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 tt tiếng anh

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

ABSTRACT OF DOCTORAL DISSERTATION OF MATHEMATICS

Hanoi – 2019


The dissertation is completed at:
Hanoi Univesity of Science and Technology

Supervisors:
1. Dr. Nguyen Thi Toan
2. Dr. Bui Trong Kien

Reviewer 1: Prof. Dr. Sc. Vu Ngoc Phat
Reviewer 2: Assoc. Prof. Dr. Cung The Anh
Reviewer 3: Dr. Nguyen Huy Chieu

The dissertation will be defended before approval committee
at Hanoi Univesity of Science and Technology

Time …….. , date….. month….. year………

The dissertation can be found at:
1. Ta Quang Buu Library - Hanoi Univesity of Science and Technology
2. Vietnam National Library


Introduction
Optimal control theory has many applications in economics, mechanic and other
fields of science. It has been systematically studied and strongly developed since
the late 1950s, when two basic principles were made: the Pontryagin Maximum Principle and the Bellman Dynamic Programming Principle. 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),...
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.
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 and in studying the stability of optimal control.
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. derived second-order necessary and sufficient optimality
conditions for problem with pure control constraint, that is,
a(x) ≤ u(x) ≤ b(x)

a.e. x ∈ Ω,

(1)

and the appearance of state constraints. In particular, E. Casas established secondorder 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 addition, C. Meyer and F. Tr¨oltzsch derived secondorder sufficient optimality conditions for Robin 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, where y is the state variable.
For boundary control problems, i.e., the control u only acts on the boundary
Γ, E. Casas et al. and F. Tr¨oltzsch derived second-order necessary and sufficient
optimality conditions with pure pointwise constraints, i.e.,
a(x) ≤ u(x) ≤ b(x)

1

a.e. x ∈ Γ.


A. R¨osch and F. Tr¨oltzsch 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 results, 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. In order to avoid this disadvantage,
B. T. Kien et al. 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 ∈ Ω
with a, b ∈ Lp (Ω) 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 with a, b ∈ Lp (Γ).
(OP 2) : Give second-order sufficient optimality conditions for optimal control problems with mixed state-control constraints when the objective function does not depend
on control 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, 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
second-order necessary optimality conditions and sufficient optimality conditions, is
a difficult problem. In some papers, 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. This result was established by basing on polyhedric
property of admissible sets and the theory of Legendre forms. However, there is an
open problem in this area. Namely, the following problem that we need to study:
(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. The study of solution stability
2


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:

F (y, u, µ) → inf,
P (µ, λ)

(2)
(y, u) ∈ Φ(λ),
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 cost function F (·, ·, µ) is strongly convex, and the
admissible set Φ(λ) is convex, then the solution map of problem (2) is single-valued.
Moreover, A. Dontchev showed that under some certain conditions, the solution
map is Lipschitz continuous w.r.t. parameters. By using techniques of implicit
function theorem, K. Malanowski 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.
When conditions mentioned above are invalid, the solution map may not be
singleton. 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. obtained the lower
semicontinuity of the solution map to a parametric optimal control problem for the
case where the cost 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. and V. H. Nhu for problems, where the cost functions may not
be convex in the both variables and the admissible sets are not convex. Notice that
the authors only considered the problems governed by ordinary differential equations
and 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.
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 < ∞;
3


(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 the solution map is upper semicontinuous and continuous in parameters.
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 optimality conditions for distributed
control problems.
Chapter 2 provides results on the no-gap optimality conditions for boundary
control problems.
The obtained results in Chapters 1 and 2 are answers for (OP 1), (OP 2) and (OP 3).
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).
The main results of the dissertation are the contents of three papers which were
published in the journals Set-Valued and Variational Analysis, SIAM Journal on Optimization, and 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.
• 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.

4


Chapter 1

No-gap optimality conditions for distributed control
problems
Let Ω be a bounded domain in RN with N ≥ 2 and the boundary Γ of class C 2 .
We consider the following distributed optimal control problem of finding a control
function u ∈ Lp (Ω) and a state function y ∈ W 2,p (Ω) ∩ W01,p (Ω) which
minimize F (y, u) =

L(x, y (x), u(x))dx,

(1.1)

Ω

(DP )

s.t.
− ∆y + h(x, y ) = u

in Ω,

y=0

on Γ,

a(x) ≤ g (x, y (x)) + λu(x) ≤ b(x) a.e. x ∈ Ω,

(1.2)
(1.3)

where L : Ω × R × R → R and g : Ω × R → R are Carath´eodory functions, h :
Ω × R → R is a continuous function of class C 2 w.r.t. the second variable such that
h(x, 0) = 0 and hy (x, y ) ≥ 0 for all y ∈ R and a.e. x ∈ Ω, a, b ∈ Lp (Ω) and λ = 0 is
a constant. Hereafter, we assume that p > N2 .

1.1
1.1.1

Second-order necessary optimality conditions
An abstract optimization problem

Let U be Banach space and E be a separable Banach space with the duals U ∗ and
E ∗ , respectively. We consider the following problem
(P )

min f (u) subject to G(u) ∈ K,
u∈U

where K is a nonempty closed and convex set in E , G : U → E and f : U → R are

second-order Frech´et differentiable on U . Put Φad := G−1 (K ).
Definition 1.1.1. A function u
¯ ∈ Φad is said to be a locally optimal solution of
problem (P ) if there exists ε > 0 such that
f (u) ≥ f (¯
u)

∀u ∈ BU (¯
u, ) ∩ Φad .

Given a point u¯ ∈ Φad , problem (P ) is said to satisfy Robinson’s constraint
qualification at u¯ if there exists ρ > 0 such that
BE (0, ρ) ⊂ ∇G(¯
u)(BU ) − (K − G(¯
u)) ∩ BE .

5

(1.4)


In this case, we also say that u¯ is regular.
Problem (P ) is associated with the following Lagrangian:
L(u, e∗ ) = f (u) + e∗ , G(u) with e∗ ∈ E ∗ .

We shall denoted by Λ(¯
u) the set of multipliers e∗ ∈ E ∗ such that
∇u L(¯
u, e∗ ) = ∇f (¯
u) + ∇G(¯

u)∗ e∗ = 0, e∗ ∈ N (K, G(¯
u)).

The set Λ(¯
u) is a non-empty, convex and weakly star compact set in E ∗ . To analyze
second-order conditions, we need the following critical cone at u¯:
C (¯
u) := {d ∈ U | ∇f (¯
u), d ≤ 0, ∇G(¯
u)d ∈ T (K, G(¯
u))}.

The set K is said to be polyhedric at z¯ ∈ K if for any v ∗ ∈ N (K, z¯), one has
T (K, z¯) ∩ (v ∗ )⊥ = cl[cone(K − z¯) ∩ (v ∗ )⊥ ],

where (v ∗ )⊥ = {v ∈ E | v ∗ , v = 0}. Moreover, problem (P ) is said to satisfy the
strongly extended polyhedricity condition at u¯ ∈ Φ if the set C0 (¯
u) is dense in C (¯
u),
where
C0 (¯
u) := {d ∈ C (¯
u) | ∇G(¯
u)d ∈ cone(K − G(¯
u))}.
Lemma 1.1.3.1 Suppose that u
¯ is regular, at which the strongly extended polyhedricity condition is fulfilled. If u¯ is a locally optimal solution, then for each
d ∈ C (¯
u), there exists a multiplier e∗ ∈ Λ(¯
u) such that

∇2uu L(¯
u, e∗ )(d, d) = ∇2 f (¯
u)(d, d) + e∗ , ∇2 G(¯
u)(d, d) ≥ 0.

