NO-GAP OPTIMALITY CONDITIONS AND SOLUTION STABILITY FOR OPTIMAL CONTROL PROBLEMS GOVERNED BY SEMILINEAR ELLIPTIC EQUATIONS

MINISTRY OF EDUCATION AND TRAININGHANOI UNIVERSITY OF SCIENCE AND TECHNOLOGYNGUYEN HAI SON NO-GAP OPTIMALITY CONDITIONS AND SOLUTION STABILITY FOR OPTIMAL CONTROL PROBLEMS GOVERNED BY SE

MINISTRY OF EDUCATION AND TRAININGHANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY

NGUYEN HAI SON

NO-GAP OPTIMALITY CONDITIONS

AND SOLUTION STABILITY FOR OPTIMAL CONTROL PROBLEMS GOVERNED BY

SEMILINEAR ELLIPTIC EQUATIONS

DOCTORAL DISSERTATION OF MATHEMATICS

Hanoi - 2019

MINISTRY OF EDUCATION AND TRAININGHANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY

NGUYEN HAI SON

NO-GAP OPTIMALITY CONDITIONS

AND SOLUTION STABILITY FOR OPTIMAL CONTROL PROBLEMS GOVERNED BY

SEMILINEAR ELLIPTIC EQUATIONS

Major: MATHEMATICS Code: 9460101

DOCTORAL DISSERTATION OF MATHEMATICS

SUPERVISORS:

1 Dr Nguyen Thi Toan

2 Dr Bui Trong Kien

Hanoi - 2019

COMMITTAL IN THE DISSERTATION

I assure that my scientific results are new and righteous Before I published theseresults, there had been no such results in any scientific document I have responsibili-ties for my research results in the dissertation

Hanoi, April 3rd, 2019

This dissertation has been carried out at the Department of Fundamental matics, School of Applied Mathematics and Informatics, Hanoi University of Scienceand Technology It has been completed under the supervision of Dr Nguyen Thi Toanand Dr Bui Trong Kien

Mathe-First of all, I would like to express my deep gratitude to Dr Nguyen Thi Toan and

Dr Bui Trong Kien for their careful, patient and effective supervision I am very lucky

to have a chance to work with them, who are excellent researchers

I would like to thank Prof Jen-Chih Yao for his support during the time I visited andstudied at Department of Applied Mathematics, Sun Yat-Sen University, Kaohsiung,Taiwan (from April, 2015 to June, 2015 and from July, 2016 to September, 2016) Iwould like to express my gratitude to Prof Nguyen Dong Yen for his encouragementand many valuable comments

I would also like to especially thank my friend, Dr Vu Huu Nhu for kind help andencouragement

I would like to thank the Steering Committee of Hanoi University of Science andTechnology (HUST), and School of Applied Mathematics and Informatics (SAMI) fortheir constant support and help

I would like to thank all the members of SAMI for their encouragement and help

I am so much indebted to my parents and my brother for their support I thank mywife for her love and encouragement This dissertation is a meaningful gift for them

Hanoi, April 3rd, 2019Nguyen Hai Son

COMMITTAL IN THE DISSERTATION i

ACKNOWLEDGEMENTS ii

CONTENTS iii

TABLE OF NOTATIONS 1

INTRODUCTION 3

Chapter 0 PRELIMINARIES AND AUXILIARY RESULTS 8 0.1 Variational analysis 8

0.1.1 Set-valued maps 8

0.1.2 Tangent and normal cones 9

0.2 Sobolev spaces and elliptic equations 13

0.2.1 Sobolev spaces 13

0.2.2 Semilinear elliptic equations 20

0.3 Conclusions 24

Chapter 1 NO-GAP OPTIMALITY CONDITIONS FOR DISTRIBUTED CONTROL PROBLEMS 25 1.1 Second-order necessary optimality conditions 26

1.1.1 An abstract optimization problem 26

1.1.2 Second-order necessary optimality conditions for optimal control problem 27

1.2 Second-order sufficient optimality conditions 40

1.3 Conclusions 57

Chapter 2 NO-GAP OPTIMALITY CONDITIONS FOR BOUNDARY CONTROL PROBLEMS 58 2.1 Abstract optimal control problems 59

2.2 Second-order necessary optimality conditions 66

2.3 Second-order sufficient optimality conditions 75

2.4 Conclusions 89

Chapter 3 UPPER SEMICONTINUITY AND CONTINUITY OF THE SOLUTION MAP TO A PARAMETRIC BOUNDARY CONTROL PROBLEM 91 3.1 Assumptions and main result 92

3.2 Some auxiliary results 94

3.2.1 Some properties of the admissible set 94

3.2.2 First-order necessary optimality conditions 98

3.3 Proof of the main result 100

3.4 Examples 104

3.5 Conclusions 109

GENERAL CONCLUSIONS 110

LIST OF PUBLICATIONS 111

REFERENCES 112

TABLE OF NOTATIONS

N:= {1, 2, } set of positive natural numbers

A × B Descartes product of the sets A and B

[x1, x2] the closed line segment between x1 and x2

X∗∗ topological bi-dual of a normed space X

B(x, δ) open ball with centered at x and radius δB(x, δ) closed ball with centered at x and radius δ

