Điều kiện tối ưu không cách biệt và tính ổn định nghiệm của các bài toán điều khiển tối ưu được cho bởi các phương trình elliptic nửa tuyến tính

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 scientific results are new and righteous. Before I published these
results, there had been no such results in any scientific document. I have responsibilities for my research results in the dissertation.
Hanoi, April 3rd , 2019
On behalf of Supervisors

Author

Dr. Nguyen Thi Toan

Nguyen Hai Son

i


ACKNOWLEDGEMENTS
This dissertation has been carried out at the Department of Fundamental Mathematics, 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 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 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.
Hanoi, April 3rd , 2019
Nguyen Hai Son

ii


CONTENTS
COMMITTAL IN THE DISSERTATION
AC
K
CO
NT
TA
BL
IN
TR

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Chapter 0.

8

PRELIMINARIES AND AUXILIARY RESULTS

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.
PR 2
OB 5
1.1 2

NO-GAP OPTIMALITY CONDITIONS FOR DISTRIBUTED CONTROL

Se 1. 62

1.1.2

1. 6

2
Se7
1.2 4
Se
1.3 05
Co 7
Chapter 2.

NO-GAP OPTIMALITY CONDITIONS FOR BOUNDARY CONTROL

58

PROBLEMS

2.1 A .
2.2 bs
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co
2.3 Se . . . . . . . . . . .
2.4 co
C . . . . . . . . . . .
on
Chapter 3. UPPER SEMICONTINUITY AND

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5
69
67
85
9

CONTINUITY OF THE SOLUTION

MAP TO A PARAMETRIC BOUNDARY CONTROL PROBLEM

91

3.1 Assumptions and main result

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92

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94

3.2 Some auxiliary results

3


3.Some properties of the admissible set . . .
2.
3.First-order necessary optimality conditions

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3.32. Proof of the main result

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3.4
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Examp

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Concl
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REF .
ERE

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RAL
LIST
OF

4


TABLE OF NOTATIONS

N := {1, 2, . . .}
R

set of real numbers

|x|

absolute value of x ∈ R

R

n

set of positive natural numbers

n-dimensional Euclidean vector space

∅

empty set

x∈A

x is in A

x ∈/ A

x is not in A

A ⊂ B(B ⊃ A)

A is a subset of B

A* B

A is not a subset of B

A∩B
A∪B
A\B

intersection of the sets A and B
union of the sets A and B
set difference of A and B

A×B

Descartes product of the sets A and B

[x1 , x2 ]

the closed line segment between x1 and x2

kxk

norm of a vector x

kxkX

norm of vector x in the space X

X

∗

X

∗∗

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

X hx∗ , xi
hx, yi

canonical pairing
canonical inner product

B(x, δ)

open ball with centered at x and radius δ

B(x, δ)

closed ball with centered at x and radius δ

BX

open unit ball in a normed space X

BX
dist(x; Ω)

closed unit ball in a normed space X
distance from x to Ω

{xk }

sequence of vectors xk

xk → x

xk converges strongly to x (in norm topology)

xk * x

xk converges weakly to x

∀x

for all x

∃x

there exists x

A := B

A is defined by B

f :X → Y

function from X to Y

f 0 (x), ∇f (x)
00

2

f (x), ∇ f (x)

Fr´echet derivative of f at x
Fr´echet second-order derivative of f at x

1


Lx , ∇x L

Fr´echet derivative of L in x

2


2
Lxy , ∇xy
L

ϕ : X → IR
function domϕ

Fr´echet second-order derivative of L in xand y
extended-real-valued
effective domain of ϕ

epiϕ

epigraph of ϕ

suppϕ

the support of ϕ

F :X ⇒ Y

multifunction from X to Y

domF
rgeF

domain of F
range of F

gphF
kerF

graph of F
kernel of F

T (K, x)

Bouligand tangent cone of the set K at x

[

adjoint tangent cone of the set K at x

2

second-order Bouligand tangent set of the set

T (K, x)
T (K, x, d)

K at x in direction d
2[

T (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

∂Ω
Ω¯

boundary of the domain Ω

0

Ω ⊂⊂ Ω

Ω0 ⊂ Ω and Ω0 is compact.

