γ N ) 0 p N−p

Ω

(︂ + diam Ω|Ω|

q−p′ qp′

′

̂︀ 1 + C ̂︀ 2 ||u||p ′ ≤C p

L (Γ N )

)︂ C γ0 p ||u||L p′ (Γ ) ||g||L q (Ω) + N N−p )︁ (︁ ̂︀ 3 1 + ||u|| p′ ||y||W 1,p (Ω;Γ D ) . +C L (Γ ) N

(3.38)

0

Finally, using the energy equality (3.13) and the standard form of the Young’s inequality, we obtain )︁ (︁ ′ p ̂︀ 1 + C ̂︀ 2 ||u||p ′ ̂︀ 3 1 + ||u|| p′ ||y||W 1,p (Ω;Γ D ) ≤ C ||y||W 1,p (Ω;Γ D ) + C L (Γ N ) L p (Γ N ) 0 )︁ (︁ + C γ0 ||u||L p′ (Γ ) + C||g||L q (Ω) ||y||W 1,p (Ω;Γ D ) N 0 )︁ (︁ ′ p ̂︀ 1 + C ̂︀ 2 ||u|| ′ ̂︀ 4 1 + ||u|| p′ ||y||W 1,p (Ω;Γ D ) ≤C + C p L (Γ ) L (Γ N ) ′

̂︀ 1 + C ̂︀ 2 ||u||p ′ ≤C p

L (Γ N )

N

0

88 | 3 Neumann Boundary Control Problem ⎡ ⎢ ̂︀ 4 ⎢ 1 ||y||p 1,p +C + ⎣ 2C W0 (Ω;Γ D ) ̂︀ 4 ′

̂︀ 1 + C ̂︀ 2 ||u||p ′ ≤C p

+

L (Γ N )

+2

2 p−1

(︃

̂︀ 4 C p

)︃p′ /p

(︃

)︁p′ ⎤ )︃p′ /p (︁ 1 + ||u||L p′ (Γ ) ⎥ ̂︀ 4 2C N ⎥ ⎦ p p′

1 p ||y|| 2 W01,p (Ω;Γ D )

)︁ 1 (︁ p′ 1 + ||u|| . ′ p L (Γ N ) p′

Hence, ||y||

p W01,p (Ω;Γ D )

̂︀ 1 + 2 ≤ 2C

p+1 p−1

(︃

̂︀ 4 C p

)︃p′ /p

⎡ 1 ⎣ ̂︀ + 2C2 + 2 p′

p+1 p−1

(︃

̂︀ 4 C p

)︃p′ /p

⎤ 1⎦ p′ ||u|| p′ ′ L (Γ N ) p

and this implies the estimate (3.34). In order to establish the estimate (3.33), it is enough to make use of (3.34) and (3.38). The proof is complete. Remark 3.3. It is worth to notice that inequality (3.24) makes sense even if we do not ′ ′ assume fulfillment of the inclusion f (y) ∈ L p (Ω) but have only that y ∈ Y, u ∈ L p (Γ N ), and g ∈ L q (Ω) with q ≥ p′ . At the same time it is unknown whether this inequality holds for an arbitrary weak solution to BVP (3.4)–(3.5). Since the existence and uniqueness of the weak solutions to the original BVP with above properties is an open question, the following result shows that a priori estimates like (3.33)–(3.34) can be derived for those ′ pairs (u, y) in L p (Γ N ) × W01,p (Ω; Γ D ) which are feasible to OCP (3.3)–(3.6). Theorem 3.2. For fixed p ≥ 2, r ≥ p′ , and q ≥

pN , pN − N + p

′

let u ∈ L p (Γ N ) and g ∈ L q (Ω) be given distributions. Let y = y(u) ∈ W01,p (Ω; Γ D ) be a weak solution to BVP (3.4)–(3.5) such that (u, y) is a feasible pair to optimal control problem (3.3)–(3.6). Then ⃒ (︃ ⃒ ]︃ )︃ [︃ ⃒ ⃒∫︁ (︁ r )︁ pr′ ⃒ ⃒ p′ ′ ′ ′ ′ (p + 1) p p p −1 ′ p −1 1− ⃒ y f (y) dx⃒ ≤ 3 + C γ0 p + 2 C γ0 |Ω| r ⃒ ⃒ p−1 α ⃒ ⃒ Ω

{︀ }︀ × max 1, J(u, y) (︂ )︂ ′ (p + 1) p′ −1 1 p′ −1 p′ + C p ||g||L q (Ω) , 3 + ′2 p p [︃ ]︃ p′ )︁ (︁ ′ {︀ }︀ r r p p′ ′ p′ −1 1− pr ≤3 ||y|| 1,p |Ω| + C γ0 p max 1, J(u, y) W0 (Ω;Γ D ) α ′

′

p′

+ 3p −1 C p ||g||L q (Ω) .

(3.39)

(3.40)

3.2 A Priori Estimates | 89

Proof. Let (u, y) be a given feasible solution. Then relation (3.22) and inequalities (3.23), (3.15), and (3.17), immediately lead to the following estimate ||y||

p−1 W01,p (Ω;Γ D )

≤ ||f (y)||L p′ (Ω) + C γ0 ||u||L p′ (Γ ) + C||g||L q (Ω) , N

where 1

||f (y)||L p′ (Ω) ≤ |Ω| p′

− 1r

(︁ r α

J(u, y)

)︁1/r

(3.41)

< +∞

by the feasibility property of the pair (u, y). Since p−1=

p p′

and

||u||

p′ L p′ (Γ N )

≤ p′ J(u, y),

the a priori estimate (3.40) is a direct consequence of (3.41). In order to establish estimate (3.39), we make use of the energy equality (3.22) and the standard form of Young’s inequality. As a result, we obtain ⃒ ⃒ ⃒ ⃒∫︁ ⃒ ⃒ ⃒ y f (y) dx⃒ ≤ ||y||p 1,p ⃒ ⃒ W0 (Ω;Γ D ) ⃒ ⃒ Ω (︁ )︁ + C γ0 ||u||L p′ (Γ ) + C||g||L q (Ω) ||y||W 1,p (Ω;Γ D ) N 0 (︂ )︂ )︁p′ (︁ 1 1 p ≤ 1+ + ′ C γ0 ||u||L p′ (Γ ) + C||g||L q (Ω) ||y|| 1,p W0 (Ω;Γ D ) N p p [︃ ]︃ ′ Cp p+1 p p′ p′ p′ −1 +2 ||y|| 1,p C γ0 J(u, y) + ′ ||g||L q (Ω) ≤ W0 (Ω;Γ D ) p p ]︃ [︃ ′ ′ p (︁ r )︁ r {︀ }︀ p′ ′ (p + 1)3p −1 + C pγ0 p ′ max 1, J(u, y) |Ω|1− r ≤ p α ]︃ [︃ ′ ′ (p + 1)3p −1 p′ p′ Cp p′ p′ p′ −1 + C ||g||L q (Ω) + 2 C γ0 J(u, y) + ′ ||g||L q (Ω) . p p After simplification, we arrive at the expected estimate (3.39). The following results reflect some interesting properties of weak solutions satisfying inequality (3.24). In particular, Proposition 3.3 can be considered as some specification of the well-known Boccardo–Murat Theorem (see L. Boccardo and F. Murat [10, Theorem 2.1]). {︀ }︀ ′ Proposition 3.3. Assume that q ≥ p′ = p/(p − 1). Let (u k , g k , y k ) k∈N ⊂ L p (Γ N ) × L q (Ω) × W01,p (Ω; Γ D ) be a sequence such that

′

u k ⇀ u weakly in L p (Γ N ), q

g k ⇀ g weakly in L (Ω), y k → y weakly in

W01,p (Ω; Γ D )

and a.e. in Ω,

(3.42) (3.43) (3.44)

90 | 3 Neumann Boundary Control Problem f (y k ) → f (y) strongly in L1 (Ω), )︁ (︁ )︁* N , ∀ k ∈ N, − div |∇y k |p−2 ∇y k = f (y k ) + g k in C∞ 0 (R ; Γ D )

(3.45)

(︁

γ0 (y k ) = 0 ⃒ ⃒ where γ1 (y) = ∂y ∂ν ⃒

ΓN

and |γ1 (y k )|p−2 γ1 (y k ) = u k ,

(3.46)

∀ k ∈ N,

(3.47)

for all y ∈ C1 (Ω) ∩ W01,p (Ω; Γ D ). Then

∇y k → ∇y strongly in L r (Ω)N for any 1 ≤ r < p.

(3.48)

Proof. As follows from (3.46)–(3.47), the functions y k are the weak solutions to the boundary value problem (3.4)–(3.5) in the sense of distributions for the corresponding ′ controls u k ∈ L p (Γ N ). Since, by Lemma 3.1, we can use in (3.8) the test function φ = T ε−1 (y k − y) ∈ W01,p (Ω; Γ D ), where T ε is the truncation at the level ε−1 , defined by (1.17), it follows that, for every k ∈ N, we have the relation ∫︁ (︁ )︁ |∇y k |p−2 ∇y k − |∇y|p−2 ∇y, ∇T ε−1 (y k − y) dx Ω

∫︁

∫︁ f (y k )T ε−1 (y k − y) dx +

=

u k γ0 (T ε−1 (y k − y)) dH N−1

ΓN

Ω

+ ⟨g k , T ε−1 (y k − y)⟩W −1,p′ (Ω;Γ );W 1,p (Ω;Γ ) D D ∫︁ (︁ )︁ p−2 − |∇y| ∇y, ∇T ε−1 (y k − y) dx = J1 + J2 + J3 − J4 .

(3.49)

Ω ′

pN Since p′ > pN−N+p and q ≥ p′ , it follows that the embedding L q (Ω) ˓→ W −1,p (Ω; Γ D ) is compact (see, for instance, [111, Theorem 11.2]). Hence, (3.43) and (3.44) imply that ′

g k → g strongly in W −1,p (Ω; Γ D ), T ε−1 (y k − y) → 0 weakly in W01,p (Ω; Γ D ) and strongly in L p (Ω). Then J3 − J4 tends to zero as k → ∞. As for the term J2 , we make use of the following ′ reasoning. By Sobolev embedding theorem, the injection W 1/p ,p (Γ N ) ˓→ L r (Γ N ) is (︀ )︀* N−1 compact for all 1 ≤ r < p N−p . Hence, by duality arguments, L r (Γ N ) is compactly (︁ )︁* ′ embedded in W 1/p ,p (Γ N ) . So, if we define (︂ r* =

N−1 p N−p

)︂′ =

N−1 ′ p N

3.2 A Priori Estimates | 91

(︁ )︁* ′ ′ then we have p′ > r* and, therefore, the injection L p (Γ N ) ˓→ W 1/p ,p (Γ N ) is compact. Thus, due to (3.42)–(3.44), we have u k → u strongly in

(︁

′

W 1/p ,p (Γ N )

)︁*

,

′

γ0 (y k ) ⇀ γ0 (y) weakly in W 1/p ,p (Γ N ). As a result, we obtain ∫︁ J2 =

u k γ0 (T ε−1 (y k − y)) dH N−1 → 0 as k → ∞.

ΓN

It remains to note that, in view of (3.45), we have J1 ≤ C||T ε−1 (y k − y) ||L∞ (Ω) ,

∀ k ∈ N.

Hence, mollifying T ε (y k − y) and the poinwise convergence y k (x) → y(x) a.e. in Ω imply that |J1 | ≤ C||T ε−1 (y k − y) ||L∞ (Ω) ≤ C1 ε, ∀ k ∈ N. (3.50) Combining all issues given above, we can finally deduce that, for a fixed ε > 0, ∫︁ (︁ )︁ |∇y k |p−2 ∇y k − |∇y|p−2 ∇y, ∇T ε−1 (y k − y) dx ≤ C1 ε. (3.51) lim sup k→∞

Ω

Let us define now the following functions (︁ )︁ d k (x) = |∇y k |p−2 ∇y k − |∇y|p−2 ∇y, ∇y k − ∇y ,

k∈N

and fix θ with 0 < θ < 1. In view of the initial assumptions, it is clear that {d k }k∈N is a bounded sequence in L1 (Ω) and (︁ )︁ |∇y k |p−2 ∇y k − |∇y|p−2 ∇y, ∇y k − ∇y ≥ 22−p |∇y k − ∇y|p (3.52) by the strict monotonicity property of the p-Laplace operator. Splitting the set Ω into {︀ }︀ S kε = x ∈ Ω : |y k (x) − y(x)| ≤ ε ,

{︀ }︀ G kε = x ∈ Ω : |y k (x) − y(x)| > ε

92 | 3 Neumann Boundary Control Problem

and using Hölder inequality, we get ∫︁ ∫︁ ∫︁ d θk dx = d θk dx + d θk dx Ω

S kε

G kε

⎞θ ⎛ ⎞θ ⎛ ∫︁ ∫︁ ⎟ ⎜ ⎟ ⎜ ≤ ⎝ d k dx⎠ |S kε |1−θ + ⎝ d k dx⎠ |G kε |1−θ G kε

S kε

⎛ ⎞θ ∫︁ (︁ )︁ ≤⎝ |∇y k |p−2 ∇y k − |∇y|p−2 ∇y, ∇T ε−1 (y k − y) dx⎠ |S kε |1−θ Ω

⎛ ⎞θ ∫︁ + ⎝ d k dx⎠ |G kε |1−θ .

(3.53)

Ω

Since, for a fixed ε, |G kε | tends to zero as k → ∞, it follows from (3.51), (3.52), and (3.53) that ∫︁ ∫︁ (︀ )︀θ |∇y k − ∇y|p dx ≤ 2θ(p−2) lim sup d θk dx ≤ 2θ(p−2) (C1 ε)θ |Ω|1−θ . lim sup k→∞

k→∞

Ω

Ω

Letting ε tend to 0 and θ tend to 1 implies that |∇y k − ∇y|p tends strongly to 0 in L1 (Ω) and thus, there exists a subsequence {k n }n∈N such that ∇y k n (x) → ∇y(x)

{︀

Since ∇y k n that

}︀ n∈N

a.e. in Ω as k n → ∞.

(3.54)

is a bounded sequence in L p (Ω)N , it follows from Vitali’s theorem ∇y k n → ∇y strongly in L r (Ω)N for any 1 ≤ r < p.

(3.55)

It remains to note that in fact we have the convergence in (3.55) for the whole sequence {∇y k }k∈N because the limit ∇y in (3.55) is independent of the subsequence {k n }n∈N . ′

Proposition 3.4. Let (u, y) be an arbitrary pair in L p (Γ N ) × W01,p (Ω; Γ D ). Also let {︀ }︀ ′ (u k , y k ) k∈N ⊂ L p (Γ N ) × Y be a sequence such that, for each k ∈ N, the pairs (u k , y k ) are related by the integral identity (3.8), satisfy inequality (3.24), and ′

(u k , y k ) ⇀ (u, y) weakly in L p (Γ N ) × W01,p (Ω; Γ D ) as k → ∞.

(3.56)

′

Then y is a weak solution to BVP (3.4)–(3.5) for the given u ∈ L p (Γ N ), the pair (u, y) satisfies the inequality (3.24), and f (y k ) → f (y) in L1 (Ω) as k → ∞.

(3.57)

3.2 A Priori Estimates | 93

Proof. By the Sobolev Embedding Theorem, the injection W01,p (Ω; Γ D ) ˓→ L p (Ω) is compact. Hence, the weak convergence y k ⇀ y in W01,p (Ω; Γ D ) implies the strong convergence in L p (Ω). Therefore, up to a subsequence, we can suppose that y k (x) → y(x) for almost every point x ∈ Ω. As a result, we have the pointwise convergence: f (y k ) → f (y) almost everywhere in Ω. Let us show that this fact implies the strong convergence (3.57). {︀ }︀ Let us show that the sequence f (y k ) k∈N is equi-integrable on Ω. To do so, we take m > 0 such that )︂ (︂ p′ p′ −1 + C3 δ−1 , (3.58) m > 2 C1 sup ||u k || p′ + C2 sup ||u k || p′ L (Γ N )

k∈N

L (Γ N )

k∈N

where the constants C i , i = 1, 2, 3, are as in (3.33). We also set τ = δ/(2f (m)). Then for every measurable set S ⊂ Ω with |S| < τ, we have ∫︁ ∫︁ ∫︁ f (y k ) dx = f (y k ) dx + f (y k ) dx S

{x∈S : y k (x)>m}

{x∈S : y k (x)≤m}

∫︁

∫︁

1 ≤ m by (3.33)

≤

by (3.58)

≤

f (m) dx

y k f (y k ) dx + {x∈S : y k (x)≤m}

{x∈S : y k (x)>m}

C1 ||u k ||

p′ L p′ (Γ N )

+ C2 ||u k ||

p′ −1 L p′ (Γ N )

+ C3

m

+ f (m)|S|

δ δ + . 2 2

As a result, the assertion (3.57) is a direct consequence of Lebesgue’s Convergence Theorem. Thus, y ∈ Y. Let us show now that the limit pair (u, y) is related by the integral identity (3.8). Indeed, in view of the initial assumptions and property (3.57), the limit passage in the right-hand side of the equality ∫︁ ∫︁ ∫︁ p−2 N−1 |∇y k | (∇y k , ∇φ) dx = f (y k )φ dx + u k φ dH Ω

ΓN

Ω

∫︁ +

N gφ dx, ∀ φ ∈ C∞ 0 (R ; Γ D )

(3.59)

Ω

becomes trivial. Taking into account Proposition 3.3, we have, up to a subsequence, the pointwise {︀ }︀ ′ convergence (3.54). Since the sequence |∇y k |p−2 ∇y k k∈N is bounded in L p (Ω)N , it follows from (3.54) that |∇y k n |p−2 ∇y k n → |∇y|p−2 ∇y almost everywhere in Ω, ′

|∇y k n |p−2 ∇y k n ⇀ |∇y|p−2 ∇y weakly in L p (Ω)N .

94 | 3 Neumann Boundary Control Problem

This allows us to pass to the limit as k n → ∞ in the left hand side of the equality (3.59). ′ Thus, we see that y is a weak solution to BVP (3.4)–(3.5) for the given u ∈ L p (Γ N ). It remains to prove that the limit pair (u, y) satisfies the inequality (3.24). With that in mind we note ∫︁ ∫︁ by (3.56) lim g (x − x0 , ∇y k ) dx = g (x − x0 , ∇y) dx, k→∞

Ω

Ω

∫︁ lim inf k→∞

′

|∇u k |p dH

N−1 by (3.56)

≥

Ω by (3.57)

F(y k ) dx

lim

∫︁ F(y) dx,

=

Ω

Ω

∫︁

by (3.56)

p

|∇y k | dx

lim inf k→∞

′

|∇u|p dH N−1 ,

Ω

∫︁ k→∞

∫︁

∫︁

≥

Ω

|∇y|p dx.

