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 elds of science. It has been systematically studied and strongly
developed since the late 1950s, when two basic principles were made: the
Pontryagin Maximum Prin-ciple and the Bellman Dynamic Programming Principle.
Up to now, optimal control theory has developed in many various research
directions such as non-smooth opti-mal control, discrete optimal control, optimal
control governed by ordinary di er-ential equations (ODEs), optimal control
governed by partial di erential 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 su cient 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 nding
out stationary points but also help us in constructing su cient optimality conditions. Let us brie y review some results on this topic.
For distributed control problems, i.e., the control only acts in the domain in
n
R , E. Casas, T. Bayen et al. derived second-order necessary and su cient
optimality conditions for problem with pure control constraint, that is,
a(x) u(x) b(x) a.e. x 2 ;
(1)
and the appearance of state constraints. In particular, E. Casas established
second-order su cient optimality conditions for Dirichlet control problems and
Neumann control problems with only constraint (1) when the objective function
does not contain control variable u. In addition, C. Meyer and F. Tr•oltzsch
derived second-order su cient optimality conditions for Robin problems with
mixed constraint of the form a(x) y(x) + u(x) b(x) a.e. x 2 and nitely many
equalities and inequalities constraints, where y is the state variable.
For boundary control problems, i.e., the control u only acts on the boundary ,
E. Casas et al. and F. Tr•oltzsch derived second-order necessary and su cient
optimality conditions with pure pointwise constraints, i.e.,
a(x) u(x) b(x) a.e. x 2
1
:
A. R•osch and F. Tr•oltzsch gave the second-order su cient optimality
conditions for the problem with the mixed pointwise constraints which has
unilateral linear form c(x) u(x) + (x)y(x) for a.e. x 2 .
1
1
We emphasize that in above results, a; b 2 L ( ) or a; b 2 L ( ). Therefore, the
1
1
control u belongs to L ( ) or L ( ). This implies that corresponding Lagrange
multipliers are measures rather than functions. 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 2
p
with a; b 2 L ( ) 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;
p
y(x)) + u(x) b(x) a.e. x 2 with a; b 2 L ( ).
(OP 2) : Give second-order su cient 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 rst goal of the dissertation.
After second-order necessary and su cient optimality conditions are established,
they should be compared to each other. According to J. F. Bonnans, if the change
between necessary and su cient second-order optimality conditions is only between
strict and non-strict inequalities, then we say that the no-gap optimality conditions are
obtained. Deriving second-order optimality conditions without a gap between secondorder necessary optimality conditions and su cient optimality conditions, is a di cult
problem. In some papers, J. F. Bonnans derived second-order necessary and su
cient optimality conditions with no-gap for an optimal control problem with pure
control constraint and the objective function is quadratic in both state variable y and
control variable u. 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 nding 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 optmal problem:
P(; )
8
2 .
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) 2 K;
u 2U
where K is a nonempty closed and convex set in E, G : U ! E and f : U ! R are
1
second-order Frechet di erentiable on U. Put ad := G (K).
De nition 1.1.1. A function u 2 ad is said to be a locally optimal solution of problem
(P ) if there exists " > 0 such that
f(u)
f(u)
8u 2 BU (u; ) \ ad:
Given a point u 2 ad, problem (P ) is said to satisfy Robinson's constraint
quali cation at u if there exists > 0 such that
BE(0; ) rG(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) + he ; G(u)i with e 2 E :
We shall denoted by (u) the set of multipliers e 2 E such that
ruL(u; e ) = rf(u) + rG(u) e = 0; e 2 N(K; G(u)):
The set (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) := fd 2 Ujhrf(u); di 0; rG(u)d 2 T (K; G(u))g:
The set K is said to be polyhedric at z 2 K if for any v 2 N(K; z), one has
[
?
T (K; z) \ (v ) = cl[cone(K
?
z) \ (v ) ];
?
where (v ) = fv 2 E j hv ; vi = 0g: Moreover, problem (P ) is said to satisfy the
strongly extended polyhedricity condition at u 2 if the set C0(u) is dense in C(u),
where
C0(u) := fd 2 C(u) j rG(u)d 2 cone(K
G(u))g:
1
Lemma 1.1.3. Suppose that u is regular, at which the strongly extended polyhedricity condition is ful lled. If u is a locally optimal solution, then for each d 2
C(u), there exists a multiplier e 2 (u) such that
2
2
2
r uuL(u; e )(d; d) = r f(u)(d; d) + he ; r G(u)(d; d)i
1.1.2
0:
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.
