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


(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

(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

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


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(¯

∇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,

j=0,j 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.



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


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:

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


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.


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

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


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

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

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

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 )

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)


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


(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

∂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)


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¯



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



(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

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

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 .


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:

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(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


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:

− 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 (¯

3.4

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.
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