1.1.2

Second-order necessary optimality conditions for optimal control problem

Recall that a couple (¯
y, u
¯) satisfying constraints (1.2)–(1.3), is said to be admissible
for (DP ). Given an admissible couple (¯
y, u
¯), symbols g [x], h[x], L[x], Ly [x], L[·], etc.,
stand respectively for g (x, y¯(x), u¯(x)), h(x, y¯(x)), L(x, y¯(x), u¯(x)), Ly (x, y¯(x), u¯(x)),
L(·, y¯(·), u
¯(·)), etc.
Definition 1.1.4. An admissible couple (¯
y, u
¯) is said to be a locally optimal solution
of (DP ) if there exists > 0 such that for all admissible couples (y, u) satisfying
¯ Lp (Ω) ≤ , one has
y − y¯ W 2,p (Ω) + u − u
F (y, u) ≥ F (¯
y, u
¯).
1

J. F. Bonnans and A. Shapiro (2000), Perturbation Analysis of Optimization Problems, Springer,

New York.

6


We now impose the following assumptions for problem (DP ) which involve (¯
y, u
¯).
(A1.1) L : Ω × R × R → R is a Carath´eodory function of class C 2 with respect to
variable (y, u), L(x, 0, 0) ∈ L1 (Ω) and for each M > 0, there exist a positive number
kLM and a function rM ∈ L∞ (Ω) such that
|Ly (x, y, u)| + |Lu (x, y, u)| ≤ kLM |y| + |u|p−1 + rM (x),
|Ly (x, y1 , u1 ) − Ly (x, y2 , u2 )| ≤ kLM (|y1 − y2 | + |u1 − u2 |),
|u1 − u2 |p−1−j |u2 |j

|Lu (x, y, u1 ) − Lu (x, y, u2 )| ≤ kLM
j=0,p−1−j>0

for all y, y1 , y2 ∈ R satisfying |y|, |yi | ≤ M and any u1 , u2 ∈ R. Also for each M > 0,
there is a number kLM > 0 such that
Lyy (x, y1 , u1 ) − Lyy (x, y2 , u2 ) ≤ kLM (|y1 − y2 | + |u1 − u2 |),
Lyu (x, y1 , u1 ) − Lyu (x, y2 , u2 ) ≤ kLM (|y1 − y2 | + ε|u1 − u2 |p−1 )

( with ε = 0 if 1 < p ≤ 2 and ε = 1 if p > 2 ) and

= 0
|Luu (x, y, u1 ) − Luu (x, y, u2 )|
p−2−j
≤ kLM
|u2 |j

j=0,j
if 1 < p ≤ 2,
if p > 2

for a.e. x ∈ Ω, for all y, ui , yi ∈ R with |y|, |yi | ≤ M and i = 1, 2.
(A1.2) The function g is a continuous function and of class C 2 w.r.t. the second
variable, and satisfies the following property: g (·, 0) ∈ Lp (Ω) and for each M > 0,
there exists a constant Cg,M > 0 such that
gy (x, y ) + gyy (x, y ) ≤ Cg,M ,
gy (x, y1 ) − gy (x, y2 ) + gyy (x, y1 ) − gyy (x, y2 ) ≤ Cg,M |y2 − y1 |

for a.e. x ∈ Ω and |y|, |y1 |, |y2 | ≤ M .
(A1.3) λ = 0 and
λhy [x] + gy [x]
≥ 0 a.e. x ∈ Ω.
λ
For each u ∈ Lp (Ω), equation (1.2) has a unique solution yu ∈ W 2,p (Ω) ∩ W01,p (Ω)
and there exists a constant C > 0 such that
yu

W 2,p (Ω)

≤C u

Lp (Ω) .

Define a mapping H : W 2,p (Ω) ∩ W01,p (Ω) × Lp (Ω) → Lp (Ω) by setting
H (y, u) = −∆y + h(·, y ) − u.

7


Then, H is of class C 2 around (¯
y, u
¯) and its derivatives at (¯
y, u
¯) are given by
hyy [·] 0
0
0

∇H (¯
y, u
¯) = (−∆ + hy [·], −I ), ∇2 H (¯
y, u
¯) =

.

¯
Since p > N/2, y¯ = y (¯
u) ∈ C (Ω).
Hence hy [·] ∈ L∞ (Ω). Therefore, for each
u ∈ Lp (Ω), the following equation has a unique solution z ∈ W 2,p (Ω) ∩ W01,p (Ω).
−∆z + hy (·, y¯)z = u

in Ω,

z=0

on Γ.

Hence the operator A := ∇y H (¯
y, u
¯) = −∆ + hy (·, y¯) is bijective. By the classical
implicit function theorem, there exist a neighborhood Y0 of y¯, a neighborhood U0 of
u
¯ and a mapping ζ : U0 → Y0 such that H (ζ (u), u) = 0 for all u ∈ U0 . Moreover, ζ
is of class C 2 and its derivatives are given by the following formulae.
Lemma 1.1.9. Assume that ζ : U0 → Y0 is the control-state mapping defined by
ζ (u) = yu . Then ζ is of class C 2 and for each u ∈ U0 , v ∈ Lp (Ω), zu,v := ζ (u)v is the
unique solution of the linearized equation

−∆z + h (·, y )z = v in Ω,
u,v
y
u u,v
(1.11)
zu,v = 0 on Γ.
In other words, ζ (¯
u) = A−1 . Moreover, for all v1 , v2 ∈ Lp (Ω), zu,v1 v2 := ζ (u)(v1 , v2 )
is the unique solution of the equation

−∆z
u,v1 v2 + hy (·, yu )zu,v1 v2 + hyy (·, yu )zu,v1 zu,v2 = 0 in Ω,
(1.12)
zu,v v = 0 on Γ.
1 2

Let us put U = E = Lp (Ω) and K := {v ∈ Lp (Ω)| a(x) ≤ v (x) ≤ b(x) a.e. x ∈ Ω}.
Then, from Lemma 1.1.8, we see that (¯
y, u
¯) is a locally optimal solution of problem
(DP ) if and only if u¯ is a locally optimal solution of the following problem which
has a form of (P ):
f (u) := F (ζ (u), u) → inf

s.t. G(u) ∈ K,

(1.13)
(1.14)

where G(u) := g (·, ζ (u)) + λu. By Φp := G−1 (K ), we denote the admissible set of
problem (1.13)–(1.14), i.e., Φp = {u ∈ Lp (Ω) | G(u) ∈ K}.
Definition 1.1.9. The function u
¯ ∈ Φp is said to be a locally optimal solution of
problem (1.13)–(1.14) if there exists ε > 0 such that
f (u) ≥ f (¯
u)

∀u ∈ BU (¯
u, ) ∩ Φp .