De nition 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 ky
ykW 2;p( ) + ku ukLp( ) , one has
F (y; u)
1
F (y; u):
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).
2
(A1:1) L : R R ! R is a Caratheodory function of class C with respect to variable
1
(y; u), L(x; 0; 0) 2 L ( ) and for each M > 0, there exist a positive number kLM and
1
a function rM 2 L ( ) such that
jLy(x; y; u)j + jLu(x; y; u)j kLM
j
L (x; y ; u )
y
1
jyj + juj
L (x; y ; u )
1
y
2
2
k
j
LM
(y
1
p 1
y
j
j
2
+ u1
u
j
);
j
2
p 1 j
X
jLu(x; y; u1) Lu(x; y; u2)j kLM
+ rM (x);
ju1 u2j
j
ju2j
j=0;p 1 j>0
for all y; y1; y2 2 R satisfying jyj; jyij M and any u1; u2 2 R. Also for each M > 0,
there is a number kLM > 0 such that
j
j
+ j" u1 uj2 p 1)
Lyu(x; y1; u1) Lyu(x; y2; u2) kLM ( y1
y2
y j +u
u );
Lyy(x; y1; u1) Lyy(x; y2; u2)
kLM ( y1
2
1
2
2 and " = 1 if p > 2 ) and
( with " = 0 if 1 < p
j
j Luu(x; y; u1)
Luu(x; y; u2) j
8
y; u ; y
i
2
u
j 1
:
i
2
j
p 2 j
=0
2
2j
y ; y ij
R with j
ju2j
M and i
jj
;:
=12
2
(A1:2) The function g is a continuous function and of class C w.r.t. the second
variable, and satis es the following property: g( ; 0) 2 Lp( ) and for each M > 0,
there exists a constant Cg;M > 0 such that
y
gy(x; y1) gy(x; y2) + g yy(x; y1) g yy(x; y2)Cg;M y2
1
C
;
j
gy(x; y) + gyy(x; y)
g;M
j
for a.e. x 2 and jyj; jy1j; jy2j
(A1:3) =6 0 and
M.
h [x] + g [x]
y
y
0 a.e. x 2
:
p
For each u 2 L ( ), equation (1.2) has a unique solution yu 2 W
and there exists a constant C > 0 such that
2;p
kyukW 2;p( ) CkukLp( ):
De ne a mapping H : W
2;p
( ) \ W0
1;p
p
( )
H(y; u) =
p
L ( ) ! L ( ) by setting
y + h( ; y) u:
7
( ) \ W0
1;p
()
Then, H is of class C2 around (y; u) and its derivatives at (y; u) are given by
2
rH(y; u) = ( + hy[ ]; I); r H(y; u) =
0 0! :
h [] 0
yy
1
Since p > N=2, y = y(u) 2 C( ). Hence hy[ ] 2 L ( ). Therefore, for each
p
u 2 L ( ), the following equation has a unique solution z 2 W
2;p
()\W
1;p
( ). 0
Hence the operator A := ryH(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 : U0 ! Y0 such that H( (u); u) = 0 for all u 2 U0. Moreover, is of class
2
C and its derivatives are given by the following formulae.
Lemma 1.1.9. Assume that
: U0 ! Y0 is the control-state mapping de ned by
2
p
0
(u) = yu. Then is of class C and for each u 2 U0; v 2 L ( ), zu;v := (u)v is the
unique solution of the linearized equation
8z + h ( ; y )z
u;v
0 such that
f(u)
f(u)
8u 2 BU (u; ) \ p:
8
Problem (1.13){(1.14) is associated with the Lagrangian:
Z
Z
L(u; e ) = f(u)+he ; G(u)i =
L(x; (u); u)dx+
e (g( ; (u))+ u)dx;
p
e 2 L 0( )
0
where p is the adjoint number of p.
Given u 2 p, the critical cone of problem (1.13){(1.14) is de ned by
C
2 Lp( )
0; r G(u)d(x) 80 if x 2
(Ly[x]zu;d + Lu[x]d)dx
p(u) = d
jZ
<0 if x 2
a
o
a:e: ;
b
:
where a := fx 2
j G(u)(x) = a(x)g;
j G(u)(x) = b(x)g:
b := fx 2
Theorem 1.1.15. Suppose that assumptions (A1:1){(A1:3) are satis ed and u is a
p0
locally optimal solution
of
(1.13){(1.14).
