MINISTRY OF EDUCATION AND TRAINING
HANOI 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 TRAINING
HANOI 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 scienti c results are new and righteous. Before I published
these results, there had been no such results in any scienti c document. I have
responsibili-ties for my research results in the dissertation.
rd

Hanoi, April 3 , 2019
Author

On behalf of Supervisors

Dr. Nguyen Thi Toan

Nguyen Hai Son

i


ACKNOWLEDGEMENTS
This dissertation has been carried out at the Department of Fundamental
Mathe-matics, School of Applied Mathematics and Informatics, Hanoi University of
Science and Technology. It has been completed under the supervision of Dr.
Nguyen Thi Toan and Dr. Bui Trong Kien.
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 e ective 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
and studied at Department of Applied Mathematics, Sun Yat-Sen University,
Kaohsiung, Taiwan (from April, 2015 to June, 2015 and from July, 2016 to
September, 2016). I would like to express my gratitude to Prof. Nguyen Dong Yen
for his encouragement and many valuable comments.
I would also like to especially thank my friend, Dr. Vu Huu Nhu for kind help and
encouragement.
I would like to thank the Steering Committee of Hanoi University of Science and
Technology (HUST), and School of Applied Mathematics and Informatics (SAMI)
for their 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 my wife for
her love and encouragement. This dissertation is a meaningful gift for them.
rd

Hanoi, April 3 , 2019
Nguyen Hai Son

ii


CONTENTS
......................
............................

COMMITTAL IN THE DISSERTATION
ACKNOWLEDGEMENTS .
CONTENTS

....................................


TABLE OF NOTATIONS
INTRODUCTION

..............................

.................................

Chapter 0. PRELIMINARIES AND AUXILIARY RESULTS
0.1 Variational analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i
ii
iii
1
3
8
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 su cient 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 su cient 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

iii


3.2.1 Some properties of the admissible set . . . . . . . . . . . . . . .
3.2.2 First-order necessary optimality conditions . . . . . . . . . . .

94
98

3.3 Proof of the main result . . . . . . . . . . . . . . . . . . . . . . . . . .

100

3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


104

3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

109

............................

110

.............................

111

GENERAL CONCLUSIONS
LIST OF PUBLICATIONS

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iv

112


N := f1; 2; : : :g
R

set of real numbers
absolute value of x 2 R


jxj

R

TABLE OF NOTATIONS
set of positive natural numbers

n

n-dimensional Euclidean vector space

;

empty set

x2A

x is in A

x 2= A
A B(B A)
A*B
A\B

x is not in A

A[B

union of the sets A and B


AnB
A B

set di erence of A and B

[x1; x2]
kxk

the closed line segment between x1 and x2
norm of a vector x

kxkX
X

norm of vector x in the space X
topological dual of a normed space X

X
hx ; xi

topological bi-dual of a normed space X

hx; yi
B(x; )

canonical inner product

B(x; )
BX

B

closed ball with centered at x and radius
open unit ball in a normed space X

dist(x; )

distance from x to
sequence of vectors xk

A is a subset of B
A is not a subset of B
intersection of the sets A and B

Descartes product of the sets A and B

canonical pairing
open ball with centered at x and radius

closed unit ball in a normed space X

X

fxkg
xk ! x
xk * x
8x

xk converges strongly to x (in norm topology)


9x
A := B
f:X!Y

there exists x

xk converges weakly to x
for all x
A is de ned by B
function from X to Y

0

f (x), rf(x)
00

2

f (x), r f(x)

Frechet derivative of f at x
Frechet second-order derivative of f at x

1


Frechet derivative of L in x

Lx, rxL
2


Lxy, rxy L
' : X ! IR
dom'
epi'

Frechet second-order derivative of L in xand y

supp'

the support of '

F:X Y

multifunction from X to Y

domF
rgeF

domain of F
range of F

gphF

graph of F

kerF

kernel of F


T (K; x)

Bouligand tangent cone of the set K at x

extended-real-valued function
e ective domain of '
epigraph of '

[

adjoint tangent cone of the set K at x

2

second-order Bouligand tangent set of the set

2[

T (K; x; d)

K at x in direction d
second-order adjoint tangent set of the set K

N(K; x)

at x in direction d
normal cone of the set K at x

@


boundary of the domain

T (K; x)
T (K; x; d)

closure of the set
0

0 and

p

0 is compact.

