MINISTRY OF EDUCATION AND TRAININGHANOI UNIVERSITY OF SCIENCE AND TECHNOLOGYNGUYEN HAI SON NO-GAP OPTIMALITY CONDITIONS AND SOLUTION STABILITY FOR OPTIMAL CONTROL PROBLEMS GOVERNED BY SE
Trang 1MINISTRY OF EDUCATION AND TRAININGHANOI 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
Trang 2MINISTRY OF EDUCATION AND TRAININGHANOI 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
Trang 3COMMITTAL IN THE DISSERTATION
I assure that my scienti c results are new and righteous Before I publishedthese results, there had been no such results in any scienti c document I haveresponsibili-ties for my research results in the dissertation
Hanoi, April 3rd, 2019
Trang 4This dissertation has been carried out at the Department of FundamentalMathe-matics, School of Applied Mathematics and Informatics, Hanoi University ofScience 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 Toanand Dr Bui Trong Kien for their careful, patient and e ective supervision I am verylucky 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 visitedand studied at Department of Applied Mathematics, Sun Yat-Sen University,Kaohsiung, Taiwan (from April, 2015 to June, 2015 and from July, 2016 toSeptember, 2016) I would like to express my gratitude to Prof Nguyen Dong Yenfor his encouragement and many valuable comments
I would also like to especially thank my friend, Dr Vu Huu Nhu for kind help andencouragement
I would like to thank the Steering Committee of Hanoi University of Science andTechnology (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, 2019Nguyen Hai Son
Trang 5COMMITTAL IN THE DISSERTATION i
ACKNOWLEDGEMENTS ii
CONTENTS iii
TABLE OF NOTATIONS 1
INTRODUCTION 3
Chapter 0 PRELIMINARIES AND AUXILIARY RESULTS 8 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 NO-GAP OPTIMALITY CONDITIONS FOR DISTRIBUTED CONTROL PROBLEMS 25 1.1 Second-order necessary optimality conditions 26
1.1.1 An abstract optimization problem 26
1.1.2 Second-order necessary optimality conditions for optimal control problem 27
1.2 Second-order su cient optimality conditions 40
1.3 Conclusions 57
Chapter 2 NO-GAP OPTIMALITY CONDITIONS FOR BOUNDARY CONTROL PROBLEMS 58 2.1 Abstract optimal control problems 59
2.2 Second-order necessary optimality conditions 66
2.3 Second-order su cient optimality conditions 75
2.4 Conclusions 89
Chapter 3 UPPER SEMICONTINUITY AND CONTINUITY OF THE SOLUTION MAP TO A PARAMETRIC BOUNDARY CONTROL PROBLEM 91 3.1 Assumptions and main result 92
3.2 Some auxiliary results 94
Trang 63.2.1 Some properties of the admissible set 94
3.2.2 First-order necessary optimality conditions 98
3.3 Proof of the main result 100
3.4 Examples 104
3.5 Conclusions 109
GENERAL CONCLUSIONS 110
LIST OF PUBLICATIONS 111
REFERENCES 112
Trang 7TABLE OF NOTATIONS
N :=f1; 2; : : :g set of positive natural numbers
[x1; x2] the closed line segment between x1 and x2
B(x; ) open ball with centered at x and radiusB(x; ) closed ball with centered at x and radius
B
Trang 8T (K; x) Bouligand tangent cone of the set K at x
T [(K; x) adjoint tangent cone of the set K at x
T 2(K; x; d) second-order Bouligand tangent set of the set
K at x in direction d
T 2[(K; x; d) second-order adjoint tangent set of the set K
at x in direction d
closure of the set
the space of nite regular Borel measures
Trang 91 Motivation
Optimal control theory has many applications in economics, mechanics andother elds of science It has been systematically studied and strongly developedsince the late 1950s, when two basic principles were made One was thePontryagin Maximum Principle which provides necessary conditions to nd optimalcontrol functions The other was the Bellman Dynamic Programming Principle, aprocedure that reduces the search for optimal control functions to nding thesolutions of partial di erential equations (the Hamilton-Jacobi equations) Up tonow, optimal control theory has developed in many various research directionssuch as non-smooth optimal control, discrete optimal control, optimal controlgoverned by ordinary di erential equations (ODEs), optimal control governed bypartial di erential equations (PDEs), (see [1, 2, 3])
In the last decades, qualitative studies for optimal control problems governed byODEs and PDEs have obtained many important results One of them is to give op-timality conditions for optimal control problems For instance, J F Bonnans et al.[4, 5, 6], studied optimality conditions for optimal control problems governed byODEs, while J F Bonnans [7], E Casas et al [8, 9, 10, 11, 12, 13, 14, 15, 16, 17],
C Meyer and F Tr•oltzsch [18], B T Kien et al [19, 20, 21, 22], A R•osch and F.Tr•oltzsch [23, 24] derived optimality conditions for optimal control problemsgoverned by el-liptic equations
It is known that if u is a local minimum of F , where F : U ! R is a di erentiablefunctional and U is a Banach space, then F 0(u) = 0 This a rst-order necessaryoptimality condition However, it is not a su cient condition in case of F is notconvex Therefore, we have to invoke other su cient conditions and should studythe second derivative (see [17])
Better understanding of second-order optimality conditions for optimal control lems governed by semilinear elliptic equations is an ongoing topic of research forseveral 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 ofnonlinear optimal control problems, and in analyzing the sequential quadraticprogramming al-gorithms (see [13, 16, 17]) and in studying the stability of optimalcontrol (see [25, 26]) Second-order necessary optimality conditions not only providecriterion of nding out stationary points but also help us in constructing su cientoptimality conditions Let us brie y review some results on this topic
Trang 10prob-For distributed control problems, i.e., the control only acts in the domain in Rn,
E Casas, T Bayen et al [11, 13, 16, 27] derived second-order necessary and sucient optimality conditions for problem with pure control constraint, i.e.,
and the appearance of state constraints More precisely, in [11] the authors gave order necessary and su cient conditions for Neumann problems with constraint
second-(1) and nitely many equalities and inequalities constraints of state variable y while thesecond-order su cient optimality conditions are established for Dirichlet problems withconstraint (1) and a pure state constraint in [13] T Bayen et al [27] derived second-order necessary and su cient optimality conditions for Dirichlet problems in the sense