8


Problem (1.13)–(1.14) is associated with the Lagrangian:
L(u, e∗ ) = f (u)+ e∗ , G(u) =

e∗ (g (·, ζ (u))+λu)dx,

L(x, ζ (u), u)dx+
Ω

e∗ ∈ Lp (Ω)

Ω

where p is the adjoint number of p.
Given u¯ ∈ Φp , the critical cone of problem (1.13)–(1.14) is defined by

≥ 0 if x ∈ Ω
a
p
Cp (¯
u) = d ∈ L (Ω) | (Ly [x]zu¯,d + Lu [x]d)dx ≤ 0, ∇G(¯
u)d(x)
≤ 0 if x ∈ Ωb
Ω

a.e. ,

where Ωa := {x ∈ Ω | G(¯
u)(x) = a(x)}, Ωb := {x ∈ Ω | G(¯
u)(x) = b(x)}.
Theorem 1.1.15. Suppose that assumptions (A1.1)–(A1.3) are satisfied and u
¯ is
a locally optimal solution of (1.13)–(1.14). There exist a unique e∗ ∈ Lp (Ω) and a
unique φ¯ ∈ W 2,p (Ω) ∩ W01,p (Ω) such that the following conditions are valid:

(i) The adjoint equation:

−∆φ¯ + h [·]φ¯ = L [·] + e∗ g [·] in Ω,
y
y
y
(1.17)
φ¯ = 0 on Γ;
(ii) The stationary conditions in u:
∇u L(¯
u, e∗ ) = φ¯ + Lu [·] + λe∗ = 0,

e∗ ∈ N (K, G(¯
u));

(1.18)

(iii) The non-negative second-order condition:
∇2uu L(¯
u, e∗ )(v, v ) =

Lyy [x]zu2¯,v + 2Lyu [x]zu¯,v v + Luu [x]v 2 + e∗ gyy [x]zu2¯,v
Ω

¯ yy [x]zu2¯,v dx ≥ 0
− φh

∀v ∈ Cp (¯
u).

Example 1.1.16 illustrates how to use necessary conditions to find stationary
points. In this example, the point (0; 0) satisfies first-order necessary optimality
conditions but it does not satisfy second-order necessary optimality conditions.

1.2

Second-order sufficient optimality conditions

To derive second-order sufficient optimality conditions for elliptic optimal control
problems we usually use two different norms. In this section, instead of using the
two-norm method we exploit the structure of the objective function in order to
derive a common critical cone to the problem for the case p = 2, N ∈ {2, 3} and the
objective function has the form
L(x, y, u) = ϕ(x, y ) + α(x)u + β (x)u2 ,

9

(1.23)


where ϕ : Ω × R → R is a Carath´eodory function and α, β ∈ L∞ (Ω).
In the sequel, we need the following assumptions.
(A1.1) Function ϕ : Ω × R → R is a Carath´eodory function of class C 2 with respect
to the second variable, ϕ(x, 0) ∈ L1 (Ω) and for each M > 0 there are a constant
Kϕ,M > 0 and a function ϕM ∈ L2 (Ω) such that
∂ϕ
∂ 2ϕ
(x, y ) ≤ ϕM (x),
(x, y ) ≤ Kϕ,M ,
∂y

∂y 2
∂ϕ
∂ 2ϕ
∂ϕ
∂ 2ϕ
(x, y1 ) −
(x, y2 ) +
(x, y1 ) − 2 (x, y2 ) ≤ Kϕ,M |y1 − y2 |
∂y
∂y
∂y 2
∂y

for a.e. x ∈ Ω and for all y1 , y2 , y ∈ R with |y|, |y1 |, |y2 | ≤ M.
(A1.2) There exists a number γ > 0 such that β (x) ≥ γ for a.e. x ∈ Ω.
Definition 1.2.1. We say that f satisfies L2 − weak quadratic growth condition at
u
¯ ∈ Φ2 if there exist > 0, α > 0 such that
f (u) ≥ f (¯
u) + α u − u
¯

for all u ∈ Φ2 satisfying u − u¯

2

2
2

≤ .

Theorem 1.2.2. Suppose that assumptions (A1.2), (A1.1) and (A1.2) are satisfied.
Let u¯ ∈ Φ2 and multipliers e∗ ∈ L2 (Ω), φ¯ ∈ W 2,2 (Ω) ∩ W01,2 (Ω) satisfy conditions
(1.17) and (1.18). Furthermore, suppose that
∇2uu L(¯
u, e∗ )(u, u) > 0

∀u ∈ C2 (¯
u) \ {0}.

Then f satisfies L2 −weak quadratic growth condition at u¯ ∈ Φ2 . In particular, u¯ is
a locally optimal solution of problem (1.13)–(1.14) in L2 (Ω).
From Theorem 1.1.14 and Theorem 1.2.2, we obtain no-gap optimality conditions
in this case.
In the rest of this section we shall derive second-order sufficient optimality conditions for the problem (DP ) for the case where L is given by (1.23) with α(x) and
β (x) may be zero. For this we need the following assumptions.
(B 1.1) Function h : R → R is of class C 2 satisfying
h(x, 0) = 0,

hy (x, y ) ≥ 0, ∀y ∈ R and a.e. x ∈ Ω

and for every M > 0 there is a constant Ch,M > 0 such that
∂ 2h
∂h
(x, y ) +
(x, y ) ≤ Ch,M ,
∂y
∂y 2

∀y ∈ R with |y| ≤ M and a.e. x ∈ Ω.

10


Moreover, for every M > 0 and > 0, there exists a positive number δ > 0 such that
∂ 2h
∂ 2h
(
x,
y
)
−
(x, y2 ) <
1
∂y 2
∂y 2

a.e. x ∈ Ω ∀y1 , y2 ∈ R with |y1 |, |y2 | ≤ M, |y1 −y2 | < δ.

(B 1.2) Function ϕ : Ω × R → R is a Carath´eodory function of class C 2 with respect
to the second variable, ϕ(x, 0) ∈ L1 (Ω) and for each M > 0 there are a constant
Cϕ,M > 0 and a function ϕM ∈ L2 (Ω) such that
∂ 2ϕ
∂ϕ
(x, y ) ≤ ϕM (x),
(x, y ) ≤ Cϕ,M
∂y
∂y 2

a.e. x ∈ Ω, ∀y ∈ R with |y| ≤ M

and for each > 0, there exists δ > 0 such that
∂ 2ϕ
∂ 2ϕ
(x, y1 ) − 2 (x, y2 ) <
∂y 2
∂y

a.e. x ∈ Ω, ∀y1 , y2 ∈ R, |y1 |, |y2 | ≤ M, |y1 − y2 | < δ.

(B 1.3) Function g : Ω × R → R is a continuous function of class C 2 with respect
to the second variable, g (·, 0) ∈ L2 (Ω) and for each M > 0 there are a constant
Cg,M > 0 and a function gM ∈ L2 (Ω) such that
∂ 2g
∂g
(x, y ) ≤ gM (x),
(x, y ) ≤ Cg,M
∂y
∂y 2

a.e. x ∈ Ω, ∀y ∈ R with |y| ≤ M.