L ( )

the space of Lebesgue measurable functions f

L ( )

and jf(x)jpdx < +1
space of bounded functions almost every

C( )

the space of continuous functions on

1

the


M( )

>
<
>

8
>

W

m;p

( );

( );

m

H ( ); H ( )
W

W

>

( ); W0

s;r


:

the space of nite regular Borel measures

m;p

m

R

Sobolev spaces
0

0
m;p

0

( )(

p 1 + p 0 1 = 1) the dual space of

W m;p ( )

X ,! Y

X is continuous embedded in Y

X ,!,! Y

i.e.
a.e.

X is compact embedded in Y

s.t.

subject to

p. 5

page 5

w.r.t

with respect to

2

The proof is complete

id est (that is)
almost every

2


INTRODUCTION

1. Motivation

Optimal control theory has many applications in economics, mechanics and
other elds of science. It has been systematically studied and strongly developed
since the late 1950s, when two basic principles were made. One was the
Pontryagin Maximum Principle which provides necessary conditions to nd optimal
control functions. The other was the Bellman Dynamic Programming Principle, a
procedure that reduces the search for optimal control functions to nding the
solutions of partial di erential equations (the Hamilton-Jacobi equations). Up to
now, optimal control theory has developed in many various research directions
such as non-smooth optimal control, discrete optimal control, optimal control
governed by ordinary di erential equations (ODEs), optimal control governed by
partial di erential equations (PDEs),...(see [1, 2, 3]).
In the last decades, qualitative studies for optimal control problems governed by
ODEs and PDEs have obtained many important results. One of them is to give optimality 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. Meyer and 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 di erentiable
0

functional and U is a Banach space, then F (u) = 0. This a rst-order necessary
optimality condition. However, it is not a su cient condition in case of F is not
convex. Therefore, we have to invoke other su cient conditions and should study
the second derivative (see [17]).
Better understanding of second-order optimality conditions for optimal control problems governed by semilinear elliptic equations is an ongoing topic of research for
several researchers. This topic is great value in theory and in applications. Secondorder su - cient optimality conditions play an important role in the numerical analysis
of nonlinear optimal 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 nding out stationary points but also help us in constructing su cient
optimality conditions. Let us brie y review some results on this topic.

3


n

For distributed control problems, i.e., the control only acts in the domain in R ,
E. Casas, T. Bayen et al. [11, 13, 16, 27] derived second-order necessary and su
cient optimality conditions for problem with pure control constraint, i.e.,
a(x) u(x) b(x) a.e. x 2 ;

(1)

and the appearance of state constraints. More precisely, in [11] the authors gave secondorder necessary and su cient conditions for Neumann problems with constraint

(1) and nitely many equalities and inequalities constraints of state variable y while the
second-order su cient optimality conditions are established for Dirichlet problems with
constraint (1) and a pure state constraint in [13]. T. Bayen et al. [27] derived secondorder necessary and su cient optimality conditions for Dirichlet problems in the sense
of strong solution. In particular, E. Casas [16] established second-order su cient
optimality conditions for Dirichlet control problems and Neumann control problems
with only constraint (1) when the objective function does not contain control variable u.
In [18], C. Meyer and F. Tr•oltzsch derived second-order su cient optimality conditions
for Robin control problems with mixed constraint of the form a(x) y(x) + u(x)

b(x) a.e. x 2

and nitely many equalities and inequalities constraints.


For boundary control problems, i.e., the control u only acts on the boundary , E.
Casas and F. Tr•oltzsch [10, 12] derived second-order necessary optimality
conditions while the second-order su cient 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 2

:

A. R•osch and F. Tr•oltzsch [23] gave the second-order su cient optimality
conditions for the problem with the mixed pointwise constraints which has
unilateral linear form c(x) u(x) + (x)y(x) for a.e. x 2 .
1

1

We emphasize that in above papers, a; b 2 L ( ) or a; b 2 L ( ). Therefore, the
1

1

control u belongs to L ( ) or L ( ). This implies that corresponding Lagrange
multipliers are measures rather than functions (see [19]). In order to avoid this disadvantage, B. T. Kien et al. [19, 20, 21] recently established second-order
necessary optimality 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 2

p

with a; b 2 L ( ), 1 < p < 1 and pure state constraints. This motivates us to develop
and study the following problems.
(OP 1) : Establish second-order necessary optimality conditions for Robin
boundary control problems with mixed state-control constraints of the form
a(x)

g(x; y(x)) + u(x)
4

b(x) a.e. x 2

;


p

where a; b 2 L ( ), 1 < p < 1.
(OP 2) : Give second-order su cient optimality conditions for optimal control
prob-lems with mixed state-control constraints when the objective function does
not depend on control variables.