of strong solution In particular, E Casas [16] established second-order su cientoptimality conditions for Dirichlet control problems and Neumann control problemswith only constraint (1) when the objective function does not contain control variable u
In [18], C Meyer and F Tr•oltzsch derived second-order su cient optimality conditionsfor Robin control problems with mixed constraint of the form a(x) y(x) + u(x)
b(x) a.e x 2 and nitely many equalities and inequalities constraints
For boundary control problems, i.e., the control u only acts on the boundary , E.Casas and F Tr•oltzsch [10, 12] derived second-order necessary optimalityconditions while the second-order su cient optimality conditions were established
by E Casas et al in [12, 13, 17] with pure pointwise constraints, i.e.,
a(x) u(x) b(x) a.e x 2 :
A R•osch and F Tr•oltzsch [23] gave the second-order su cient optimalityconditions for the problem with the mixed pointwise constraints which hasunilateral linear form c(x) u(x) + (x)y(x) for a.e x 2
We emphasize that in above papers, a; b 2 L1( ) or a; b 2 L1( ) Therefore, thecontrol u belongs to L1( ) or L1( ) This implies that corresponding Lagrangemultipliers are measures rather than functions (see [19]) In order to avoid this dis-advantage, B T Kien et al [19, 20, 21] recently established second-ordernecessary optimality conditions for distributed control of Dirichlet problems withmixed state-control constraints of the form
a(x) g(x; y(x)) + u(x) b(x) a.e x 2
with a; b 2 Lp( ), 1 < p < 1 and pure state constraints This motivates us to developand study the following problems
(OP 1) : Establish second-order necessary optimality conditions for Robinboundary control problems with mixed state-control constraints of the form
a(x) g(x; y(x)) + u(x) b(x) a.e x 2 ;
Trang 11where a; b 2 Lp( ), 1 < p < 1.
(OP 2) : Give second-order su cient optimality conditions for optimal controlprob-lems with mixed state-control constraints when the objective function doesnot depend on control variables
Solving problems (OP 1) and (OP 2) is therst goal of the dissertation
After second-order necessary and su cient optimality conditions are established,they should be compared to each other According to J F Bonnans [4], if the changebetween necessary and su cient second-order optimality conditions is only betweenstrict and non-strict inequalities, then we say that the no-gap optimality conditions areobtained Deriving second-order optimality conditions without a gap between second-order necessary optimality conditions and su cient optimality conditions, is a di cultproblem which requires to nd a common critical cone under which both second-ordernecessary optimality conditions and su cient optimality conditions are satis ed In [7],
J F Bonnans derived second-order necessary and su cient optimality conditions withno-gap for an optimal control problem with pure control constraint and the objectivefunction 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 optimalcon-trol problems governed by semilinear elliptic equations with mixed pointwiseconstraints Solving problem (OP 3) is the second goal of this dissertation
Solution stability of optimal control problem is also an important topic inoptimiza-tion and numerical method of nding solutions (see [25, 29, 30, 31, 32, 33,
34, 35, 36, 37, 38, 39, 40, 41]) An optimal control problem is called stable if theerror of the output data is small in some sense for a small change in the inputdata The study of solution stability is to investigate continuity properties of solutionmaps in parameters such as lower semicontinuity, upper semicontinuity, H•oldercontinuity and Lipschitz continuity
Let us consider the following parametric optimal problem:
It is well-known that if the objective function F ( ; ; ) is strongly convex, and the admissible set ( ) is convex, then the solution map of problem (2) is single-valued (see [29], [30], [31]) Moreover, A Dontchev [30] showed that under some certain conditions,
Trang 12the solution map is Lipschitz continuous w.r.t parameters By using implicit functiontheorems, K Malanowski [35]-[40] proved that the solution map of problem (2) is also
a Lipschitz continuous function in parameters if weak second-order optimalityconditions and standard constraint quali cations are satis ed at the reference point.Notice that the obtained results in [37]-[40] are for problems with pure stateconstraints, while the one in [35] is for problems with pure control constraints
When the conditions mentioned above are invalid, the solution map may not besingleton (see [32, 33]) In this situation, we have to use tools of set-valued analysisand 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 optimalcontrol problem for the case where the objective function is convex in both variablesand the admissible sets are also convex Recently, the upper semicontinuity of thesolution 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 theadmissible sets are not convex Notice that in [34] the authors considered the problemgoverned by ordinary di erential equations meanwhile in [42] the author investigatedthe problem governed by semilinear elliptic equation with distributed control From theabove, one may ask to study the following problem:
(OP 4) : Establish su cient conditions under which the solution map ofparametric boundary control problem is upper semicontinuous and continuous.Giving a solution for (OP 4) is the third goal of this dissertation
2 Objective
The objective of this dissertation is to study no-gap second-order optimalitycon-ditions and stability of solution to optimal control problems governed bysemilinear elliptic equations with mixed pointwise constraints Namely, the maincontent 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( ), 1 < p < 1;
(ii) deriving second-order su cient optimality conditions for distributed controlprob-lems and boundary control problems when objective functions arequadratic forms in the control variables, and showing that no-gap optimalitycondition holds in this case;
(iii) deriving second-order su cient optimality conditions for distributed controlprob-lems and boundary control problems when objective functions areindependent of the control variables, and showing that in general theory ofno-gap conditions does not hold;
(iv) giving su cient conditions for a parametric boundary control problem under which
Trang 13the solution map is upper semicontinuous and continuous in parameters.