There
exist
a
unique
e
2
L
( ) and a
0
0
2;p
1;p
2W
unique
( ) \ W0 ( ) such that the following conditions are valid:
(i) The adjoint equation:
8+ hy[ ] = Ly[ ] + e gy[ ] in ;
(1.17)
<= 0 on ;
:
(ii) The stationary conditions in u:
u; e ) =
+ L u[ ] + e
= 0; e
ruL(
(iii) The non-negative second-order condition:
2
ruu L(u; e )(v; v) =
Z
2 NK; G u
(
( ));
2
2
Lyy[x]zu;v + 2Lyu[x]zu;vv + Luu[x]v + e gyy[x]zu;v
hyy[x]zu;v
2
(1.18)
2
dx 0 8v 2 Cp(u):
Example 1.1.16 illustrates how to use necessary conditions to nd stationary
points. In this example, the point (0; 0) satis es rst-order necessary optimality
conditions but it does not satisfy second-order necessary optimality conditions.
1.2
Second-order su cient optimality conditions
To derive second-order su cient optimality conditions for elliptic optimal control
problems we usually use two di erent 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 f2; 3g
and the objective function has the form
2
L(x; y; u) = '(x; y) + (x)u + (x)u ;
9
(1.23)
1
where ' :
R ! R is a Caratheodory function and ; 2 L ( ).
In the sequel, we need the following assumptions.
(A1:1)0 Function ' : R ! R is a Caratheodory function of class C 2 with respect to the
second variable, '(x; 0) 2 L1( ) and for each M > 0 there are a constant K ';M > 0 and
a function 'M 2 L2( ) such that
2
K ;
@'
'M (x);
@ ' (x; y) ';M
@2 '
2 '
@'
@y
for a.e. x
@y
and for
@y 2
(x; y)
@'
+
(x; y 1)
@y (x; y 2)
@
(
@y2
x; y
1
K';M jy1 y2j
)
@y
with
all y1 ; y2; y
2
2
y ; y1
2R
M:
(x; y 2)
; y2
j jj
jj
j
2
for a.e. x
0
(A1:2) There exists a number > 0 such that (x)
.
2
De nition 1.2.1. We say that f satis es L weak quadratic growth condition at u 2 2
if there exist > 0; > 0 such that
f(u)
for all u 2
f(u) + ku
2 satisfying ku uk2
uk
2
2
:
0
0
Theorem 1.2.2. Suppose that assumptions (A1:2), (A1:1) and (A1:2) are satis ed.
Let
2
that
and multipliers
2
2
( ),
2
2;2
( ) \
1;2
( ) satisfy conditions
u 2 e L W W0 (1.17) and (1.18). Furthermore, suppose
2
u; e u; u ) > 0 8u
u
:
ruuL( )(
2 C2( ) n f0g
2
Then f satis es L weak quadratic growth condition at u 2 2. In particular, u is a
2
locally optimal solution of problem (1.13){(1.14) in L ( ).
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 su cient optimality conditions for the problem (DP ) for the case where L is given by (1.23) with (x) and
(x) may be zero. For this we need the following assumptions.
2
(B1:1) Function h : R ! R is of class C satisfying
h(x; 0) = 0;
hy(x; y)
0; 8y 2 R and a.e. x 2
and for every M > 0 there is a constant Ch;M > 0 such that
@h
(x; y) + 2 (x; y) Ch;M ; 8y 2 R with jyj M and a.e. x 2 :
@y
@ h
@y 2
10
C
';M
Moreover, for every M > 0 and > 0; there exists a positive number > 0 such that
2
2 (x; y ) < a.e. x 2 8y ; y 2 R with jy j; jy j M; jy y j < :
2
1 2
1
2
1 2
@ h
@y 2(x; y1)
@ h
@y 2
2
(B1:2) Function ' : R ! R is a Caratheodory function of class C with respect to the
1
second variable, '(x; 0) 2 L ( ) and for each M > 0 there are a constant
> 0 and a function 'M 2 L2( ) such that
2
C
@'@y (x; y) 'M (x);
@ ' (x; y) ';M
a.e. x 2 ; 8y 2 R with jyj M
@y 2
and for each > 0, there exists > 0 such that
2
2
@ ' (x; y1) @ ' (x; y2) <
a.e. x 2 ; 8y1; y2 2 R; jy1j; jy2j M; jy1 y2j < :
@y 2
@y 2
2
(B1:3) Function g : R ! R is a continuous function of class C with respect to the
2
second variable, g( ; 0) 2 L ( ) and for each M > 0 there are a constant Cg;M > 0
2
and a function gM 2 L ( ) such that
(x; y) Cg;M
2
@y
@g
@g
2
a.e. x 2 ; 8y 2 R with jyj M:
(x; y) gM (x);
@y
Moreover, for every M > 0 and > 0; there exists a positive number > 0 such that
2
2 (x; y )
< a.e. x 2 ; 8y1; y2 2 R with jy1j; jy2j M; jy1 y2j < :
2
@g
@g
2
@y 2 (x; y1)@y
;
(B1:4)
2
0 for a.e. x 2
(x)
L1( ) and
1
: Besides, the following is veri ed
1
a 2 L ( a);
(1.26)
b 2 L ( b):
Note that condition (1.26) holds whenever one of the following conditions is veri ed:
1
(i) a; b 2 L ( );
1
(ii) u 2 L ( );
(iii)
= 0:
Based on Casas, we enlarge C2(u) by de ning the following critical cone.