Solving problems (OP 1) and (OP 2) is the rst goal of the dissertation.
After second-order necessary and su cient optimality conditions are established,
they should be compared to each other. According to J. F. Bonnans [4], if the change
between necessary and su cient second-order optimality conditions is only between
strict and non-strict inequalities, then we say that the no-gap optimality conditions are
obtained. Deriving second-order optimality conditions without a gap between secondorder necessary optimality conditions and su cient optimality conditions, is a di cult
problem which requires to nd a common critical cone under which both second-order
necessary optimality conditions and su cient optimality conditions are satis ed. In [7],
J. F. Bonnans derived second-order necessary and su cient optimality conditions with
no-gap for an optimal control problem with pure control constraint and the objective
function 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
optimiza-tion and numerical method of nding solutions (see [25, 29, 30, 31, 32, 33,
34, 35, 36, 37, 38, 39, 40, 41]). An optimal control problem is called stable if the
error of the output data is small in some sense for a small change in the input
data. The study of solution stability is to investigate continuity properties of solution
maps in parameters such as lower semicontinuity, upper semicontinuity, H•older
continuity and Lipschitz continuity.
Let us consider the following parametric optimal problem:
P(; )

8

:(y; u) 2 ( );

(2)

where y 2 Y; u 2 U are state and control variables, respectively; 2 ; 2 are
parameters, F : Y U ! R is an objective function on Banach space Y U and ( ) is an
admissible set of the problem.
It is well-known that if the objective function F ( ; ; ) is strongly convex, and the
admissible set ( ) 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,
5


the solution map is Lipschitz continuous w.r.t. parameters. By using implicit function
theorems, K. Malanowski [35]-[40] proved that the solution map of problem (2) is also
a Lipschitz continuous function in parameters if weak second-order optimality
conditions and standard constraint quali cations are satis ed at the reference point.
Notice that the obtained results in [37]-[40] are for problems with pure state
constraints, while the one in [35] is for problems with pure control constraints.
When the conditions mentioned above are invalid, the solution map may not be
singleton (see [32, 33]). In this situation, we have to use tools of set-valued analysis
and variational 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
control problem for the case where the objective function is convex in both variables
and the admissible sets are also convex. Recently, the upper semicontinuity of the
solution map has been given by B. T. Kien et al. [34] and V. H. Nhu [42] for problems,
where the objective functions may not be convex in the both variables and the
admissible sets are not convex. Notice that in [34] the authors considered the problem
governed by ordinary di erential equations meanwhile in [42] the author investigated
the problem governed by semilinear elliptic equation with distributed control. From the

above, one may ask to study the following problem:

(OP 4) : Establish su cient conditions under which the solution map of
parametric boundary 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
con-ditions and stability of solution to optimal control problems governed by
semilinear elliptic equations with mixed pointwise constraints. Namely, the main
content of the dissertation is to concentrate on
(i) establishing second-order necessary optimality conditions for boundary
p

control problems with the control variables belong to L ( ), 1 < p < 1;
(ii) deriving second-order su cient optimality conditions for distributed control
prob-lems and boundary control problems when objective functions are
quadratic forms in the control variables, and showing that no-gap optimality
condition holds in this case;
(iii) deriving second-order su cient optimality conditions for distributed control
prob-lems and boundary control problems when objective functions are
independent of the control variables, and showing that in general theory of
no-gap conditions does not hold;
(iv) giving su cient conditions for a parametric boundary control problem under which
6


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,

Sobolev spaces and partial di erential equations.
Chapter 1 presents results on the no-gap second-order optimality conditions for
distributed control problems.
Chapter 2 provides results on the no-gap second-order optimality conditions for
boundary 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
solution map 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 the List
of publications which were published in the journals Set-Valued and Variational
Analysis and SIAM Journal on Optimization, respectively. The results of Chapter 3
were 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
of Science and Technology in November 2016.
The 15th Conference on Optimization and Scienti c Computation, Ba Vi in
April 2017.
The 7th International Conference on High Performance Scienti c Computing in
March 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, Vietnam
Academy of Science and Technology.