T (K, x) Bouligand tangent cone of the set K at x

T[(K, x) adjoint tangent cone of the set K at x

T2(K, x, d) second-order Bouligand tangent set of the set

K at x in direction d

T2[(K, x, d) second-order adjoint tangent set of the set K

at x in direction d

N (K, x) normal cone of the set K at x

1 Motivation

Optimal control theory has many applications in economics, mechanics and otherfields of science It has been systematically studied and strongly developed since thelate 1950s, when two basic principles were made One was the Pontryagin MaximumPrinciple which provides necessary conditions to find optimal control functions Theother was the Bellman Dynamic Programming Principle, a procedure that reducesthe search for optimal control functions to finding the solutions of partial differentialequations (the Hamilton-Jacobi equations) Up to now, optimal control theory hasdeveloped in many various research directions such as non-smooth optimal control,discrete optimal control, optimal control governed by ordinary differential equations(ODEs), optimal control governed by partial differential equations (PDEs), (see [1, 2,3])

In the last decades, qualitative studies for optimal control problems governed byODEs and PDEs have obtained many important results One of them is to give op-timality conditions for optimal control problems For instance, J F Bonnans et al.[4, 5, 6], studied optimality conditions for optimal control problems governed by ODEs,while J F Bonnans [7], E Casas et al [8, 9, 10, 11, 12, 13, 14, 15, 16, 17], C Meyerand F Tr¨oltzsch [18], B T Kien et al [19, 20, 21, 22], A R¨osch and F Tr¨oltzsch[23, 24] derived optimality conditions for optimal control problems governed by el-liptic equations

It is known that if ¯u is a local minimum of F , where F : U →R is a differentiablefunctional and U is a Banach space, then F0(¯u) = 0 This a first-order necessaryoptimality condition However, it is not a sufficient condition in case of F is notconvex Therefore, we have to invoke other sufficient conditions and should study thesecond derivative (see [17])

Better understanding of second-order optimality conditions for optimal control lems governed by semilinear elliptic equations is an ongoing topic of research for severalresearchers This topic is great value in theory and in applications Second-order suffi-cient optimality conditions play an important role in the numerical analysis of nonlinearoptimal control problems, and in analyzing the sequential quadratic programming al-gorithms (see [13, 16, 17]) and in studying the stability of optimal control (see [25, 26]).Second-order necessary optimality conditions not only provide criterion of finding outstationary points but also help us in constructing sufficient optimality conditions Let

prob-us briefly review some results on this topic

For distributed control problems, i.e., the control only acts in the domain Ω inRn,

E Casas, T Bayen et al [11, 13, 16, 27] derived second-order necessary and sufficientoptimality conditions for problem with pure control constraint, i.e.,

and the appearance of state constraints More precisely, in [11] the authors gavesecond-order necessary and sufficient conditions for Neumann problems with constraint(1) and finitely many equalities and inequalities constraints of state variable y whilethe second-order sufficient optimality conditions are established for Dirichlet problemswith constraint (1) and a pure state constraint in [13] T Bayen et al [27] derivedsecond-order necessary and sufficient optimality conditions for Dirichlet problems in thesense of strong solution In particular, E Casas [16] established second-order sufficientoptimality conditions for Dirichlet control problems and Neumann control problemswith only constraint (1) when the objective function does not contain control variable u

In [18], C Meyer and F Tr¨oltzsch derived second-order sufficient optimality conditionsfor Robin control problems with mixed constraint of the form a(x) ≤ λy(x) + u(x) ≤b(x) a.e x ∈ Ω and finitely many equalities and inequalities constraints

For boundary control problems, i.e., the control u only acts on the boundary Γ, E.Casas and F Tr¨oltzsch [10, 12] derived second-order necessary optimality conditionswhile the second-order sufficient optimality conditions were established by E Casas et

al in [12, 13, 17] with pure pointwise constraints, i.e.,

a(x) ≤ u(x) ≤ b(x) a.e x ∈ Γ

A R¨osch and F Tr¨oltzsch [23] gave the second-order sufficient optimality conditionsfor the problem with the mixed pointwise constraints which has unilateral linear formc(x) ≤ u(x) + γ(x)y(x) for a.e x ∈ Γ

We emphasize that in above papers, a, b ∈ L∞(Ω) or a, b ∈ L∞(Γ) Therefore,the control u belongs to L∞(Ω) or L∞(Γ) This implies that corresponding Lagrangemultipliers are measures rather than functions (see [19]) In order to avoid this dis-advantage, B T Kien et al [19, 20, 21] recently established second-order necessaryoptimality conditions for distributed control of Dirichlet problems with mixed state-control constraints of the form

a(x) ≤ g(x, y(x)) + u(x) ≤ b(x) a.e x ∈ Ωwith a, b ∈ Lp(Ω), 1 < p < ∞ and pure state constraints This motivates us to developand study the following problems

(OP 1) : Establish second-order necessary optimality conditions for Robin boundarycontrol problems with mixed state-control constraints of the form

a(x) ≤ g(x, y(x)) + u(x) ≤ b(x) a.e x ∈ Γ,

where a, b ∈ Lp(Γ), 1 < p < ∞.