3 The structure and results of the dissertation The
dissertation has four chapters and a list of references
Chapter 0 collects several basic concepts and facts on variational analysis,Sobolev spaces and partial di erential equations
Chapter 1 presents results on the no-gap second-order optimality conditions fordistributed control problems
Chapter 2 provides results on the no-gap second-order optimality conditions forboundary control problems
The obtained results in Chapters 1 and 2 are answers for problems (OP 1); (OP2) and (OP 3), respectively
Chapter 3 presents results on the upper semicontinuity and continuity of thesolution map to a parametric boundary control problem, which is a positive answerfor 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 VariationalAnalysis and SIAM Journal on Optimization, respectively The results of Chapter 3were content of article [3] in the List of publications which is published in Optimization.These results have been presented at:
The Conference on Applied Mathematics and Informatics at Hanoi University
of Science and Technology in November 2016
The 15th Conference on Optimization and Scienti c Computation, Ba Vi in April 2017
The 7th International Conference on High Performance Scienti c Computing inMarch 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
Trang 14Chapter 0PRELIMINARIES AND AUXILIARY RESULTS
In this chapter, we review some background on Variational Analysis, Sobolev spaces, and facts of partial di erential equations relating to solutions of linear elliptic equations and semilinear elliptic equations For more details, we refer the reader to [1], [2], [3], [27], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], and [56]
0.1 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 2 X a subset F (x) Y F (x) is
called the image or the value of F at x
Let F : X Y be a set-valued map between topological spaces X and Y We callthe sets
the graph, the domain and the range of F , respectively
The inverse F 1 : Y X of F is the set-valued map, de ned by
F 1(y) := f x 2 X j y 2 F (x)g for all y 2 Y:
The set-valued map F is called proper if dom(F ) 6= ;
De nition 0.1.1 ([46, p 34]) Let F : X Y be a set-valued map between topolog-ical spaces X and Y
(i) If gph(F ) is a closed subset of the topological space X Y then F is called closed map (or graph-closed map)
(ii) If X; Y are linear topological spaces and gph(F ) is a convex subset of the topo-logical space X Y then F is called convex set-valued map
(iii) If F (x) is a closed subset of Y for all x 2 X then F is called closed-valued map.(iv) If F (x) is a compact subset of Y for all x 2 X then F is called compact-valued map
Trang 15The concepts of semicontinuous set-valued maps had been introduced in 1932 by G.
Bouligand and K Kuratowski (see [44])
De nition 0.1.2 ([45, De nition 1, p 108] and [44, De nition 1.4.1, p.38]) Let
F : X Y be a set-valued map between topological spaces and x0 2 dom(F )
(i) F is said to be upper semicontinuous at x0 if for any open set W in Y
satisfying F (x0) W , there exists a neighborhood V of x0 such that
F (x) W for all x 2 V:
(ii) F is said to be lower semicontinuous at x0 if for any open set W in Y satisfying
F (x0) \ W 6= ;, there exists a neighborhood V of x0 such that
F (x) \ W 6= ; for all x 2 V \ dom(F ):
(iii) F is continuous at x0 if it is both lower semicontinuous and upper
semicontinuous at x0
The map F is called upper semicontinuous (resp., lower semicontinuous, continuous)
if it is upper semicontinuous (resp., lower semicontinuous, continuous) at every point
x 2 dom(F )
Notice that in case of single-valued map F : X ! Y , the above concepts arecoincident
When X; Y are metric spaces, set-valued map F : X Y is lower semicontinuous
at x 2 dom(F ) if and only if for all y 2 F (x) and sequence fxng 2 dom(F ), xn ! x,there exists a sequence fyng Y , yn 2 F (xn) such that yn ! y
0.1.2 Tangent and normal cones