Lp (Ω)

the space of Lebesgue measurable functions f

closure of the set Ω

and
L∞ (Ω)
C (Ω¯ )



m,p

Ω |f

(x)|p dx < +∞

the space of bounded functions almost every Ω
the space of continuous functions on Ω¯

¯)
M(Ω



W

R

the space of finite regular Borel measures
m,p

(Ω), W0

(Ω),

W s,r (Γ),

Sobolev spaces

H m (Ω), H 0m (Ω)

m,p

W −m,p (Ω)(p−1 + p0−1 = 1)

the dual space of W0

X ,→ Y

X is continuous embedded in Y

X ,→,→ Y

i.e.

X is compact embedded in Y
id est (that is)

a.e.

almost every

s.t.

subject to

p. 5

page 5

w.r.t
✷

with respect to
The proof is complete

0

3

(Ω)

INTRODUCTION

1. Motivation
Optimal control theory has many applications in economics, mechanics and other
fields 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 find optimal control functions. The
other was the Bellman Dynamic Programming Principle, a procedure that reduces
the search for optimal control functions to finding the solutions of partial differential
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 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 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. Troltzsch [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

differentiable functional and U is a Banach space, then F 0 (u¯) = 0. This a firstorder necessary optimality condition. However, it is not a sufficient condition in
case of F is not convex. Therefore, we have to invoke other sufficient 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. Second-order sufficient optimality conditions play an important role in the numerical analysis of nonlinear
optimal control problems, and in analyzing the sequential quadratic programming algorithms (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 out
stationary points but also help us in constructing sufficient optimality conditions. Let
us briefly review some results on this topic.


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

(1)

and the appearance of state constraints. More precisely, in [11] the authors gave
second-order necessary and sufficient conditions for Neumann problems with constraint
(1) and finitely many equalities and inequalities constraints of state variable y while
the second-order sufficient optimality conditions are established for Dirichlet problems
with constraint (1) and a pure state constraint in [13]. T. Bayen et al. [27] derived
second-order necessary and sufficient optimality conditions for Dirichlet problems in the
sense of strong solution. In particular, E. Casas [16] established second-order sufficient
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 sufficient optimality
conditions for 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
conditions while 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. Rosch and F. Tr¨oltzsch [23] gave the second-order sufficient optimality
conditions for the problem with the mixed pointwise constraints which has unilateral
linear form c(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 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 statecontrol constraints of the form
a(x) ≤ g(x, y(x)) + u(x) ≤ b(x) a.e x ∈ Ω
with a, b ∈ Lp (Ω), 1 < p < ∞ 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) ≤ b(x) a.e. x ∈ Γ,


where a, b ∈ Lp (Γ), 1 < p <
∞.
(OP 2) : Give second-order sufficient optimality conditions for optimal control problems with mixed state-control constraints when the objective function
depend on control variables.

does not

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 change
between necessary and sufficient 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 sufficient optimality conditions, is a difficult
problem which requires to find a common critical cone under which both second-order
necessary optimality conditions and sufficient optimality conditions are satisfied. In [7],
J. F. Bonnans derived second-order necessary and sufficient 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 control 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 optimization and numerical method of finding 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, Holder continuity and Lipschitz
continuity.
Let us consider the following parametric optimal problem:




P (µ, λ)

F (y, u, µ) → inf,
(y, u) ∈ Φ(λ),

(2)

where y ∈ Y, u ∈ U are state and control variables, respectively; µ ∈ Π, λ ∈ Λ
are parameters, F : Y × U × Π → R is an objective function on Banach space Y × U
× Π and Φ(λ) is an admissible set of the problem.
It is well-known that if the objective function F (·, ·, µ) is strongly convex, and the
admissible 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 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 qualifications are satisfied 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 differential 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 sufficient 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. Ob jective
The objective of this dissertation is to study no-gap second-order optimality conditions 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 control
problems with the control variables belong to Lp (Γ), 1 < p < ∞;
(ii) deriving second-order sufficient optimality conditions for distributed control problems 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 sufficient optimality conditions for distributed control problems 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 sufficient 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 differential 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 Scientific Computation, Ba Vi in April
2017.
• The 7th International Conference on High Performance Scientific 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 differential 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