(OP 2) : Give second-order sufficient optimality conditions for optimal control lems with mixed state-control constraints when the objective function does not depend

prob-on cprob-ontrol variables

Solving problems (OP 1) and (OP 2) is the first goal of the dissertation

After second-order necessary and sufficient optimality conditions are established,they should be compared to each other According to J F Bonnans [4], if the changebetween necessary and sufficient second-order optimality conditions is only betweenstrict and non-strict inequalities, then we say that the no-gap optimality conditions areobtained Deriving second-order optimality conditions without a gap between second-order necessary optimality conditions and sufficient optimality conditions, is a difficultproblem which requires to find a common critical cone under which both second-ordernecessary optimality conditions and sufficient optimality conditions are satisfied In [7],

J F Bonnans derived second-order necessary and sufficient optimality conditions withno-gap for an optimal control problem with pure control constraint and the objectivefunction is quadratic in both state variable y and control variable u The result in[7] was established by basing on polyhedric property of admissible sets and the theory

of Legendre forms Recently, the result has been extended by [27] and [28] However,there is an open problem in this area Namely, we need to study the following problem:(OP 3) : Find a theory of no-gap second-order optimality conditions for optimal con-trol problems governed by semilinear elliptic equations with mixed pointwise constraints.Solving problem (OP 3) is the second goal of this dissertation

Solution stability of optimal control problem is also an important topic in tion and numerical method of finding solutions (see [25, 29, 30, 31, 32, 33, 34, 35, 36,

optimiza-37, 38, 39, 40, 41]) An optimal control problem is called stable if the error of theoutput data is small in some sense for a small change in the input data The study ofsolution stability is to investigate continuity properties of solution maps in parameterssuch as lower semicontinuity, upper semicontinuity, H¨older continuity and Lipschitzcontinuity

Let us consider the following parametric optimal problem:

(2)

where y ∈ Y, u ∈ U are state and control variables, respectively; µ ∈ Π, λ ∈ Λ areparameters, F : Y × U × Π →R is an objective function on Banach space Y × U × Πand Φ(λ) is an admissible set of the problem

It is well-known that if the objective function F (·, ·, µ) is strongly convex, and theadmissible set Φ(λ) is convex, then the solution map of problem (2) is single-valued (see[29], [30], [31]) Moreover, A Dontchev [30] showed that under some certain conditions,

the solution map is Lipschitz continuous w.r.t parameters By using implicit functiontheorems, K Malanowski [35]-[40] proved that the solution map of problem (2) is also aLipschitz continuous function in parameters if weak second-order optimality conditionsand standard constraint qualifications are satisfied at the reference point Notice thatthe obtained results in [37]-[40] are for problems with pure state constraints, while theone in [35] is for problems with pure control constraints.

When the conditions mentioned above are invalid, the solution map may not besingleton (see [32, 33]) In this situation, we have to use tools of set-valued analysis andvariational analysis to deal with the problem In 2012, B T Kien et al [32] and [33]obtained the lower semicontinuity of the solution map to a parametric optimal controlproblem for the case where the objective function is convex in both variables and theadmissible sets are also convex Recently, the upper semicontinuity of the solution maphas been given by B T Kien et al [34] and V H Nhu [42] for problems, where theobjective functions may not be convex in the both variables and the admissible setsare not convex Notice that in [34] the authors considered the problem governed byordinary differential equations meanwhile in [42] the author investigated the problemgoverned by semilinear elliptic equation with distributed control From the above, onemay ask to study the following problem:

(OP 4) : Establish sufficient conditions under which the solution map of parametricboundary control problem is upper semicontinuous and continuous

Giving a solution for (OP 4) is the third goal of this dissertation

2 Objective

The objective of this dissertation is to study no-gap second-order optimality ditions and stability of solution to optimal control problems governed by semilinearelliptic equations with mixed pointwise constraints Namely, the main content of thedissertation is to concentrate on

con-(i) establishing second-order necessary optimality conditions for boundary controlproblems with the control variables belong to Lp(Γ), 1 < p < ∞;

(ii) deriving second-order sufficient optimality conditions for distributed control lems and boundary control problems when objective functions are quadratic forms

prob-in the control variables, and showprob-ing that no-gap optimality condition holds prob-inthis case;

(iii) deriving second-order sufficient optimality conditions for distributed control lems and boundary control problems when objective functions are independent ofthe control variables, and showing that in general theory of no-gap conditions doesnot hold;

prob-(iv) giving sufficient conditions for a parametric boundary control problem under which

the solution map is upper semicontinuous and continuous in parameters.

3 The structure and results of the dissertation

The dissertation has four chapters and a list of references

Chapter 0 collects several basic concepts and facts on variational analysis, Sobolevspaces and partial differential equations

Chapter 1 presents results on the no-gap second-order optimality conditions fordistributed control problems

Chapter 2 provides results on the no-gap second-order optimality conditions forboundary control problems

The obtained results in Chapters 1 and 2 are answers for problems (OP 1), (OP 2)and (OP 3), respectively

Chapter 3 presents results on the upper semicontinuity and continuity of the solutionmap to a parametric boundary control problem, which is a positive answer for problem(OP 4)

Chapter 1 and Chapter 2 are based on the contents of papers [1] and [2] in theList of publications which were published in the journals Set-Valued and VariationalAnalysis and SIAM Journal on Optimization, respectively The results of Chapter 3were content of article [3] in the List of publications which is published in Optimization.These results have been presented at:

• The Conference on Applied Mathematics and Informatics at Hanoi University ofScience and Technology in November 2016

• The 15th Conference on Optimization and Scientific Computation, Ba Vi in April2017

• The 7th International Conference on High Performance Scientific Computing inMarch 2018 at Vietnam Institute for Advanced Study in Mathematics (VIASM)