Let X be a normed space with the norm k k For each x0 2 X and > 0, we denote
by B(x0; ) the open ball fx 2 X j kx x0k < g, and by B(x0; ) the correspondingclosed ball We will write BX and BX for B(0X ; 1) and B(0X ; 1), respectively Let D
be a nonempty subset of X The distance from x 2 X to D is de ned by
dist(x; D) = inf kx uk:
Trang 16From De nition 0.1.3, it follows that T (D; x) is a closed cone and T (D; x)cone(D x); where cone(A) := f a j 0; a 2 Ag is the cone generated by the set A:Moreover, the following property characterizes the Bouligand cone:
T (D; x) = fv 2 X j 9tn ! 0+; 9vn ! v s.t x + tnvn 2 D for all n 2 Ng:
De nition 0.1.4 ([44, De nition 4.1.5, p 126]) Let D X be a subset of normed space
X and x 2 D: The adjoint tangent cone or the intermediate cone T [(K; x) of D at x
is de ned by
T [(D; x) := v X lim dist(x + tv; D) = 0 :The Clarke tangent cone TC (D; x) of D at x is de ned by
where x0 ! x means that x0 2 D and x0 ! x:
From De nition 0.1.4, we have the following characters of the adjoint cones and theClarke tangent cones (see [44, p 128]):
Trang 17TC (D; x) = T [(D; x) = T (D; x) = cone(D x):
The above tangent cones has important roles in the study of rst-order optimalityconditions for optimal control problems with constraints However, in order toobtain second-order optimality conditions for optimal control problems, we need touse second-order tangent sets
De nition 0.1.8 ([44, De nition 1.1.1, p 17]) Let X be a normed space and (Dt)t2T X
be a sequence of sets depend on parameters t 2 T; where T is a metric space.Suppose that t0 2 T: The set
Limsup D t := fx 2X j limtinf dist(x; Dt0 t ) = 0g
is called Painlev -Kuratowski lower limit of (Dt) as t ! t0:
De nition 0.1.9 ( [44, De nition 4.7.1 and 4.7.2, p 171]) Let D be a subset in thenormed space X and x 2 D; v 2 X:
T 2(D; x; v) := Limsup
t2t!0 +
is called Bouligand second-order tangent set of D at x in direction v:
Trang 18Obviously, T 2(D; x; v) and T 2[(D; x; v) are closed and
fx1;kg It is easily seen that x1;k > x1;k+1 > 0 and x1;k ! 0 as k ! 1
Taking K = f(x1; x2) 2 R2 j x2 '(x1)g and x = (0; 0); v = (1; 0), we have
T (K; u0) = T [(K; u0)
n o = v 2 Lp( ) j v(x) 2 T [([a(x); b(x)]; u0(x)) a.e x 2 :
Trang 19In the sequel, we shall use concept normal cone which is dual concept of Clarketangent cones We denote by X the dual space of the normed space X, i.e., the space
of all continuous linear functionals on X; the (dual) norm on X is de ned by
kfkX = supff(x) j x 2 X; kxk 1g:
Then X is a Banach space, i.e., X is complete even if X is not (see [48, p.3]) Let
us denote by h ; i the canonical pairing between X and X
De nition 0.1.13 ([44, De nition 4.4.2, p 157]) Let X be a Banach space, a subset
D X and a point x 2 D: We shall say that the polar cone
N(D; x) := TC (D; x) = fp 2 X j hp; vi 0 8v 2 TC (D; x)g
is (Clarke) normal cone of D at x:
When D is convex, N(D; x) coincides with the normal cone of D at x in ConvexAnalysis, i.e.,
Let be an open subset in RN : For each function u : ! R; we call suppu :=
fx 2 : u(x) =6 0g the support of u:
For each non-negative integer number m; we have the following classicalfunction spaces:
Trang 20De nition 0.2.1 ([43, Chapter 2] and [49, De nition 2.1, p 14]) Let be an open set
with respectively norms
kukLp ( ) := ju(x)jpdx ;
u(x)v(x)dx 8u; v 2 L2( ):
It is noted that C0( ) is dense in Lp( ) for 1 p < +1: The topological dual spaces of
Lp spaces for (1 p < +1) are Lp space too, namely, Lp( ) = Lp0( ); 1 < p < +1
and L1( ) = L1( ) (see [43, Chapter 2] and [48, Section 4.3])
In the sequel, we will write 0 if 0 is included in and compact We denoted by
L1loc( ) the space of local integrable functions on ; i.e.,
p0 :=
For p 2 [1; +1]; let us denote by p0
kukL1 ( ) := inf fC j ju(x)j C a:e: x 2 g :
Trang 21for all measurable subset 0 :Then, for any open set in RN and for all p 2 [1; +1]; we have Lp( ) L1loc( ) (see [43,Chapter 2, p 26]).