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

x ∈ X | F (x) = ∅ ,
y ∈ Y | y ∈ F (x) for some x ∈ X

[
:=

F (x)

x∈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 ) = ∅.
Definition 0.1.1. ([46, p. 34]) Let F : X ⇒ Y be a set-valued map between topological 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 topological space X × Y then F is called convex set-valued map.
(iii) If F (x) is a closed subset of Y for all x ∈ X then F is called closed-valued map.

8


(iv) If F (x) is a compact subset of Y for all x ∈ X then F is called compactvalued map.

9


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 = ∅, there exists a neighborhood V of x0 such that
F (x) ∩ W = ∅ 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 are
coincident.
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 − x0 k < δ}, and by B(x0 , δ) the
corresponding closed ball. We will write BX and B X 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

10


dist(x; D) = inf kx − uk.
u∈D

Definition 0.1.3. ([44, Definition 4.1.1, p. 121]) Let D ⊂ X be a subset of a normed
space X and a point x ∈ D. The set
T (D, x) := v ∈ X | lim inf
+
t→0

dist(x + tv, D)
=0 .

t

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

11


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 + tn vn ∈ D for all n ∈ N}.
Definition 0.1.4. ([44, Definition 4.1.5, p. 126]) Let D ⊂ X be a subset of normed
space X and x ∈ D. The adjoint tangent cone or the intermediate cone T [ (K, x) of
D at x is defined by


T [ (D, x) := v ∈ X | lim+
t→0

dist(x + tv, D)
=0 .
t

The Clarke tangent cone TC (D, x) of D at x is defined by



TC (D, x) :=

v ∈ X | lim+

t→0
D0
x

d(x0



+ tv , D)
=0
t

,

−→x
D
0 −

where x

→ x means that x0 ∈ D and x0 → x.

From Definition 0.1.4, we have the following characters of the adjoint cones and the
Clarke tangent cones (see [44, p. 128]):
T [ (D, x) = {v ∈ X | ∀tn → 0+ , ∃vn → v s.t. x + tn vn ∈ D ∀n ∈
N},
and
D

TC (D, x) = {v ∈ X | ∀tn → 0+ , ∀xn −→ x, ∃vn → v s.t. x n + tn v ∈ D ∀n ∈ N}.

It is nclear that
TC (D, x) ⊂ T [ (D, x) ⊂ T (D, x) ⊂ cone(D − x).
Example 0.1.5. ( TC (D, x) = T [ (D, x) = T (D, x) = cone(D − x))
Putting D = {(x1 , x2 ) | x2 = x12 } ⊂ R2 and taking x = (0, 0), we
have
TC (D, x) = {(0, 0)},
T [ (D, x) = T (D, x) = {(x1 , 0) | x1 ∈ R},
cone(D − x) = {(x1 , x2 ) | x2 ≥ 0}.
Example 0.1.6. ( TC (D, x) = T [ (D, x) = T (D, x) = cone(D − x))
Putting D = { 1 | n = 1, 2, ...} ⊂ R and taking x = 0 ∈ D¯ , we have
n