• The 9th Vietnam Mathematical Congress, Nha Trang in August 2018

• Seminar ”Optimization and Control” at the Institute of Mathematics, VietnamAcademy of Science and Technology

Chapter 0PRELIMINARIES AND AUXILIARY RESULTS

In this chapter, we review some background on Variational Analysis, Sobolev spaces,and facts of partial differential equations relating to solutions of linear elliptic equationsand semilinear elliptic equations For more details, we refer the reader to [1], [2], [3],[27], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], and [56]

0.1 Variational analysis

0.1.1 Set-valued maps

Let X and Y be nonempty sets A set-valued map/multifunction F from X to Y ,denoted by F : X ⇒ Y , which assigns for each x ∈ X a subset F (x) ⊂ Y F (x) iscalled the image or the value of F at x

Let F : X ⇒Y be a set-valued map between topological spaces X and Y We callthe sets

gph(F ) :=(x, y) ∈ X × Y | y ∈ F (x) dom(F ) :=x ∈ X | F (x) 6= ∅ ,

rge(F ) :=y ∈ Y | y ∈ F (x) for some x ∈ X := [

x∈X

F (x)

the graph, the domain and the range of F , respectively

The inverse F−1 : Y ⇒ X of F is the set-valued map, defined by

F−1(y) := {x ∈ X | y ∈ F (x)} for all y ∈ Y

The set-valued map F is called proper if dom(F ) 6= ∅

Definition 0.1.1 ([46, p 34]) Let F : X ⇒Y be a set-valued map between ical spaces X and Y

topolog-(i) If gph(F ) is a closed subset of the topological space X × Y then F is called closedmap (or graph-closed map)

(ii) If X, Y are linear topological spaces and gph(F ) is a convex subset of the logical space X × Y then F is called convex set-valued map

topo-(iii) If F (x) is a closed subset of Y for all x ∈ X then F is called closed-valued map.(iv) If F (x) is a compact subset of Y for all x ∈ X then F is called compact-valuedmap

The concepts of semicontinuous set-valued maps had been introduced in 1932 by G.Bouligand and K Kuratowski (see [44]).

Definition 0.1.2 ([45, Definition 1, p 108] and [44, Definition 1.4.1, p.38]) Let

F : X ⇒Y be a set-valued map between topological spaces and x0 ∈ dom(F )

(i) F is said to be upper semicontinuous at x0 if for any open set W in Y satisfying

F (x0) ⊂ W , there exists a neighborhood V of x0 such that

F (x) ⊂ W for all x ∈ V

(ii) F is said to be lower semicontinuous at x0 if for any open set W in Y satisfying

F (x0) ∩ W 6= ∅, there exists a neighborhood V of x0 such that

F (x) ∩ W 6= ∅ for all x ∈ V ∩ dom(F )

(iii) F is continuous at x0 if it is both lower semicontinuous and upper semicontinuous

at x0

The map F is called upper semicontinuous (resp., lower semicontinuous, continuous)

if it is upper semicontinuous (resp., lower semicontinuous, continuous) at every point

x ∈ dom(F )

Notice that in case of single-valued map F : X → Y , the above concepts arecoincident

When X, Y are metric spaces, set-valued map F : X ⇒ Y is lower semicontinuous

at x ∈ dom(F ) if and only if for all y ∈ F (x) and sequence {xn} ∈ dom(F ), xn → x,there exists a sequence {yn} ⊂ Y , yn ∈ F (xn) such that yn → y

0.1.2 Tangent and normal cones

Let X be a normed space with the norm k · k For each x0 ∈ X and δ > 0, we denote

by B(x0, δ) the open ball {x ∈ X | kx − x0k < δ}, and by B(x0, δ) the correspondingclosed ball We will write BX and BX for B(0X, 1) and B(0X, 1), respectively Let D

be a nonempty subset of X The distance from x ∈ X to D is defined by

From Definition 0.1.3, it follows that T (D, x) is a closed cone and T (D, x) ⊂cone(D − x), where cone(A) := {λa | λ ≥ 0, a ∈ A} is the cone generated by the set A.Moreover, the following property characterizes the Bouligand cone:

T (D, x) = {v ∈ X | ∃tn → 0+, ∃vn → v s.t x + tnvn ∈ D for all n ∈ N}.Definition 0.1.4 ([44, Definition 4.1.5, p 126]) Let D ⊂ X be a subset of normedspace X and x ∈ D The adjoint tangent cone or the intermediate cone T[(K, x) of D

where x0 D−→ x means that x0 ∈ D and x0→ x

From Definition 0.1.4, we have the following characters of the adjoint cones and theClarke tangent cones (see [44, p 128]):

Example 0.1.6 ( TC(D, x) = T[(D, x) 6= T (D, x) = cone(D − x))

Putting D = {n1 | n = 1, 2, } ⊂R and taking x = 0 ∈ ¯D, we have

TC(D, x) = T[(D, x) = {0},

T (D, x) = cone(D − x) =R+

D is convex then

TC(D, x) = T[(D, x) = T (D, x) = cone(D − x)

The above tangent cones has important roles in the study of first-order optimalityconditions for optimal control problems with constraints However, in order to obtainsecond-order optimality conditions for optimal control problems, we need to use second-order tangent sets

Definition 0.1.8 ([44, Definition 1.1.1, p 17]) Let X be a normed space and(Dt)t∈T ⊂ X be a sequence of sets depend on parameters t ∈ T, where T is a metricspace Suppose that t0 ∈ T The set