Ω

Then we can pass to the limit in the inequality (3.24) to finally obtain (︂ )︂ ∫︁ )︂ (︂ ∫︁ N N p |∇y| dx ≤ −1 − 1 lim inf |∇y k |p dx p p k→∞ Ω Ω ⎡ ⎤ ∫︁ ∫︁ ≤ lim inf ⎣N F(y k ) dx − g (x − x0 , ∇y k ) dx⎦ k→∞

Ω

Ω

⎤ ′ β p N−1 ⎦ |u k | dH + lim inf ⎣ ′ p k→∞ ΓN ∫︁ ∫︁ ∫︁ ′ β ≤ N F(y) dx − g (x − x0 , ∇y) dx + ′ |u|p dH N−1 . p ⎡

∫︁

Ω

ΓN

Ω

The proof is complete. This result can be slightly specified for the case of feasible solutions. {︀ }︀ Proposition 3.5. Assume that q ≥ p′ and r ≥ p′ . Let (u k , y k ) k∈N ⊂ Ξ be a sequence of feasible solutions such that sup J(u k , y k ) < +∞,

(3.60)

k∈N

′

(u k , y k ) ⇀ (u, y) weakly in L p (Γ N ) × W01,p (Ω; Γ D ) as k → ∞.

(3.61)

Then (u, y) ∈ Ξ and f (y k ) → f (y) strongly in L1 (Ω) and weakly in L r (Ω) as k → ∞.

(3.62)

Proof. By the Sobolev Embedding Theorem, the injection W01,p (Ω; Γ D ) ˓→ L p (Ω) is compact. Hence, the weak convergence y k ⇀ y in W01,p (Ω; Γ D ) implies the strong

3.2 A Priori Estimates | 95

convergence in L p (Ω). Therefore, up to a subsequence, we can suppose that y k (x) → y(x) for almost every point x ∈ Ω. As a result, we have the pointwise convergence: f (y k ) → f (y) almost everywhere in Ω. Let us show that this fact implies the strong convergence (3.62). {︀ }︀ Let us show that the sequence f (y k ) k∈N is equi-integrable on Ω. To do so, we take m > 0 such that m > 2Lδ−1 , (3.63) where ]︃ [︃ )︃ p′ ′ (︁ + 1) r )︁ r p′ ′ p′ −1 p′ 1− pr L := 3 + C γ0 p + 2 C γ0 |Ω| p−1 α )︂ {︂ }︂ (︂ ′ (p + 1) p′ −1 1 p′ −1 p′ 3 + ′2 C p ||g||L q (Ω) . × max 1, sup J(u k , y k ) + p p k∈N (︃

p′ −1 (p

We also set τ = δ/(2f (m)). Then for every measurable set S ⊂ Ω with |S| < τ, we have ∫︁ ∫︁ ∫︁ f (y k ) dx = f (y k ) dx + f (y k ) dx S

{x∈S : y k (x)>m}

{x∈S : y k (x)≤m}

∫︁

∫︁

1 ≤ m

{x∈S : y k (x)≤m}

{x∈S : y k (x)>m} by (3.39)

≤

f (m) dx

y k f (y k ) dx + L + f (m)|S| m

by (3.63)

≤

δ δ + . 2 2

As a result, the assertion (3.62) is a direct consequence of Lebesgue’s Convergence Theorem. Let us show now that the limit pair (u, y) is a feasible pair to optimal control problem (3.3)–(3.6). Indeed, in view of the initial assumptions and property (3.62), the limit passage in the right-hand side of the equality ∫︁ ∫︁ ∫︁ N−1 p−2 |∇y k | (∇y k , ∇φ) dx = f (y k )φ dx + u k φ dH Ω

ΓN

Ω

∫︁ +

N gφ dx, ∀ φ ∈ C∞ 0 (R ; Γ D )

(3.64)

Ω

becomes trivial. Taking into account Proposition 3.3, we have, up to a subsequence, the pointwise {︀ }︀ ′ convergence (3.54). Since the sequence |∇y k |p−2 ∇y k k∈N is bounded in L p (Ω)N , it follows from (3.54) that |∇y k n |p−2 ∇y k n → |∇y|p−2 ∇y almost everywhere in Ω, ′

|∇y k n |p−2 ∇y k n ⇀ |∇y|p−2 ∇y weakly in L p (Ω)N .

96 | 3 Neumann Boundary Control Problem This allows us to pass to the limit as k n → ∞ in the left hand side of the equality (3.64). ′ Thus, y is a weak solution to BVP (3.4)–(3.5) for the given u ∈ L p (Γ N ). ′ Since the set Aad is convex and closed in L p (Γ N ), it follows that this set is sequen′ tially weakly closed in L p (Γ N ) by the Mazur theorem. Therefore, the weak convergence (3.61) implies that u ∈ Aad . It remains to prove that the limit pair (u, y) satisfies the condition J(u, y) < +∞. With that in mind we take into account the lower semi-continuity of the norm in ′ ′ L p (Γ N ) × L2 (Ω) with respect to the weak convergence in L p (Γ N ) × W01,p (Ω; Γ D ) and property (3.62). This yields ∫︁ ∫︁ by (3.61) |y − y d |2 dx, (3.65) lim |y k − y d |2 dx = k→∞

Ω

∫︁ lim inf k→∞

Ω ′

|u k |p dH

N−1 by (3.61)

∫︁

≥

Ω

′

|u|p dH N−1 .

(3.66)

Ω

In view of condition (3.60), we have sup ||f (y k )||L r (Ω) < +∞. k∈N

Utilizing this fact together with the pointwise convergence f (y k ) → f (y) a.e. in Ω which is a consequence of the property (3.62), we get f (y k ) ⇀ f (y) in L r (Ω). Hence, ∫︁ ∫︁ (3.67) lim inf |f (y k )|r dx ≥ |f (y)|r dx. k→∞

Ω

Ω

As a result, we deduce from (3.65), (3.66), and (3.67) that J(u, y) ≤ lim inf J(u k , y k ) < sup J(u k , y k ) < +∞. k→∞

k∈N

Thus, (u, y) is a feasible solution to the problem (3.3)–(3.6) in the sense of Definition 3.1. The proof is complete.

3.3 Existence of optimal boundary controls The aim of this section is to study the existence of a solution for the optimal control problem (3.3)–(3.6). As was mentioned before, there is no reason to suppose that there ′ exists a weak solution to (3.4)–(3.5) for a given boundary control u ∈ L p (Γ N ). As for the last term in the cost functional (3.3), it plays rather special role and it is unknown whether this OCP is consistent without this stabilizing term. Let us show that in contrast to the BVP (3.4)–(3.5), the corresponding optimal control problem (3.3)–(3.6) is wellposed and consistent.

3.3 Existence of Optimal Boundary Controls | 97

Theorem 3.3. Let p ≥ 2, r ≥ p′ , and q ≥ p′ be given exponents. Assume that for given f ∈ C loc (R), g ∈ L q (Ω), and Aad , the set of feasible solutions Ξ is nonempty. Then, for any y d ∈ L2 (Ω), the optimal control problem (3.3)–(3.6) has at least one solution. Proof. Since J(u, y) ≥ 0 for all (u, y) ∈ Ξ, it follows that there exists a non-negative {︀ }︀ value μ ≥ 0 such that μ = inf (u,y)∈Ξ J(u, y). Let (u k , y k ) k∈N be a minimizing sequence to the problem (3.3)–(3.6), i.e. (u k , y k ) ∈ Ξ

∀k ∈ N

lim J(u k , y k ) = μ.

and

k→∞

So, we can suppose that J(u k , y k ) ≤ μ + 1 for all k ∈ N.

(3.68)

Then taking into account the implicit form of the cost functional (3.3), Theorem 3.2, pN , we deduce the following estimates and the fact that q ≥ p′ > pN−N+p p sup ||y k || 1,p W0 (Ω;Γ D ) k∈N

≤3

[︃

p′ −1

′

1− pr

|Ω|

′

(︁ r )︁ pr′ α

′

+

′ C pγ0 p ′

by (3.68)

p′

]︃

{︂ }︂ max 1, sup J(u k , y k ) k∈N

′

′

p′

+ 3p −1 C pem ||g||L q (Ω) ≤ 3p −1 C pem ||g||L q (Ω) ]︃ [︃ p′ ′ (︁ r )︁ r p′ ′ p′ −1 1− pr + C γ0 p (μ + 1), +3 |Ω| α ||u k ||

p′ L p′ (Γ N ) r

(3.69)

≤ p′ sup J(u k , y k ) ≤ p′ (μ + 1),

(3.70)

r r sup J(u k , y k ) ≤ (μ + 1). α k∈N α

(3.71)

||f (y k )||L r (Ω) ≤

k∈N

Thus, without loss of generality, we can suppose that there exists a subsequence {︀ }︀ of the minimizing sequence (u k , y k ) k∈N (still denoted by the same index) and a pair ′

(u0 , y0 ) ∈ L p (Γ N ) × W01,p (Ω; Γ D ) such that ′

(u k , y k ) ⇀ (u0 , y0 ) weakly in L p (Γ N ) × W01,p (Ω; Γ D ) as k → ∞, 0

y k (x) ⇀ y (x) a.e. in Ω.

(3.72) (3.73)

Let us show now that the limit pair (u0 , y0 ) is feasible to the problem (3.3)–(3.6). {︀ }︀ ′ Indeed, since (u k , y k ) k∈N ⊂ Ξ, it follows that f (y k ) ∈ L p (Ω) for all k ∈ N. Hence, for every k ∈ N, (u k , y k ) satisfies the inequality (3.24) by Proposition 3.2. Moreover, Proposition 3.4 implies that in this case y0 is a weak solution to BVP (3.4)–(3.5) for ′ u = u0 . Since the set Aad is convex and closed in L p (Γ N ), it is sequentially weakly ′ closed in L p (Γ N ) by the Mazur theorem [48, Appendix D.4]. Therefore, (3.72) implies that ′ u0 ∈ Aad . To deduce the condition (u0 , y0 ) ∈ Ξ, it remains to show that f (y0 ) ∈ L p (Ω). With that in mind, we make use of the pointwise convergence (3.73). Then we have that

98 | 3 Neumann Boundary Control Problem }︀ {︀ f (y k ) → f (y) almost everywhere in Ω. Since the sequence f (y k ) k∈N is bounded in ′

L p (Ω) (see (3.71)), it follows that ′

f (y k ) ⇀ f (y0 ) weakly in L p (Ω),

(3.74)

′

and, hence, f (y0 ) ∈ L p (Ω). Thus, the limit pair (u0 , y0 ) is feasible to the problem (3.3)–(3.6). To conclude the proof, we take into account the lower semi-continuity of the cost ′ functional J : L p (Γ N ) × W01,p (Ω; Γ D ) → R with respect to the weak convergence in ′

L p (Γ N ) × W01,p (Ω; Γ D ) and property (3.74). This yields μ = inf J(u, y) = lim J(u k , y k ) ≥ J(u0 , y0 ). (u,y)∈Ξ

k→∞

Thus, (u0 , y0 ) ∈ Ξ is an optimal pair to the problem (3.3)–(3.6).

3.4 On approximation of optimal boundary control problem Following in many aspect the previous chapters (see also [23, 47, 81, 82, 108]), in this section we introduce a special family of optimization problems with fictitious controls and show that an optimal pair to the original optimal control problem can be attained by optimal solutions to the approximating ones provided the original problem admits at least one bounded feasible solution. With that in mind we consequently provide the well-posedness analysis for the perturbed partial differential equations as well as for the corresponding fictitious optimal control problems. After that we pass to the limits as k → ∞ and ε → 0. Since the fictitious optimization problems are stated for the quasilinear elliptic equations with coercive and monotone operators, the approximation and regularization approach is not only considered to be useful for the rigorous analysis of the original problem, but also for the purpose of its numerical simulations. Let us introduce the following two-parametric family of perturbed optimal control problems ∫︁ ∫︁ {︁ ′ k 1 |y − y d |2 dx + ′ Minimize I ε,k (u, v, y) = |v − T ε (f (y))|p dx 2 p Ω Ω ∫︁ ∫︁ }︁ ′ α 1 p′ N−1 |u| dH + ′ |v|p dx + ′ (3.75) p p ΓN

Ω

subject to the constraints (︁ )︁ − div |∇y|p−2 ∇y = v + g y=0 ′

v ∈ L p (Ω),

on Γ D ,

|∇y|

p−2 ′

Ω,

(3.76)

on Γ N ,

(3.77)

y ∈ W01,p (Ω; Γ D ).

(3.78)

in

∂ν y = u

u ∈ Aad ⊂ L p (Γ N ),

3.4 On Approximation of OCP | 99

′

We consider here the function v ∈ L p (Ω) as a fictitious control. Since the operator (︁ )︁ ′ −∆ p (·) = − div |∇ · |p−2 ∇· : W01,p (Ω; Γ D ) → W0−1,p (Ω; Γ D ) is bounded, strictly monotone, semi-continuous, and coercive (see [113, Section II.1.1]), ′ it follows from the general theory of monotone operators that for each u ∈ L p (Γ N ), ′ v ∈ L p (Ω), and g ∈ L q (Ω) with p ≥ 2 and q ≥ p′ , boundary value problem (3.76)–(3.77) admits a unique weak solution y ∈ W01,p (Ω; Γ D ) satisfying the integral identity ∫︁

|∇y|p−2 (∇y, ∇φ) dx =

Ω

∫︁ vφ dx Ω

∫︁ +

uφ dH N−1 +

ΓN

∫︁

N gφ dx, ∀ φ ∈ C∞ 0 (R ; Γ D )

Ω

and the energy equality ∫︁ ∫︁ p = vy dx + uγ0 (y) dH N−1 + ⟨g, y⟩W −1,p′ (Ω;Γ ||y|| 1,p W0 (Ω;Γ D )

Ω

(3.79)

1,p D );W 0 (Ω;Γ D )

.

(3.80)

ΓN

Taking into account the inequalities (3.15), (3.19), and (3.16), it is easy to deduce from (3.80) the following a priory estimate ||y||

p−1 W01,p (Ω;Γ D )

q−p′

≤ |Ω| qp′ ||g||L q (Ω) + diam Ω||v||L p′ (Ω) + C γ0 ||u||L p′ (Γ ) . N

(3.81)

Thus, for every positive value ε > 0 and integer k ∈ N, the set of feasible solutions to the problem (3.75)–(3.78), ⎧ ⃒ ⎫ ′ ′ ⃒ ⎪ ⎪ u ∈ Aad ⊂ L p (Γ N ), v ∈ L p (Ω), ⃒ ⎨ ⎬ ⃒ 1,p Ξ0 = (u, v, y) ⃒ (3.82) y ∈ W0 (Ω; Γ D ), ⃒ ⎪ ⎪ ⎩ ⃒ (u, v, y) are related by integral identity (3.79) ⎭ is nonempty. Let us show that the optimal control problem (3.75)–(3.78) is solvable. Theorem 3.4. Let p ≥ 2 and q ≥ p′ . Then, for a given y d ∈ L2 (Ω) and for every positive value ε > 0 and integer k ∈ N, the approximating optimal control problem (3.75)–(3.78) has at least one solution. Proof. Since I ε,k (u, v, y) ≥ 0

for all

′

′

(u, v, y) ∈ L p (Γ N ) × L p (Ω) × W01,p (Ω; Γ D ),

it follows that there exists a non-negative value μ ε,k such that μ ε,k =

inf

(u,v,y)∈Ξ0

I ε,k (u, v, y).

100 | 3 Neumann Boundary Control Problem {︀ }︀ Let (u m , v m , y m ) m∈N be a minimizing sequence, i.e. (u m , v m , y m ) ∈ Ξ0 ∀ m ∈ N and

lim I ε,k (u m , v m , y m ) = μ ε,k .

m→∞

Then, without loss of generality, we can suppose that ∫︁ ∫︁ ′ 1 k I ε,k (u m , v m , y m ) = |v m − T ε (f (y m ))|p dx |y m − y d |2 dx + ′ 2 p Ω Ω ∫︁ ∫︁ ′ α 1 p′ N−1 |u m | dH + ′ |v m |p dx + ′ p p ΓN

Ω

≤ μ ε,k + 1

for all m ∈ N.

(3.83)

Since T ε (f (y m )) ∈ L∞ (Ω), it follows from (3.83) that the sequence of fictitious ′ controls {v m }k∈N is uniformly bounded in L p (Ω). The similar conclusion can be made ′

for the sequence {u m }k∈N with respect to the norm of L p (Γ N ). So, we can admit the ′

′

existence of elements v0 ∈ L p (Ω) and u0 ∈ L p (Γ N ) such that (up to a subsequence) ′

′

v m ⇀ v0 in L p (Ω) and u m ⇀ u0 in L p (Γ N ) as m → ∞.

(3.84)

Moreover, in view of estimate (3.81), we see that the sequence {y m }k∈N is bounded in W01,p (Ω; Γ D ). As a result, we deduce the existence of a subsequence of {y m }k∈N , denoted in the same way, and an element y0 ∈ W01,p (Ω; Γ D ) such that y m ⇀ y0 in W01,p (Ω; Γ D ) as m → ∞. Let us prove that y0 is the solution of (3.76)-(3.77) with v = v0 and u = u0 . To do so, we fix an arbitrary test function φ ∈ C∞ 0 (Ω; Γ D ) and pass to the limit in the integral identity ∫︁ ∫︁ ∫︁ ∫︁ |∇y m |p−2 (∇y m , ∇φ) dx = v m φ dx + u m φ dH N−1 + gφ dx (3.85) Ω

ΓN

Ω

Ω ′

′

as m → ∞. In view of the convergences v m ⇀ v0 in L p (Ω) and u m ⇀ u0 in L p (Γ N ), we have ∫︁ ∫︁ ∫︁ ∫︁ lim v m φ dx = v0 φ dx, lim u m φ dH N−1 = u0 φ dH N−1 . m→∞

m→∞

Ω

Ω

ΓN

ΓN

As for the limit in the left-hand side of (3.85), we make use of Proposition 3.3. Then ∇y m → ∇y0 strongly in L r (Ω)N for any 1 ≤ r < p and, therefore, up to a subsequence, we have the pointwise convergence ∇y m (x) → ∇y0 (x) almost everywhere in Ω. Since {︀ }︀ ′ the sequence |∇y m |p−2 ∇y m k∈N is bounded in L p (Ω)N , it follows that |∇y m |p−2 ∇y m → |∇y0 |p−2 ∇y0 almost everywhere in Ω, ′

|∇y m |p−2 ∇y m ⇀ |∇y0 |p−2 ∇y0 weakly in L p (Ω)N .