Recall that C01( ) the space of functions in nitely di erentiable in with compactsupport in : We introduce a notion of convergence in the space C01( ) which can
be de ned by a topology on C01( ) Then C01( ) is denoted by D( )
De nition 0.2.2 ([43, Chapter 1, p 19] and [49, De nition 2.3, p 18]) Let ('i); i 2 N
be a sequence of functions in D( ): We say that ('i) converges to ' in D( ) when
14
Trang 22i ! +1; if there exists a compact set K satisfying supp' K; supp'i K for all i 2 N and
D 'i ! D ' uniformly in K 8 2 NN ;i.e.,
lim sup jD ' x) D ' x)j = 0 NN :
i !+ 1
Functions ' 2 D( ) is called test functions
De nition 0.2.3 ([43, Chapter 1, p 19] and [49, De nition 2.4, p 19]) A distribu-tion
T on is a continuous linear form on D( ), i.e., T : D( ) ! R is a linear map such that
de nes a distribution on : Thus, we have L1loc( ) D0( ) (see [49, Example, p 22])
De nition 0.2.4 ([43, Chapter 1, p 20] and [49, De nition 2.5, p 20]) For = ( 1;
2; :::; N ) 2 NN and T 2 D0( ); the map
De nition 0.2.5 ([49, De nition 2.6, p 21]) Let (Ti) be a sequence of distributions in
Trang 23distribu-Proposition 0.2.6 ([43, Chapter 1, p 20] and [49, distribu-Proposition 2.5, p 22]) The operator D with 2 NN is continuous on D0( ); i.e., if Ti ! T in D0( ) then
D Ti ! D T in D0( ):
We now give de nition of weak partial derivative for locally integrable functions
De nition 0.2.7 ([43, Chapter 1, p 21] and [50, Chapter 5]) Let u; v 2 L1loc( ) and
be a multiindex We say that v is -order weak partial derivative (or -order generalpartial derivative) of u, written by v = D u; if
u(x)D '(x)dx = ( 1)j j v(x)'(x)dx 8' 2 D( ):
From De nition 0.2.4 and 0.2.7, it is easily seen that if v = D u is -order weak partial derivative then v is -order partial derivative in the distributional sense of u:Next, we give de nition of Sobolev spaces
De nition 0.2.8 ([43, Chapter 3, p 44] and [50, Chapter 5]) Let m 2 N; p 2 [1; +1]:
We consider the space
W m;p( ) := fu 2 Lp( ) j D u 2 Lp( ) with 0 j j m
and D u is -order weak partial derivative of ug
with corresponding norm
We call W m;p( ) and W0m;p( ) Sobolev spaces
Remark 0.2.9 (i) In case of p = 2; we write Hm( ) := W m;2( ) and H0m( ) :=
W0m;2( ):
(ii) In case of m = 0; we have W 0;p( ) = Lp( ): Moreover, if is bounded and p 2 [1; +1) then we have W00;p( ) = Lp( ) (see [43, Chapter 3, p 44])
subspace of W m;p( ): Moreover, Hm( ); Hm( ) are Hilbert spaces with scalar product
(iv) Sobolev spaces W m;p( )
and strictly convex) if p 2 (1;
+1); is in [43])
W0m;p( ) are re exive and uniformly convex (and so, separable if p 2 [1; +1) (see Theorems 1.21 and 3.5
Trang 24The following is de nition on the regularity of boundary of domain
De nition 0.2.10 ([52, De nition 1.2.1.1, p 5] and [3, Subsection 2.2.2, p.26]) Let
be an open set in RN : Boundary of is called continuous (respectively Lipschitz,continuously di erential, of class Ck;l, m times continuously di erential) if for each
x 2 ; there exist a neighborhood V RN of x and a new orthogonal coordinate fy1;
j'(y0)j aN for every y0 := (y1; y2; :::; yN 1) 2 V 0;
P 2 E, there is y 2 V 0 such that P = (y; '(y)) Let D := ' 1(E) V 0 Then we say that
E is measurable if D is measurable with respect to (N 1) dimensional Lebesguemeasure The measure of E is de ned by
Let us introduce Sobolev spaces on the set , (see [2, Chapter 2, p 75] and [52,
De nition 1.3.2.1 and De nition 1.3.3.2]) For s 2 (0; 1); p 1 and u 2 C1( ); we consider the norm
kukW s;p ( ) := Z ju(x)jpd (x) + Z
jjx x 0 j N 1+0sp j d (x)d (x0) 1=p ; (1)
u(x) u(x )
where d is measure on : Let us denoted by W s;p( ) the closed space generated by
C1( ) under norm (1) Thus, W s;p( ) is a Banach space
We denote by W m;p( ) ans W r;s( ) the dual spaces of the spaces W m;p0( ) and
W r;s0 ( ), respectively, where 1 + 1 = 1 + 1 = 1:
p p 0 s s 0
Trang 25De nition 0.2.11 ([43, Chapter 1, p 9]) Let X; Y be the normed spaces We saythat X is imbedded in Y and write X ,! Y; if there a linear continuous injection j : X !Y:
Moreover, if j is compact then we say that X compactly imbedded in Y and write
X ,!,! Y:
We are ready to present some imbedding results for Sobolev spaces
Theorem 0.2.12 (Sobolev and Rellich embedding theorem, [43, Theorem 5.4, p
97 and Theorem 6.2, p 144], [48, Theorem 9.16, p 285] and [52, Chapter 1, p.27]) Let RN be a bounded Lipschitz domain, 1 p +1 and 1 p +1:
Remark 0.2.13 (i) The injection W 1;p( ) ,! Lp ( ), p = even if
is bounded and smooth (see [48])
(ii) If W 1;p( ) is replaced by W01;p( ) then Theorem 0.2.12 is valid even if is not Lipschitz
This results can be extended for the spaces W m;p( ) with m is a non negative integer
Theorem 0.2.14 (see [3, Theorem 7.1, p 355]) Let RN be a bounded Lipschitz
N p , is never compact
N p
Trang 26If mp < N then N p
W m;p( ) ,! Lq( ) 81 q N mpand this embedding is compact for 1 q < N p :
Trang 27The following theorem was proved by Gagliardo in 1975 and it is called trace theorem
Moreover, the map T is surjective and has a right inverse which does not depend on
p, i.e., there exists a unique bounded linear map : W 1 1=p;p( ) ! W 1;p( ) such that
T ( ( ))( ) = ( ); 8 ( ) 2 W 1 1=p;p( ):
We shall call T the trace operator and T u the trace of u on
The below results represent the relation between Sobolev spaces W01;p( ) andthe set Ker(T ) of trace operator T :
Theorem 0.2.16 ([52, Theorem 1.5.1.5, p 38] and [50, Chapter 5]) Suppose that
is a bounded open subset of RN with Lipschitz boundary Then
u 2 W01;p( ) if and only if T (u) = 0 on :
Theorem 0.2.17 ([48, Theorem 9.17, p 288]) Let be of class C1 and u 2 W 1;p( )\
1;p
C( ): Then u = 0 on if and only if u 2 W0 ( ):
The smoothness of boundary plays an important role in the following results.Theorem 0.2.18 ([3, Theorem 7.2, p.355 ]) Let m 2 N with m > 0, and let be ofclass Cm1;1 Then for all mp < N the trace operator T is continuous from W m;p( )
into Lr( ), provided by 1 r (N 1)p If mp = N then T is continuous for all
Trang 28To nish this subsection, we recall some of classical inequalities concerning Lpspaces and Sobolev spaces which can be found in [3, 43, 48, 49, 50].