C
2 L2( ) j hr uf(u); v i
k z k 2; gy[x]zu;v(x)+ v(x) 8
if x 2 a;
2 (u) = v
u;v
0
n
Obviously, C 2(u) =
<
C
2
0
(u) and
C
2 (u )
2
(u) for all > 0:
C
Theorem 1.2.4. Suppose that assumptions (A1:3) and (B1:1)
there exist multipliers
2 2( ) and
2
2;2
( )\
e L
W
and (1.18). If there exist positive constants ;
1;2
( ) satisfy conditions (1.17)
W0
> 0 such that
0
if x 2 b:
:
(B1:4) are ful lled,
:
o
2
u; e )(v; v )
kz u;v
ruuL(
then there are constants ; r > 0 such that
f(u) f(u) + rkzu;u
uk2
2
k2
2
11
8v
u ;
2 C2 ( )
8u 2 BL2( )(u; ) \ 2:
Chapter 2
No-gap optimality conditions for boundary control
problems
N
1;1
Let be a bounded domain in R with the boundary of class C
and N 2. We
q
consider the problem of nding a control function u 2 L ( ) and a corresponding
1;r
state function y 2 W ( ) which
minimize F (y; u) = Z
(BP )
s.t.
L(x; y(x))dx +
8 Ay + h(x; y) = 0
in ;
< @ y + b0 y = u
on ;
Z
`(x; y(x); u(x))d ;
(2.1)
(2.2)
(2.3)
:
a (x) g(x; y(x)) + u(x) b(x) a:e: x 2 ;
where L; h : R ! R and ` : R R ! R are Caratheodory functions, g : R ! R is
q
1
continuous, a; b 2 L ( ), a(x) < b(x) for a.e. x 2 , b0 2 L ( ), b0 0, A denotes a
second-order elliptic operator of the form
X
N
Ay(x) =
Dj(aij(x)Diy(x)) + a0(x)y(x);
i;j=1
2
2
0;1
1
coe cients aij C ( ) satisfy aij(x) = aji(x), a0 L ( ); a0(x) 0 for a.e. x 2 , a0 6 0 and
there exists m > 0 such that
X
mk k
2
N
aij i j 8 2 R
N
i;j=1
for a.e.
x2
and @ denote the conormal-derivative associated with A. Moreover, we assume that
N >r >q
:
1 N
1
2.1
1
1
1
(2.4)
Abstract optimal control problems
Let Y; U; V and E be either separable or re exive Banach spaces with the dual spaces
Y ; U ; V and E ; respectively. We consider the following problem:
Min F (y; u);
s.t. H(y; u) = 0;
(2.5)
(2.6)
G(y; u) 2 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 de ne the space Z := Y U and the set Q := fz = (y; u) 2 Z j H(y; u) = 0g: De
1
ne ad := Q \ G (K): An couple (y; u) 2 ad is said to be a locally optimal solution of
problem (2.5){(2.7) if there exists > 0 such that for all (y; u) 2 ad satisfying ky ykY
+ ku ukU , one has F (y; u) F (y; u):
For a given point z = (y; u) 2 ad, we need the following assumptions:
2
(H2:1) The mappings F; H; G are of class C around z.
(H2:2) ryH(z) : Y ! V is bijective.
(H2:3) The regularity condition is veri ed at z, i.e., there is a number > 0 satsfying
\
z2BZ ( \
0 2 int
z; ) Q
(2.8)
rG(z)(T (Q; z) \ BZ ) (K G(z)) \ BE :
[
(H2:4) rG(z)(T (Q; z)) = E:
De nition 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 satis ed:
[
(i) Fy(z)y+Fu(z)u
0;
(ii) Hy(z)y+Hu(z)u = 0;
(iii) rG(z)z 2 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) + hv ; H(z)i + he ; G(z)i ;
(2.9)
where z = (y; u) 2 Z; e 2 E ; v 2 V .