3.4 On Approximation of OCP |

101

Thus, passing to the limit in relation (3.85) as m → ∞, we arrive at the integral identity ∫︁ ∫︁ ∫︁ ∫︁ (︁ )︁ |∇y0 |p−2 ∇y0 , ∇φ dx = v0 φ dx + u0 φ dH N−1 + gφ dx Ω

ΓN

Ω

Ω

1,p 0 which holds true for every φ ∈ C∞ 0 (Ω; Γ D ). Hence, y ∈ W 0 (Ω; Γ D ) is a weak solution 0 to the boundary value problem (3.76)-(3.77) for v = v and u = u0 . Since the solution of (3.76)-(3.77) is unique, the whole sequence {y m }m∈N converges weakly to y0 in W01,p (Ω; Γ D ). It remains to note that by the Mazur theorem the set Aad is sequentially ′

weakly closed in L p (Γ N ). Therefore, (3.84) implies that u0 ∈ Aad . As a result, we have )︁ (︁ u0 , v0 , y0 ∈ Ξ0 . (︀ )︀ To deduce the fact that u0 , v0 , y0 is an optimal solution to the problem (3.75)– (3.78), we notice that due to the strong convergence y m → y0 in L p (Ω), we have 1

1

T ε (f (y m )) → T ε (f (y0 )) a.e. in Ω and sup ||T ε (f (y m ))||L q (Ω) ≤ ε− q |Ω| q m∈N

for each q > p′ . Then the Vitali’s lemma implies that T ε (f (y m )) → T ε (f (y0 )) strongly ′ in L p (Ω). Combining this fact with the weak convergence (3.84) and the lower semicontinuity of the norm || · ||L p′ (Ω) + || · ||L p′ (Γ ) with respect to the weak convergence in ′

N

′

L p (Ω) × L p (Γ N ), we finally obtain inf

(u,v,y)∈Ξ0

(︁ )︁ I ε,k (u, v, y) = lim I ε,k (u m , v m , y m ) ≥ I ε,k u0 , v0 , y0 . m→∞

(︀ )︀ Thus, u0 , v0 , y0 is an optimal solution to the problem (3.75)–(3.78). Theorem 3.5. Assume that 2 ≤ p < N, r = p′ in (3.3), and for given f ∈ C loc (R), g ∈ L q (Ω), and Aad , there exists a pair ]︁ [︁ ′ (̃︀ u, ̃︀ y) ∈ L p (Γ N ) × W01,p (Ω; Γ D ) ∩ L∞ (Ω) such that (̃︀ u, ̃︀ y) ∈ Ξ. {︀ )︀ Let p ≥ 2 and q ≥ p′ be given exponents. Let (u0ε,k , v0ε,k , y0ε,k ) } ε>0 be an arbik∈N

trary sequence of optimal solutions to the approximating problems (3.75)–(3.78). Then the sequence {︁ )︁ (u0ε,k , v0ε,k , y0ε,k ) } ε>0 k∈N

′

′

is bounded in L p (Γ N ) × L p (Ω) × respect to the weak topology of ′

W01,p (Ω; Γ D )

and any its cluster triple (u0 , v0 , y0 ) with

′

L p (Γ N ) × L p (Ω) × W01,p (Ω; Γ D ) is such that v0 = f (y0 ) and (u0 , y0 ) is a feasible solution of the OCP (3.3)–(3.6).

102 | 3 Neumann Boundary Control Problem

u, ̃︀ y) ∈ Ξ Proof. In view of the initial assumptions, for a given p ≥ 2, there exists a pair (̃︀ such that ̃︀ y) ∈ L∞ (Ω) and, therefore, y ∈ L∞ (Ω). Since f ∈ C loc (R), it follows that f (̃︀ (̃︀ u, ̃︀ v, ̃︀ y) := (̃︀ u, f (̃︀ y), ̃︀ y) ∈ Ξ0 for all k ∈ N and ε > 0. Then, for sufficiently small ε > 0, (︀ )︀ we can suppose that ̃︀ v ε,k = T ε f (̃︀ y) almost everywhere in Ω. Taking this fact into account, we see that (︁ )︁ )︀ (︀ u, ̃︀ v, ̃︀ y I ε,k u0ε,k , v0ε,k , y0ε,k = inf I ε,k (u, v, y) ≤ I ε,k ̃︀ (u,v,y)∈Ξ0 ∫︁ ∫︁ ∫︁ ′ ′ 1 α 1 2 = |̃︀ u|p dH N−1 + ′ |̃︀ v|p dx, (3.86) |̃︀ y − y d | dx + ′ 2 p p ΓN

Ω

Ω

where ∫︁

′

|̃︀ v|p dx =

Ω

∫︁

⃒ ⃒p′ p′ ⃒f (̃︀ y)⃒ dx ≤ ||f (̃︀ y)||L∞ (Ω) |Ω|.

(3.87)

Ω

Then, utilizing the estimates (3.86) and (3.87), we obtain (︁ [︁ 1 ∫︁ )︁ sup I ε,k u0ε,k , v0ε,k , y0ε,k = sup |y0ε,k − y d |2 dx 2 ε>0 ε>0 k∈N k∈N Ω ∫︁ k 0 0 p′ |v ε,k − T ε (f (y ε,k ))| dx + ′ p Ω ∫︁ ∫︁ ]︁ ′ 1 α 0 q N−1 + ′ |u ε,k | dH + ′ |v0ε,k |p dx p p ΓN Ω ∫︁ ∫︁ 1 1 α p′ ≤ y)||L∞ (Ω) |Ω| < +∞ |̃︀ y − y d |2 dx + |̃︀ u|q dx + ′ ||f (̃︀ 2 q p Ω

Ω

and, as a consequence, we can deduce the existence of a constant C* > 0 independent on ε and k such that sup ||v0ε,k || ε>0 k∈N

p′ L p′ (Ω)


0 k∈N

and sup ||v0ε,k − T ε (f (y0ε,k ))|| ε>0

p′ L p′ (Γ N )

p′ L p′ (Ω)

< p′ C* ,

< p′ C* k−1 .

(3.88)

As a result, we can derive from (3.81) the following estimate sup ||y0ε,k ||W 1,p (Ω;Γ D ) ε>0 k∈N

0

(︂ ≤ sup |Ω| ε>0 k∈N

≤ |Ω|

q−p′ qp′

q−p′ qp′

||g||L q (Ω) + diam Ω||v0ε,k ||L p′ (Ω) (︂

||g||L q (Ω) + diam Ω

p′ * C α

)︂1/p′

+

C γ0 ||u0ε,k ||L p′ (Γ ) N

1 )︂ p−1

(︁ )︁1/p′ = C1 < +∞, + C γ0 p ′ C *

(3.89)

3.4 On Approximation of OCP |

103

}︀ {︀ i.e. the sequence y0ε,k ε>0 is bounded in W01,p (Ω; Γ D ) and, therefore, we can suppose k∈N

that the sequence of optimal solutions to the approximating problems {︁(︁ )︁}︁ u0ε,k , v0ε,k , y0ε,k ) ε>0 k∈N ′

′

is compact with respect to the weak convergence in L p (Γ N ) × L p (Ω) × W01,p (Ω; Γ D ). Let (u0 , v0 , y0 ) be its any cluster triplet, i.e. up to a subsequence, we have ′

y0ε,k ⇀ y0 in W01,p (Ω; Γ D ), u0ε,k ⇀ u0 in L p (Γ N ),

(3.90)

v0ε,k ⇀ v0 in L p′ (Ω).

Then applying the arguments of the proof of Theorem 3.4, it can be shown that y0 = y(u0 , v0 ) is a weak solution of (3.76)-(3.77) with v = v0 and u = u0 . Moreover, for any φ ∈ C∞ 0 (Ω; Γ D ) we have ∫︁ ∫︁ ⃒ ⃒ ⃒ ∫︁ ⃒ ∫︁ ⃒ ⃒ ⃒ ⃒ 0 0 0 0 ⃒ v φ dx − f (y )φ dx⃒ ≤ ⃒ v ε,k φ dx − v φ dx⃒ Ω

Ω

Ω

Ω

∫︁ ⃒ ∫︁ ⃒ ⃒ ⃒ 0 + ⃒ v ε,k φ dx − f (y0 )φ dx⃒ Ω

Ω

⃒ ∫︁ ⃒ ⃒ ∫︁ (︁ (︁ )︁⃒ )︁ ⃒ ⃒ ⃒ ⃒ v0ε,k − v0 φ dx⃒ + ⃒v0ε,k − T ε f (y0ε,k ) ⃒ |φ| dx ≤⃒ Ω

Ω

∫︁ ⃒ ∫︁ ⃒ (︁ ⃒ ⃒ )︁ ⃒ ⃒ ⃒ ⃒ + ⃒T ε f (y0ε,k ) − f (y0ε,k )⃒ |φ| dx + ⃒f (y0ε,k ) − f (y0 )⃒ |φ| dx Ω

Ω

= J0 + J1 + J2 + J3 ,

(3.91)

⃒ by (3.90)3 ⃒ ∫︁ (︁ )︁ ⃒ ⃒ v0ε,k − v0 φ dx⃒ → 0, J0 = ⃒

(3.92)

where

Ω

(︁ )︁ J1 ≤ ||v0ε,k − T ε f (y0ε,k ) ||L p′ (Ω) ||φ||L p (Ω)

by (3.88)

(︁ )︁ J2 ≤ ||T ε f (y0ε,k ) − f (y0ε,k )||L1 (Ω) ||φ||L∞ (Ω)

→

0,

(3.93)

by properties of T ε

→

0,

(3.94)

J3 ≤ ||f (y0ε,k ) − f (y0 )||L1 (Ω) ||φ||L∞ (Ω) as ε → 0 and k → ∞. Let us show that f (y0ε,k ) → f (y0 ) strongly in L1 (Ω) as ε → 0 and k → ∞. With that in mind, we note that by compactness of the injection W01,p (Ω; Γ D ) ˓→ L p (Ω),

(3.95)

104 | 3 Neumann Boundary Control Problem }︀ {︀ we can deduce the existence of a subsequence of y0ε,k ε>0 , denoted in the same way, k∈N

such that f (y0ε,k ) → f (y0 ) almost everywhere in Ω. So, in order to conclude (3.95), it {︀ (︀ )︀}︀ remains to establish the equi-integrability on Ω of the sequence f y0ε,k ε>0 . For k∈N

this purpose, we make use of the following relation, coming from the energy equality (3.80), ∫︁ (︁ ∫︁ (︁ )︁)︁ (︁ )︁ p v0ε,k − f y0ε,k y0ε,k dx − y0ε,k f y0ε,k dx = ||y0ε,k || 1,p W0 (Ω;Γ D )

Ω

Ω

∫︁ −

⟨ ⟩ u0ε,k γ0 (y0ε,k ) dH N−1 − g, y0ε,k

ΓN

W −1,p′ (Ω;Γ D );W01,p (Ω;Γ D )

.

(3.96)

Then ||y0ε,k || ∫︁

by (3.89) p ≤ W01,p (Ω;Γ D )

C1p ,

(3.97)

u0ε,k γ0 (y0ε,k ) dx ≤ C γ0 ||u0ε,k ||L p′ (Γ ) ||y0ε,k ||W 1,p (Ω;Γ D ) N

0

ΓN by (3.97), (3.88)

≤

(︁ )︁1/p′ C2 := C γ0 p′ C* C1

(3.98)

and ⃒ ⃒ ⃒ ∫︁ (︁ ⃒ ∫︁ (︁ (︁ (︁ )︁)︁)︁ (︁ )︁)︁ ⃒ ⃒ ⃒ ⃒ v0ε,k − T ε f y0ε,k y0ε,k dx⃒ v0ε,k − f y0ε,k y0ε,k dx⃒ ≤ ⃒ ⃒ Ω

Ω

⃒ ⃒ ∫︁ (︁ (︁ )︁ (︁ (︁ )︁)︁)︁ ⃒ ⃒ f y0ε,k − T ε f y0ε,k y0ε,k dx⃒ +⃒ Ω by (3.97)

[︁ (︁ (︁ )︁)︁ C1 diam Ω ||v0ε,k − T ε f y0ε,k ||L p′ (Ω) (︁ )︁ (︁ (︁ ]︁ )︁)︁ + ||f y0ε,k − T ε f y0ε,k ||L p′ (Ω) , ≤

(3.99)

where (see (3.93) and (3.94)) (︁ (︁ (︁ )︁)︁ )︁ (︁ (︁ )︁)︁ ||v0ε,k − T ε f y0ε,k ||L p′ (Ω) + ||f y0ε,k − T ε f y0ε,k ||L p′ (Ω) → 0 as ε → 0 and k → ∞. Utilizing estimates (3.99), (3.98), and (3.97), it follows from (3.96) that there exists a constant M > 0 independent of ε and k such that ⃒ ⃒ ⃒∫︁ (︁ )︁ ⃒⃒ ⃒ (3.100) sup ⃒⃒ y0ε,k f y0ε,k dx⃒⃒ ≤ M. ε>0 ⃒ ⃒ k∈N Ω

{︀ (︀ )︀}︀ So, in order to show that the sequence f y0ε,k ε>0 is equi-integrable on Ω, we take k∈N

m > 0 such that m > 2Mδ−1 ,

(3.101)

3.4 On Approximation of OCP |

105

where δ > 0 is a given value. We also set τ = δ/(2f (m)). Then for every measurable set S ⊂ Ω with |S| < τ, we have ∫︁ ∫︁ ∫︁ f (y0ε,k ) dx = f (y0ε,k ) dx + f (y0ε,k ) dx S

{︁

≤

x∈S : y0ε,k (x)>m

1 m {︁

{︁

∫︁

x∈S : y0ε,k (x)≤m

}︁

{︁

}︁

∫︁

y0ε,k f (y0ε,k ) dx +

x∈S : y0ε,k (x)>m

by (3.100)

≤

}︁

f (m) dx x∈S : y0ε,k (x)≤m

}︁

by (3.101) δ δ M + f (m)|S| ≤ + . m 2 2

As a result, the assertion (3.95) is a direct consequence of Lebesgue’s Convergence Theorem. Thus, J3 ≤ ||f (y0ε,k ) − f (y0 )||L1 (Ω) ||φ||L∞ (Ω)

by (3.95)

→

0 as ε → 0 and k → ∞.

Combining this fact with properties (3.92)–(3.94), we deduce from (3.91) that v0 = f (y0 ) ′ almost everywhere on Ω. Hence, by (3.90), we have f (y0 ) ∈ L p (Ω) and v0ε,k ⇀ f (y0 ) in ′

L p (Ω). Thus, (u0 , y0 ) is a feasible solution of the OCP (3.3)–(3.6) for r = p′ . The proof is complete. We are now in a position to show that optimal solutions to the approximating OCP (3.75)–(3.78) lead in the limit to optimal pairs of the original OCP (3.3)–(3.6). Theorem 3.6. Assume that for given f ∈ C loc]︁(R), g ∈ L q (Ω), and Aad , there exists a [︁ ′ 1,p pair (̃︀ u, ̃︀ y) ∈ L p (Γ N ) × W0 (Ω; Γ D ) ∩ L∞ (Ω) such that (̃︀ u, ̃︀ y) ∈ Ξ. Let p ≥ 2, r = p′ , {︀ )︀ and q ≥ p′ be given exponents, and let (u0ε,k , v0ε,k , y0ε,k ) } ε>0 be an arbitrary sequence k∈N

of optimal solutions to the approximating problems (3.75)–(3.78). Then, this sequence is ′ ′ bounded in L p (Γ N ) × L p (Ω) × W01,p (Ω; Γ D ) and any its cluster point (u0 , v0 , y0 ) with respect to the weak topology is such that v0 = f (y0 ) and (u0 , y0 ) is solution of the OCP (3.3)–(3.6) with r = p′ . Moreover, if for one subsequence we have y0ε,k ⇀ y0 in ′

′

W01,p (Ω; Γ D ), u0ε,k ⇀ u0 in L p (Γ N ), and v0ε,k ⇀ v0 in L p (Ω), then the following properties hold ′

′

u0ε,k → u0 in L p (Γ N ), v0ε,k → f (y0 ) in L p (Ω), ∫︁ ′ k |v0ε,k − T ε (f (y0ε,k ))|p dx → 0, p′

(3.102) (3.103)

Ω

lim I ε,k (u0ε,k , v0ε,k , y0ε,k ) = lim J(u0ε,k , y0ε,k ) = J(u0 , y0 ). ε→0 k→∞

ε→0 k→∞

Proof. The boundedness of the sequence {︁ )︁ ′ ′ (u0ε,k , v0ε,k , y0ε,k ) } ε>0 ⊂ L p (Γ N ) × L p (Ω) × W01,p (Ω; Γ D ) k∈N

(3.104)

106 | 3 Neumann Boundary Control Problem has been proved in Theorem 3.5. Let (u0 , v0 , y0 ) be its any cluster point with respect ′ ′ to the weak topology of L p (Γ N ) × L p (Ω) × W01,p (Ω; Γ D ). Let us take a subsequence, denoted in the same way, satisfying the property (3.90). Then Theorem 3.5 implies that ′ v0 = f (y0 ), f (y0 ) ∈ L p (Ω), and y0 is a weak solution of (3.4)-(3.5) with u = u0 . Let us prove that (u0 , y0 ) is an optimal pair to the problem (3.3)–(3.6). Given a feasible pair (u, y) ∈ Ξ such that y ∈ L∞ (Ω), we define u ε = u, v ε = T ε (f (y)), and y ε as ′ the solution of the boundary value problem (3.76)-(3.77). Since v ε ∈ L p (Ω), it follows that (u ε , v ε , y ε ) ∈ Ξ0 . By definition of the cut-off operator T ε , we have v ε → f (y)

′

strongly in L p (Ω) as ε → 0.

(3.105)

Then using the arguments of the proof of Theorem 3.4, we can deduce the existence of an element y* ∈ W01,p (Ω) such that y ε → y* in W01,p (Ω; Γ D ), where y* is a weak solution to the following problem (where indicated equalities are fulfilled in the sense of distributions) (︁ )︁ − div |∇y* |p−2 ∇y* = f (y) + u in Ω, y* = 0

on Γ D ,

|∇y* |p−2 ∂ ν y* = u

on Γ N .

On the other hand, the condition (u, y) ∈ Ξ leads to the relations (︁ )︁ − div |∇y|p−2 ∇y = f (y) + u in Ω, y=0

on Γ D ,

|∇y|p−2 ∂ ν y = u

on Γ N .

Hence, (︁ )︁ (︁ )︁ − div |∇y* |p−2 ∇y* + div |∇y|p−2 ∇y = 0 y* − y = 0

on Γ D ,

in Ω,

|∇y* |p−2 ∂ ν y* − |∇y|p−2 ∂ ν y = 0

on Γ N

and, as a result, we obtain ⟨ (︁ )︁ ⟩ 0 = − div |∇y* |p−2 ∇y* , y* − y W −1,p′ (Ω;Γ D );W01,p (Ω;Γ D ) ⟩ )︁ ⟨ (︁ + div |∇y|p−2 ∇y0 , y* − y W −1,p′ (Ω;Γ D );W01,p (Ω;Γ D ) ∫︁ [︁ ]︁ = |∇y* |p−2 ∂ ν y* − |∇y|p−2 ∂ ν y γ0 (y* − y) dH N−1 ∂Ω

+

∫︁ (︁

|∇y* |p−2 ∇y* − |∇y|p−2 ∇y, ∇y* − ∇y

)︁

dx

Ω

=

∫︁ (︁

|∇y* |p−2 ∇y* − |∇y|p−2 ∇y, ∇y* − ∇y

Ω

)︁

dx.