Proposition 0.2.21 (H•older inequality) Let p 2 (1; +1): If u 2 Lp( ) and v 2 0
kuvkL1 ( ) kukLp ( )kvkLp 0
( ):Proposition 0.2.22 (General H•older inequality) Let p1; :::; pk 2 (1; +1) such that
P k p 1 = 1: If u Lpi ( ) for all i 1; 2; :::; k thenQk u L1 ( ): Moreover,
Theorem 0.2.23 (Poincare inequality, [52, Theorem 1.4.3.4, p 26]) Suppose that
is a bounded domain in RN with Lipschitz boundary, p 2 [1; +1) There exists a constant C( ) such that
kukLp ( ) C( )krukLp ( ) 8u 2 W01;p( ):
Theorem 0.2.24 (Gereralized Poincare inequality, [3, p 35]) Suppose that is abounded domain in RN with Lipschitz boundary , and 1 is a positive measure set,i.e., j 1j > 0 Then there exists a constant C( 1) > 0, which is independent of y 2 H1(), such that
Z
0.2.2 Semilinear elliptic equations
Throughout this subsection, let us denote by p0; q0; r0 be adjoint numbers of positive numbers p; q; r respectively, i.e., p1 + p10 = 1q + q10 = 1r + r10 = 1:
Let is a bounded domain in RN with the boundary We denote by A the order elliptic operator of the form
second-N
X
Ay = D j (a ij D i y);
i;j=1
Trang 29where the coe cients aij 2 L1( ) satisfying aij = aji; i; j = 1; 2; :::; N:
The Dirichlet problems
Let us consider semilinear Dirichlet problem of the form:
where h : R ! R is a Caratheodory function andu2Lp( ); p > 1:
De nition 0.2.26 Let u 2 W 1;p( ) for p > 1: A function y is called (weak or
variational) solution of (2) if y 2 W01;p( ) and
We will need the following assumptions:
(A0:1) The matrix of coe cients (aij(x))i;j=1;2;:::;N are strongly elliptic, i.e., there existconstant m; M > 0 such that
C ; aij C ( ) for all i; j = 1; 2; :::; N; h(x; y) = a0(x)y with a0 L ( ); a0
0 a:e: x 2 and that assumption (A0:1) is ful lled Then, for u 2 Lp( ); equation(2) has a unique solution yu 2 W01;p( ) \ W 2;p( ): Moreover, there exists a constant
C > 0 independent of u such that
ky
ukW 2;p ( ) Cku
k L p ( ):
21
Trang 30The following theorem gives the existence and the uniqueness of weak solution
to equation (2) when boundary is Lipschitz and N 2 f2; 3g:
Theorem 0.2.28 ([13, Theorem 2.1] and [15, Theorem 2.1]) Suppose that boundary
is Lipschitz, N 2 f2; 3g, and that assumption (A0:1) is satis ed and assumption(A0:2) is valid for p = 2: Then for each u 2 L2( ); equation (2) has a unique solution
u 2 Lp( ); equation (2) has a unique solution yu 2 W01;p( ) \ W 2;p( ): Moreover, there exists a constant C > 0 independent of u such that
ky
ukW 2;p ( ) Cku
k L p ( ):
The Robin problems
We consider Robin problem of the form:
8
< @ y + b 0 (x)y + k(x; y) = v on ;
:
where h : R ! R; k : R ! R are Caratheodory functions, and (u; v) 2 Lp( ) Lq( ); p; q >
1, and @ denote the conormal-derivative associated with A, i.e.,
Trang 31De nition 0.2.31 Let 1 < r < 1 For each given u 2 (W 1;r0( )) ; v 2 W 1r ;r( ), afunction y 2 W 1;r
( ) is said to be a (weak or variational) solution of (3) if
We will need the following assumptions:
(A0:4) a0 : ! R and b0 : ! R are bounded, measurable and almost everywherenonnegative Moreover, a0 6 0:
(A0:5) h : R ! R and k : are Caratheodory functions and of class C1 w.r.t thesecond variable and satisfying the following conditions