Let z be a locally optimal solution of problem (2.5){(2.7) and denote by ( z)
the set of Lagrange multipliers (e ; v ) 2 E V which satisfy
rzL(z; e ; v ) = 0; e 2 N(K; G(z)):
Lemma 2.1.4. Suppose that the assumptions (H2:1) (H2:3) are ful lled and z is a
locally optimal solution of (2.5){(2.7). Then (z) is nonempty and bounded. In
addition, if (H2:4) is ful lled then (z) is singleton.
When K is polyhedric at G(z), we have the following result.
Lemma 2.1.5. Suppose that the assumptions (H2:1){(H2:4) are ful lled and let z be
a locally optimal solution of problem (2.5){(2.7). Then, the set of critical directions
C(z) satis es
[
C(z) = fd 2 Z j rF (z)d = 0; rH(z)d = 0; rG(z)d 2 T (K; G(z))g:
In addition, if K is polyhedric at G(z) then C(z) = C0(z), where
?
1
C0(z) := (rF (z)) \ KerrH(z) \ rG(z) (cone(K
13
G(z))):
Theorem 2.1.7. Let z be a locally optimal solution of problem (2.5)-(2.7). Suppose
that assumptions (H2:1){(H2:4) are ful lled and K is polyhedric at G(z). Then there
exists (e ; v ) 2 (z) such that
2
2
2
2
2
2
2
r zzL(z; e ; v )(d; d) = r F (z)d + he ; r G(z)d i + hv ; r H(z)d i
0
for all d 2 C(z).
2.2
Second-order necessary optimality conditions
De nition 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 ky
ykW 1;r( ) + ku ukLq( ) , one has F (y; u) F (y; u).
Let us impose some assumptions for problem (BP ) which involve (y; u).
2
(A2:1) L : R ! R is a Caratheodory function of class C with respect to second
1
variable, L(x; 0) 2 L ( ) and for each M > 0, there exists a positive number kLM
such that
jLy(x; y)j + jLyy(x; y)j kLM ;
jLy(x; y1) Ly(x; y2)j + Lyy(x; y1) Lyy(x; y2) kLM jy1 y2j
M, i = 1; 2:
for a.e. x 2 , for all y; yi 2 R with j y ; yi j
jj
(A2:2) ` : R R ! R is a Caratheodory function of class C 2 with respect to variable (y;
u), `(x; 0; 0) 2 L1( ) and for each M > 0, there exist a positive number k `M and a
function rM 2 L1( ) such that
q 1
j`y(x; y; u)j + j`u(x; y; u)j k`M
jyj + juj
+ rM (x);
1 1
y
2 2 j
j y
j
j
y j
j y1
` (x; y ; u )
` (x; y ; u )
2
+ u1 u2
X
q 1 j
j
ju1 u2j
ju2j
j 0; q 1 j>0
;
j` (x; y ; u ) ` (x; y ; u )j k
jy y j +
k
u
1 1
`yu(x; y1; u1)
u
`yy(x; y1; u1)
and `uu(x; y1; u1)
for a.e .
q
j
2
2 2
`yu(x; y2; u2)
`uu(x; y2; u2)
i
x
y; u ; y
2 and "q = 1 if q > 2.
);
(
`M
1
k`M (
`yy(x; y2; u2)
, for all
`M
2
y1
j
2 j
+
j
" q u1
k`M ( y1
y2 + u1 u2j);
k
y2 +"q
j
R
j
y
`M
i
j
y2
1
j
j
q 1);
u2
j
i
satisfying y ; y
j
P
j 0; q
,
j
2 j>0
= 1 2 and
M i
;
2
u
1
jj
u
q
2
q
"
= 0 if 1
2 j
u
j
2
j j
(A2:3) h :
R ! R is a Caratheodory function and of class C w.r.t. the second
variable, and satis es the following property:
h( ; 0) 2 L
N r=(N+r)
( ); hy(x; y) 0
14
a:e: x 2
<
and for each M > 0, there exists a constant Ch;M > 0 such that
C ;
hy(x; y) + hyy(x; y)
hyy(x; y1) hyy(x; y2)Ch;M jy2 y1j
h;M
for a.e. x 2 and jyj; jy1j; jy2j M.