3.4 On Approximation of OCP |

107

Since the p-Laplace operator is strictly monotone, it follows that y* = y as element of W01,p (Ω; Γ D ). Thus, we get that Ξ0 ∋ (u ε , v ε , y ε ) −→ (u, f (y) , y) weakly in L p′ (Γ N ) × L p′ (Ω) × W01,p (Ω; Γ D ).

(3.106)

Further, we make use of the following observation. Since f ∈ C loc (R), it follows from (3.105) that ′ v ε − T ε (f (y ε )) −→ 0 strongly in L p (Ω) as ε → 0. (3.107) Utilizing now (3.106) and (3.107), we have ∫︁ ∫︁ ]︁ [︁ 1 ∫︁ ′ α 1 p′ N−1 2 |v ε |p dx |u ε | dH + ′ |y ε − y d | dx + ′ lim ε→0 2 p p k→∞ ΓN Ω Ω ∫︁ ∫︁ [︁ 1 ∫︁ ]︁ ′ ′ α 1 = lim |v ε |p dx |u ε |p dH N−1 + ′ |y ε − y d |2 dx + ′ ε→0 2 p p ΓN Ω Ω ∫︁ ∫︁ ∫︁ ′ 1 α 1 p′ N−1 2 = |u| dH |f (y)|p dx. + ′ |y − y d | dx + + ′ 2 p p Ω

Ω

(3.108)

Ω

Since ⎤

⎡ k 0 ≤ lim ⎣ ′ ε→0 p k→∞

∫︁

′

|v ε − T ε (f (y ε ))|p dx⎦

Ω

⎡

⎤

k ≤ lim sup ⎣ ′ lim sup p ε→0 k→∞

∫︁

′

|v ε − T ε (f (y ε ))|p dx⎦ = 0,

(3.109)

[︂ ]︂ lim I ε,k (u ε , v ε , y ε ) = lim lim I ε,k (u ε , v ε , y ε ) = J(u, y).

(3.110)

Ω

it follows from (3.108) and (3.109) that

ε→0 k→∞

k→∞ ε→0

Now, using (3.88), the above identity, and the fact that (u0ε , v0ε , y0ε ) is an optimal solution to the approximating problem (3.75)–(3.78), we get J(u0 , y0 ) ≤ lim inf I ε,k (u0ε,k , v0ε,k , y0ε,k ) ≤ lim inf I ε,k (u ε , v ε , y ε ) = J(u, y). ε→0 k→∞

ε→0 k→∞

(3.111)

Since (u, y) is an arbitrary pair in Ξ, this implies that (u0 , y0 ) is a solution of the original optimal control problem (3.3)–(3.6). Moreover, taking (u, y) = (u0 , y0 ) in the inequality (3.111), we arrive at the relations (3.104). To end of the proof it remains to notice that (3.102) and (3.103) are the direct consequences of the property (3.104) and the weak convergence (3.90) established before.

108 | 3 Neumann Boundary Control Problem

3.5 On existence of bounded feasible solutions The existence of bounded feasible solutions to the Dirichlet-Nuemann boundary value problem (3.4)–(3.5) is a crucial characteristic for the wide spectrum of investigations related with this problem: differentiability of the state y(u) with respect to the boundary control u, deriving and substantiation of optimality conditions, existence of appropriate approximations in the form that was proposed in the previous section, and many others (see, for instance, [21]). In view of this, our main concern in this section is to discuss the existence of bounded feasible solutions to optimal control problem (3.3)–(3.6). In particular, we are focused on the following question. Let (u, y) be a feasible solution to the problem (3.3)–(3.6). Which conditions should be imposed on p, r, q, Ω, Γ N , ′ u ∈ L p (Γ N ), and g ∈ L q (Ω) in order to guarantee the inclusions y ∈ L∞ (Ω) and/or y ∈ L∞ (∂Ω)? As was shown in the previous section, the existence of at least one feasible pair (u, y) with the extra property y ∈ W01,p (Ω; Γ D ) ∩ L∞ (Ω) plays a crucial role for the substantiation of attainability of optimal pairs to the problem (3.3)–(3.6) by optimal solutions of some fictitious optimization problem for quasi-linear elliptic equations with coercive and monotone operators. The key result of this section can be stated as follows. Theorem 3.7. Let p, q, r be exponents such that {︂ }︂ {︂ }︂ N N p p 1 ≤ p < N, q > max ; ; , and r > max . p p−1 p p−1

(3.112)

Let (u, y) be a feasible solution to the problem (3.3)–(3.6) and let u ∈ L t (Γ N ) for some {︂ }︂ N−1 p t > max ; . (3.113) p−1 p−1 Then y ∈ W01,p (Ω; Γ D ) ∩ L∞ (Ω)

and

′

γ0 (y) ∈ W 1/p ,p (Γ N ) ∩ L∞ (Γ N ),

′

where γ0 : W 1,p (Ω; Γ D ) → W 1/p ,p (Γ N ) stands for the trace operator. Theorem 3.8. Let p, q, r be exponents such that p > N,

q≥

p p−1

and

r≥

p . p−1

(3.114)

Let (u, y) be a feasible solution to the problem (3.3)–(3.6). Then y ∈ W01,p (Ω; Γ D ) ∩ L∞ (Ω). Before proceeding with the proof of these results, we begin with some preliminaries.

3.5 On Existence of Bounded Feasible Solutions |

Lemma 3.2. Let 1 ≤ p < N and let s* =

(N−1)p N−p .

109

Then the following norms

⎞1/p

⎛ ∫︁ ||y||W 1,p (Ω;Γ D ) := ⎝ 0

|∇y|p dx⎠

,

Ω

⎛ ⎞1/p ⎛ ⎞1/s* ∫︁ ∫︁ * ||y||* := ⎝ |∇y|p dx⎠ + ⎝ |γ0 (y)|s dH N−1 ⎠ ΓN

Ω

are equivalent for W01,p (Ω; Γ D ). Proof. Since the inequality ||y||W 1,p (Ω;Γ D ) ≤ ||y||* is obvious, we focus on the reverse one. 0

With that in mind we remind that by continuity of the trace operator γ0 : W 1,p (Ω; Γ D ) → ′ W 1/p ,p (Γ N ), we have ||γ0 (y)||W 1/p′ ,p (Γ

N)

≤ C γ0 ||y||W 1,p (Ω;Γ D ) ,

∀ y ∈ W 1,p (Ω; Γ D ).

′

Since, for p < N, the Sobolev space W 1/p ,p (Γ N ) is continuously embedded in L s (Γ N ) for all s ∈ [1, s* ], it follows existence of a constant C s > 0 such that ||γ0 (y)||L s* (Γ N ) ≤ C s ||γ0 (y)||W 1/p′ ,p (Γ

N)

≤ C s C γ0 ||y||W 1,p (Ω;Γ D ) ,

(3.115)

for all y ∈ W 1,p (Ω; Γ D ). Hence, (︁ )︁ 1 ||γ0 (y)||L s* (Γ N ) + ||y||W 1,p (Ω;Γ D ) ≤ ||y||W 1,p (Ω;Γ D ) . 0 0 1 + C s C γ0 Thus, the indicated norms are equivalent on W01,p (Ω; Γ D ). For our further analysis, we make use of another representation for the last estimate. As immediately follows from (3.115), we have ⎡ ⎤ ∫︁ ∫︁ 1⎣ 1 p p p + |∇y| dx⎦ . ||γ (y)|| s* (3.116) |∇y| dx ≥ L (Γ N ) 2 C ps C pγ0 0 Ω

Ω

The next result reflexes some special properties of composition of W01,p (Ω; Γ D )functions with regular functions and is a direct consequence of the well-know Stampacchia Lemma (see [75]). Lemma 3.3. Let G : R → R be a Lipschitz continuous function such that G(0) = 0. Then for every function y ∈ W01,p (Ω; Γ D ) we have: (i) G(y) ∈ W01,p (Ω; Γ D ); (ii) ∇G(y) = G′ (y)∇y almost everywhere in Ω.

110 | 3 Neumann Boundary Control Problem

We note that at the first glance the equality in (ii) is not valid because a Lipschitz continuous function G : R → R is only almost everywhere differentiable, so that the right-hand side in (ii) may not be defined. On the other hand, we have two possible cases: {︀ }︀ if k ∈ R is a value such that G′ (k) does not exist, then either the set x ∈ Ω : y(x) = k {︀ }︀ has zero measure or the set x ∈ Ω : y(x) = k has positive measure. In the first case, since the identity ∇G(y) = G′ (y)∇y only holds almost everywhere, this value does not give any problems. In this latter case, however, we have both ∇y = 0 and ∇G(y) = 0 almost everywhere, so that the identity ∇G(y) = G′ (y)∇y still holds. In what follows, we will use the composition of functions of Sobolev space W01,p (Ω; Γ D ) with the following Lipschitz continuous function G k (z) = z − T k−1 (z) = (|z| − k|)+ sign (z),

(3.117)

where k > 0 is a given value. Here, T k−1 (z) stands for the truncation operator (see (1.17)). Then Lemma 3.3 implies the following equality for W01,p (Ω; Γ D )-functions ∇G k (y) = ∇y χ{x∈Ω : |y(x)|≥k} almost everywhere in Ω,

(3.118)

where χ A denotes the characteristic function of the set A (for the details we refer to L. Orsina [127]). The first result concerning the boundedness of the weak solutions of Dirichlet boundary value problem for elliptic equations comes from Stampacchia classical work [75]. Theorem 3.9. Let y ∈ W01,p (Ω) be the weak solution of the following BVP (︁ )︁ − div |∇y|p−2 ∇y = g in Ω, y = 0 on ∂Ω, where g ∈ W −1,q (Ω) and q >

N p−1 .

Then y ∈ L∞ (Ω).

The proof of this result essentially based on the following technical lemma [148, Lemma 4.1]. Lemma 3.4. Let ψ : R+ → R+ be a nonincreasing function such that ψ(h) ≤

Mψ δ (k) , (h − k)γ

∀ h > k > 0,

(3.119) δγ

where M > 0, δ > 1, and γ > 0. Then ψ(d) = 0, where d γ = Mψ δ−1 (0)2 δ−1 . For the reader’s convenience, we cite the proof of this lemma. Proof. We define the numerical sequence {d k }k∈N as follows d k = d(1 − 2−k ) for each k ∈ N. Let us show that kγ ψ(d k ) ≤ ψ(0)2− δ−1 , (3.120)

3.5 On Existence of Bounded Feasible Solutions | 111

where ψ possesses the property (3.119). Indeed, inequality (3.120) is clearly true if k = 0. If we suppose, by the induction, that it is true for some k, then (3.119) implies ψ(d k+1 ) ≤

(k+1)γ kγδ Mψ δ (d k ) ≤ Mψ δ (0)2− δ−1 2(k+1)γ d−γ = ψ(0)2− δ−1 . (d k+1 − d k )γ

Since (3.120) holds for every k, and since ψ is a non-increasing function, it follows that kγ

0 ≤ ψ(d) ≤ lim inf ψ(d k ) ≤ lim ψ(0)2− δ−1 = 0. k→∞

k→∞

The proof is complete. We are now in a position to prove the main result of this section that has been announced in Theorem 3.7. Proof. Let k > 0 and let (u, y) ∈ Ξ be a feasible solution to the original optimal control problem. We define the set Ω k as the bigest closed subset of Ω such that Ω k ⊆ {x ∈ Ω : |∇y| ≤ k} . Hereinafter, we suppose that the parameter k varies within a strictly increasing sequence of positive real numbers tending to ∞ and such that A k := Ω \ Ω k

(3.121)

is an open set with Lipschitz boundary for each k and {A k }k>0 form a strictly monotone by inclusion (i.e. A h ⊂ A k for h > k) sequence such that limk→∞ |A k | = 0. We also set {︀ }︀ Γ N,k := σ ∈ Γ N : |γ0 (y)(σ)| ≥ k .

(3.122) ′

By definition of the trace operator γ0 : W 1,p (Ω; Γ D ) → W 1/p ,p (Γ N ), we can suppose that Γ N,k ⊂ ∂A k for each k ∈ N within a subset of Γ N,k with zero Hausdorff surface (N − 1)-dimensional measure. Since the integral identity (3.8) is valid for each function φ ∈ W 1,p (Ω; Γ D ), we chose φ = G k (y) as the test function in (3.8). Here, G k (z) is defined in (3.117). Then G k (y) = G k (y)χ A k a.e. in Ω, and, by Lemma 3.3, ∇G k (y) = ∇yχ A k for almost all x ∈ Ω. Moreover, the inclusion Γ N,k ⊂ ∂A k implies the following relations γ0 (G k (y)) = G k (γ0 (y)) and G k (γ0 (y)) = G k (γ0 (y))χ Γ N,k a.e. on Γ N . Using the fact that g ∈ L q (Ω) and q > p′ (see (3.112)), we deduce from (3.8) that ∫︁ ⟨︀ ⟩︀ g, G k (y) W −1,p′ (Ω;Γ );W 1,p (Ω;Γ ) = gG k (y) dx D

D

Ω

112 | 3 Neumann Boundary Control Problem

and, therefore, ∫︁ ∫︁ ∫︁ |∇G k (y)|p dx = |∇y|p−2 (∇y, ∇y)χ A k dx = f (y)G k (y) dx Ak

Ω

Ω

∫︁ γ0 (G k (y))u dH

+

N−1

⟨︀

⟩︀ + g, G k (y) W −1,p′ (Ω;Γ

D );W

1,p (Ω;Γ

D)

ΓN

∫︁

∫︁

γ0 (G k (y))u dH N−1 +

f (y)G k (y) dx +

=

gG k (y) dx Ak

Γ N,k

Ak

∫︁

= I1 + I2 + I3 .

(3.123)

In order to estimate the terms I i , we make use of the Hölder inequality and the following ′ * * facts: W01,p (Ω; Γ D ) ˓→ L p (Ω) and W 1/p ,p (Γ N ) ˓→ L s (Γ N ) with continuous embedding (N−1)p Np for p* = N−p and s* = N−p , respectively. As a result, we have ⎞ 1* ⎞ p1 ⎛ ⎛ p * ∫︁ ∫︁ ⎟ ⎟ ⎜ ⎜ p* p* I1 ≤ ⎝ |f (y)| dx⎠ ⎝ |G k (y)| dx⎠ ⎞ s1 ⎛

⎛

⎞ 1* s

*

∫︁ ⎜ I2 ≤ ⎝

(3.124)

Ak

Ak

s*

⎟ |u| dx⎠

Γ N,k

∫︁ ⎜ ⎝

s*

⎟ |γ0 (G k (y))| dx⎠

,

(3.125)

Γ N,k

⎞ 1* ⎞ p1 ⎛ ⎛ p * ∫︁ ∫︁ ⎟ ⎟ ⎜ ⎜ p* p* I3 ≤ ⎝ |g| dx⎠ ⎝ |G k (y)| dx⎠ ,

(3.126)

Ak

Ak

p N−1 where s* = (s* )′ = p−1 N and p * is defined by (3.14). To estimate the left-hand side of (3.123), we make use of the well-known Sobolev inequality. Namely, in view of the Sobolev embedding theorem there exists a constant S p (depending only on N and p) such that

||G k (y)||L p* (A

k)

⎞ 1p ⎛ ∫︁ ⎟ ⎜ ≤ S p ⎝ |∇G k (y)|p dx⎠

provided 1 ≤ p < N.

(3.127)

Ak

Then utilizing (3.127), Lemma 3.2 (see (3.116)), and our assumptions with respect to the set A k and its boundary, we obtain ⎡ ⎤ ∫︁ ∫︁ 1⎢ 1 ⎥ p |∇G k (y)|p dx ≥ ⎣ p p ||γ0 (G k (y))|| s* + |∇G k (y)|p dx⎦ L (Γ N,k ) 2 C s C γ0 Ak

Ak

3.5 On Existence of Bounded Feasible Solutions | 113

[︃

1 1 p p + p ||G k (y)|| p* ||γ0 (G k (y))|| s* L (A k ) L (Γ N,k ) C ps C pγ0 Sp }︃ {︃ 1 1 1 ≥ p min , 2 C ps C pγ0 S pp [︁ ]︁p × ||γ0 (G k (y))||L s* (Γ ) + ||G k (y)||L p* (A ) N,k k [︁ ]︁p ̂︀ = C ||γ0 (G k (y))||L s* (Γ ) + ||G k (y)||L p* (A ) .

1 ≥ 2

N,k

]︃

(3.128)

k

Combining this issue with estimates (3.124)–(3.126), we see from (3.123) that [︁ ̂︀ ||γ0 (G k (y))|| s* C L (Γ

N,k

+ ||G k (y)||L p* (A )

]︁p−1 k)

≤ ||f (y)||L p* (A k ) + ||g||L p* (A k ) + ||u||L s* (Γ N,k ) .

(3.129)

We now take h > k so that Ah ⊆ Ak Γ N,h ⊆ Γ N,k

G k (y) ≥ h − k on A h ,

and and

γ0 (G k (y)) ≥ h − k on Γ N,h .

Then we have ⎞1/p*

⎛ ∫︁ ⎜ ||G k (y)||L p* (A ) = ⎝

*

|G k (y)|p dx⎠

⎟

k

⎞1/p*

⎛ ∫︁ ⎜ ≥⎝

*

|G k (y)|p dx⎠

⎟

Ah

Ak 1/p*

||γ0 (G k (y))||L s* (Γ

N,k )

≥ (h − k)|A h | , ⎞1/s* ⎛ ∫︁ * ⎟ ⎜ = ⎝ |G k (y)|s dH N−1 ⎠

(3.130)

Γ N,k

⎞1/s*

⎛ ∫︁ ⎜ ≥⎝

*

|G k (y)|s dH N−1 ⎠

⎟

*

≥ (h − k)|Γ N,h |1/s .

(3.131)

Γ N,h

Since

it follows that [︁ ̂︀ ||γ0 (G k (y))|| s* C L (Γ

1 N 1 N−p = = , s* (N − 1)p N − 1 p*

N,k )

+ ||G k (y)||L p* (A

]︁p−1 k)

by (3.130)–(3.131)

≥

[︁ N 1 ]︁ p−1 ̂︀ − k)p−1 |A h |1/p* + |Γ N,h | N−1 p* C(h

[︀ ]︀ p−1 ̂︀ − k)p−1 ψ(h) p* , ≥ C(h

(3.132)

114 | 3 Neumann Boundary Control Problem

where N

ψ(h) := |A h | + |Γ N,h | N−1 .