Moreover, for every M > 0 and > 0, there exists a positive number δ > 0 such that
∂ 2g
∂ 2g
(x, y1 ) − 2 (x, y2 ) <
∂y 2
∂y

a.e. x ∈ Ω, ∀y1 , y2 ∈ R with |y1 |, |y2 | ≤ M, |y1 −y2 | < δ.

(B 1.4) α, β ∈ L∞ (Ω) and β (x) ≥ 0 for a.e. x ∈ Ω. Besides, the following is verified
βa ∈ L∞ (Ωa ),

βb ∈ L∞ (Ωb ).

(1.26)

Note that condition (1.26) holds whenever one of the following conditions is verified:
(i) a, b ∈ L∞ (Ω);
(ii) u¯ ∈ L∞ (Ω);
(iii) β = 0.
Based on Casas, we enlarge C2 (¯
u) by defining the following critical cone.

≥ 0 if x ∈ Ω ,
a
τ
2
C2 (¯
u) = v ∈ L (Ω) | ∇u f (¯
u), v ≤ τ zu¯,v 2 , gy [x]zu¯,v (x)+λv (x)
.
≤ 0 if x ∈ Ωb .
Obviously, C2 (¯
u) = C20 (¯
u) and C2 (¯
u) ⊂ C2τ (¯
u) for all τ > 0.
Theorem 1.2.4. Suppose that assumptions (A1.3) and (B 1.1) − (B 1.4) are fulfilled,

there exist multipliers e∗ ∈ L2 (Ω) and φ¯ ∈ W 2,2 (Ω)∩W01,2 (Ω) satisfy conditions (1.17)
and (1.18). If there exist positive constants γ, τ > 0 such that
∇2uu L(¯
u, e∗ )(v, v ) ≥ γ zu¯,v

2
2

∀v ∈ C2τ (¯
u),

then there are constants ρ, r > 0 such that
f (u) ≥ f (¯
u) + r zu¯,u−¯u

2
2

11

∀u ∈ BL2 (Ω) (¯
u, ρ) ∩ Φ2 .


Chapter 2

No-gap optimality conditions for boundary control
problems
Let Ω be a bounded domain in RN with the boundary Γ of class C 1,1 and N ≥ 2. We
consider the problem of finding a control function u ∈ Lq (Γ) and a corresponding

state function y ∈ W 1,r (Ω) which

(BP )

minimize F (y, u) =
L(x, y (x))dx +
Ω

Ay + h(x, y ) = 0 in Ω,
s.t.
∂ν y + b0 y = u
on Γ,

(x, y (x), u(x))dσ,

(2.1)

Γ

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

(2.2)
(2.3)

where L, h : Ω × R → R and : Γ × R × R → R are Carath´eodory functions,
g : Γ × R → R is continuous, a, b ∈ Lq (Γ), a(x) < b(x) for a.e. x ∈ Γ, b0 ∈ L∞ (Γ),
b0 ≥ 0, A denotes a second-order elliptic operator of the form
N

Ay (x) =

Dj (aij (x)Di y (x)) + a0 (x)y (x);
i,j=1

¯ satisfy aij (x) = aji (x), a0 ∈ L∞ (Ω), a0 (x) ≥ 0 for a.e.
coefficients aij ∈ C 0,1 (Ω)
x ∈ Ω, a0 ≡ 0 and there exists m > 0 such that
N

m ξ

2

aij ξi ξj ∀ξ ∈ RN

≤

for a.e. x ∈ Ω

i,j=1

and ∂ν denote the conormal-derivative associated with A. Moreover, we assume that
1
1 1
1
> >
1−
N
r
q

N

2.1

.

(2.4)

Abstract optimal control problems

Let Y, U, V and E be either separable or reflexive Banach spaces with the dual spaces
Y ∗ , U ∗ , V ∗ and E ∗ , respectively. We consider the following problem:
Min F (y, u),

(2.5)

s.t. H (y, u) = 0,

(2.6)

G(y, u) ∈ K,

(2.7)

12


where F : Y × U → R, H : Y × U → V , G : Y × U → E are given mappings and K
is a nonempty closed convex subset in E .
We define the space Z := Y × U and the set Q := {z = (y, u) ∈ Z | H (y, u) = 0}.

Define Φad := Q ∩ G−1 (K ). An couple (¯
y, u
¯) ∈ Φad is said to be a locally optimal
solution of problem (2.5)–(2.7) if there exists > 0 such that for all (y, u) ∈ Φad
satisfying y − y¯ Y + u − u¯ U ≤ , one has F (y, u) ≥ F (¯
y, u
¯).
For a given point z¯ = (¯
y, u
¯) ∈ Φad , we need the following assumptions:
(H 2.1) The mappings F, H, G are of class C 2 around z¯.
(H 2.2) ∇y H (¯
z ) : Y → V is bijective.
(H 2.3) The regularity condition is verified at z¯, i.e., there is a number δ > 0 satisfying
0 ∈ int

∇G(¯
z )(T (Q, z ) ∩ BZ ) − (K − G(¯
z )) ∩ BE .

(2.8)

z∈BZ (¯
z ,δ)∩Q

(H 2.4) ∇G(¯
z )(T (Q, z¯)) = E.
Definition 2.1.3. A couple z = (y, u) is called a critical direction of problem (2.5)–
(2.7) at z¯ = (¯
y, u

¯) if the following conditions are satisfied:
(i) Fy (¯
z )y +Fu (¯
z )u ≤ 0;
(ii) Hy (¯
z )y +Hu (¯
z )u = 0;
(iii) ∇G(¯
z )z ∈ T (K, G(¯
z )).
The set of such critical directions will be denoted by C (¯
z ).
Problem (2.5)–(2.7) is associated with the Lagrangian
L(z, e∗ , v ∗ ) := F (z ) + v ∗ , H (z ) + e∗ , G(z ) ,

(2.9)

where z = (y, u) ∈ Z, e∗ ∈ E ∗ , v ∗ ∈ V ∗ .
Let z¯ be a locally optimal solution of problem (2.5)–(2.7) and denote by Λ(¯
z ) the
∗ ∗
∗
∗
set of Lagrange multipliers (e , v ) ∈ E × V which satisfy
∇z L(¯
z , e∗ , v ∗ ) = 0, e∗ ∈ N (K, G(¯
z )).
Lemma 2.1.4. Suppose that the assumptions (H 2.1) − (H 2.3) are fulfilled and z¯ is
a locally optimal solution of (2.5)–(2.7). Then Λ(¯
z ) is nonempty and bounded. In

addition, if (H 2.4) is fulfilled then Λ(¯
z ) is singleton.
When K is polyhedric at G(¯
z ), we have the following result.
Lemma 2.1.5. Suppose that the assumptions (H 2.1)–(H 2.4) are fulfilled and let
z¯ be a locally optimal solution of problem (2.5)–(2.7). Then, the set of critical
directions C (¯
z ) satisfies
C (¯
z ) = {d ∈ Z | ∇F (¯
z )d = 0, ∇H (¯
z )d = 0, ∇G(¯
z )d ∈ T (K, G(¯
z ))}.