7

Chapter 0
PRELIMINARIES AND AUXILIARY RESULTS

In this chapter, we review some background on Variational Analysis, Sobolev spaces,
and facts of partial di erential equations relating to solutions of linear elliptic equations and
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
0.1.1

Variational analysis
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 2 X a subset F (x) Y . F (x) is
called 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 call
the sets

2

dom(F ) := x X F (x) =
;
X
Y
gph(F ) := (x; y)
j y 2 F (x)
2

rge( ) :=
F

y

2

j
Y

j

y

6;
2

( )
Fx

for some x

[
2

X :=

F (x)

x2X

the graph, the domain and the range of F , respectively.
The inverse F

1

:Y

X of F is the set-valued map, de ned by
F

1

(y) :=

f x 2 X j y 2 F (x)g for all y 2 Y:

The set-valued map F is called proper if dom(F ) 6= ;.
De nition 0.1.1. ([46, p. 34]) Let F : X Y be a set-valued map between topolog-ical
spaces X and Y .
(i) If gph(F ) is a closed subset of the topological space X Y then F is called
closed map (or graph-closed map).
(ii) If X; Y are linear topological spaces and gph(F ) is a convex subset of the
topo-logical space X Y then F is called convex set-valued map.
(iii) If F (x) is a closed subset of Y for all x 2 X then F is called closed-valued map.
(iv) If F (x) is a compact subset of Y for all x 2 X then F is called compact-valued
map.
8

The concepts of semicontinuous set-valued maps had been introduced in 1932 by G.

Bouligand and K. Kuratowski (see [44]).
De nition 0.1.2. ([45, De nition 1, p. 108] and [44, De nition 1.4.1, p.38]) Let
F : X Y be a set-valued map between topological spaces and x 0 2 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 2 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 2 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 2 dom(F ).
Notice that in case of single-valued map F : X ! Y , the above concepts are
coincident.
When X; Y are metric spaces, set-valued map F : X Y is lower semicontinuous
at x 2 dom(F ) if and only if for all y 2 F (x) and sequence fx ng 2 dom(F ), xn ! x,
there exists a sequence fyng Y , yn 2 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 x 0 2 X and > 0, we denote
by B(x0; ) the open ball fx 2 X j kx

x0k < g, and by B(x0; ) the corresponding

closed 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 2 X to D is de ned by
dist(x; D) = inf kx
uk:
u2D

De nition 0.1.3. ([44, De nition 4.1.1, p. 121]) Let D X be a subset of a normed
space X and a point x 2 D: The set
T (D; x) := v

2

dist(x + tv; D)
X j limt inf
0+

!

is called Bouligand (contingent) cones of D at x.

9

t

=0 :


From De nition 0.1.3, it follows that T (D; x) is a closed cone and T (D; x)
cone(D x); where cone(A) := f a j 0; a 2 Ag is the cone generated by the set A:
Moreover, the following property characterizes the Bouligand cone:
+

T (D; x) = fv 2 X j 9tn ! 0 ; 9vn ! v s.t. x + tnvn 2 D for all n 2 Ng:
De nition 0.1.4. ([44, De nition 4.1.5, p. 126]) Let D X be a subset of normed space
[

X and x 2 D: The adjoint tangent cone or the intermediate cone T (K; x) of D at x is
de ned by
2 j t!0+
t
dist(x + tv; D) = 0 :
[
v
X
lim
T (D; x) :=
The Clarke tangent cone TC (D; x) of D at x is de ned by
TC (D; x) := 8 v
>
<

D

lim+ d(x0 + tv; D)

X

2

>

x 0 !x

9;
>

!

j

t 0

=

t

>

D

:

0

=0

0

;

0

where x ! x means that x 2 D and x ! x:
From De nition 0.1.4, we have the following characters of the adjoint cones and the

Clarke tangent cones (see [44, p. 128]):
[

+

T (D; x) = fv 2 X j 8tn ! 0 ; 9vn ! v s.t. x + tnvn 2 D 8n 2 Ng;
and
D

! v s.t. xn + tnvn 2 D 8n 2 Ng:

+

TC (D; x) = fv 2 X j 8tn ! 0 ; 8xn ! x; 9vn
It is clear that
TC (D; x)

[

T (D; x)

T (D; x)

cone(D

x):

[

Example 0.1.5. ( TC (D; x) 6= T (D; x) = T (D; x) 6= cone(D x))
2

2

Putting D = f(x1; x2) j x2 = x 1g R and taking x = (0; 0), we have
TC (D; x) = f(0; 0)g;
[

T (D; x) = T (D; x) = f(x1; 0) j x1 2 Rg;
cone(D

x) = f(x1; x2) j x2

0g:

[

Example 0.1.6. ( TC (D; x) = T (D; x) 6= T (D; x) = cone(D
Putting D = f

1
n

x))

j n = 1; 2; :::g R and taking x = 0 2 D, we have
[

TC (D; x) = T (D; x) = f0g;
T (D; x) = cone( D x) = R+

10


[

Example 0.1.7. ( TC (D; x) 6= T (D; x) = T (D; x) = cone(D

x))