(3.133)

For our further analysis, we make use of the following observations. Since, by the initial assumptions, we have Np Np − N + p

p′ ≥ p* =

and

q, r ≥ p′ ,

(3.134)

it follows by the Hölder inequality that

||g||L p* (A k )

⎞1/p* ⎛ ∫︁ 1 ⎟ ⎜ ≤ ||g||L q (Ω) |A k | p* = ⎝ |g |p* dx⎠

q−p* q

,

(3.135)

Ak

||f (y)||L p* (A k )

⎞1/p* ⎛ ∫︁ 1 ⎟ ⎜ ≤ ||f (y)||L r (Ω) |A k | p* = ⎝ |f (y)|p* dx⎠

r−p* r

.

(3.136)

Ak

As for the term ||u||L s* (Γ N,k ) in (3.129), following the similar arguments and taking into account the inclusion u ∈ L t (Γ N ) for t satisfying condition (3.113), we get ⎞ s1

⎛

*

∫︁ ||u||L s* (Γ N,k )

t−s*

s*

≤ |Γ N,k | ts* ||u||L t (Γ N ) .

⎟ |u| dx⎠

⎜ =⎝

(3.137)

Γ N,k

Since p* /(p − 1) > 1, it follows from (3.129), (3.132), and (3.135)–(3.137) that p* [︁ (︀ )︀]︁ p−1 * ̂︀ −1 ||f (y)|| r + ||g|| q + ||u|| t (h − k)p ψ(h) ≤ C L (Ω) L (Ω) L (Γ N ) ⏟ ⏞

D

×3

p* −p+1 p−1

[︂

1 p*

|A k | (

p 1− r*

)

p* p−1

1

+ | A k | p*

)︁ * (︁ p p 1− q* p−1

1

+ |Γ N,k | s* (

1−

s* t

]︂ p* ) p−1 .

(3.138)

We also see that [︀ ]︀ Np r(Np − N + p − Np + p2 + N − p) − Np (N − p)(Np − N + p) [︁ ]︁ by (3.112) Np > 0. p2 r − Np = (N − p)(Np − N + p)

p* (r − p* ) p* r(p − 1) =

Hence, δ1 :=

1 (︁ p )︁ p* > 1. 1− * p* r p−1

(3.139)

p* >1 p−1

(3.140)

By analogy it can be shown that δ2 :=

1 p*

(︂ 1−

p* q

)︂

3.5 On Existence of Bounded Feasible Solutions | 115

provided inequality (3.112)1 holds true. As for the third exponent in (3.138), we see that 1 (︁ N s )︁ p* = δ , 1− * s* t p−1 N−1 3 where N − 1 1 (︁ s )︁ p* 1− * N s* t p−1 [︀ ]︀ (N − 1)p − N + p t − (N − 1)p >1 = (N − p)(p − 1)t

δ3 =

(3.141)

provided the parameter t satisfies inequality (3.113). Since |Γ N,k | < 1 and |A k | < 1 for k large enough, it follows from (3.138) that ψ(h) ≤ 3

p* −p+1 p−1

[︁ (︁ )︁]︁min{δ1 ;δ2 ;δ3 } Mψ δ (k) N D N−1 2 | A | + | Γ | , = k N,k * (h − k)p (h − k)p*

(3.142)

where δ = min {δ1 ; δ2 ; δ3 } M=3

p* −p+1 p−1

by (3.139)–(3.141)

>

1, *

2

δ

[︁

p (︀ )︀]︁ p−1 ̂︀ −1 ||f (y)|| r + ||g|| q + ||u|| t C . L (Ω) L (Ω) L (Γ N )

Therefore, by Lemma 3.4 we finally deduce that N

ψ(d) := |A d | + |Γ N,d | N−1 = 0 for

]︁δ−1 δp* [︁ N 2 δ−1 . d = M |Ω| + |Γ N | N−1

Thus, for the given feasible pair (u, y) ∈ Ξ, the following inference is valid: conditions (3.112)–(3.113) imply that y ∈ L∞ (Ω) and γ0 (y) ∈ L∞ (∂Ω). The proof of Theorem 3.7 is complete. As for the proof of Theorem 3.8, its validity immediately follows from Sobolev embedding theorem saying that the injection W01,p (Ω; Γ D ) ˓→ C(Ω) is compact if p > N.

4 Asymptotic Analysis of Optimal Neumann Boundary Control Problem in Domain with Boundary Oscillation for Elliptic Equation with Exponential Non-Linearity Let ε be a small parameter varying within a strictly decreasing sequence of positive real numbers which converge to zero. Let Ω ε and Ω be given bounded open subsets of R N (N > 2) with C1,1 -boundaries confined in a fixed bounded domain D. Then the unit outward normal ν = ν(x) is well-defined for almost all x ∈ ∂Ω and x ∈ ∂Ω ε with respect to the (N − 1)-dimensional Hausdorff measure H N−1 . We assume that Ω lies locally on one side of ∂Ω and the boundary ∂Ω consists of two disjoint parts ∂Ω = Γ D ∪ Γ N such that the sets Γ D and Γ N have positive (N − 1)-dimensional measures, Γ D ⊂ ∂D. We make the similar assumption with respect to the boundary of Ω ε , though each of domains Ω ε may have rather rough part of the boundary ∂Ω ε = Γ D ∪ Γ N,ε , where Γ D ∩ Γ N,ε = ∅ and Γ D -part does not depend on ε > 0. Since we consider Ω ε as some perturbation of Ω that excludes the case of perforated domains, we assume that the family {Ω ε }0 2, g d ∈ L q (D) and y d ∈ L2 (D) are given distributions, q ≥ N+2 , ′ 2 r > N, f (y) = F (y), f : R → (0, ∞) is a strictly convex function, and F ∈ C (K) for any https://doi.org/10.1515/9783110668520-005

4 Asymptotic Analysis of Optimal Neumann Boundary Control Problem | 117

compact set K ⊂ R, where F is a non-decreasing positive function such that ⃒ ⃒ 0 ⃒ ⃒ ∫︁ ⃒ ⃒ ′ ′ ⃒ F (z) ≥ C F F(z), ∀ z ∈ R and ⃒ zF (z) dz⃒⃒ < +∞. ⃒ ⃒

(4.6)

−∞

for some constant C F > 0. Here, W01,2 (Ω ε ; Γ D ) stands for the Banach space which is defined as the closure of {︁ }︁ N ∞ N C∞ 0 (R ; Γ D ) = φ ∈ C 0 (R ) : φ = 0 on Γ D with respect to the norm ⎞1/2

⎛ ∫︁ ⎜ ||y||W 1,2 (Ω ε ;Γ D ) = ⎝ 0

|∇y|2 dx⎠

⎟

.

Ωε

)︁* Let W −1,2 (Ω ε ; Γ D ) := W01,2 (Ω ε ; Γ D ) be the dual space to W 1,2 (Ω ε ; Γ D ). Hereinafter (︁

in this Chapter, we consider the element g ∈ L q (D) as a fictitious control that we intend to chose as close as possible to the given distribution g d ∈ L q (D) with respect to the norm of L q (Ω ε ). Our main intention is to discuss the existence of optimal pairs to OCP (4.1)–(4.2), derive the corresponding optimality conditions, and provided the asymptotic analysis of this problem as ε → 0. The main characteristic feature of the indicated boundary value problem (BVP) (4.3)–(4.5) is again, as in the previous Chapters, the fact that because of the specificity of non-linearity f (y), we can not assert that this problem admits at least one solution for a given pair of controls u ∈ Aad (Γ N,ε ) and g ∈ L q (D). Instead of this we will show that the original BVP possesses a special type of weak solutions satisfying some extra state constraints. In fact, this circumstance will allow us to prove the consistency of the original optimal control problem, establish its solvability, and show that an optimal solution is unique (in spite of the fact that we deal with ill-posed boundary value problem for strongly nonlinear elliptic equation). The main novelty of OCP we consider, is the fact that we have two different types of controls — distributed and boundary, and the control zone for one of them is supported along of a rough part of the boundary of Ω ε . The ill-posedness of strongly nonlinear elliptic equation with L1 -type of nonlinearity motivates us to begin with the following notion. Definition 4.1. We say that (u, g, y) ∈ L2 (Γ N,ε ) × L q (D) × W01,2 (Ω ε ; Γ D ) is a feasible solution to the problem (4.1)–(4.5) if – (u, g) are admissible controls, i.e., u ∈ Aad (Γ N,ε ) and g ∈ L q (D); – f (y) ∈ L1 (Ω ε ); N – for every test function φ ∈ C∞ 0 (R ; Γ D ), the following integral identity holds ∫︁ ∫︁ ∫︁ ∫︁ uφ dH N−1 + gφ dx; (4.7) (∇y, ∇φ) dx = f (y)φ dx + Ωε

Ωε

Γ N,ε

Ωε

118 | 4 Asymptotic Analysis of Optimal Neumann Boundary Control Problem –

there exists x0 ∈ int Ω ∩ Ω ε such that ∫︁ ∫︁ ∫︁ 2N 3 diam D |∇y|2 dx ≤ F(y) dx + |u|2 dH N−1 N−2 N−2 Ωε

Γ N,ε

Ωε

2 − N−2

∫︁ g (x − x0 , ∇y) dx.

(4.8)

Ωε

We denote by Ξ ε the set of all feasible solutions to the problem (4.1)–(4.5). At the first glance the role of extra constraint (4.8) on the class of feasible solutions looks rather vague. Moreover, it is unknown whether the inequality (4.8) holds for an arbitrary weak solution to the boundary value problem (4.2)–(4.3). However, as we will see later on, such definition of feasible solutions is absolutely plausible and makes the setting of the original optimal control problem fully consistent. Remark 4.1. To the best knowledge of authors, the existence of the weak solutions to the original BVP is an open question for nowadays. Moreover, in the context of the optimal control problem that we deal with in this chapter, there is no reason to suppose that there exists a weak solution to (4.2)–(4.3) for given u ∈ Aad (Γ N,ε ) and g ∈ L q (Ω ε ), and even if it exists, this solution satisfies extra property (4.8). In view of this, we adopt the following non-triviality assumption: 2N Hypothesis A. For given ε > 0, q N+2 , Ω ε ⊂ D, f ∈ C1loc (R), and Aad (Γ N,ε ), the ≥ feasible solutions Ξ ε is nonempty. set of We also assume that there exists a point x0 ∈ int Ω ε such that x0 ∈ Ω ε for all ε > 0 and Ω ε satisfies the so-called Γ D -star-shaped property with respect to x0 , i.e. (︀ )︀ σ − x0 , ν(σ) ≥ 0 for H N−1 -a.a. σ ∈ Γ D . (4.9) It is clear that the standard star-shaped property of Ω ε implies that this domain is also Γ D -star-shaped. However, the inverse statement, especially because of the rough part Γ N,ε of boundary ∂Ω ε , is not true in general. Before proceeding further, it is worth to note that the study of elliptic partial differential equations in domains with rugose boundary is a subject that has been treated in the literature by different authors (see, for instance [6, 19, 31, 38, 36, 37, 101, 102, 126, 128, 145]). At the same time, the analogous results for the case of optimal control problems for essentially nonlinear elliptic equations (4.2) with mixed boundary conditions (4.3) in the presence of boundary oscillations along the control zone remain arguably open.

4.1 Previous analysis of optimal control problem As we mentioned before, it is unknown whether the original BVP admits at least one weak solution for any admissible controls u ∈ Aad (Γ N,ε ) ⊂ C(Γ N,ε ) and g ∈ L q (D).

4.1: Previous analysis of optimal control problem

| 119

Hence, it is not easy to embark directly on the set of feasible solutions Ξ to the original optimal control problem because its structure and the main topological properties remain unknown in general. We begin with the following result (for the proof we refer to Proposition 3.1, Lemma 3.1 and its Corollary 3.1). Proposition 4.1. Let u ∈ L2 (Γ N,ε ) and g ∈ L q (D) be given functions, and let y = y(u, g) ∈ W01,2 (Ω ε ; Γ D ) be a weak solution to BVP (4.2)–(4.3) in the sense of distributions. Then (i) f (y) ∈ W −1,2 (Ω ε ; Γ D ) and z f (y) ∈ L1 (Ω ε ) for every element z ∈ W01,2 (Ω ε ; Γ D ), where ⟨·, ·⟩W −1,2 (Ω ε ;Γ D );W 1,2 (Ω ε ;Γ D ) : W −1,2 (Ω ε ; Γ D ) × W01,2 (Ω ε ; Γ D ) → R 0

stands for the duality pairing between W −1,2 (Ω ε ; Γ D ) and W01,2 (Ω ε ; Γ D ); (ii) the inequality ⃒ ⃒ ⎞1/2 ⎛ ⃒ ⃒∫︁ ∫︁ ⃒ ⃒ ⎟ ⎜ ⃒ ⃒ 2 ⃒ f (y)φ dx⃒ ≤ c(y) ⎝ |∇φ| dx⎠ , ⃒ ⃒ ⃒ ⃒Ω ε Ωε

N ∀ φ ∈ C∞ 0 (R ; Γ D )

holds with some constant depending arguably on y; (iii) y satisfies the energy equality ∫︁ ∫︁ ∫︁ γ0 (y)u dH N−1 + ⟨g, y⟩W −1,2 (Ω ε ;Γ D );W 1,2 (Ω ε ;Γ D ) . |∇y|2 dx = yf (y) dx + 0

Ωε

(4.10)

(4.11)

Γ N,ε

Ωε

Here, γ0 : W01,2 (Ω ε ; Γ D ) → W 1/2,2 (∂Ω ε ) is the trace operator (see [115, Theorem 8.3]), γ0 (y) = y|∂Ω ε ,

∀ y ∈ W01,2 (Ω ε ; Γ D ) ∩ C(Ω ε ).

Remark 4.2. We note that duality pairing ⟨g, y⟩W −1,2 (Ω ε ;Γ D );W 1,2 (Ω ε ;Γ D ) is well defined for all y ∈ W 1,2 (Ω ε ; Γ D ) provided g ∈ L q (Ω ε ) with q ≥

0

2N N+2 . Indeed, by Sobolev embedding * 2N . Hence, by in L2 (Ω ε ) with 2* = N−2

theorem, W01,2 (Ω ε ; Γ D ) is continuously embedded (︁ * )︁* duality arguments, L2 (Ω ε ) is continuously embedded in W −1,2 (Ω ε ; Γ D ). So, if we define 2N 2* = (2* )′ = , N+2 then we have L q (Ω ε ) ⊂ L2* (Ω ε ) ⊂ W −1,2 (Ω ε ; Γ D ), for all q ≥

2N N+2 .

Therefore,

⃒ ⃒ ⃒ ⃒ ⃒⟨g, y⟩W −1,2 (Ω ε ;Γ D );W 1,2 (Ω ε ;Γ D ) ⃒ ≤ ||g||W −1,2 (Ω ε ;Γ D ) ||y||W 1,2 (Ω ε ;Γ D ) 0

0

≤ C||g||L q (Ω ε ) ||y||W 1,2 (Ω ε ;Γ D ) , ∀ y ∈ W01,2 (Ω ε ; Γ D ). 0

(4.12)

120 | 4 Asymptotic Analysis of Optimal Neumann Boundary Control Problem We also note that the injection W 1/2,2 (Γ N,ε ) ˓→ L2 (Γ N,ε ) is compact by Sobolev embedding theorem. Hence, by continuity of the trace operator γ0 : W01,2 (Ω ε ; Γ D ) → W 1/2,2 (Γ N,ε ), ||γ0 (y)||L2 (Γ N,ε ) ≤ C γ0 ||y||W 1,2 (Ω ε ;Γ D ) , 0

∀ y ∈ W01,2 (Ω ε ; Γ D ),

we have ⃒ ⃒ ⃒ ⃒ ∫︁ ⃒ ⃒ ⃒ N−1 ⃒ uγ0 (y) dH ⃒ ≤ ||u||L2 (Γ N,ε ) ||y||L2 (Γ N,ε ) ≤ C γ0 ||u||L2 (Γ N,ε ) ||y||W 1,2 (Ω ε ;Γ D ) < +∞. ⃒ 0 ⃒ ⃒ ⃒ ⃒Γ N,ε

(4.13)

(4.14)

It is worth to emphasize that energy equality (4.11) makes sense if only the triplet (u, g, y) is feasible and it is unknown whether we can guarantee the fulfilment of this relation for an arbitrary weak solution (u, g, y(u, g)) to BVP (4.2)–(4.3). Moreover, since it is unknown whether the operator −∆y − f (y) is monotone on the set W01,2 (Ω ε ), we cannot use the energy equality (4.11) in order to derive a priori estimate in || · ||W 1,2 (Ω ε ) 0 norm for the weak solutions. Nevertheless, following in many aspects the proof line of Proposition 3.2, we can deduce the following result. Proposition 4.2. Let f (y) ∈ L2 (Ω ε ), u ∈ L2 (Γ N,ε ), g ∈ L q (D) with q ≥ 2, and let y = y(u, g) ∈ W01,2 (Ω ε ; Γ D ) be a weak solution to BVP (4.2)–(4.3). Assume that Ω ε is a starshaped domain, i.e., there exists x0 ∈ int Ω ε such that (︀

)︀ σ − x0 , ν(σ) ≥ 0 for H N−1 -a.a. σ ∈ ∂Ω ε .

Then (︂

)︂ ∫︁ ∫︁ ∫︁ N |∇y|2 dx ≤ N F(y) dx − g (x − x0 , ∇y) dx −1 2 Ωε

Ωε

Ωε

3 diam D + 2

∫︁

|u|2 dH N−1 .

Γ N,ε

Remark 4.3. Since 2 >

2N N+2

and q ≥ 2, it follows that the term ∫︁ g (x − x0 , ∇y) dx Ωε

in (4.15) is well defined. Moreover, using Hölder inequality, we get ∫︁ ∫︁ g (x − x0 , ∇y) dx ≤ diam Ω ε |g ||∇y| dx Ωε

Ωε

≤ diam D||g||L2 (D) ||∇y|L2 (Ω ε )N q−2

≤ diam D|D| 2q ||g||L q (D) ||y||W 1,p (Ω ε ;Γ D ) < +∞. 0

(4.15)

4.1: Previous analysis of optimal control problem | 121

Proof. We divide this proof onto several steps. Step 1. In view of the initial assumptions and Remark 4.3, it is easy to see that ∆y ∈ L2 (Ω ε ).

(4.16)

Hence, we can multiply the equation (4.2) by any function φ ∈ L2 (Ω ε ) and make the integration over Ω ε . Let us consider φ := (x − x0 , ∇y) ∈ L2 (Ω ε ) as the test function. Then (4.7) implies the relation ∫︁ ∫︁ (︀ )︀ ∂ ν y(σ) σ − x0 , ∇y(σ) dH N−1 (∇y, ∇ (x − x0 , ∇y)) dx − Ωε

∂Ω ε

∫︁

∫︁ f (y) (x − x0 , ∇y) dx +

= Ωε

g (x − x0 , ∇y) dx.