h( ; 0) = 0; h
y(x; y) 0 a.e x 2 and for all y 2 R:
k( ; 0) = 0; k
y(x0; y) 0 a.e x0 2 and for all y 2 R:
For each M > 0; there is a positive constant Ch;M such that
hy(x; y) Ch;M andky(x0; y) C
h;M
for a.e x 2, x0 2 and all jy M:
j(A0:6) The numbers p; q; r satisfy the following inequalities
Lemma 0.2.32 ([3, Theorem 4.7]) Suppose that is a Lipschitz domain, p > N2 ; q >
N 1 and assumptions (A0:1); (A0:4); (A0:5) are ful lled Then the equation (3) hasfor any pair (u; v) 2 Lp( ) Lq( ) a unique weak solution y 2 H1( )\C( ) Moreoverthere exists a positive constant C independent of a0; b0; h; k such that
kykH1 ( ) + kykL1 ( ) C(kukLp ( ) + kvkLq ( )):
In case of linear equations, i.e., h 0, and k 0, M Mateos [57] showed thatequation (3) has a unique solution in W 1;r( )
Lemma 0.2.33 ([11, Lemma 2.4] and [57, Theorem 2.13]) Suppose that is of class
C1, 1 < r < 1 and h 0, and k 0; and that assumptions (A0:1), (A0:4) are ful lled.Then for each u 2 (W 1;r0( )) ; v 2 W 1
Trang 32where C > 0 is a constant only depending on r, the dimension N, the coe cients aij,and the domain
The following theorem is an extension of Lemma 0.2.33
Theorem 0.2.34 Suppose that is of class C1;1 and that assumptions (A0:1), (A0:4)A(1:6) are ful lled Then for any u 2 Lp( ); v 2 Lq( ), the equation (3) has a uniquesolution y 2 W 1;r( ) Moreover, there exists positive constants C1; C2 independent
Since W 1;r( ) ,! H1( ) and W 1;r( ) ,! C( ), fyig converges strongly to y in H1( ) \ C() as i ! 1, too
Corollary 0.2.35 ([8, Theorem 3.1]) Suppose that is of class C1;1, p > N2 ; q > N 1and that assumptions (A0:1), (A0:4), A(1:5) are ful lled Then for any u 2 Lp( ); v 2
Lq( ), the equation (3) has a unique solution y 2 H1( ) \ C( ) In addition, if fujg and
fvjg converge weakly to u in Lp( ) and v in Lq( ), respectively then fyjg convergesstrongly to y in H1( ) \ C( )
Remark 0.2.36 In case of N = 2, then we may choose p = q = 2 This case will beconsidered in Chapter 3
0.3 Conclusions
This chapter presents several basic concepts and facts on set-valued maps,tangent and normal cones, Sobolev spaces, imbedding theorems and results onthe existence and uniqueness of weak solution to semilinear elliptic equations
Trang 33Chapter 1NO-GAP OPTIMALITY CONDITIONS FOR DISTRIBUTED
CONTROL PROBLEMS
Let be a bounded domain in RN with N 2 and the boundary of class C2 We
consider the following distributed optimal control problem of nding a controlfunction u 2 Lp( ) and a state function y 2 W 2;p( ) \ W01;p( ) which
minimize F (y; u) = Z
L(x; y(x); u(x))dx; (1.1)(DP ) s.t
y + h(x; y) = u in ; y = 0 on ; (1.2)a(x) g(x; y(x)) + u(x) b(x) a:e: x 2 ; (1.3)where L : R R ! R and g : R ! R are Caratheodory functions, h : R ! R is acontinuous function of class C2 w.r.t the second variable such that h(x; 0) = 0 and
hy(x; y) 0 for all y 2 R and a.e x 2 , a; b 2 Lp( ) and 6= 0 is a constant Hereafter,
we assume that p > N2
A standard example for problem (DP ) is the case, where the integrand is given by
L(x; y; u) := 1
2jy y (x)j2 + 2 juj2and the constraint (1.3) has the form
a(x) y(x) + u(x) b(x) a:e: x 2 ;with y 2 L2( ), and are constants.