2

Putting D = f(x1; x2) j x2 = 0g [ f(x1; x2) j x1 = 0g R and taking x = (0; 0), we have
TC (D; x) = f(0; 0)g;
[

T (D; x) = T (D; x) = cone(D

x) = D:

Let x1; x2 2 X: The segment [x1; x2] connect x1 and x2 is de ned by
[x1; x2] := fx 2 X j x =

x1 + (1

)x2; 0

1g:

A subset K of X is said to be convex if [x 1; x2] K for all x1; x2 2 K. (see [1,
section 0.3.1, p. 45]). According to [44, Chapter 4], the Clarke cone T C (D; x) is a
[

convex and closed cone while the adjoint cone T (D; x) is a closed cone.
Moreover, if 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 rst-order optimality
conditions for optimal control problems with constraints. However, in order to
obtain second-order optimality conditions for optimal control problems, we need to
use second-order tangent sets.
De nition 0.1.8. ([44, De nition 1.1.1, p. 17]) Let X be a normed space and (D t)t2T X
be a sequence of sets depend on parameters t 2 T; where T is a metric space.

Suppose that t0 2 T: The set
Limsup D t := f

x

2X

inf dist(x; D

j limtt0

t

)=0

g

!

t!t0

is called Painlev -Kuratowski upper limit of (D t) as t ! t0:
The set
Liminf D t := f x 2X j tlimt0 dist( x; D t) = 0g
t

t0

!

!

is called Painlev -Kuratowski lower limit of (Dt) as t ! t0:
De nition 0.1.9. ( [44, De nition 4.7.1 and 4.7.2, p. 171]). Let D be a subset in the
normed space X and x 2 D; v 2 X:
The set
D x tv
2
T (D; x; v) := Limsup
2
t!0

+

t

is called Bouligand second-order tangent set of D at x in direction v:
The set
D x tv
2[
T (D; x; v) := Liminf
2
t !

0+

t

is called second-order adjoint tangent set D at x in direction v:
11

2

2[

Obviously, T (D; x; v) and T (D; x; v) are closed and
2

2[

[

T (D; x; 0) = T (D; x); T (D; x; 0) = T (D; x):
Moreover, we have
2

+

2

2[

+

2

T (D; x; v) = fwj9tn ! 0 ; 9wn ! w; x + tnv + t nwn 2 Dg;
T (D; x; v) = fwj8tn ! 0 ; 9wn ! w; x + tnv + t nwn 2 Dg:
2

2[

Notice that T (D; x; v) and T (D; x; v) are nonempty only if v 2 T (D; x) and v 2 T
[
2[
(D; x), respectively. Moreover, when D is convex, T (D; x; v) is convex but T (D; x;
v) may not be convex (see [47, Subsection 3.2.1]).
2

2[

The following example shows that in general (T (D; x; v) is di erent from T (D;
x; v)) (see [47, Example 3.31]).
2

2[

Example 0.1.10. (T (D; x; v) 6= T (D; x; v))
Let us rst construct a convex piecewise linear function x 2 = '(x1), x1 2 R, oscil-lating
2
2
between two parabolas x2 = x 1 and x2 = 2x 1 in the following way: '(x 1) = '( x1); '(0) =
2
0 and the function '(x1) is linear on every interval [x 1;k+1; x1;k], '(x1;k) = x 1;k and its
2
graph on [x1;k+1; x1;k] is tangent to the curve x 2 = 2x 1 for some monotonically
decreasing to zero sequence fx1;kg. It is evident how such a function can be
constructed. Indeed, for a given point x 1;k > 0 consider the straight line passing
2

2
through the point (x1;k; x 1;k) and is tangent to the curve x2 = 2x 1. It intersects the
2
curve x2 = x 1 at a point x1;k+1. We can iterate this process and obtain a sequence

fx1;kg. It is easily seen that x1;k > x1;k+1 > 0 and x1;k ! 0 as k ! 1.
2

Taking K = f(x1; x2) 2 R j x2

2

T (D; x; v) = f(x1; x2) j x2

'(x1)g and x = (0; 0); v = (1; 0), we have

2[

2g and T (D; x; v) = f(x1; x2) j x2

4g:

The following result allows us to compute tangent cones of a convex and closed
p

p

subset K in L ( ) with 1 p < +1 (see De nition L ( ) in next section).
p

Theorem 0.1.11. ([44, Theorem 8.5.1, p. 324]). Let K be a subset of L ( ) such that
M(x) := fu(x) j u 2 Kg is measurable and closed in R for a.e. x 2 : Then for all
u0 2 K; one has

o

n
p

[

v 2 L ( ) j v(x) 2 T (M(x); u0(x)) a.e. x 2 T
[

(K; u0) T (K; u0)
p

fv 2 L ( ) j v(x) 2 T (M(x); u0(x)) a.e. x 2 g :
p

Corollary 0.1.12. ([27, Lemma 4.11] Let 1 p < +1, and K := fu 2 L ( ) j a(x) u(x) b(x)
p

a.e. x 2 g; with a; b 2 L ( ) and u0 2 K: Then
[

T (K; u0) = T (K; u0)
p

[

n o = v 2 L ( ) j v(x) 2 T ([a(x); b(x)]; u0(x)) a.e. x 2 :
12


In the sequel, we shall use concept normal cone which is dual concept of Clarke
tangent 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 de ned by

kfkX = supff(x) j x 2 X; kxk 1g:
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.
De nition 0.1.13. ([44, De nition 4.4.2, p. 157]). Let X be a Banach space, a subset
D X and a point x 2 D: We shall say that the polar cone
N(D; x) := TC (D; x)

= fp 2 X j hp; vi

0 8v 2 TC (D; x)g

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 Convex
Analysis, i.e.,
N(D; x) = fx 2 X j hx ; d
xi
0 8d 2 Dg:

0.2
0.2.1

Sobolev spaces and elliptic equations
Sobolev spaces

First, we recall some relative concepts and properties which are introduced in
many books on Sobolev spaces, elliptic equations and partial di erential equations.
N

For any multiindex := ( 1; 2; :::; N ) 2 N , let us denote by x := x1 1 x2 2 xNN ; with x = (x1; x2; :::;
PN
N
xN ) 2 R ; a monomial of order j j =
i; and denote by
i=1

D := D1 1 D2 2

DNN
@

a di erential operator of order j j, where D j = @x j for 1 j N. We adopt the
(0;:::;0)
N
convention that D
u = u for all function u de ned on R :
N

Let be an open subset in R : For each function u :
fx 2
: u(x) =6 0g the support of u:

! R; we call suppu :=

For each non-negative integer number m; we have the following classical
function spaces:
m

1

C ( ) := fu : ! R j D u is continuous on ; 8j j mg; C ( ) := \
1

m
m=0C ( );
0

C0( ) := fu 2 C ( ) j suppu is a compact subset in g; 1

1
0

Notice that C ( )

C( ):
13


De nition 0.2.1. ([43, Chapter 2] and [49, De nition 2.1, p. 14]) Let be an open set
N

in R ; N 1; and p 1.

kukL1( ) := inf fC j ju(x)j C a:e: x 2 g :
For p 2 [1; +1]; let us denote by p

0

the adjoint number of p; i.e.,
8

p

0

p :=

if p 2 (1; 1);

>

p1

+1 if p = 1;
:1 if p = +1:

p

p

The spaces L ( ); 1 p 1; are Banach spaces. Moreover, L ( ) with 1 < p < +1 are re
1

2
exive and separable, while L ( ) is separable. Besides, L ( ) is a Hilbert space with
the scalar product
Z

(u; v)L2( ) :=

Z

p

L ( ) :=

u:

! R j u is Lebesgue measurable and

u:

! R j u is Lebesgue measurable and

n

1

L ( ) :=

p

ju(x)j dx < +1 ;

9C > 0 such that ju(x)j

C a:e: x 2

o

;

with respectively norms

Z

1=p

p

kukLp( ) :=

ju(x)j dx

;

u(x)v(x)dx

2

8u; v 2 L ( ):

p

It is noted that C0( ) is dense in L ( ) for 1 p < +1: The topological dual spaces of
p
p
p
p
L spaces for (1 p < +1) are L space too, namely, L ( ) = L 0( ); 1 < p < +1
1
1
and L ( ) = L ( ) (see [43, Chapter 2] and [48, Section 4.3]).
0

0

In the sequel, we will write if is included in and compact. We denoted by
1
L loc( ) the space of local integrable functions on ; i.e.,
1

Z

n

L loc( ) := u :

! R j u is Lebesgue measurable and

ju(x)jdx < +1
0

for all measurable subset

0

o

:


N

p

1

Then, for any open set in R and for all p 2 [1; +1]; we have L ( ) L loc( ) (see [43,
Chapter 2, p. 26]).
1

Recall that C0 ( ) the space of functions in nitely di erentiable in with compact
1

support in : We introduce a notion of convergence in the space C 0 ( ) which can
1

1

be de ned by a topology on C0 ( ). Then C0 ( ) is denoted by D( ).
De nition 0.2.2. ([43, Chapter 1, p. 19] and [49, De nition 2.3, p. 18]) Let (' i); i 2 N
be a sequence of functions in D( ): We say that (' i) converges to ' in D( ) when

14


i ! +1; if there exists a compact set K satisfying supp' K; supp' i K for all i 2 N and
N

8 2N ;

D 'i ! D ' uniformly in K
i.e.,
lim sup j D '
i !+ 1

i(

x2K

N

D ' x)j = 0
(

x)

8 2

N :

Functions ' 2 D( ) is called test functions.
De nition 0.2.3. ([43, Chapter 1, p. 19] and [49, De nition 2.4, p. 19]). A distribu-tion

T on is a continuous linear form on D( ), i.e., T : D( ) ! R is a linear map such that
lim T ('i) = T (')
i!+1

for every sequence 'i ! ' in D( ) when i ! +1: T (') will be denoted by hT; 'i and the
0

space of distributions on by D ( ):
1

For example, for each T 2 L loc( ); the equality

Z

hT; 'i :=

T (x)'(x)dx

8' 2 D( )

1

0

de nes a distribution on : Thus, we have L loc( ) D ( ) (see [49, Example, p. 22]).
De nition 0.2.4. ([43, Chapter 1, p. 20] and [49, De nition 2.5, p. 20]). For = ( 1;
2; :::; N ) 2

N

0

N and T 2 D ( ); the map
jj

' 7!( 1) hT; D 'i
de nes a distribution on

which we denoted by D T: Distribution D T called the

derivative in the distributional sense of T: Moreover, we have
jj

hD T; 'i = ( 1) hT; D 'i

8' 2 D( ):

It can show that if T is a k-time di erentiable function on then the classical
derivative D T of T coincides with the derivative in the distributional sense of T for
N

any multiindex 2 N with j j k. Therefore, notion of the derivative in the
distributional sense is an extension of notion of the derivative in the usual sense.
De nition 0.2.5. ([49, De nition 2.6, p. 21]) Let (T i) be a sequence of distributions in
0

D ( ). We say that

lim T

i

=T

i!+1

i
i +

lim

i=h
;
'
T; '
T

hi

in D

0

( );

i 8 2 D( )
'

:

!1

The following proposition shows continuity of derivative operator in the distributional sense.
15


Proposition 0.2.6. ([43, Chapter 1, p. 20] and [49, Proposition 2.5, p. 22]). The
N

0

0

operator D with 2 N is continuous on D ( ); i.e., if Ti ! T in D ( ) then
D Ti ! D T

0

in

D ( ):

We now give de nition of weak partial derivative for locally integrable functions.
1

De nition 0.2.7. ([43, Chapter 1, p. 21] and [50, Chapter 5]) Let u; v 2 L loc( ) and
be a multiindex. We say that v is -order weak partial derivative (or -order general
partial derivative) of u, written by v = D u; if

Z

u(x)D '(x)dx = ( 1)

Z

jj

v(x)'(x)dx

8' 2 D( ):

From De nition 0.2.4 and 0.2.7, it is easily seen that if v = D u is -order weak
partial derivative then v is -order partial derivative in the distributional sense of u:
Next, we give de nition of Sobolev spaces.
De nition 0.2.8. ([43, Chapter 3, p. 44] and [50, Chapter 5]) Let m 2 N; p 2 [1; +1]:
We consider the space
W

m;p

p

p

( ) := fu 2 L ( ) j D u 2 L ( ) with 0
jj
m
and D u is -order weak partial derivative of ug

with corresponding norm

8n

u W m;p( ) :=

0 jjm

>

kk

>

<

m;p

0 jjm

kD ukL

P

k

p

p(

o

)

1=p

if 1

u

max D

if p = +1

L1( )

k

p < +1;

>

:

and a subspace of W

>

( ) ;
m;p

W0

We call W

m;p

m;p

( ) and W0

1

( ) := Closure of C0 ( ) in W

m;p

( ):