Limsupt→t 0

Dt:= {x ∈ X | lim

t→t 0

dist(x, Dt) = 0}

is called Painlev´e-Kuratowski lower limit of (Dt) as t → t0

Definition 0.1.9 ( [44, Definition 4.7.1 and 4.7.2, p 171]) Let D be a subset in thenormed space X and x ∈ ¯D, v ∈ X

Obviously, T2(D, x, v) and T2[(D, x, v) are closed and

T2(D, x, 0) = T (D, x), T2[(D, x, 0) = T[(D, x)

Moreover, we have

T2(D, x, v) = {w|∃tn → 0+, ∃wn → w, x + tnv + t2nwn ∈ D},

T2[(D, x, v) = {w|∀tn → 0+, ∃wn → w, x + tnv + t2nwn ∈ D}

Notice that T2(D, x, v) and T2[(D, x, v) are nonempty only if v ∈ T (D, x) and v ∈

T[(D, x), respectively Moreover, when D is convex, T2[(D, x, v) is convex but T (D, x, v)may not be convex (see [47, Subsection 3.2.1])

The following example shows that in general (T2(D, x, v) is different from T2[(D, x, v))(see [47, Example 3.31])

Example 0.1.10 (T2(D, x, v) 6= T2[(D, x, v))

Let us first construct a convex piecewise linear function x2 = ϕ(x1), x1 ∈ R, lating between two parabolas x2 = x21 and x2 = 2x21 in the following way: ϕ(x1) =ϕ(−x1), ϕ(0) = 0 and the function ϕ(x1) is linear on every interval [x1,k+1, x1,k],ϕ(x1,k) = x21,k and its graph on [x1,k+1, x1,k] is tangent to the curve x2 = 2x21 for somemonotonically decreasing to zero sequence {x1,k} It is evident how such a functioncan be constructed Indeed, for a given point x1,k > 0 consider the straight line passingthrough the point (x1,k, x21,k) and is tangent to the curve x2 = 2x21 It intersects thecurve x2 = x21 at a point x1,k+1 We can iterate this process and obtain a sequence{x1,k} It is easily seen that x1,k > x1,k+1> 0 and x1,k → 0 as k → ∞

oscil-Taking K = {(x1, x2) ∈ R2 | x2 ≥ ϕ(x1)} and x = (0, 0), v = (1, 0), we have

T2(D, x, v) = {(x1, x2) | x2 ≥ 2} and T2[(D, x, v) = {(x1, x2) | x2 ≥ 4}

The following result allows us to compute tangent cones of a convex and closedsubset K in Lp(Ω) with 1 ≤ p < +∞ (see Definition Lp(Ω) in next section)

Theorem 0.1.11 ([44, Theorem 8.5.1, p 324]) Let K be a subset of Lp(Ω) such that

M (x) := {u(x) | u ∈ K} is measurable and closed in R for a.e x ∈ Ω Then for all

T (K, u0) = T[(K, u0)

=nv ∈ Lp(Ω) | v(x) ∈ T[([a(x), b(x)], u0(x)) a.e x ∈ Ωo

In the sequel, we shall use concept normal cone which is dual concept of Clarketangent cones We denote by X∗ the dual space of the normed space X, i.e., the space

of all continuous linear functionals on X; the (dual) norm on X∗ is defined by

kf kX∗ = sup{f (x) | x ∈ X, kxk ≤ 1}

Then X∗ is a Banach space, i.e., X∗ is complete even if X is not (see [48, p.3]) Let

us denote by h·, ·i the canonical pairing between X∗ and X

Definition 0.1.13 ([44, Definition 4.4.2, p 157]) Let X be a Banach space, a subset

D ⊂ X and a point x ∈ D We shall say that the polar cone

N (D, x) := TC(D, x)− = {p ∈ X∗ | hp, vi ≤ 0 ∀v ∈ TC(D, x)}

is (Clarke) normal cone of D at x

When D is convex, N (D, x) coincides with the normal cone of D at x in ConvexAnalysis, i.e.,

i=1αi, and denote by

Dα := Dα1

1 Dα2

2 · · · DαN

N

a differential operator of order |α|, where Dj = ∂x∂

j for 1 ≤ j ≤ N We adopt theconvention that D(0, ,0)u = u for all function u defined on RN

Let Ω be an open subset in RN For each function u : Ω → R, we call suppu :={x ∈ Ω : u(x) 6= 0} the support of u

For each non-negative integer number m, we have the following classical functionspaces:

Cm(Ω) := {u : Ω →R| Dαu is continuous on Ω, ∀|α| ≤ m},

C∞(Ω) := ∩∞m=0Cm(Ω),

C0(Ω) := {u ∈ C0(Ω) | suppu is a compact subset in Ω},

C0∞(Ω) := {u ∈ C∞(Ω) | suppu is a compact subset in Ω}

Notice that C0(Ω) ≡ C(Ω)

Definition 0.2.1 ([43, Chapter 2] and [49, Definition 2.1, p 14]) Let Ω be an openset in RN, N ≥ 1, and p ≥ 1.