In addition, if K is polyhedric at G(¯
z ) then C (¯
z ) = C0 (¯
z ), where
C0 (¯
z ) := (∇F (¯
z ))⊥ ∩ Ker∇H (¯
z ) ∩ ∇G(¯
z )−1 (cone(K − G(¯
z ))).

13


Theorem 2.1.7. Let z¯ be a locally optimal solution of problem (2.5)-(2.7). Suppose

that assumptions (H 2.1)–(H 2.4) are fulfilled and K is polyhedric at G(¯
z ). Then
there exists (e∗ , v ∗ ) ∈ Λ(¯
z ) such that
∇2zz L(¯
z , e∗ , v ∗ )(d, d) = ∇2 F (¯
z )d2 + e∗ , ∇2 G(¯
z )d2 + v ∗ , ∇2 H (¯
z )d2 ≥ 0

for all d ∈ C (¯
z ).

2.2

Second-order necessary optimality conditions

Definition 2.2.1. An admissible couple (¯
y, u
¯) is said to be a locally optimal solution
of (BP ) if there exists > 0 such that for all admissible couples (y, u) satisfying
y − y¯ W 1,r (Ω) + u − u
¯ Lq (Γ) ≤ , one has F (y, u) ≥ F (¯
y, u
¯).

Let us impose some assumptions for problem (BP ) which involve (¯
y, u
¯).
(A2.1) L : Ω × R → R is a Carath´eodory function of class C 2 with respect to second

variable, L(x, 0) ∈ L1 (Ω) and for each M > 0, there exists a positive number kLM
such that
|Ly (x, y )| + |Lyy (x, y )| ≤ kLM ,
|Ly (x, y1 ) − Ly (x, y2 )| + Lyy (x, y1 ) − Lyy (x, y2 ) ≤ kLM |y1 − y2 |

for a.e. x ∈ Ω, for all y, yi ∈ R with |y|, |yi | ≤ M , i = 1, 2.
(A2.2) : Γ × R × R → R is a Carath´eodory function of class C 2 with respect to
variable (y, u), (x, 0, 0) ∈ L1 (Γ) and for each M > 0, there exist a positive number
k M and a function rM ∈ L∞ (Γ) such that
| y (x, y, u)| + | u (x, y, u)| ≤ k M |y| + |u|q−1 + rM (x),
| y (x, y1 , u1 ) −

y (x, y2 , u2 )|

≤ k M (|y1 − y2 | + |u1 − u2 |),

| u (x, y1 , u1 ) −

u (x, y2 , u2 )|

≤ k M |y1 − y2 | +

|u1 − u2 |q−1−j |u2 |j ,
j≥0, q−1−j>0

yy (x, y1 , u1 )

−

yy (x, y2 , u2 )

≤ k M (|y1 − y2 | + |u1 − u2 |),

yu (x, y1 , u1 )

−

yu (x, y2 , u2 )

≤ k M (|y1 − y2 | + εq |u1 − u2 |q−1 ),

and | uu (x, y1 , u1 ) − uu (x, y2 , u2 )| ≤ k M |y1 −y2 | + εq j≥0, q−2−j>0 |u1 −u2 |q−2−j |u2 |j
for a.e. x ∈ Γ, for all y, ui , yi ∈ R satisfying |y|, |yi | ≤ M , i = 1, 2 and εq = 0 if 1 <
q ≤ 2 and εq = 1 if q > 2.
(A2.3) h : Ω × R → R is a Carath´eodory function and of class C 2 w.r.t. the second
variable, and satisfies the following property:
h(·, 0) ∈ LN r/(N +r) (Ω), hy (x, y ) ≥ 0

14

a.e. x ∈ Ω


and for each M > 0, there exists a constant Ch,M > 0 such that
hy (x, y ) + hyy (x, y ) ≤ Ch,M ,

hyy (x, y1 ) − hyy (x, y2 ) ≤ Ch,M |y2 − y1 |

for a.e. x ∈ Ω and |y|, |y1 |, |y2 | ≤ M .
(A2.4) g : Γ × R → R is a Carath´eodory function and of class C 2 w.r.t. the second

variable, g (·, 0) ∈ Lq (Γ) a.e. x ∈ Γ and for each M > 0, there exists a constant
Cg,M > 0 such that
gy (x, y ) + gyy (x, y ) ≤ Cg,M ,

gyy (x, y1 ) − gyy (x, y2 ) ≤ Cg,M |y2 − y1 |

for a.e. x ∈ Γ and |y|, |y1 |, |y2 | ≤ M .
(A2.5) b0 + gy [x] ≥ 0 a.e. x ∈ Γ.
Let us define the mappings
H : Z → V,

H (z ) = H (y, u) := (Ay + h(x, y ); ∂ν y + b0 y − u),

G : Z → E,

G(z ) = G(y, u) := g (., y ) + u,

and set K := {v ∈ Lq (Γ) : a(x) ≤ v (x) ≤ b(x) a.e. x ∈ Γ}.
Then problem (BP ) reduces to the following problem:

s.t.

Min F (z )

(2.14)

H (z ) = 0,

(2.15)

G(z ) ∈ K.

(2.16)

Denote by Φq := Q ∩ G−1 (K ) the admissible set of problem (2.14)-(2.16), where
Q := {z = (y, u) ∈ Z | H (z ) = 0}.

We now use Theorem 2.1.6 to derive second-order necessary optimality conditions
for problem (BP ). For this we have to show that under assumptions (A2.1)–(A2.4)
all of hypotheses (H 2.1)-(H 2.4) are satisfied.
Lemma 2.2.2. Suppose that assumptions (A2.1)–(A2.4) are fulfilled. Then F, H and
G are of class C 2 .
Lemma 2.2.3. Under assumption (A2.3), ∇y H (ˆ
y, u
ˆ) is bijective for all (ˆ
y, u
ˆ) ∈ Z.
Lemma 2.2.4. Suppose that assumptions (A2.3)–(A2.5) are fulfilled. Then, the following assertions are valid:
(i) (the regularity condition) for some constant δ > 0, one has

0∈

z )(T (Q, z ) ∩ BZ ) − (K − G(¯
z )) ∩ BE ] .
[∇G(¯
z∈BZ (¯
z ,δ)∩Q

(ii) ∇G(¯
z )(T (Q, z¯)) = Lq (Γ).

15

(2.18)


From Lemmas 2.2.3 and 2.2.4, we see that hypotheses (H 2.2)-(H 2.4) are valid.
Let us introduce the Lagrangian associated with problem (BP ).
L(z, ψ, v ∗ ) =F (z ) + v ∗ H (z ) + ψG(z )
N

L(·, y )dx +

=
Ω

(·, y, u)dσ +
Γ

h(·, y )v1 dx −

+
Ω

aij (·)Di yDj v1 + a0 (·)yv1 dx
Ω

(∂ν y + b0 y − u)v2 dσ +

∂ν yv1 dσ +
Γ

i,j=1

Γ

(g (·, y ) + u)ψdσ,
Γ

1

where v ∗ = (v1 , v2 ) ∈ V ∗ = W 1,r (Ω) × W r ,r (Γ), ψ ∈ Lq (Γ)∗ = Lq (Γ). Here, we use
the fact (X × Y )∗ = X ∗ × Y ∗ . In case of v1 = φ, v2 = T φ, we denote
L(z, ψ, φ) := L(z, ψ, v ∗ ) =

(·, y, u)dσ +

L(x, y )dx +
Ω

Γ

h(·, y )φdx
Ω

N

aij (·)Di yDj φ + a0 (·)yφ dx +

+
Ω

(b0 y − u)T φdσ +
Γ

i,j=1

(g (·, y ) + u)ψdσ,
Γ

Let us consider the set-valued map K : Γ ⇒ R, defined by K(x) = [a(x), b(x)] a.e.
x in Γ. Then K = {v ∈ Lq (Γ) | v (x) ∈ K(x) a.e. x ∈ Γ}. Let us set
Γa = {x ∈ Γ | G(¯
z )(x) = g (x, y¯(x)) + u
¯(x) = a(x)},
Γb = {x ∈ Γ | G(¯
z )(x) = g (x, y¯(x)) + u
¯(x) = b(x)}.
Definition 2.2.6. A pair z = (y, u) ∈ W 1,r (Ω) × Lq (Γ) is said to be a critical
direction for problem (BP ) at z¯ = (¯
y, u
¯) if the following conditions hold:
(i) ∇F
z )z = Ω (Ly [x]y (x)dx + Γ ( y [x]y (x) + u [x]u(x)) dσ ≤ 0;
 (¯
− N D (a (·)D y ) + a (·)y + h [·]y = 0
in Ω,
i
0
y
i,j=1 j ij

(ii)
∂ν y + b0 y = u
on Γ;

≥ 0 a.e. x ∈ Γ ,
a
(iii) gy [x]y (x) + u(x)
≤ 0 a.e. x ∈ Γb .

We shall denote by Cq (¯
z ) the set of such critical directions.
Theorem 2.2.7. Suppose that assumptions (A2.1)-(A2.5) are fulfilled and z¯ is a
local optimal solution of problem (BP ). There exists a unique couple (φ, ψ ) ∈
W 1,r (Ω) × Lq (Γ) with r ∈ (1, NN−1 ) such that the following hold:
(i) The adjoint equation:

A∗ φ + h [·]φ = −L [·]
in Ω,
y
y
(2.21)
∂ν ∗ φ + b0 φ = − y [·] − gy [·]∗ ψ on Γ,
A

16


where A∗ is the formal adjoint operator to A, and
N

∂νA∗ φ =

aij (x)Dj φ(x)νi (x);
i,j=1

(ii) The stationary conditions in u:
∇u L(¯
z , ψ, φ) =

u [·]

−φ+ψ =0

on Γ;

(iii) The complement condition with ψ :




≤ 0 a.e. x ∈ Γa ,
ψ (x) ≥ 0 a.e. x ∈ Γb ,



= 0 otherwise;

(2.22)

(2.23)

(iv) The second-order nonnegativity condition:
∇2zz L(¯
z , ψ, φ)(y, u)2 ≥ 0

∀z = (y, u) ∈ Cq (¯
z ),

where
∇2zz L(¯
z , ψ, φ)(y, u)2 =

Lyy [x]y (x)2 + φhyy [x]y (x)2 dx
Ω

2
yy [x]y (x)

+

+2

yu [x]y (x)u(x)

+

2
uu [x]u(x)

+ ψ (x)gyy [x]y (x)2 dσ.

Γ

2.3

Second-order sufficient optimality conditions

We consider (BP ) for the case p = 2 and the objective function has the form
F (y, u) :=

[ϕ(x, y (x)) + α(x)u(x) + β (x)u2 (x)]dσ,

L(x, y (x))dx +
Ω

(2.30)

Γ

where ϕ : Γ × R → R is a Carath´eodory function and α, β ∈ L∞ (Γ). It is noted that
by p > N − 1, we have N = 2. In addition, we need the following assumption.
(A2.2) The function ϕ satisfies assumption (A2.1) with ϕ substituted for L and Γ
substituted for Ω. Moreover, there exists γ > 0 such that β (x) ≥ γ for a.e. x ∈ Γ.
Definition 2.3.1. The function F is said to satisfy the quadratic growth condition
at z¯ ∈ Φ if there exist > 0, δ > 0 such that
F (z ) ≥ F (¯
z ) + δ z − z¯

for all z ∈ Φ2 satisfying z − z¯

Z

≤ .

17

2
Z


Theorem 2.3.2. Suppose that assumptions (A2.1), (A2.2) , (A2.3), (A2.4) are fulfilled, N = 2, z¯ ∈ Φ2 , there exist multipliers φ ∈ W 1,r (Ω), ψ ∈ L2 (Γ), r ∈ (1, 2)
satisfying conditions (2.21)– (2.23) of Theorem 2.2.7, and
∇2zz L(¯
z , ψ, φ)(d, d) > 0,

∀d ∈ C2 (¯
z ) \ {0}.

(2.31)

Then F satisfies the quadratic growth condition at z¯. In particular, z¯ is a locally
optimal solution of problem (BP ).
From Theorems 2.2.7 and 2.3.2, we obtain no-gap conditions in this case.
Remark 2.3.4. The above result holds if the objective function is of the form
F (y, u) =

[ϕ(x, y (x)) + α(x)u(x) + β (x)u2 (x) + α0 (x)y (x)u(x)]dσ,

L(x, y (x))dx +
Ω

Γ

where α0 ∈ L∞ (Γ).
Example 2.3.5 illustrates how to use necessary and sufficient optimality conditions
to find extremal points.
In the rest of this section, we shall derive second-order sufficient optimality conditions for problem (BP ) in the case, where F (y, u) is given by (2.30), where α(x)
and β (x) may be zero.
(B 2.1) Function L : Ω × R → R is a Carath´eodory function of class C 2 with respect
to the second variable, L(x, 0) ∈ L1 (Ω) and for each M > 0 there are a constant
CL,M > 0 and a function LM ∈ L2 (Ω) such that
|Ly (x, y )| ≤ LM (x), |Lyy (x, y )| ≤ CL,M , for a.e. x ∈ Ω, ∀y ∈ R, |y| ≤ M

and for each > 0, there exists δ > 0 such that
|Lyy (x, y1 ) − Lyy (x, y2 )| <

a.e. x ∈ Ω, ∀y1 , y2 ∈ R with |y1 |, |y2 | ≤ M, |y1 − y2 | < δ.

(B 2.2) The function ϕ satisfies assumption (B 2.1), where ϕ and Γ are replaced by
L and Ω, respectively.
(B 2.3) Function h : R → R is of class C 2 satisfying
h(x, 0) = 0,

hy (x, y ) ≥ 0

for a.e. x ∈ Ω, ∀y ∈ R

and for every M > 0 there is a constant Ch,M > 0 such that
|hy (x, y )| + |hyy (x, y )| ≤ Ch,M

for a.e. x ∈ Ω, ∀y ∈ R, |y| ≤ M.