(4.17)

Ωε

By analogy with the proof of Proposition 3.2, we note that the boundary ∂Ω ε consists of two disjoint parts ∂Ω ε = Γ D ∪ Γ N,ε and we have y(σ) = 0 on Γ D . Hence, the tangential component of ∇y(σ) vanish on the Γ D -part of boundary. Therefore, ∇y(σ) = ±|∇y(σ)|ν(σ) on Γ D .

Then (︀

)︀ (︀ )︀ σ − x0 , ∇y(σ) = ±|∇y(σ)| σ − x0 , ν(σ) and (︀ )︀ ∂ ν y(σ) = ∇y(σ), ν(σ) = ±|∇y(σ)| on Γ D .

As for the Γ N,ε -part of ∂Ω ε , we can assert the existence of a couple of functions ζ1,ε , ζ2,ε ∈ L∞ (Γ N,ε ) with |ζ i,ε (σ)| ≤ 1 H N−1 -a.e. on Γ N,ε such that (︀

)︀ σ − x0 , ∇y(σ) = ζ1,ε (σ) diam Ω ε |∇y(σ)|, (︀ )︀ ∂ ν y(σ) = ∇y(σ), ν(σ) = ζ2,ε (σ)|∇y(σ)| H N−1 -a.e. on Γ N,ε . Taking this fact into account, we can rewrite the equality (4.17) as follows ∫︁ ∫︁ (︀ )︀ σ − x0 , ν(σ) |∇y(σ)|2 dH N−1 (∇y, ∇ (x − x0 , ∇y)) dx − ΓD

Ωε

∫︁ −

ζ1,ε (σ)ζ2,ε (σ) diam Ω ε |∇y(σ)|2 dH N−1

Γ N,ε

∫︁

∫︁ f (y) (x − x0 , ∇y) dx +

= Ωε

g (x − x0 , ∇y) dx. Ωε

(4.18)

122 | 4 Asymptotic Analysis of Optimal Neumann Boundary Control Problem Step 2. We apply the integration by parts to the first term in left hand side of (4.17). This yields ⎡ ⎤ ∫︁ N ∫︁ N ∑︁ ∑︁ ∂y ∂ ⎣ ∂y ⎦ dx (x j − x0j ) (∇y, ∇ (x − x0 , ∇y)) dx = ∂x i ∂x i ∂x j i=1 Ω

Ωε

j=1

ε

⃒ ∫︁ ∑︁ ∫︁ ∑︁ N N ⃒ ⃒ ∂y ⃒2 ∂y ∂2 y ⃒ ⃒ (x j − x0j ) dx = ⃒ ∂x i ⃒ dx + ∂x i ∂x i ∂x j Ω ε i,j=1

Ω ε i=1

∫︁

|∇y|2 dx +

=

∫︁ N 1 ∑︁ 2

Ω ε j=1

Ωε

∫︁

∫︁

1 |∇y| dx + 2 2

= Ωε

(x j − x0j )

|∇y(σ)|2

N 2

−

N ∑︁

(σ j − x0j )ν j (σ) dH N−1

j=1

∂Ω ε

∫︁

∂ |∇y|2 dx ∂x j

|∇y|2 dx

Ωε

(︂ =

N 1− 2

)︂ ∫︁

|∇y|2 dx +

Ωε

1 2

∫︁

(︀

)︀ σ − x0 , ν(σ) |∇y(σ)|2 dH N−1 .

∂Ω ε

Since, by the star-shaped property of Ω ε , we have (︀

)︀ σ − x0 , ν(σ) ≥ 0 for H N−1 -a.a. σ ∈ Γ D and |∇y|2 = |u|2 on Γ N,ε ,

it follows that 1 2

∫︁

(︀

)︀ σ − x0 , ν(σ) |∇y(σ)|2 dH N−1

∂Ω ε

∫︁

ζ1,ε (σ)ζ2,ε (σ) diam Ω ε |∇y(σ)|2 dH N−1

− Γ N,ε

∫︁

(︀

−

)︀ σ − x0 , ν(σ) |∇y(σ)|2 dH N−1

ΓD

1 ≤− 2

∫︁

(︀

)︀ σ − x0 , ν(σ) |∇y(σ)|2 dH N−1

ΓD

+

3 diam D 2

∫︁

Γ N,ε

(4.9)

|u|2 dH N−1 ≤

β 2

∫︁ Γ N,ε

|u|2 dH N−1

4.1: Previous analysis of optimal control problem | 123

with β = 3 diam D. Hence, ∫︁ ∫︁ (︀ )︀ σ − x0 , ν(σ) |∇y(σ)|2 dH N−1 (∇y, ∇ (x − x0 , ∇y)) dx − ΓD

Ωε

∫︁

ζ1,ε (σ)ζ2,ε (σ) diam Ω ε |∇y(σ)|2 dH N−1

− Γ N,ε

(︂ ≤

1−

N 2

)︂ ∫︁ Ωε

|∇y|2 dx +

β 2

∫︁

|u|2 dH N−1 .

(4.19)

Γ N,ε

Step 3. Let us show that the relation ∫︁ ∫︁ ∫︁ (︀ )︀ f (y) (∇y, ψ) dx = F(γ0 (y(σ))) ν(σ), ψ(σ) dσ − F(y) div ψ dx Ωε

∂Ω ε

(4.20)

Ωε

holds true for any vector-valued test function ψ ∈ C1 (Ω ε )N provided y ∈ W01,p (Ω ε ; Γ D ) is a weak solution to (4.2)–(4.3). To do so, it is enough to show that (see the proof of Proposition 3.2) f (y)∇y = ∇F(y) as elements of L1 (Ω ε )N . (4.21) Then the equality (4.20) is a direct consequence of the formula of integration by parts. Let T η : R → R be the truncation operator defined by the rule (1.17). Then we can suppose that T η (y) → y strongly in W01,2 (Ω ε ; Γ D ) and a.e. in Ω ε as η → 0.

(4.22)

Since f (y) ∈ L2 (Ω ε ) and f ∈ C1loc (R), it follows from (4.22) that (︀ )︀ f T η (y) → f (y) in L2 (Ω ε ) and a.e. in Ω ε . Then, utilizing the fact that ∇y ∈ L p (Ω ε )N and ∇T η (y) → ∇y a.e. in Ω ε ,

the Lebesgue Dominated Theorem implies: (︀ )︀ f T η (y) ∇T η (y) → f (y)∇y in L1 (Ω ε )N , (︀ )︀ ∇F T η (y) → ∇F (y) in L1 (Ω ε )N . Taking into account that (︀ )︀ (︀ )︀ f T η (y) ∇T η (y) = ∇F T η (y) ,

∀ η > 0,

and passing to the limit in both parts of this relation as η → 0, we arrive at the desired equality (4.21).

124 | 4 Asymptotic Analysis of Optimal Neumann Boundary Control Problem

Step 4. Now we are in a position to transform the right hand side in (4.17). Indeed, due to relation (4.20) and the star-shaped property of Ω ε , we have ∫︁ ∫︁ (︀ )︀ by (4.21) f (y) (x − x0 , ∇y) dx = x − x0 , ∇F(y) dx Ωε

Ωε by (4.20)

=

∫︁ F(y) div(x − x0 ) dx

− Ωε

∫︁

)︀ (︀ F(γ0 (y(σ))) ν(σ), σ − x0 dH N−1

+ ∂Ω ε

∫︁

∫︁ F(y)div(x − x0 ) dx + F(0)

=−

(︀

)︀ ν(σ), σ − x0 dH N−1

ΓD

Ωε

∫︁ +

)︀ F(γ0 (y(σ))) ν(σ), σ − x0 dH N−1 (︀

Γ N,ε

∫︁ F(y) dx.

≥ −N Ωε

Combining this inequality with (4.17) and (4.19), we arrive at the desired relation (4.15). The proof is complete. The reasonings that were applied for the proof of Proposition 4.2 allow us to specify this assertion as follows. Corollary 4.1. Assume that, instead of the star-shaped property of Ω ε , we have the following ones: Ω ε is just a Γ D -star-shaped domain and the inequalities ∫︁ (︀ )︀ σ − x0 , ν(σ) |∇y(σ)|2 dH N−1 ≥ 0, ΓD

∫︁ Γ N,ε

(︀ )︀ F(γ0 (y(σ))) ν(σ), σ − x0 dH N−1 ≥ −F(0)

∫︁

(︀

)︀ ν(σ), σ − x0 dH N−1 .

ΓD

hold true. Then relation (4.15) remains valid for given y = y(u, g) ∈ W01,2 (Ω ε ; Γ D ), u ∈ L2 (Γ N,ε ), g ∈ L q (D) with q ≥ 2, and f (y) ∈ L2 (Ω ε ). The next results play a crucial role in the sequel and they can be considered as some specification of Theorem 3.1 and Proposition 3.3 (for the proof we refer to Chapter 3). Theorem 4.1. Let u ∈ L2 (Γ N,ε ), g ∈ L q (D) with q ≥ 2, and let y = y(u) ∈ W01,2 (Ω ε ; Γ D ) be a weak solution to BVP (4.2)–(4.3) such that y satisfies inequality (4.15). Then there

4.1: Previous analysis of optimal control problem |

exist positive constants C i , 1 ≤ i ≤ 6, independent of u, y, and ε, such that ∫︁ 2 2 y f (y) dx ≤ C1 ||u||L2 (Γ N,ε ) + C2 ||g||L q (D) + C3 ,

125

(4.23)

Ωε

||y||W 1,2 (Ω ε ;Γ D ) ≤ C4 ||u||L2 (Γ N,ε ) + C5 ||g||L q (D) + C6 . 0

(4.24)

{︀ }︀ Proposition 4.3. Let (u k , g k , y k ) k∈N ⊂ L2 (Γ N,ε ) × L q (D) × W01,2 (Ω ε ; Γ D ), with q ≥ 2, be a sequence such that u k ⇀ u weakly in L2 (Γ N,ε ), q

g k ⇀ g weakly in L (D), y k → y weakly in

W01,2 (Ω ε ; Γ D )

and a.e. in Ω ε , 1

f (y k ) → f (y) strongly in L (Ω ε ), (︁ )︁* N , ∀ k ∈ N, −∆y k = f (y k ) + g k in C∞ 0 (R ; Γ D ) γ0 (y k ) = 0 ⃒ ⃒ where γ1 (y) = ∂y ∂ν ⃒

Γ N,ε

and γ1 (y k ) = u k ,

∀ k ∈ N,

(4.25) (4.26) (4.27) (4.28) (4.29) (4.30)

for all y ∈ C1 (Ω ε ) ∩ W01,2 (Ω ε ; Γ D ). Then

∇y k → ∇y strongly in L r (Ω ε )N for any 1 ≤ r < 2.

(4.31)

Remark 4.4. It is worth to notice that inequality (4.15) makes sense even if we do not assume fulfillment of the inclusion f (y) ∈ L2 (Ω) but have only that y ∈ W01,2 (Ω ε ; Γ D ), f (y) ∈ L1 (Ω ε ), u ∈ L2 (Γ N,ε ), and g ∈ L q (Ω) with q ≥ 2. At the same time it is unknown whether this inequality holds for an arbitrary weak solution to BVP (4.2)–(4.3). Since the existence of the weak solutions to the original BVP satisfying the integral inequality (4.8) is an open question for arbitrary given controls u ∈ L2 (Γ N,ε ) and g ∈ L q (D), Theorem 4.1 gives the main motivation to introduce the notion of feasible solutions in the form of Definition 4.1. In particular, as follows from Definition 4.1, if for a given ε > 0, (u, g, y) ∈ Ξ ε is a feasible solution to the problem (4.1)–(4.5) then a priori estimates (4.23)–(4.24) remain true. The following result highlights rather important properties of the set of feasible solutions and plays a key role in our further analysis. {︀ }︀ Proposition 4.4. Let q ≥ 2 and let (u k , g k , y k ) k∈N ⊂ Ξ ε be a sequence of feasible solutions to the optimal control problem (4.1)–(4.5) for a given value ε > 0. Assume that {g k }k∈N is a bounded sequence in L q (D). Then there exists a triple (u, g, y) ∈ L2 (Γ N,ε ) × {︀ }︀ L q (D) × W01,2 (Ω ε ; Γ D ) and a subsequence of (u k , g k , y k ) k∈N , still denoted by the same

126 | 4 Asymptotic Analysis of Optimal Neumann Boundary Control Problem

index k ∈ N, such that u k → u strongly in L2 (Γ N,ε ) as k → ∞, q

g k ⇀ g weakly in L (D) as k → ∞, y k ⇀ y weakly in

W01,2 (Ω ε ; Γ D )

as k → ∞,

(u, g, y) ∈ Ξ ε ,

(4.32) (4.33) (4.34) (4.35)

1

f (y k ) → f (y) strongly in L (Ω ε ) as k → ∞.

(4.36)

{︀ }︀ Proof. Let (u k , g k , y k ) k∈N ⊂ Ξ ε be a given sequence. Then assertion (4.33) together with inclusion g ∈ L q (D) is a direct consequence of the initial assumptions. Since u k ∈ Aad (Γ N,ε ) for all k ∈ N, it follows from (4.5) that there exists a sequence of {︀ }︀ prototypes v ∈ W 1,r (D) k∈N such that sup ||v k ||W 1,r (D) ≤ β k∈N

and

u k = v k |Γ N,ε ∀ k ∈ N.

(4.37)

Moreover, using the fact that the injection W 1,r (D) ˓→ C(D) is compact for r > N, we can deduce from (4.37) the existence of element v ∈ W 1,r (D) such that, up to a subsequence, ||v||W 1,r (D) ≤ β, v k ⇀ v in W 1,r (D), v k |Γ N,ε → v|Γ N,ε in C(Γ N,ε ), sup || v k |Γ N,ε ||C(Γ N,ε ) ≤ C r sup ||v k ||W 1,r (D) ≤ C r β, k∈N

k∈N

and, therefore, v k |Γ N,ε → v|Γ N,ε in L2 (Γ N,ε ). On the other hand, each of the triplets (u k , g k , y k ) is a feasible solution to the optimal control problem (4.1)–(4.5). Hence, these tuples are related by integral inequality (4.8). Then taking into account Theorem 4.1, it is easy to deduce the estimate sup ||y k ||W 1,p (Ω ε ;Γ D ) ≤ C4 sup ||u k ||L2 (Γ N,ε ) + C5 sup ||g k ||L q (D) + C6 k∈N

0

k∈N

≤ C4 C r β

√︁

k∈N

H N−1 (Γ N,ε ) + C5 sup ||g k ||L q (D) + C6 .

(4.38)

k∈N

Thus, we can suppose the existence of element y ∈ W01,p (Ω ε ; Γ D ) such that, within a subsequence, (4.39) y k ⇀ y in W01,p (Ω ε ; Γ D ). As a result, we see that properties (4.32)–(4.34) holds true with u = v|Γ N,ε . It remains to establish the rest properties (4.35)–(4.36). By the Sobolev Embedding Theorem, the injection W01,2 (Ω ε ; Γ D ) ˓→ L2 (Ω ε ) is compact. Hence, the weak convergence y k ⇀ y in W01,2 (Ω ε ; Γ D ) implies the strong convergence in L2 (Ω ε ). Therefore, up

4.1: Previous analysis of optimal control problem | 127

to a subsequence, we can suppose that y k (x) → y(x) for almost every point x ∈ Ω ε . As a result, we have the pointwise convergence: f (y k ) → f (y) almost everywhere in Ω ε . Arguing as in the proof of Proposition 1.3, it can be shown that this fact implies the strong convergence (4.36) and, as a consequence, f (y) ∈ L1 (Ω ε ). Let us show now that the limit triple (u, g, y) is related by the integral identity (4.7). With that in mind we note that, in view of the initial assumptions and properties (4.36) and (4.39), the limit passage in the equality ∫︁ ∫︁ ∫︁ u k φ dH N−1 (∇y k , ∇φ) dx = f (y k )φ dx + Γ N,ε

Ωε

Ωε

∫︁

N g k φ dx, ∀ φ ∈ C∞ 0 (R ; Γ D )

+

(4.40)

Ωε

becomes trivial. As a result, we see that y is a weak solution to BVP (4.2)–(4.3) for the limit functions u ∈ L2 (Γ N,ε ) and g ∈ L q (D). It remains to prove that the triplet (u, g, y) satisfies the inequality (4.15). Since the sequence {∇y k }k∈N is bounded in L2 (Ω ε )N , it follows from (4.38) and Proposition 4.3 that ′

∇y k n → ∇y strongly in L q (Ω ε )N with q′ = q/(q − 1).

(4.41)

Then ∫︁ g k (x − x0 , ∇y k ) dx

lim

k→∞

by (4.34),(4.41)

∫︁ k→∞ Γ N,ε

Ωε

|∇u k |2 dH N−1

F(y k ) dx

lim

by (4.36)

∫︁ k→∞

Ωε

∫︁

=

|∇u|2 dH N−1 ,

∫︁ F(y) dx ≤

=

Ωε

lim inf

by (4.32)

Γ N,ε

∫︁ k→∞

g (x − x0 , ∇y) dx,

=

Ωε

lim

∫︁

C−1 F

Ωε

|∇y k |2 dx

by (4.34)

∫︁

≥

∫︁ f (y) dx < +∞, Ωε

|∇y|2 dx.

Ωε

Here, we essentially used the strong convergence (4.33) and the fact that the sequence {︀ }︀ f (y k ) k∈N is equi-integrable and F(z) ≤ C−1 F f (z) for all z ∈ R.

128 | 4 Asymptotic Analysis of Optimal Neumann Boundary Control Problem Then passing to the limit in inequality (4.15), we finally obtain (︂ )︂ ∫︁ )︂ (︂ ∫︁ N N |∇y|2 dx ≤ −1 − 1 lim inf |∇y k |2 dx 2 2 k→∞ Ωε

Ωε

⎤

⎡ ∫︁

∫︁

⎢ ≤ lim inf ⎣N

⎥ g k (x − x0 , ∇y k ) dx⎦

F(y k ) dx −

k→∞

Ωε

Ωε

3 diam D + lim sup 2 k→∞

∫︁

|u k |2 dH N−1

Γ N,ε

∫︁ =N

∫︁ F(y) dx −

Ωε

g (x − x0 , ∇y) dx +

3 diam D 2

∫︁

|u|2 dH N−1 .