L∞(Ω) :=nu : Ω →R| u is Lebesgue measurable and

∃C > 0 such that |u(x)| ≤ C a.e x ∈ Ωo,with respectively norms

1 if p = +∞

The spaces Lp(Ω), 1 ≤ p ≤ ∞, are Banach spaces Moreover, Lp(Ω) with 1 < p < +∞are reflexive and separable, while L1(Ω) is separable Besides, L2(Ω) is a Hilbert spacewith the scalar product

(u, v)L2 (Ω) :=

Z

Ωu(x)v(x)dx ∀u, v ∈ L2(Ω)

It is noted that C0(Ω) is dense in Lp(Ω) for 1 ≤ p < +∞ The topological dual spaces of

Lp−spaces for (1 ≤ p < +∞) are Lp−space too, namely, Lp(Ω)∗= Lp0(Ω), 1 < p < +∞and L1(Ω)∗ = L∞(Ω) (see [43, Chapter 2] and [48, Section 4.3])

In the sequel, we will write Ω0 ⊂⊂ Ω if Ω0is included in Ω and compact We denoted

by L1loc(Ω) the space of local integrable functions on Ω, i.e.,

L1loc(Ω) :=nu : Ω →R| u is Lebesgue measurable and

Z

Ω 0

|u(x)|dx < +∞

for all measurable subset Ω0 ⊂⊂ Ωo

Then, for any open set Ω in RN and for all p ∈ [1, +∞], we have Lp(Ω) ⊂ L1loc(Ω) (see[43, Chapter 2, p 26])

Recall that C0∞(Ω) the space of functions infinitely differentiable in Ω with compactsupport in Ω We introduce a notion of convergence in the space C0∞(Ω) which can bedefined by a topology on C0∞(Ω) Then C0∞(Ω) is denoted by D(Ω)

Definition 0.2.2 ([43, Chapter 1, p 19] and [49, Definition 2.3, p 18]) Let (ϕi), i ∈N

be a sequence of functions in D(Ω) We say that (ϕi) converges to ϕ in D(Ω) when

i → +∞, if there exists a compact set K ⊂⊂ Ω satisfying suppϕ ⊂ K, suppϕi ⊂ Kfor all i ∈N and

Dαϕi → Dαϕ uniformly in K ∀α ∈NN,i.e.,

limi→+∞supx∈K

|Dαϕi(x) − Dαϕ(x)| = 0 ∀α ∈NN.Functions ϕ ∈ D(Ω) is called test functions

Definition 0.2.3 ([43, Chapter 1, p 19] and [49, Definition 2.4, p 19]) A tion T on Ω is a continuous linear form on D(Ω), i.e., T : D(Ω) →R is a linear mapsuch that

distribu-limi→+∞T (ϕi) = T (ϕ)for every sequence ϕi→ ϕ in D(Ω) when i → +∞ T (ϕ) will be denoted by hT, ϕi andthe space of distributions on Ω by D0(Ω)

For example, for each T ∈ L1loc(Ω), the equality

hT, ϕi :=

Z

Ω

T (x)ϕ(x)dx ∀ϕ ∈ D(Ω)defines a distribution on Ω Thus, we have L1loc(Ω) ⊂ D0(Ω) (see [49, Example, p 22]).Definition 0.2.4 ([43, Chapter 1, p 20] and [49, Definition 2.5, p 20]) For α =(α1, α2, , αN) ∈NN and T ∈ D0(Ω), the map

ϕ 7→ (−1)|α|hT, Dαϕidefines a distribution on Ω which we denoted by DαT Distribution DαT called thederivative in the distributional sense of T Moreover, we have

hDαT, ϕi = (−1)|α|hT, Dαϕi ∀ϕ ∈ D(Ω)

It can show that if T is a k-time differentiable function on Ω then the classicalderivative DαT of T coincides with the derivative in the distributional sense of T forany multiindex α ∈ NN with |α| ≤ k Therefore, notion of the derivative in thedistributional sense is an extension of notion of the derivative in the usual sense.Definition 0.2.5 ([49, Definition 2.6, p 21]) Let (Ti) be a sequence of distributions

in D0(Ω) We say that

limi→+∞Ti= T in D0(Ω),iff

limi→+∞hTi, ϕi = hT, ϕi ∀ϕ ∈ D(Ω)

The following proposition shows continuity of derivative operator in the tional sense

distribu-Proposition 0.2.6 ([43, Chapter 1, p 20] and [49, distribu-Proposition 2.5, p 22]) Theoperator Dα with α ∈NN is continuous on D0(Ω), i.e., if Ti→ T in D0(Ω) then

Z

Ωv(x)ϕ(x)dx ∀ϕ ∈ D(Ω)

From Definition 0.2.4 and 0.2.7, it is easily seen that if v = Dαu is α-order weakpartial derivative then v is α-order partial derivative in the distributional sense of u.Next, we give definition of Sobolev spaces

Definition 0.2.8 ([43, Chapter 3, p 44] and [50, Chapter 5]) Let m ∈N, p ∈ [1, +∞]

We consider the space

and a subspace of Wm,p(Ω),

W0m,p(Ω) := Closure of C0∞(Ω) in Wm,p(Ω)

We call Wm,p(Ω) and W0m,p(Ω) Sobolev spaces

Remark 0.2.9 (i) In case of p = 2, we write Hm(Ω) := Wm,2(Ω) and H0m(Ω) :=

W0m,2(Ω)

(ii) In case of m = 0, we have W0,p(Ω) = Lp(Ω) Moreover, if Ω is bounded and

p ∈ [1, +∞) then we have W00,p(Ω) = Lp(Ω) (see [43, Chapter 3, p 44])

(iii) Sobolev spaces Wm,p(Ω) and W0m,p(Ω) are Banach spaces, W0m,p(Ω) is a closedsubspace of Wm,p(Ω) Moreover, Hm(Ω), H0m(Ω) are Hilbert spaces with scalar product

The following is definition on the regularity of boundary Γ of domain Ω.