Moreover, for every M > 0 and > 0, there exists a positive number δ > 0 such that
|hyy (x, y1 ) − hyy (x, y2 )| <

18

a.e. x ∈ Ω,


for all y1 , y2 ∈ R with |y1 |, |y2 | ≤ M, |y1 − y2 | < δ.
(B 2.4) Function g : Γ × R → R is a continuous function of class C 2 with respect to
the second variable, g (·, 0) ∈ L2 (Γ), and for each M > 0 there is a constant Cg,M > 0
such that
|gy (x, y )| + |gyy (x, y )| ≤ Cg,M for a.e. x ∈ Γ and ∀y ∈ R with |y| ≤ M.

Moreover, for every M > 0 and > 0, there exists a positive number δ > 0 such that
|gyy (x, y1 ) − gyy (x, y2 )| <

for a.e. x ∈ Γ and for all y1 , y2 ∈ R with |y1 |, |y2 | ≤ M, |y1 − y2 | < δ.
(B 2.5) α, β ∈ L∞ (Γ) and β (x) ≥ 0 for a.e. x ∈ Γ. In addition, the following is
verified
βa ∈ L∞ (Γa ), βb ∈ L∞ (Γb ).
In what follows, we define · ∗ := · L2 (Ω) + · L2 (Γ) .
Associated with z¯, for each τ ≥ 0 we define the following critical cone by
C2τ (¯
z ) = z = (y, u) ∈ Z1 | z satisfies (C 2.1) − (C 2.3)

(C 2.1)
(C 2.2)

(C 2.3)

Ly [x]y (x)dx +

(ϕy [x]y (x) + αu(x) + 2β u¯u(x)) dσ ≤ τ y ∗ ,

Γ
Ω
N
−
i,j=1 Dj (aij (·)Di y ) + a0 (·)y + hy [·]y = 0
∂ν y + b0 y = u

≥ 0 if x ∈ Γ ,
a
gy [x]y (x) + u(x)
≤ 0 if x ∈ Γb .

in Ω,
on Γ,

Obviously, C2 (¯
z ) ⊂ C2τ (¯
z ) for all τ ≥ 0.
Theorem 2.3.6. Suppose that N = 2, z¯ ∈ Φ∗ and assumption (A2.5) and assumptions (B 2.1)–(B 2.5) are fulfilled and that there exist multipliers ψ ∈ L2 (Γ) and
φ ∈ W 1,r (Ω), s ∈ (1, 2) satisfying conditions (2.21)–(2.23), and positive constants
γ, τ > 0 such that
∇zz L(¯
z , ψ, φ)(z, z ) ≥ γ y

2
∗

∀z = (y, u) ∈ C2τ (¯
z ).

Then there are constants ρ, ε > 0 such that
F (z ) ≥ F (¯
z ) + ε y − y¯

2
∗

∀z = (y, u) ∈ BZ1 (¯
z , ρ) ∩ Φ∗ ,

where Φ∗ := {(y, u) ∈ Z1 | (y, u) satisfies (2.2) and (2.3)}.
19


Chapter 3

Upper semicontinuity and continuity of the solution
map to a parametric boundary control problem
Let Ω be a bounded domain in R2 with the boundary Γ of class C 1,1 . We consider
the following parametric elliptic optimal control problem (P ). Determine a control
¯ which
function u ∈ L2 (Γ) and a corresponding state function y ∈ H 1 (Ω) ∩ C (Ω),
minimize the cost function

L(x, y (x), µ(1) (x))dx +

F (y, u, µ) =
Ω

(x, y (x), u(x), µ(2) (x))dσ,

(3.1)

Γ

subject to

Ay + f (x, y ) = 0
∂ν y = u + λ(1)

in Ω,

(3.2)

on Γ,

a(x) ≤ g (x, y ) + u(x) + λ(2) ≤ b(x) a.e. x ∈ Γ,

(3.3)

where L : Ω × R × R → R, l : Γ × R × R × R → R, f : Ω × R → R and g : Γ × R → R
are functions, a, b ∈ L2 (Γ), a(x) < b(x) for a.e. x ∈ Γ, (µ, λ) ∈ (L∞ (Ω) × L∞ (Γ)) ×
(L2 (Γ))2 is a vector of parameters with µ = (µ(1) , µ(2) ) and λ = (λ(1) , λ(2) ). The
second-order elliptic operator A is defined as in Chapter 2 with N = 2. Let us put

¯ , U := L2 (Γ), Π := L∞ (Ω) × L∞ (Γ), Λ := (L2 (Γ))2 .
Y := H 1 (Ω) ∩ C (Ω)
For each λ ∈ Λ, the admissible set Φ(λ) := {(y, u) ∈ Y ×U |(3.2) and (3.3) are satisfied}.
Then problem (3.1)-(3.3) can be written in the form

F (y, u, µ) → inf,
(3.4)
P (µ, λ)
(y, u) ∈ Φ(λ).
Let us denote by S (µ, λ) the solution set of (3.1)-(3.3) or P (µ, λ) corresponding
¯ ) the reference point and by P (¯
¯ ) the unperturbed problem.
to (µ, λ), and by (¯
µ, λ
µ, λ

3.1

Assumptions and main result

¯ ) ∈ Π × Λ and
Fix (¯
µ, λ
given function h.

0

> 0. Notation hz stands for the derivative w.r.t. z of a

(A3.1) L : Ω ×R×R → R and l : Γ ×R×R×R → R are Carath´eodory functions such

that y → L(x, y, µ(1) ) and (y, u) → (x , y, u, µ(2) ) are Fr´echet continuous differential
20


functions for a.e. x ∈ Ω and x ∈ Γ, respectively and for all µ(1) , µ(2) ∈ R with
|µ(1) − µ
¯(1) (x)| + |µ(2) − µ
¯(2) (x )| ≤ 0 . Furthermore, for each M > 0 there exist CLM ,
ClM > 0 and r1M ∈ L1 (Ω), r2M ∈ L1 (Γ), r3M ∈ L∞ (Ω), r4M ∈ L∞ (Γ) such that
|L(x, y, µ(1) )| ≤ r1M (x),
|Ly (x, y, µ(1) )| ≤ r3M (x),

| (x , y, u, µ(2) )| ≤ r2M (x ) + ClM (1 + |u|2 ),
|Ly (x, y1 , µ(1) ) − Ly (x, y2 , µ(1) )| ≤ CLM |y1 − y2 |,

| y (x , y, u, µ(2) )|+| u (x , y, u, µ(2) )| ≤ ClM (|y| + |u|) + r4M (x ),
| y (x , y1 , u1 , µ(2) )− y (x , y2 , u2 , µ(2) )| ≤ ClM (|y1 − y2 | + δ|u1 − u2 |),
| u (x , y1 , u1 , µ(2) )− u (x , y2 , u2 , µ(2) )| ≤ ClM |y1 − y2 | + |u1 − u2 |

for some δ ≥ 0, a.e. x ∈ Ω, x ∈ Γ, for all µ(1) , µ(2) , y, ui , yi ∈ R satisfying
|µ(1) − µ
¯(1) (x)| + |µ(2) − µ
¯(2) (x )| ≤ 0 and |y|, |yi | ≤ M , i = 1, 2.
(A3.2) The function u → (x , y, u, µ(2) ) is convex for all (x , y, µ(2) ) ∈ Γ × R2 and
|µ(2) − µ
¯(2) (x )| ≤ 0 . Moreover, for each M > 0 there exist functions aM ∈ L2 (Γ)
and bM ∈ L1 (Γ) satisfying
(x , y, u, µ(2) ) ≥ aM (x )u + bM (x )
for a.e. x ∈ Γ and for all y, u, µ(2) ∈ R with |y| ≤ M and |µ(2) − µ
¯(2) (x )| ≤ 0 .