Γ N,ε

Ωε

that is, (u, g, y) is a feasible solution to the problem (4.1)–(4.5). The proof is complete. We are now in a position to prove the existence result for optimal control problem (4.1)–(4.5). Theorem 4.2. Let ε > 0 and q ≥ 2 be given values. Assume that Hypothesis A is fulfilled. Then, for any y d ∈ L2 (Ω ε ) and g d ∈ L q (D), the optimal control problem (4.1)–(4.5) has at least one solution. Proof. Since Ξ ε = ̸ ∅ and J ε (u, g, y) ≥ 0 for all (u, g, y) ∈ Ξ ε , it follows that there exists a {︀ }︀ non-negative value μ ε ≥ 0 such that μ ε = inf (u,g,y)∈Ξ ε J ε (u, g, y). Let (u k , g k , y k ) k∈N be a minimizing sequence to the problem (4.1)–(4.5), i.e. (u k , g k , y k ) ∈ Ξ ε

∀k ∈ N

and

lim J ε (u k , g k , y k ) = μ ε .

k→∞

So, we can suppose that J(u k , g k , y k ) ≤ μ + 1 for all k ∈ N and, hence, )︀1/q (︀ )︀1/q (︀ < +∞. ≤ q(μ + 1) sup ||g k ||L q (Ω ε ) ≤ sup qJ ε (u k , g k , y k ) k∈N

k∈N

Since each of the functions g k can be zero extended outside of Ω ε , we see that the sequence {g k }k∈N is bounded in L q (D). Then by Proposition 4.4, there exists a feasible {︀ }︀ solution (u0ε , g 0ε , y0ε ) ∈ Ξ ε and a subsequence of (u k , g k , y k ) k∈N , still denoted by the same index, such that u k → u0ε strongly in L2 (Γ N,ε ), yk ⇀

y0ε

weakly in

gk ⇀ as k → ∞.

g 0ε

W01,2 (Ω ε ; Γ D ), q

weakly in L (D)

(4.42) (4.43) (4.44)

4.2 On consistency of optimal control problem | 129

To conclude the proof, it remains to take into account the continuity of the cost functional J ε : L2 (Γ N,ε ) × L q (D) × W01,2 (Ω ε ; Γ D ) → R with respect to the strong convergence in L2 (Γ N,ε ) × L q (D) × L2 (Ω ε ) and properties (4.42)–(4.44). This yields με = Thus,

(u0ε , g 0ε , y0ε )

inf

(u,g,y)∈Ξ ε

J ε (u, g, y) = lim J ε (u k , g k , y k ) = J ε (u0ε , g 0ε , y0ε ). k→∞

∈ Ξ ε is an optimal triplet to the problem (4.1)–(4.5).

4.2 On consistency of optimal control problem As we mentioned before, it is unknown whether the original boundary value problem admits at least one weak solution for any admissible controls (u, g) ∈ Aad (Γ N,ε )× L q (D). Moreover, even if for some (u, g) ∈ Aad (Γ N,ε ) × L q (D) there exists a weak solution y = y(u, g) to the problem (4.2)–(4.3), we cannot guarantee the fulfillment of integral inequality (4.8) for the given triplet (u, g, y). Hence, it is not an easy matter to touch directly the set of feasible solutions Ξ ε to the original optimal control problem because its structure and the main topological properties are unknown in general. To clarify this option, we adopt the following concept. Definition 4.2. We say that, for a given ε > 0, the optimal control problem (4.1)–(4.5) is consistent if: (i) the set of feasible solutions Ξ ε is non-empty; (ii) for any triplet (u, g, y), where u ∈ Aad (Γ N,ε ), g ∈ L q (D), and y = y(u, g) ∈ W01,2 (Ω ε ; Γ D ) is a weak solution to the problem (4.2)–(4.3), there exists a distribution z such that z ∈ W01,2 (Ω ε ; Γ D ) and (u, g, z) ∈ Ξ ε . Thus, if the original optimal control problem is consistent in the above mentioned sense then the description of its feasible solutions in the form of Definition 4.1 is not restrictive from the control point of view. In order to establish this property, we begin with the following technical result. Lemma 4.1. Let (u, g) ∈ Aad (Γ N,ε ) × L q (D), be a pair of admissible controls. Assume that, for given ε > 0 and (u, g), boundary value problem (4.2)–(4.3) admits a solution. Then for every k ∈ N there exists an element z k ∈ W01,2 (Ω ε ; Γ D ) such that −∆z k = T k (f (z k )) + g zk = 0

on Γ D ,

in Ω ε ,

∂ν zk = u

(4.45)

on Γ N,ε ,

(4.46)

where T k (s) = min {s, k}. Moreover, z k ≤ z k+1 and the sequence {z k }k∈N is bounded in W01,2 (Ω ε ; Γ D ). Proof. Let y be a solution of (4.2)–(4.3) associated to the controls (u, g) ∈ Aad (Γ N,ε ) × L q (D). Let us define y0 ∈ W01,2 (Ω ε ; Γ D ) as a unique solution to the problem −∆y0 = g

in Ω ε ,

y0 = 0 on Γ D ,

and

∂ ν y0 = u

on Γ N,ε .

130 | 4 Asymptotic Analysis of Optimal Neumann Boundary Control Problem Then we have the following obvious relations −∆y0 ≤ T k (f (y0 )) + g −∆(y − y0 ) ≥ 0 and

− ∆y ≥ T k (f (y)) + g

and in Ω ε ,

in Ω ε ,

y = y0 = 0 on Γ D ,

∂ ν y = ∂ ν y0 = 0

on Γ N,ε .

From this we deduce that y0 is a subsolution and y is a supersolution of the problem − ∆z = T k (f (z)) + u in Ω ε , z = 0 and ∂ ν z = u on Γ N,ε .

on Γ D ,

(4.47)

Moreover, by the strong maximum principle [56], we conclude that y0 ≥ y in Ω ε . Thus, following the classical techniques [146], we can deduce that for given controls (u, g) ∈ Aad (Γ N,ε ) × L q (D) there exists a solution z k to the boundary value problem (4.47) and it is such that y0 (x) ≤ z k (x) ≤ y(x) and

z k (x) ≤ z k+1 (x)

a.e. in Ω ε .

(4.48)

Besides, as follows from (4.47), we get to obtain the following relations ∫︁ ∫︁ ∫︁ ∫︁ γ0 (z k )u dH N−1 + gz k dx |∇z k |2 dx = z k T k (f (z k )) dx + Ωε

Γ N,ε

Ωε

∫︁ ≤

[︀

Ωε

]︀

f (z k ) + |g | |z k | dx + C γ0 ||u||L2 (Γ N,ε ) ||z k ||W 1,2 (Ω ε ;Γ D ) 0

Ωε

∫︁ f (y)|z k | dx

≤ Ωε

)︁ (︁ + ||g||W −1,2 (Ω ε ;Γ D ) + C γ0 ||u||L2 (Γ N,ε ) ||z k ||W 1,2 (Ω ε ;Γ D ) 0 )︁ (︁ ≤ ||z k ||W 1,2 (Ω ε ;Γ D ) ||f (y)||W −1,2 (Ω ε ;Γ D ) + C||g||L q (Ω ε ) + C γ0 ||u||L2 (Γ N,ε ) . 0

Hence, (︁ )︁ sup ||z k ||W 1,2 (Ω ε ;Γ D ) ≤ ||f (y)||W −1,2 (Ω ε ;Γ D ) + C||g||L q (D) + C γ0 ||u||L2 (Γ N,ε ) < +∞, k∈N

0

(4.49)

i.e. the sequence {z k }k∈N is bounded in W01,2 (Ω ε ; Γ D ). Our next intention is to prove the following result. Proposition 4.5. Assume that, for a given ε > 0, Ω ε is star-shaped with respect to some point x0 ∈ int Ω ε , i.e. (︀ )︀ σ − x0 , ν(σ) ≥ 0 for H N−1 -a.a. σ ∈ ∂Ω ε . (4.50) Assume also that boundary value problem (4.2)–(4.3) has a solution for some (u, g) ∈ Aad (Γ N,ε ) × L q (D) with q ≥ 2. Then there exists an element z ∈ W01,2 (Ω ε ; Γ D ) such that (u, g, z) ∈ Ξ ε .

4.2 On Consistency of Optimal Control Problem | 131

Proof. Let {z k }k∈N be the sequence given by Lemma 4.1. Then, in view of estimate (4.49), there exists an element z ∈ W01,2 (Ω ε ; Γ D ) such that, up to a subsequence, we have z k ⇀ z in W01,2 (Ω ε ; Γ D ) and z k (x) → z(x) a.e. in Ω ε . (4.51) Hence, T k (f (z k (x))) → f (z(x)) for almost all x ∈ Ω ε . Since the relations (4.48) and Proposition 4.1 imply the property T k (f (z k )) ≤ f (z k ) ≤ f (y) ∈ L1 (Ω ε ), it follows from the Lebesgue dominated convergence theorem that T k (f (z k )) → f (z) strongly in L1 (Ω ε ). Using this fact and the weak convergence (4.51), it is easy to pass to the limit as k → ∞ in the integral identity ∫︁ ∫︁ ∫︁ φT k (f (z k )) dx + φu dH N−1 (∇z k , ∇φ) dx = Γ N,ε

Ωε

Ωε

∫︁ gφ dx,

+

N ∀ φ ∈ C∞ 0 (R ; Γ D )

(4.52)

Ωε

and show that z ∈ H f is the weak solution to boundary value problem (4.2)–(4.3). To deduce the inclusion (u, g, z) ∈ Ξ ε it remains to prove that this triplet satisfies the integral inequality (4.8). Since T k (f (z k )) ∈ L∞ (Ω ε ) and g ∈ L q (D), it follows that ∆z k ∈ L2 (Ω ε ) and, therefore, we can multiply the equality (4.45) by any function of φ ∈ L2 (Ω ε ) and make the integration over Ω ε . Let us consider φ := (x − x0 , ∇z k ) ∈ L2 (Ω ε ) as this function. Arguing as in the proof of Proposition 4.2, after integration in (4.45) over Ω ε , we get (︂ )︂ ∫︁ ∫︁ N 3 diam D 1− |∇z k |2 dx + |u|2 dH N−1 2 2 Γ N,ε

Ωε

∫︁

∫︁

g (x − x0 , ∇z k ) dx.

T k (f (z k )) (x − x0 , ∇z k ) dx +

≥ Ωε

Ωε

Let us define the function F k : R → R as follows {︃ F(t) F k (t) = F(f −1 (k)) + k(t − f −1 (k))

if f (t) ≤ k, if f (t) > k.

Then F k (t) is the primitive of T k (f (t)) and, moreover, because of the relation F(f −1 (k)) + k(t − f −1 (k)) = F(f −1 (k)) + F ′ (f −1 (k))(t − f −1 (k)) ≤ F(t) which is obviously valid for all t > f −1 (k) by convexity of F, we have F k (t) ≤ F(t) ∀ t ∈ R.

(4.53)

132 | 4 Asymptotic Analysis of Optimal Neumann Boundary Control Problem Now we are in a position to transform the right hand side in (4.53). Due to relation (4.20) and the star-shaped property of Ω ε , we have ∫︁ ∫︁ (︀ )︀ T k (f (z k )) (x − x0 , ∇z k ) dx= x − x0 , ∇F k (z k ) dx Ωε

Ωε

∫︁ F k (z k ) div(x − x0 ) dx

=− Ωε

∫︁

)︀ (︀ F k (γ0 (z k (σ))) ν(σ), σ − x0 dH N−1

+ ∂Ω ε

∫︁

∫︁ F k (z k ) dx ≥ −N

≥ −N Ωε

F(z k ) dx. Ωε

Combining this inequality with (4.53), we finally get ∫︁ ∫︁ ∫︁ 2N 2 |∇z k |2 dx ≤ F(z k ) dx − g (x − x0 , ∇y) dx N−2 N−2 Ωε

Ωε

Ωε

3 diam D + N−2

∫︁

2

|u| dH

N−1

.

(4.54)

Γ N,ε

It remains to pass to the limit in the last relation as k → ∞ taking into account the following properties ∫︁ ∫︁ ∫︁ ∫︁ by (4.48) by (4.51) F(z) dx ≤ F(y) dx ≤ C−1 f (y) dx, lim F(z k ) dx = F k→∞

Ωε

∫︁ lim inf k→∞

Ωε

|∇z k |2 dx

Ωε by (4.51)

∫︁

≥

Ωε

Ωε

|∇z|2 dx.

Ωε

Thus, the integral constraint (4.8) holds true with y = z and, hence, the triplet (u, g, z) is feasible to optimal control problem (4.1)–(4.5). We are now in a position to prove the main result of this section. Namely, let us show that for the star-shaped domains the consistency property of the original optimal control problem is a direct consequence of Proposition 4.5. Theorem 4.3. Let y d ∈ L2 (D), f ∈ C1loc (R), and g d ∈ L q (D) with q ≥ 2 be arbitrary distributions. If, for a given ε > 0, the domain Ω ε is star-shaped with respect to some point x0 ∈ int Ω ε , then optimal control problem (4.1)–(4.5) is consistent in the sense of Definition 4.2. Proof. As immediately follows from Definition 4.2 and Proposition 4.5, to prove this assertion it is enough to show that the set of feasible solutions Ξ ε is nonempty for given y d ∈ L2 (D), f ∈ C1loc (R), g d ∈ L q (D), and ε > 0.

4.3 On uniqueness of optimal solution and optimality conditions | 133

Let ̃︀ such that ||̃︀ y||W 1,r (D) ≤ ̂︀ β. Since this supposition is quite y ∈ C∞ 0 (D) be a function ⃒ (︀ )︀ plausible, let us set ̃︀ y⃒Γ and ̃︀ u := ∂ ν ̃︀ g := −∆̃︀ y − f (̃︀ y), where ∂ ν y(σ) = ∇y(σ), ν(σ) N,ε stands for the outward normal derivative of ̃︀ y to Γ N,ε at the point σ. Then ̃︀ u ∈ Aad (Γ N,ε ), 1,2 2 q ̃︀ y) ∈ L (Ω ε ). Hence, it follows from Proposition 4.2 g ∈ L (D), ̃︀ y ∈ W0 (Ω ε ; Γ D ), and f (̃︀ that the triplet (̃︀ u, ̃︀ g, ̃︀ y) is related by integral identity (4.7) and satisfies the inequality (4.8). Hence, (̃︀ u, ̃︀ g, ̃︀ y) ∈ Ξ ε . As an obvious consequence of this assertion, we can give the following one. Corollary 4.2. Under conditions of Theorem 4.3, the non-triviality assumption in the form of Hypothesis A is fulfilled and it can be omitted in the statement of existence result (see Theorem 4.2).

4.3 On uniqueness of optimal solution and optimality conditions Our main intention in this section is to show that the original optimal control problem has a unique solution. To begin with, we fix the parameter ε > 0 and assume that the corresponding optimal control problem (4.1)–(4.5) is consistent in the sense of Definition 4.2. For our further analysis we need to define the following set {︀ }︀ Λ ε = (u, g) ∈ Aad (Γ N,ε ) × L q (D) : ∃ y ∈ H f s. t. (u, g, y) ∈ Ξ ε .

(4.55)

The main characteristic properties of the set Λ ε are summarized in the following assertion. Proposition 4.6. Assume that, in addition to the property (4.6), the function F ∈ C2loc (R) is such that its derivative f = F ′ : R → (0, ∞) is a strictly convex function and Ω ε is a star-shaped domain with respect to some point x0 ∈ int Ω ε . Then the set Λ ε is nonempty, convex, and closed in L2 (Γ N,ε ) × L q (Ω ε ). Proof. Since the set of feasible solution Ξ ε is nonempty, it follows from (4.55) that Λ ε ̸= ∅. Let us establish the convexity of Λ ε . Let (u1 , g1 , y1 ) and (u2 , g2 , y2 ) be two different triplets of Ξ ε . It is clear that in this case we have y1 ̸= y2 and (u1 , g1 ), (u2 , g2 ) ∈ Λ ε . Let λ ∈ (0, 1). We set u = λu1 + (1 − λ)u2 ,

g = λg1 + (1 − λ)g2 ,

y = λy1 + (1 − λ)y2 .

Our aim is to show that (u, g) ∈ Λ. Since for Lipschitz domain Ω ε the mapping −∆ : W01,2 (Ω ε ; Γ D ) → W −1,2 (Ω ε ; Γ D )

134 | 4 Asymptotic Analysis of Optimal Neumann Boundary Control Problem is an isomorphism, we can define in a unique way a distribution z such that z ∈ W01,2 (Ω ε ; Γ D ), −∆z = g = λg1 + (1 − λ)g2 in Ω ε , z = 0 on Γ D , and

∂z = λu1 + (1 − λ)u2 on Γ N,ε . ∂ν

By the initial assumptions, we have ∂y1 = u1 on Γ N,ε , ∂ν ∂y2 = u2 on Γ N,ε , −∆y2 = f (y2 ) + g2 , ∂ν

−∆y1 = f (y1 ) + g1 ,

and

y1 = 0 on Γ D ,

and

y2 = 0 on Γ D .

Hence, − ∆y = λf (y1 ) + (1 − λ)f (y2 ) + u in Ω ε , ∂y y = 0 on Γ D , and = u on Γ N,ε . ∂ν

(4.56)

Taking into account that f = F ′ (z) > 0 almost everywhere in Ω ε (see (4.6)) and f satisfies the Jensen’s inequality f (y) ≤ λf (y1 ) + (1 − λ)f (y2 ), (4.57) we obtain − ∆z ≤ f (z) + g

and

− ∆y

by (4.56) and (4.57)

≥

f (y) + g,

(4.58)

i.e. z is a subsolution to the boundary value problem −∆ψ = f (ψ) + g in Ω ε , ψ = 0 on Γ D , and

∂ψ = u on Γ N,ε , ∂ν

and y is its supersolution. Moreover, since − ∆y ≥ f (y) + g

by (4.6)

≥

g = −∆z in Ω ε ,

(4.59)

it follows that −∆(y − z) ≥ 0. Hence, by the strong maximum principle [152] (see also Chapter 2 in [56]), we conclude that y ≥ z in Ω. Thus, following the classical techniques introduced by D.H. Sattinger [144], we deduce that for given controls u = λu1 + (1 − λ)u2 and g = λg1 + (1 − λ)g2 there exists a solution ψ to the above boundary value problem such that z(x) ≤ ψ(x) ≤ y(x) almost everywhere in Ω ε . (4.60) Moreover, as follows from (4.60) and the fact that z, y ∈ W01,2 (Ω ε ; Γ D ), we have ψ ∈ W01,2 (Ω ε ; Γ D ) and ∫︁ ∫︁ by (4.42) and (4.6) ||f (ψ)||L1 (Ω ε ) := |f (ψ)| dx ≤ |f (y)| dx Ωε by (4.57)

≤

Ωε

λ||f (y1 )||L1 (Ω ε ) + (1 − λ)||f (y2 )||L1 (Ω ε ) < +∞.

4.3 Uniqueness and optimality system | 135

Then, by Proposition 4.5 there exists an element z ∈ W01,2 (Ω ε ; Γ D ) such that (u, g, z) is a feasible solution to (4.1)–(4.5). Thus, (u, g) ∈ Λ ε and, hence, Λ ε is a convex set. Now let us prove the closedness of Λ ε in L2 (Γ N,ε ) × L q (Ω ε ). By Mazur’s theorem, it is enough to show that this set is sequentially closed with respect to the product of the strong convergence in L2 (Γ N,ε ) and the weak convergence in L q (Ω ε ). Let {︀ }︀ (u k , g k ) k∈N ⊂ Λ ε be a sequence such that u k → u in L2 (Γ N,ε )

and

g k ⇀ g in L q (Ω ε ).