Definition 0.2.10 ([52, Definition 1.2.1.1, p 5] and [3, Subsection 2.2.2, p.26]) Let

Ω be an open set in RN Boundary Γ of Ω is called continuous (respectively Lipschitz,continuously differential, of class Ck,l, m times continuously differential) if for each

x ∈ Γ, there exist a neighborhood V ⊂ RN of x and a new orthogonal coordinate{y1, y2, , yN} such that

(i) V is a hypercube in the new coordinate {y1, y2, , yN} :

V = {(y1, y2, , yN) | −ai < yi< ai, 1 ≤ i ≤ N };

ii) there exists a continuous (respectively Lipschitz, continuously differential, of class

Ck,l, m times continuously differential) function ϕ, defined in

P ∈ E, there is y ∈ V0 such that P = (y, ϕ(y)) Let D := ϕ−1(E) ⊂ V0 Then we saythat E is measurable if D is measurable with respect to (N − 1)−dimensional Lebesguemeasure The measure of E is defined by

C∞(Γ) under norm (1) Thus, Ws,p(Γ) is a Banach space

We denote by W−m,p(Ω) ans W−r,s(Γ) the dual spaces of the spaces Wm,p0(Ω) and

Wr,s0(Γ), respectively, where 1p +p10 = 1s +s10 = 1

Definition 0.2.11 ([43, Chapter 1, p 9]) Let X, Y be the normed spaces We saythat X is imbedded in Y and write X ,→ Y, if there a linear continuous injection

j : X → Y

Moreover, if j is compact then we say that X compactly imbedded in Y and write

X ,→,→ Y

We are ready to present some imbedding results for Sobolev spaces

Theorem 0.2.12 (Sobolev and Rellich embedding theorem, [43, Theorem 5.4,

p 97 and Theorem 6.2, p 144], [48, Theorem 9.16, p 285] and [52, Chapter 1, p 27])Let Ω ⊂RN be a bounded Lipschitz domain, 1 ≤ p ≤ +∞ and 1 ≤ p ≤ +∞

• If 1 ≤ p < N then

W1,p(Ω) ,→ Lq(Ω) ∀1 ≤ q ≤ N p

N − pand this embedding is compact for 1 ≤ q < N −pN p

The following theorem was proved by Gagliardo in 1975 and it is called trace theorem.

Theorem 0.2.15 ([52, Theorem 1.5.1.3, p 38]) Suppose that Ω is a bounded opensubset of RN with Lipschitz boundary Γ Then there is a unique linear bounded map

We shall call T the trace operator and T u the trace of u on Γ

The below results represent the relation between Sobolev spaces W01,p(Ω) and theset Ker(T ) of trace operator T

Theorem 0.2.16 ([52, Theorem 1.5.1.5, p 38] and [50, Chapter 5]) Suppose that Ω

is a bounded open subset of RN with Lipschitz boundary Γ Then

u ∈ W01,p(Ω) if and only if T (u) = 0 on Γ

Theorem 0.2.17 ([48, Theorem 9.17, p 288]) Let Ω be of class C1 and u ∈ W1,p(Ω)∩C( ¯Ω) Then u = 0 on Γ if and only if u ∈ W01,p(Ω)

The smoothness of boundary Γ plays an important role in the following results.Theorem 0.2.18 ([3, Theorem 7.2, p.355 ]) Let m ∈ N with m > 0, and let Γ be ofclass Cm−1,1 Then for all mp < N the trace operator T is continuous from Wm,p(Ω)into Lr(Γ), provided by 1 ≤ r ≤ (N −1)pN −mp If mp = N then T is continuous for all

To finish this subsection, we recall some of classical inequalities concerning Lp−spacesand Sobolev spaces which can be found in [3, 43, 48, 49, 50].

Proposition 0.2.21 (H¨older inequality) Let p ∈ (1, +∞) If u ∈ Lp(Ω) and v ∈

Lp0(Ω) then uv ∈ L1(Ω) Moreover, one has

kuvkL1 (Ω)≤ kukLp (Ω)kvkLp0 (Ω).Proposition 0.2.22 (General H¨older inequality) Let p1, , pk ∈ (1, +∞) such that

Ω is a bounded domain in RN with Lipschitz boundary, p ∈ [1, +∞) There exists aconstant C(Ω) such that

kukLp (Ω) ≤ C(Ω)k∇ukLp (Ω) ∀u ∈ W01,p(Ω)

Theorem 0.2.24 (Gereralized Poincar´e inequality, [3, p 35]) Suppose that Ω is abounded domain in RN with Lipschitz boundary Γ, and Ω1 ⊂ Ω is a positive measureset, i.e., |Ω1| > 0 Then there exists a constant C(Ω1) > 0, which is independent of

0.2.2 Semilinear elliptic equations

Throughout this subsection, let us denote by p0, q0, r0 be adjoint numbers of positivenumbers p, q, r respectively, i.e., 1p +p10 = 1q + q10 = 1r + r10 = 1

Let Ω is a bounded domain in RN with the boundary Γ We denote by A thesecond-order elliptic operator of the form

where the coefficients aij ∈ L∞(Ω) satisfying aij = aji, i, j = 1, 2, , N.