(A3.3) There exist a continuous function η : Γ × R3 → R and constants 0 ≤ θ ≤ 1,
α, β > 0 such that
| u (x , y, u, µ(2) ) −
u (x

u (x

, y, u, µ
¯(2) (x )| ≤ η (x , |y|, |µ(2) |, |µ
¯(2) (x )|)|u|θ |µ(2) − µ
¯(2) (x )|α ,

, y¯(x ), u, µ
¯(2) (x )) −

u (x

, y¯(x ), u
¯, µ
¯(2) (x )), u − u¯(x ) ≥ β|u − u¯(x )|2

for a.e. x ∈ Γ, for all y, u, µ(2) ∈ R with |µ(2) − µ
¯(2) (x )| ≤ 0 .
(A3.4) f : Ω × R → R and g : Γ × R → R are Carath´eodory functions of class C 1
w.r.t. the second variable and satisfy the following properties:
f (·, 0) = 0, fy (x, y ) ≥ 0

a.e. x ∈ Ω,

g (·, 0) = 0, gy (x , y ) ≥ 0

a.e. x ∈ Γ

and for each M > 0, there exist constants Cf M , CgM > 0 such that
fy (x, y ) ≤ Cf M ,

fy (x, y1 ) − fy (x, y2 ) ≤ Cf M |y1 − y2 |,

gy (x , y ) ≤ CgM ,

gy (x , y1 ) − gy (x y2 ) ≤ CgM |y1 − y2 |,

for a.e. x ∈ Ω, x ∈ Γ and for all y, y1 , y2 ∈ R with |y|, |y1 |, |y2 | ≤ M.
We are now ready to state our main result of this chapter.
Theorem 3.1.1. Suppose that assumptions (A3.1)–(A3.4) are fulfilled. Then the
following assertions are valid:

21


(i) S (µ, λ) = ∅ for all (µ, λ) ∈ Π × Λ;
¯ );
(ii) S : Π × Λ → Y × U is upper semicontinuous at (¯
µ, λ
¯ ) is singleton then S (·, ·) is continuous at (¯
¯ ).
(iii) if, in addition, S (¯
µ, λ
µ, λ

3.2

Some auxiliary results

3.2.1

Some properties of the admissible set

Lemma 3.2.2. Under assumption (A3.4), the admissible set Φ(λ) is a nonempty and
closed set for any λ ∈ Λ.
Lemma 3.2.3. Suppose that assumption (A3.4) is satisfied and {λn } is a sequence
ˆ strongly in Λ. Then the following assertions are valid:
converging to λ
ˆ ), there is a sequence {(yn , un )}, (yn , un ) ∈ Φ(λn ) which con(i) For any (ˆ
y, u
ˆ) ∈ Φ(λ
verges to (ˆ
y, u
ˆ) strongly in Z.
(ii) For any sequence {(yn , un )}, (yn , un ) ∈ Φ(λn ), there exist a subsequence {(ynk , unk )}
ˆ ) such that
and (ˆ
y, u
ˆ) ∈ Φ(λ
ynk → yˆ and

3.2.2

unk

u
ˆ.

First-order necessary optimality conditions

Lemma 3.2.5. Suppose that (¯
y, u
¯) is a local optimal solution of problem (3.1)–(3.3).
Then there exists a unique element φ ∈ H 1 (Ω) such that the following conditions are
fulfilled:
(i) The adjoint equation:

A∗ φ + f (·, y¯)φ = L (·, y¯, µ(1) )
in Ω,
y
y
∂n ∗ φ + gy (·, y¯)φ = y (·, y¯, u
¯, µ(2) ) − u (·, y¯, u¯, µ(2) )gy (·, y¯) on Γ,
A

where A∗ is the formal adjoint operator to A, that is,
2
∗

A y (x) = −

Di (aij (x)Dj y (x)) + a0 (x)y (x);
i,j=1

(ii) The weak minimum principle:

(φ(x)+ u (x, y¯(x), u¯(x), µ(2) (x)))(g (x, y¯(x)) + u¯(x) + λ(2) (x))
=

min
v∈[a(x),b(x)]

(φ(x) +

¯(x), u¯(x), µ(2) (x)))v
u (x, y
22

a.e. x ∈ Γ.

(3.14)


Let us define
K := {v ∈ L2 (Γ)| a(x) ≤ v (x) ≤ b(x)

a.e. x ∈ Γ}.

(3.15)

Then from (3.14), we get
(φ(x) +

¯(x), u¯(x), µ(2) (x)))(v (x)
u (x, y

− g (x, y¯) − u
¯(x)) − λ(2) (x))dσ ≥ 0,

Γ

¯ that
for all v ∈ K . Furthermore, it follows from assumption (A3.1) and y¯ ∈ C (Ω)
Ly (·, y¯, µ(1) ) ∈ L∞ (Ω), y (·, y¯, u
¯, µ(2) ) − u (·, y¯, u¯, µ(2) )gy (·, y¯) ∈ L2 (Γ). Therefore, we
¯
get φ ∈ H 1 (Ω) ∩ C (Ω).

3.3

Proof of the main result

(i) The non-emptiness of S (µ, λ)
(ii) Upper semicontinuity of S (·, ·)
¯ ).
We argue by contradiction. Assume that S (·, ·) is not upper semicontinuous at (¯
µ, λ
Then there exist open sets W1 in Y , W2 in U and sequences {(µn , λn )} ⊂ Π × Λ,
{(yn , un )} ⊂ Y × U such that


¯ ) ⊂ W1 × W2 ,

µ, λ

 S (¯

¯ ),
(3.16)
(µn , λn ) → (¯
µ, λ
µn − µ Π ≤ 0 ,



 (yn , un ) ∈ S (µn , λn ) \ (W1 × W2 ), ∀n ≥ 1.

By Lemma 3.2.3, we can assume after choosing a subsequence that
yn → y¯ in Y

and un

u
¯ in U

¯ ). If we can show that (¯
¯ ) and un → u¯ strongly
for some (¯
y, u
¯) ∈ Φ(λ
y, u
¯) ∈ S (¯
µ, λ
in U as n → +∞ then (yn , un ) ∈ W1 × W2 for n large enough. This contradicts to
(3.16) and the proof is complete.
(iii) The continuity of S (·, ·)

3.4

Examples

In this section, we give some examples illustrating Theorem 3.1.1. One shows that
¯ ) is singleton and the solution map S (·, ·) is continuous at (¯
¯ ). Other says
S (¯
µ, λ
µ, λ
that although the unperturbed problem has a unique solution, the perturbed problems may have several solutions and solution map is continuous at a reference point.
23