Then, there exists a sequence of states {y k }k∈N such that (u k , g k , y k ) ∈ Ξ ε for all k ∈ N. Thanks to Theorem 4.1, we know that this sequence is bounded in W01,p (Ω ε ; Γ D ). Then Proposition 4.4 claims that for any weak limit y of {y k }k∈N in W01,p (Ω ε ; Γ D ), we have (u, g, y) ∈ Ξ ε . Thus, by convexity of Λ ε , this set is closed in L2 (Γ N,ε ) × L q (Ω ε ). The above mentioned properties of the set Λ ε allow to specify the main result of Theorem 4.2 as follows. Theorem 4.4. Let y d ∈ L2 (D) and g d ∈ L q (D) with q ≥ 2 be arbitrary distributions. Assume that, in addition to the property (4.6), the function F ∈ C2loc (R) is such that its derivative f = F ′ : R → (0, ∞) is a strictly convex function and Ω ε is a star-shaped domain with respect to some point x0 ∈ int Ω ε . Then optimal control problem (4.1)–(4.5) has a unique solution. Proof. Let us assume the converse. Namely, in view of Theorem 4.2, we can suppose that there exists at least two different feasible solutions (u ε,1 , g ε,1 , y ε,1 ) and (u ε,2 , g ε,2 , y ε,2 ) such that y ε,1 ̸= y ε,2 and J ε (u ε,1 , g ε,1 , y ε,1 ) = J ε (u ε,2 , g ε,2 , y ε,2 ) =

inf

(u,g,y)∈Ξ ε

J ε (u, g, y).

(4.61)

We set u ε = (u ε,1 + u ε,2 ) /2 and g ε = (g ε,1 + g ε,2 ) /2. Arguing as in the proof of Proposition 4.6, it is easy to show that there exists a distribution ψ ε ∈ W01,2 (Ω ε ; Γ D ) such that (u ε , g ε , ψ ε ) ∈ Ξ ε and ψ ε ≤ (y ε,1 + y ε,2 ) /2 a.e. in Ω ε .

136 | 4 Asymptotic Analysis of Optimal Neumann Boundary Control Problem In fact, because of the strict convexity of f , it can be shown that this inequality is strict in Ω ε . Indeed, by the Jensen’s inequality, )︂ (︂ )︀ 1 (︀ )︀ 1 1 (︀ 1 f (y ε,1 ) + g ε,1 + f (y ε,2 ) + g ε,2 y + y − ψε = −∆ 2 ε,1 2 ε,2 2 2

(︂ ∂ ∂ν

(︂

1 1 y + y − ψε 2 ε,1 2 ε,2 1 1 y + y − ψε 2 ε,1 2 ε,2

− f (ψ ε ) − g ε )︀ 1 (︀ = f (y ε,1 ) + f (y ε,2 ) − f (ψ ε ) 2 (︁ y + y )︁ )︀ 1 (︀ ε,1 ε,2 ≥ ≥ 0, f (y ε,1 ) + f (y ε,2 ) − f 2 2 )︂ = 0 on Γ D , )︂ = 0 on Γ N,ε ,

(︀ )︀ (︀ )︀ ε,2 and 12 f (y ε,1 ) + f (y ε,2 ) − f y ε,1 +y ̸= 0 on Ω ε because of the strict convexity of f . 2 Hence, (y ε,1 + y ε,2 ) /2 > ψ ε in Ω ε by the strong maximum principle [56, 152]. As a result, we arrive at the inequality ∫︁ 1 |ψ ε − y d |2 dx J ε (u ε , g ε , ψ ε ) = 2 Ωε

1 + 2

∫︁

2

|u ε | dH

N−1

1 + q

Γ N,ε

∫︁

|g ε − g d |q dx

Ωε

⎤

⎡
N2 and t > N − 1. Moreover, it is easy to deduce from Sobolev embedding theorem that * L s (Ω ε ) is continuously injected in W −1,s (Ω ε ; Γ D ) (4.66) Ns with s* = N−s provided

N 2

< s < N. Moreover, taking into account the following properties

[︁ ]︁* * * ′ W −1,s (Ω ε ; Γ D ) = W01,(s ) (Ω ε ; Γ D ) , where (s* )′ = * ′

1

Ns , Ns − N + s

* ′

γ0 : W01,(s ) (Ω ε ; Γ D ) −→ W s* ,(s ) (Γ N,ε ) is the trace operator, ]︁* [︁ 1 * ′ * 1 W s* ,(s ) (Γ N,ε ) = W − s* ,s (Γ N,ε ); 1

*

L p (Γ N,ε ) ˓→ W − s* ,s (Γ N,ε ) with continuous injection ∀ p ≥ and the fact that

s(N − 1) , N−s

s(N − 1) N > N −1 ∀s : < s < N, N−s 2

we get 1

*

L t (Γ N,ε ) is continuously injected in W − s* ,s (Γ N,ε )

(4.67)

provided t≥

s(N − 1) N−s

and

N < s < N. 2

(4.68)

Combining this fact with (4.65) and (4.66), we finally deduce the existence of a constant C*s,t > 0 independent of h1 and h2 such that ]︂ [︂ * . (4.69) ||z||L∞ (Ω ε ) ≤ C s,t ||h1 ||W −1,s* (Ω ε ;Γ D ) + ||γ0 (h2 )|| − 1 ,s* W

s*

(Γ N,ε )

For the simplicity, we assume that the constant ̂︀ β > 0 in (4.5) is large enough and the restriction ||v||W 1,r (D) ≤ ̂︀ β is not active for an optimal solution. Theorem 4.5. Assume that Ω ε ⊂ D is a star-shaped domain with respect to some of its interior point x0 . Let F : R → (0, +∞) be a mapping of the class C2loc (R) such that F satisfies estimate (4.6), its derivative f = F ′ : R → (0, ∞) is a strictly convex function, the second derivative F ′′ is bounded from below, and f is log-quasi-additive, i.e. there exists a constant C* > 0 such that f (v + z) ≤ C* f (v)f (z) ∀ v, z ∈ R.

(4.70)

138 | 4 Asymptotic Analysis of Optimal Neumann Boundary Control Problem For a fixed ε > 0, let (u0ε , g 0ε , y0ε ) ∈ Ξ ε be an optimal solution to the problem (4.1)–(4.5) with an extra property f (y0ε ) ∈ L q (Ω ε ) with q ≥ 2. (4.71) * ′

Then there exists a distribution φ ε ∈ W01,(s ) (Ω ε ; Γ D ), with (s* )′ = such that the tuple

Ns Ns−N+s

and

N 2

< s < N,

* ′

(u0ε , g 0ε , y0ε , φ ε ) ∈ L2 (Γ N,ε ) × L q (Ω ε ) × W01,2 (Ω ε ; Γ D ) × W01,(s ) (Ω ε ; Γ D ) satisfies the following optimality system −∆y0ε = f (y0ε ) + g 0ε in Ω ε , y0ε

on Γ D ,

=0 |g 0ε

∂ ν y0ε

− gd | u0ε

q−2

=

on Γ N,ε ,

− g d ) = −φ ε in Ω ε ,

= −γ0 (φ ε ) on Γ N,ε ,

−∆φ ε = f φε = 0

(g 0ε

u0ε

′

(y0ε )φ ε

on Γ D ,

+

y0ε

− y d in Ω ε ,

∂ ν φ ε = 0 on Γ N,ε

(4.72) (4.73) (4.74) (4.75) (4.76) (4.77)

Proof. For an arbitrary distribution z ∈ W01,2 (Ω ε ; Γ D ) ∩ L∞ (Ω ε ) such that ∂z ∈ L∞ (Γ N,ε ), ∂ν

∆z ∈ L q (Ω ε ) with q ≥ 2,

and every λ ∈ R (λ ̸= 0), we set y λε = y0ε + λz, g ελ u λε

=

g 0ε

− λ∆z +

=

u0ε

+ λ∂ ν z,

(4.78) f (y0ε ) −

f (y0ε

+ λz),

(4.79) (4.80)

and, in view of the initial assumptions, we can suppose that there exists λ0 ̸= 0 such that u λε ∈ Aad (Γ N,ε ) for all |λ| ≤ λ0 . Since z ∈ L∞ (Ω ε ) =⇒ z ∈ L q (Ω ε ) and (1 − C* f (λz)) ∈ L∞ (Ω ε ), ⃒ ⃒ by (4.70) ⃒ ⃒ ⃒ 0 by (4.71) q ⃒ ⃒ ⃒ 0 0 ⃒f (y ε ) − f (y ε + λz)⃒ ≤ ⃒1 − C* f (λz)⃒ f (y ε ) ∈ L (Ω ε ), it follows from our original assumptions and (4.78)–(4.80) that g ελ ∈ L q (D),

u λε ∈ Aad (Γ N,ε ),

−∆y λε = f (y λε ) + g ελ in Ω ε , y λε = 0

on Γ D ,

∂ ν y λε = u λε

J ε (u λε , g ελ , y λε ) < +∞.

on Γ N,ε ,

(4.81)

4.3 Uniqueness and optimality system | 139

Moreover, condition f (y λε ) ∈ L q (Ω ε ), Proposition 4.2, and the star-shaped property of Ω ε imply that the following integral inequality ∫︁ ∫︁ ∫︁ (︁ )︁ 2N 2 |∇y λε |2 dx ≤ F(y λε ) dx − g ελ x − x0 , ∇y λε dx N−2 N−2 Ωε

Ωε

Ωε

∫︁

3 diam D + N−2

|u λε |2 dH N−1

Γ N,ε

holds true. Hence, (u λε , g ελ , y λε ) is a feasible solution to the problem (4.1)–(4.5) for all λ ∈ [0, λ0 ]. As a result, we have the following inequality for the increment of the cost functional (︁ )︁ (︁ )︁ ∆J ε (u0ε , g 0ε , y0ε ) := J ε u λε , g ελ , y λε − J ε u0ε , g 0ε , y0ε ≥ 0, ∀ |λ| ≤ λ0 . (4.82) Using Lebesgue’s convergence theorem, we get (︁ )︁ (︀ )︀ ∫︁ J ε u λε , g ελ , y λε − J ε u0ε , g 0ε , y0ε |y λε − y d |2 − |y0ε − y d |2 1 = lim dx lim λ 2 λ→0 λ λ→0 Ωε

+

1 lim q λ→0

∫︁

|g ελ

− gd |

q

− |g 0ε

− gd |

λ

q

dx

Ωε

+

1 lim 2 λ→0

∫︁

|u λε |2 − |u0ε |2

λ

dH N−1 =

Γ N,ε

∫︁ +

∫︁

(y0ε − y d )z dx +

∫︁

u0ε ∂ ν z dH N−1

Γ N,ε

Ωε

(︁ )︁ |g 0ε − g d |q−2 (g 0ε − g d ) −∆z − f ′ (y0ε )z dx.

Ωε

Then condition (4.82) and the linearity of this relation with respect to z imply ∫︁ ∫︁ 0 (y ε − y d )z dx + u0ε ∂ ν z dH N−1 Γ N,ε

Ωε

∫︁ +

(︁

|g 0ε − g d |q−2 (g 0ε − g d ) −∆z − f ′ (y0ε )z

)︁

dx = 0.

Ωε

Setting φ ε = −|g 0ε − g d |q−2 (g 0ε − g d ) and ψ ε = u0ε , we see that ∫︁ ∫︁ ∫︁ (︁ )︁ ψ ε ∂ ν z dH N−1 = (y0ε − y d )z dx. φ ε −∆z − f ′ (y0ε )z dx − Ωε

Γ N,ε

Ωε

(4.83)

140 | 4 Asymptotic Analysis of Optimal Neumann Boundary Control Problem 1,2 Given h1 , h2 ∈ C∞ 0 (D), let z ∈ W 0 (Ω ε ; Γ D ) be the solution of the boundary value problem (4.62)–(4.63). Combining this fact with relation (4.83), we obtain ∫︁ ∫︁ ψ ε h2 |Γ N,ε dH N−1 φ ε h1 dx − Γ N,ε

Ωε

∫︁

∫︁ φ ε (−∆z) dx −

=

Γ N,ε

Ωε

∫︁ =

ψ ε h2 |Γ N,ε dH N−1

(y0ε − y d )z dx +

Ωε

∫︁

|g 0ε − g d |q−2 (g 0ε − g d )f ′ (y0ε )z dx.

(4.84)

Ωε

Let us estimate the right-hand side of this equality. Taking into account the convexity property (︁ (︁ )︁ )︁ (︁ )︁ f y0ε + λz − f y0ε ≥ λf ′ y0ε z, estimate (4.81), and the boundedness of f ′ from below, we see that (︁ )︁ f ′ y0ε ∈ L q (Ω ε ).

(4.85)

Hence, ∫︁

|g 0ε − g d |q−2 (g 0ε − g d )f ′ (y0ε )z dx ≤

Ωε

∫︁

⃒ ⃒

⃒ ⃒

|g 0ε − g d |q−1 ⃒f ′ (y0ε )⃒ dx||z||L∞ (Ω ε )

Ω q−1

≤ ||g 0ε − g d ||L q (Ω ε ) ||f ′ (y0ε )||L q (Ω ε ) ||z||L∞ (Ω ε ) and ∫︁

(y0ε − y d )z dx ≤ ||y0ε − y d ||L1 (Ω ε ) ||z||L∞ (Ω ε )

Ωε

≤

|D|||y0ε − y d ||L2 (Ω ε ) ||z||L∞ (Ω ε ) .

√︀

Combining the above estimates with (4.67) and (4.69), we finally deduce from (4.84) ∫︁ ∫︁ ψ ε h2 |Γ N,ε dH N−1 φ ε h1 dx − Ωε

Γ N,ε

]︁ [︁ √︀ q−1 ≤ C*s,t ||g 0ε − g d ||L q (Ω ε ) ||f ′ (y0ε )||L q (Ω ε ) + |D|||y0ε − y d ||L2 (Ω ε ) [︂ ]︂ × ||h1 ||W −1,s* (Ω ε ;Γ D ) + ||γ0 (h2 )|| − 1 ,s* . W

s*

(Γ N,ε )

Since )︁ (︁ * ′ 1 * ′ W s* ,(s ) (Γ N,ε ) = γ0 W01,(s ) (Ω ε ; Γ D ) , [︁ 1 * ′ ]︁* * 1 W s* ,(s ) (Γ N,ε ) = W − s* ,s (Γ N,ε ),

(4.86)

4.4 Description of the domain perturbations | 141

1

*

*

−1,s and C∞ (Ω ε ; Γ D ) and W − s* ,s (Γ N,ε ), it follows from inequality 0 (D) is dense in W (4.86) that

[︁ ]︁* * ′ * φ ε ∈ W −1,s (Ω ε ; Γ D ) = W01,(s ) (Ω ε ; Γ D ), ]︁* [︁ * * ′ 1 1 ψ ε ∈ W − s* ,s (Γ N,ε ) = W s* ,(s ) (Γ N,ε ),

(4.87) (4.88)

Ns where (s* )′ = Ns−N+s , for every N2 < s < N. The fact that φ ε satisfies the boundary value problem (4.76)–(4.77), immediately follows from (4.83) and definition of φ ε and ψ ε . Indeed, making use of the formula of integration by parts and using the fact that γ0 (φ ε ) = 0 along the boundary Γ D , we get ∫︁ ∫︁ ∫︁ (y0ε − y d )z dx + |g 0ε − g d |q−2 (g 0ε − g d )f ′ (y0ε )z dx + ψ ε ∂ ν z dH N−1 Ωε

Γ N,ε

Ωε

∫︁ γ0 (φ ε )∂ ν z dH

=−

N−1

∫︁ +

γ0 (z)∂ ν φ ε dH N−1

Γ N,ε

Γ N,ε

∫︁ (−∆φ ε ) z dx.

+ Ωε

To guarantee the fulfilment of this equality, it is enough to define φ ε as the solution to the boundary value problem (4.76)–(4.77), and ψ ε as the trace of φ ε on Γ N,ε (see (4.75) and properties (4.87)–(4.88) ensuring correctness of such definition).

4.4 Description of the domain perturbations The main object of our concern in this section is a family of domains {Ω ε }0 0,

k=1

(︃ Ω∆Ω ε =

=

=

m ⋃︁

)︃ (︃ k

Π N (ξ ) ∩ Ω

∆

m ⋃︁

)︃ k

Π N (ξ ) ∩ Ω ε

k=1

k=1 m (︁ ⋃︁

)︁ (︁ )︁ Π N (ξ k ) ∩ Ω ∆ Π N (ξ k ) ∩ Ω ε

k=1 m [︁ (︁ ⋃︁

)︁ (︁ )︁ ]︁ Π N (ξ k ) ∩ Ω \ Ω ε ∪ Π N (ξ k ) ∩ Ω ε \ Ω .

k=1

Then we see that sup

inf |x − y|RN ≤

x∈Ω\Ω ε y∈∂Ω

≤

≤

m ∑︁

≤

m ∑︁

sup

inf

y∈∂Ω∩Π N (ξ k ) k k=1 x∈(Π N (ξ )∩Ω)\Ω ε

| x − y |R N

|x N − ϕ k (x1 , . . . , x N−1 )|

sup

k=1 x∈(Π N m ∑︁ k=1

m ∑︁

(ξ k )∩Ω

)\Ω ε |x N − ϕ k (x1 , . . . , x N−1 )|

sup

x=(x1 ,...,x N−1 ,x N )∈Π N (ξ k ) ϕ kε (x1 ,...,x N−1 ) 0 ε→0

small enough. Since α1 ≤ α2 + α3 + α4 and for each σ′ ∈ Π N−1 (ξ k ) there can be found an (︀ )︀ index k ∈ {1, . . . , m} such that σ ε − x0 , ν(σ ε ) = |σ ε − x0 |Jϕ kε cos α1 , it follows that the star-shaped property for the domain Ω ε with respect to the point x0 ∈ int Ω holds true if only α3 ≤ 2π − α2 − α4 for each σ′ ∈ Π N−1 (ξ k ). From this, we finally deduce (︁ )︁ ∇Φ kε (σ ε ), ∇Φ k (σ) = cos α3 ≥ sin(α2 + α4 ) Jϕ k (σ) Jϕ kε (σ ε ) (σ ε − x0 , σ − x0 ) √︀ ≥ 1 − (γ* )2 . |σ ε − x0 ||σ − x0 | Then (4.99) is a straightforward consequence of the last inequality. Remark 4.5. As follows from (4.99), the star-shaped property of the family of domains {Ω ε }0 0 small enough can be induced by the the enhanced star-shaped property of non-perturbed domain Ω provided the surface area of the graph Γ N,ε over Γ N does not go to infinity locally in measure as ε tends to zero. Otherwise, we come into conflict with the inequality (4.99). Taking this observation into account, we restrict our further consideration by the following type of ’rugosity’ along Γ N,ε -part of the boundary ∂Ω ε . Definition 4.4. We say that a sequence {Ω ε }0