The Dirichlet problems

Let us consider semilinear Dirichlet problem of the form:

where h : Ω ×R→R is a Carath´eodory function and u ∈ Lp(Ω), p > 1

Definition 0.2.26 Let u ∈ W−1,p(Ω) for p > 1 A function y is called (weak orvariational) solution of (2) if y ∈ W01,p(Ω) and

for all ϕ ∈ W1,p

0

0 (Ω)

We will need the following assumptions:

(A0.1) The matrix of coefficients (aij(x))i,j=1,2, ,N are strongly elliptic, i.e., there existconstant m, M > 0 such that

(A0.2) h : Ω ×R→Ris a Carath´eodory function and satisfies the following conditions

h(·, 0) ∈ Lp(Ω), hy(x, y) ≥ 0 a.e x ∈ Ω and ∀y ∈R.Besides, for each M > 0, there is a positive constant Ch,M such that

hy(x, y) ≤ Ch,Ma.e x ∈ Ω and for all |y| ≤ M

C1,1, aij ∈ C0,1( ¯Ω) for all i, j = 1, 2, , N, h(x, y) = a0(x)y with a0 ∈ L∞(Ω), a0 ≥

0 a.e x ∈ Ω and that assumption (A0.1) is fulfilled Then, for u ∈ Lp(Ω), equation(2) has a unique solution yu ∈ W01,p(Ω) ∩ W2,p(Ω) Moreover, there exists a constant

C > 0 independent of u such that

kyukW2,p (Ω)≤ CkukLp (Ω)

The following theorem gives the existence and the uniqueness of weak solution toequation (2) when boundary Γ is Lipschitz and N ∈ {2, 3}.

Theorem 0.2.28 ([13, Theorem 2.1] and [15, Theorem 2.1]) Suppose that boundary

Γ is Lipschitz, N ∈ {2, 3}, and that assumption (A0.1) is satisfied and assumption(A0.2) is valid for p = 2 Then for each u ∈ L2(Ω), equation (2) has a unique solution

yu∈ H1

0(Ω) ∩ C( ¯Ω)

Moreover, if uk * u in L2(Ω) then yuk → yu in H01(Ω) ∩ C( ¯Ω) when k → +∞.For case N ≥ 2 and Γ is of class C1, E Casas [14] showed the existence and theregularity of solution to Dirichlet problems

Theorem 0.2.29 ([14, Theorem 2.4]) Suppose that Ω is of class C1, aij ∈ C1( ¯Ω) forall i, j = 1, 2, , N, and that assumptions (A0.1), (A0.2), and (A0.3) are fulfilled Then,for each u ∈ W−1,r(Ω), equation (2) has a unique solution yu ∈ W01,r(Ω) Moreover,for any bounded subset U ∈ W−1,r(Ω), there exists a constant CU such that

kyukW1,r

0 (Ω) ≤ CUfor all u ∈ U Besides, if un → u in W−1,r(Ω) then yun → yu trong W01,r(Ω) when

n → +∞

Corollary 0.2.30 Suppose that p > N2, Ω of class C1,1, aij ∈ C0,1( ¯Ω) for all i, j =

1, 2, , N, h(x, 0) = 0 and that assumptions (A0.1), (A0.2) are fulfilled Then, for

u ∈ Lp(Ω), equation (2) has a unique solution yu ∈ W01,p(Ω) ∩ W2,p(Ω) Moreover,there exists a constant C > 0 independent of u such that

kyukW2,p (Ω)≤ CkukLp (Ω)

The Robin problems

We consider Robin problem of the form:

where h : Ω × R → R, k : Γ × R → R are Carath´eodory functions, and (u, v) ∈

Lp(Ω) × Lq(Γ), p, q > 1, and ∂ν denote the conormal-derivative associated with A, i.e.,

Definition 0.2.31 Let 1 < r < ∞ For each given u ∈ (W1,r (Ω))∗, v ∈ W−r ,r(Γ), afunction y ∈ W1,r(Ω) is said to be a (weak or variational) solution of (3) if

Z

Ωu(x)v0(x)dx +

Z

Γv(x)T v0(x)dσfor all v0∈ W1,r0(Ω)

We will need the following assumptions:

(A0.4) a0 : Ω → R and b0 : Γ → R are bounded, measurable and almost everywherenonnegative Moreover, a06≡ 0

(A0.5) h : Ω ×R→R and k : Γ are Carath´eodory functions and of class C1 w.r.t thesecond variable and satisfying the following conditions

h(·, 0) = 0, hy(x, y) ≥ 0 a.e x ∈ Ω and for all y ∈R.k(·, 0) = 0, ky(x0, y) ≥ 0 a.e x0 ∈ Γ and for all y ∈R.For each M > 0, there is a positive constant Ch,M such that

hy(x, y) ≤ Ch,M and ... concepts and facts on set-valued maps, tangentand normal cones, Sobolev spaces, imbedding theorems and results on the existenceand uniqueness of weak solution to semilinear elliptic equations. .. no-gap optimality conditions for problem (DP ).Namely, we shall develop the results in [27], [7] and [18] by giving a correct critical coneunder which both the second-order necessary conditions and. .. necessary optimality conditions for optimal control

problem

Recall that a couple (¯y, ¯u) satisfying constraints (1.2)–(1.3), is said to be admissiblefor problem (DP ) By Corollary