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

R+


Example 0.1.7. ( TC (D, x) = T [ (D, x) = T (D, x) = cone(D − x))
Putting D = {(x1 , x2 ) | x2 = 0} ∪ {(x1 , x2 ) | x1 = 0} ⊂ R2 and taking x = (0, 0),
we have
TC (D, x) = {(0, 0)},
T [ (D, x) = T (D, x) = cone(D − x) = D.
Let x1 , x2 ∈ X. The segment [x1 , x2 ] connect x1 and x2 is defined by
[x1 , x2 ] := {x ∈ X | x = αx1 + (1 − α)x2 , 0 ≤ α ≤
1}.
A subset K of X is said to be convex if [x1 , x2 ] ⊂ K for all x1 , x2 ∈ K . (see
[1, section 0.3.1, p. 45]). According to [44, Chapter 4], the Clarke cone TC (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 first-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 secondorder 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
metric space. Suppose that t0 ∈ T . The set
Limsup Dt := {x ∈ X | lim inf dist(x, Dt ) = 0}


t→t0

t→t0

is called Painlev´e-Kuratowski upper limit of (Dt ) as t → t0 .
The set
Liminf Dt := {x ∈ X | lim dist(x, Dt ) = 0}
t→t0

t→t0

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 the
normed space X and x ∈ D¯ , v ∈ X.
The set

T 2 (D, x, v) := Limsup D −tvx −
t→0+

t2

is called Bouligand second-order tangent set of D at x in direction v.
The set

D− x−
T 2[ (D, x, v) := Liminf
tv
t→0+

t2
is called second-order adjoint tangent set D at x in direction v.


Obviously, T 2 (D, x, v) and T 2[ (D, x, v) are closed and
T 2 (D, x, 0) = T (D, x), T 2[ (D, x, 0) = T [ (D,
x).
Moreover, we have
T 2 (D, x, v) = {w|∃tn → 0+ , ∃wn → w, x + tn v + t2n wn ∈ D},
T 2[ (D, x, v) = {w|∀tn → 0+ , ∃wn → w, x + tn v +n t2 wn ∈
D}.
Notice that T 2 (D, x, v) and T 2[ (D, x, v) are nonempty only if v ∈ T (D, x) and v
∈ T [ (D, x), respectively. Moreover, when D is convex, T 2[ (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 (T 2 (D, x, v) is different from T 2[ (D, x,
v)) (see [47, Example 3.31]).
Example 0.1.10. (T 2 (D, x, v) = T 2[ (D, x, v))
Let us first construct a convex piecewise linear function x2 = ϕ(x1 ), x1 ∈ R, oscil1


lating between two parabolas x2 = x12 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 ],

2
ϕ(x1,k ) = x1,k
and its graph on [x1,k+1 , x1,k ] is tangent to the curve x2 = 2x2 for some

monotonically decreasing to zero sequence {x1,k }. It is evident how such a function
can be constructed. Indeed, for a given point x1,k > 0 consider the straight line passing
2
through the point (x1,k , x1,k
) and is tangent to the curve x2 = 2x21. It intersects the

curve x2 = x12 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 → ∞.
Taking K = {(x1 , x2 ) ∈ R2 | x2 ≥ ϕ(x1 )} and x = (0, 0), v = (1, 0), we have
T 2 (D, x, v) = {(x1 , x2 ) | x2 ≥ 2} and T 2[ (D, x, v) = {(x1 , x2 ) | x2 ≥
4}.
The following result allows us to compute tangent cones of a convex and closed
subset 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 u0 ∈ K, one has

n

p

o

[

v ∈ L (Ω) | v(x) ∈ T (M (x), u0 (x)) a.e. x ∈ Ω

⊂ T [ (K, u0 ) ⊂ T (K, u0 )
⊂ {v ∈ Lp (Ω) | v(x) ∈ T (M (x), u0 (x)) a.e. x ∈ Ω} .
Corollary 0.1.12. ([27, Lemma 4.11] Let 1 ≤ p < +∞, and K := {u ∈ Lp (Ω) |
a(x) ≤ u(x) ≤ b(x) a.e. x ∈ Ω}, with a, b ∈ Lp (Ω) and u0 ∈ K.
Then
T (K, u0 ) = T [ (K, u0 )

n

=

p

[

o

v ∈ L (Ω) | v(x) ∈ T ([a(x), b(x)], u0 (x)) a.e. x ∈ Ω .