2
(A2:4) g :
R ! R is a Caratheodory function and of class C w.r.t. the second
q
variable, g( ; 0) 2 L ( ) a:e: x 2 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 jy2 y1j
for a.e. x 2
and jyj; jy1j; jy2j M.
(A2.5) b0 + gy[x]
0 a:e: x 2 :
Let us de ne the mappings
H : Z ! V;
H(z) = H(y; u) := (Ay + h(x; y); @ y + b0y
G : Z ! E;
G(z) = G(y; u) := g(:; y) + u;
u);
q
and set K := fv 2 L ( ) : a(x) v(x) b(x) a:e: x 2 g. Then
problem (BP ) reduces to the following problem:
s.t.
Denote by
q := Q \ G
1
Min F (z)
H(z) = 0;
(2.14)
(2.15)
G(z) 2 K:
(2.16)
(K) the admissible set of problem (2.14)-(2.16), where
Q := fz = (y; u) 2 Z j H(z) = 0g:
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 (H2:1)-(H2:4) are satis ed.
Lemma 2.2.2. Suppose that assumptions (A2:1){(A2:4) are ful lled. Then F; H and
2
G are of class C .
Lemma 2.2.3. Under assumption (A2:3), ryH(^y; u^) is bijective for all (^y; u^) 2 Z.
Lemma 2.2.4. Suppose that assumptions (A2:3){(A2:5) are ful lled. Then, the following assertions are valid:
(i) (the regularity condition) for some constant > 0, one has
\
0 2 z2BZ ( \
(2.18)
[rG(z)(T (Q; z) \ BZ ) (K G(z)) \ BE] :
z; ) Q
[
q
(ii) rG(z)(T (Q; z)) = L ( ):
15
From Lemmas 2.2.3 and 2.2.4, we see that hypotheses ( H2:2)-(H2:4) are valid.
Let us introduce the Lagrangian associated with problem (BP ).
L(z; ; v ) =F (z) + v H(z) + G(z)
=
Z
L( ; y)dx +
Z
`( ; y; u)d +
Z
N
+ a0( )yv1 dx
ij ij1
a
i ;j =1
()D
yD
v
X
+
Z
Z
h( ; y)v1dx
@ yv1d +
where v = (v1; v2) 2 V = W
1;r
Z
(@ y + b0y u)v2d +
0
1 0
;r
q(
( ),
Z
)
(g( ; y) + u) d ;
( ) Wr
2L
=L
the fact (X Y ) = X Y . In case of v1 = ; v2 = T , we denote
L(z; ; ) := L(z; ; v ) =
+
Z
Z
L(x; y)dx +
Z
N
i ;j =1
a
Z
( )D y D + a ( )y d x +
ij
i j
0
`( ; y; u)d +
(b0y u)T d +
Z
Z
q
0
( ). Here, we use
h( ; y) dx
(g( ; y) + u) d ;
X
Let us consider the set-valued
K :R, de ned by K(x) = [a(x); b(x)] a.e.
q
map x in . Then K = fv 2 L ( ) j v(x) 2 K(x) a.e. x 2 g. Let us set
a
= fx 2
j G(z)(x) = g(x; y(x)) + u(x) = a(x)g;
b
= fx 2
j G(z)(x) = g(x; y(x)) + u(x) = b(x)g:
De nition 2.2.6.
A pair z = (y; u) 2 W
1;r
q
( ) L ( ) is said to be a critical
direction for problem (BP ) at z = (y; u) if the following conditions hold:
R
R
(i) rF (z)z = (Ly[x]y(x)dx + (`y[x]y(x) + `u[x]u(x)) d 0; 8
N
(ii)
i;j =1 Dj(aij( )Diy) + a0( )y + hy[ ]y = 0in ;
<@
P
0
by
+
: y
=
on ;
u 8
(iii) gy[x]y(x) + u(x)
<
We shall denote
by
0
a.e.
x 2 a;
0
a.e.
x 2 b:
(z) the set of such critical dir ections.
:Cq
Theorem 2.2.7. Suppose that assumptions (A2:1)-(A2.5) are ful lled and z is a
local optimal solution of problem (BP ). There exists a unique couple ( ; ) 2 W
q
0
1;r0
()
N
L 0( ) with r 2 (1; N 1) such that the following hold:
(i) The adjoint equation:
8A + hy[ ] = Ly[ ]
<@
:
A
+ b0 = `y[ ]
gy [ ]
in ;
on ;
(2.21)
16
- Xem thêm -
.DOC 
31 
59 
95 