The aim of this chapter is to give no-gap optimality conditions for problem (DP ).Namely, we shall develop the results in [27], [7] and [18] by giving a correct criticalcone under which both the second-order necessary conditions and the second-order su cient conditions for problem (DP ) are valid whenever the integrand isquadratic in u In order to obtain the results, we shall reduce the problem to amathematical program-ming problem where the admissible set satis es thestrongly extended polyhedricity condition and satis es Robinson's constraint qualication (see [58]) Note that when the integrand does not depend on controlvariables, it is not quadratic in u To deal with second-order su cient conditions forthe problem, we need to enlarge the critical cone Based on this critical cone andtechnique in [13], we give second-order su cient conditions for this case
The content of this chapter is based on article [1] in the List of publications
Trang 341.1 Second-order necessary optimality conditions
1.1.1 An abstract optimization problem
Let U be a Banach space and E be a separable Banach space with the duals Uand E , respectively We consider the following problem
(P ) min f(u) subject to G(u) 2 K;
u2U
where K is a nonempty closed and convex set in E, G : U ! E and f : U ! R aresecond-order Frechet di erentiable on U By ad := G 1(K), we denote theadmissible set of problem (P )
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:
De nition 1.1.2 Given a point u 2 ad, problem (P ) is said to satisfy Robinson'sconstraint quali cation at u if there exists > 0 such that
BE(0; ) rG(u)(BU ) (K G(u)) \ BE : (1.4)
In this case, we also say that u is regular
According to [59, Theorem 2.1] (see also [19, Theorem 2.5]), condition (1.4) isequiv-alent to the following:
Under the Robinson constraint quali cation, we have from [60, Theorem 3.1] thefol-lowing formulae:
T 2[( ; u; d) = rG(u) 1[T 2[(K; G(u); rG(u)d) 1r2 G(u)(d; d)]; 8d 2 U:
2
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)):
By [59, Theorem 4.1], 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:
Trang 35From now on, given a continuously linear mapping A, we shall denote by A theadjoint operator of A For v 2 E , we de ne
(v )? = fv 2 E j hv ; vi = 0g:
In the sequel we shall need the so-called polyhedricity property of K This propertyplays an important role in deriving second-order optimality conditions It is known that,the sigma term usually appears in second-order necessary optimality conditions andthis term may be negative However, when the constraint K has polyhedricity property,the sigma term vanishes According to Bonnans and Shapiro [47, Chapter 3], the set
K is said to be polyhedric at u 2 K if for any v 2 N(K; z), one has
r2uuL(u; e )(d; d) = r2f(u)(d; d) + he ; r2G(u)(d; d)i 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 admissiblefor problem (DP ) By Corollary 0.2.30, for each u 2 Lp( ), equation (1.2) has a uniquesolution yu 2 W 2;p( ) \ W01;p( ) and there exists a constant C > 0 such that
ky
ukW 2;p ( ) Cku
k L p ( ):Given an admissible couple (y; u), symbols g[x]; h[x]; L[x]; Ly[x]; L[ ]; etc., standrespectively 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 ukL p ( ) , one has
F (y; u) F (y; u):
Trang 36We now impose the following assumptions for problem (DP ) which involve (y; u).(A1:1) L : R R ! R is a Caratheodory function of class C2 with respect to variable (y;u), L(x; 0; 0) 2 L1( ) and for each M > 0, there exist a positive number kLM and afunction rM 2 L1
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
(A1:2) The function g is a continuous function and of class C2 w.r.t the second
variable, and satis es the following properties: g( ; 0) 2 Lp( ) and for each M > 0,there exists a constant Cg;M > 0 such that
gy(x; y) + gyy(x; y)Cg;M ;
g y (x; y 1 ) g y (x; y 2 ) + g yy (x; y 1 ) g yy (x; y 2 ) C
g;M jy 2 y 1jfor a.e x and y ; y1 ;y2 M
Trang 37j=0;p 1 j>0
where 0 < 1(x); 2(x) < 1 a.e x 2 , M = kyk1 + kh1k1 and kfkk is the norm of f in Lk( ).Use the H•older inequality and the fact that kfk1 c:kfkp for all f 2 Lp( ), there exists aconstant C > 0 such that
Trang 38Fix h = (h1; h2) with khkZ 1; we then have
Trang 39Note that assumption (A1:2) guarantees that G(y; u) is second-order Frechet ferentiable on (W 2;p( ) \ W 1;p( ))Lp( ) Meanwhile, (A1:3) guarantees that the 0
dif-Robinson condition is satis ed In contrast with the case where the control variable
u 2 L1( ), the objective function F is di cult to be second-order Frechet di eren-tiable on(W 2;p( )\W01;p( )) Lp( ) However, when L is a polynomial of variables y and u thenassumption (A1:1) is easily ful lled Let us give some illustrative examples
showing that L satis es (A1:1)
Trang 40Example 1.1.6 The following integrands satisfy (A1:1):
(a) p = 2 and L(x; y; u) := (y y (x)) + 2u , where > 0 and y 2 L ( );
(b) p 2, p 2 N and
L(x; y; u) := yp + yp 1 + + y2 + yu + u2 + + up:Example 1.1.7 Let p = 7=3 and L(x; y; u) := p3 .
u7 Then L satis es assumption
(A1:1).
7 p
4
f(x) = p3 x is H•older continuous with order = 1 , there exists a constant k such that
From the inequality jab(a b)j ja3 b3j for all a; b 2 R, it follows that
ja bj3 = ja3 b3 3ab(a b)j ja3 b3j + 3jab(a b)j 4ja3 b3j:
p
Therefore with C = 3 56, inequality (1.9) is valid and so is (1.8) We obtain the desired conclusion
The following example illustrates assumption (A1:3)
Example 1.1.8 Assume that h(x; y) = y3 + y, g(x; y) = y and the constraint (1.3) has the form
a(x) y(x) + u(x) b(x) a:e: x 2 ;where > 0 or 1 Then (A1:3) is ful lled Note that this constraint is the same with the constraint which was considered in [18]
De ne a mapping H : W 2;p( ) \ W01;p( ) Lp( ) ! Lp( ) by setting
H(y; u) = y + h( ; y) u: