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

MINISTRY OF EDUCATION AND TRAINING HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY NGUYEN HAI SON NO-GAP OPTIMALITY CONDITIONS AND SOLUTION STABILITY FOR OPTIMAL CONTROL PROBLEMS GOVERNED BY SEMILINEAR ELLIPTIC EQUATIONS DOCTORAL DISSERTATION OF MATHEMATICS Hanoi - 2019 MINISTRY OF EDUCATION AND TRAINING HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY NGUYEN HAI SON NO-GAP OPTIMALITY CONDITIONS AND SOLUTION STABILITY FOR OPTIMAL CONTROL PROBLEMS GOVERNED BY SEMILINEAR ELLIPTIC EQUATIONS Major: MATHEMATICS Code: 9460101 DOCTORAL DISSERTATION OF MATHEMATICS SUPERVISORS: Dr Nguyen Thi Toan Dr Bui Trong Kien Hanoi - 2019 COMMITTAL IN THE DISSERTATION I assure that my scienti c results are new and righteous Before I published these results, there had been no such results in any scienti c document I have responsibili-ties for my research results in the dissertation rd Hanoi, April , 2019 Author On behalf of Supervisors Dr Nguyen Thi Toan Nguyen Hai Son i ACKNOWLEDGEMENTS This dissertation has been carried out at the Department of Fundamental Mathe-matics, School of Applied Mathematics and Informatics, Hanoi University of Science and Technology It has been completed under the supervision of Dr Nguyen Thi Toan and Dr Bui Trong Kien First of all, I would like to express my deep gratitude to Dr Nguyen Thi Toan and Dr Bui Trong Kien for their careful, patient and e ective supervision I am very lucky to have a chance to work with them, who are excellent researchers I would like to thank Prof Jen-Chih Yao for his support during the time I visited and studied at Department of Applied Mathematics, Sun Yat-Sen University, Kaohsiung, Taiwan (from April, 2015 to June, 2015 and from July, 2016 to September, 2016) I would like to express my gratitude to Prof Nguyen Dong Yen for his encouragement and many valuable comments I would also like to especially thank my friend, Dr Vu Huu Nhu for kind help and encouragement I would like to thank the Steering Committee of Hanoi University of Science and Technology (HUST), and School of Applied Mathematics and Informatics (SAMI) for their constant support and help I would like to thank all the members of SAMI for their encouragement and help I am so much indebted to my parents and my brother for their support I thank my wife for her love and encouragement This dissertation is a meaningful gift for them rd Hanoi, April , 2019 Nguyen Hai Son ii CONTENTS COMMITTAL IN THE DISSERTATION ACKNOWLEDGEMENTS CONTENTS TABLE OF NOTATIONS INTRODUCTION Chapter PRELIMINARIES AND AUXILIARY RESULTS 0.1 Variational analysis i ii iii 8 0.1.1 Set-valued maps 0.1.2 Tangent and normal cones 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 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 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 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 iii 3.2.1 Some properties of the admissible set 3.2.2 First-order necessary optimality conditions 94 98 3.3 Proof of the main result 100 3.4 Examples 104 3.5 Conclusions 109 110 111 GENERAL CONCLUSIONS LIST OF PUBLICATIONS REFERENCES iv 112 N := f1; 2; : : :g R set of real numbers absolute value of x R jxj R TABLE OF NOTATIONS set of positive natural numbers n n-dimensional Euclidean vector space ; empty set x2A x is in A x 2= A A B(B A) A*B A\B x is not in A A[B union of the sets A and B AnB A B set di erence of A and B [x1; x2] kxk the closed line segment between x1 and x2 norm of a vector x kxkX X norm of vector x in the space X topological dual of a normed space X X hx ; xi topological bi-dual of a normed space X hx; yi B(x; ) canonical inner product B(x; ) BX B closed ball with centered at x and radius open unit ball in a normed space X dist(x; ) distance from x to sequence of vectors xk A is a subset of B A is not a subset of B intersection of the sets A and B Descartes product of the sets A and B canonical pairing open ball with centered at x and radius closed unit ball in a normed space X X fxkg xk ! x xk * x 8x xk converges strongly to x (in norm topology) 9x A := B f:X!Y there exists x xk converges weakly to x for all x A is de ned by B function from X to Y f (x), rf(x) 00 f (x), r f(x) Frechet derivative of f at x Frechet second-order derivative of f at x Frechet derivative of L in x Lx, rxL Lxy, rxy L ' : X ! IR dom' epi' Frechet second-order derivative of L in xand y supp' the support of ' F:X Y multifunction from X to Y domF rgeF domain of F range of F gphF graph of F kerF kernel of F T (K; x) Bouligand tangent cone of the set K at x extended-real-valued function e ective domain of ' epigraph of ' [ adjoint tangent cone of the set K at x second-order Bouligand tangent set of the set 2[ T (K; x; d) K at x in direction d second-order adjoint tangent set of the set K N(K; x) at x in direction d normal cone of the set K at x @ boundary of the domain T (K; x) T (K; x; d) closure of the set 0 and p is compact L ( ) the space of Lebesgue measurable functions f L ( ) and jf(x)jpdx < +1 space of bounded functions almost every C( ) the space of continuous functions on the M( ) > < > > W m;p ( ); ( ); m H ( ); H ( ) W W > ( ); W0 s;r : the space of nite regular Borel measures m;p m R Sobolev spaces 0 m;p ( )( p + p = 1) the dual space of W m;p ( ) X ,! Y X is continuous embedded in Y X ,!,! Y i.e a.e X is compact embedded in Y s.t subject to p page w.r.t with respect to The proof is complete id est (that is) almost every INTRODUCTION Motivation Optimal control theory has many applications in economics, mechanics and other elds of science It has been systematically studied and strongly developed since the late 1950s, when two basic principles were made One was the Pontryagin Maximum Principle which provides necessary conditions to nd optimal control functions The other was the Bellman Dynamic Programming Principle, a procedure that reduces the search for optimal control functions to nding the solutions of partial di erential equations (the Hamilton-Jacobi equations) Up to now, optimal control theory has developed in many various research directions such as non-smooth optimal control, discrete optimal control, optimal control governed by ordinary di erential equations (ODEs), optimal control governed by partial di erential equations (PDEs), (see [1, 2, 3]) In the last decades, qualitative studies for optimal control problems governed by ODEs and PDEs have obtained many important results One of them is to give optimality conditions for optimal control problems For instance, J F Bonnans et al [4, 5, 6], studied optimality conditions for optimal control problems governed by ODEs, while J F Bonnans [7], E Casas et al [8, 9, 10, 11, 12, 13, 14, 15, 16, 17], C Meyer and F 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 problems governed by el-liptic equations It is known that if u is a local minimum of F , where F : U ! R is a di erentiable functional and U is a Banach space, then F (u) = This a rst-order necessary optimality condition However, it is not a su cient condition in case of F is not convex Therefore, we have to invoke other su cient conditions and should study the second derivative (see [17]) Better understanding of second-order optimality conditions for optimal control problems governed by semilinear elliptic equations is an ongoing topic of research for several researchers This topic is great value in theory and in applications Secondorder su - cient optimality conditions play an important role in the numerical analysis of nonlinear optimal control problems, and in analyzing the sequential quadratic programming al-gorithms (see [13, 16, 17]) and in studying the stability of optimal control (see [25, 26]) Second-order necessary optimality conditions not only provide criterion of nding out stationary points but also help us in constructing su cient optimality conditions Let us brie y review some results on this topic n For distributed control problems, i.e., the control only acts in the domain in R , E Casas, T Bayen et al [11, 13, 16, 27] derived second-order necessary and su cient optimality conditions for problem with pure control constraint, i.e., a(x) u(x) b(x) a.e x ; (1) and the appearance of state constraints More precisely, in [11] the authors gave secondorder necessary and su cient conditions for Neumann problems with constraint (1) and nitely many equalities and inequalities constraints of state variable y while the second-order su cient optimality conditions are established for Dirichlet problems with constraint (1) and a pure state constraint in [13] T Bayen et al [27] derived secondorder necessary and su cient optimality conditions for Dirichlet problems in the sense of strong solution In particular, E Casas [16] 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 [18], C Meyer and F Tr•oltzsch derived second-order su cient optimality conditions for Robin control problems with mixed constraint of the form a(x) y(x) + u(x) b(x) a.e x 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 optimality conditions 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 : A R•osch and F Tr•oltzsch [23] 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 1 We emphasize that in above papers, a; b L ( ) or a; b L ( ) Therefore, the 1 control u belongs to L ( ) or L ( ) This implies that corresponding Lagrange multipliers are measures rather than functions (see [19]) In order to avoid this disadvantage, B T Kien et al [19, 20, 21] recently established second-order necessary optimality conditions for distributed control of Dirichlet problems with mixed state-control constraints of the form a(x) g(x; y(x)) + u(x) b(x) a.e x p with a; b L ( ), < p < and pure state constraints This motivates us to develop and study the following problems (OP 1) : Establish second-order necessary optimality conditions for Robin boundary control problems with mixed state-control constraints of the form a(x) g(x; y(x)) + u(x) b(x) a.e x ; n!+1 n!+1 (3.32) lim sup Kn 0: n!+1 However, from (3.5) we have From this and (3.32), we have lim sup kun ukL2( ) = and so un ! u in L2( ) The proof of lemma is complete From Lemmas 3.3.1 and 3.3.2, we obtain assertion (ii) of Theorem 1.1 (iii) The continuity of S( ; ) It remains to prove that S( ; ) is lower semicontinuous at ( ; ) In fact, let W1 an open set in Y and W2 be an open set in U such that S( ; ) \ W1 W2 S( ; ) is singleton, S( ; ) W1 V2 of such that S( ; ) W1 be 6= ; Since W2 Hence there exist neighborhoods V1 of and W2 the fact S( ; ) 6= ; that S( ; ) \ W1 for all ( ; ) V1 V2 It follows from this and W2 6= ; for all ( ; ) V1 V2 and so S( ; ) is lower semicontinuous at ( ; ) Therefore, S( ; ) is continuous at ( ; ) The proof of Theorem 3.1.2 is complete 3.4 Examples In this section, we give some examples illustrating Theorem 3.1.2 The rst example shows that S( ; ) is singleton and the solution map S( ; ) is continuous at ( ; ) 104 Example 3.4.1 Let be the open unit ball in R with boundary We consider the following problem Z F (y; u; ) = Z 4y(x) + t(x)y(x) + t(x)y (x) + 1(x) dx s.t [ 2(u(x) + y(x) + 2(x)) + 4y(x) 2u(x)]d ! inf in ; y+y+y =0 < @y=u+ on ; : y(x) + u(x) + 2(x) a.e x ; where t(x) := x1 2 Here we suppose that0;0 We x2, x = (x1; x2) R have L(x; y; 1) = 4y + t(x)y + t(x)y + 1; `(x; y; u; 2) = (u + y + 2) 4y 2u; Ly(x; y; 1) = + t(x) + 3t(x)y ; `y(x; y; u; 2) = u + y + 4; `u(x; y; u; 2) = u + y + 2; f(x; y) = y ; g(x; y) = y; fy(x; y) = 3y ; gy(x; y) = 1: Notice that F is convex in u and is not convex in y Firstly, we show that assumptions (A3:1){(A3:4) are ful lled In fact, it is easy to check that (A3:2) and (A3:4) are satis ed For assumption (A3:1), we have 3 jL(x; y; 1)j 4jyj + jyj + jy j + j 1j 5M + M + 0; j`(x ; y; 2 u; 2)j 2:2(juj + jy + 2j ) + 4jyj + + juj 2 2juj + (M + 0) + 4M + 1; jLy(x; y; 1)j + 3M ; jLy(x; y1; 1) Ly(x; y2; 1)j = j3t(x)(y1 2 y2 )j 6Mjy1 j`y(x ; y; u; 2)j + j`u(x ; y; u; 2)j = ju + y + y2j; 2j + ju + y + 4j 2(jyj + juj) + + 6; 0 j`y(x ; y1; u1; 2) `y(x ; y2; u2; 2)j = j`u(x ; y1; u1; 2) `u(x ; y2; u2; 2)j = jy1 y2j + ju1 u2j for a.e x ; x , for all 2; y; u; ui; yi R satisfying jyj; jyij M, i = 1; and j 1j + j 2j Hence assumption (A3:1) is satis ed For assumption (A3:3), we have j`u(x ; y; u; 2) `u(x ; y; u; 2)j = j 2j; h`u(x; y;^ u; 2) `u(x; y;^ u;^ 2); u u^i = hu u;^ u u^i = ju 105 u^j ; 2R j j for a.e x , for all 2; y; u satisfying assumption (A3:3) is ful lled Now, we suppose that (y; u) 2H and (^y; u^) S( ; ) Therefore, S( ; ) By Lemma 3.2.5, there exists a function ( ) \ C( ) such that the following conditions are valid: (i) the adjoint equation: (ii) the weakly 8+ + 3y2 = + t(x) + 3t(x)y2 in ; (3.33) sign(v) = < if v = 0; :1 if v < 0: > > It is obvious that F (y; u; ) Let us take = 0; = We shall show that the following assertions are valid: (i) P ( ; ) has unique solution (y; u) = (0; 0) (ii) P ( ; ) satis es assumptions (A3:1){(A3:4) (iii) If j (x)j then S( ; ) f(y( ; ); s y( ; )); < s < 1g; where y( ; ) is a solution of the equation 2+ 8y + y + y3 = (u + y + ) > if > :(u + y 2)2 107 : 2 ; u + y ; if u + y > Hence `y(x; y; u; ) = `u(x; y; u; ) = > < 82(u + y + if u + ) 2 y< ; ifu + y; > > : ) 2(u + y if u + y > : It is clear that assumptions ( 2) > A : j`(x; y; u; )j a nd (A3:4) hold For a ssum ption (A3:1), we ve (u + y + 22 22 ) + (u + y 22 ) 22 2[u + (y + ) + u + (y ) ] 4u + 4y + j`u(x; y; u; ) + `y(x; y; u; )j 2 2ju + y + 2 j + 2ju + y 4(juj + jyj) + R satisfying jyj M; j j for a.e x , for all y; 4u + 4M + 0; j 4(juj + jyj) + 0 For any y1; y2; u1; u2 R, let us put T1 := `u(x; y1; u1; ) `u(x; y2; u2; ) = `y(x; y1; u1; ) `y(x; y2; u2; ): )j In case of y1 + u1 , we get j 2(y + u T1 = j j + ) 2(y > < j2(y1 + u1 + 2 + u2 + 2 < if y + u 2 2 ; ify2 + u2; j2(y + u1 + )j > ) 2(y2 > = : if y + u > y j + j u1 u2 ) ; if y2 + u2 < if j 2; 2jy1 + u1 y2 u2j y2 + u2 > < 2(jy1 y2j + ju1 u2j) + 2 if y2 > + u2 > > : 2(jy1 y j + j u1 u2 ) if y2 + u2 < if j > if y2 + u2 > > j 2j(y1 y2) + (u1 u2) + : 2(jy1 2; y2 + u2 > if y2 + u2 > > : 2(jy1 ; 2(jy1 y2j + ju1 u2j) > < 2(jy1 y2j + ju1 u2j) + 2jy2 + u2 y1 u1j2 ; ify2 + u2 2jy1 + u1 + j > < > if y2 + u2 > + u2 > )j y j + u1 j u2 ) j if y2 + u2 < if ; ; > 2(jy1 y2j + ju1 u2j) y2 + u2 > < 4(jy1 y2j + ju1 u2j) > > if y2 + u2 > : 4(jy1 y2j + ju1 u2j): 108 Similarly, for the other cases, we also have jT1j 4(jy1 y2j + ju1 u2j): Hence assumption (A3:1) is satis ed Notice that `y(x; y; u; 0) = 2(y + u) for all x; y; u; Putting T2 := `u(x; y; u; ) we get T2 = j2( > < j j j 2(y + u)j > + + ) 2( + y u y y u y >j2(2 + > :2 >2 ; y + u; ; 2; y+u if y + u >2 2 if y+u< if > > 2 if :2 82 u u )j if y + u < > > )j ) 2( + = 82jy + uj > < `u(x; y; u; 0); ; ; y+u if y + u > > > : 2 for all x; u; y; On the other hand, we obtain h`u(x; 0; u; 0) `u(x; 0; 0; 0); u 0i = h2u; ui = 2juj = 2ju 0j : Thus, assumption (A3:3) is ful lled Finally, if y( ; ) is a solution of equation (3.35) and u( ; ) = s u( ; )) ( ) Moreover, since y( ; ) + u( ; ) = s This implies that (y( ; ); u( ; )) S( ; ) 3.5 2 y( ; ) then (y( ; ); with < s < 1, we get F (y; u; ) = Conclusions In this chapter, we present a result on the solution stability of boundary optimal control problems Theorem 3.1.2 provides suitable conditions under which the solution map S( ; ) is upper semicontinuous and continuous in parameters The obtained result is an answer for (OP 4) and is proved by the direct method, techniques of variational analysis and necessary optimality conditions 109 GENERAL CONCLUSIONS The main results of this dissertation include: Second-order necessary optimality conditions for boundary control problems with mixed pointwise constraints No-gap second-order optimality conditions for distributed control problems and boundary control problems governed by semilinear elliptic equations with mixed pointwise constraints, where objective functions are quadratic forms in the control variables Second-order su cient optimality conditions for distributed control problems and boundary control problems governed by semilinear elliptic equations with mixed pointwise constraints when objective functions may not depend on the control variables A set of conditions for a parametric boundary optimal control problems in twodimensional space, under which the solution map is upper semicontinuous and continuous in parameters in the case where the objective function is not convex, and the admissible set is not convex Some open problems related to the dissertation are continued to study: Second-order optimality conditions and stability of solutions for optimal control problems governed by partial di erential equations It is an ongoing topic No-gap second-order optimality conditions for optimal control problems governed by semilinear elliptic equations with mixed pointwise constraints and pure state constraints The role of second-order su cient conditions in the study of solution stability for boundary optimal control problems 110 LIST OF PUBLICATIONS [1] N H Son, B T Kien and A R•osch (2016), Second-order optimality conditions for boundary control problems with mixed pointwise constraints, SIAM J Optim., 26, pp 1912{1943 [2] B T Kien, V H Nhu and N H Son (2017), Second-order optimality conditions for a semilinear elliptic optimal control problem with mixed pointwise constraints, Set-Valued Var Anal., 25, pp 177{210 [3] N H Son (2017), On the 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(2005), On the lower semicontinuity of optimal solution sets Optimiza-tion, 54, pp 123{130 116 ... 0.2 Sobolev spaces and elliptic equations 13 0.2.1 Sobolev spaces 13 0.2.2 Semilinear elliptic equations 20... 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],... 0.2.1 Sobolev spaces and elliptic equations Sobolev spaces First, we recall some relative concepts and properties which are introduced in many books on Sobolev spaces, elliptic equations and partial