( ) Sobolev spaces.
m

Remark 0.2.9. (i) In case of p = 2; we write H ( ) := W

m;2

m

( ) and H0 ( ) :=

m;2

W0

( ):
0;p

p

(ii) In case of m = 0; we have W ( ) = L ( ): Moreover, if is bounded and p 2 [1;
0;p
p
+1) then we have W0 ( ) = L ( ) (see [43, Chapter 3, p. 44]).
(iii) Sobolev spaces W
subspace of W

m;p

m;p

( ) and W0

m;p
m

( ) are Banach spaces, W0
m

m;p

( ) is a closed

( ): Moreover, H ( ); H ( ) are Hilbert spaces with scalar product

X

(u; v) H m (

)

=

0

jj

0

(D u; D v) 2
m

m;p

L

( )

:

m;p
(iv) Sobolev spaces W
( ) W0 ( ) are re exive and uniformly convex (and so,
and strictly convex) if p 2 (1; separable if p 2 [1; +1) (see Theorems 1.21 and 3.5
+1); is in [43]).

16


The following is de nition on the regularity of boundary

of domain .

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

be an open set in R : Boundary of is called continuous (respectively Lipschitz,
k;l
continuously di erential, of class C , m times continuously di erential) if for each
N

x 2 ; there exist a neighborhood V R of x and a new orthogonal coordinate fy1;
y2; :::; yN g such that
(i) V is a hypercube in the new coordinate fy1; y2; :::; yN g :
V = f(y1; y2; :::; yN ) j ai < yi < ai; 1

i

Ng;

ii) there exists a continuous (respectively Lipschitz, continuously di erential, of
k;l

class C , m times continuously di erential) function '; de ned in
V

0

:= f(y1; y2; :::; yN 1) j ai < yi < ai; 1

i

N

1g

and such that
0

aN for every y0 := (y1; y2; :::; yN 1) 2 V 0;
2

j'(y )j

0

0

\ V = fy = (y ; yN ) 2 V j yN < '(y )g; \
0

0

V = fy = (y ; yN ) 2 V j yN = '(y )g:
In other words, in a neighborhood of x 2 ; is below the graph of ' and the
boundary is the graph of ':

Using new locally coordinate systems, we can derive Lebesgue measure on in
natural way (see [3, Subsection 2.2.2]). Assume that the set E can be completely
represented by a locally coordinate system S = fy1; y2; ; yN g, i.e., for every point
0

1

0

P 2 E, there is y 2 V such that P = (y; '(y)). Let D := ' (E) V . Then we say that
E is measurable if D is measurable with respect to (N 1) dimensional Lebesgue
measure. The measure of E is de ned by

Z
jEj :=

p

2

1 + jr'(y1; y2;

D

; yN 1)j dy1dy2:::dyN 1:

We denoted by d is measure on .
Let us introduce Sobolev spaces on the set , (see [2, Chapter 2, p. 75] and [52,
1

De nition 1.3.2.1 and De nition 1.3.3.2]). For s 2 (0; 1); p 1 and u 2 C ( ); we
consider the norm
Z
Z j
0
p
0 j
1=p
kukW s;p( ) :=
ju(x)j d (x) +
d (x)d (x )
j x x0jN 1+ sp
u(x) u(x )

s;p

; (1)

where d is measure on : Let us denoted by W ( ) the closed space generated by
1
s;p
C ( ) under norm (1). Thus, W ( ) is a Banach space.
m;p
r;s
m;p0
We denote by W
( ) ans W
( ) the dual spaces of the spaces W
( ) and
W

r;s0

1

1

1

1

( ), respectively, where + = + = 1:
p

p0

s

s0

17


De nition 0.2.11. ([43, Chapter 1, p. 9]) Let X; Y be the normed spaces. We say
that X is imbedded in Y and write X ,! Y; if there a linear continuous injection j : X !
Y:

Np
N p

, is never compact

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.
N

27]) Let R be a bounded Lipschitz domain, 1 p +1 and 1 p +1:
If 1 p < N then
W

1;p

and this embedding is compact for 1 q <

p If p = N then W

1;p

Np

:N

q

( ) ,!,! L ( ) 8q 2 [1; +1):

Remark 0.2.13. (i) The injection W

is bounded and smooth (see [48]).
1;p

Np
N p

q

( ) ,! L ( ) 81 q

(ii) If W ( ) is replaced by W0
Lipschitz.

1;p

1;p

p

( ) ,! L ( ), p = even if

( ) then Theorem 0.2.12 is valid even if is not

This results can be extended for the spaces W
integer.

m;p

( ) with m is a non negative
N

Theorem 0.2.14. (see [3, Theorem 7.1, p. 355]) Let R be a bounded Lipschitz